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By
Genick Bar–Meir
Illustration of a typical design of cold and hot chambers die casting.
Fundamentals of Die Casting
Design
Genick Bar–Meir, Ph. D.
8118 Kolmar Ave
Skokie, IL 60076
email:barmeir at gmail dot com
Copyright © 2023, 2022, 2009, 2008, 2007, and 1999 by Genick Bar-Meir
current version 0.4.5
doi: https://doi.org/10.5281/zenodo.5523594
See the file copying.fdl or copyright.tex for copying conditions.
Version (0.4.5 pre
July 20, 2023)
‘We are like dwarfs sitting on the shoulders of giants”
from The Metalogicon by John in 1159
Please Update
This book became victim of its own successes. More than 90% of downloads of this book are
for the old versions because the search engines keep track the previous downloads. That is,
when the old version with many downloads from one web site for example, like researchgate.net, the new version cannot surface up. The book is released on a rolling fashion. It
means that it released several times during the year. In other words, if you have a copy of
the book and it is older than a month, the chances are that you have an old version. Please
do yourself a favor and download a new version. You can get the last version from zenodo https://zenodo.org/record/5521908#.Y2bVrBxOlH0. While you are there you can
download several items:
• “Stability of Ships and Other bodies”.
https://zenodo.org/record/5784893#.Yd1uuYpME-0.
• “Fundamentals of Compressible Flow”,
https://zenodo.org/record/5523349#.YhxNZ1RMFhE.
• the world largest gad dynamics tables (over 600 pages).
https://zenodo.org/record/5523532#.YhxOD1RMFhE
• “Basics of Fluid Mechanics”
https://zenodo.org/record/5523594#.Y2bWERxOlH0
• “The Aquatic Bodies Locomotion serious” exposing the violations of first and second
laws of the thermodynamics by the establishment’s models and showing how to do to
correct , and
• other material like “15 Years Experience Creating Open Content Engineering Material”
describing the depth of the great depth analytically.
Like the largest gas table in world published by Potto Project NFP. All these materials are
authored by the undersigned. If you would like to learn more about this author you can
grab the article “15 Years Experience Creating Open Content Engineering Material.” https:
//zenodo.org/record/5791182#.Yd1v3YpME-0 In the near future the article “20 years of
producing open content engineering.”
Thank you for using this book, Genick
iii
iv
PLEASE UPDATE
Abstract
Die-casting engineers have to compete not only with other die–casting companies, but also
against other industries such as plastics, and composite materials. Clearly, the "black art" approach, which has been an inseparable part of the engineer’s tools, must be replaced by a scientific approach. Excuses that “science has not and never will work” need to be replaced with
“science does work and we use it”. All technologies developed in recent years are described
in a clear, simple manner in this book. All the errors of the old models and the violations of
physical laws are shown and described. For example, the “common” pQ2 diagram violates
many physical laws, such as the first and second laws of thermodynamics. Furthermore, the
“common” pQ2 diagram produces trends that are the opposite of reality, which are described
in this book. The concepts such as the critical vent area must be used. In addition, the critical
plunger velocity must be calculated and implemented.
The die casting engineer’s job is to produce maximum profits for the company. In order
to achieve this aim, the engineer must design high quality products at a minimum cost with
the available tools. Thus, understanding the economics of the die casting design and process
are essential in the design process. These requirements are described in a mathematical form
for the first time in this volume. Many new concepts and ideas are also introduced. For
instance, how to minimize the scrap/cost due to the runner system, and to match the size of
die casting machine with a specific project.
The die–casting industry is undergoing a revolution, and this book is the main part of
it. One reason (if one reason can describe the situation) is companies such as Doehler Jorvis
(which used to be the biggest die caster in the world) and Shelby went bankrupt because they
did not know how to calculate and reduce their production costs. It is hoped that die–casters
will adopt these technologies which are presented in this book. Proper calculations is the only
way to keep the die casting professionals and the industry itself, from being “left in the dust.”
The change that die–casting industry undergoes is not a linear one. There was some
progress and it is hoped these indoctrinated individuals will leave and new crop of new engineers replace them so there is no regression. Recently there was regression in the knowledge
probably due to NADCA production of flawed and indoctrinated engineers. For example,
the results of the Ohio State university education produce a student who works in Marquette
University which advised a fresh new student. The thesis of the student was stated to use
surface tension and strangely later retracted from this claim of use. Yet the main problem is
no proper literature review was not done and hence no awareness of the concepts like critical
area and proper dimensional analysis. They use turbulence without knowing if it should be
v
vi
ABSTRACT
applied or not. Clearly this work is not worth the paper (computer memory) it was done. The
problem is that the work propagates misconception and reinforces what can be described as
garbage.
Hence, this book started to include material that introductory such as open channel
flow, basics of compressible flow. As dimensional analysis is important it will be extensively
presented. It will be flowed with several chapters dealing with economy, pQ2 , critical plunger
velocity, critical vent area.
Prologue For This Book
This author has developed many breakthroughs for which he attribute to the fact that there
are so many Money Paddlers in the field (see below what is Money Paddlers). Thus, one
eye individual seems genius in the land of the blinds (money paddler). Intrinsically, these
breakthroughs caused many schemes (of make money) of organization and individuals to be
exposed like the story of the critical plunger velocity story. Yet, the control of the establishment (individuals and/or organizations) like NADCA responsible to continue the spread of
fake science. This hampering has to be stopped.
vii
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PROLOGUE FOR THIS BOOK
Categories of Scientists and Engineers
The quality of engineers or scientists broadly is categorized as Free Thinker, Cathedral
Builder, Research Manager, Dust Collection, Money Paddler.
Free Thinkers
Most of the time, the main research work is carried by the free thinkers who make
revolutions in the process of science and technology e.g. Einstein.
Cathedral Builder
The Cathedral builders are like those of European cathedral builders who where part
in a long line of builders who made these wonderful building e.g Darcy, Hunter Rose
and the famous plagiarizer Moody (Moody diagram).
Research Manager
The next in line are the research managers, these days the system promotes them with
large reach grands, in which they produce many “research” papers with little (that can
be measured) or no significance.
Dust Collector
The dust collect are those who make contributions that cannot be measured like dust
that is very little. Yet after extremely large amount papers, like a dust, their work shows
some progress.
Money Paddler
The money paddlers are good in obtaining grants and producing erroneous research
papers which at time violate basics and fundamentals principles. In die casting is full
individuals which are belong to the last category.
Version 0.4.5 July 20, 2023
pages 393 size 4.2M (font change) old 406p 4.6M
The intensification pressure solution was finally published after hiding the material for sometime. Now this solution is back ported into the book. With about 6 out 10 open questions
solved by this author it seem to be time to reflect on the progression in die casting. While
the vent locations and number still open questions, now considerable design decisions can be
made. The connection between the pQ2 and cost has to be postponed since this research is
not sponsored and the most it not covered by research grants, which, thanks the NADCA, are
going to the money paddlers. It is strange to see that research grants that producing no real
science getting grants while the real science production is done else where.
Considerable attempts were made to reduce the size of the file and page count of these
VERSION 0.3.5 MARCH 20, 2023
ix
books. It can be noticed that the page count is reduced every time the major change made in
style file (such a font change) while the material is increased.
Version 0.3.5 March 20, 2023
pages 345 size 4.2M selected publication
During the last two decades, three additional books were written by this author which include
“Fundamentals of compressible Flow Mechanics” with the world largest gas dynamics tables,
“Basics of Fluid Mechanics”, and “Stability of Ships and Other Bodies”. During that time, this
book went through a limited very minor change because large amount work was dedicated
to breakthroughs in other areas. Hence, all these works exhibited breakthroughs in different
fields. For example, the revamping of the understanding ship stability and thus rending the
almost the research work done in the area as erroneous.
Bar-Meir’s ideas are spreading around, as can be evidenced by several things: On the
one hand, Dr. Jiarong Hong, from the University of Minnesota, alleged that the solution
for the deep ocean is useless or cumbersome. On the other hand, Dr. Sandip Ghosal, from
Northwestern University, plagiarized Bar-Meir’s work on deep ocean pressure calculations,
and he spent almost a week teaching this topic, and even copied Bar-Meir’s nomenclature. In
fact, one–third of the mid-quarter exam that he gave to his students was taken from this topic.
It is also interesting to point out that Dr. Hong also suggested to “burn” the Eckert (Schmidt)
dimensional analysis method, which Eckert introduced to heat and mass transfer education.
100 years of use of Schmidt/Eckert method now and it is currently almost exclusive use, the
genius Hong think that shout be ignored.
The adherence that Dr. Ghosal displayed to Bar-Meir’s work was probably because he
did not think that he would be caught. On the other hand, Kostas J. Spyrou, when he gave his
keynote lecture, basically copied many concepts that were developed by Bar-Meir (such as the
stability dome) and presented them as his ideas. These concepts had never appeared in the
literature before. It is interesting to point out that Dr. Spyrou has been observed downloading
Bar-Meir’s book. Dr. Spyrou’s behavior is not unique to him only. Dr. Kenneth Brezinsky
went out of his way to declare that Bar-Meir’s calculations of compressible gas potential were
useless and baseless. However, Brian J. Cantwell from Stanford University recently decided to
copy Bar-Meir’s idea and to dedicate a whole section (2.9) to it in his book (and probably also in
his classes). Except that Dr. Cantwell oversimplified the idea (he removed the dimensionless
presentation), maybe because he did not fully understand it. There are more cases of this
copying/plagiarizing, but normally they try to make the appearance that there is nothing to
see here while the using the Bar–Meir’s ideas.
These books where written initially as chapters in one book and grown to be a full
book. Now, these books are used as a bases for the some chapters for this book. In addition,
several ideas were added to these books which made it more effective. In an article published
by this author published, (Bar-Meir 1997) discussed a solution to the critical plunger velocity
while not providing the actual solution. Hence, over 300 experienced CFD (Computational
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PROLOGUE FOR THIS BOOK
Fluid Mechanics) groups from over the world with loads of research grants attempted to
solve the critical plunger velocity problem and failed (Bar-Meir 2002). The main reason that
these groups failed is lack of understating and accounting for the physical phenomena which
cannot be replaced by just numerical techniques of solving random equations and erroneous
boundary equations.
It is similar to the situation that occurred in the ship stability, in which people assumed
that ships are like a giant pendulums. In fact, these individuals had abundant experimental
“evidence” to prove their theory where they ignored the added properties can be ignored.
Later it become clear that the liquid added moment of inertial has to be accounted for which
rendered the previous “abundantly clear solution” erroneous by a factor of the very least of 2
(according to their model the actual value is more than 4 in many cases and solution does not
exist in many other cases). Now the very same individuals (figuratively) have again abundant
“evidence” for the new conflicting theory and yet their theory does not conform the observed
physics.
In the same pattern of ignoring the physics, hundreds of groups from all over the world
attempted to solve the question what is the physics of what the intensive pressure die casting.
The intensive pressure period is the added pressure applied to the cavity after the cavity was
filled. Intensive pressure phase is intended to compensate for the errors for the locations of
the vent and thermal shrinkage. This topic is tackled in this book.
The question is in what category all these superior intelligent groups according to the
table above. It reasonable to assume that many of them are in the money paddler category or at
best in the dust collection. It they are the establishment at this moment. It is the same situation
that occurred in basics electricity that Georg Ohm faced before. Ohm was persecuted by the
money paddlers who could not stand his discoveries.
Version 0.1 January 12, 2009
pages 213 size 1.5M
The author Ph.D. thesis was the focus which admittedly was not his preferred choice. Dr. E.R.G Eckert, his advisor, asked him to work on die casting per the request of his administrative advisor Dr. Goldstein. Thus, with an advisor of around the 90 and administrative advisor who
is dealing with politics of science (president of ASME, department head choosing university
of Minnesota president etc) was left most completely on his own to developed knowledge of
die casting industry. The first thing that he has done is a literature review which force his to
realize that there is very little scientific known about how to design the die casting process.
Works/papers by him reviewed from of Ohio State University by A. Miller, Brevick, J. Wallace from Case Western, Murry from Australia are either erroneous or meaningless (these
works not even dust collection). Scientists are categorized in the following categories, Free
thinkers, Cathedral builder, research managers, dust collectors, (important work but minor),
and Money Peddlers and thus who should not be in science at all. This author feels that he,
in same sense, very lucky that die casting research is infested with thus who should not be
VERSION 0.0.3 OCTOBER 9, 1999
xi
in science at all. Being in a field of these individuals has two aspects one positive and one
negative. The negative is that all the research grants are sucked to produce “negative” science
and positive that he become the main scientist in the area. All these discoveries are achieved
because no real scientist was in the area. It is like being a dwarf in sand country.
This book is used as a tool in his efforts to convert die casting design process from
the stone age to modern time based on real scientific principles. It was found that found
that the book early version (0.0.3) have been downloaded over 50,000. Many were using the
economical part of this book to explain many other the economical problems of large scale
manufacturing processes. As this author is drifting toward a different field (renewal energy
hopefully), the author interest still in this material but with a different emphasis. Subjects
like Fanno Flow that was as written as appendix will be expanded. Moreover, material like
the moving shock issue will be explained and add to process description was omitted in the
previous version. While this topic is not directly affecting die casting, the issue of future value
will be discussed.
Version 0.0.3 October 9, 1999
pages 178 size 3.2M
This book is the first and initial book in the series of POTTO project books. This book started
as a series of articles to answer both specific questions that I have been asked, as well as questions that I was curious about myself. While addressing these questions, I realized that many
commonly held "truths" about die-casting were scientifically incorrect. Because of the importance of these results, I have decided to make them available to the wider community of
die-casting engineers. However, there is a powerful group of individuals who want to keep
their monopoly over “knowledge” in the die-casting industry and to prevent the spread of this
information.1 Because of this, I have decided that the best way to disseminate this information
is to write a book. This book is written in the spirit of my adviser and mentor E.R.G. Eckert.
Eckert, aside from his research activity, wrote the book that brought a revolution in the education of the heat transfer. Up to Eckert’s book, the study of heat transfer was without any
dimensional analysis. He wrote his book because he realized that the dimensional analysis
utilized by him and his adviser (for the post doc), Ernst Schmidt, and their colleagues, must
be taught in engineering classes. His book met strong criticism in which some called to “burn”
his book. Today, however, there is no known place in world that does not teach according
to Eckert’s doctrine. It is assumed that the same kind of individual(s) who criticized Eckert’s
work will criticize this work. As a wise person says “don’t tell me that it is wrong, show me
what is wrong”; this is the only reply. With all the above, it must be emphasized that this book
is not expected to revolutionize the field but change some of the way things are taught.
The approach adapted in this book is practical, and more hands–on approach. This
statement really meant that the book is intent to be used by students to solve their exams and
1 Please read my correspondence with NADCA editor Paul Bralower and Steve Udvardy. Also, please read the
references and my comments on pQ2 .
xii
PROLOGUE FOR THIS BOOK
also used by practitioners when they search for solutions for practical problems. So, issue of
proofs so and so are here only either to explain a point or have a solution of exams. Otherwise,
this book avoids this kind of issues.
This book is divided into two parts. The first discusses the basic science required by a
die–casting engineer; the second is dedicated to die-casting–specific science. The die-casting
specific is divided into several chapters. Each chapter is divided into three sections: section
1 describes the “commonly” believed models; section 2 discusses why this model is wrong or
unreasonable; and section 3 shows the correct, or better, way to do the calculations. I have
made great efforts to show what existed before science “came” to die casting. I have done this
to show the errors in previous models which make them invalid, and to “prove” the validity
of science. I hope that, in the second edition, none of this will be needed since science will
be accepted and will have gained validity in the die casting community. Please read about my
battle to get the information out and how the establishment react to it.
Plea for LATEX usage
Is it only an accident that both the quality of the typesetting of papers in die casting congress
and their technical content quality is so low? I believe there is a connection. All the major
magazines of the scientific world using TEX or LATEX, why? Because it is very easy to use and
transfer (via the Internet) and, more importantly, because it produces high quality documents.
NADCA has continued to produce text on a low quality word processor. Look for yourself;
every transaction is ugly.
Linux has liberated the world from the occupation and the control of Microsoft OS. We
hope to liberate the NADCA Transaction from such a poor quality word processor. TEX and
all (the good ones) supporting programs are free and available every where on the web. There
is no reason not to do it. Please join me in improving NADCA’s Transaction by supporting
the use of LATEX by NADCA.
Will I Be in Trouble?
Initial part
Many people have said I will be in trouble because I am telling the truth. Those with a vested
interest in the status quo (North American Die Casting Association, and thus research that
this author exposed there poor and or erroneous work). will try to use their power to destroy
me. In response, I challenge my opponents to show that they are right. If they can do that, I
will stand wherever they want and say that I am wrong and they are right. However, if they
cannot prove their models and practices are based on solid scientific principles, nor find errors
with my models (and I do not mean typos and English mistakes), then they should accept my
results and help the die–casting industry prosper.
People have also suggested that I get life insurance and/or good lawyer because my opponents are very serious and mean business; the careers of several individuals are in jeopardy
VERSION 0.0.3 OCTOBER 9, 1999
xiii
because of the truths I have exposed. If something does happen to me, then you, the reader,
should punish them by supporting science and engineering and promoting the die–casting industry. By doing so, you prevent them from manipulating the industry and gaining additional
wealth.
For the sake of my family, I have, in fact, taken out a life insurance policy. If something
does happen to me, please send a thank you and work well done card to my family.
The Continued Struggle
It was exposed that second reviewer that appear in this book is Brevick from Ohio. It strange
that in a different correspondence he say that he cannot wait to get this author future work.
This part is holding for some juicy details.
xiv
PROLOGUE FOR THIS BOOK
Prologue For The POTTO Project
2023
Mahatma Gandhi
First they ignore you, then they laugh at you, then they fight you, then you win.
Years after this project started perhaps one can wonder at what stage this project is
currently. It is very hard to say exactly but there are some objective evidence that show the
popularity and relevancy.
Bar-Meir’s ideas are spreading around, as can be evidenced by several things: On the
one hand, Dr. Jiarong Hong, from the University of Minnesota, alleged that the solution
for the deep ocean is useless or cumbersome. On the other hand, Dr. Sandip Ghosal, from
Northwestern University, plagiarized Bar-Meir’s work on deep ocean pressure calculations,
and he spent almost a week teaching this topic, and even copied Bar-Meir’s nomenclature. In
fact, one–third of the mid–quarter exam that he gave to his students was taken from this topic.
It is also interesting to point out that Dr. Hong also suggested to “burn” the Eckert (Schmidt)
dimensional analysis method, which Eckert introduced to heat and mass transfer education.
100 years of use of Schmidt/Eckert method now and it is currently almost exclusive use, the
genius Hong think that shout be ignored.
The adherence that Dr. Ghosal displayed to Bar-Meir’s work was probably because he
did not think that he would be caught. On the other hand, Kostas J. Spyrou, when he gave his
keynote lecture, basically copied many concepts that were developed by Bar-Meir (such as the
stability dome) and presented them as his ideas. These concepts had never appeared in the
literature before. It is interesting to point out that Dr. Spyrou has been observed downloading
Bar-Meir’s book. Dr. Spyrou’s behavior is not unique to him only. Dr. Kenneth Brezinsky
went out of his way to declare that Bar-Meir’s calculations of compressible gas potential were
useless and baseless. However, Brian J. Cantwell from Stanford University recently decided to
copy Bar-Meir’s idea and to dedicate a whole section (2.9) to it in his book (and probably also in
his classes). Except that Dr. Cantwell oversimplified the idea (he removed the dimensionless
presentation), maybe because he did not fully understand it. There are more cases of this
copying/plagiarizing, but normally they try to make the appearance that there is nothing to
see here while the using the Bar–Meir’s ideas.
xv
xvi
PROLOGUE FOR THIS BOOK
Pre 2014
This books series was born out of frustrations in two respects. The first issue is the enormous
price of college textbooks. It is unacceptable that the price of the college books will be over
$150 per book (over 10 hours of work for an average student in The United States).
The second issue that prompted the writing of this book is the fact that we as the public
have to deal with a corrupted judicial system. As individuals we have to obey the law, particularly the copyright law with the “infinite2 ” time with the copyright holders. However, when
applied to “small” individuals who are not able to hire a large legal firm, judges simply manufacture facts to make the little guy lose and pay for the defense of his work. On one hand,
the corrupted court system defends the “big” guys and on the other hand, punishes the small
“entrepreneur” who tries to defend his or her work. It has become very clear to the author
and founder of the Potto Project that this situation must be stopped. Hence, the creation of
the POTTO Project. As R. Kook, one of this author’s sages, said instead of whining about arrogance and incorrectness, one should increase wisdom. This project is to increase wisdom
and humility.
The Potto Project has far greater goals than simply correcting an abusive Judicial system
or simply exposing abusive judges. It is apparent that writing textbooks especially for college
students as a cooperation, like an open source, is a new idea3 . Writing a book in the technical
field is not the same as writing a novel. The writing of a technical book is really a collection
of information and practice. There is always someone who can add to the book. The study
of technical material isn’t only done by having to memorize the material, but also by coming
to understand and be able to solve related problems. The author has not found any technique
that is more useful for this purpose than practicing the solving of problems and exercises. One
can be successful when one solves as many problems as possible. To reach this possibility the
collective book idea was created/adapted. While one can be as creative as possible, there are
always others who can see new aspects of or add to the material. The collective material is
much richer than any single person can create by himself.
The following example explains this point: The army ant is a kind of carnivorous ant
that lives and hunts in the tropics, hunting animals that are even up to a hundred kilograms in
weight. The secret of the ants’ power lies in their collective intelligence. While a single ant is
not intelligent enough to attack and hunt large prey, the collective power of their networking
creates an extremely powerful intelligence to carry out this attack4 . When an insect which
is blind can be so powerful by networking, so can we in creating textbooks by this powerful
tool.
Why would someone volunteer to be an author or organizer of such a book? This is the
2 After the last decision of the Supreme Court in the case of Eldred v. Ashcroff (see http://cyber.law.harvard.
edu/openlaw/eldredvashcroft for more information) copyrights practically remain indefinitely with the holder
(not the creator).
3 In some sense one can view the encyclopedia Wikipedia as an open content project (see http://en.wikipedia.
org/wiki/Main_Page). The wikipedia is an excellent collection of articles which are written by various individuals.
4 see also in Franks, Nigel R.; "Army Ants: A Collective Intelligence," American Scientist, 77:139, 1989 (see for information http://www.ex.ac.uk/bugclub/raiders.html)
PRE 2014
xvii
first question the undersigned was asked. The answer varies from individual to individual. It
is hoped that because of the open nature of these books, they will become the most popular
books and the most read books in their respected field. For example, the books on compressible flow and die casting became the most popular books in their respective area. In a way, the
popularity of the books should be one of the incentives for potential contributors. The desire
to be an author of a well–known book (at least in his/her profession) will convince some to
put forth the effort. For some authors, the reason is the pure fun of writing and organizing
educational material. Experience has shown that in explaining to others any given subject,
one also begins to better understand the material. Thus, contributing to these books will help
one to understand the material better. For others, the writing of or contributing to this kind
of books will serve as a social function. The social function can have at least two components.
One component is to come to know and socialize with many in the profession. For others the
social part is as simple as a desire to reduce the price of college textbooks, especially for family members or relatives and those students lacking funds. For some contributors/authors,
in the course of their teaching they have found that the textbook they were using contains
sections that can be improved or that are not as good as their own notes. In these cases, they
now have an opportunity to put their notes to use for others. Whatever the reasons, the undersigned believes that personal intentions are appropriate and are the author’s/organizer’s
private affair.
If a contributor of a section in such a book can be easily identified, then that contributor
will be the copyright holder of that specific section (even within question/answer sections).
The book’s contributor’s names could be written by their sections. It is not just for experts
to contribute, but also students who happened to be doing their homework. The student’s
contributions can be done by adding a question and perhaps the solution. Thus, this method
is expected to accelerate the creation of these high quality books.
These books are written in a similar manner to the open source software process.
Someone has to write the skeleton and hopefully others will add “flesh and skin.” In this
process, chapters or sections can be added after the skeleton has been written. It is also hoped
that others will contribute to the question and answer sections in the book. But more than
that, other books contain data5 which can be typeset in LATEX. These data (tables, graphs and
etc.) can be redone by anyone who has the time to do it. Thus, the contributions to books can
be done by many who are not experts. Additionally, contributions can be made from any part
of the world by those who wish to translate the book.
It is hoped that the books will be error–free. Nevertheless, some errors are possible
and expected. Even if not complete, better discussions or better explanations are all welcome
to these books. These books are intended to be “continuous” in the sense that there will be
someone who will maintain and improve the books with time (the organizer(s)).
These books should be considered more as a project than to fit the traditional definition
of “plain” books. Thus, the traditional role of author will be replaced by an organizer who will
be the one to compile the book. The organizer of the book in some instances will be the main
author of the work, while in other cases only the gate keeper. This may merely be the person
5 Data are not copyrighted.
xviii
PROLOGUE FOR THIS BOOK
who decides what will go into the book and what will not (gate keeper). Unlike a regular book,
these works will have a version number because they are alive and continuously evolving.
In the last 5 years three textbooks have been constructed which are available for download. These books contain innovative ideas which make some chapters the best in the world.
For example, the chapters on Fanno flow and Oblique shock contain many original ideas such
as the full analytical solution to the oblique shock, many algorithms for calculating Fanno flow
parameters which are not found in any other book. In addition, Potto has auxiliary materials
such as the gas dynamics tables (the largest compressible flow tables collection in the world),
Gas Dynamics Calculator (Potto-GDC), etc.
The combined number downloads of these books is over half a million (December 2009)
or in a rate of 20,000 copies a month. Potto books on compressible flow and fluid mechanics
are used as the main textbook or as a reference book in several universities around the world.
The books are used in more than 165 different countries around the world. Every month
people from about 110 different countries download these books. The book on compressible
flow is also used by “young engineers and scientists” in NASA according to Dr. Farassat, NASA
Langley Research Center.
The undersigned of this document intends to be the organizer/author/coordinator of
the projects in the following areas:
Table 1 – Books under development in Potto project.
Project
Name
Progress
Compressible Flow
Die Casting
Dynamics
Fluid Mechanics
Heat Transfer
beta
alpha
NSY
alpha
NSY
Mechanics
Open Channel Flow
Statics
NSY
NSY
early
alpha
NSY
early
alpha
NSY
Strength of Material
Thermodynamics
Two/Multi
flow
phases
NSY = Not Started Yet
The meaning of the progress is as:
Remarks
Based on
Eckert
first
chapter
TelAviv’notes
Version
0.4.8.2
0.0.3
0.0.0
0.1.1
0.0.0
Availability
for
Public
Download
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PRE 2014
xix
• The Alpha Stage is when some of the chapters are already in a rough draft;
• in Beta Stage is when all or almost all of the chapters have been written and are at least
in a draft stage;
• in Gamma Stage is when all the chapters are written and some of the chapters are in a
mature form; and
• the Advanced Stage is when all of the basic material is written and all that is left are
aspects that are active, advanced topics, and special cases.
The mature stage of a chapter is when all or nearly all the sections are in a mature stage and
have a mature bibliography as well as numerous examples for every section. The mature stage
of a section is when all of the topics in the section are written, and all of the examples and
data (tables, figures, etc.) are already presented. While some terms are defined in a relatively
clear fashion, other definitions give merely a hint on the status. But such a thing is hard to
define and should be enough for this stage.
The idea that a book can be created as a project has mushroomed from the open source
software concept, but it has roots in the way science progresses. However, traditionally books
have been improved by the same author(s), a process in which books have a new version
every a few years. There are book(s) that have continued after their author passed away, i.e.,
the Boundary Layer Theory originated6 by Hermann Schlichting but continues to this day.
However, projects such as the Linux Documentation project demonstrated that books can be
written as the cooperative effort of many individuals, many of whom volunteered to help.
Writing a textbook is comprised of many aspects, which include the actual writing
of the text, writing examples, creating diagrams and figures, and writing the LATEX macros7
which will put the text into an attractive format. These chores can be done independently
from each other and by more than one individual. Again, because of the open nature of this
project, pieces of material and data can be used by different books.
6 Originally authored by Dr. Schlichting, who passed way some years ago. A new version is created every several
years.
7 One can only expect that open source and readable format will be used for this project. But more than that,
only LATEX, and perhaps troff, have the ability to produce the quality that one expects for these writings. The text
processes, especially LATEX, are the only ones which have a cross platform ability to produce macros and a uniform
feel and quality. Word processors, such as OpenOffice, Abiword, and Microsoft Word software, are not appropriate
for these projects. Further, any text that is produced by Microsoft and kept in “Microsoft” format are against the
spirit of this project In that they force spending money on Microsoft software.
xx
PROLOGUE FOR THIS BOOK
CONTRIBUTORS LIST
How to contribute to this book
As a copylefted work, this book is open to revisions and expansions by any interested parties.
The only "catch" is that credit must be given where credit is due. This is a copyrighted work:
it is not in the public domain!
If you wish to cite portions of this book in a work of your own, you must follow the
same guidelines as for any other GDL copyrighted work.
Credits
All entries have been arranged in alphabetical order of surname, hopefully. Major contributions are listed by individual name with some detail on the nature of the contribution(s),
date, contact info, etc. Minor contributions (typo corrections, etc.) are listed by name only
for reasons of brevity. Please understand that when I classify a contribution as "minor," it is
in no way inferior to the effort or value of a "major" contribution, just smaller in the sense
of less text changed. Any and all contributions are gratefully accepted. I am indebted to all
those who have given freely of their own knowledge, time, and resources to make this a better
book!
• Date(s) of contribution(s): 1999 to present
• Nature of contribution: Original author.
• Contact at: genick at potto.org
Steven from artofproblemsolving.com
• Date(s) of contribution(s): June 2005, Dec, 2009
• Nature of contribution: LaTeX formatting, help on building the useful equation and
important equation macros.
• Nature of contribution: In 2009 creating the exEq macro to have different counter
for example.
xxi
xxii
PROLOGUE FOR THIS BOOK
Dan H. Olson
• Date(s) of contribution(s): April 2008
• Nature of contribution: Some discussions about chapter on mechanics and correction of English.
Richard Hackbarth
• Date(s) of contribution(s): April 2008
• Nature of contribution: Some discussions about chapter on mechanics and correction of English.
John Herbolenes
• Date(s) of contribution(s): August 2009
• Nature of contribution: Provide some example for the static chapter.
Eliezer Bar-Meir
• Date(s) of contribution(s): Nov 2009, Dec 2009
• Nature of contribution: Correct many English mistakes Mass.
• Nature of contribution: Correct many English mistakes Momentum.
Henry Schoumertate
• Date(s) of contribution(s): Nov 2009
• Nature of contribution: Discussion on the mathematics of Reynolds Transforms.
Your name here
• Date(s) of contribution(s): Month and year of contribution
• Nature of contribution: Insert text here, describing how you contributed to the book.
• Contact at: my_email@provider.net
CREDITS
xxiii
Typo corrections and other "minor" contributions
• R. Gupta, January 2008, help with the original img macro and other ( LaTeX issues).
• Tousher Yang April 2008, review of statics and thermo chapters.
• Correction to equation (2.40) by Michal Zadrozny. (Nov 2010)
• Seon-Kyu Kim, April 2021, Correction of typo in equation for Rayleigh stagnation
pressure ratio (P0 ).
xxiv
PROLOGUE FOR THIS BOOK
The Book Change Log
Version 0.4.5
July 17, 2023 (4.5 M 401 pages)
• English
• port back the intensification solution
• part of the runner design
Version 0.3.5
March 20, 2023 (4.2 M 345 pages)
• Changing to 2023 standard especially ipe.
• English
• Added open channel chapter added
• part of the runner design
Version 0.2.5
Nov 27, 2022 (2.0 M 273 pages)
• Changing to 2022 standard
Version 0.1.4
Nov 27, 2012 (1.9M 269 pages)
• Additional discussion on the economics chapter marginal profits.
xxv
xxvi
PROLOGUE FOR THIS BOOK
Version 0.1.3
Nov 8, 2012 (1.9M 265 pages)
• Improvements to some of the figures of dimensional analysis chapter (utilizing blender).
• Add an analysis of the minimum cost ordering supply. The minimum cost ordering
refers to the analysis dealing with the minimum cost achieved by finding the optimum
number of ordering.
Version 0.1.2
April 1, 2009 (1.9M 263 pages)
• Irene Tan provided many English corrections to the dimensional analysis chapter.
Version 0.1.1
Feb 8, 2009 (1.9M 261 pages)
• Add Steve Spurgeon (from Dynacast England) corrections to pQ2 diagram.
• Minor English corrections to pQ2 diagram chapter (unfinished).
• Fix some figures and captions issues.
• Move to potto style file.
Version 0.1
Jan 6, 2009 (1.6M 213 pages)
• Change to modern Potto format.
• English corrections
• Finish some examples in Dimensionless Chapter (manometer etc)
Version 0.0.3
Nov 1, 1999 (3.1 M 178 pages)
• Initial book of Potto project.
• Start of economy, dimensional analysis, pQ2 diagram chapters.
Contents
Please Update
iii
Abstract
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Prologue For This Book
Version 0.4.5 July 20, 2023 . . . . . . . . . . . . . . . . . . . .
pages 393 size 4.2M (font change) old 406p 4.6M . . .
Version 0.3.5 March 20, 2023 . . . . . . . . . . . . . . . . . .
pages 345 size 4.2M selected publication . . . . . . . .
Version 0.1 January 12, 2009 . . . . . . . . . . . . . . . . . . .
pages 213 size 1.5M . . . . . . . . . . . . . . . . . . . .
Version 0.0.3 October 9, 1999 . . . . . . . . . . . . . . . . . .
pages 178 size 3.2M . . . . . . . . . . . . . . . . . . . .
2023 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pre 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How to contribute to this book . . . . . . . . . . . . . . . .
Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steven from artofproblemsolving.com . . . . . . . .
Dan H. Olson . . . . . . . . . . . . . . . . . . . . . . .
Richard Hackbarth . . . . . . . . . . . . . . . . . . . .
John Herbolenes . . . . . . . . . . . . . . . . . . . . .
Eliezer Bar-Meir . . . . . . . . . . . . . . . . . . . . .
Henry Schoumertate . . . . . . . . . . . . . . . . . . .
Your name here . . . . . . . . . . . . . . . . . . . . .
Typo corrections and other "minor" contributions . .
GNU Free Documentation License . . . . . . . . . . . . . .
1. APPLICABILITY AND DEFINITIONS . . . . . . .
2. VERBATIM COPYING . . . . . . . . . . . . . . . .
3. COPYING IN QUANTITY . . . . . . . . . . . . . .
4. MODIFICATIONS . . . . . . . . . . . . . . . . . .
5. COMBINING DOCUMENTS . . . . . . . . . . . .
6. COLLECTIONS OF DOCUMENTS . . . . . . . .
7. AGGREGATION WITH INDEPENDENT WORKS
xxvii
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vii
. viii
. viii
. ix
. ix
. x
. x
. xi
. xi
. xv
. xvi
. xxi
. xxi
. xxi
. xxii
. xxii
. xxii
. xxii
. xxii
. xxii
. xxiii
. xxxix
. xl
. xli
. xli
. xlii
. xliv
. xliv
. xliv
xxviii
CONTENTS
8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . .
9. TERMINATION . . . . . . . . . . . . . . . . . . . . . . . .
10. FUTURE REVISIONS OF THIS LICENSE . . . . . . . .
ADDENDUM: How to use this License for your documents
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How This Book Was Written
li
Preface
liii
Properties . . . . . . .
Inviscid Flow . . . . .
Internal Viscous Flow
Open Channel Flow .
pQ2 improvements .
1
xlv
xlv
xlv
xlv
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Introduction
1.1
The Importance of Reducing Production Costs . . .
1.2 Designed/Undesigned Scrap/Cost . . . . . . . . . .
1.3 Linking the Production Cost to the Product Design
1.4 Historical Background . . . . . . . . . . . . . . . . .
1.4.1
The Main Challenges Faced By Die Casting
1.4.2
Gate Velocity and Shape . . . . . . . . . . .
1.4.3
Filling time . . . . . . . . . . . . . . . . . .
1.5 Numerical Simulations . . . . . . . . . . . . . . . .
1.6 “Integral” Models . . . . . . . . . . . . . . . . . . . .
1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . .
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I
Bases of Sciences
2
Basic Fluid Mechanics
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Thermodynamics and mechanics concepts . . . . . . . . . .
2.2.1
Thermodynamics . . . . . . . . . . . . . . . . . . . .
2.3 Fundamentals of Fluid Mechanics . . . . . . . . . . . . . . .
2.3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.3.2
What is Shear Stress? . . . . . . . . . . . . . . . . . .
2.3.3
Mass Conservation . . . . . . . . . . . . . . . . . . .
2.3.4
Reynolds Transport Theorem . . . . . . . . . . . . .
2.3.5
Momentum Equation . . . . . . . . . . . . . . . . . .
2.3.6
Momentum Equation in Acceleration System . . . .
2.3.7
Differential Analysis . . . . . . . . . . . . . . . . . .
2.3.8
Conservation of General Quantity . . . . . . . . . .
2.3.9
Examples for Differential Equation (Navier-Stokes)
1
3
5
6
6
8
12
12
14
18
19
21
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23
. 23
. 23
. 23
. 31
. 31
. 31
. 33
. 37
. 37
. 39
. 40
. 44
. 53
xxix
CONTENTS
2.4
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. 78
Dimensional Analysis
3.1 Basics of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1
How The Dimensional Analysis Work . . . . . . . . . . . . . . .
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Nusselt Schmidt method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1
Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2
Theory Behind Dimensional Analysis . . . . . . . . . . . . . . .
3.4 Nusselt’s Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Summary of Dimensionless Numbers . . . . . . . . . . . . . . . . . . . .
3.5.1
The Significance of these Dimensionless Numbers . . . . . . . .
3.5.2
Relationship Between Dimensionless Numbers . . . . . . . . . .
3.5.3
Examples for Dimensional Analysis . . . . . . . . . . . . . . . .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Appendix summary of Dimensionless Form of Navier–Stokes Equations
3.7.1
The ratios of various time scales . . . . . . . . . . . . . . . . . .
3.8 Similarity applied to Die cavity . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1
Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2
Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Summary of dimensionless numbers . . . . . . . . . . . . . . . . . . . . .
3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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81
81
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2.5
2.6
3
Compressible Flow . . . . . . . . . . .
2.4.1
Speed of Sound . . . . . . . .
2.4.2
Choked Flow . . . . . . . . .
2.4.3
Shock Wave . . . . . . . . . .
Solution of the Governing Equations
2.5.1
Informal Model . . . . . . . .
2.5.2
Formal Model . . . . . . . . .
2.5.3
Prandtl’s Condition . . . . . .
Operating Equations and Analysis . .
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4 The Die Casting Process Stages
4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
4.1.1
Filling The Shot Sleeve . . . . . . . . . . .
4.1.2
Plunger Slow Moving Part . . . . . . . . .
4.1.3
Runner System . . . . . . . . . . . . . . .
4.1.4
The Mold . . . . . . . . . . . . . . . . . .
4.1.5
Intensification Period . . . . . . . . . . . .
4.1.6
Concluding Remarks For Intensification .
4.2 Special Topics . . . . . . . . . . . . . . . . . . . . .
4.2.1
Is the Flow in Die Casting Turbulent? . .
4.2.2
Dissipation effect on the temperature rise
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xxx
CONTENTS
4.3
4.4
4.5
5
4.2.3
Gravity effects . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimates of the time scales in die casting . . . . . . . . . . . . . . .
4.3.1
Utilizing semi dimensional analysis for characteristic time
4.3.2
The ratios of various time scales . . . . . . . . . . . . . . .
Similarity applied to Die cavity . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fundamentals of Pipe Flow
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
5.2 Universality of the loss coefficients . . . . . . . . . . .
5.3 A simple flow in a straight conduit . . . . . . . . . . .
5.3.1
Examples of the calculations . . . . . . . . . .
5.4 Typical Components in the Runner and Vent Systems
5.4.1
bend . . . . . . . . . . . . . . . . . . . . . . .
5.4.2
Y connection . . . . . . . . . . . . . . . . . .
5.4.3
Expansion/Contraction . . . . . . . . . . . .
5.5 Putting it all to Together . . . . . . . . . . . . . . . . .
5.5.1
Series Connection . . . . . . . . . . . . . . .
5.5.2
The Parallel Connection . . . . . . . . . . . .
6 Runner Design
6.1 Introduction . . . . . . . . . . . . . .
6.1.1
Backward Design . . . . . . .
6.1.2
Connecting runner segments
6.1.3
Resistance . . . . . . . . . . .
7
pQ2
7.1
7.2
7.3
7.4
7.5
7.6
7.7
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Diagram Calculations
Introduction . . . . . . . . . . . . . . .
The “common” pQ2 diagram . . . . . .
The validity of the “common” diagram .
7.3.1
Is the “Common” Model Valid?
7.3.2
Are the Trends Reasonable? . .
7.3.3
Variations of the Gate area, A3
The Correct pQ2 Diagram . . . . . . .
7.4.1
The reform model . . . . . . .
7.4.2
Examining the solution . . . . .
7.4.3
Poor design effects . . . . . . .
7.4.4
Transient effects . . . . . . . . .
Design Process . . . . . . . . . . . . . .
The Intensification Consideration . . .
Summary . . . . . . . . . . . . . . . . .
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151
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163
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xxxi
CONTENTS
8
Critical Slow Plunger Velocity
8.1 Introduction . . . . . . . . . . . . . . .
8.2 The “common” models . . . . . . . . . .
8.2.1
Garber’s model . . . . . . . . .
8.2.2
Brevick’s Model . . . . . . . . .
8.2.3
Brevick’s circular model . . . .
8.2.4
Miller’s square model . . . . . .
8.3 The validity of the “common” models .
8.3.1
Garber’s model . . . . . . . . .
8.3.2
Brevick’s models . . . . . . . .
8.3.3
Miller’s model . . . . . . . . . .
8.3.4
EKK’s model (numerical model)
8.4 The Reformed Model . . . . . . . . . .
8.4.1
The reformed model . . . . . .
8.4.2
Design process . . . . . . . . .
8.5 Summary . . . . . . . . . . . . . . . . .
8.6 Questions . . . . . . . . . . . . . . . . .
9 Venting System Design
9.1 Introduction . . . . . . .
9.2 The “common” models . .
9.2.1
Early (etc.) model
9.2.2
Miller’s model . .
9.3 General Discussion . . . .
9.4 The Analysis . . . . . . .
9.5 Results and Discussion . .
9.6 Summary . . . . . . . . .
9.7 Questions . . . . . . . . .
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201
201
201
202
204
205
205
206
206
206
206
206
207
207
209
209
209
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211
211
212
212
212
213
215
217
220
220
10 Clamping Force Calculations
11 Analysis of Die Casting Economy
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
11.2 The “common” model, Miller’s approach . . . . . .
11.3 The Validity of Miller’s Price Model . . . . . . . . .
11.4 The combined Cost of the Controlled Components
11.5 Die Casting Machine Capital Costs . . . . . . . . . .
11.6 Operational Cost of the Die Casting Machine . . .
11.7 Runner Cost (Scrap Cost) . . . . . . . . . . . . . . .
11.8 Start–up and Mold Manufacturing Cost . . . . . .
11.9 Personnel Cost . . . . . . . . . . . . . . . . . . . . .
11.10 Uncontrolled components . . . . . . . . . . . . . .
11.11 Minimizing Cost of Single Operation . . . . . . . .
221
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223
223
224
224
226
226
227
227
230
230
231
232
xxxii
CONTENTS
11.12 Introduction to Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
11.12.1 Marginal Profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
11.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
A
Fanno Flow
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
A.2 Fanno Model . . . . . . . . . . . . . . . . . . . . . .
A.3 Non–Dimensionalization of the Equations . . . . .
A.4 The Mechanics and Why the Flow is Choked? . . .
A.4.1 Why the flow is choked? . . . . . . . . . . .
A.4.2 The Trends . . . . . . . . . . . . . . . . . . .
A.5 The Working Equations . . . . . . . . . . . . . . . .
A.6 Examples of Fanno Flow . . . . . . . . . . . . . . . .
A.7 Supersonic Branch . . . . . . . . . . . . . . . . . . .
A.8 Working Conditions . . . . . . . . . . . . . . . . . .
A.8.1 Variations of The Tube Length ( 4fL
D ) Effects
A.8.2 The Pressure Ratio, P2 / P1 , effects . . . . .
A.8.3 Entrance Mach number, M1 , effects . . . .
A.9 Practical Examples for Subsonic Flow . . . . . . . .
A.10 The Table for Fanno Flow . . . . . . . . . . . . . .
A.11 Appendix – Reynolds Number Effects . . . . . . .
B Flow in Open Channels
B.1 Introduction . . . . . . . . . . . . . . . . . .
B.2 What is Open Channel Flow? . . . . . . . . .
B.2.1
Introduction . . . . . . . . . . . . . .
B.2.2 Open Channel “Intuition’ . . . . . .
B.2.3 Energy Line . . . . . . . . . . . . . .
B.3 Energy conservation . . . . . . . . . . . . . .
B.3.1
Some Design Considerations . . . .
B.3.2 Expansion and Contraction . . . . .
B.3.3 Summery . . . . . . . . . . . . . . .
B.4 Hydraulic Jump . . . . . . . . . . . . . . . . .
B.4.1 Poor Man Dimensional Analysis . .
B.4.2 Velocity Profile . . . . . . . . . . . .
B.5 Cross Section Area . . . . . . . . . . . . . . .
B.5.1
Introduction . . . . . . . . . . . . . .
B.6 Energy For Non–Rectangular Cross–Section
B.6.1 Triangle Channel . . . . . . . . . . .
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241
241
242
243
246
246
246
247
252
256
257
257
262
265
270
270
272
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275
275
276
276
277
278
281
288
290
294
295
298
300
301
301
303
306
xxxiii
CONTENTS
C What The Establishment’s Scientists Say
C.1 Summary of Referee positions . . . .
C.2 Referee 1 (from hand written notes) .
C.3 Referee 2 . . . . . . . . . . . . . . . . .
C.4 Referee 3 . . . . . . . . . . . . . . . . .
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309
310
311
312
315
D My Relationship with Die Casting Establishment
321
Index
337
Bibliography
337
xxxiv
CONTENTS
List of Figures
1.1
1.2
1.3
1.4
The profits as a function of the amount of the scrap
Increase of profits as reduction of scrap reduction. .
Filling the test bars via two different designs . . . .
Filling time according to Weishan et all . . . . . . .
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2
5
13
13
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
The velocity distribution in Couette flow . . . . . . . . . . . . . . . . .
The deformation of fluid due to shear stress as progression of time. . .
Control volume and system in motion . . . . . . . . . . . . . . . . . . .
Piston control volume . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematics of velocities at the interface . . . . . . . . . . . . . . . . . .
Schematics of flow in a pipe with varying density . . . . . . . . . . . .
The explanation for the direction relative to surface . . . . . . . . . . .
Mass flow due to temperature difference . . . . . . . . . . . . . . . . .
Mass flow in coating process . . . . . . . . . . . . . . . . . . . . . . . .
Control volume at different times under continuous angle deformation
Shear stress at two coordinates in 45◦ orientations . . . . . . . . . . . .
1–Dimensional free surface . . . . . . . . . . . . . . . . . . . . . . . . .
Flow driven by surface tension . . . . . . . . . . . . . . . . . . . . . . .
Flow in candle with a surface tension gradient . . . . . . . . . . . . . .
Flow between two plates when the top moving . . . . . . . . . . . . . .
One dimensional flow with shear between plates . . . . . . . . . . . . .
caption top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The control volume of liquid element in “short cut” . . . . . . . . . . .
Mass flow due to gravity . . . . . . . . . . . . . . . . . . . . . . . . . . .
Liquid flow due to gravity . . . . . . . . . . . . . . . . . . . . . . . . . .
A very slow moving piston in a still gas . . . . . . . . . . . . . . . . . .
Stationary sound wave and gas moves relative to the pulse. . . . . . . .
Gas flow through a converging–diverging nozzle. . . . . . . . . . . . .
The stagnation properties as a function of the Mach number, k=1.4 . .
A shock wave inside a tube . . . . . . . . . . . . . . . . . . . . . . . . .
The intersection of Fanno flow and Rayleigh flow . . . . . . . . . . . .
The Mexit and P0 as a function Mupstream . . . . . . . . . . . . . .
The ratios of the static properties of the two sides of the shock. . . . .
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31
32
33
34
35
36
38
41
43
45
46
50
52
53
53
55
56
58
60
62
65
66
67
68
71
73
77
79
xxxv
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xxxvi
LIST OF FIGURES
3.1
3.2
3.3
Rod into the hole example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Description of the boat crossing river . . . . . . . . . . . . . . . . . . . . . . . 111
Rigid body brought into rest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
Hydraulic jump in the shot sleeve . . . . . . . . . . . . . . . . . . . .
Filling of the shot sleeve . . . . . . . . . . . . . . . . . . . . . . . . .
Jet dragging air into the liquid medium . . . . . . . . . . . . . . . .
Heat transfer processes in the shot sleeve . . . . . . . . . . . . . . .
Solidification process in the shot sleeve time estimates . . . . . . .
Interface instability interface of liquid into gas . . . . . . . . . . . .
2–D of Velocity Profile to explain the nomenclature . . . . . . . . .
Entrance of liquid metal to the runner . . . . . . . . . . . . . . . . .
Flow in runner when during pressurizing process. . . . . . . . . . .
Typical flow pattern in die casting, jet entering into empty cavity. .
Thermal expansion of Aluminum . . . . . . . . . . . . . . . . . . . .
Transition to turbulent flow in instantaneous flow after Wygnanski
Flow patterns in the shot sleeve . . . . . . . . . . . . . . . . . . . . .
Two streams of fluids into a medium . . . . . . . . . . . . . . . . . .
Schematic of heat transfer processes in the die. . . . . . . . . . . . .
The oscillating manometer for the example 4.5 . . . . . . . . . . . .
Mass Balance on the lest side of the manometer . . . . . . . . . . .
Rigid body brought into rest. . . . . . . . . . . . . . . . . . . . . . .
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126
127
129
129
130
134
138
140
140
142
143
148
148
150
153
156
157
160
5.1
5.2
5.3
5.4
5.5
Friction of orifice as a function velocity.
General simple conduit description . .
General simple conduit description. . .
A sketch of the bend in die casting. . . .
3 Parallel Pipes in parallel connections .
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164
165
165
167
168
6.1
6.2
A geometry of runner connection . . . . . . . . . . . . . . . . . . . . . . . . . 170
y connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.1
7.2
7.3
7.4
. 173
. 174
. 176
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Schematic of typical die casting machine . . . . . . . . . . . . . . . . . . . .
A typical trace on a cold chamber machine . . . . . . . . . . . . . . . . . . .
The “common” pQ2 version . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pmax and Qmax as a function of the plunger diameter according to “common” model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Hydralic piston schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 P̄ as A3 to be relocated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 P1 as a function of Pmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8 KF as a function of gate area, A3 . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 Gate velocity area ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10 Gate velocities as a function of the area ratio for constant power . . . . . . .
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176
177
179
181
186
187
187
xxxvii
LIST OF FIGURES
7.11
7.12
7.13
7.14
7.15
7.16
7.17
Die casting characteristics. . . . . . . . . . . . . . . . . . . . . .
Various die casting machine performances . . . . . . . . . . . .
Reduced pressure performances as a function of Ozer number. .
Schematic of the plunger and piston balance forces . . . . . . .
Metal pressure at the plunger tip. . . . . . . . . . . . . . . . . . .
The gate velocity, U3 as a function of the plunger area, A1 . . .
The reduced power as a function of the normalized flow rate. .
8.1
8.2
8.3
A schematic of wave formation in stationary coordinates . . . . . . . . . . . . 202
The two kinds in the sleeve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
A schematic of the wave with moving coordinates . . . . . . . . . . . . . . . . 203
9.1
9.2
9.3
The relative shrinkage porosity as a function of the casting thickness. . . . . . 211
A simplified model for the venting system. . . . . . . . . . . . . . . . . . . . . 216
The pressure ratios for air and vacuum venting at end. . . . . . . . . . . . . . 218
11.1
11.2
11.3
Production cost as a function of the runner hydraulic diameter . . . . . . . . 224
The reduced power as a function of the normalized flow rate . . . . . . . . . . 229
Supply and Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
A.1
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.10
A.11
A.12
A.13
A.14
A.15
A.16
A.17
Control volume of the gas flow in a constant cross section . . .
Various parameters in fanno flow . . . . . . . . . . . . . . . . .
Schematic of Example A.1 . . . . . . . . . . . . . . . . . . . . . .
The schematic of Example (A.2) . . . . . . . . . . . . . . . . . . .
fL
The effects of increase of 4D
on the Fanno line . . . . . . . . .
fL
The effects of the increase of 4D
on the Fanno Line . . . . . .
Min and ṁ as a function of the 4fL
D . . . . . . . . . . . . . . . .
M1 as a function M2 for various 4fL
D . . . . . . . . . . . . . . .
M1 as a function M2 . . . . . . . . . . . . . . . . . . . . . . . .
fL
The pressure distribution as a function of 4D
. . . . . . . . .
4fL
Pressure as a function of long D . . . . . . . . . . . . . . . . .
The effects of pressure variations on Mach number profile . . .
fL
fL
Pressure ratios as a function of 4D
when the total 4D
= 0 .3
Schematic of a “long” tube in supersonic branch . . . . . . . .
The extra tube length as a function of the shock location . . .
The maximum entrance Mach number as a function of 4fL
D . .
“Moody” diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
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241
251
252
253
257
257
258
260
261
262
263
264
265
265
266
267
272
B.1
B.2
B.3
B.4
B.5
Equilibrium of Forces in an open channel . . . . . . . . .
Open channel flow in die casting . . . . . . . . . . . . . .
What is open channel flow . . . . . . . . . . . . . . . . .
Change of the height of the bottom . . . . . . . . . . . .
Flow on an include plane to change the bottom direction
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275
276
277
277
278
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189
190
191
191
193
196
197
xxxviii
B.6
B.7
B.8
B.9
B.10
B.11
B.12
B.13
B.14
B.15
B.16
B.17
B.18
B.19
B.20
B.21
B.22
B.23
B.24
B.25
B.26
B.27
B.28
B.29
LIST OF FIGURES
Uniform flow on include plane . . . . . . . . . . . . . . . . . . . .
Control volume from the front . . . . . . . . . . . . . . . . . . . .
Force balance in the flow direction open channel . . . . . . . . . .
Constant cross section in general . . . . . . . . . . . . . . . . . . .
Small control volume to ascertain shear stress . . . . . . . . . . .
to explain the transition from Shear stress to velocity function . .
a
Height lines for open channel as a function of the energy
b
Energy line with the effects of elevation change . . . . . .
Energy lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transition from subcritical to supercritical . . . . . . . . . . . . .
Downstream flow height as a function of the step height . . . . .
Energy Diagram q=10[m2 . . . . . . . . . . . . . . . . . . . . . . .
Energy line for flow rate 1.5 . . . . . . . . . . . . . . . . . . . . . .
Flow rate as a function of the energy or the height . . . . . . . . .
Flow Rate as Function of height, h, for various, H . . . . . . . . .
Flow Rate as Function of height, h, for various, H . . . . . . . . .
expansion and contraction top view in gradual and abrupt . . . .
Flow in contraction subcritical flow . . . . . . . . . . . . . . . . .
Flow in contraction subcritical energy Diagram . . . . . . . . . .
The Flow Rate for Contraction Exercise . . . . . . . . . . . . . . .
Flow in Contraction Supercritical Energy Diagram . . . . . . . .
The energy line and surface line with energy lost . . . . . . . . . .
Schematic of hydraulic jump . . . . . . . . . . . . . . . . . . . . .
Optimal angle for triangular cross section . . . . . . . . . . . . . .
Specific energy lines for non–rectangular channel . . . . . . . . .
Open channel flow in an isosceles triangular shape . . . . . . . . .
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Notice of Copyright For This Document:
This document published Modified FDL. The change of the license is to prevent from situations where the author has to buy his own book. The Potto Project License isn’t long apply
to this document and associated docoments.
GNU Free Documentation License
The modification is that under section 3 “copying in quantity” should be add in the end.
"If you print more than 200 copies, you are required to furnish the author with two (2) copies
of the printed book.”
Version 1.2, November 2002
Copyright ©2000,2001,2002 Free Software Foundation, Inc.
51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
Everyone is permitted to copy and distribute verbatim copies of this license document, but
changing it is not allowed.
Preamble
The purpose of this License is to make a manual, textbook, or other functional
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This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public
License, which is a copyleft license designed for free software.
We have designed this License in order to use it for manuals for free software,
because free software needs free documentation: a free program should come with manuals
providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether
it is published as a printed book. We recommend this License principally for works whose
purpose is instruction or reference.
xxxix
xl
LIST OF FIGURES
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LIST OF FIGURES
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ADDENDUM: How to use this License for your documents
xlvi
LIST OF FIGURES
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in the document and put the following copyright and license notices just after the title page:
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GNU General Public License, to permit their use in free software.
About This Author
Dr. Genick Bar-Meir is a world renowned scientist who made breakthroughs in several
areas which include die casting, compressible flow, oceanography, ship stability, mathematics
etc. He especially known for demonstrating that strongly and fundamental held “scientific”
believes to be false and correcting them. He further introduce many new scientific concepts
that revitalize several areas. For instance, most the ship stability research carried by professors
from all around the world including MIT, University of Iowa, major European Universities,
and from Japanese conflicts with the physical observations. This fact was explained in Bar–
Meir’s book on ship stability.
The author’s work in die casting is presented in the introduction chapter and will
not be discussed there. The Author limited list of achievements includes the following:
• Develop the connection between the caudal (tail) fin shape and propulsion of the fish.
• Develop the connection between the pectoral fin and the power/force
• Demonstrate that von Karman’s vortices do no propel the fish but indication of the
energy transfer
• Developed the equation of the ocean great depth for pressure, and speed of sound
• Developed the equation to calculate the circle segment centroid
• Create new method to calculate the ship stability
• Made major progress in the parallel axis for the added moment of inertia
• Discover the transfer properties
• Locate the ship rotation point
• Discover the transfer mechanisms ship the movement modes
• Built the ship stability dome
• Developed the tool to ascertain the Energy in Shock Tubes
• Explain some moving shock dynamics
• Make progress in real gas understanding
xlvii
xlviii
LIST OF FIGURES
• Oblique shock analytical solution and new solution
Bar–Meir holds a Ph.D. in Mechanical Engineering from University of Minnesota
and a Master in Fluid Mechanics from Tel Aviv University. Dr. Bar-Meir was the last student
of the late Dr. R.G.E. Eckert. He started his reach in the area heat and mass transfer where
he found analytical solution to equation which mathematicians could only prove that the
solution exist. Currently, solutions without practical application seems less interesting to
this author.
Bar-Meir’s books are used by millions of peoples as it shown by independent web
sites which publish his books. His books are used in many universities like Purdue, Caltech,
Queens University in Canada, Singapore etc. According to Google his books are very popular. Bar–Meir’s ideas are important as evident from the fact that people Dr. Sandip Ghosal
from Northwestern University Engineering plagiarizing it. Additionally there other who are
claiming Bar–Meir’s ideas unspecified the source after they download Bar–Meir’s books.
In his early part of his professional life, Bar-Meir was mainly interested in elegant models whether they have or not a practical applicability. Now, this author’s views had
changed and the virtue of the practical part of any model becomes the prominent part of his
ideas, books and software. The change occurred when Bar-Meir developed models for several
manufacturing processes.
In the area of compressible flow, it was commonly believed and taught that there is
only weak and strong shock and it is continue by Prandtl–Meyer function. Bar–Meir discovered the analytical solution for oblique shock and showed that there is a quiet buffer between
the oblique shock and Prandtl–Meyer. He also build analytical solution to several moving
shock cases. He described and categorized the filling and evacuating of chamber by compressible fluid in which he also found analytical solutions to cases where the working fluid
was ideal gas. The common explanation to Prandtl–Meyer function shows that flow can turn
in a sharp corner. Engineers have constructed design that based on this conclusion. Bar-Meir
demonstrated that common Prandtl–Meyer explanation violates the conservation of mass
and therefor the turn must be around a finite radius. The author’s explanations on missing
diameter and other issues in Fanno flow and ““naughty professor’s question”” are used in the
industry.
In his book “Basics of Fluid Mechanics”, Bar-Meir demonstrated that fluids must
have wavy surface when two different materials flow together. All the previous models for the
flooding phenomenon did not have a physical explanation to the dryness. He built a model to
explain the flooding problem (two phase flow) based on the physics. He also constructed and
explained many new categories for two flow regimes.
Bar-Meir’s ideas are spreading around, as can be evidenced by several things: On
the one hand, Dr. Jiarong Hong, from the University of Minnesota, alleged that the solution
for the deep ocean is useless or cumbersome. On the other hand, Dr. Sandip Ghosal, from
Northwestern University, plagiarized Bar-Meir’s work on deep ocean pressure calculations,
and he spent almost a week teaching this topic, and even copied Bar-Meir’s nomenclature. In
fact, one-third of the mid-quarter exam that he gave to his students was taken from this topic.
It is also interesting to point out that Dr. Hong also suggested to “burn” the Eckert (Schmidt)
GNU FREE DOCUMENTATION LICENSE
xlix
dimensional analysis method, which Eckert introduced to heat and mass transfer education.
100 years of use of Schmidt/Eckert method now and it is currently almost exclusive use, the
genius Hong think that shout be ignored.
The adherence that Dr. Ghosal displayed to Bar-Meir’s work was probably because
he did not think that he would be caught. On the other hand, Kostas J. Spyrou, when he gave
his keynote lecture, basically copied many concepts that were developed by Bar-Meir (such as
the stability dome) and presented them as his ideas. These concepts had never appeared in the
literature before. It is interesting to point out that Dr. Spyrou has been observed downloading
Bar-Meir’s book. Dr. Spyrou’s behavior is not unique to him only. Dr. Kenneth Brezinsky
went out of his way to declare that Bar-Meir’s calculations of compressible gas potential were
useless and baseless. However, Brian J. Cantwell from Stanford University recently decided to
copy Bar-Meir’s idea and to dedicate a whole section (2.9) to it in his book (and probably also
in his classes). Except that B. Cantwell oversimplified the idea (he removed the dimensionless
presentation), maybe because he did not fully understand it. There are more cases of this
copying/plagiarizing, but normally they try to make the appearance that there is nothing to
see here while the using the Bar–Meir’s ideas.
l
LIST OF FIGURES
How This Book Was Written
This book started because the author was frustrated with the system made of NADCA and
associates that promote erroneous research simply for economical reasons. Then, the author
believed that the book cannot be “stolen” if they are under open content. Well it turn out
to be wrong. The die casting process is interesting enough to insert my contributions. Bar–
Meir have found that works or models in this area are lack of any serious scientific principles.
Thus, he started to write class notes to his clients and add his research work to create this
book. During the writing, he added the material on economy which he felt was missing piece
of knowledge in the die casting engineering.
Later, the author wrote several books on fluid mechanics, compressible flow, ship
stability etc. In these books the author made several breakthroughs in different areas for example in the added mass. These material were back ported into this book to make this book
more complete. For example, the open channel flow chapter in “Basics of Fluid Mechanics”
is summered and back ported to here with emphasis on the die casting aspect. Now the discussion on the mixing in the runner will be back ported and enlarged for the fluid mechanics
book with discussion for filling and emptying containers.
Of course, this book was written on Linux (Micro$oftLess book). This book was
written using the vim editor for editing (sorry never was able to be comfortable with emacs).
The graphics were done before by TGIF currently which now is replaced with IPE. The figures
were done by GRAP and family but are modified to be done GLE. The spell checking was done
by ispell, and hope to find a way to use gaspell, a program that currently cannot be used on
new Linux systems. The cover page figure was created by Genick Bar-Meir, and is copylefted
by him.
li
lii
HOW THIS BOOK WAS WRITTEN
Preface
"In the beginning, the POTTO project was without form, and
void; and emptiness was upon the face of the bits and files. And
the Fingers of the Author moved upon the face of the keyboard.
And the Author said, Let there be words, and there were words."
8.
This book, “Fundamentals of Die Casting Design,” describes the fundamentals of die casting
process design and economics for engineers and others. This book is designed to fill the gap
of the missing materials on economy and scientific principles and replace erroneous material
promoted by NADCA. It is hoped that the book could be used as a reference book for people
who have at least some basics knowledge of science areas such as calculus, physics, etc. It has
to be realized the some materials are very advance and required knowledge of fluid mechanics particularly compressible flow and open channel flow. This author’s popular books on
compressible flow and fluid mechanics should provide the introduction to these areas. The
readers’ reactions to this book and the usage of the book as a textbook suggested that the
chapter which deals with economy should be expand. In the following versions this area will
strength and expended.
The structure of this book is such that many of the chapters could be usable independently. For example, if you need information about, say, economy of the large scale productions, you can read just chapter 11. I hope this makes the book easier to use as a reference
manual. However, this manuscript is first and foremost a textbook, and secondly a reference
manual only as a lucky coincidence.
It have been attempted to describe why the theories the way they are, rather than just
listing “seven easy steps” for each task. This means that a lot of information is presented
which is not necessary for everyone. These explanations have been marked as such and can be
skipped. Reading everything will, naturally, increase your understanding of the many aspects
of fluid mechanics.
This book is written and maintained on a volunteer basis. Like all volunteer work, there
is a limit on how much effort I was able to put into the book and its organization. Moreover,
due to the fact that English is my third language and time limitations, the explanations are not
as good as if I had a few years to perfect them. Nevertheless, I believe professionals working in
many engineering fields will benefit from this information. This book contains many worked
examples, which can be very useful for many.
Some issues have been left with unsatisfactory explanations in the book, marked with
a Mata mark. It is hoped to improve or to add to these areas in the near future. Furthermore,
8 To the power and glory of the mighty God.
This book is only attempt to explain his power.
liii
liv
PREFACE
it is hoped that many others will participate of this project and will contribute to this book
(even small contributions such as providing examples or editing mistakes are needed).
Attempt were made to make this text of the highest quality possible and interested
in the users comments and ideas on how to make text better are welcome which include
Incorrect language, errors, ideas for new areas to cover, rewritten sections, more fundamental
material, more mathematics (or less mathematics). Users are encouraged to comment on in
the best arrangement of the book. If you want to be involved in the editing, graphic design,
or proofreading, please contact via Email at barmeir at gmail dot com.
Naturally, this book contains material that never was published before (sorry it cannot
avoid). This material never went through a close content review. The close content review and
publication in a professional publication is excellent idea in theory. In practice, this process
leaves a large room to blockage of novel ideas and plagiarism. For example, Brevick from
Ohio State is one the individual who attempted to block this author idea on pQ2 diagram. If
you would like to critic to my new ideas please send the author your comment(s). However,
please do not hide your identity, it clouds your motives.
Several people have helped with this book, directly or indirectly. The author would
like to especially thank to his adviser, Dr. E. R. G. Eckert, whose work was the inspiration for
this book. I also would like to thank to Jannie McRotien (Open Channel Flow chapter) and
Tousher Yang for their advice, ideas, and assistance.
Anyone with a penchant for writing, editing, graphic ability, LATEX knowledge, and material knowledge and a desire to provide open content textbooks and to improve them encourage to join the author in this project. If you have Internet e-mail access, you can contact
the author at “barmeir@gmail.com”.
To Do List and Road Map
This book isn’t complete and probably never will be completed. There will always new problems to add or to polish the explanations or include more new materials. Also issues that
associated with the book like the software has to be improved. It is hoped the changes in TEX
and LATEX related to this book in future will be minimal and minor. It is hoped that the style
file will be converged to the final form rapidly. Nevertheless, there are specific issues which
are on the “table” and they are described herein.
At this stage, some chapters are missing. Specific missing parts from every chapters
are discussed below. These omissions, mistakes, approach problems are sometime appear in
the book when possible. You are always welcome to add a new material: problems, questions, illustrations or photos of experiment(s). Material can be further illuminate. Additional
material can be provided to give a different angle on the issue at hand.
Properties
The chapter in beta stage and will be boosted in the future.
Inviscid Flow
To add the unsteady Bernoulli in moving frames To add K–J condition and Add properties.
Internal Viscous Flow
To add this Chapter. – done
Open Channel Flow
The chapter isn’t in the development stage yet. Some parts were taken from Fundamentals of
Die Casting Design book and are in a process of improvement. – almost done
pQ2 improvements
Missing the economic aspect of the pQ2 .
1
Introduction
In the recent years, many die casting companies have gone bankrupt (Doehler–Jarvis and
Shelby to name a few) and many other die casting companies have been sold (St. Paul Metalcraft, Tool Products, OMC etc.). What is/are the reason/s for this situation? Some blame poor
management. Others blame bad customers (which is mostly the automobile industry). Perhaps there is something to these claims. Nevertheless, one can see that the underlying reasons
are the lake of knowledge on how to calculate parameters required to increase the profits. To
demonstrate how the absurd situation is the fact that there is not even one company today
that can calculate the actual price of any product that they are producing. Moreover, if a company is able to produce a specific product, no one in that company looks at the redesign (mold
or process) in order to reduce the cost systematically.
Here is challenge, If there is a company which does such a thing, the author will be
more than glad to learn about it. You will get a reward. With this in mind, book cannot be
written only for the engineers but also for the academic community that should support the
die casting industry. This conflicting demands insert additional complications in the writing.
As simple engineering would mostly would like to have seven step solutions, the researchers
will must know why. Thus, this book must dive into complicated material occasionally. Additionally, not everything can be solved by one individual. Unfortunately, simple numerical
simulations are useless if they do not describe the physics. It is hope quality researchers who
will join and put the required time the problems in die casting.
In order to compete with other industries and other companies, the die casting indusmust
try
reduce the cost as much as possible (20% to 40%) and lead time significantly (by 1/2
or more). To achieve these goals, the engineer must learn to connect mold design to the cost
1
2
CHAPTER 1. INTRODUCTION
Cost
Prof its
Maximum
Investment
of production (charged to the customer) and to use the correct scientific principals involved
in the die casting process to reduce/eliminate the guess work. This book is part of the revolution in die casting by which science is replacing the black art of design. For the first time,
a link between the cost and the design is spelled out. Many new concepts, based on scientific
principles, are introduced. The old erroneous models, which was plagued by the die casting
industry for many decades, are analyzed, their errors are explained and the old models are
superseded.
“Science is good, but it is not useful in the floor of our plant!!” George Reed, the former
president of SDCE, in 1999 announced in a meeting in the local chapter (16) of NADCA. He
does not believe that there is a relationship between “science” and what he does with the die
casting machine. He stated that because he does not follow NADCA recommendations, he
achieves good castings. For instance, he stated that the common and NADCA supported, recommendation in order to increase the gate velocity, plunger diameter needs to be decreased.
He said that because he does not follow this recommendation, and/or others, that is the reason
his succeeds in obtaining good castings. He is right and wrong. He is right not to follow the
NADCA recommendations since their models violate many basic scientific principles. One
should expect that models violating scientific principles would produce unrealistic results.
When such results occur, this should actually strengthen the idea that science has validity. The
fact that models which appear in books today are violating scientific principals and therefore
they do not work should actually convince him, and others, that science does have validity.
Mr. Reed is right (in certain ranges) to increase the diameter in order to increase the gate
velocity as will be covered in Chapter 7.
The above example is but one of all the significant models that are errant and in need of
major corrections. To this date, the author has not found so much as a single
“commonly” used model that has been correct
in its conclusions, trends, and/or assumptions.
Prof its
%
23
The wrong models/methods that have plagued
20
23
the industry are: 1) critical slow plunger veloc0
ity, 2) pQ2 diagram, 3) plunger diameter calBreakeven
culations, 4) runner system design, 5) vent sysPoint
tem design, etc. These incorrect models are
∼ 65 − 75 78 80
0
100
P roduct
the reasons that “science” does not work. The
scrap 0
100
models presented in this book are here for the
Fig. 1.1 – The profits as a function of the amount
purpose of answering the questions of design
of the scrap. The drawing is not to scale.
in a scientific manner which will result in reduction of costs and increased product quality.
This author would like to find/learn about even about a single model presently used in the
industry that is correct in any area of the process modeling (understanding) supported or
sponsored or associated with or by NADCA.
Once the reasons for why “science” does not work are clear, one should learn the models
that based on real physical principles for improving quality, reducing lead time and reduce
the production cost. The main underlying reason people are in the die casting business is to
1.1. THE IMPORTANCE OF REDUCING PRODUCTION COSTS
3
make money. One has to use science to examine what components of production cost/scrap
are and how to minimize or eliminate each of them to increase profitability. The underlying
purpose of this book is to help the die caster to achieve this target.
As writing this revision Lordan et al (2022) publish summary on how he and his co–
workers keep up the “no science” in our backyard. This author contacted them in the past
so they are aware that real science have been done like the concepts like critical vent area,
critical plunger velocity etc. Yet, they refuse to use any concept of how the air escapes from
the cavity as it is a mystery to them. The key word in their vocabulary is that they have experience. However, the simulations that they carry are wrong because they simulate imaginary
situations with wrong boundary conditions.
1.1
The Importance of Reducing Production Costs
This author observed that NADCA’s indoctrinated engineers that the scrap is irrelevant in
the economics product. The same as the critical plunger velocity, the establishment with its
head NADCA did not study this topic (Chandrasekaran, Campilho, and Silva 2019). Outside,
organizations, mostly from the Asia, recently realized that something has to be done use statistical methods without gaining any real knowledge. If you do not know what to effects are
important, how you can get any information (see Taguchi’s method). It is analogous to study
examining what cause water to boil by looking if you are drinking water in the morning or
the evening but ignoring if there is a heat source. Well, boiling water and drinking water
both water. The examination was motivated by the energy cost and not by the economy of
the production. There are research papers like Neto (2009) that looks on the environmental
issues none with economics. However, there is no serious paper or book that deals with the
scrap and economics scientifically was found. This section is to fill this gap.
The amount of scrap created by each product is a major factor in the economy of the
industry. Contrary to popular belief, a reduction of a few percentage points of the production
cost/scrap does not translate into the same percentage of increase in profits. The increase is a
more complicated function. To study the relationship further, (see Fig. 1.1) to where profits are
plotted as a function of the scrap. Assuming that a linear function describes the relationship
when the secondary operations such as the trimming are neglected. This linear assumption
just simplifies the explanation. Naturally, the maximum loss occurs when all the material
turned to be scrap and it is referred to as the “investment cost” per unit. On the other hand, the
maximum profits occurs when all the material becomes products (no scrap of any kind (again
see Figure 1.1). While the linearly assumption stated above is not essential for the analysis, it
makes the analysis simpler. Regardless the function chosen the functionally of the relationship
is monotonic increase. It means increase of percent of the product must follow by increase of
the profit. The breakeven point (BEP) has to exist somewhere between these two extremes.
It has to be point out that it is physically impossible to make die casting operation
without scrap for several reasons. These reasons include some material has to push out the
air at the vent to minimize the porosity and cold shot, some material has to be in runners
to allow the intensification pressure to be transfer the pressure, and mismatch between ideal
4
CHAPTER 1. INTRODUCTION
plunger diameter to available diameter. Additionally, the instability interface of liquid metal
discovered and explained by this author requires some flushing. There is possible scrap due
to the leakage but this scrap is unintentional. The reject project and initial run also should be
considered but during the typical run, this effect is neglected.
Typically for the die casting industry the breakeven point lies within the range of 55%–
75% product (or 25%-45% scrap). Typical profits in the die casting industry are or should be
about 20% or more. When the profits fails below 15% or typical profit in the stock exchange
then the production should stop. If it is below 15%, the owner should consider investing in
the stock exchange instead being in die casting business. There is a possibility to make more
profits this way. Perhaps this is the reason why so many die casting companies are currently
not doing well in business.
Under the linear assumption, the relative profits of the operation depends on BEP as
it by definition a point where no profit and no lost. The new profit is difference between the
percent
From Fig. 1.1 it can be observed that
relative change in profits% =
new product percent − BEP
− 1 × 100
old product percent − BEP
(1.1)
As a demonstration Ex. 1.1 a check of the effect for a small change in the scrap on the profits.
Example 1.1: What Profit
Level: Basics
What would be the effect on the profits of a small change (2%) in a amount of scrap
for a job with 22% scrap (78% product) and with breakeven point of 65%?
Solution
Utilizing Eq. (1.1) which is applicable in this case reads
80 − 65
− 1 × 100 = 15.3%
78 − 65
(1.1.a)
A reduction of 2% in a amount of the scrap to be 20% or 80% product from 22% (78% product)
results in increase of more than 15.3% in the profits. This mount is a very substantial difference.
Therefore, a much bigger reduction in scrap will result in larger increase in the profits.
To analysis the change of BEP on the relative profit, consider Fig. 1.2 which exhibits two
sub–figures are built for two “old” scrap values. In these figures, for every line BEP remains
constant while the scrap percent is reduced (product increase). The nomenclature on the left
stats the BEP for each line based on the color and for the whole sub–figure the initial delta
(old product percent - BEP) is constant as stated in the figure. The figure on the left exhibits
relative profit for the case of 10% delta (old product percent - BEP). on the right figure is for
20%. The two sub-figures (left and right) in Fig. 1.2 demonstrate that higher BEP the change
in reduction of scrap is more important. The smaller the old scrap difference is, the more
important the reduction is (see fig. A larger changes as compared to fig. B.).
5
1.2. DESIGNED/UNDESIGNED SCRAP/COST
500
400
350
160
140
Old Scrap = 10
300
250
200
100
80
60
100
40
50
Old Scrap = 20
120
150
0
40
50
60
70
180
Profit percent
Profit percent
200
40
50
60
70
80
450
20
40
50
60
70
80
Scrap percent
90
100
Fig a. For BEP= 10%
0
40
50
60
70
80
Scrap percent
90
100
Fig b. For BEP= 20%
Fig. 1.2 – The left graph depicts the increase of profits as reduction of the scrap for 10% + BEP. The right graph
depicts same for for 20% + BEP.
1.2
Designed/Undesigned Scrap/Cost
There can be many definitions of scrap. The best suited definition for the die casting industry
should be defined as all the metal that did not become a product. There are two kinds of
scrap/cost: 1) those that can be eliminated, and 2) those that can only be minimized. The first
kind is referred to here as the undesigned scrap and the second is referred as designed scrap.
What is the difference between the two scrap? It is desired not to have rejection of any part
(the rejection should be zero) and of course it is not designed. Therefore, this is the undesigned
scrap/cost. However, it is impossible to eliminate the runner completely and it is desirable
to minimize its size in such a way that the cost will be minimized. This minimization of cost
and this minimum scrap is the designed scrap/cost. The die casting engineer must distinguish
between these two scrap components in order to be able to determine what should be done
and what cannot be done.
Science can make a significant difference; for example, it is possible to calculate the
critical slow plunger velocity and thereby eliminating (almost) air entrainment in the shot
sleeve in order to minimize the air/gas porosity. As opposed to Chandrasekaran (2019), here it
is advocating to use the understanding of the process physics has to enter into the calculation.
This claim means that air porosity will be reduced and marginal products (even poor products
in some cases) will be converted into good quality products. In this way, the undesigned
scrap can be eliminated or minimized. Additional way of minimizing the scrap is changing
of several parameters. The minimum scrap/cost can be achieved when a combination of the
smallest possible runner volume and the cheapest die casting machine are selected for a single
cavity. Similar analysis can be done for multiply cavity molds. This topic will be studied
further in Chapter 11.
There is a possibility that a parameter which reduces the designed scrap/cost will reduce the undesigned scrap/cost. An example of such a parameter is the venting system design.
6
CHAPTER 1. INTRODUCTION
It will be shown that there is a critical design above which air/gas is exhausted easily and below which air is trapped. In the later case, the air/gas pressure builds up and results in a poor
casting (a large amount of porosity) The meaning of the critical design and above and below
critical design will be presented in Chapter 9. The analysis of the vent system demonstrates
that a design much above the critical design and design just above the critical design yielding
has almost the same results– small amount of air entrapment. One can design the vent just
above the critical design so the design scrap/cost is reduced to a minimum amount possible.
Now both targets have been achieved: less rejections (undesigned scrap) and less vent system volume (designed scrap). It also possible to have an opposite case in which reduction of
designed scrap results in poor design. The engineer has to be aware of these points.
1.3
Linking the Production Cost to the Product Design
It is a sound accounting practice to tie the cost of every aspect of production to the cost to
be charged to the customer. Unfortunately, the practice today is such that the price of the
products are determined by some kind of average based on the part weight plus geometry
and not on the actual design and production costs. Furthermore, this idea is also perpetuated
by researchers who do not have any design factor (El-Mehalawi, Liu, and Miller 1997). Here
it is advocated to price according to the actual design and production costs. It is believed
that a better pricing results from such a practice. In today’s practice, even after the project is
finished, no one calculates the actual cost of production, let alone calculating the actual profits.
The consequences of such a practice are clear: it results in no push for better design and with
no idea which jobs make profits and which do not. Furthermore, considerable financial cost
is incurred which could easily be eliminated. Several chapters in this book are dedicated to
linking the design to the cost (end-price).
The scrap is controlled by several parameters which include: the left over material in
the way to mold and over flow. There is no way to create a situation in which the mold is filled
without some material is left over to freezing in the way. Additionally, some material has to
spill for two main reasons, one to expel some of the mixed liquid metal and two remove to
cooled liquid metal (due phase change e.g. freezing). The amount of the material left over in
the supply line is referred as the runner system. The flushed out material is located in the vent
system. The scrap runner system is include material that is in plunger area and this referred
as the biscuit. The biscuit size is controlled by the critical plunger velocity and size plunger
size.
1.4
Historical Background
Die casting is relatively speaking a very forgiving process in which after tuning several variables one can obtain a medium quality casting with a large machine. Due to lack of knowledge, the prevail approach in die casting is throwing money (bigger machine) is used. For this
reason there has not been any real push toward doing a good research. Hence, all the major
advances in the understanding of the die casting process were not sponsored by any of die
1.4. HISTORICAL BACKGROUND
7
casting institutes/associations or governments. In fact, the idea that science is not important
and anyone can do the research is echo by NADCA in which one of board of the directors
state to the effect “we sure must find competent person to proof that the Bar–Meir is wrong.”
Thirty years later they did not find anyone who has the ability to prove that Bar–Meir was
wrong.
An example why it is not that easy. Favi et al (2017) suggested a method to price casting
products. They asserted that all the methods can be categorized by the following four classifications: (i) knowledge-based methods grounded on the estimator experience, (ii) analogical
methods based on the similarity with existing products, (iii) analytical methods based on elementary tasks decomposition and, (iv) parametric methods founded on the relations between
product characteristics and their cost. This idea is based on Chougule (2006). It is amazing
that none of the suggested method actually check the real cost of production of the past. Before going over these methods, one cannot wonder where the strange die casting machine
shown in the paper figure 1 came from. The exhibited machine simply cannot work because
the gravity. Putting this observation aside, clearly methods (i) and (ii) are identical because it
depends on the engineer ( or technician?). Method (iii) is based on the experience of operator
(engineer?) which none exist. Method (iv) is basically based on the product weight the most
common in the industry. Why none of these methods even check what is the scrap? The
main thing in the method should be the incentive to improve the design so that the cost will
be reduced. This research was published in 2017! Why researchers did not conduct a proper
literature review? This concept was described in the early version of this book. The above
ruminant of nonsense that has been debunked is an example why the science cannot move
forward.
As indication of the current and poor state in die casting four months ago Martinez et
al (2022) to advocate a method to incorporating some solutions to a statistical method. For example, he suggesting to use “Karni’s pQ2 diagram” (actually it is well known that Karni did not
invented as oppose to Martinez’s allegations) with his group erroneous critical plunger velocity with Bar–Meir venting system. Essentially he argues that the different locations (processes)
cannot be decoupled. As fact, he argues and point to poor die casting results as indication for
the decoupling cannot occur. Hence, there must be divorce from the unknown and to some
extend of the known science. In a way somehow incorporating some elements of known
models like Bar–Meir’s venting system and his group on the critical plunger velocity into a
statistical method. Basically Martinez accept as facts that all these methods are correct. Even
the semi analytical model such as Karni (Karni 1991b) (not including the secret data massaging)
erroneous physic explanation. Another prime example of lack of physics understanding of by
Lo´ pez et al (2003) where the plunger velocity was calculated based on the shallow–water
approximation (energy conservation) while in realty explanation is totaly different and based
on the hydraulic jump (energy lost). The fact that many models pointed and used by Martinez
and alleged to be correct while they are really erroneous is no reason to advocate to use statistical method but to promote real science. As a side note it should be pointed out that during
their publication, the correct solution to critical plunger velocity (Bar-Meir 1997; Bar-Meir
and Brauner 2021) was published for sometime with the correct pQ2 diagram (Bar-Meir and
8
CHAPTER 1. INTRODUCTION
Brauner 1999) which they ignored.
Many of the people in important positions in the die casting industry suffer from what
is known as the “Detroit’s attitude”, which is very difficult to change. “We are making a lot of
money so why change? and if do not, the Government will pay for it.”. Moreover, the controlling
personnel on the research funds believe that the die casting is a metallurgical/manufacturing
process and therefore, the research has to be carried out by either Metallurgical Engineers
or Industrial Engineers none of which have the background to carry proper research. For
example, the vent system requires understanding compressible flow.
Furthermore, should come as no surprise – that people–in–charge of the research
funding fund their own research. One cannot stop wondering if there is a relationship between so many erroneous models which have been produced and the personnel controlling
the research funding. A highlight of the major points of the progress of the understanding is
described herein.
1.4.1
The Main Challenges Faced By Die Casting
The over arching mission of the die casting engineer is to make money for his/her company.
To achieve this target, the engineer has to make better products with less cost. This statement
should not be controversial and yet it is. This book ignores the controversy and considered
this idea to be correct. These two aims are not conflicting but complementing each other. The
areas the engineer control are: the operational and the design parameters. These parameters
are: runner design, vent design, operational design (shot sleeve, pQ2 etc), The shape of the
product should be negotiated with clients with general guidelines.
There is relatively few die casting researchers with different backgrounds than industrial or metallurgical. In die casting and other areas such as fish locomotion are dominated
researchers who are clueless about fluid mechanics. Even those who have some fluid mechanics background are numerical guys and hence they know how to handle numerical scheme
but lack the understanding of the physics of fluid mechanics. Hence, the phenomenon of fake
science where violations of physics laws such as the volume equal diameter times the length
(see for example (Gašpár, Coranič, Majerník, Husár, Knapčíková, Gojdan, and Paško 2021))
appears as a “valid science”. This author observed a similar situation in different areas such as
the ship stability where almost the whole field believes that the rotation point does not affect
the moment of inertia.
1.4.1.1
Runner Design
In the area of runner design, there are about 40k articles about possible design guidelines and
understanding. For example, (Kan, Ipek, and Koru 2022) suggested “and the runner channels
should be as short as possible”. This citation is indication of two things, how poor the understanding and how horrible the researchers in the area. This paper (citation) was published in
2022 with rehashing recommendations from 1980. Another example of avoid science at any
cost is of (Niu, Liu, Li, and Ji 2022). The conclusion in their research on the runner design
9
1.4. HISTORICAL BACKGROUND
is that the more pressure creates larger velocity (really? Do you have to be researcher/engineer to claim this concept?). In recent paper by Rajkumar and Rajini (Rajkumar and Rajini
2021) applied a numerical simulation to examine the effects of the runner design. Beside many
mistakes to describe the physics, main conclusion is the numerical simulation work. It can
be assumed that word “work” means the simulation converge to a solution regardless that if
it does not solve the real problem. Gašpár et al (2021) accepted the formula for the filling time
suggested by Zhang (2013) as the maximum time as
Ti − Tf + S Z
t=C
T
(1.2)
Tf − Td
where
t
C
T
Tf
Ti
Td
S
Z
time
experimental Constant
the lowest characteristic average thickness of the die-cast part wall (mm)
liquid temperature (K) where?,
melt temperature in the ingate (K),
temperature of the mold cavity surface prior
to pressuring (K) where? Averaged?,
solidification percentage at the end of loading,
conversion factors of stable units connected with the range of solidification.
Note that only the constant nomenclature is deviated from the original nomenclature and
the value of this constant depends the liquid metal and the mold material. The values of the
solidification percentage depends on the liquid metal. The lake of knowledge in dimensional
analysis and heat transfer is evident when on the checking the dimensions of the equation.
For example, it emphasizes to temperature should be in K while it does not mater if it given
in [◦ C]. To demonstrate how ridicules/unrealistic this equation is, consider relatively long
and narrow mold. Using the same kind material and changing the orientation of flow in the
mold yield different filling time in real world while the above formula it is not accounted for.
Clearly this filling time does not describe the physics of the phenomenon die casting.
The irrelevant/incorrect filling time (discussed above) is used to calculate the velocity
of the ingate. Typically in fluid mechanics the velocity related to area not the diameter, in
other words q = A U. Ironically, this idea does not apply to die casting as a bunch of die
casting engineers (Gašpár, Coranič, Majerník, Husár, Knapčíková, Gojdan, and Paško 2021)
citing (Knapcikova, Dupláková, Radchenko, and Hatala 2017) which uses the following
U1 =
m
ρ t d 0.785
(1.3)
where 0.785 is a “magical” number and d is the diameter of the channel (runner?). The mass,
m is unknown and/or not defined and t is the time. The authors pulled dubious recommendations for the volume flow rate based (Konopka, Zyska, Łagiewka, and Nadolski 2015). How
does it happened that the flow rate equation is not the area times the average velocity becomes
controversial? Is the mass conservation not applicable in die casting?
10
CHAPTER 1. INTRODUCTION
On the other hand, Huang et al (2021) suggest to use 40 [m/sec] velocity and the gate
area derived based on this velocity when the filling time is based on (Pan 2006). While the
filling time seems almost like arbitrary value, the equation has, at least, the integrity from
the dimensional analysis point of view. Of course, these researchers cannot avoid including
counter engineering idea. Huang determined that the slow plunger velocity should be based
on the gate velocity. One can only wonder if the processes in the plunger zone (air/gases
entrapment) is determined by the gate velocity. Is it in real life the way around? The physics
seems to be hyperbolic problem. The flow in a water fall in a far way mountain like the
Himalayan is not effected by flow in a small creek in Africa. A better design will be obtained
from values selected by a random machine. Later they explained why their analysis did not
work. One can see that so many reasons for the problems, but basically it is boils to the
fact that the whole thing is erroneous (wrong critical velocity, wrong vent area, wrong gate
velocity etc).
One of the secrets of the black art of design was that there is a range of gate velocity
which creates good castings depending on the alloy properties being casted. The existence
of a minimum velocity hints that a significant change in the liquid metal flow pattern occurs.
Veinik (1962) linked the gate velocity to the flow pattern (atomization) and provided a qualitative physical explanation for this occurrence. Experimental work (Maier 1974) showed that
liquid metals, like other liquids, flow in three main patterns: a continuous flow jet, a coarse
particle jet, and an atomized particle jet. Other researchers utilized the water analogy method
to study flow inside the cavity for example, Bochvar et all (1946). At present, the concept of
(minimum) required gate velocity is supported by experimental evidence which is related to
the flow patterns. However, the numerical value is unknown because the experiments were
poorly conducted for example, (1966) in which the differential equations that “solved” are not
typical to die casting. Discussion about this poor research is presented in Chapter 3.
As a review indicts there is really very little known about the runner design. However,
some things can be said and they are common sense based tradition thermofluids tradition
and it will be discussed in that chapter.
1.4.1.2
Vent System Design
The vent system design requirements were studied by some researchers, for example Suchs
(1952), Veinik (1962), and Draper (1967) and others. These models, however, are unrealistic and
do not provide no relation to the physics or realistic picture of the real requirements or of the
physical situation since they ignore the major point, the air compressibility. Another model
was suggested by Karni (1991a) which full with many mistakes. For example, the statement
When pressure increases over a critical value, shock waves develop at the vent’s
outlet limiting the pressure differential, and maximum air velocity.
This statement is wrong. The critical value does not required as simple pushing a piston or
a plunger create a moving shock. What Karni referred to is a standing shock as the standing
shock wave requires converging diverging nozzle which in exist at the vent system. Additionally, Karni assumed that the flow is isothermal which wrong assumption (too short conduit),
1.4. HISTORICAL BACKGROUND
11
and he made several other errors the procedure of calculations. The analysis of his data seems
that he massaged the data to match with his calculations. Karni conflated chocked with shock.
Since lack knowledge or not doing proper literature review for example Dougan et all
(Doğan, Kenar, Erdil, and Altuncu 2018) use arbitrary area because “experience”. They and
others start to use a device to stop the liquid metal and so to have a large area vent. Yet this
device added additional complications as it adds additional causes to failure and increase the
secondary machining. This idea is trading the cost for knowledge and it could work in some
cases.
Another example of poor research is by Lee et al (2012) in which the verification of
numerical simulation with questionable equations and initial and boundary conditions to
study the vent system. Clearly, the selected critical plunger velocity was erroneous and the
verification experiment lack any logic. They concluded that their simulation was a success
because “[t]he volume of porosity in the casting was found to be significantly reduced using
vacuum assistance during die casting”. Regardless to their suggestions (which were what?)
the vacuum system improve the quality of the casting has nothing with the process at hand.
It is good assumption that vacuum will improve the casting (in most cases) regardless the
“analysis” of the flow.
Actual solution to the size of the vent was done by this author which treats this situation
to emptying container from compressed gas. This research is presented in this book later on.
1.4.1.3
pQ2 Diagram
In the late 70’s, an Australian group (Davis 1975) suggested adopting the pQ2 diagram for die
casting in order to fit the machine size (capability) to required capacity by the product (mold).
As with all the previous models they missed the major points of the calculations. As will
be shown in Chapter 7, the Australian’s model produce incorrect results and predict trends
opposite to reality. This model took root in die casting industry for the last 25 years. Yet, one
can only wonder why this well established method (the supply and demand theory which was
introduced by Fanno (the brother of other famous Fanno from Fanno flow), which was used
in fluid mechanics in the early of this century, reached the die casting only in the late 70’s and
was then erroneously implemented.
It interesting that the erroneous model was limited success in the research. There is
only 131 results on google scholar search. This method now properly build for the first time for
the die casting industry in this book. Guerra (1997) attempted to expand Bar–Meir’s method.
1.4.1.4
Critical Plunger Velocity
Until the 1980’s there was no model based on real scientific principle that assisted the understanding air entrapment in the shot sleeve. Garber (1982) described the hydraulic jump in the
shot sleeve and called it the “wave”, probably because lake of familiarity with this study area.
Yet, he deserve to get the credit for the recognition of the critical plunger velocity. Garber
also developed the erroneous model which took root in the industry in spite the fact that
it never works
works. One can only wonder why any die casting institutes/associations have not
12
CHAPTER 1. INTRODUCTION
published this fact even though they know that it is wrong. Moreover, NADCA and other
institutes continue to funnel large sums of money to the researchers (for example, Brevick
from Ohio State) who used Garber’s model even after they knew that Garber’s model was
totally wrong.
The turning point of the understanding was when Prof. Eckert, the father of modern heat transfer, introduced the dimensional analysis and applied to the die casting process.
This established a scientific approach provides an uniform schemata for uniting experimental
work with the actual situations in the die casting process. Dimensional analysis demonstrates
that the fluid mechanics processes, such as filling of the cavity with liquid metal and evacuation/extraction of the air from the mold, can be dealt when the heat transfer is assumed to
be negligible. However, the fluid mechanics has to be taken into account in the calculations
of the heat transfer process (the solidification process).
This realization provided an excellent opportunity for “simple” models to predict the
many parameters in the die casting process, which will be discussed later in this book. Here,
two examples of new ideas that mushroomed in the inspiration of prof. Eckert’s work. It
has been shown that (Bar-Meir 1995b) the net effect of the reactions is negligible. This fact
is contradictory to what was believed at that stage. The development of the critical vent
area concept provided the major guidance for 1) the designs to the venting system, and 2)
criterion when the vacuum system needs to be used. In this book, many of the new concepts
and models, such as economy of the runner design, plunger diameter calculations, minimum
runner design, etc, are described for the first time.
1.4.2
Gate Velocity and Shape
These two parameters are important and there independent. There are some rough suggestions without scientific rigorous. These suggestions include covering the area (volume) so
that liquid in reach all the corners. Due to the fact, these suggestions did get the minimum
maturity in which one cannot even ridicule them is how bad the situation in this area.
There two conflicting demands on the gate(s) which are one hand, it to be large so the
liquid has less resistance to the flow on the other hand, it has to be narrow so gate provides a
trimming point. Furthermore, the velocity has to change the flow regime from a continuous
flow to a spray flow. Thus, the interaction of the gate thickness and gate area has significance
important. This issue will be discussed later on in the book.
1.4.3
Filling time
While the filling time is not a direct problem of die casting, it indirectly affects many other
design considerations. In fact Fu et al (2008) recommend to abandoned the high pressure
and move to the law pressure die casting in which the filling time is extended. While this
suggestion is radical here it suggest simply improving the die casting process.
The common approach as suggested by Eq. (1.2). This approach attempts to capture
a rough estimate for the filling time which without considering geometry and other effects
without real scientific base. Consider the most standard thing to case which is the test bar.
13
1.4. HISTORICAL BACKGROUND
V ents
Gates
Gates
Fig a. Filling the test bar long way
Fig b. Filling the test bar short way
Fig. 1.3 – Filling the test bars via two different designs. The first design is the traditional method while the
second is innovative in which the feeding is from the side. The vent, not shown is from the other side.
The images are only for illustration to demonstrate the change of the path for the liquid
This design exhibited in Fig. 1.3a for which the flow start at one end finish at the other end.
The filling time for the long design should be larger than other design. The exhibits on the
right show much shorter filling time.
Fig. 1.4 – Filling time according to Weishan et all where velocity is related to pressure and not to flow rate.
After Weishan et al (Weishan, Shoumei, and Baicheng 1997)
The review how poor understating of filling is obvious from the literature review. For example, Weishan et al (1997) depicted a pQ2 diagram to calculate the filling time as demonstration
of no science in our place in which basic physics has no influence on the understanding. The
constant velocity lines are shown as horizontal lines. The velocity is related to the flow rate
(Q = A Uavarged basic geometry). Yet here the velocity is related to pressure. On the another hand, Pinto et al (2019) believe this number should be arbitrary selected.
More serious consideration was given in cold weather studies on freezing of pipes.
The similarity of these two physical phenomena that energy has withdrawn from the flow
and hence solidify the liquid. This topic will discussed in dimensional analysis chapter.
14
1.5
CHAPTER 1. INTRODUCTION
Numerical Simulations
Numerical simulations have been found to be very useful in many areas which lead many
researchers attempting to implement them into die casting process. Considerable research
work has been carried out on the problem of solidification including fluid flow which is
known also as Stefan problems (Hu and Argyropoulos 1996). Minaie et al in one of the pioneered work (1991) use this knowledge and simulated the filling and the solidification of the
cavity using finite difference method. The Enthalpy method was exploited by Swaminathan
and Voller (1993) and others to study the filling and solidification problem. Other develop
techniques that involve modifying the grid by modifying the shape and number of elements
during calculations. The above works are mentioned more about the improvements to the
numerical schemes rather the physical understanding.
While numerical simulation looks very promising, all the methods (finite difference,
finite elements, or boundary elements etc) including all Commercial or academic versions
suffer from several major drawbacks which prevents them from yielding reasonable results.
Perhaps why these methods useless currently because they do not describe the physics of the
situations appeared in die casting.
The ship stability is another case where numerical analysis is extensively used. In the
198x people neglected the added properties (added moment of inertia and the added mass)
on this “fact” produced models to describe the situation. In fact they have “strong and convincing evidence” that their models working spectacularly and describe beyond adequately
the ship frequency. It is stunning that the some people group 20 years later claimed that the
added properties have to be accounted and their models now accounting for it. The added
properties change the results by factor of 4 or more. Clearly the physics did not change, thus
this situation forces one to conclude that these models are pure garbage. In fact, these models
errors magnitude are by thousands percents range and other words these models have very
little to do with reality.
In die casting the situation is not much different. For example, over 300 research teams
attempted to examine the flow in the shot sleeve using numerical techniques. And of course,
they have “clear and compelling evidence” to prove that their model is working wonderfully.
The only problem is that their model never work. There is no real case that their model indeed
calculate accurately to predict the critical plunger velocity. Not using the right equations and
boundary conditions is the main reason that their analysis is wrong.
Reviewing several of the recent works show a similar pattern no matter what equations and boundary conditions the results are “full agreement” the experiments. These work
recently include from example Kohlstädta (2021) use extreme sophisticate mechanism to calculate turbulence where it does not exist. One should not expect results of laminar or plug
flow to be identical for turbulent flow. Kohlstädta at el also assumed that velocity or the gradient at the wall is zero which is not physically possible. These boundaries demonstrate a
strong lack of any physical understanding. An example of such research see Dou et al (2020)
which were the unknown pressure (in other words totally wrong) is used to calculated the
velocity. This work is epitome of how not do research work.
On the other hand, several other people realize these research works are in the twi-
1.5. NUMERICAL SIMULATIONS
15
light zone. These research work are good only in the eyes of researchers themselves who
carry them. People realize that something else has to be done. This realization is the point
when nothing is known to be wrong then some industrial engineer like Chandrasekaran
(2019) presents work on examine what parameters effecting the die casting process. These
approaches are mushroomed when design experiment were the physics of focus material is
unknown. For example, Balikai (2018) (Karthik, Karunanithi, Srinivasan, and Prashanth 2020)
advance the Taguchi method. Taguchi’s method is used when you are clueless about the process and try like a blind person to find what is effecting the process. As most of these methods, is more indication how much less is known more on the fact that they do illuminate the
physics and explain it. By the way, these papers are done by people who cannot even conduct
proper literature review.
Here what are unknowns in die casting:
• There is no theory (model) that explains the heat transfer between the mold walls and
the liquid metal. The lubricant sprayed on the mold change the characteristics of the
heat transfer. The difference in the density between the liquid phase and solid phase
creates a gap during the solidification process between the mold and the ingate which
depends on the geometry. For example, Osborne et al (1993) showed that a commercial
software (MAGMA) required fiddling with the heat transfer coefficient to get the numerical simulation match the experimental results. Actually they attempted to prove
that the software is working very well. However, the fact that coefficient is needed to
fiddled with is excellent proof why this work meaningless.
• As it was mentioned earlier, it is not clear when the liquid metal flows as a spray and
when it flows as continuous liquid in the die cavity and the runner. Experimental work
has demonstrated that the flow, for a large part of the filling time, is atomized (Bar-Meir
1995d).
• The pressure in the mold cavity in all the commercial codes are calculated without taking into account the resistance to the air flow out. Thus, built–up pressure in the cavity
is poorly estimated, or even not realistic, and therefore the characteristic flow of the
liquid metal in the mold cavity is poorly estimated as well. In other words, there is no
real relationship between model and the physics. This error is significant in determining the calculations results.
• The flow in all the simulations is assumed to be turbulent flow. However, time and
space are required to achieved a fully turbulent flow. For example, if the flow at the
entrance to a pipe with the typical conditions in die casting is laminar (actually it is a
plug flow) it will take a runner with a length of about 10[m] to achieved fully developed
flow. With this in mind, clearly some part of the flow is laminar. Additionally, the
solidification process is faster compared to the dissipation process in the initial stage,
so it is also a factor in changing the flow from a turbulent (in case the flow is turbulent)
to a laminar flow.
16
CHAPTER 1. INTRODUCTION
• The liquid metal velocity at the entrance to the runner is assumed for the numerical
simulation and not calculated. In reality this velocity has to be calculated utilizing the
pQ2 diagram.
• If turbulence exists in the flow field, what is the model that describes it adequately?
Clearly, model such k − ϵ are based on isentropic homogeneous with mild change in
the properties cannot describe situations where the flow changes into two-phase flow
(solid-liquid flow) etc.
• The heat extracted from the die is done by cooling liquid (oil or water). In most models
(all the commercial models) the mechanism is assumed to be by “regular cooling”. In
actuality, some part of the heat is removed by boiling heat transfer. As it seems to
expedite the heat transfer.
• The governing equations in all the numerical models, that have examined, neglect the
dissipation term in during the solidification. The dissipation term is the most important
term in that case. This dissipation is what hamper the flow.
• All the numerical work that were examined do not have slip condition of the velocity
near the plunger. In reality the velocity is close to plug flow which conflicts with models
in the literature.
One wonders how, with unknown flow pattern (or correct flow pattern), unrealistic
pressure in the mold, wrong heat removal mechanism (cooling method), erroneous governing
equation in the solidification phase, and inappropriate heat transfer coefficient, a simulation
could produce any realistic results. Clearly, much work is need to be done in these areas before any realistic results should be expected from any numerical simulation. Furthermore,
to demonstrate this point, there are numerical studies that assume that the flow is turbulent,
continuous, no air exist (or no air leaving the cavity) and proves with their experiments that
their model simulate “reality” (Kim and Sant 1995). On the other hand, other numerical studies assumed that the flow does not have any effect on the solidification and of course have
their experiments to support their claim (Davey and Bounds 1997). Clearly, this contradiction
suggest several options:
• Both of the them are right and the model itself does not matter.
• One is right and the other one is wrong.
• Both of them are wrong.
The third research that provide an example how the calculations can be shown to be totally wrong and yet the researchers have “experimental proofs” to back them up. Viswanathan
et al (1997) studied a noble process in which the liquid metal is poured into the cavity and direct
pressure is applied to the cavity. In their calculations the authors assumed that liquid metal
enters to the cavity and fills the whole entrance (gate) to the cavity. Based on this assumption, their model predicts defects in a certain geometry. A critical examination of this model
1.5. NUMERICAL SIMULATIONS
17
present the following abused conclusion of no air flow out. The authors “explained” privately
that air amount is a insignificant and therefore not important. This assumption is very critical
as will be shown later that make this research to be in the twilight zone. The volumetric air
flow rate into the cavity has to be on average equal to liquid metal flow rate (conservation of
volume for constant density). Hence, air velocity has to be approximately infinite to achieve
zero vent area. Conversely, if the assumption that the air flows in the same velocity as the
liquid entering the cavity, liquid metal flow area is a half what is assume in the researchers
model. In realty, the flow of the liquid metal is in the two phase region and in this case, it
is like turning a bottle full of water over and liquid inside flows as “blobs.” The check this
research, the authors should fill a bottle and turn it upside and examine what happen. More
information can be found on reversible flow multiphase flow by this author book in Potto
series of “Basics of Fluid Mechanics.” In this case the whole calculations do not have much to
do with reality since the velocity is not continuous and different from what was calculated.
The study of the flow in the shot sleeve by Backer and Sant from EKK (Backer and
Sant 1997)1 . The researchers assumed that the flow is turbulent and they justified it because
they calculated and found a “jet” with extreme velocity. Unfortunately, all the experimental
evidence demonstrate that there is no such jet (Madsen and Svendsen 1983). It seems that
this jet results from the “poor” boundary and initial conditions. The boundary and initial
conditions were not spelled out in the paper! However they were implicitly stated in the
presentation. In the presentation, the researchers stated that results they obtained for laminar
and turbulent flow were the same. So why to use the complicate flow model is it is not needed?
Simple analysis can demonstrate the difference is very large for different flow pattern. Also,
one can wonder how liquid with zero velocity to be turbulent. With these results one can
wonder if the code is of any value or the implementation is at fault.
The bizarre belief that the numerical simulations are a panacea to all the design problem is very popular in the die casting industry. Any model has to describe and account for the
effecting physical parameters in order to be useful. Experimental evidence which is supporting wrong models as a real evidence is nonsense. Clearly something(s) wrong must be there.
For example, see the paper by Murray from Australia and colleagues in which they use the
fact that two unknown companies (somewhere in the outer space maybe?) were using their
model to claim that it is correct. A proper way can be done by numerical calculations based
on real physics principles which produce realistic results.
Until that point come, the reader should be suspicious about any numerical model and
its supporting evidence.2
1 It was suggested by several people that the paper was commissioned by NADCA to counter Bar-Meir’s equation
to shot sleeve. This fact is up to the reader to decide.
2 With all these harsh words, this author would like to take the opportunity for the record, to state he think that
work by Davey’s group is a good one. They have inserted more physics (for example the boiling heat transfer) into
their models which it is hoped in the future, lead to more realistic numerical models.
18
1.6
CHAPTER 1. INTRODUCTION
“Integral” Models
Unfortunately, the numerical simulations of the liquid metal flow and solidification process
do not yield reasonable results at the present time. This problem has left the die casting engineers with the usage of the “integral approach” method. In this method the calculations are
broken into “simplified” models. One of the most important tool in this approach is the pQ2
diagram, one of the manifestations of the supply and demand theory. In this diagram, an engineer insures that die casting machine ability can fulfill the die mold design requirements; the
liquid metal is injected at the right velocity range and the filling time is small enough o prevent
premature freezing. One can, with the help of the pQ2 diagram and by utilizing experimental values for desired filling time and gate velocities improve the quality of the casting. The
gate velocity has to be above a certain value to assure atomization and below a critical value
to prevent erosion of the mold. These two values are experimental and no reliable theory is
available today known to predict these values. The correct model for the pQ2 diagram has
been developed and will be presented in Chapter 7. A by–product of the above model is the
plunger diameter calculations and it is discussed in Chapter 7.
It turned out that many of the design parameters in die casting have a critical point
above which good castings are produced and below which poor castings are produced. Furthermore, much above and just above the critical point do not change much the quality but
costs much more. This fact is where the economical concepts play a significant role. Using
these concepts, one can increase the profitability significantly, and obtain very good quality
casting and reduce the leading time. Additionally, the main cost components like machine
cost and other are analyzed which have to be taken into considerations when one chooses to
design the process will be discussed in the Chapter 11.
1.6.0.1
Porosity
Porosity can be divided into two main categories; shrinkage porosity and gas/air entrapment.
The porosity due to entrapped gases constitutes a large part of the total porosity. The creation
of gas/air entrainment can be attributed to at least four categories: 1) lubricant evaporation
and reaction processes. Some researchers view the chemical reactions (e.g. release of nitrogen
during solidification process as category by itself.), 2) vent locations (last place to be filled), 3)
mixing processes, and 4) vent/gate area. The effects of lubricant evaporation have been found
to be insignificant. The vent location(s) can be considered partially solved since only qualitative explanation exist. The mixing mechanisms are divided into two zones: the mold, and the
shot sleeve. Some mixing processes have been investigated and can be considered solved. The
requirement on the vent/gate areas is discussed in Chapter 9. When the mixing processes are
very significant in the mold, other methods are used and they include: evacuating the cavities (vacuum venting), Pore Free Technique (in zinc and aluminum casting), low pressure die
casting, and squeeze casting. The first two techniques are used to extract the gases/air from
the shot sleeve and die cavity before the gases have the opportunity to mix with the liquid
metal. The squeeze casting is used to increase the capillary forces (actually surface tension)
and therefore, to minimize the mixing processes. The low pressure die casting is used to in-
1.7. SUMMARY
19
crease the window for filling and thus reducing the mixing process. All these solutions are
cumbersome and more expensive and should be avoided if possible.
The mixing processes in the runners, where the liquid metal flows vertically against
gravity in relatively large conduit, are considered to be insignificant compare to the other
process. Yet, they cannot be neglected. Some work has been carried out practical ideas will be
presented here. The enhanced air entrapment in the shot sleeve is attributed to operational
conditions for which a blockage of the gate by a liquid metal wave occurs before the air is
exhausted. Consequently, the residual air is forced to be mixed into the liquid metal in the shot
sleeve. With Bar-Meir’s formula, one can calculate the correct critical slow plunger velocity
and this will be discussed in Chapter 8.
1.7
Summary
It is an exciting time in the die casing industry because for the first time, an engineer can
start using real science in designing the runner/mold and the die casting process. Many new
models have been build and many old techniques mistake have been exposed. It is the new
revolution in the die casting industry.
20
CHAPTER 1. INTRODUCTION
Part I
Bases of Sciences
21
2
Basic Fluid Mechanics
2.1
Introduction
The discussion in this chapter is presented to fill the void in basic fluid mechanics to the
die casting community. It was observed that knowledge in this area cannot be avoided. The
design of the process as well as the properties of casting (especially magnesium alloys) are
determined by the fluid mechanics/heat transfer processes. It is hoped that others will join
to spread this knowledge. There are numerous books for introductory fluid mechanics but
the Potto series book “Basic of Fluid Mechanics” is a good place to look and might be over
kill. This chapter is a summary of that book plus some pieces from the “Fundamentals of
Compressible Flow Mechanics.” It is hoped that the reader will find this chapter interesting
and will further continue expanding his knowledge by reading the full Potto books on fluid
mechanics and compressible flow.
Many aspects of the die casting involve fluid mechanics and thermodynamics. Hence
process design which affects the monetary aspects is controlled by these topics and hence
a review is provided. Thermodynamics is the foundation for fluid mechanics and hence a
review of it is provided first.
2.2
2.2.1
Thermodynamics and mechanics concepts
Thermodynamics
In this section, a review of several definitions of common thermodynamics terms is presented.
This introduction is provided to bring the student back up this this point or build a minimum
23
24
CHAPTER 2. BASIC FLUID MECHANICS
base.
2.2.1.1
Basic Definitions
The following basic definitions are common to thermodynamics and will be used in this book.
2.2.1.1.1 Work
In mechanics, the work was defined as
Z
Z
mechanical work = F • dℓ = P dV
(2.1)
This definition can be expanded to include two issues. The first issue that must be addressed
is the sign, that is the work done on the surroundings by the system boundaries is considered
positive. Two, there is distinction between transfer of energy and work. Heat transfer can
cause work but it is not necessarily so. For example, the electrical current is pure work while
pure conductive conductive heat transfer isn’t. Several terms have to find so analysis can be
made.
2.2.1.1.2 System
The term “system” is used in this book to define region of volume as a continuous (at least
partially) and fixed quantity of matter. The dimensions of this material can be changed. In
this definition, it is assumed that the system speed is significantly lower than that of the speed
of light. So, the mass can be assumed constant even though the true conservation law applied
to the combination of mass energy (see Einstein’s law). In fact for almost all engineering purposes, this law is reduced to two separate laws of mass conservation and energy conservation.
The system can receive loss energy, work, etc as long the mass remain constant the definition
is not broken.
2.2.1.1.3 Control Volume, c.v.
The control volume was introduced by L. Euler to simplify the calculations. In the control
volume, the specific volume is examined which mass can enter and leave. The simplest c.v. is
when the boundary is fixed and it is referred to as the Non–deformable c.v. The conservation
of mass to such system can be very good approximated by
d
dt
Z
Z
ρ dV = −
Vc.v.
ρ Vrn dA
Sc.v.
(2.2)
The change in the control volume is results of difference of enter or leave the control volume.
For deformable control volume mass conservation can be written as
Z
Z
Z
d
dρ
ρ dV =
dV +
ρ Vrn dA
(2.3)
dt Vc.v.
Vc.v. dt
Sc.v.
2.2. THERMODYNAMICS AND MECHANICS CONCEPTS
2.2.1.2
25
Thermodynamics First Law
In thermodynamics there are three laws and here only two laws will be introduced.
The first law refers to conservation of energy in a non accelerating system. Since all
the systems can be calculated in a non accelerating systems, the conservation is applied to all
systems. The statement is describing the law as the following:
Q12 − W12 = E2 − E1
(2.4)
The energy transferred to the system between state 1 and state 2 The work done by the
system between state 1 and state 2 The system energy is a state property. From the first law it
directly implies that for process without heat transfer (adiabatic process) the following is true
W12 = E1 − E2
(2.5)
Interesting results of Eq. (2.5) is that the way the work is done and/or intermediate states
are irrelevant to final results. There are several definitions/separations of the kind of works
and they include kinetic energy, potential energy (gravity), chemical potential, and electrical
energy, etc. The internal energy is the energy that depends on the other properties of the
system. For example for pure/homogeneous and simple gases it depends on two properties
like temperature and pressure. The internal energy is denoted in this book as EU and it will
be treated as a state property.
The potential energy of the system is depended on. the body force A common body
force is the gravity. For such body force, the potential energy is m g z where g is the gravity
force (acceleration), m is the mass and the z is the vertical height from a datum. The kinetic
energy is
K.E. =
mU2
2
(2.6)
Thus the energy equation can be written as
Total Energy Equation
2
mU2 2
mU1
+ m g z1 + EU1 + Q =
+ m g z2 + EU2 + W
2
2
(2.7)
For the unit mass of the system equation (2.7) is transformed into
Specific Energy Equation
2
U1
U 2
+ g z1 + Eu1 + q = 2 + g z2 + Eu2 + w
2
2
(2.8)
Where q is the energy per unit mass and w is the work per unit mass. The “new” internal
energy, Eu , is the internal energy per unit mass.
26
CHAPTER 2. BASIC FLUID MECHANICS
Since the above equations are true between arbitrary points, choosing any point in time
will make it correct. Thus differentiating the energy equation with respect to time yields the
rate of change energy equation. The rate of change of the energy transfer is
DQ
= Q̇
Dt
(2.9)
In the same manner, the work change rate transferred through the boundaries of the system
is
DW
= Ẇ
Dt
(2.10)
Since the system is with a fixed mass, the rate energy equation is
Q̇ − Ẇ =
D EU
DU
D Bf z
+mU
+m
Dt
Dt
Dt
(2.11)
For the case were the body force, Bf , is constant with time like in the case of gravity equation
(2.11) reduced to
Time Dependent Energy Equation
Q̇ − Ẇ =
D EU
DU
Dz
+mU
+mg
Dt
Dt
Dt
(2.12)
The time derivative operator, D/Dt is used instead of the common notation because
it referred to system property derivative.
2.2.1.3
Thermodynamics Second Law
There are several definitions of the second law. No matter which definition is used to describe
the second law it will end in a mathematical form. The most common mathematical form is
Clausius inequality which state that
I
δQ
⩾0
(2.13)
T
The integration symbol with the circle represent integral of cycle (therefor circle) in with
system return to the same condition. If there is no lost, it is referred as a reversible process
and the inequality change to equality.
I
δQ
=0
(2.14)
T
The last integral can go though several states. These states are independent of the path the
system goes through. Hence, the integral is independent of the path. This observation leads
to the definition of entropy and designated as S and the derivative of entropy is
δQ
(2.15)
ds ≡
T rev
2.2. THERMODYNAMICS AND MECHANICS CONCEPTS
Performing integration between two states results in
Z2 Z2
δQ
S2 − S1 =
=
dS
T rev
1
1
27
(2.16)
One of the conclusions that can be drawn from this analysis is for reversible and adiabatic process dS = 0. Thus, the process in which it is reversible and adiabatic, the entropy
remains constant and referred to as isentropic process. It can be noted that there is a possibility that a process can be irreversible and the right amount of heat transfer to have zero change
entropy change. Thus, the reverse conclusion that zero change of entropy leads to reversible
process, isn’t correct.
For reversible process equation (2.14) can be written as
(2.17)
δQ = T dS
and the work that the system is doing on the surroundings is
(2.18)
δW = P dV
Substituting equations (2.17) (2.18) into (2.12) results in
(2.19)
T dS = d EU + P dV
Even though the derivation of the above equations were done assuming that there is
no change of kinetic or potential energy, it still remain valid for all situations. Furthermore,
it can be shown that it is valid for reversible and irreversible processes.
2.2.1.3.1 Enthalpy
It is a common practice to define a new property, which is the combination of already defined
properties, the enthalpy of the system.
(2.20)
H = EU + P V
The specific enthalpy is enthalpy per unit mass and denoted as, h.
Or in a differential form as
dH = dEU + dP V + P dV
(2.21)
Combining equations (2.20) the Eq. (2.19) yields
(one form of) Gibbs Equation
(2.22)
T dS = dH − V dP
For isentropic process, equation (2.19) is reduced to dH = VdP. The equation (2.19) in mass
unit is
T ds = du + P dv = dh −
dP
ρ
when the density enters through the relationship of ρ = 1/v.
(2.23)
28
CHAPTER 2. BASIC FLUID MECHANICS
2.2.1.3.2 Specific Heats
The change of internal energy and enthalpy requires new definitions. The first change of the
internal energy and it is defined as the following
Specific Volume Heat
∂Eu
Cv ≡
∂T
(2.24)
And since the change of the enthalpy involve some kind of boundary work is defined as
Specific Pressure Heat
∂h
Cp ≡
∂T
(2.25)
The ratio between the specific pressure heat and the specific volume heat is called the
ratio of the specific heat and it is denoted as, k.
Specific Heats Ratio
k≡
Cp
Cv
(2.26)
For solid, the ratio of the specific heats is almost 1 and therefore the difference between
them is almost zero. Commonly the difference for solid is ignored and both are assumed to
be the same and therefore referred as C. This approximation less strong for liquid but not by
that much and in most cases it applied to the calculations. The ratio the specific heat of gases
is larger than one.
2.2.1.3.3 Equation of state
Equation of state is a relation between state variables. Normally the relationship of temperature, pressure, and specific volume define the equation of state for gases. The simplest
equation of state referred to as ideal gas. And it is defined as
P = ρRT
(2.27)
Application of Avogadro’s law, that "all gases at the same pressures and temperatures have
the same number of molecules per unit of volume," allows the calculation of a “universal gas
constant.” This constant to match the standard units results in
kj
kmol K
(2.28)
R̄
M
(2.29)
R̄ = 8.3145
Thus, the specific gas can be calculate as
R=
29
2.2. THERMODYNAMICS AND MECHANICS CONCEPTS
Table 2.1 – Properties of Various Ideal Gases [300K]
Chemical
Formula
Gas
Molecular
Weight
h
i
kj
R KgK
CP
h
kj
KgK
i
Cv
h
kj
KgK
i
k
Air
Argon
Butane
Carbon
Dioxide
Carbon
Monoxide
Ar
C4 H10
28.970
39.948
58.124
0.28700
0.20813
0.14304
1.0035
0.5203
1.7164
0.7165
0.3122
1.5734
1.400
1.667
1.091
CO2
44.01
0.18892
0.8418
0.6529
1.289
CO
28.01
0.29683
1.0413
0.7445
1.400
Ethane
Ethylene
Helium
Hydrogen
Methane
Neon
C2 H6
C2 H4
He
H2
CH4
Ne
30.07
28.054
4.003
2.016
16.04
20.183
0.27650
0.29637
2.07703
4.12418
0.51835
0.41195
1.7662
1.5482
5.1926
14.2091
2.2537
1.0299
1.4897
1.2518
3.1156
10.0849
1.7354
0.6179
1.186
1.237
1.667
1.409
1.299
1.667
Nitrogen
Octane
Oxygen
Propane
Steam
N2
C8 H18
O2
C3 H8
H2 O
28.013
114.230
31.999
44.097
18.015
0.29680
0.07279
0.25983
0.18855
0.48152
1.0416
1.7113
0.9216
1.6794
1.8723
0.7448
1.6385
0.6618
1.4909
1.4108
1.400
1.044
1.393
1.126
1.327
The specific constants for select gas at 300K is provided in table 2.1.
From equation (2.27) of state for perfect gas it follows
d (P v) = R dT
(2.30)
dh = dEu + d(Pv) = dEu + d(R T ) = f(T ) (only)
(2.31)
For perfect gas
From the definition of enthalpy it follows that
d(Pv) = dh − dEu
(2.32)
Utilizing equation (2.30) and subsisting into equation (2.32) and dividing by dT yields
Cp − Cv = R
(2.33)
30
CHAPTER 2. BASIC FLUID MECHANICS
This relationship is valid only for ideal/perfect gases.
The ratio of the specific heats can be expressed in several forms as
Cv to Specific Heats Ratio
Cv =
R
k−1
(2.34)
Cp to Specific Heats Ratio
Cp =
kR
k−1
(2.35)
The specific heat ratio, k value ranges from unity to about 1.667. These values depend on
the molecular degrees of freedom (more explanation can be obtained in Van Wylen “F. of
Classical thermodynamics.” The values of several gases can be approximated as ideal gas and
are provided in Table 2.1.
The entropy for ideal gas can be simplified as the following
Z2 dh dP
−
(2.36)
s2 − s1 =
T
ρT
1
Using the identities developed so far one can find that
Z2
Z2
R dP
T
P
dT
s 2 − s1 =
Cp
−
= Cp ln 2 − R ln 2
T
P
T
P1
1
1
1
(2.37)
Or using specific heat ratio equation (2.37) transformed into
k
T
P
s2 − s1
=
ln 2 − ln 2
R
k−1
T1
P1
(2.38)
For isentropic process, ∆s = 0, the following is obtained
T
ln 2 = ln
T1
P2
P1
k−1
k
(2.39)
There are several famous identities that results from equation (2.39) as
Ideal Gas Isentropic Relationships
k−1 k−1
T2
P2 k
V1
=
=
T1
P1
V2
(2.40)
The ideal gas model is a simplified version of the real behavior of real gas. The real gas
has a correction factor to account for the deviations from the ideal gas model. This correction
factor referred as the compressibility factor and defined as
Z deviation from the Ideal Gas Model
PV
Z=
RT
(2.41)
2.3. FUNDAMENTALS OF FLUID MECHANICS
2.3
2.3.1
31
Fundamentals of Fluid Mechanics
Introduction
First, the nature of fluids and basic concepts from thermodynamics will be introduced. Later
the integral analysis will be discussed in which it will be divided into introduction of the control volume concept and Continuity equations.
The energy equation will be
explained in the next section. Later, the momentum equation will be discussed. Lastly, the
U
chapter will be dealing with the compressible
flow gases. Here it will be refrained from dealing with topics such boundary layers, non–
viscous flow, machinery flow etc which are not
essential to understand the rest of this book.
Fig. 2.1 – The velocity distribution in Couette
Nevertheless, they are important and it is adflow when the top plate is moving.
visable that the reader will read on these topics
as well.
Fluid is considered as a substance that “moves” continuously and permanently when
exposed to a shear stress. The liquid metals are an example of such substance. However, the
liquid metals do not have to be in the liquidus phase to be considered liquid. Aluminum at
approximately 4000 C is continuously deformed when shear stress are applied. The whole
semi–solid die casting area deals with materials that “looks” solid but behaves as liquid.
2.3.1.1
What is Fluid?
The fluid is mainly divided into two categories: liquids and gases. The main difference between the liquids and gases state is that gas will occupy the whole volume while liquids has
an almost fixed volume. This difference can be, for most practical purposes considered, sharp
even though in reality this difference isn’t sharp. The difference between a gas phase to a liquid
phase above the critical point are practically minor. But below the critical point, the change
of water pressure by 1000% only change the volume by less than 1 percent. For example, a
change in the volume by more than 5% will require tens of thousands percent change of the
pressure. So, if the change of pressure is significantly less than that, then the change of volume
is at best 5%. Hence, the pressure will not affect the volume. In gaseous phase, any change in
pressure directly affects the volume. The gas fills the volume and liquid cannot. Gas has no
free interface/surface (since it does fill the entire volume).
2.3.2
What is Shear Stress?
The shear stress is part of the pressure tensor. However, here it will be treated as a separate
issue. In solid mechanics, the shear stress is considered as the ratio of the force acting on area
in the direction of the forces perpendicular to area. Different from solid, fluid cannot pull
32
CHAPTER 2. BASIC FLUID MECHANICS
directly but through a solid surface. Consider liquid that undergoes a shear stress between a
short distance of two plates as shown in Figure (2.2).
The upper plate velocity generally will be
(2.42)
U = f(A, F, h)
Where A is the area, the F denotes the force, h is the distance between the plates. From solid
mechanics study, it was shown that when the force per area increases, the velocity of the plate
increases also. Experiments show that the increase of height will increase the velocity up to
a certain range. Consider moving the plate with a zero lubricant (h ∼ 0) (results in large
force) or a large amount of lubricant (smaller force). In this discussion, the aim is to develop
differential equation, thus the small distance analysis is applicable.
For cases where the dependency is linear, the following can be written
hF
A
(2.43)
F
U
∝
h
A
(2.44)
F
A
(2.45)
U∝
Equation (2.43) can be rearranged to be
Shear stress is defined as
τxy =
The force, F is the force component perpendicular to the area. From equations (2.44) and (2.45)
it follows that ratio of the velocity to height is proportional to shear stress. Hence, applying
the coefficient to obtain a new equality as
τxy = µ
U
h
(2.46)
Where µ is called the absolute viscosity or dynamic viscosity. In steady state, the distance the
upper plate moves after small amount of time, δt is
t0 <
dℓ = U δt
t1 < t2 <
t3
(2.47)
From Fig. 2.2 it can be noticed that for a small
angle, the regular approximation provides
geometry
dℓ = U δt =
z}|{
h δβ
(2.48)
Fig. 2.2 – The deformation of fluid due to shear
stress as progression of time.
From equation (2.48) it follows that
U=h
δβ
δt
(2.49)
33
2.3. FUNDAMENTALS OF FLUID MECHANICS
Combining equation (2.49) with equation (2.46) yields
τxy = µ
δβ
δt
(2.50)
If the velocity profile is linear between the plate (it will be shown later that it is consistent
with derivations of velocity), then it can be written for small angle that
δβ
dU
=
δt
dy
(2.51)
Materials which obey Eq. (2.50) are referred to as Newtonian fluid.
For typical liquid metal used in the die casting industry is considered as Newtonian
fluid.
2.3.3
Mass Conservation
2.3.3.1
Introduction
This section presents a discussion on the control volume and will be focused on the conservation of the mass. When the fluid system moves or changes, one wants to find or predict the velocities in the system. The main target of such analysis is to find the value of
certain variables. This kind of analysis is reasonable. Even though this system looks reasonable, the Lagrangian system turned out
system
to be difficult to solve and to analyze. This
ii
method applied and used in very few cases.
iii
The main difficulty lies in the fact that every
control
particle has to be traced to its original state
volume
Leonard. Euler (1707–1783) suggested an alterFig. 2.3 – Control volume and system before and
native approach. In Euler’s approach the focus
after motion.
is on a defined point or a defined volume. This
methods is referred as Eulerian method.
i
The Eulerian method focuses on a defined area or location to find the needed information. The use of the Eulerian methods leads to a set differentiation equations that is referred
to as Navier–Stokes equations which are commonly used. Additionally, the Eulerian system
leads to integral equations which are the focus in this section.
Lagrangian equations are associated with the system while the Eulerian equations are
associated with the control volume. The difference between the system and the control volume is shown in Fig. 2.3. The green lines in Fig. 2.3 represent the system. The red dotted lines
are the control volume. At certain time the system and the control volume are identical location. After a certain time, some of the mass in the system exited the control volume which
are marked “i” in Fig. 2.3. The material that remained in the control volume is marked as “ii”.
At the same time, the control gains some material which is marked as “iii ”.
34
CHAPTER 2. BASIC FLUID MECHANICS
The Eulerian method requires to define a control volume (some times more than one).
The control volume is a defined volume that was discussed earlier. The control volume is
differentiated into two categories of control volumes, non–deformable and deformable.
Non–deformable control volume is a control volume which is fixed in space
relatively to an one coordinate system. This coordinate system may be in a
relative motion to another (almost absolute) coordinate system.
Deformable control volume is a volume having part of all of its boundaries in
motion during the process at hand.
out
In the case where no mass crosses the
boundaries, the control volume is a system.
Every control volume is the focus of the certain
interest and will be dealt with the basic equain
tions, mass, momentum, energy, entropy etc.
Two examples of control volume are
presented to illustrate difference between a deformable control volume and non–deformable
control volume.
Flow in conduits can
Fig. 2.4 – Control volume of a moving piston
be analyzed by looking in a control volwith in and out flow.
ume between two locations. The coordinate system could be fixed to the conduit.
The control volume chosen is non-deformable control volume. The control volume should be
chosen so that the analysis should be simple and dealt with as less as possible issues which are
not in question. When a piston pushing gases a good choice of control volume is a deformable
control volume that is a head the piston inside the cylinder as shown in Fig. 2.4.
2.3.3.2
Continuity Equation
In this subsection and the next three sections, the conservation equations will be applied to
the control volume. In this subsection the mass conservation will be discussed. The system
mass change is
Z
D msys
D
=
ρdV = 0
(2.52)
Dt
Dt Vsys
The system mass after some time, according Fig. 2.3, is made of
msys = mc.v. + ma − mc
(2.53)
35
2.3. FUNDAMENTALS OF FLUID MECHANICS
The change of the system mass is by definition is zero. The change with time (time derivative
of Eq. (2.53)) results in
D msys
d mc.v. d ma d mc
=0=
+
−
Dt
dt
dt
dt
(2.54)
The first term in Eq. (2.54) is the derivative of the mass in the control volume and at any given
time is
Z
d
d mc.v. (t)
=
ρ dV
(2.55)
dt
dt Vc.v.
and is a function of the time.
The interface of the control volume can
move. The actual velocity of the fluid leaving
the control volume is the relative velocity (see
Fig. 2.5). The relative velocity is
−
→ −
→ −→
Ur = Uf − Ub
(2.56)
Where Uf is the liquid velocity and Ub is the
boundary velocity (see Figure 2.5). The velocity
component that is perpendicular to the surface
is
Ub n̂
Uf
C ont
rolV
olum
e Uf − Ub
θ
−Ub
Fig. 2.5 – Schematics of velocities at the interface.
−
→
Urn = −n̂ · Ur = Ur cos θ
(2.57)
Where n̂ is an unit vector perpendicular to the surface. The convention of direction is taken
positive if flow out the control volume and negative if the flow is into the control volume.
The mass flow out of the control volume is the system mass that is not included in the control
volume. Thus, the flow out is
Z
d ma
=
ρs Urn dA
(2.58)
dt
Scv
It has to be emphasized that the density is taken at the surface thus the subscript s. In the
same manner, the flow rate in is
Z
d mb
=
ρs Urn dA
(2.59)
dt
Sc.v.
It can be noticed that the two equations (2.59) and (2.58) are similar and can be combined,
taking the positive or negative value of Urn with integration of the entire system as
Z
d ma d mb
−
=
ρs Urn dA
(2.60)
dt
dt
Scv
36
CHAPTER 2. BASIC FLUID MECHANICS
applying negative value to keep the convention. Substituting equation (2.60) into equation
(2.54) results in
d
dt
Continuity
Z
ρs dV = −
ρ Urn dA
Z
c.v.
(2.61)
Scv
Equation (2.61) is essentially accounting of the
mass. Again notice the negative sign in surface integral. The negative sign is because
flow out marked positive which reduces of the
mass (negative derivative) in the control volume. The change of mass change inside the
control volume is net flow in or out of the control system.
dx
x
L
Fig. 2.6 – Schematics of flow in in pipe with
varying density as a function time.
2.3.3.2.1 One–Dimensional Control Volume
Additional simplification of the continuity equation is of one dimensional flow. This simplification provides very useful description for many fluid flow phenomena. The main assumption made in this model is that the proprieties in the across section are only function of x
coordinate . This assumptions leads
Z
Z
d
ρ2 U2 dA −
ρ1 U1 dA =
dt
A2
A1
Z
dV
z }| {
ρ(x) A(x) dx
V(x)
(2.62)
When the density can be considered constant equation (2.62) is reduced to
Z
Z
U2 dA −
A2
U1 dA =
A1
d
dt
Z
A(x)dx
(2.63)
For steady state but with variations of the velocity and variation of the density reduces equation (2.62) to become
Z
Z
ρ2 U2 dA =
ρ1 U1 dA
(2.64)
A2
A1
For steady state and uniform density and velocity equation (2.64) reduces further to
ρ1 A1 U1 = ρ2 A2 U2
(2.65)
For incompressible flow (constant density), continuity equation is at its minimum form of
U1 A1 = A2 U2
(2.66)
2.3. FUNDAMENTALS OF FLUID MECHANICS
2.3.4
37
Reynolds Transport Theorem
It can be noticed that the same derivations carried for the density can be carried for other
intensive properties such as specific entropy, specific enthalpy. Suppose that g is intensive
property (which can be a scalar or a vector) undergoes change with time. The change of
accumulative property will be then
Z
Z
Z
d
D
f ρdV =
f ρdV +
f ρ Urn dA
(2.67)
Dt sys
dt c.v.
c.v
This theorem named after Reynolds, Osborne, (1842-1912) which is actually a three dimensional generalization of Leibniz integral rule. To make the previous derivation clearer, the
Reynolds Transport Theorem will be reproofed and discussed. The ideas are the similar but
extended some what.
Leibniz integral rule1 is an one dimensional and it is defined as
Z
Z x2 (y)
d x2 (y)
∂f
dx
dx
f(x, y) dx =
dx + f(x2 , y) 2 − f(x1 , y) 1
(2.68)
dy x1 (y)
dy
dy
x1 (y) ∂y
A proof is provided in the Fluid Mechanics book.
Reynolds Transport theorem is a generalization of the Leibniz rule and thus the same
arguments are used. The only difference is that the velocity has three components and only
the perpendicular component enters into the calculations.
D
DT
Reynolds Transport
Z
Z
d
f ρdV =
f ρ dV +
f ρ Urn dA
dt c.v
sys
Sc.v.
Z
2.3.5
Momentum Equation
2.3.5.1
Introduction to Momentum
(2.69)
In the previous section, the Reynolds Transport Theorem (RTT) was applied to mass conservation. Mass is a scalar (quantity without magnitude). This section deals with momentum
conservation which is a vector. The Reynolds Transport Theorem (RTT) is applicable to any
quantity and the discussion here deals with forces that acting on the control volume. Newton’s second law for a single body is as the following
F=
U)
d(mU
dt
(2.70)
It can be noticed that the bold notation for the velocity is U (and not U) to represent that the
velocity has a direction. For several bodies (n), Newton’s law becomes
n
X
i=1
Fi =
n
X
U )i
d(mU
dt
i=1
1 This material is unnecessary but is added here for completeness.
(2.71)
38
CHAPTER 2. BASIC FLUID MECHANICS
The fluid can be broken into infinitesimal elements which turn the above equation (2.71) into
a continuous form of small bodies which results in
n
X
Fi =
i=1
Z
D
Dt
element mass
U
z }| {
ρ dV
(2.72)
sys
Note that the notation D/Dt is used and not the regular operator d/dt to signify that the
operator refers to a derivative of the system. The Reynold’s Transport Theorem (RTT) has to
be used on the right hand side of Eq. (2.72).
2.3.5.2
External Forces
First, the terms on the left hand side, or the forces, have to be discussed. The forces, excluding
the external forces, are divided to the body forces, and the surface forces as the following
(2.73)
F total = F b + F s
In this book (at least in this discussion), the main body force is the gravity. The gravity acts
on all the system elements. The total gravity force is
X
Z
Fb =
element mass
z }| {
ρ dV
g
(2.74)
sys
which acts through the mass center towards the center of earth. After infinitesimal time the
gravity force acting on the system is the same for control volume, hence,
Z
Z
g ρ dV =
g ρ dV
(2.75)
sys
cv
The integral yields a force trough the center mass which has to be found separately.
In this chapter, the surface forces are divided
into two categories: one perpendicular to the
surface and one with the surface direction (in
the surface plane see Figure 2.7.). Thus, it can
be written as
Z
Z
X
Fs =
Sn dA +
τ dA
(2.76)
c.v.
c.v.
with the n̂
surface perpendicular
to the surface
Fig. 2.7 – The explanation for the direction relative to surface perpendicular and with the
surface.
Where the surface “force”, Sn , is in the
surface direction, and τ are the shear stresses perpendicular to the surface. The surface “force”,
Sn , is made out of two components, one due to the viscosity and two consequence of the fluid
pressure. Here for simplicity, only the pressure component is used which is reasonable for
most situations. Thus,
∼0
z}|{
P n̂ + Sν
S n = −P
(2.77)
2.3. FUNDAMENTALS OF FLUID MECHANICS
39
Where Sν is perpendicular stress due to viscosity. Again, n̂ is an unit vector outward of
element area and the negative sign is applied so that the resulting force acts on the body.
2.3.5.3
Momentum Governing Equation
The right hand side, according Reynolds Transport Theorem (RTT), is
Z
Z
Z
D
t
ρ U dV =
ρ U dV +
ρ UU rn dA
Dt sys
dt c.v.
c.v.
(2.78)
The liquid velocity, U , is measured in the frame of reference and U rn is the liquid relative
velocity to boundary of the control volume measured in the same frame of reference.
Thus, the general form of the momentum equation without the external forces is
Integral Momentum Equation
Z
Z
g ρ dV −
P dA +
τ · dA
c.v.
c.v.
Z c.v.
Z
t
=
ρ U dV+
ρ U Urn dV
dt c.v.
c.v.
Z
(2.79)
With external forces Eq. (2.79) is transformed to
X
Integral Momentum Equation & External Forces
Z
Z
Z
F ext +
g ρ dV−
P · dA +
τ · dA =
c.v.
c.vZ.
c.v.Z
t
ρ U dV +
ρ U Urn dV
dt c.v.
c.v.
(2.80)
The external forces, Fext , are the forces resulting from support of the control volume by non–
fluid elements. These external forces are commonly associated with pipe, ducts, supporting
solid structures, friction (non-fluid), etc.
Eq. (2.80) is a vector equation which can be broken into its three components. In Cartesian coordinate, for example in the x coordinate, the components are
Z
Z
Z
X
Fx +
g · î ρ dV −
P cos θx dA +
τx · dA =
c.v.
c.v.
Z c.v.
Z
t
ρ U x dV +
ρ U x · U rn dA (2.81)
dt c.v.
c.v.
where θx is the angle between n̂ and î or the resulting of (n̂ · î).
2.3.6
Momentum Equation in Acceleration System
For accelerate system, the right hand side has to include the following acceleration
a acc = ω × (rr × ω) + 2 U × ω + r × ω̇ − a 0
(2.82)
40
CHAPTER 2. BASIC FLUID MECHANICS
Where r is the distance from the center of the frame of reference and the add force is
Z
F acc =
a acc ρ dV
(2.83)
Vc.v.
Integral of Uniform Pressure on Body
In this kind of calculations, it common to obtain a situation where one of the
term will be an integral of the pressure over the body surface. This situation
is a similar to the idea that was shown in buoyancy. In this case the resulting
force due to the pressure is zero to all directions.
2.3.7
Differential Analysis
2.3.7.1
Introduction
The integral analysis has a limited accuracy, which leads to a different approach of differential
analysis. The differential analysis allows the flow field investigation in greater details. In
differential analysis, the emphasis is on infinitesimal scale and thus the analysis provides better
accuracy. These equations based on the relationship between stress and rate–of–strain.
Navier–Stokes equations are non–linear and there are more than one possible solution
in many cases (if not most cases) e.g. the solution is not unique. Historically, complexity of the
equations, on one hand, leads to approximations and consequently to the ideal flow approximation (equations) and on the other hand experimental solutions of Navier–Stokes equations.
The connection between these two ideas or fields was done via introduction of the boundary
layer theory by Prandtl which will be discussed as well.
Even for simple situations, there are cases when complying with the boundary conditions leads to a discontinuity (shock or choked flow). These equations cannot satisfy the
boundary conditions in other cases and in way the fluid pushes the boundary condition(s)
further downstream (choked flow). These issues are discussed in other books like the Open
Channel Flow and Compressible Flow chapters in this author “Basics of Fluid Mechanics” and
“Compressible Flow”. Sometimes, the boundary conditions create instability which alters the
boundary conditions itself which is known as Interfacial instability. The choked flow is associated with a single phase flow (even the double choked flow) while the Interfacial instability
associated with the Multi–Phase flow.
However for a control volume using Reynolds Transport Theorem (RTT), the following
can be written
Z
Z
Z
D
d
ρdV =
ρdV +
Urn ρ dA = 0
(2.84)
Dt V
dt V
A
For a constant control volume, the derivative can enter into the integral and hence
Z
Z
dρ
dV +
Urn ρ dA = 0
V dt
A
(2.85)
41
2.3. FUNDAMENTALS OF FLUID MECHANICS
The first term in Eq. (2.85) for the infinitesimal volume is expressed, neglecting higher order
derivatives, as
Z
z
dV
V
dρ z }| {
dρ
dV =
dx dy dz + f
dt
dt
∼0
}|
{
d2 ρ
+···
dt2
(2.86)
The second term in the LHS of Eq. (2.84) is expressed as
Z
A
dAyz
z }| { Urn ρ dA = dy dz (ρ Ux )|x − (ρ Ux )|x+dx +
dA
dA
i z }|xz{ z }|xz{ h
dx dz (ρ Uy )|y − (ρ Uy )|y+dy + dx dy (ρ Uz )|z − (ρ Uz )|z+dz
(2.87)
The difference between point x and x + dx can be obtained by developing Taylor series as
(ρ Ux )|x+dx = (ρ Ux )|x +
∂ (ρ Ux )
∂x
(2.88)
dx
x
And after some additional manipulations, one can get
Continuity in Cartesian Coordinates
∂ρ ∂ρ Ux ∂ρ Uy ∂ρ Uz
+
+
+
=0
∂t
∂x
∂y
∂z
2.3.7.2
(2.89)
Mass Conservation Examples
Example 2.1: Liquid Layer
Level: Basic
A layer of liquid has an initial height of
ρ
T1
H0 with an uniform temperature of T0 .
At time, t0 , the upper surface is exposed
H0(t)
to temperature T1 (see Figure ??). AsT0
y
ρ
sume that the actual temperature is exponentially approaches to a linear temperature profile as depicted in Figure ??.
T(t = 0) T(t > 0) T(t = ∞)
The density is a function of the temperFig.
2.8
–
Mass flow due to temperature difature according to
ference for example 2.1.
T − T0
ρ − ρ0
=α
(2.1.a)
T1 − T0
ρ1 − ρ0
where ρ1 is the density at the surface and where ρ0 is the density at the bottom.
Assume that the velocity is only a function of the y coordinate. Calculates the
velocity of the liquid. Assume that the velocity at the lower boundary is zero at all
1
0
42
CHAPTER 2. BASIC FLUID MECHANICS
End of Ex. 2.1
times. Neglect the mutual dependency of the temperature and the height.
Solution
The situation is unsteady state thus the unsteady state and one dimensional continuity equation
has to be used which is
∂ρ ∂ (ρUy )
(2.1.b)
+
=0
∂t
∂y
with the boundary condition of zero velocity at the lower surface Uy (y = 0) = 0. The
expression that connects the temperature with the space for the final temperature as
The exponential decay is 1 − e
T − T0
H −y
=α 0
T1 − T0
H0
−β t
(2.1.c)
and thus the combination (with equation (2.1.a)) is
ρ − ρ0
H −y 1 − e−β t
=α 0
ρ1 − ρ0
H0
(2.1.d)
Equation (2.1.d) relates the temperature with the time and the location was given in the question
(it is not the solution of any model). It can be noticed that the height H0 is a function of time.
For this question, it is treated as a constant. Substituting the density, ρ, as a function of time
into the governing equation (2.1.b) results in
z
αβ
∂ρ
∂t
}| {
H0 − y
−β t
e
+
H0
z ∂ Uy α
∂ρ Uy
∂y
}|
H0 −y
H0
1 − e−β t
∂y
{
(2.1.e)
=0
Equation (2.1.e) is first order ODE with the boundary condition Uy (y = 0) = 0 which can be
arranged as
∂ Uy α
H0 −y
H0
1 − e−β t
∂y
= −α β
H0 − y
H0
e−β t
(2.1.f)
Uy is a function of the time but not y. Equation (2.1.f) holds for any time and thus, it can be
treated for the solution of equation (2.1.f) as a constant2 . Hence, the integration with respect to
y yields
Uy α
H0 − y 1 − e−β t
H0
= −α β
2 H0 − y
2 H0
e−β t y + c
(2.1.g)
Utilizing the boundary condition Uy (y = 0) = 0 yields
2 H0 − y
H −y Uy α 0
1 − e−β t
= −α β
e−β t (y − 1)
H0
2 H0
or the velocity is
Uy = β
2 H0 − y
2 (H0 − y)
e−β t
(1 − y)
1 − e−β t
It can be noticed that indeed the velocity is a function of the time and space y.
2 Since the time can be treated as a constant for
y integration.
(2.1.h)
(2.1.i)
43
2.3. FUNDAMENTALS OF FLUID MECHANICS
2.3.7.3
Simplified Continuity Equation
A simplified equation can be obtained for a steady state in which the transient term is eliminated as (in a vector form)
∇ · (ρ U ) = 0
(2.90)
If the fluid is incompressible then the governing equation is a volume conservation as
Continuity Eq
∇·U = 0
Note that this equation appropriate only for a single phase case.
(2.91)
Example 2.2: Coating Process
Level: Intermediate
In many coating processes a thin film is created by a continuous process in
which liquid injected into a moving belt which carries the material out as exhibited in Fig. 2.9. The temperature and mass transfer taking place which reduces (or increases) the thickness of the film. For this example, assume that
no mass transfer occurs or can be neglected and the main mechanism is heat
transfer. Assume that the film temperature is only a function of the distance from
the extraction point. Calculate the film
velocity field if the density is a function
T0
of the temperature. The relationship beH0
tween the density and the temperature is
T0
T(x)
T∞
x
linear as
x
ρ − ρ∞
T − T∞
Fig. 2.9 – Mass flow in coating process for
=α
(2.2.a)
Ex. 2.2.
ρ0 − ρ∞
T0 − T∞
State your assumptions.
Solution
At any point the governing equation in coordinate system that moving with the belt is
∂ (ρ Ux ) ∂ (ρ Uy )
+
=0
∂x
∂y
(2.2.b)
At first, it can be assumed that the material moves with the belt in the x direction in the same
velocity. This assumption is consistent with the first solution (no stability issues). If the frame
of reference was moving with the belt then there is only velocity component in the y direction.
Hence Eq. (2.2.b) can be written as
Ux
Where Ux is the belt velocity.
∂ (ρ Uy )
∂ρ
=−
∂x
∂y
(2.2.c)
44
CHAPTER 2. BASIC FLUID MECHANICS
End of Ex. 2.2
See the resembles to equation (2.1.b). The solution is similar to the previous Example 2.1
for a general function T = F(x).
∂ρ
α ∂F(x)
(ρ0 − ρ∞ )
=
∂x
Ux ∂x
(2.2.d)
Substituting this relationship in equation (2.2.d) into the governing equation results in
∂Uy ρ
α ∂F(x)
(ρ0 − ρ∞ )
=
∂y
Ux ∂x
(2.2.e)
The density is expressed by equation (2.2.a) and thus
Uy =
α ∂F(x)
(ρ0 − ρ∞ ) y + c
ρ Ux ∂x
(2.2.f)
Notice that ρ could “come” out of the derivative (why?) and move into the RHS. Applying the
boundary condition Uy (t = 0) = 0 results in
Uy =
∂F(x)
α
(ρ0 − ρ∞ ) y
ρ(x) Ux ∂x
2.3.8
Conservation of General Quantity
2.3.8.1
Generalization of Mathematical Approach for Derivations
(2.2.g)
In this section a general approach for the derivations for conservation of any quantity e.g.
scalar, vector or tensor, is presented. Suppose that the property ϕ is under a study which is
a function of the time and location as ϕ(x, y, z, t). The total amount of quantity that exist in
arbitrary system is
Z
Φ=
ϕ ρ dV
(2.92)
sys
Where Φ is the total quantity of the system which has a volume V and a surface area of A
which is a function of time. A change with time is
Z
DΦ
D
=
ϕ ρ dV
(2.93)
Dt
Dt sys
Using RTT to change the system to a control volume yields
Z
Z
Z
D
d
ϕ ρ dV =
ϕ ρ dV +
ρ ϕ U · dA
Dt sys
dt cv
A
(2.94)
The last term on the RHS can be converted using the divergence theorem from a surface
integral into a volume integral (alternatively, the volume integral can be changed to the surface
integral) as
Z
Z
ρ ϕ U · dA =
∇ · (ρ ϕ U ) dV
(2.95)
A
V
45
2.3. FUNDAMENTALS OF FLUID MECHANICS
Substituting Eq. (2.95) into Eq. (2.94) yields
Z
Z
Z
d
D
ϕ ρ dV =
ϕ ρ dV +
∇ · (ρ ϕ U ) dV
Dt sys
dt cv
cv
(2.96)
Since the volume of the control volume remains independent of the time, the derivative can
enter into the integral and thus combining the two integrals on the RHS results in
D
Dt
Z
Z
ϕ ρ dV =
sys
cv
d (ϕ ρ)
+ ∇ · (ρ ϕ U ) dV
dt
(2.97)
The definition of equation (2.92) LHS can be changed to simply the derivative of Φ. The
integral is carried over arbitrary system. For an infinitesimal control volume the change is
DΦ ∼
=
Dt
z dV
}| {
d (ϕ ρ)
+ ∇ · (ρ ϕ U ) dx dy dz
dt
2.3.8.2
Momentum Conservation
2.3.8.3
Derivations of the Momentum Equation
(2.98)
At time t, the control volume is at a square shape and at a location as depicted in
Fig. 2.10 (by the blue color). At time t + dt
the control volume undergoes three different
changes. The control volume moves to a new
location, rotates and changes the shape (the
=
+
+
purple color in in Fig. 2.10). The translational
∂U
U
+
dt
movement is referred to a movement of body
y
∂y
D
without change of the body and without roB
@ t + dt
tation. The rotation is the second movement
that referred to a change in of the relative ori∂U
U dt
U +
dt
∂x
entation inside the control volume. The third
@t C
A
A
U dt
change is the misconfiguration or control volx
ume (deformation). The deformation of the
45
control volume has several components (see
the top of Fig. 2.10). The shear stress is related
Fig. 2.10 – Control volume at t and t + dt
under continuous angle deformation Notice
to the change in angle of the control volume
the three combinations of the deformation
lower left corner. The angle between x to the
shown by purple color relative to blue color.
new location of the control volume can be approximate for a small angle as


y

y

y

y

y
dγx
= tan
dt
Uy +
dUy
dx dx − Uy
dx
!
= tan
dUy
dx
x’
y’
x
∼ dUy
=
dx
◦
(2.99)
46
CHAPTER 2. BASIC FLUID MECHANICS
The total angle deformation (two sides x and y) is
Dγxy
dUy dUx
=
+
Dt
dx
dy
(2.100)
dU
x
In these derivatives, the symmetry dxy ̸= dU
dy was not assumed and or required because
rotation of the control volume. However, under isentropic material it is assumed that all the
shear stresses contribute equally.
Dγxy
dUy dUx
τxy = µ
=µ
+
(2.101)
Dt
dx
dy
where, µ is the “normal” or “ordinary” viscosity coefficient which relates the linear coefficient
of proportionality and shear stress. This deformation angle coefficient is assumed to be a
property of the fluid. In a similar fashion it can be written to other directions for x z as
dUz dUx
+
dx
dz
and for the directions of y z as
Dγyz
dUz dUy
τyz = µ
=µ
+
Dt
dy
dz
(2.102)
B
D
y
τx y
τxx
(2.103)
Note that the viscosity coefficient (the linear coefficient3 ) is assumed to be the same regardless of the direction. This assumption is referred as isotropic viscosity. It can be noticed at this stage, the relationship for
the two of stress tensor parts was established. The only
missing thing, at this stage, is the diagonal component
which to be dealt below.
’
’
τx x
’
τxy
A
’
C
τyx τyy
x
x’
Dγxz
=µ
Dt
y’
τxz = µ
45◦
Fig. 2.11 – Shear stress at two coordinates in 45◦ orientations.
“mechanical” pressure
τx’x’
z
}|
{
τx’x’ + τy’y’ + τz’z’
∂Ux’ 2
∂Ux’ ∂Uy’ ∂Uz’
=
+2 µ
− µ
+
+
3
∂x’
3
∂x’
∂y’
∂z’
(2.104)
The “mechanical” pressure, Pm , is defined as the (negative) average value of pressure in directions of x’–y’–z’. This pressure is a true scalar value of the flow field since the propriety is
averaged or almost4 In situations where the main diagonal terms of the stress tensor are not
the same in all directions (in some viscous flows) this property can be served as a measure of
the local normal stress. The mechanical pressure can be defined as averaging of the normal
stress acting on a infinitesimal sphere. It can be shown that this two definitions are “identical”
in the limits. With this definition and noticing that the coordinate system x’–y’ has no special
3 The first assumption was mentioned above.
4 It identical only in the limits to the mechanical measurements.
2.3. FUNDAMENTALS OF FLUID MECHANICS
47
significance and hence equation (2.104) must be valid in any coordinate system thus equation
(2.104) can be written as
τxx = −Pm + 2 µ
∂Ux 2
+ µ∇·U
∂x
3
(2.105)
Again where Pm is the mechanical pressure and is defined as
Mechanical Pressure
τxx + τyy + τzz
Pm = −
3
(2.106)
Commonality engineers like to combined the two difference expressions into one as
τxy
=0
z}|{
2
∂Ux ∂Uy
= − Pm + µ∇ · U δxy +µ
+
3
∂y
∂x
(2.107)
τxx
=1
z}|{
2
∂Ux ∂Uy
U
= − Pm + µ∇ ·
δxy +µ
+
3
∂x
∂y
(2.108)
or
where δij is the Kronecker delta what is δij = 1 when i = j and δij = 0 otherwise. While this
expression has the advantage of compact writing, it does not add any additional information.
This expression suggests a new definition of the thermodynamical pressure is
Thermodynamic Pressure
2
P = Pm + µ∇ · U
3
(2.109)
2.3.8.3.1 Summary of The Stress Tensor
The above derivations were provided as a long mathematical explanation. To reduced one
unknown (the shear stress). The relationship between the stress tensor and the velocity were
established. First, connection between τxy and the deformation was constructed. Then the
association between normal stress and perpendicular stress was constructed. Using the coordinates transformation, this association was established. The linkage between the stress in
the rotated coordinates to the deformation was established.
The explicit form of the strain is


∂Uy
∂Ux
∂Ux
∂Uz
x
2 ∂U
+
+
∂x
∂y
∂x
∂z
∂x



1
∂Uy
∂Uy
∂Uy
∂Ux
∂Uz


ε =  ∂y + ∂x
(2.110)
2 ∂y
+ ∂y

∂z
2 

∂Ux
∂Uz
∂Ux
∂Uz
z
2 ∂U
∂z + ∂x
∂z + ∂x
∂z
This matrix Eq. (2.110) referred in literature as the shear rate which is similar to strain in solid
mechanics. These changes in the angles and element’s geometry can be obtained from the
velocity field.
48
CHAPTER 2. BASIC FLUID MECHANICS
Example 2.3: Given Velocity to Strain
Level: Intermediate
Calculate the shear rate of the hypothetical flow field given as
(2.3.a)
U = A y2 z3 î + A x2 ey ĵ + A x2 sin(x) k̂
Solution
The process is simply applying the formulas. The strain the xx direction is
εxx =
∂ A y2 z3
∂Ux
=
=0
∂x
∂x
(2.3.b)
in the xy or yx which are the same is
εxy =
1
2
∂Ux ∂Uy
+
∂y
∂x
=
in the xz or zx which are the same is
εxz =
1
2
∂Ux ∂Uz
+
∂z
∂x
(2.3.c)
(2.3.d)
= 3 A y2 z2 + 2 A x sin(x) + A x2 cos(x)
The next main strains is
εyy =
∂Uy
= A x2 e y
∂y
(2.3.e)
∂Uz
=0
∂z
(2.3.f)
The last main strain is
εzz =
The last mix strain is yz
εyz =
1 A 2 y z 3 + A 2 x ey
2
1
2
∂Uy ∂Uz
+
∂z
∂y
(2.3.g)
=0
There is no special significance for these results. In certain direction the strain can be zero
while it can have value in others.
For the total effect can be now written as
∂ P + 23 µ − λ ∇ · U
DUx
=−
+
ρ
Dt
∂x
µ
or in a vector form as
∂2 Ux ∂2 Ux ∂2 Ux
+
+
∂x2
∂y2
∂z2
+ f Bx
N-S in Stationary Coordinates
U
DU
1
ρ
= −∇P +
µ + λ ∇ (∇ · U ) + µ ∇2U + f B
Dt
3
For in index form as
D Ui
∂
2
∂
∂Ui ∂Uj
ρ
=−
P+
µ−λ ∇·U +
µ
+
+ f Bi
Dt
∂xi
3
∂xj
∂xj
∂xi
(2.111)
(2.112)
(2.113)
2.3. FUNDAMENTALS OF FLUID MECHANICS
49
For incompressible flow the term ∇ · U vanishes, thus equation (2.112) is reduced to
Momentum for Incompressible Flow
ρ
U
DU
= −∇P + µ ∇2U + f B
Dt
(2.114)
or in the index notation it is written
ρ
D Ui
∂P
∂2U
=−
+µ
+ f Bi
Dt
∂xi
∂xi ∂xj
(2.115)
The momentum equation in Cartesian coordinate can be written explicitly for x coordinate
as
∂Ux
∂Ux
∂Ux
∂Ux
=
+ Ux
+ Uy
+ Uz
ρ
∂t
∂x ∂y
∂z
(2.116)
∂P
∂2 Ux ∂2 Ux ∂2 Ux
−
+µ
+
+
+
ρ
g
x
∂x
∂x2
∂y2
∂z2
Where gx is the body force in the x direction (bi · g ). In the y coordinate the momentum
equation is
∂Uy
∂Uy
∂Uy
∂Uy
ρ
+ Ux
+ Uy
+ Uz
=
∂t
∂x ∂y
∂z (2.117)
∂P
∂2 v ∂2 v ∂2 v
−
+µ
+
+
+
ρ
g
y
∂y
∂x2 ∂y2 ∂z2
in z coordinate the momentum equation is
∂Uz
∂Uz
∂Uz
∂Uz
+ Ux
+ Uy
+ Uz
=
ρ
∂t
∂x ∂y
∂z
2
2
2
∂P
∂ Uz ∂ Uz ∂ Uz
−
+µ
+
+
+ ρ gz
∂z
∂x2
∂y2
∂z2
2.3.8.4
(2.118)
Boundary Conditions and Driving Forces
2.3.8.4.1 Boundary Conditions Categories
The governing equations that were developed earlier requires some boundary conditions and
initial conditions. These conditions described physical situations that are believed or should
exist or approximated. These conditions can be categorized by the velocity, pressure, or in
more general terms as the shear stress conditions (mostly at the interface). For this discussion,
the shear tensor will be separated into two categories, pressure (at the interface direction) and
shear stress (perpendicular to the area). A common velocity condition is that the liquid has
the same value as the solid interface velocity. In the literature, this condition is referred as
the “no slip” condition. The solid surface is rough thus the liquid participles (or molecules)
are slowed to be at the solid surface velocity. This boundary condition was experimentally
50
CHAPTER 2. BASIC FLUID MECHANICS
observed under many conditions yet it is not universal true. The slip condition (as oppose to
“no slip” condition) exist in situations where the scale is very small and the velocity is relatively
very small. The slip condition is dealing with a difference in the velocity between the solid (or
other material) and the fluid media. The difference between the small scale and the large scale
is that the slip can be neglected in the large scale while the slip cannot be neglected in the small
scale. In another view, the difference in the velocities vanishes as the scale increases.
Another condition which affects whether the
tb n
c
slip condition exist is how rapidly of the velocity change. The slip condition cannot be igf (x) flow
direction
y
nored in some regions, when the flow is with
x
a strong velocity fluctuations. Mathematically
Fig. 2.12 – 1–Dimensional free surface describing
the “no slip” condition is written as
b and tb.
n
bt · U fluid − U boundary = 0
(2.119)
b
where n is referred to the area direction (perpendicular to the area see Figure 2.12). While this
condition (2.119) is given in a vector form, it is more common to write this condition as a given
velocity at a certain point such as
U(ℓ) = Uℓ
(2.120)
Note, the “no slip” condition is applicable to the ideal fluid (“inviscid flows”) because this kind
of flow normally deals with large scales. The "slip" condition is written in similar fashion to
equation (2.119) as
bt · U fluid − U boundary = f(Q, scale, etc)
(2.121)
As oppose to a given velocity at particular point, a requirement on the acceleration
(velocity) can be given in unknown position. The condition (2.119) can be mathematically represented in another way for free surface conditions. To make sure that all the material is
accounted for in the control volume (does not cross the free surface), the relative perpendicular velocity at the interface must be zero. The location of the (free) moving boundary can be
given as f(b
r , t) = 0 as the equation which describes the bounding surface. The perpendicular
relative velocity at the surface must be zero and therefore
Df
=0
Dt
on the surface f(b
r , t) = 0
(2.122)
This condition is called the kinematic boundary condition. For example, the free surface in
the two dimensional case is represented as f(t, x, y). The condition becomes as
0=
∂f
∂f
∂f
+ Ux
+ Uy
∂t
∂x
∂y
(2.123)
The solution of this condition, sometime, is extremely hard to handle because the location
isn’t given but the derivative given on unknown location. In this book, this condition will not
be discussed (at least not plane to be written).
51
2.3. FUNDAMENTALS OF FLUID MECHANICS
The free surface is a special case of moving surfaces where the surface between two
distinct fluids. In reality the interface between these two fluids is not a sharp transition but
only approximation (see for the surface theory). There are situations where the transition
should be analyzed as a continuous transition between two phases. In other cases, the transition is idealized an almost jump (a few molecules thickness). Furthermore, there are situations
where the fluid (above one of the sides) should be considered as weightless material. In these
cases the assumptions are that the transition occurs in a sharp line, and the density has a jump
while the shear stress are continuous (in some cases continuously approach zero value). While
a jump in density does not break any physical laws (at least those present in the solution), the
jump in a shear stress (without a jump in density) does break a physical law. A jump in the
shear stress creates infinite force on the adjoin thin layer. Off course, this condition cannot
be tolerated since infinite velocity (acceleration) is impossible. The jump in shear stress can
appear when the density has a jump in density. The jump in the density (between the two
fluids) creates a surface tension which offset the jump in the shear stress. This condition is
expressed mathematically by equating the shear stress difference to the forces results due to
the surface tension. The shear stress difference is
∆τ(n) = 0 = ∆τ(n) upper − ∆τ(n) lower
surface
surface
(2.124)
where the index (n) indicate that shear stress are normal (in the surface area). If the surface
is straight there is no jump in the shear stress. The condition with curved surface are out the
scope of this book yet mathematically the condition is given as without explanation as
1
1
b · τ(n) = σ
n
+
(2.125)
R1 R2
b
t · τ(t) = −b
t · ∇σ
(2.126)
b is the unit normal and b
where n
t is a unit tangent to the surface (notice that direction pointed
out of the “center” see Figure 2.12) and R1 and R2 are principal radii. One of results of the free
surface condition (or in general, the moving surface condition) is that integration constant is
unknown). In same instances, this constant is determined from the volume conservation. In
index notation equation (2.125) is written5 as
1
1
(2)
(1)
τij nj + σ ni
+
= τij nj
(2.127)
R1 R2
where 1 is the upper surface and 2 is the lower surface. For example in one dimensional6
b=
n
b
t=
(−f′ (x), 1)
q
1 + (f′ (x))2
(1, f′ (x))
q
1 + (f′ (x))2
(2.128)
5 There is no additional benefit in this writing, it just for completeness and can be ignored for most purposes.
6A
one example of a reference not in particularly important or significant just a random example. Jean, M. Free
surface of the steady flow of a Newtonian fluid in a finite channel. Arch. Rational Mech. Anal. 74 (1980), no. 3, 197–217.
52
CHAPTER 2. BASIC FLUID MECHANICS
the unit vector is given as two vectors in x and y and the radius. The equation is given by
∂f
∂f
+ Ux
= Uy
∂t
∂x
2.3.8.4.2
(2.129)
The Pressure Condition
The second condition that commonality prescribed at the interface is the static pressure at
a specific location. The static pressure is measured perpendicular to the flow direction. The
last condition is similar to the pressure condition of prescribed shear stress or a relationship
to it. In this category include the boundary conditions with issues of surface tension which
were discussed earlier. It can be noticed that the boundary conditions that involve the surface
tension are of the kind where the condition is given on boundary but no at a specific location.
2.3.8.4.3
Gravity as A Driving Force
The body forces, in general and gravity in a particular, are the condition that given on the
flow beside the velocity, shear stress (including the surface tension) and the pressure. The
gravity is a common body force which is considered in many fluid mechanics problems. The
gravity can be considered as a constant force in most cases (see for dimensional analysis for
the reasons).
2.3.8.4.4
Shear Stress and Surface Tension as Driving Forces
If the fluid was solid material, pulling the side will pull
all the material. In fluid (mostly liquid) shear stress
pulling side (surface) will have limited effect and yet
sometime is significant and more rarely dominate. Consider, for example, the case shown in Fig. 2.13. The shear
stress carry the material as if part of the material was
a solid material. For example, in the kerosene lamp the
burning occurs at the surface of the lamp top and the liquid is at the bottom. The liquid does not move up due
the gravity (actually it is against the gravity) but because
the surface tension.
Fig. 2.13 – Surface tension as driving
force in a kerosene lamp.
The physical conditions in Fig. 2.13 are used to idealize the flow around an inner rode to
understand how to apply the surface tension to the boundary conditions. The fluid surrounds
the rode and flows upwards. In that case, the velocity at the surface of the inner rode is zero.
The velocity at the outer surface is unknown. The boundary condition at outer surface given
by a jump of the shear stress. The outer diameter is depends on the surface tension (the larger
53
2.3. FUNDAMENTALS OF FLUID MECHANICS
surface tension the smaller the liquid diameter). The
surface tension is a function of the temperature therefore the gradient in surface tension is result of temperature gradient. In this book, this effect is not discussed. However, somewhere downstream the temperature gradient is insignificant. Even in that case, the surface tension gradient remains. It can be noticed that, under the assumption presented here, there are two principal radii of the flow. One radius toward the center
of the rode while the other radius is infinite (approximately). In that case, the contribution due to the curvature is zero in the direction of the flow (see Figure 2.14).
The only (almost) propelling source of the flow is the
∂σ
).
surface gradient ( ∂n
2.3.9
U(ri) = 0
µ
∂U ∂σ
=
∂r
∂h
}
}
}
temperature
gradent
mix zone
constant
T
Fig. 2.14 – Flow in a candle with a surface tension gradient.
Examples for Differential Equation (Navier-Stokes)
Examples of an one-dimensional flow driven by the shear stress and pressure are presented.
For further enhance the understanding some of the derivations are repeated. First, example
dealing with one phase are present. Later, examples with two phase are presented.
Example 2.4: Flow Between Two Plates
Level: Simple
Uℓ
y
flow direction
x
dy
z
Fig. 2.15 – Flow between two plates, top plate is moving at speed of Uℓ to the right (as positive).
The control volume shown in darker colors.
0.7 Incompressible liquid flows between two infinite plates from the left to the right
(as shown in Figure 2.15). The distance between the plates is ℓ. The static pressure per
length is given as ∆P (The difference is measured at the bottom point of the plate.)
The upper surface is moving in velocity, Uℓ (The right side is defined as positive).
Solution
In this example, the mass conservation yields
=0
z Z }|
{
Z
d
ρdV = −
ρ Urn dA = 0
dt cv
cv
(2.4.a)
54
CHAPTER 2. BASIC FLUID MECHANICS
continue Ex. 2.4
The momentum is not accumulated (steady state and constant density). Further because no
change of the momentum thus
Z
ρ Ux Urn dA = 0
A
(2.4.b)
Thus, the flow in and the flow out are equal. It can concluded that the velocity in and out are
the same (for constant density). The momentum conservation leads
Z
Z
P dA +
−
cv
τxy dA = 0
cv
(2.4.c)
The reaction of the shear stress on the lower surface of control volume based on Newtonian
fluid is
dU
τxy = −µ
(2.4.d)
dy
On the upper surface is different by Taylor explanation as
τxy

∼
=0
z
}|
{

 dU d2 U
d3 U


+
dy +
dy2 + · · ·
= µ
2
3

 dy
dy
dy

(2.4.e)
The net effect of these two will be difference between them
µ
dU d2 U
dU ∼ d2 U
+
dy
−µ
dy
=µ
2
dy
dy
dy
dy2
(2.4.f)
The assumptions is that there is no pressure difference in the z direction. The only difference
in the pressure is in the x direction and thus
dP
dP
P− P+
dx = −
dx
dx
dx
(2.4.g)
A discussion why ∂P
∂y ∼ 0 will be presented later. The momentum equation in the x direction
(or from equation (2.116)) results (without gravity effects) in
−
d2 U
dP
=µ
dx
dy2
(2.4.h)
Equation (2.4.h) was constructed under several assumptions which include the direction
of the flow, Newtonian fluid. No assumption was imposed on the pressure distribution.
Equation (2.4.h) is a partial differential equation but can be treated as ordinary differential equation in the z direction of the pressure difference is uniform. In that case, the
left hand side is equal to constant. The “standard” boundary conditions is non–vanishing
pressure gradient (that is the pressure exist) and velocity of the upper or lower surface
or both. It is common to assume that the “no–slip” condition on the boundaries condition7 .
The boundaries conditions are
Ux (y = 0) = 0
Ux (y = ℓ) = Uℓ
(2.4.i)
55
2.3. FUNDAMENTALS OF FLUID MECHANICS
End of Ex. 2.4
The solution of the “ordinary” differential equation (2.4.h) after the integration becomes
Ux = −
1 dP 2
y + c2 y + c3
2 µ dx
(2.4.j)
Applying the first boundary condition results in C3 = 0. The second boundary condition
results in
1 dP 2
U
1 dP
Uℓ = −
ℓ + c2 ℓ −−−→ C2 = ℓ +
ℓ
(2.4.k)
2 µ dx
ℓ
2 µ dx
Velocity distributions in one dimensional flow
For the case where the pressure gradient is zero the velocity is linear
as was discussed earlier in chapter 1.
The pressure gradient also affects the
boundary condition. Inserting the
coefficients into Eq. (2.4.j) reads
1.2
Ψ = −1.75
1.0
Ψ = −0.25
Ψ = −1.25
Ψ = −0.75
Ψ = 0.25
Ψ = 0.75
Ψ = 1.25
0.8
Ψ = 1.75
Ψ = 2.25
Ux
Uℓ
Ψ = 2.75
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
y
ℓ
1 dP 2
y +
2µ dx
Uℓ
1 dP
+
ℓ y
ℓ
2 µ dx
October 4, 2010
Ux (y) = −
Fig. 2.16 – One dimensional flow with a shear between two plates when Ψ change value between
-1.75 green line to 3 the blue line.
(2.4.l)
Rearranging Eq. (2.4.l) yields

{
 y
Ux (y) 
y

 ℓ2 dP
1−
+ 1
=
 ℓ
 2 µ Uℓ dx
Uℓ
ℓ
What happens when

∂P
∂y
z
Φ
}|
(2.4.m)
∼ 0?
When more than one liquid is following in the conduit the mathematics because more
complected but the principle is the same. The following problem was inspired by a stability
question of two fluids transition in die casting.
7 A discussion about the boundary will be presented later.
56
CHAPTER 2. BASIC FLUID MECHANICS
Example 2.5: Two layers velocity profiles
Two fluids flow as two layers one above
each other in conduit is examined here. Ignoring the stability issue at this stage, calculate the velocity profiles of these fluids.
The properties of the fluids are given in
this problem. The height of fluid A and B
are given. Calculate the flow rate for both
liquid. Assume no–slip boundaries conditions. What are the relationship between
the flow rates and the the pressure gradient?
Level: Intermediate
hB
yh
A
ρB , µB
B
ρA , µA
xA
Fig. 2.17 – caption here.
Solution
The governing equation for both fluids is
d2 U
dP
=µ
dx
dy2
(2.5.a)
UA (y = 0) = 0
(2.5.b)
−
with boundaries conditions
The shear stress has to match on both sides thus
µA
dUA
dy
(y=h1 )
= µB
dUB
dy
(y=h1 )
(2.5.c)
The no–slip condition between the liquid must be obey
UA (y = h1 ) = UB (y = h1 )
(2.5.d)
and the no–slip condition on the upper surface reads
UB (y = h2 ) = 0
(2.5.e)
The shear stress requirement force a jump in the abrupt change in the velocity profile. These
conditions Eqs. (2.5.b)–(2.5.e) need to be augmented with two more equation to deal with expected 4 unknowns. These four unknowns are the result of the solution of ODE Eq. (2.5.a) (two
ranges thus two times two). This combination is sufficient solve the problem. This author is
not aware of a single and ultimate solution to the problem. Thus, any method is valid. The
general solution of of the governing equation is
U(A ,B
B) = −
1 dP 2
y + C(A ,B
B) 1 y + C(A ,B
B) 2
µA ,B
B dx
(2.5.f)
where C(A ,B
B) 1 and C(A ,B
B) 2 are the integration constants. Applying condition Eq. (2.5.b)
results in
1 dP 2
0=−
0 + CA 1 0 + CA 2
(2.5.g)
µA dx
57
2.3. FUNDAMENTALS OF FLUID MECHANICS
End of Ex. 2.5
which dictates that CA 2 = 0. Similarly, the upper condition can be written as
0=−
1 dP
(h + hB )2 + CB 1 (hA + hB ) + CB 2
µA dx A
(2.5.h)
at the interface between the two fluids the velocities are the same
−
1 dP
1 dP
h 2 + CA 1 hA = −
h 2 + CB 1 hA + CB 2
µA dx A
µB dx A
(2.5.i)
And the shear stress are the same at the interface
µA
1 dP
1 dP
hA + CA 1 = µB −
hA + CB 1
−
µA dx
µB dx
(2.5.j)
or after rearrangement it can be written as
Z
Z
dP
dP
ZhA + µA CA 1 = − dxZhA + µB CB 1
dx Z
Z
−
(2.5.k)
which relates the two coefficients as
CA 1 =
µB
C
µA B 1
(2.5.l)
Combining or substituting Eq. (2.5.l) into Eq. (2.5.k) yields
−
µ
1 dP
1 dP
h 2 + B CB 1 hA = −
h 2 + CB 1 hA + CB 2
µA dx A
µA
µB dx A
or
−
1
1
−
µA
µB
dP
h 2 = CB 1
dx A
µ
1 − B hA + CB 2
µA
(2.5.m)
(2.5.n)
Eq. (2.5.n) and Eq. (2.5.h) provides a linear set of equation to solve for CB 1 for CB 2 . Eq. (2.5.h)
slightly rearranged to be
1 dP
(h + hB )2 = CB 1 (hA + hB ) + CB 2
µA dx A
The solution is
µ
(hA + hB )2 B − hA 2
dP µA
µA
CB 1 =
(hA µB + hB µA )
dx µB
(2.5.o)
(2.5.p)
The value of CB 1 can be either positive or negative which could effect of the stability of flow
(solution) which depend the viscosity ratio. Notice that Eq. (2.5.l) also dictate that CB 2 has the
same sign. This topic is above the scope of the example. The solution for the second coefficient
is
CB 2 =
dP µA
dx µB
µB
− 1 (hB µB + hA µB − hA µA )
µA
µA µB (hA µB + hB µA )
hA (hB + hA )
(2.5.q)
These solutions are not fully non–dimensionalized as the heights can be pulled out and some
additional manipulations and create an universal solution to this problem.
58
CHAPTER 2. BASIC FLUID MECHANICS
2.3.9.0.1 Cylindrical Coordinates
Similarly the problem of one dimensional flow
can be constructed for cylindrical coordinates.
The problem is still one dimensional because
the flow velocity is a function of (only) radius. This flow referred as Poiseuille flow after Jean Louis Poiseuille a French Physician
who investigated blood flow in veins. Thus,
Poiseuille studied the flow in a small diameters (he was not familiar with the concept of
Reynolds numbers). Rederivation are carried
out for a short cut.
r
θ
rθ
flo
direc w
tion
dz
r z
Fig. 2.18 – The control volume of liquid element
in cylindrical coordinates.
The momentum equation for the control volume depicted in the Figure 2.18a is
Z
Z
Z
− P dA + τdA = ρ Uz Urn dA
(2.130)
The shear stress in the front and back surfaces do no act in the z direction. The shear stress
on the circumferential part small dark blue shown in Figure 2.18a is
Z
dA
τ dA = µ
dUz z }| {
2 π r dz
dr
(2.131)
The pressure integral is
Z
∂P
∂P
2
P dA = Pzd z − Pz π r = Pz +
dz − Pz π r2 =
dz π r2
∂z
∂z
(2.132)
The last term is
Z
Z
ρ Uz Urn dA = ρ Uz Urn dA =
ρ
Z
Z
2
2
Uz+dz dA −
z+dz
Uz dA
z
Z =ρ
Uz+dz 2 − Uz 2 dA
z
(2.133)
The term Uz+dz 2 − Uz 2 is zero because Uz+dz = Uz because mass conservation conservation for any element. Hence, the last term is
Z
ρ Uz Urn dA = 0
(2.134)
Substituting equation (2.131) and (2.132) into equation (2.130) results in
µ
∂P
dUz
= − π r2
2π
r
dz
dz
dr
∂z (2.135)
59
2.3. FUNDAMENTALS OF FLUID MECHANICS
Which shrinks to
∂P
2 µ dUz
=−
r dr
∂z
(2.136)
Equation (2.136) is a first order differential equation for which only one boundary condition
is needed. The “no slip” condition is assumed
Uz (r = R) = 0
(2.137)
Where R is the outer radius of pipe or cylinder. Integrating equation (2.136) results in
Uz = −
1 ∂P 2
r + c1
µ ∂z
(2.138)
It can be noticed that asymmetrical element8 was eliminated due to the smart short cut. The
integration constant obtained via the application of the boundary condition which is
1 ∂P 2
R
µ ∂z
(2.139)
r 2 1 ∂P 2
R 1−
µ ∂z
R
(2.140)
c1 = −
The solution is
Uz =
While the above analysis provides a solution, it has several deficiencies which include the ability to incorporate different boundary conditions such as flow between concentering cylinders.
This problem study by Nusselt9 which developed the basics equations. This problem
is related to many industrial process and is fundamental in understanding many industrial
processes. Furthermore, this analysis is a building bloc for heat and mass transfer understanding10 .
Example 2.6: Thin Film
Level: Simple
In many situations in nature and many industrial processes liquid flows downstream on inclined plate at θ as shown in Figure 2.19. For this example, assume
8 Asymmetrical element or function is −f(x)
9 German
= f(−x)
mechanical engineer, Ernst Kraft Wilhelm Nusselt born November 25, 1882 September 1, 1957 in
Munchen
10 Extensive discussion can be found in this author master thesis. Comprehensive discussion about this problem
can be found this author Master thesis.
that the gas density is zero (located
outside the liquid domain). Assume
that “scale” is large enough so that the
“no slip” condition prevail at the plate
(bottom). For simplicity, assume that
the flow is two dimensional. Assume
that the flow obtains a steady state after some length (and the acceleration
vanished). The dominate force is the
gravity. Write the governing equations for this situation. Calculate the
velocity profile. Assume that the flow
is one dimensional in the x direction.
CHAPTER 2. BASIC FLUID MECHANICS
continue Ex. 2.6
h
g co
sθ
60
g si
nθ
θ
g
Fig. 2.19 – Mass flow due to gravity difference for
example 2.6
Solution
This problem is suitable to Cartesian coordinates in which x coordinate is pointed in the flow
direction and y perpendicular to flow direction (depicted in Figure 2.19). For this system, the
gravity in the x direction is g sin θ while the direction of y the gravity is g cos θ. The governing
in the x direction is

̸=f(t)
=0

z }| { =0
z }| {
−0
 ∂U
z}|{ ∂Ux z}|{ ∂Ux
∂U

x
x
=
ρ
+ Ux
+ Uy
+ Uz
 ∂t
∂x
∂y
∂z


=0
=0
∼0
z }| {
z }| {
z}|{
g sin θ
 ∂2 U
z}|{
∂2 Ux ∂2 Ux 
∂P


x
+µ 
+
+
−
 + ρ gx
2
2
2
 ∂x
∂x
∂y
∂z 
(2.6.a)
The first term of the acceleration is zero because the flow is in a steady state. The first term of
the convective acceleration is zero under the assumption of this example flow is fully developed
and hence not a function of x (nothing to be “improved”). The second and the third terms in
the convective acceleration are zero because the velocity at that direction is zero (Uy = Uz =
0). The pressure is almost constant along the x coordinate. As it will be shown later, the
pressure loss in the gas phase (mostly air) is negligible. Hence the pressure at the gas phase is
almost constant hence the pressure at the interface in the liquid is constant. The surface has
no curvature and hence the pressure at liquid side similar to the gas phase and the only change
in liquid is in the y direction. Fully developed flow means that the first term of the velocity
x
Laplacian is zero ( ∂U
∂x ≡ 0). The last term of the velocity Laplacian is zero because no velocity
in the z direction.
Thus, equation (2.6.a) is reduced to
0=µ
∂2 Ux
+ ρ g sin θ
∂y2
(2.6.b)
61
2.3. FUNDAMENTALS OF FLUID MECHANICS
End of Ex. 2.6
With boundary condition of “no slip” at the bottom because the large scale and steady state
(2.6.c)
Ux (y = 0) = 0
The boundary at the interface is simplified to be
∂Ux
∂y
= τair (∼ 0)
y=0
(2.6.d)
If there is additional requirement, such a specific velocity at the surface, the governing equation
can not be sufficient from the mathematical point of view. Integration of equation (2.6.b) yields
∂Ux
ρ
= g sin θ y + c1
∂y
µ
(2.6.e)
The integration constant can be obtain by applying the condition (2.6.d) as
y
τair
∂Ux
=µ
∂y
Solving for c1 results in
c1 =
z}|{
= −ρ g sin θ h +c1 µ
(2.6.f)
h
1
τair
+
g sin θ h
µ
ν
|{z}
(2.6.g)
µ
ρ
The second integration applying the second boundary condition yields c2 = 0 results in
Ux =
τ
g sin θ 2 y h − y2 − air
ν
µ
(2.6.h)
g sin θ 2 h y − y2
ν
(2.6.i)
When the shear stress caused by the air is neglected, the velocity profile is
Ux =
The flow rate per unit width is
Q
=
W
Z
Ux dA =
A
Zh 0
τ
g sin θ 2 h y − y2 − air dy
ν
µ
(2.6.j)
Where W here is the width into the page of the flow. Which results in
Q
g sin θ 2 h3 τair h
=
−
W
ν
3
µ
(2.6.k)
Q
g sin θ 2 h2 τair
W
Ux =
=
−
h
ν
3
µ
(2.6.l)
The average velocity is then
Note the shear stress at the interface can be positive or negative and hence can increase or
decrease the flow rate and the averaged velocity.
62
CHAPTER 2. BASIC FLUID MECHANICS
In the following following example the issue of driving force of the flow through curved
interface is examined. The flow in the kerosene lamp is depends on the surface tension. The
flow surface is curved and thus pressure is not equal on both sides of the interface.
2.3.9.1
Interfacial Instability
In Ex. 2.6 no requirement was made as for the
sa
velocity at the interface (the upper boundm
e
air
so ve
(
lu loc
ga
ary). The vanishing shear stress at the intertio it
s)
n y
y
face was the only requirement was applied. If
x w
ate
the air is considered two governing equations
r(
liq
uid
)
must be solved one for the air (gas) phase and
ah
one for water (liquid) phase. Two boundary
h
conditions must be satisfied at the interface.
For the liquid, the boundary condition of “no
slip” at the bottom surface of liquid must be
satisfied. Thus, there is total of three boundary conditions11 to be satisfied. The solution
Fig. 2.20 – Flow of liquid in partially filled duct.
to the differential governing equations provides only two constants. The second domain (the gas phase) provides another equation with
two constants but again three boundary conditions need to satisfied. However, two of the
boundary conditions for these equations are the identical and thus the six boundary conditions are really only 4 boundary conditions.
The governing equation solution12 for the gas phase (h ⩾ y ⩾ a h) is
Uxg =
g sin θ 2
y + c1 y + c2
2 νg
(2.141)
Note, the constants c1 and c2 are dimensional which mean that they have physical units
(c1 −→ [1/sec]. The governing equation in the liquid phase (0 ⩾ y ⩾ h) is
Uxℓ =
g sin θ 2
y + c3 y + c4
2 νℓ
(2.142)
The gas velocity at the upper interface is vanished thus
Uxg [(1 + a) h] = 0
(2.143)
At the interface the “no slip” condition is regularly applied and thus
Uxg (h) = Uxℓ (h)
(2.144)
11 The author was hired to do experiments on thin film (gravity flow). These experiments were to study the formation of small and big waves at the interface. The phenomenon is explained by the fact that there is somewhere
instability which is transferred into the flow. The experiments were conducted on a solid concrete laboratory and
the flow was in a very stable system. No matter how low flow rate was small and big occurred. This explanation
bothered this author, thus current explanation was developed to explain the wavy phenomenon occurs.
12 This equation results from double integrating of equation (2.6.b) and subtitling ν = µ/ρ.
63
2.3. FUNDAMENTALS OF FLUID MECHANICS
Also at the interface (a straight surface), the shear stress must be continuous
µg
∂Uxg
∂Uxℓ
= µℓ
∂y
∂y
(2.145)
Assuming “no slip” for the liquid at the bottom boundary as
(2.146)
Uxℓ (0) = 0
The boundary condition (2.143) results in
0=
g sin θ 2
h (1 + a)2 + c1 h (1 + a) + c2
2 νg
(2.147)
The same can be said for boundary condition (2.146) which leads
(2.148)
c4 = 0
Applying equation (2.145) yields
ρg
ρ
ℓ
z}|{
z}|{
µg
µℓ
g sin θ h + c1 µg =
g sin θ h + c3 µℓ
νg
νℓ
(2.149)
Combining boundary conditions equation(2.144) with (2.147) results in
g sin θ 2
g sin θ 2
h + c3 h
h + c1 h + c2 =
2 νg
2 νℓ
(2.150)
Advance material can be skipped
The solution of equation (2.147), (2.149) and (2.150) is obtained by computer algebra (see
in the code) to be
sin θ (g h ρg (2 ρg νℓ ρℓ + 1) + a g h νℓ )
ρg (2 a νℓ + 2 νℓ )
c1
=
−
c2
=
sin θ g h2 ρg (2 ρg νℓ ρℓ + 1) − g h2 νℓ
2 ρg νℓ
c3
=
sin θ (g h ρg (2 a ρg νℓ ρℓ − 1) − a g h νℓ )
ρg (2 a νℓ + 2 νℓ )
(2.151)
End Advance material
When solving this kinds of mathematical problem the engineers reduce it to minimum
amount of parameters to reduce the labor involve. So equation (2.147) transformed by some
simple rearrangement to be
z
C1
C
}| { z }|2 {
2
ν
2 c2 νg
g c1
(1 + a)2 =
+
g h sin θ g h2 sin θ
(2.152)
64
CHAPTER 2. BASIC FLUID MECHANICS
And equation (2.149)
1
2
1
2 C1
z
µℓ
µg
C3
}| {
}|
{
z
νg c1
µℓ νg c3
ρℓ
1+
=
+
g h sin θ
ρg µg g h sin θ
(2.153)
and equation (2.150)
1+
2 νg h
c1
2 g sin θ
h
+
2 νg c2
νg
2 νg h
c3
=
+
2
2 sin θ
νℓ
h g sin θ
g h
(2.154)
Or rearranging equation (2.154)
C1
z
C
C
}| { z }|2 { z }|3 {
νg
2 νg c1
2 νg c2
2 νg c3
−1 =
+
−
νℓ
h g sin θ h2 g sin θ g h sin θ
(2.155)
This presentation provide similarity and it will be shown in the Dimensional analysis
chapter better physical understanding of the situation. Equation (2.152) can be written as
(1 + a)2 = C1 + C2
(2.156)
ρℓ
C
µ C
−1 = 1 − ℓ 3
ρg
2
µg 2
(2.157)
νg
− 1 = C1 + C2 − C3
νℓ
(2.158)
Further rearranging equation (2.153)
and equation (2.155)
This process that was shown here is referred as non–dimensionalization13 . The ratio of the
dynamics viscosity can be eliminated from equation (2.158) to be
µ g ρℓ
− 1 = C1 + C2 − C3
µ ℓ ρg
(2.159)
The set of equation can be solved for the any ratio of the density and dynamic viscosity. The
solution for the constant is
µg
µg
ρg
− 2 + a2 + 2 a
+2
(2.160)
C1 =
ρℓ
µℓ
µℓ
−
C2 =
µg ρℓ
+a
µℓ ρg
µg
µg
µg
2
−2 +3
+ a2
−1 −2
µℓ
µℓ
µℓ
µg
µℓ
13 Later it will be move to the Dimensional Chapter
(2.161)
65
2.4. COMPRESSIBLE FLOW
C3 = −
µ g ρℓ
+ a2 + 2 a + 2
µℓ ρg
(2.162)
The two different fluids14 have a solution as long as the distance is a finite reasonable
similar. What happen when the lighter fluid, mostly the gas, is infinite long. This is one of the
source of the instability at the interface. The boundary conditions of flow with infinite depth
is that flow at the interface is zero, flow at infinite is zero. The requirement of the shear stress
in the infinite is zero as well. There is no way obtain one dimensional solution for such case
and there is a component in the y direction. Combining infinite size domain of one fluid with
finite size on the other one side results in unstable interface.
2.4
Compressible Flow
The die casting process involve several compressible flow concepts that need to be addressed. Yet ,this material is extensive and
velocity sound wave
dU
requires a semester or more for student to
U
= dU
have good understanding of this complex maP
P + dP
ρ
ρ + dρ
terial. It realized that to whole field suffer from lack of knowledge even the top
Fig. 2.21 – A very slow moving piston in a still
researchers. Yet to give very minimal ingas.
formation is seems to be essential to the
understanding of the venting design. The
summary material here is derived from the book “Fundamentals of Compressible Flow Mechanics.”
These several concepts are abase no Mach which in turn depends on the speed of sound.
Hence, first what is speed of sound and how it is calculated. Later two phenomena the choke
and shock are explained. Later Fanno flow, which is pertaining to die casting, is explained.
2.4.1
Speed of Sound
The speed of sound is a very important parameter in the die casting process because it effects and explains the choking in the die casting process. What is the speed of the small
disturbance as it travels in a “quiet” medium? This velocity is referred to as the speed of
14 This topic will be covered in dimensional analysis in more extensively. The point here the understanding issue
related to boundary condition not per se solution of the problem.
66
CHAPTER 2. BASIC FLUID MECHANICS
sound. To answer this question, consider a piston moving from the left to the right at a relatively small velocity (see Fig. 2.21). The information that the piston is moving passes thorough a single “pressure pulse.” It is assumed
that if the velocity of the piston is infinitesimally small, the pulse will be infinitesimally
small. Thus, the pressure and density can be
assumed to be continuous.
−c
c − dU
P + dP
ρ + dρ
U = −c
P
ρ
Fig. 2.22 – Stationary sound wave and gas moves
relative to the pulse.
It is convenient to look at a control volume which is attached to a pressure pulse. Applying the mass balance (the area is canceled) yields
ρ c = (ρ + dρ)(c − dU)
(2.163)
or when the higher term dUdρ is neglected yields
ρ dU = c dρ =⇒ dU =
c dρ
ρ
(2.164)
From the energy equation (Bernoulli’s equation), Bernoulli’s equation assuming isentropic
flow and neglecting the gravity results
(c − dU)2 − c2 dP
+
=0
2
ρ
(2.165)
neglecting second term (dU2 ) yield
−cdU +
dP
=0
ρ
(2.166)
Substituting the expression for dU from Eq. (2.164) into Eq. (2.166) yields
c2
Speed of Sound – P
dρ
dP
dP
=
=⇒ c2 =
ρ
ρ
dρ
(2.167)
67
2.4. COMPRESSIBLE FLOW
2.4.2
Choked Flow
In this section a discussion on a steady state
flow through a smooth and continuous variable area flow rate is presented which include
the flow through a converging–diverging nozzle. The isentropic flow models are important
because of two main reasons: it provide way to
approximate the flow in short distance where
the resistance is insignificant. It provide a way
to explain the non intuitive the phenomenon
of choking. It phenomenon is counter to everything that one experienced and it puzzle
scientists for a long time. For this reason scientists invent the concept of stagnation state.
2.4.2.1
P
P0
PB = P0
subsonic
M <1
M >1
supersonic
distance, x
Fig. 2.23 – Flow of a compressible substance (gas)
through a converging–diverging nozzle.
Stagnation State for Ideal Gas Model
It is assumed that the flow is one–dimensional. Figure (2.23) describes a gas flow through a
converging–diverging nozzle. It has been found that a theoretical state known as the stagnation state is very useful in which the flow is brought into a complete motionless condition in
isentropic process without other forces (e.g. gravity force). Several properties can be represented by this theoretical process which include temperature, pressure, and density etc and
denoted by the subscript “0.”
A dimensionless velocity and it is referred as Mach number for the ratio of velocity to
speed of sound as
M≡
U
c
(2.168)
The temperature ratio reads
k−1 2
T0
= 1+
M
T
2
(2.169)
The ratio of stagnation pressure to the static pressure can be expressed as the function
of the temperature ratio because of the isentropic relationship as
P0
=
P
T0
T
k
k−1
k
k − 1 2 k−1
= 1+
M
2
(2.170)
In the same manner the relationship for the density ratio is
ρ0
=
ρ
T0
T
1
k−1
=
1
k − 1 2 k−1
1+
M
2
(2.171)
68
CHAPTER 2. BASIC FLUID MECHANICS
Static Properties As A Function of Mach Number
1
0.9
P/P0
ρ/ρ0
T/T0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Mon Jun 5 17:39:34 2006
4
5
Mach number
6
7
8
9
Fig. 2.24 – The stagnation properties as a function of the Mach number, k=1.4
A new useful definition is introduced for the case when M = 1 and denoted by superscript
“∗.” The special case of ratio of the star values to stagnation values are dependent only on the
heat ratio as the following:
2
T∗
c∗ 2
= 2 =
T0
k+1
c0
(2.172)
and
P∗
=
P0
2
k+1
k
k−1
Density Ratio
1
k−1
ρ∗
2
=
ρ0
k+1
(2.173)
(2.174)
The definition of the star Mach is ratio of the velocity and star speed of sound at M = 1.
The flow in a converging–diverging nozzle has two models: First is isentropic and
adiabatic model. Second is isentropic and isothermal model. Clearly, the stagnation temperature, T0 , is constant through the adiabatic flow because there isn’t heat transfer. Therefore,
69
2.4. COMPRESSIBLE FLOW
the stagnation pressure is also constant through the flow because of the isentropic flow. Conversely, in mathematical terms, Eq. (2.169) and Eq. (2.170) are the same. If the right hand side
is constant for one variable, it is constant for the other. In the same argument, the stagnation
density is constant through the flow. Thus, knowing the Mach number or the temperature
will provide all that is needed to find the other properties. The only properties that need to
be connected are the cross section area and the Mach number. Examination of the relation
between properties can then be carried out.
2.4.2.2
The Properties in the Adiabatic Nozzle
When there is no external work and heat transfer, the energy equation, reads
dh + UdU = 0
(2.175)
Differentiation of continuity equation, ρAU = ṁ = constant, and dividing by the continuity equation reads
dρ dA dU
+
+
=0
ρ
A
U
(2.176)
The thermodynamic relationship between the properties can be expressed as
T ds = dh −
dP
ρ
(2.177)
For isentropic process ds ≡ 0 and combining equations (2.175) with (2.177) yields
dP
+ UdU = 0
ρ
(2.178)
Differentiation of the equation state (perfect gas), P = ρ R T , and dividing the results by the
equation of state (ρ R T ) yields
dρ dT
dP
=
+
P
ρ
T
(2.179)
Obtaining an expression for dU/U from the mass balance equation (2.176) and using it in
equation (2.178) reads
z
dU
U
}|
{
dρ
dP
2 dA
−U
+
=0
ρ
A
ρ
(2.180)
Rearranging Eq. (2.180) so that the density, ρ, can be replaced by the static pressure, dP/ρ
yields


1
c2
z}|{


 dA
dP
dA dρ dP
dρ dP 

(2.181)
= U2
+
= U2 
+
 A
ρ
A
ρ dP
dP ρ 


70
CHAPTER 2. BASIC FLUID MECHANICS
Recalling that dP/dρ = c2 and substitute the speed of sound into equation (2.181) to obtain
"
2 #
dP
U
dA
1−
(2.182)
= U2
ρ
c
A
Or in a dimensionless form
dP dA
1 − M2 = U2
ρ
A
(2.183)
Eq. (2.183) is a differential equation for the pressure as a function of the cross section area. It
is convenient to rearrange Eq. (2.183) to obtain a variables separation form of
dP =
ρ U2 dA
A 1 − M2
(2.184)
Before going further in the mathematical derivation it is worth while to look at the physical
meaning of Eq. (2.184). The term ρU2 /A is always positive (because all the three terms can be
only positive). Now, it can be observed that dP can be positive or negative depending on the
dA and Mach number. The meaning of the sign change for the pressure differential is that the
pressure can increase or decrease. It can be observed that the critical Mach number is one.
If the Mach number is larger than one than dP has opposite sign of dA. If Mach number is
smaller than one dP and dA have the same sign. For the subsonic branch M < 1 the term
1/(1 − M2 ) is positive hence
dA > 0 =⇒ dP > 0
dA < 0 =⇒ dP < 0
From these observations the trends are similar to those in incompressible fluid. An increase
in area results in an increase of the static pressure (converting the dynamic pressure to a static
pressure). Conversely, if the area decreases (as a function of x) the pressure decreases. Note
that the pressure decrease is larger in compressible flow compared to incompressible flow.
For the supersonic branch M > 1, the phenomenon is different. For M > 1 the term
1/1 − M2 is negative and change the character of the equation.
dA > 0 ⇒ dP < 0
dA < 0 ⇒ dP > 0
This behavior is opposite to incompressible flow behavior.
For the special case of M = 1 (sonic flow) the value of the term 1 − M2 = 0 thus
mathematically dP → ∞ or dA = 0. Since physically dP can increase only in a finite amount
it must be that dA = 0.It must also be noted that when M = 1 occurs only when dA = 0.
However, the opposite, not necessarily means that when dA = 0 that M = 1. In that case, it
is possible that dM = 0 thus the diverging side is in the subsonic branch and the flow isn’t
choked.
71
2.4. COMPRESSIBLE FLOW
2.4.3
Shock Wave
In this section the relationship between the two sides of normal shock are presented. In die
casting, the shock appears in some of the processes such as the air/gas flows out (and in) the
runner especially in the initial phase. In this discussion, the flow is assumed to be in a steady
state, and the thickness of the shock is assumed to be very small.
A shock can occur in at least two different mechanisms. The first is when a large
flow
P
Py
difference (above a small minimum value) bedirection ρ x
ρ
x
y
tween the two sides of a membrane, and
c.v.
T
T
when the membrane bursts. Of course, the
x
y
shock travels from the high pressure to the
Fig. 2.25 – A shock wave inside a tube, but it can
low pressure side. The second is when many
also be viewed as a one–dimensional shock
sound waves “run into” each other and accuwave.
mulate (some refer to it as “coalescing”) into
a large difference, which is the shock wave.
In fact, the sound wave can be viewed as an extremely weak shock. In the speed of sound analysis, it was assumed the medium is continuous, without any abrupt changes. This assumption
is no longer valid in the case of a shock. Here, the relationship for a perfect gas is constructed.
Fig. 2.25 exhibits a control volume for this analysis, and the gas flows from left to right.
The conditions are assumed to be uniform. The conditions to the right of the shock wave are
uniform, but different from the left side. The transition in the shock is abrupt and in a very
narrow width.
The chemical reactions (even condensation) are neglected, and the shock occurs at a
very narrow section. Clearly, the isentropic transition assumption is not appropriate in this
case because the shock wave is a discontinued area. Therefore, the increase of the entropy
is fundamental to the phenomenon and the understanding of it. It is further assumed that
there is no friction or heat loss at the shock (because the heat transfer is negligible due to the
fact that it occurs on a relatively small surface). It is customary in this field to denote x as the
upstream condition and y as the downstream condition.
The mass flow rate is constant from the two sides of the shock and therefore the mass
balance is reduced to
ρx Ux = ρy Uy
(2.185)
In a shock wave, the momentum is the quantity that remains constant because there
are no external forces. Thus, it can be written that
(2.186)
Px − Py = ρx Uy 2 − ρy Ux 2
The process is adiabatic, or nearly adiabatic, and therefore the energy equation can be written
as
C p Tx +
Uy 2
Ux 2
= C p Ty +
2
2
(2.187)
72
CHAPTER 2. BASIC FLUID MECHANICS
The equation of state for perfect gas reads
(2.188)
P = ρRT
If the conditions upstream are known, then there are four unknown conditions downstream. A system of four unknowns and four equations is solvable. Nevertheless, one can note
that there are two solutions because of the quadratic of equation (2.187). These two possible
solutions refer to the direction of the flow. Physics dictates that there is only one possible
solution. One cannot deduce the direction of the flow from the pressure on both sides of the
shock wave. The only tool that brings us to the direction of the flow is the second law of
thermodynamics. This law dictates the direction of the flow, and as it will be shown, the gas
flows from a supersonic flow to a subsonic flow. Mathematically, the second law is expressed
by the entropy. For the adiabatic process, the entropy must increase. In mathematical terms,
it can be written as follows:
(2.189)
sy − sx > 0
Note that the greater–equal signs were not used. The reason is that the process is irreversible,
and therefore no equality can exist. Mathematically, the parameters are P, T , U, and ρ, which
are needed to be solved. For ideal gas, equation (2.189) is
ln
Ty
Tx
− (k − 1)
Py
>0
Px
(2.190)
It can also be noticed that entropy, s, can be expressed as a function of the other parameters. Now one can view these equations as two different subsets of equations. The first set
is the energy, continuity, and state equations, and the second set is the momentum, continuity, and state equations. The solution of every set of these equations produces one additional
degree of freedom, which will produce a range of possible solutions. Thus, one can have a
whole range of solutions. In the first case, the energy equation is used, producing various
resistance to the flow. This case is refered to as Fanno flow, and appendix A deals extensively
with this topic. Instead of solving all the equations that were presented, one can solve only
four (4) equations (including the second law), which will require additional parameters. If the
energy, continuity, and state equations are solved for the arbitrary value of the Ty , a parabola
in the T − −s diagram will be obtained. On the other hand, when the momentum equation
is solved instead of the energy equation, the degree of freedom is now energy, i.e., the energy
amount “added” to the shock. This situation is similar to a frictionless flow with the addition
of heat, and this flow is known as Rayleigh flow. This flow is dealt with in greater detail in
Fundamentals of Compressible flow by this author.
73
2.5. SOLUTION OF THE GOVERNING EQUATIONS
Since the shock has no heat transfer (a special
case of Rayleigh flow) and there isn’t essentially any momentum transfer (a special case
of Fanno flow), the intersection of these two
curves is what really happened in the shock.
In Fig. 2.26, the intersection is displayed where
two solutions are intersected. Clearly, the increase of the entropy determines the direction
of the flow. The entropy increases from point x
to point y. It is also worth noting that the temperature at M = 1 on Rayleigh flow is larger
than that on the Fanno line.
2.5
2.5.1
M <1M=
T
Ty , Py , sy
√1
k
subsonic
flow
supersonic
flow
M =1
shock jump
M =1
Rayleigh
line
Fanno
line
M >1
Tx , Px , sx
s
Fig. 2.26 – The intersection of Fanno flow and
Rayleigh flow produces two solutions for the
shock wave.
Solution of the Governing Equations
Informal Model
Accepting the fact that the shock is adiabatic or nearly adiabatic requires that total energy is
conserved, T0x = T0y . The relationship between the temperature and the stagnation temperature provides the relationship of the temperature for both sides of the shock.
Ty
k−1
1+
Mx 2
T0y
Ty
2
=
=
Tx
k−1
Tx
1+
My 2
T0x
2
(2.191)
All the other relationships are essentially derived from this equation. The only issue left
to derive is the relationship between Mx and My . Note that the Mach number is a function
of temperature, and thus for known Mx all the other quantities can be determined, at least,
numerically. The analytical solution is discussed in the next section.
2.5.2
Formal Model
Equations (2.185), (2.186), and (2.187) can be converted into a dimensionless form. The reason
that dimensionless forms are heavily used in this book is because by doing so it simplifies and
clarifies the solution. It can also be noted that in many cases the dimensionless equations set
is more easily solved.
From the continuity equation (2.185) substituting for density, ρ, the equation of state
yields
Py
Px
Ux =
Uy
R Tx
R Ty
(2.192)
74
CHAPTER 2. BASIC FLUID MECHANICS
Squaring equation (2.192) results in
Px 2
2
R Tx 2
Ux 2 =
Py 2
2
R Ty 2
Uy 2
(2.193)
Multiplying the two sides by the ratio of the specific heat, k, provides a way to obtain the speed
of sound definition/equation for perfect gas, c2 = k R T to be used for the Mach number
definition, as follows:
Py 2
Px 2
Uy 2
Ux 2 =
Tx k R Tx
Ty k R Ty
| {z }
| {z }
cx 2
(2.194)
cy 2
Note that the speed of sound on the different sides of the shock is different. Utilizing the
definition of Mach number results in
Py 2
Px 2
Mx 2 =
My 2
Tx
Ty
(2.195)
Rearranging equation (2.195) results in
Ty
=
Tx
Py
Px
2 My
Mx
2
(2.196)
Energy equation (2.187) can be converted to a dimensionless form which can be expressed as
k−1
k−1
Ty 1 +
My 2 = Tx 1 +
Mx 2
(2.197)
2
2
It can also be observed that equation (2.197) means that the stagnation temperature is the same,
T0y = T0x . Under the perfect gas model, ρU2 is identical to kPM2 because
M2
ρ
z }| {
z}|{
2 
P 
2
 U  k R T = k P M2
ρU =
R T k
R T} 
| {z
(2.198)
c2
Using the identity (2.198) transforms the momentum equation (2.186) into
Px + k Px Mx 2 = Py + k Py My 2
(2.199)
Rearranging equation (2.199) yields
Py
1 + k Mx 2
=
Px
1 + k My 2
(2.200)
2.5. SOLUTION OF THE GOVERNING EQUATIONS
75
The pressure ratio in equation (2.200) can be interpreted as the loss of the static pressure.
The loss of the total pressure ratio can be expressed by utilizing the relationship between the
pressure and total pressure (see equation (??)) as
P0y
P0x
k
k−1
My 2 k − 1
Py 1 +
2
=
k
k−1
Px 1 +
Mx 2 k − 1
2
(2.201)
The relationship between Mx and My is needed to be solved from the above set of equations.
This relationship can be obtained from the combination of mass, momentum, and energy
equations. From equation (2.197) (energy) and equation (2.196) (mass) the temperature ratio
can be eliminated.
k−1
1+
Mx 2
Py My 2
2
=
(2.202)
k−1
Px Mx
1+
My 2
2
Combining the results of (2.202) with equation (2.200) results in
1 + k Mx 2
1 + k My 2
!2
=
Mx
My
2 1 + k − 1 Mx 2
2
k−1
1+
My 2
2
(2.203)
Equation (2.203) is a symmetrical equation in the sense that if My is substituted with Mx and
Mx substituted with My the equation remains the same. Thus, one solution is
My = Mx
(2.204)
It can be observed that equation (2.203) is biquadratic. According to the Gauss Biquadratic
Reciprocity Theorem this kind of equation has a real solution in a certain range15 which will
be discussed later. The solution can be obtained by rewriting equation (2.203) as a polynomial (fourth order).
It is also possible to cross–multiply equation (2.203) and divide it by
Mx 2 − My 2 results in
k−1 1+
My 2 + My 2 − k My 2 My 2 = 0
(2.205)
2
Equation (2.205) becomes
Shock Solution
2
Mx 2 +
k
−
1
2
My =
(2.206)
2k
2
Mx − 1
k−1
15 Ireland, K. and Rosen, M. "Cubic and Biquadratic Reciprocity." Ch. 9 in A Classical Introduction to Modern
Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108-137, 1990.
76
CHAPTER 2. BASIC FLUID MECHANICS
The first solution (2.204) is the trivial solution in which the two sides are identical and no
shock wave occurs. Clearly, in this case, the pressure and the temperature from both sides of
the nonexistent shock are the same, i.e. Tx = Ty , Px = Py . The second solution is where
the shock wave occurs.
The pressure ratio between the two sides is a function of only a single Mach number,
for example, Mx . Utilizing Eq. (2.200) and equation Eq. (2.206) provides the pressure ratio as
only a function of the upstream Mach number as
Py
2k
k−1
=
Mx 2 −
Px
k+1
k+1
or
Shock Pressure Ratio
Py
2k = 1+
Mx 2 − 1
Px
k+1
(2.207)
The density and upstream Mach number relationship can be obtained in the same fashion to became
Shock Density Ratio
ρy
Ux
(k + 1)Mx 2
=
=
ρx
Uy
2 + (k − 1)Mx 2
(2.208)
The fact that the pressure ratio is a function of the upstream Mach number, Mx , provides
additional way of obtaining an additional useful relationship. And the temperature ratio, as a
function of pressure ratio, is transformed into
Shock Temperature Ratio


k + 1 Py
+

Ty
Py 
 k − 1 Px 
=

k + 1 Py 
Tx
Px
1+
k − 1 Px
(2.209)
In the same way, the relationship between the density ratio and pressure ratio is
Shock P − ρ
Py
k+1
1+
ρx
k−1
P
x
= Py
k+1
ρy
+
k−1
Px
which is associated with the shock wave.
(2.210)
77
2.5. SOLUTION OF THE GOVERNING EQUATIONS
Shock Wave relationship
My and P0y/P0x as a function of Mx
1
0.9
0.8
My
0.7
P0y/P0x
My
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
Fri Jun 18 15:47:34 2004
4
5
6
7
8
9
10
Mx
Fig. 2.27 – The exit Mach number and the stagnation pressure ratio as a function of upstream Mach number.
2.5.2.1
The Maximum Conditions
The maximum speed of sound is when the highest temperature is achieved. The maximum
temperature that can be achieved is the stagnation temperature
r
2k
Umax =
R T0
(2.211)
k−1
The stagnation speed of sound is
c0 =
p
k R T0
(2.212)
Based on this definition a new Mach number can be defined
M0 =
2.5.2.2
U
c0
(2.213)
The Star Conditions
The speed of sound at the critical condition can also be a good reference velocity. The speed
of sound at that velocity is
√
c∗ = k R T ∗
(2.214)
78
CHAPTER 2. BASIC FLUID MECHANICS
In the same manner, an additional Mach number can be defined as
M∗ =
2.5.3
U
c∗
(2.215)
Prandtl’s Condition
It can be easily observed that the temperature from both sides of the shock wave is discontinuous. Therefore, the speed of sound is different in these adjoining mediums. It is therefore
convenient to define the star Mach number that will be independent of the specific Mach
number (independent of the temperature).
M∗ =
c U
c
U
= ∗ = ∗M
c∗
c c
c
(2.216)
The jump condition across the shock must satisfy the constant energy.
c2
U2
c∗ 2
c∗ 2
k + 1 ∗2
+
=
+
=
c
k−1
2
k−1
2
2 (k − 1)
(2.217)
Dividing the mass equation by the momentum equation and combining it with the perfect
gas model yields
c 2
c1 2
+ U1 = 2 + U2
k U1
k U2
Combining equation (2.217) and (2.218) results in
1
1
k + 1 ∗2 k − 1
k + 1 ∗2 k − 1
c −
U1 + U1 =
c −
U2 + U2
k U1
2
2
k U2
2
2
(2.218)
(2.219)
After rearranging and dividing equation (2.219) the following can be obtained:
U1 U2 = c∗ 2
(2.220)
M∗ 1 M∗ 2 = c∗ 2
(2.221)
or in a dimensionless form
2.6
Operating Equations and Analysis
In Figure 2.27, the Mach number after the shock, My , and the ratio of the total pressure,
P0y /P0x , are plotted as a function of the entrance Mach number. The working equations
were presented earlier. Note that the My has a minimum value which depends on the specific heat ratio. It can be noticed that the density ratio (velocity ratio) also has a finite value
regardless of the upstream Mach number.
79
2.6. OPERATING EQUATIONS AND ANALYSIS
The typical situations in which these equations can be used also include the moving
shocks. The equations should be used with the Mach number (upstream or downstream)
for a given pressure ratio or density ratio (velocity ratio). This kind of equations requires
examining Table (??) for k = 1.4 or utilizing Potto-GDC for for value of the specific heat
ratio. Finding the Mach number for a pressure ratio of 8.30879 and k = 1.32 and is only a
few mouse clicks away from the following table.
Shock Wave relationship
Py/Py, ρy/ρx and Ty/Tx as a function of Mx
120.0
110.0
100.0
Py/Px
90.0
Ty/Tx
ρy/ρx
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0
1
2
3
Fri Jun 18 15:48:25 2004
4
5
6
7
8
9
10
Mx
Fig. 2.28 – The ratios of the static properties of the two sides of the shock.
To illustrate the use of the above equations, an example is provided.
Example 2.7: Air Flows
Level: Basic
Air flows with a Mach number of Mx = 3, at a pressure of 0.5 [bar] and a temperature of 0◦ C goes through a normal shock. Calculate the temperature, pressure, total
pressure, and velocity downstream of the shock. Assume that k = 1.4.
Solution
Analysis:
First, the known information are Mx = 3, Px = 1.5[bar] and Tx = 273K. Using these data,
the total pressure can be obtained (through an isentropic relationship in Table by this author
“Fundamentals of Compressible Flow Table” i.e., P0x is known). Also with the temperature,
Tx , the velocity can readily be calculated. The relationship that was calculated will be utilized
x
= 0.0272237 =⇒ P0x =
to obtain the ratios for the downstream of the normal shock. PP0x
1.5/0.0272237 = 55.1[bar]
√
√
cx = kRTx = 1.4 × 287 × 273 = 331.2m/sec
80
CHAPTER 2. BASIC FLUID MECHANICS
End of Ex. 2.7
Mx
My
Ty
Tx
y
3.0000
0.47519
2.6790
3.8571
x
P0 y
P0 x
Py
Px
10.3333
0.32834
Ux = Mx × cx = 3 × 331.2 = 993.6[m/sec]
Now the velocity downstream is determined by the inverse ratio of ρy /ρx = Ux /Uy =
3.85714.
Uy = 993.6/3.85714 = 257.6[m/sec]
P0y =
P0y
P0x
× P0x = 0.32834 × 55.1[bar] = 18.09[bar]
“The shear, S, at the ingate is determined by the average velocity, U, of the
liquid and by the ingate thickness, t. Dimensional analysis shows that is directly proportional to (U/ℓ). The constant of proportionality is difficult to
determine, . . .1 ”
Murray, CSIRO Australia
3
Dimensional Analysis
3.1
Basics of Dimensional Analysis
One of the important tools to understand the die casting process is dimensional analysis.
Fifty years ago, this method transformed the fluid mechanics/heat transfer into a “uniform”
understanding. This book attempts to introduce to the die casting industry this established
method2 . Experimental studies can be “expanded/generalized” as it was done in convective
heat transfer. It is hoped that as a result, separate sections for aluminum, zinc, and magnesium will not exist anymore in die casting conferences. This chapter is based partially on Dr.
Eckert’s book, notes, and the article on dimensional analysis (Eckert 1989) applied to die casting. Several conclusions are derived from this analysis and they will be presented throughout
this chapter. This material can deliver great benefit for researchers who want to built their
research on a solid foundation. For those who are dealing with the numerical research/cal1 Citing Murray, M. T., and Griffiths,.J. R. “The Design of feed systems for thin walled zinc high pressure die
castings,” Metallurgical and materials transactions B Vol. 27B, February 1996, pp. 115–118. This excerpt is an excellent
example of poor research and poor understanding. This “unknown” constant is called viscosity (see Basics of Fluid
Mechanics in Potto Project Publishing series. Here, a discussion on some specific mistakes were presented in that
paper (which are numerous). Dimensional analysis is a tool which can take “cluttered” and meaningless paper such as
the above and turn them into something with a real value. As proof of their model, the researchers have mentioned
two unknown companies that their model is working. What a nice proof! Are the physics laws really different in
Australia? Also one avoid wondering how dimensional analysis shows that that shear stress is proportional. No
one does know how to demonstrate such functionality. In fact, this is the main point Buckingham’s pie theory that
functionality is unknown and should be found by experimental means. One can wonder how gave Murray a Ph. D?
2 Actually, Prof. E. R. G. Eckert introduced the dimensional analysis to the die casting long before. The author is
his zealous disciple, all the credit should go to Eckert. Of course, all the mistakes are the author’s and none of Dr.
Eckert’s. All the typos in Eckert’s paper were this author’s responsibility for which he apologizes.
81
82
CHAPTER 3. DIMENSIONAL ANALYSIS
culation, it is useful to learn when some parameters should be taken into account and why.
For example, Crowley (2021) assumed that surface tension is important without doing any dimensional analysis. Later in a private conversion he confess that he could not figure out how
to incorporate it into the physics of the shot sleeve even having any serious clue about it.
3.1.1
How The Dimensional Analysis Work
In dimensional analysis, the number of the effecting parameters is reduced to a minimum by
replacing the dimensional parameters by dimensionless parameters. Some researchers point
out that the chief advantage of this analysis is “to obtain experimental results with a minimum amount of labor, results in a form having maximum utility” (Hansen 1967, pp. 395). The
dimensional analysis has several other advantages which include; 1) increase of understanding, 2) knowing what is important and what not, and 3) compacting the presentation3 . The
advantage of compact of presentation allows one to “see” the big picture with a minimal effort.
Dimensionless parameters are parameters representing a ratio which does not have a
physical dimension. The experimental study assists to solve problems when the solution of
the governing equation cannot be obtained. To achieve this, experiments are designed to be
“similar” to the situations which need to be solved or simulated. The bases for this concept
are mathematical and physical. Two different sets of phenomena will produce a similar result
if the governing differential equations with boundaries conditions are similar. The actual experiments are difficult to carry out in many cases. Thus, design experiments with the same
governing differential equations as the actual phenomenon is the solution. This similarity
does not necessarily mean that the experiments have to be carried exactly as studied phenomena. It is enough that the main dimensionless parameters are similar, since the minor
dimensional parameters, in many cases, are insignificant. For example, a change in Reynolds
number is insignificant since a change in Reynolds number in a large range does not affect
the friction factor.
An example of the similarity applied to the die cavity is given in the section 3.8. Researchers in casting in general and die casting in particular do not utilize this method. For
example, after the Russians (Bochvar, Notkin, Spektorova, and Sadchikova 1946) introduced
the water analogy method (in casting) in the 40s all the experiments such as Wallace, CSIRO,
etc. conducted poorly designed experiments. For example, Wallace recorded the Reynolds
and Froude number without attempting to match the governing equations. Another example
is the experimental study of Gravity Tiled Die Casting (low pressure die casting) performed
by Nguyen’s group in 1986 comparing two parameters Re and We. Flow of ”free” falling, the
√
velocity is a function of the height (U ∼ gH). Hence, the equation Remodel = Reactual
should lead only to Hmodel ≡ Hactual and not to any function of Umodel /Uactual . The
value of Umodel /Uactual is actually constant for the same height ratio. The Wallace’s experiments with Reynolds number matching does not lead to matching of similar governing
equations. Many other important parameters which control the governing equations are not
3 The importance of compact presentation is attributed to Prof. M. Bentwitch who was mentor to many including
this author during his masters studies.
83
3.2. INTRODUCTION
simulated (Nguyen and Carrig 1986). The governing equations in these cases include several
other important parameters which have not been controlled or even measured, monitored,
and simulated4 . Moreover, the Re number is controlled by the flow rate and the characteristics of the ladle opening and not as in the pressurized pipe flow as the authors assumed.
3.2
Introduction
Take a trivial example of fitting a rode into a circular hole (see Fig. 3.1). To solve this problem,
it is required to know two parameters; 1) the diameter of the rode and 2) the diameter of
the hole. Actually, it is required to have only one parameter, the ratio of the rode diameter
to the hole diameter. The ratio is a dimensionless number and with this number one can
say that for a ratio larger than one, the rode will not enter the hole; and ratio smaller than
one, the rod is too small. Only when the ratio is
equal to one, the rode is said to be fit. This fact
allows one to draw conclusion on the situation
by using only one coordinate. Furthermore, if
one wants to deal with tolerances, the non dimensional presentation can easily be extended
to say that when the ratio is equal from 0.99 to
1.0 the rode is fitting, and etc. If one were to use
the two diameters description, one will need
more than this simple sentence to describe it.
D
Fig. 3.1 – Rod into the hole example.
In the preceding simplistic example, the shown of advantages of dimensional analogy
were very minimal. In many real problems, including the die casting process, this approach
can remove clattered views and put the problem into focus. It also helps to use and dig information from different problems from a “similar” situation. Throughout this book the reader
will notice that the systems/equations considerable efforts were made to convert to a dimensionless form to augment the understanding.
Recently this author was interested in woodworking for which a question of how much
lumber can be exacted from a tree. This question can be answered by basically two parameters
the volume of the wood and percentage of usable material which depend on the type of the
tree. Thus, the minimum parameters are two (notice the first one is dimensional for which
there is a trick to make it dimensionless.)
4 Besides many conceptual physical mistakes, the authors have a conceptual mathematical mistake. They tried to
achieve the same Re and Fr numbers in the experiments as in reality for low pressure die casting. They derived
an equation for the velocity ratio based on equal Re numbers (model and actual). They have done the same for
Fr numbers. Then they equate the velocity ratio based on equal Re to velocity ratio based on equal Fr numbers.
However, velocity ratio based on equal Re is a constant and does vary with the tunnel dimension (as opposed to
distance from the starting point). The fact that these ratios have the same symbols do not mean that they are really
the same. These two ratios are different and cannot be conflated and equated.
84
3.3
CHAPTER 3. DIMENSIONAL ANALYSIS
Nusselt Schmidt method
The dimensional analysis was independently developed by Buckingham (one method) and
Nusselt (another method). The last method was improved by Nusselt’s students/co–workers
(Schmidt, and E. R. G. Eckert) in which the governing equations are used as well. More information is inserted into the problem and thus a better understanding on the dimensionless
parameters are extracted. The advantage or disadvantage of these similar methods depend
on the point of view. The Buckingham–π technique is simpler while Nusselt’s technique produces a much better result. Sometime, the simplicity of Buckingham’s technique yields insufficient knowledge or simply becomes useless. However, when no governing equations are
found, Buckingham’s method has usefulness while Nusselt’s method cannot be used. It can be
argued that these situations really do not exist in the Thermo–Fluid field. Nusselt’s technique
is more cumbersome but more precise and provide more useful information. The advantage
of the Nusselt’s technique are: a) compact presentation, b)knowledge what parameters affect
the problem, and c) easier to extent the solution to more general situations. In very complex
problems both techniques suffer from in inability to provide a significant information on the
effective parameters such multi–phase flow etc.
It has to be recognized that the dimensional analysis provides answer to what group of
parameters affecting the problem. In fact, there are fields in thermo–fluid and others where
dimensional analysis, is recognized as very useless. For example, the area of multiphase flows
there is no solution based on dimensionless parameters (with the exception of the rough solution of Martinelli). In the Buckingham’s approach it merely suggests the number of dimensional parameters based on a guess of all parameters affecting the problem. Nusselt’s
technique provides the form of these dimensionless parameters, and the relative relationship
of these parameters.
3.3.1
Brief History
The idea of experimentation with a different, rather than the actual, dimension was suggested
by several individuals independently. Some attribute it to Newton (1686) who coined the
phrase of “Great Principle of Similitude.” Later, Maxwell a Scottish Physicist played a major role in establishing the basic units of mass, length, and time as building blocks of all other
units. Another example, John Smeaton (8 June 1724–28 October 1792) was an English civil and
mechanical engineer who study relation between propeller/wind mill and similar devices to
the pressure and velocity of the driving forces.
Jean B. J. Fourier (1768-1830) first attempted to formulate the dimensional analysis theory. This idea was extend by William Froude (1810-1871) by relating the modeling of open
channel flow and actual body but more importantly the relationship between drag of models
to actual ships. While the majority of the contributions were done by thermo–fluid guys the
concept of the equivalent or similar propagated to other fields. Aiméem Vaschy, a German
Mathematical Physicist (1857–1899), suggested using similarity in electrical engineering and
suggested the Norton circuit equivalence theorems. Rayleigh probably was the first one who
used dimensional analysis (1872) to obtain the relationships between the physical quantities
85
3.3. NUSSELT SCHMIDT METHOD
(see the question why the sky is blue story.).
Osborne Reynolds (1842–1912) was the first to derive and use dimensionless parameters to analyze experimental data. Riabouchunsky (1915) proposed of relating temperature by
molecules velocity and thus creating dimensionless group with the byproduct of compact
solution (solution presented in a compact and simple form).
Buckingham culminated the dimensional analysis and similitude and presented it in
a more systematic form. In about the same time (1915, Wilhelm Nusselt (November 25, 1882
– September 1, 1957), a German engineer, developed the dimensional analysis (proposed the
principal parameters) of heat transfer without knowledge about previous work of Buckingham.
3.3.2
Theory Behind Dimensional Analysis
In chemistry it was recognized that there are fundamental elements that all the material is
made from (the atoms). That is, all the molecules are made from a combination of different
atoms. Similarly to this concept, it was recognized that in many physical systems there are
basic fundamental units which can describe all the other dimensions or units in the system.
For example, isothermal single component systems (which does not undergo phase change,
temperature change and observed no magnetic or electrical effect) can be described by just
basic four physical units. The units or dimensions are, time, length, mass, quantity of substance (mole). For example, the dimension or the units of force can be constructed utilizing
Newton’s second law i. e. mass times acceleration −→ m a = M L/t2 . Increase of degree
of freedom, allowing this system to be non–isothermal will increase only by one additional
dimension of temperature, θ. These five fundamental units are commonly the building blocks
for most of the discussion in fluid mechanics (see Table of basic units 3.1).
Table 3.1 – Basic Units of Two Common Systems
Standard System
Old System
Name
Letter
Units
Name
Letter
Units
Mass
M
[kg]
Force
F
[N]
Length
L
[m]
Length
L
[m]
Time
t
[sec]
Time
t
[sec]
Temperature
θ
[◦ C]
Temperature
T
[ ◦ C]
Additional Basic Units for Magnetohydrodynamics
Electric
Current
A
[A]mpere
Electric
Current
A
[A]mpere
Continued on next page
86
CHAPTER 3. DIMENSIONAL ANALYSIS
Table 3.1 – Basic Units of Two Common Systems (continue)
Standard System
Name
Luminous
Intensity
Old System
Letter
Units
cd
[cd] candle
Name
Luminous
Intensity
Letter
Units
cd
[cd] candle
M
mol
Chemical Reactions
Quantity of
substance
M
Quantity of
substance
mol
The choice of these basic units is not unique and several books and researchers suggest
a different choice of fundamental units. One common selection is substituting the mass with
the force in the previous selection (F, t, L, mol, Temperature). This author is not aware of
any discussion on the benefits of one method over the other method. Yet, there are situations
in which first method is better than the second one while in other situations, it can be the
reverse. In this book, these two selections are presented. Other selections are possible but
not common and, at the moment, will not be discussed here.
Example 3.1: Force Basic Units
Level: Basic
What are the units of force when the basic units are: mass, length, time, temperature
(M, L, t, θ)? What are the units of mass when the basic units are: force, length, time,
temperature (F, L, t, T)? Notice the different notation for the temperature in the two
systems of basic units. This notation has no significance but for historical reasons
remained in use.
Solution
These two systems are related as the questions are the reversed of each other. The connection
between the mass and force can be obtained from the simplified Newton’s second law F = m a
where F is the force, m is the mass, and a is the acceleration. Thus, the units of force are
F=
ML
t2
(3.1.a)
For the second method the unit of mass are obtain from Equation (3.1.a) as
M=
F t2
L
(3.1.b)
87
3.3. NUSSELT SCHMIDT METHOD
Table 3.2 – Physical units for two common systems. Note the second (time) in large size units appear as “s”
while in small units as “sec.”
Standard System
Old System
Name
Letter
Units
Name
Letter
Units
Area
L2
[m2 ]
Area
L2
[m2 ]
Volume
L3
Volume
L3
Angular
velocity
1
t
L
t2
1
t2
[m3 ]
1 Angular
velocity
h
m
sec2
Angular
acceleration
h
m
sec2
ML
t2
h
1
t
L
t2
1
t2
[m3 ]
1 h
kg m
sec2
Mass
F t2
L
N s2
m
Density
F t2
L4
h
Momentum
Ft
[N sec]
Acceleration
Angular
acceleration
Force
Density
M
L3
sec
1
sec2
h
h
kg
m3
i
i
i
i
i
Acceleration
sec
h
h
1
sec2
N s2
m4
i
i
i
i
Momentum
ML
t
Angular
Momentum
M L2
t
m
[ kg
sec ]
Angular
Momentum
LFt
[m N s]
Torque
M L2
t2
Torque
LF
[m N]
Absolute
Viscosity
M
L1 t1
h
Absolute
Viscosity
tF
L2
Kinematic
Viscosity
L2
t
h
Volume
Flow Rate
L3
t1
Mass
flow rate
Ft
L1
Pressure
F
L2
Kinematic
Viscosity
L2
t1
Volume
Flow Rate
L3
t1
Mass
flow rate
M
t1
Pressure
M
L t2
kg m
sec
2
kg m2
sec2
h
h
h
h
h
kg
ms
m2
sec
m3
sec
kg
sec
i
i
i
i
i
kg
m s2
i
h
h
Ns
m2
m3
sec
m3
sec
i
i
i
Ns
m
h
N
m2
i
Continued on next page
88
CHAPTER 3. DIMENSIONAL ANALYSIS
Table 3.2 – Basic Units of Two Common System (continue)
Standard System
Name
Old System
Letter
Surface
Tension
M
t2
Work or
Energy
M L2
t2
Power
M L2
t3
Thermal
Conductivity
ML
t3 θ
Specific
Heat
L2
t2 θ
Entropy
M L2
t2 θ
Specific
Entropy
L2
t2 θ
Molar
Specific
Entropy
M L2
t2 M θ
Enthalpy
M L2
t2
Specific
Enthalpy
L2
t2
Thermodynamic M L
t2 M
Force
Catalytic
Activity
M
t
Gravity
Constant
Heat
Transfer
Rate
L3
M t2
M L2
t3
Units
i
h
h
h
kg
sec2
kg m2
sec2
kg m2
sec3
h
kg m
s3 K
h
h
h
m2
s2 K
i
i
i
i
kg m2
s2 K
h
i
i
kg m2
s2 K mol
h
h
m2
s2 K
kg m2
sec2
h
m2
sec2
kg m
sec2 mol
mol sec
h
h
m3
kg s2
i
kg m2
sec3
i
i
i
i
Letter
Units
Surface
Tension
F
L
Work or
Energy
N
FL
[N m]
Power
FL
t1
Thermal
Conductivity
F
tT
Nm
Specific
Heat
L2
t2 T
Entropy
FL
T
Specific
Entropy
L2
t2 T
Molar
Specific
Entropy
FL
TM
Enthalpy
FL
[N m]
Specific
Enthalpy
L2
t2
h
Name
i
Thermodynamic N
M
Force
m
sec
h
N
sK
m2
s2 K
i
Nm
K
h
m2
s2 K
i
Nm
K mol
h
m2
sec2
m2
sec2
i
i
mol Catalytic
Activity
M
t
Gravity
Constant
Heat
Transfer
Rate
L4
t4 F
h
LF
t
mN
sec
m4
s4 N
i
sec
The number of fundamental or basic dimensions determines the number of the combi-
89
3.4. NUSSELT’S TECHNIQUE
nations which affect the physical situations. The dimensions or units which affect the problem
at hand can be reduced because these dimensions are repeating or reoccurring. All the dimensional analysis methods such as Buckingham method are based on the fact that all equations
must be consistent with their units. That is the left hand side and the right hand side of the
equations have to have the same units. The equations can be divided to create unitless equations because they have the same units. Buckingham method alludes to the fact that these
unitless parameters can be provide a minimum parameters without any knowledge of the
governing equations. Thus, the arrangement of the effecting parameters in unitless groups
yields a minimum of affecting parameters. These unitless parameters are the dimensional
parameters. The following trivial example demonstrates the consistency of units
Example 3.2: Force Second Term Units
Level: Simple
Newton’s equation has two terms that related to force F = m a + ṁ U. Where F
is force, m is the mass, a is the acceleration and dot above ṁ indicating the mass
derivative with respect to time. In particular case, this equation get a form of
F = ma+7
(3.2.a)
where 7 represent the second term. What are the requirements on Eq. (3.2.a)?
Solution
Clearly, the units of [F], m a and 7 have to be same. The units of force are [N] which is defined
by first term of the right hand side. The same units force has to be applied to 7 thus it must be
in [N].
3.4
Nusselt’s Technique
The Nusselt’s method is a bit more labor intensive, in that the governing equations with the
boundary and initial conditions are used to determine the dimensionless parameters. In this
method, the boundary conditions together with the governing equations are taken into account as opposed to Buckingham’s method. A common mistake is to ignore the boundary
conditions or initial conditions. The parameters that results from this process are the dimensional parameters which control the problems. An example comparing the Buckingham’s
method with Nusselt’s method is presented.
In this method, the governing equations, initial conditions and boundary conditions are
normalized resulting in a creation of dimensionless parameters which govern the solution. It
is recommended, when the reader is out in the real world to simply abandon Buckingham’s
method all together. This point can be illustrated by example of flow over inclined plane. For
comparison reasons Buckingham’s method presented and later the results are compared with
the results from Nusselt’s method.
90
CHAPTER 3. DIMENSIONAL ANALYSIS
Example 3.3: 2-D Inclined Plane
Level: Intermediate
Utilize the Buckingham’s method to analyze a two dimensional flow in incline plane.
Assume that the flow infinitely long and thus flow can be analyzed per width which
is a function of several parameters. The potential parameters are the angle of inclination, θ, liquid viscosity, ν, gravity, g, the height of the liquid, h, the density, ρ, and
liquid velocity, U. Assume that the flow is not affected by the surface tension (liquid),
σ. You furthermore are to assume that the flow is stable. Develop the relationship
between the flow to the other parameters.
Solution
Under the assumptions in the example presentation leads to following
(3.1)
ṁ = f (θ, ν, g, ρ, U)
The number of basic units is three while the number of the parameters is six thus the difference
is 6 − 3 = 3. Those groups (or the work on the groups creation) further can be reduced the
because angle θ is dimensionless. The units of parameters can be obtained in table 3.2 and
summarized in the following table.
Parameter
Units
Parameter
Units
Parameter
Units
ν
L2 t−1
g
L1 t−2
U
L1 t−1
ṁ
M t−1 L−1
θ
none
ρ
M L3
The basic units are chosen as for the time, U, for the mass, ρ, and for the length g. Utilizing
the building blocks technique provides
 ρ a  g b  U c
ṁ
z}|{
z}|{
z}|{
z}|{
M  L   L 
M






= 3   2  
tL
t 
L
t
(3.3.a)
The equations obtained from Eq. (3.3.a) are
or
Mass, M
a=
Length, L
−3a + b + c =
time, t
−2b − c =
1






ṁ g
−1  =⇒ π1 = ρ U3




−1
 ρ a  g b  U c
ν
z}|{
z}|{
z}|{
z}|{
2
M  L   L 
L






= 3   2  
t
t 
L
t
(3.3.b)
(3.3.c)
91
3.4. NUSSELT’S TECHNIQUE
End of Ex. 3.3
The equations obtained from equation (3.3.a) are
Mass, M
a=
0
Length, L
−3a + b + c =
2
time, t
−2b − c =
−1











=⇒ π2 =
νg
U3
(3.3.d)
Thus governing equation and adding the angle can be written as
0=f
ṁ g ν g
,
,
θ
ρ U3 U3
(3.3.e)
The conclusion from this analysis are that the number of controlling parameters totaled in
three and that the initial conditions and boundaries are irrelevant.
A small note, it is well established that the combination of angle gravity or effective
body force is significant to the results. Hence, this analysis misses, at the very least, the issue of
the combination of the angle gravity. Nusselt’s analysis requires that the governing equations
along with the boundary and initial conditions to be written. While the analytical solution
for this situation exist, the parameters that effect the problem are the focus of this discussion.
In book, “Basics of Fluid Mechanics” by this author, the Navier–Stokes equations were
developed. These equations along with the energy, mass or the chemical species of the system, and second laws governed almost all cases in thermo–fluid mechanics. This author is
not aware of a compelling reason that this fact should be used in this chapter. The two dimensional NS equation can obtained from Eq. (2.6.a) as
ρ
∂Ux
∂Ux
∂Ux
∂Ux
+ Ux
+ Uy
+ Uz
∂t
∂x
∂y
∂z
=
∂2 Ux ∂2 Ux ∂2 Ux
+
+
∂x2
∂y2
∂z2
∂Uy
∂Uy
∂Uy
∂Uy
+ Ux
+ Uy
+ Uz
∂t
∂x
∂y
∂z
=
∂P
−
+µ
∂x
∂2 Uy ∂2 Uy ∂2 Uy
+
+
∂x2
∂y2
∂z2
−
∂P
+µ
∂x
+ ρg sin θ
(3.2)
+ ρg sin θ
(3.3)
and
ρ
With boundary conditions
Ux (y = 0) = U0x f(x)
(3.4a)
∂Ux
(y = h) = τ0 f(x)
∂x
(3.4b)
92
CHAPTER 3. DIMENSIONAL ANALYSIS
The value U0 x and τ0 are the characteristic and maximum values of the velocity or the shear
stress, respectively. and the initial condition of
(3.5)
Ux (x = 0) = U0y f(y)
where U0y is characteristic initial velocity.
These sets of equations (3.2)–(3.5) need to be converted to a dimensionless form. It can
be noticed that the boundary and initial conditions are provided in a special form were the
representative velocity multiply a function. Any function can be presented by this form (normal function).
In the process of transforming the equations into a dimensionless form associated with
some intelligent guess work. However, no assumption is made or required about whether or
not the velocity, in the y direction. The only exception is that the y component of the velocity
vanished on the boundary. No assumption is required about the acceleration or the pressure
gradient etc.
The boundary conditions have typical velocities which can be used. The velocity is
selected according to the situation or the needed velocity. For example, if the effect of the
initial condition is under investigation than the characteristic of that velocity should be used.
Otherwise the velocity at the bottom should be used. In that case, the boundary conditions
are
Ux (y = 0)
= f(x)
U0x
µ
(3.6a)
∂Ux
(y = h) = τ0 g(x)
∂x
(3.6b)
Now it is very convenient to define several new variables:
U=
where :
x=
Ux (x)
U0x
x
h
(3.7a)
y=
y
h
(3.7b)
The length h is chosen as the characteristic length since no other length is provided. It can
be noticed that because the units consistency, the characteristic length can be used for “normalization” (see Example 3.4). Using these definitions the boundary and initial conditions
becomes
Ux (y=0)
U0x
′
= f (x)
h µ ∂Ux
′
(y = 1) = τ0 g (x)
U0x ∂x
(3.8)
93
3.4. NUSSELT’S TECHNIQUE
It commonly suggested to arrange the second part of equation (3.8) as
′
∂Ux
τ U
(y = 1) = 0 0x g (x)
∂x
hµ
(3.9)
Where new dimensionless parameter, the shear stress number is defined as
τ0 =
τ0 U0x
hµ
(3.10)
With the new definition equation (3.9) transformed into
′
∂Ux
(y = 1) = τ0 g (x)
∂x
(3.11)
Example 3.4: Boundary Conditons
Level: Intermediate
Non–dimensionalize the following boundary condition. What are the units of the
coefficient in front of the variables, x. What are relationship of the typical velocity,
U0 to Umax ?
Ux (y = h) = U0 a x2 + b exp(x)
(3.4.a)
Solution
The coefficients a and b multiply different terms and therefore must have different units. The
results must be unitless thus a
x2
z}|{
1
L0 = a L2 =⇒ a =
L2
(3.4.b)
From equation (3.4.b) it clear the conversion of the first term is Ux = a h2 x. The exponent
appears a bit more complicated as
x
x
L0 = b exp h
= b exp (h) exp
= b exp (h) exp (x)
h
h
Hence defining
b=
1
exp h
(3.4.c)
(3.4.d)
With the new coefficients for both terms and noticing that y = h −→ y = 1 now can be
written as
a
b
z}|{
z }| {
Ux (y = 1)
= a h2 x2 + b exp (h) exp (x) = a x2 + b exp x
U0
(3.4.e)
Where a and b are the transformed coefficients in the dimensionless presentation.
After the boundary conditions the initial condition can undergo the non–dimensional process. The initial condition (3.5) utilizing the previous definitions transformed into
U0y
Ux (x = 0)
=
f(y)
U0x
U0x
(3.12)
94
CHAPTER 3. DIMENSIONAL ANALYSIS
Notice the new dimensionless group of the velocity ratio as results of the boundary condition. This dimensionless number cannot be obtained by using the Buckingham’s technique.
The physical significance of this number is an indication to the “penetration” of the initial
(condition) velocity.
The main part of the analysis if conversion of the governing equation into a dimensionless form uses previous definition with additional definitions. The dimensionless time is
defined as t = t U0x /h. This definition based on the characteristic time of h/U0x . Thus, the
derivative with respect to time is
Ux
U
0x
z}|{
∂Ux
∂ Ux U0x
U0x 2 ∂Ux
=
=
∂t
h
∂t
∂ |{z}
t Uh0x
(3.13)
t U0x
h
Notice that the coefficient has units of acceleration. The second term
Ux
U0x
Ux
U
0x
z}|{
z}|{
∂Ux
∂ Ux U0x
∂Ux
U 2
Ux
= Ux U0x
= 0x Ux
∂x
∂ |{z}
x h
h
∂x
(3.14)
x
h
The pressure is normalized by the same initial pressure or the static pressure as (P − P∞ ) / (P0 − P∞ )
and hence
P−P∞
P0 −P∞
z}|{
(P − P∞ ) ∂P
∂ P
∂P
(P0 − P∞ ) = 0
=
∂x
∂xh
h
∂x
(3.15)
The second derivative of velocity looks like
∂2 Ux
U0x ∂2 Ux
∂ ∂ Ux U0x
=
=
∂ (xh) ∂ (xh)
∂x2
h2 ∂x2
(3.16)
The last term is the gravity g which is left for the later stage. Substituting all terms and dividing by density, ρ result in
U0x 2
h
∂Ux
∂Ux
∂Ux
∂Ux
+ Ux
+ Uy
+ Uz
∂x
∂y
∂z
∂t
−
P0 − P∞ ∂P U0x µ
+ 2
h ρ ∂x
h ρ
=
∂2 Ux ∂2 Ux ∂2 Ux
+
+
∂x2
∂y2
∂z2
ρg
+ sin θ
ρ
(3.17)
95
3.4. NUSSELT’S TECHNIQUE
Dividing equation (3.17) by U0x 2 /h yields
∂Ux
∂Ux
∂Ux
∂Ux
+ Ux
+ Uy
+ Uz
∂z
∂x
∂y
i
∂t
µ
P − P∞ ∂P
+
− 0 2
∂x
U
U0x ρ
0x h ρ
=
∂2 Ux ∂2 Ux ∂2 Ux
+
+
∂x2
∂y2
∂z2
+
gh
sin θ
U0x 2
(3.18)
Defining “initial” dimensionless parameters as
Re =
U0x h ρ
µ
U
Fr = √ 0x
gh
Eu =
P0 − P∞
U0x 2 ρ
(3.19)
Substituting definition of equation (3.19) into equation (3.18) yields
∂Ux
∂Ux
∂Ux
∂Ux
+ Ux
+ Uy
+ Uz
∂x
∂y
∂z
∂t
− Eu
1
∂P
+
∂x Re
=
∂2 Ux ∂2 Ux ∂2 Ux
+
+
∂x2
∂y2
∂z2
+
1
sin θ
Fr2
(3.20)
Equation (3.20) show one common possibility of a dimensionless presentation of governing
equation. The significance of the large and small value of the dimensionless parameters will
be discuss later in the book. Without actually solving the problem, Nusselt’s method provides
several more parameters that were not obtained by the Buckingham pi method. The solution
of the governing equation is a function of all the parameters present in that equation and
boundaries condition as well the initial condition. Thus, the solution is
U0y
Ux = f x, y, Eu, Re, Fr, θ, τ0 , fu , fτ ,
(3.21)
U0x
The values of x, y depend on h and hence the value of h is an important parameter.
It can be noticed with Buckingham’s method, the number of parameters obtained was
only three (3) while Nusselt’s method yields 12 dimensionless parameters. This is a very significant difference between the two methods. In fact, there are numerous examples in the literature that showing people doing experiments based on Buckingham’s methods. In these experiments, major parameters are ignored rendering these experiments useless in many cases
and deceiving.
Common Transformations
Fluid mechanics in particular and Thermo–Fluid field in general have several common transformations that appear in boundary conditions, initial conditions and equations5 .
It recognized that not all the possibilities can presented in the example shown above. Several
5 Many of these tricks spread in many places and fields. This author is not aware of a collection of this kind of
transforms.
96
CHAPTER 3. DIMENSIONAL ANALYSIS
common boundary conditions which were not discussed in the above example are presented
below. As an initial matter, the results of the non dimensional transformation depends on the
selection of what and how is nondimensionalization carried. This section of these parameters depends on what is investigated. Thus, one of the general nondimensionalization of the
Navier–Stokes and energy equations will be discussed at end of this chapter.
Boundary conditions are divided into several categories such as a given value to the
function6 , given derivative (Neumann b.c.), mixed condition, and complex conditions. The
first and second categories were discussed to some degree earlier and will be expanded later.
The third and fourth categories were not discussed previously. The nondimensionalization
of the boundary conditions of the first category requires finding and diving the boundary
conditions by a typical or a characteristic value. The second category involves the nondimensionalization of the derivative. In general, this process involve dividing the function by a
typical value and the same for length variable (e.g. x) as
U
∂
U0
∂U
ℓ ∂U
ℓ
=
=
(3.22)
∂x
U0 ∂ xℓ
U0 ∂x
In the Thermo–Fluid field and others, the governing equation can be of higher order than
second order7 . It can be noticed that the degree of the derivative boundary condition cannot
exceed the derivative degree of the governing equation (e.g. second order equation has at
most the second order differential boundary condition.). In general “nth” order differential
equation leads to
U
n
n
∂
U0
∂ U
U0
U ∂n U
n = n0
=
(3.23)
n
n
x
∂x
ℓ
ℓ ∂xn
∂ ℓ
The third kind of boundary condition is the mix condition. This category includes
combination of the function with its derivative. For example a typical heat balance at liquid
solid interface reads
∂T
(3.24)
h(T0 − T ) = −k
∂x
This kind of boundary condition, since derivative of constant is zero, translated to
T − T0
( −∂
((
)
T0 − T
k (T (−(Tmax
T0 − Tmax
(
((
x
= − (0
(3.25)
h(
(T0(−(Tmax
)
T0 − Tmax
ℓ
∂
ℓ
or
T − T0
∂
T0 − T
k
1 ∂Θ
T0 − Tmax
x
=
=⇒ Θ =
(3.26)
T0 − Tmax
hℓ
Nu
∂x
∂
ℓ
6 Mathematicians like to refer to it as Dirichlet conditions
7 This author aware of fifth order partial differential governing equations in some cases. Thus, the highest derivative can be fifth order derivative.
97
3.4. NUSSELT’S TECHNIQUE
Where Nusselt Number and the dimensionless temperature are defined as
Nu =
hℓ
k
Θ=
T − T0
T0 − Tmax
(3.27)
and Tmax is the maximum or reference temperature of the system.
The last category is dealing with some non–linear conditions of the function with its
derivative. For example,
σ r1 + r2
1
1
∆P ≈ σ
=
+
(3.28)
r1 r2
r1
r2
Where r1 and r2 are the typical principal radii of the free surface curvature, and, σ, is the
surface tension between the gas (or liquid) and the other phase. The surface geometry (or
the radii) is determined by several factors which include the liquid movement instabilities etc
chapters of the problem at hand. This boundary condition (3.28) can be rearranged to be
r + r2
r + r2
∆P r1
≈ 1
=⇒ Av ≈ 1
σ
r2
r2
(3.29)
Where Av is Avi number . The Avi number represents the geometrical characteristics combined with the material properties. The boundary condition (3.29) can be transferred into
∆P r1
= Av
σ
(3.30)
Where ∆P is the pressure difference between the two phases (normally between the liquid
and gas phase).
One of advantage of Nusselt’s method is the Object–Oriented nature which allows one
to add additional dimensionless parameters for addition “degree of freedom.” It is common
assumption to initially assume that liquid is incompressible. If greater accuracy is needed
than this assumption is removed. In that case, a new dimensionless parameters is introduced
as the ratio of the density to a reference density as
ρ=
ρ
ρ0
(3.31)
In case of ideal gas model with isentropic flow this assumption becomes
ρ
ρ̄ =
=
ρ0
P0
P
1
n
(3.32)
The power n depends on the gas properties.
Characteristics Values
Normally, the characteristics values are determined by physical. values e.g The diameter of cylinder as a typical length . There are several situations where the characteristic
length, velocity, for example, are determined by the
p physical properties of the fluid(s). The
characteristic velocity can determined from U0 = 2P0 /ρ. The characteristic length can be
determined from ratio of ℓ = ∆P/σ.
98
CHAPTER 3. DIMENSIONAL ANALYSIS
Example 3.5: Renewable Energy
Level: Intermediate
One idea of renewable energy is to use and to utilize the high concentration of of
brine water such as in the Salt Lake and the Salt Sea (in Israel). This process requires
analysis the mass transfer process. The governing equation is non–linear and this
example provides opportunity to study nondimensionalizing of this kind of equation.
The conversion of the species yields a governing nonlinear equation8 for such process
is
∂ DAB ∂CA
∂C
U0 A =
(3.5.a)
∂x
∂y (1 − XA ) ∂y
Where the concentration, CA is defended as the molar density i.e. the number of
moles per volume. The molar fraction, XA is defined as the molar fraction of species
A divide by the total amount of material (in moles). The diffusivity coefficient, DAB
is defined as penetration of species A into the material. What are the units of the
diffusivity coefficient? The boundary conditions of this partial differential equation
are given by
∂CA
(y = ∞) = 0
(3.5.b)
∂y
(3.5.c)
CA (y = 0) = Ce
Where Ce is the equilibrium concentration. The initial condition is
(3.5.d)
CA (x = 0) = C0
Select dimensionless parameters so that the governing equation and boundary and
initial condition can be presented in a dimensionless form. There is no need to discuss
the physical significance of the problem.
Solution
This governing equation requires to work with dimension associated with mass transfer and
chemical reactions, the “mole.” However, the units should not cause confusion or fear since
it appear on both sides of the governing equation. Hence, this unit will be canceled. Now
the units are compared to make sure that diffusion coefficient is kept the units on both sides
the same. From units point of view, equation (3.5.a) can be written (when the concentration is
simply ignored) as
U
∂C
∂x
∂
∂y
DAB
(1−X)
∂C
∂y
z}|{ z}|{ z}|{ z }| { z}|{
L C
1 DAB C
=
t L
L
1
L
(3.5.e)
It can be noticed that X is unitless parameter because two same quantities are divided.
1
1
L2
(3.5.f)
= 2 DAB =⇒ DAB =
t
t
L
Hence the units of diffusion coefficient are typically given by m2 /sec (it also can be observed
that based on Fick’s laws of diffusion it has the same units).
99
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
End of Ex. 3.5
The potential of possibilities of dimensionless parameter is large. Typically, dimensionless parameters are presented as ratio of two quantities. In addition to that, in heat and mass transfer
(also in pressure driven flow etc.) the relative or reference to certain point has to accounted
for. The boundary and initial conditions here provides the potential of the “driving force” for
the mass flow or mass transfer. Hence, the potential definition is
Φ=
CA − C0
Ce − C0
(3.5.g)
With almost “standard” transformation
x=
x
ℓ
y=
y
ℓ
(3.5.h)
Hence the derivative of Φ with respect to time is
∂Φ
=
∂x
∂
0
CA − C0
>
C
∂ CA − 0
ℓ
ℓ
∂CA
Ce − C0
=
=
x
C
−
C
∂x
Ce − C0 ∂x
e
0
∂
ℓ
(3.5.i)
In general a derivative with respect to x or y leave yields multiplication of ℓ. Hence, equation
(3.5.a) transformed into
;
(C
∂ DAB e − C0 ) ∂Φ
∂y (1 − XA )
ℓ
∂y
1 ∂ DAB ∂Φ
= 2
ℓ ∂y (1 − XA ) ∂y
∂Φ
0 )
(C −C
U0 e
ℓ
=
∂x
U0 ∂Φ
ℓ ∂x
1
ℓ
(3.5.j)
Equation (3.5.j) like non–dimensionalized and proper version. However, the term XA , while is
dimensionless, is not proper. Yet, XA is a function of Φ because it contains CA . Hence, this
term, XA has to be converted or presented by Φ. Using the definition of XA it can be written
as
C
C − C0 1
XA = A = (Ce − C0 ) A
(3.5.k)
C
Ce − C0 C
Thus the transformation in equation (3.5.k) another unexpected dimensionless parameter as
XA = Φ
Ce − C0
C
(3.5.l)
0
Thus number, Ce −C
was not expected and it represent ratio of the driving force to the height
C
of the concentration which was not possible to attend by Buckingham’s method.
3.5
Summary of Dimensionless Numbers
This section summarizes all the major dimensionless parameters which are commonly used
in the fluid mechanics field.
8 More information how this equation was derived can be found in (Bar–Meir (Meyerson) 1991).
100
CHAPTER 3. DIMENSIONAL ANALYSIS
Table 3.3 – Common Dimensionless Parameters of Thermo–Fluid in the Field
Name
Symbol
Archimedes
Ar
Number
Equation
g ℓ3 ρ
f (ρ − ρf )
µ2
Interpretation
Application
buoyancy forces
viscous force
in nature and force
convection
buoyancy forces
“penetration” force
in stability of liquid layer a over b
Rayleigh–Taylor instability etc.
Atwood
Number
A
(ρa − ρb )
ρa + ρb
Bond
Number
Bo
ρ g ℓ2
σ
gravity forces
surface tension force
in open channel flow,
thin film flow
Brinkman
Number
Br
µU2
k ∆T
heat dissipation
heat conduction
during dissipation
problems
Capillary
Number
Ca
µU
σ
viscous force
surface tension force
For small Re and
surface tension involve problem
Cauchy
Number
Cau
ρ U2
E
inertia force
elastic force
Cavitation
Number
σ
Pl − Pv
1
2
2 ρU
pressure difference
inertia energy
For large Re and
surface
tension
involve problem
pressure difference
to vapor pressure
to the potential of
phase change (mostly
to gas)
Courant
Number
Co
∆t U
∆x
wave distance
typical distance
A requirement in numerical schematic to
achieve stability
Dean
Number
D
Re
p
R/h
inertia forces
viscous deviation forces
related to radius of
channel with width
h stability
stress relaxation time
observation time
the ratio of the fluidity of material primary used in rheology
Deborah
Number9
De
tc
tp
Drag Coefficient
CD
D
1
2
2 ρU A
drag force
inertia effects
Aerodynamics,
hydrodynamics, note
this coefficient has
many definitions
Eckert
Number
Ec
U2
Cp ∆T
inertia effects
thermal effects
during dissipation
processes
Continued on next page
101
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
Table 3.3 – Common Dimensionless Parameters of Fluid Mechanics (continue)
Standard System
Name
Symbol
Equation
Interpretation
Application
Ekman
Number
Ek
ν
2ℓ2 ω
viscous forces
Coriolis forces
geophysical flow like
atmospheric flow
Euler
Number
Eu
P0 − P∞
1
2
2 ρU
Froude
Number
Fr
U
√
gℓ
inertia effects
gravitational effects
open channel flow
and two phase flow
Galileo
Number
Ga
ρ g ℓ3
µ2
gravitational effects
viscous effects
open channel flow
and Stokes flow
Grashof
Number
Gr
β ∆T g ℓ3 ρ2
µ2
buoyancy effects
viscous effects
Knudsen
Number
Kn
λ
ℓ
LMFP
characteristic length
length of mean free
path, LMFP, to characteristic length
Laplace
Constant
La
2σ
g(ρ1 − ρ2 )
surface force
gravity effects
liquid raise, surface
tension
problem,
also
ref:Capillary
constant
Lift Coefficient
CL
L
1
2A
ρ
U
2
lift force
inertia effects
Aerodynamics,
hydrodynamics, note
this coefficient has
many definitions
Mach
Number
M
U
c
velocity
sound speed
compressibility
and propagation of
disturbances
“thermal” surface tension
surface
tension
caused by thermal
gradient
Marangoni
Ma
Number
Morton
Number
Ozer
Number
Mo
Oz
r
−
dσ ℓ ∆T
dT να
pressure
potential
effects
inertia effects
viscous force
potential
of
resistance problems
natural convection
gµ4c ∆ρ
ρ2c σ3
viscous force
surface tension force
bubble and drop flow
CD 2 Pmax
ρ
Qmax 2
A
“maximum” supply
“maximum” demand
supply and demand
analysis such pump
& pipe system, economy
Continued on next page
102
CHAPTER 3. DIMENSIONAL ANALYSIS
Table 3.3 – Common Dimensionless Parameters of Fluid Mechanics (continue)
Standard System
Name
Symbol
Equation
Interpretation
Prandtl
Number
Pr
ν
α
viscous diffusion rate
thermal diffusion rate
Prandtl
number
is fluid property
important in flow
due to thermal forces
Reynolds
Number
Re
ρUℓ
µ
inertia forces
viscous forces
In most fluid mechanics issues
Rossby
Number
Ro
U
ω ℓ0
inertia forces
Coriolis forces
In rotating fluids
Shear
Number
Sn
τc ℓc
µc Uc
actual shear
“potential” shear
Stokes
Number
Stk
tp
tK
particle
relaxation
Application
shear flow
time
Kolmogorov time
in aerosol flow dealing with penetration
of particles
Strouhal
Number
St
ωℓ
U
“unsteady” effects
inertia effect
The effects of natural or forced frequency in all the field
that is how much the
“unsteadiness” of the
flow is
Taylor
Number
Ta
ρ2 ωi 2 ℓ4
µ4
centrifugal forces
viscous forces
Stability of rotating
cylinders Notice ℓ
has special definition
Weber
Number
We
ρ U2 ℓ
σ
inertia force
surface tension force
For large Re and
surface
tension
involve problem
The dimensional parameters that were used in the construction of the dimensionless
parameters in Table 3.3 are the characteristics of the system. Therefore there are several definition of Reynolds number. In fact, in the study of the physical situations often people refers
to local Re number and the global Re number. Keeping this point in mind, there several typical dimensions which need to be mentioned. The typical body force is the gravity g which
has a direction to center of Earth. The elasticity E in case of liquid phase is BT , in case of solid
phase is Young modulus. The typical length is denoted as ℓ and in many cases it is referred
to as the diameter or the radius. The density, ρ, is referred to the characteristic density or
9 This number is named by Reiner, M. (1964), “The Deborah Number”, (Reiner 1964). Reiner, a civil engineer who
is considered the father of Rheology, named this parameter because theological reasons perhaps since he was living
in Israel.
103
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
density at infinity. The area, A, in drag and lift coefficients is referred normally to projected
area.
The frequency ω or f is referred to as the “unsteadiness” of the system. Generally,
the periodic effect is enforced by the boundary conditions or the initial conditions. In other
situations, the physics itself instores or forces periodic instability. For example, flow around
cylinder at first looks like symmetrical situation. And indeed in a low Reynolds number it is
a steady state. However after a certain value of Reynolds number, vortexes are created in an
infinite parade and this phenomenon is called Von Karman vortex street (see (Bar-Meir 2021a,
Dimenstional Analsyis Chapter)) which named after von Karman. These vortexes are created
in a non–symmetrical way and hence create an unsteady situation. When Reynolds number
increases, these vortexes are mixed and the flow becomes turbulent which, can be considered
a steady state10 .
The pressure P is the pressure at infinity or when the velocity is at rest. c is the speed
of sound of the fluid at rest or characteristic value. The value of the viscosity, µ is typically
some kind averaged value. The inability to define a fix value leads also to new dimensionless
numbers which represent the deviations of these properties.
3.5.1
The Significance of these Dimensionless Numbers
Reynolds number, named in the honor of Reynolds, represents the ratio of the momentum
forces to the viscous forces. Historically, this number was one of the first numbers to be
introduced to fluid mechanics. This number determines, in many cases, the flow regime.
Example 3.6: Eckert Number
Level: Intermediate
Eckert number (Bird, Stewart, and Lightfoot 1960) determines whether the role of
the momentum energy is transferred to thermal energy is significant to affect the
flow. This effect is important in situations where high speed is involved. This fact
suggests that Eckert number is related to Mach number. Determine this relationship
and under what circumstances this relationship is true.
Solution
In Table 3.3 Mach and Eckert numbers are defined as
Ec =
U2
Cp ∆T
U
M= r
P
ρ
(3.6.a)
The material which obeys the ideal flow model11 (P/ρ = R T and P = C1 ρk ) can be written
that
,s
M=U
P
U
= √
ρ
kRT
(3.6.b)
10 This is an example where the more unsteady the situation becomes the situation can be analyzed as a steady
state because averages have a significance.
104
CHAPTER 3. DIMENSIONAL ANALYSIS
End of Ex. 3.6
For the comparison, the reference temperature used to be equal to zero. Thus Eckert number
can be written as
√
√
U
k−1U √
U
√
Ec = p
=
= k−1M
= v
u Rk Cp T
kRT
u
u k−1 T
t
| {z }
(3.6.c)
Cp
The Eckert number and Mach number are related under ideal gas model and isentropic relationship.
Brinkman number measures of the importance of the viscous heating relative the conductive heat transfer. This number is important in cases when a large velocity change occurs
over short distances such as lubricant, supersonic flow in rocket mechanics creating large heat
effect in the head due to large velocity (in many place it is a combination of Eckert number
with Brinkman number. The Mach number is based on different equations depending on the
property of the medium in which pressure disturbance moves through. Cauchy number and
Mach number are related as well and see Example 3.8 for explanation.
Example 3.7: Historical Reason
Level: Simple
For historical reason some fields prefer to use certain numbers and not others. For
example in Mechanical engineers prefer to use the combination Re and We number while Chemical engineers prefers to use the combination of Re and the Capillary
number. While in some instances this combination is justified, other cases it is arbitrary. Show what the relationship between these dimensionless numbers.
Solution
The definitions of these number in Table 3.3
We =
ρ U2 ℓ
σ
Re =
ρUℓ
µ
Ca =
µU
U
= σ
σ
µ
(3.7.a)
Dividing Weber number by Reynolds number yields
ρ U2 ℓ
We
U
σ
= σ = Ca
=
ρUℓ
Re
µ
µ
(3.7.b)
Euler number is named after Leonhard Euler (1707 1783), a German Physicist who pioneered so many fields that it is hard to say what and where are his greatest contributions.
Euler’s number and Cavitation number are essentially the same with the exception that these
numbers represent different driving pressure differences. This difference from dimensional
11 See for more details http://www.potto.org/gasDynamics/node70.html
105
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
analysis is minimal. Furthermore, Euler number is referred to as the pressure coefficient, Cp .
This confusion arises in dimensional analysis because historical reasons and the main focus
area. The cavitation number is used in the study of cavitation phenomena while Euler number
is mainly used in calculation of resistances.
Example 3.8: Mach and Cauchy
Level: Intermediate
Explained under what conditions and what are relationship between the Mach number and Cauchy number?
Solution
Cauchy number is defined as
ρ U2
E
(3.8.a)
√
U
Cau = r
E
ρ
(3.8.b)
Cau =
The square root of Cauchy number is
In the liquid phase the speed of sound is approximated as
c=
E
ρ
(3.8.c)
Using equation (3.8.b) transforms equation (3.8.a) into
√
U
=M
Cau =
c
(3.33)
Thus the square root of Cau is equal to Mach number in the liquid phase. In the solid phase
equation (3.8.c) is less accurate and speed of sound depends on the direction of the grains.
However, as first approximation, this analysis can be applied also to the solid phase.
3.5.2
Relationship Between Dimensionless Numbers
The Dimensionless numbers tend to be duplicated since many of them have been formulated
in a certain field. For example, the Bond number is referred in Europe as Eotvos number.
In addition to the above confusion, many dimensional numbers expressed the same things
under certain conditions. For example, Mach number and Eckert Number under certain circumstances are same.
106
CHAPTER 3. DIMENSIONAL ANALYSIS
Example 3.9: Galileo Number
Level: Intermediate
Galileo Number is a dimensionless number which represents the ratio of gravitational forces and viscous forces in the system as
Ga =
ρ2 g ℓ3
µ2
(3..a)
The definition of Reynolds number has viscous forces and the definition of Froude
number has gravitational forces. What are the relation between these numbers?
Solution
Submit your answer.
Example 3.10: Laplace Number
Level: Intermediate
Laplace Number is another dimensionless number that appears in fluid mechanics
which related to Capillary number. The Laplace number definition is
La =
ρσℓ
µ2
(3.10.a)
Show what are the relationships between Reynolds number, Weber number and
Laplace number.
Solution
Submit your answer.
Example 3.11: Rotating Froude Number
Level: Intermediate
The Rotating Froude Number is a somewhat a similar number to the regular Froude
number. This number is defined as
FrR =
ω2 ℓ
g
What is the relationship between two Froude numbers?
Solution
Submit your answer.
(3.11.a)
107
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
Example 3.12: Ohnesorge Number
Level: Intermediate
Ohnesorge Number is another dimensionless parameter that deals with surface tension and is similar to Capillary number and it is defined as
µ
Oh = √
ρσℓ
(3..b)
Defined Oh in term of We and Re numbers.
Solution
3.5.3
Examples for Dimensional Analysis
Example 3.13: Pump Similarity
Level: Intermediate
The similarity of pumps is determined by comparing several dimensional numbers
among them are Reynolds number, Euler number, Rossby number etc. Assume that
the only numbers which affect the flow are Reynolds and Euler number. The flow
rate of the imaginary pump is 0.25 [m3 /sec] and pressure increase for this flow rate
is 2 [Bar] with 2500 [kw]. Due to increase of demand, it is suggested to replace the
pump with a 4 times larger pump. What is the new estimated flow rate, pressure
increase, and power consumption?
Solution
It provided that the Reynolds number controls the situation. The density and viscosity remains
the same and hence
Rem = Rep =⇒ Um Dm = Up Dp =⇒ Up =
Dm
Um
DP
(3.13.a)
It can be noticed that initial situation is considered as the model and while the new pump is
the prototype. The new flow rate, Q, depends on the ratio of the area and velocity as
Qp
Ap Up
Ap Up
Dp 2 Up
=
=⇒ Qp = Qm
= Qm
Qm
Am Um
Am Um
Dm 2 Um
(3.13.b)
Thus the prototype flow rate is
Qp = Qm
Dp
Dm
3
= 0.25 × 43 = 16
m3
sec
(3.13.c)
The new pressure is obtain by comparing the Euler number as
Eup = Eum =⇒
∆P
1
2
2ρU
!
=
p
∆P
1
2
2ρU
!
(3.13.d)
m
108
CHAPTER 3. DIMENSIONAL ANALYSIS
End of Ex. 3.13
Rearranging equation (3.13.d) provides
(∆P)p
(∆P)m
Utilizing equation (3.13.a)
2
2
ρU p
U p
= 2 =
2
ρ
U
U
m
m
∆Pp = ∆Pm
The power can be obtained from the following
Ẇ =
Dp
Dm
2
Fℓ
= FU = PAU
t
(3.13.e)
(3.13.f)
(3.13.g)
In this analysis, it is assumed that pressure is uniform in the cross section. This assumption
is appropriate because only the secondary flows in the radial direction (to be discussed in this
book section on pumps.). Hence, the ratio of power between the two pump can be written as
(P A U)p
Ẇp
=
(P A U)m
Ẇm
(3.13.h)
Utilizing equations above in this ratio leads to
Pp /Pm
Ẇp
Ẇm
Ap /Am
/Um
z }| { z }| { zUp}|
2 2 { Dp
Dp
Dp
Dp 5
=
=
Dm
Dm
Dm
Dm
(3.13.i)
Example 3.14: Simulating Water by Air
Level: Intermediate
The flow resistance to flow of the water in a pipe is to be simulated by flow of air.
Estimate the pressure loss ratio if Reynolds number remains constant. This kind
of study appears in the industry in which the compressibility of the air is ignored.
However, the air is a compressible substance that flows the ideal gas model. Water is
a substance that can be considered incompressible flow for relatively small pressure
change. Estimate the error using the averaged properties of the air.
Solution
For the first part, the Reynolds number is the single controlling parameter which affects the
pressure loss. Thus it can be written that the Euler number is function of the Reynolds number.
Eu = f(Re)
(3.14.a)
Thus, to have a similar situation the Reynolds and Euler have to be same.
Rep = Rem
Hence,
Eum = Eup
ℓp ρ µp
Um
=
Up
ℓm ρm µm
(3.14.b)
(3.14.c)
109
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
End of Ex. 3.14
and for Euler number
∆Pm
ρm Um
=
∆Pp
ρp Up
(3.14.d)
and utilizing equation (3.14.c) yields
∆Pm
=
∆Pp
ℓp
ℓm
2 µm
µp
2 ρp
ρm
(3.14.e)
Inserting the numerical values results in
∆Pm
= 1 × 1000×
∆Pp
(3.14.f)
It can be noticed that the density of the air changes considerably hence the calculations produce
a considerable error which can render the calculations useless (a typical problem of Buckingham’s method). Assuming a new variable that effect the problem, air density variation. If that
variable is introduced into problem, air can be used to simulate water flow. However as a first
approximation, the air properties are calculated based on the averaged values between the entrance and exit values. If the pressure reduction is a function of pressure reduction (iterative
process).
to be continue
Example 3.15: Boat Model
Level: Intermediate
A device operating on a surface of a liquid to study using a model with a ratio 1:20.
What should be ratio of kinematic viscosity between the model and prototype so that
Froude and Reynolds numbers remain the same. Assume that body force remains the
same and velocity is reduced by half.
Solution
The requirement is that Reynolds
Rem = Rp =⇒
Uℓ
ν
=
p
Uℓ
ν
(3.15.a)
m
The Froude needs to be similar so
Frm = Frp =⇒
U
√
gℓ
=
p
Uℓ
ν
/
(3.15.b)
m
dividing equation (3.15.a) by equation (3.15.b) results in
or
Uℓ
ν
p
/
U
√
gℓ
p
=
Uℓ
ν
m
√ √ ℓ gℓ
ℓ gℓ
=
ν
ν
p
m
U
√
gℓ
m
(3.15.c)
(3.15.d)
110
CHAPTER 3. DIMENSIONAL ANALYSIS
End of Ex. 3.15
If the body forcea , g, The kinematic viscosity ratio is then
νp
=
νm
ℓm
ℓp
3/2
= (1/20)3/2
(3.15.e)
It can be noticed that this can be achieved using Ohnesorge Number like this presentation.
a The body force does not necessarily have to be the gravity.
Example 3.16: AP Physics
Level: Intermediate
In AP physics in 2005 the first question reads “A ball of mass M is thrown vertically
upward with an initial speed of U0 . Does it take longer for the ball to rise to its maximum height or to fall from its maximum height back to the height from which it was
thrown? It also was mentioned that resistance is proportional to ball velocity (Stoke
flow). Justify your answer.” Use the dimensional analysis to examine this situation.
Solution
The parameters that can effect the situation are (initial) velocity of the ball, air resistance (assuming Stokes flow e.g. the resistance is function of the velocity), maximum height, and gravity.
Functionality of these parameters can be written as
t = f(U, k, H, m, g)
(3.16.a)
The time up and/or down must be written in the same fashion since fundamental principle of
Buckingham’s π theorem the functionally is unknown but only dimensionless parameters are
dictated. Hence, no relationship between the time up and down can be provided.
However, Nusselt’s method provides first to written the governing equations. The governing
equation for the ball climbing up is
m
dU
= −m g − k U
dt
(3.16.b)
when the negative sign indicates that the positive direction is up. The initial condition is that
(3.16.c)
U(0) = U0
The governing equation the way down is
m
dU
= −m g + k U
dt
with initial condition of
(3.16.d)
(3.16.e)
U(0) = 0
Equation (3.16.d) has no typical velocity (assuming at this stage that solution was not solved
ever before). Dividing equation (3.16.d) by m g and inserting the gravitation constant into the
derivative results in
dU
kU
= −1 +
(3.16.f)
d (g t)
mg
111
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
End of Ex. 3.16
The gravity constant, g, could be inserted because it is constant. Equation suggests that velocity should be normalized by as dimensionless group, k U/ m g. Without solving the equations,
it can be observed that value of dimensionless group above or below one change the characteristic of the governing equation (positive slop or negative slop). Non–dimensioning of initial
condition (3.16.c) yields
k U(0)
k U0
=
mg
mg
(3.16.g)
In this case, if the value k U0 / m g is above one change the characteristic of the situation. This
exercise what not to solve this simple Physics mechanics problem but rather to demonstrate the
power of dimensional analysis power. So, What this information tell us? In the case the supper
critical initial velocity, the ball can be above critical velocity
k U0
> 1 on the up. However the
mg
ball never can be above the critical velocity and hence the time up will shorter the time done.
For the initial velocity below the critical velocity, while it is know that the answer is the same,
the dimensional analysis does not provide a solution. On the way up ball can start
Example 3.17: Sail Boats
Level: Simple
Two boats sail from the opposite
sides of river (see Figure 3.2). They
meet at a distance ℓ1 (for example
1000) meters from bank A . The
boats reach the opposite side respectively and continue back to
their original bank. The boats
meet for the second time at ℓ2 (for
example 500) [m] from bank B .
What is the river width? What are
the dimensional parameters that
control the problem?
A
ℓ1
ℓ2
B
Fig. 3.2 – Description of the boat crossing river.
Solution
The original problem was constructed so it was suitable to the 11 years old author’s daughter
who was doing her precalculus. However, it appears that this question can be used to demonstrate some of the power of the dimensional analysis. Using the Buckingham’s method it is
assumed that diameter is a function of the velocities and lengths. Hence, the following can be
written
(3.17.a)
D = f(ℓ1 , ℓ2 , UA , UB )
Where D is the river width. Hence, according basic idea the following can be written
D = ℓ1 a ℓ1 b UA c UB d
(3.17.b)
112
CHAPTER 3. DIMENSIONAL ANALYSIS
continue Ex. 3.17
The solution of equation (3.17.b) requires that
D = [L]a [L]b
c d
L
L
T
T
(3.17.c)
The time has to be zero hence it requires that
0 = c+d
(3.17.d)
1 = a+b+c+d
(3.17.e)
The units length requires that
Combined equation (3.17.d) with equation (3.17.e) results in
(3.17.f)
1 = a+b
It canpnoticed that symmetry arguments require that a and b must be identical. Hence, a =
√
1/2 and the solutions is of the form D = f( ℓ1 ℓ2 ). From the analytical solution it was
found that this solution is wrong.
Another approach utilizing the minimized Buckingham’s approach reads
b=
D = f(ℓ1 , UA )
(3.17.g)
b
L
T
(3.17.h)
In the standard form this leads to
D = [L]a
Which leads to the requirements of b = 0 and a = 1. Which again conflict with the actual
analytical solution.
Using Nusselt’s method requires to write the governing equation. The governing equations
are based equating the time traveled to first and second meeting as the following
ℓ1
D − ℓ1
=
UA
UB
(3.17.i)
2 D − ℓ2
D + ℓ2
=
UA
UB
(3.17.j)
At the second meeting the time is
Equations (3.17.i) and (3.17.j) have three unknowns D, UA and UB . The non–dimensioning
process can be carried by dividing governing equations by D and multiply by UB to become
ℓ1 = 1 − ℓ1
UA
UB
1 + ℓ2 = 2 − ℓ2
UA
UB
(3.17.k)
(3.17.l)
Equations (3.17.k) and (3.17.l) have three unknowns. However, the velocity ratio is artificial
parameter or dependent parameter. Hence division of the dimensionless governing equations
yield one equation with one unknown as
ℓ1
1 − ℓ1
=
1 + ℓ2
2 − ℓ2
(3.17.m)
113
3.5. SUMMARY OF DIMENSIONLESS NUMBERS
End of Ex. 3.17
Equation 3.17.m determines that ℓ1 is a function of ℓ2 . It can be noticed that D, ℓ1 and ℓ2 are
connected. Hence, knowing two parameters leads to the solution of the missing parameter.
From dimensional analysis it can be written that the
ℓ1
−1
1 − ℓ1
ℓ2 = f(ℓ1 ) =
ℓ1
1+
1 − ℓ1
2
(3.17.n)
It can be concluded that river width is a function of implicit of ℓ1 and ℓ2 . Clearly the Nusselt’s
technique provided write based to obtain the dimensionless parameters. A bit smarter selection of the normalizing parameter can provide explicit solution. An alternative definition of
e = D/ℓ1 and ℓe2 = ℓ2 /ℓ1 can provide the need path. Equation
dimensionless parameters of D
(3.17.m) can be converted quadratic equation for D as
1
e − ℓe2
D
=
e −1
D
e − ℓe2
2D
(3.17.o)
Equation (3.17.o) is quadratic which can be solved analytically. The solution can be presented
as
D = ℓ1 f
ℓ2
ℓ1
(3.17.p)
Example 3.18: Lumped capacity System
Level: Intermediate
Lumped Capacity System refers to a systems were the heat conduction is faster then
the heat convection process. This situation is typical when to small metal is placed
into cooling air. This situations can be approximated by Newton Law of cooling.
Assume that dimensional analysis indeed show that the situation for Newton law of
cooling. The temperature of the metal object is measured at two different times and
the temperature was recorded. Find what parameters effect the temperature ratio by
using the two methods: Buckingham and Nusselt.
Solution
The Buckingham method requires that the parameters should be guessed. In this situation
some knowledge of the problem can be helpful. It is logical to assume that the heat conduction
coefficient, k, surface area, A, volume, V , density, ρ, heat capacity, Cv , the convection coefficient, h and temperature difference are the effecting parameters of the time. Thus it can be
written that
(3.18.a)
t = f(k, A, V , ρ, Cv , h, ∆T )
Later it can be shown that these parameters are indeed affecting the time. The number of basic
parameters
in this problem is four which
are, length,
L, M, t, and θ.
k=
M
L2
L2
ML
A = L2 V = L3 ρ =
Cv = 2
h= 2
J/(L2 K)
3
3
t θ
L
t θ
t θ
unfinished.
114
3.6
CHAPTER 3. DIMENSIONAL ANALYSIS
Summary
The two dimensional analysis methods or approaches were presented in this chapter. Buckingham’s π technique is a quick “fix approach” which allow rough estimates and relationship
between model and prototype. Nusselt’s approach provides an heavy duty approach to examine what dimensionless parameters effect the problem. It can be shown that these two
techniques in some situations provide almost similar solution. In other cases, these technique
proves different and even conflicting results. The dimensional analysis technique provides a
way to simplify models (solving the governing equation by experimental means) and to predict effecting parameters.
3.7
Appendix summary of Dimensionless Form of Navier–Stokes
Equations
In a vector form Navier–Stokes equations can be written and later can be transformed into
dimensionless form which will yield dimensionless parameters. First, the typical or characteristics values of scaling e parameters has to b presented and appear in the following table
Parameter Symbol
Parameter Description
Units
h
characteristic length
U0
characteristic velocity
f
characteristic frequency
ρ0
characteristic density
Pmax − P∞
maximum pressure drive
[L]
L
t
1
t
M
L3
M
L t2
Basic non–dimensional form of the parameters
t̃ = ft
P − P∞
P=
P̃
Pmax − P∞
into
r̃r =
⃗r
h
˜ = h∇
∇
U=
Ũ
⃗
U
U0
ρ
ρ̃ =
ρ0
(3.34)
For the Continuity Equation (??) for non–compressible substance can be transformed
0
∂ρ
+ ∇ · (ρ̃ U ) = 0
∂t
(3.35)
3.7. APPENDIX SUMMARY OF DIMENSIONLESS FORM OF NAVIER–STOKES EQUATIONS 115
For the N-S equation, every additive term has primary dimensions m1 L−2 t−2 . To non
nondimensionalization, we multiply every term by L/(V 2 ), which has primary dimensions
m−1 L2 t2 , so that the dimensions cancel.
Using these definitions equation (??) results in
U
f h ∂Ũ
Pmax − P∞ ˜
1 ˜2
1
˜
U
U
P + 2 f⃗g +
U
+ Ũ · ∇ Ũ = −
∇P̃
∇ Ũ
(3.36)
U0 ∂t̃
U
U
ρ Ũ
ρ
Ũ
h
U
Ũ
µ
gh
Or after using the definition of the dimensionless parameters as
St
U
1
1 ˜2
∂Ũ
˜ Ũ
˜ P̃
U·∇
U = −Eu∇
P + 2 f⃗g +
U
+ Ũ
∇ Ũ
Re
Fr
∂t̃
(3.37)
The definition of Froude number is not consistent in the literature. In some places Fr is defined as the square of Fr = U2 /g h.
The Strouhal number is named after Vincenz Strouhal (1850–1922), who used this parameter in his study of “singing wires.” This parameter is important in unsteady, oscillating
flow problems in which the frequency of the oscillation is important.
Example 3.19: Constant Accelarated
Level: Intermediate
A device is accelerated linearly by a constant value B . Write a new N–S and continuity equations for incompressible substance in the a coordinate system attached
to the body. Using these equations developed new dimensionless equations so the
new “Froude number” will contain or “swallow” by the new acceleration. Measurement has shown that the acceleration to be constant with small sinusoidal on top the
constant such away as
f
a = B + ϵ sin
(3..c)
2π
Suggest a dimensionless parameter that will take this change into account.
Solution
Under construction
Example 3.20: Rotating Cylinder
Level: Advance
A thin (t/DL1) cylinder full with liquid is rotating. in a velocity, ω The rigid body
is brought to a stop. Assuming no secondary flows (Bernard’s cell, etc.), describe the
flow as a function of time. Utilize the ratio 1Gt/D.
d2 X µ dX
+ 2
+X = 0
dt
dt2
ℓ
Discuss the case of rapid damping, and the case of the characteristic damping
(3.20.a)
116
CHAPTER 3. DIMENSIONAL ANALYSIS
D
t
Fig. 3.3 – Rigid body brought into rest.
End of Ex. 3.20
Solution
These examples illustrate that the characteristic time of dissipation can be assessed by
∼ µ(du/d“y ′′ )2 thus given by ℓ2 /ν. Note the analogy between ts and tdiss , for which ℓ2
appears in both of them, the characteristic length, ℓ, appears as the typical die thickness.
3.7.1
The ratios of various time scales
The ratio of several time ratios can be examined for typical die casting operations. The ratio
of solidification time to the filling time
ρlm
kd
L
tf Lkd ∆TMB
Ste
∼
=
(3.38)
ts
Pr Re
ρs
klm
Uρs hsl ℓL̃
L̃
where
Re
Reynolds number
Ste
Stefan number
cp
Uℓ
νlm
∆TMB
hsl
lm
the discussion is augmented on the importance of Eq. (3.38). The ratio is extremely important
since it actually defines the required filling time.
ρlm
kd
L
Ste
tf = C
(3.39)
ρs
klm
L̃ Pr Re
At the moment, the “constant”, C, is unknown and its value has to come out from experiments. Furthermore, the “constant” is not really a constant and is a very mild function of the
3.8. SIMILARITY APPLIED TO DIE CAVITY
117
geometry. Note that this equation is also different from all the previously proposed filling
time equations, since it takes into account solidification and filling process12 .
The ratio of liquid metal conduction characteristic time to characteristic filling time is
given by
tc d UL̃2
Uℓ ν L̃2
L̃2
=
= Re Pr
∼
tf
Lα
ν α Lℓ
Lℓ
The solidification characteristic time to conduction characteristic time is given by
! cp lm
ts
ρs hsl ℓL̃αd
ρs
ℓ
1
∼
=
tc kd ∆TMB L̃2
Ste ρd
cp d
L̃
The ratio of the filling time and atomization is
ta viscosity
νℓU
ℓ
∼ 6 × 10−8
≈
= Ca
tf
σL
L
Note that ℓ, in this case, is the thickness of the gate and not of the die cavity.
ta momentum
ρℓ2 U2
ℓ
≈
= We
∼ 0.184
tf
σL
L
(3.40)
(3.41)
(3.42)
(3.43)
which means that if atomization occurs, it will be very fast compared to the filling process.
The ratio of the dissipation time to solidification time is given by
tdiss
ℓ2 kd ∆TMB
Ste
kd
ρlm
ℓ
∼
=
∼ 100
(3.44)
ts
νlm ρs hsl ℓL̃
Pr
klm
ρs
L̃
this equation yields typical values for many situations in the range of 100 indicating that the
solidification process is as fast as the dissipation. It has to be noted that when the solidification
progress, the die thickness decreases. The ratio, ℓ/L̃, reduced as well. As a result, the last
stage of the solidification can be considered as a pure conduction problem as was done by the
“English” group.
3.8
Similarity applied to Die cavity
This section is useful for those who are dealing with research on die casting and or other
casting process.
3.8.1
Governing Equations
The filling of the mold cavity can be divided into two periods. In the first period (only fluid
mechanics; minimum heat transfer/solidification) and the second period in which the solidification and dissipation occur. This discussion deals with how to conduct experiments in die
casting13 . It has to be stressed that the conditions down–stream have to be understood prior
12 In this book, this equation because of its importance is referred to as Eckert–BarMeir’s equation. If you have
good experimental work, your name can be added to this equation.
13 Only minimal time and efforts was provided how to conduct experiments on the filling of the die. In the future,
other zones and different processes will be discussed.
118
CHAPTER 3. DIMENSIONAL ANALYSIS
to the experiment with the die filling. The liquid metal velocity profile and flow pattern are
still poorly understood at this stage. However, in this discussion we will assume that they are
known or understood to same degree14 .
The governing equations are given in the preceding sections and now the boundary
conditions will be discussed. The boundary condition at the solid interface for the gas/air
and for the liquid metal are assumed to be “no–slip” condition which reads
ug = vg = wg = ulm = vlm = wlm = 0
(3.45)
where the subscript g is used to indicate the gas phase. It is noteworthy to mention that this
can be applied to the case where liquid metal is mixed with air/gas and both are touching the
surface. At the interface between the liquid metal and gas/air, the pressure jump is expressed
as
σ
≈ ∆p
r1 + r2
(3.46)
where r1 and r2 are the principal radii of the free surface curvature, and, σ, is the surface
tension between the gas and the liquid metal. The surface geometry is determined by several
factors which include the liquid movement15 instabilities etc.
Now on the difficult parts, the velocity at gate has to be determined from the pQ2 diagram or previous studies on the runner and shot sleeve. The difficulties arise due to the fact
that we cannot assign a specific constant velocity and assume only liquid flow out. It has to
be realized that due to the mixing processes in the shot sleeve and the runner (especially in
a poor design process and runner system, now commonly used in the industry), some portions at the beginning of the process have a significant part which contains air/gas. There are
several possibilities that the conditions can be prescribed. The first possibility is to describe
the pressure variation at the entrance. The second possibility is to describe the velocity variation (as a function of time). The velocity is reduced during the filling of the cavity and is a
function of the cavity geometry. The change in the velocity is sharp in the initial part of the
filling due to the change from a free jet to an immersed jet. The pressure varies also at the
entrance, however, the variations are more mild. Thus, it is a better possibility16 to consider
the pressure prescription. The simplest assumption is constant pressure
P = P0 =
1
ρ U0 2
2
(3.47)
We also assume that the air/gas obeys the ideal gas model.
ρg =
14 Again the die casting process is a parabolic process.
P
RT
(3.48)
15 Note, the liquid surface cannot be straight, for unsteady state, because it results in no pressure gradient and
therefore no movement.
16 At this only an intelligent guess is possible.
119
3.8. SIMILARITY APPLIED TO DIE CAVITY
where R is the air/gas constant and T is gas/air temperature. The previous assumption of
negligible heat transfer must be inserted and further it has to be assumed that the process is
polytropic17 . The dimensionless gas density is defined as
ρ′ =
ρ
=
ρ0
P0
P
1
n
(3.49)
The subscript 0 denotes the atmospheric condition.
The air/gas flow rate out the cavity is assumed to behave according to the model in
Chapter 9. Thus, the knowledge of the vent relative area and 4fL
D are important parameters.
For cases where the vent is well designed (vent area is near the critical area or above the
density, ρg can be determined as was done by (Bar-Meir 1995b)).
To study the controlling parameters, the equations are dimensionless–ed. The mass
conservation for the liquid metal becomes
∂ρlm ∂ρlm u ′ lm ∂ρlm v ′ lm ∂ρlm w ′ lm
+
+
+
=0
∂t ′
∂x ′
∂y ′
∂z ′
(3.50)
where x ′ = xℓ , y ′ = y/ℓ , z ′ = z/ℓ , u ′ = u/U0 , v ′ = v/U0 , w ′ = w/U0 and the
p
0
dimensionless time is defined as t ′ = tU
2P0 /ρ.
ℓ , where U0 =
Eq. (3.50) can be similar under the assumption for constant density to read
∂u ′ lm ∂v ′ lm ∂w ′ lm
+
+
=0
∂x ′
∂y ′
∂z ′
(3.51)
Please note that this simplification can be used for the gas phase. The momentum equation
for the liquid metal in the x-coordinate assuming constant density and no body forces reads
′
′
′
∂ρlm u ′ lm
′ ∂ρlm u lm
′ ∂ρlm u lm
′ ∂ρlm u lm
+
u
+
v
+
w
=
∂t ′
∂x ′
∂y ′
∂z ′
∂p ′ lm
1
−
+
∂x ′
Re
′
∂2 ulm
∂x ′ 2
+
∂2 v ′
′ 2
∂ylm
+
∂2 w ′
′ 2
∂zlm
!
(3.52)
where Re = U0 ℓ/νlm and p ′ = p/P0 .
The gas phase continuity equation reads
∂ρg′ vg′
∂ρg′ wg′
∂ρ ′ g ∂ρg′ ug′
+
+
+
=0
∂t ′
∂x ′
∂y ′
∂z ′
17 There are several possibilities, this option is chosen only to obtain the main controlling parameters.
(3.53)
120
CHAPTER 3. DIMENSIONAL ANALYSIS
The gas/air momentum equation18 is transformed into
∂ρg′ u ′ g
∂t ′
+ u′
∂ρg′ u ′ g
∂x ′
+ v′
∂ρg′ u ′ g
∂y ′
+ w′
∂ρg′ u ′ g
∂z ′
∂p ′ g νlm ρg0 1
−
+
∂x ′
νg ρlm Re
|
=
∂2 ug′
∂x ′ 2
{z
+
∂2 vg′
∂y ′ 2
+
∂2 wg′
∂z ′ 2
∼0
!
(3.54)
}
Note that in this equation, additional terms were added, (νlm /νg )(ρg0 /ρlm ).
The “no-slip” conditions are converted to:
′
′
′
ug′ = vg′ = wg′ = ulm
= vlm
= wlm
=0
(3.55)
The surface between the liquid metal and the air satisfy
1
p ′ r1′ + r2′ =
We
where the p ′ , r1′ , and r2′ /ℓ are defined as r ′ 1 = r1 r2′ = r2 /ℓ
The solution to equations has the form of
ρg νlm
A 4fL
,
, D , n,
u′ =
fu x ′ , y ′ , z ′ , Re, We,
Ac
ρlm νg
ρg νlm
A
, 4fL
,
n
,
,
v′ =
fv x ′ , y ′ , z ′ , Re, We,
Ac D
ρlm νg
ρg νlm
A 4fL
′
′
′ ′
w =
fw x , y , z , Re, We,
, n,
,
,
Ac D
ρlm νg
ρg νlm
A 4fL
p′ =
fp x ′ , y ′ , z ′ , Re, We,
, D , n,
,
Ac
ρlm νg
(3.56)
(3.57)
If it will be found that equation Eq. (3.54) can be approximated19 by
′
′
′
∂p ′ g
∂u ′ g
′ ∂u g
′ ∂u g
′ ∂u g
+
u
+
v
+
w
≈
−
∂t ′
∂x ′
∂y ′
∂z ′
∂x ′
then the solution is reduced to
u′ =
v′ =
w′ =
p′ =
A 4fL
fu x ′ , y ′ , z ′ , Re, We,
, D ,n
Ac
A 4fL
fv x ′ , y ′ , z ′ , Re, We,
, D ,n
Ac
A 4fL
′
′ ′
fw x , y , z , Re, We,
,
,n
Ac D
A 4fL
fp x ′ , y ′ , z ′ , Re, We,
, D ,n
Ac
(3.58)
(3.59)
18 In writing this equation, it is assumed that viscosity of the air is independent of pressure and temperature.
19 This topic is controversial in the area of two phase flow.
3.9. SUMMARY OF DIMENSIONLESS NUMBERS
121
At this stage, it is not known if it is the case and if it has to come out from the experiments.
The density ratio can play a role because two phase flow characteristic is a major part of the
filling process.
3.8.2
Design of Experiments
Under Construction
3.9
Summary of dimensionless numbers
This section summarizes all the major dimensionless parameters and what effects they have
on the die casting process.
Reynolds number
Re =
ρU2 /ℓ
internal Forces
=
viscous forces
νU/ℓ2
Reynolds number represents the ratio of the momentum forces to the viscous forces. In die
casting, Reynolds number plays a significant role which determines the flow pattern in the
runner and the vent system. The discharge coefficient, CD , is used in the pQ2 diagram is
determined largely by the Re number through the value of friction coefficient, f, inside the
runner.
Eckert number
Ec =
1/2ρ U2
inertial energy
=
1/2ρcp ∆T
thermal energy
Eckert number determines if the role of the momentum energy transferred to thermal energy
is significant.
Brinkman number
Br =
µU2 /ℓ2
heat production by viscous dissipation
=
heat transfer transport by conduction
k∆T/ℓ2
Brinkman number is a measure of the importance of the viscous heating relative the conductive heat transfer. This number is important in cases where large velocity change occurs
over short distances such as lubricant flow (perhaps, the flow in the gate). In die casting, this
number has small values indicating that practically the viscous heating is not important.
Mach number
Ma = q
U
γ∂p
∂ρ
122
CHAPTER 3. DIMENSIONAL ANALYSIS
For ideal gas (good assumption for the mixture of the gas leaving the cavity). It becomes
∼ √ U = characteristic velocity
M=
gas sound velocity
γRT
Mach number determines the characteristic of flow in the vent system where the air/gas velocity is reaching to the speed of sound. The air is chocked at the vent exit and in some cases
other locations as well for vacuum venting. In atmospheric venting the flow is not chocked for
large portion of the process. Moreover, the flow, in well design vent system, is not chocked.
Yet the air velocity is large enough so that the Mach number has to be taken into account for
reasonable calculation of the CD .
Ozer number
CD 2 Pmax
ρ
2
Qmax
A3
Oz = =
A3
Qmax
2
CD 2
Pmax
effective static pressure energy
=
ρ
average kinematic energy
One of the most important number in the pQ2 diagram calculation is Ozer number. This
number represents how good the runner is designed.
Froude number
Fr =
ρU2 /ℓ
inertial forces
=
ρg
gravity forces
Fr number represent the ratio of the gravity forces to the momentum forces. It is very im-
portant in determining the critical . slow plunger velocity This number is determined by. the
height of the liquid metal in the shot sleeve . The Froude number does not play a significant
role in the filling of the cavity.
Capillary number
Ca =
ρU2 /ℓ
inertial forces
=
ρg
gravity forces
Capillary number (Ca) determine when the flow during the filling of the cavity is atomized or is continuous flow (for relatively low Re number).
Weber number
We =
1/2ρU2
inertial forces
=
1/2σ/ℓ
surface forces
We number is the other parameter that govern the flow pattern in the die. The flow in die
casting is atomized and, therefore, We with combinations of the gate design also determine
the drops sizes and distribution.
123
3.10. SUMMARY
Critical vent area
Ac =
V(0)
ctmax mmax
The critical area is the area for which the air/gas is well vented.
3.10
Summary
The dimensional analysis demonstrates that the fluid mechanics process, such as the filling of
the cavity with liquid metal and evacuation/extraction of the air from the mold, can be dealt
with when heat transfer is neglected. This provides an excellent opportunity for simple models to predict many parameters in the die casting process. It is recommended for interested
readers to read Eckert’s book “Analysis of Heat and Mass transfer” to have better and more
general understanding of this topic.
3.11
Questions
Under construction
124
CHAPTER 3. DIMENSIONAL ANALYSIS
4
The Die Casting Process Stages
4.1
Introduction
This chapter deals with the various parts of the die casting processes. These processes are
examined and their physical explained. At this stage, no discussion offered on topics such the
die heat treatment. The die casting processed can be broken into many separated processes
which are controlled by different parameters. The simplest division of the process for a cold
chamber is the following: 1) filling the shot sleeve, 2) slow plunger velocity, 3) filling the runner
system 4) filling the cavity and overflows, and 5) solidification process (also referred as intensification process). This division into such sub–processes results in a clear picture on each
process. On one hand, in processes 1 to 3, it is desirable to have a minimum heat transfer/solidification to take place for obvious reasons. On the other hand, in the rest of the processes,
the solidification is the major concern. Hence for the filling of the cavity also known as the
filling time is important in avoiding defects in the process and it will discuss down.
In die casting, the information and conditions do not travel upstream. For example,
the turbulence does not travel from some point at the cavity to the runner and of–course,
to the shot sleeve. This kind of relationship is customarily denoted as a parabolic process
(because in mathematics the differential equations describe these kind of cases as parabolic or
hyperbolic). To a very larger extent it is true in die casting. The pressure in the cavity does not
affect the flow in the sleeve or the runner if the vent system is well designed. This point has to
be emphasized. (Crowley 2021) and his advisor Domblesky build a model which basically the
vent system is not existence and the pressure in cavity is irrelevant. This absurd model was
125
126
CHAPTER 4. THE DIE CASTING PROCESS STAGES
build in 2022!1 In other words, the design of the pQ2 diagram is not controlled by down–
stream conditions. Another example, the critical slow plunger velocity is not affected by the
air/gas flow/pressure in the cavity in the initial part (up to filling the runner). In general, the
turbulence generated down–stream does not travel up–stream in this process. One has to
restrict this characterization to some points. One point is particularly mentioned here: the
poor design of the vent system affects the pressure in the cavity (see Domblesky’s work again)
and therefore the effects do travel down stream. For example, the pQ2 diagram calculations
are affected by a poor vent system design.
4.1.1
Filling The Shot Sleeve
In
st
ab
ili
tie
s
The flow from the ladle to the shot sleeve did not receive much attention in the die casting
research because it is believed that the flow
does play an insignificant role. The importance of the understanding of this process
shows how to minimize the heat transfer,
the layer created on the sleeve (solidification
layer), and sleeve protection from; a) erosion
b) plunger problem. After the hydraulic jump
Hydralic
Jump
period there is a jet entrance into large body
problem. In this range a gas boundary layer
attached itself to the jet and create mixing
in the liquid metals. The jet itself has no
smooth surface and two kinds of instability ocBubles
curs. The first instability is of Bernoulli’s efFig. 4.1 – Hydraulic jump in the shot sleeve.
fect and second effect is Bar-Meir’s instability
that boundary conditions cannot be satisfied
for two phase flow. Yet, for die casting process, these two effects (see Fig. 4.1) do not change
the global flow in the sleeve. At first, the hydraulic jump is created when the liquid metal
enters the sleeve. The typical time scale for hydraulic jump creation is almost instant and
extremely short as can be shown by the characteristic methods. As the liquid metal level in
the sleeve rises, the location of the jump moves closer to the impinging center. At a certain
point, the liquid depth level is over the critical depth level and the hydraulic jump disappears.
The critical depth depends on the liquid properties and the ratio of impinging momentum or
velocity to the hydraulic static pressure. The impinging momentum impact is proportional to
(ρ U2 π r2 ) and hydraulic “pressure” is proportional to (ρ g h 2 π r h). Where r is the radius
of the impinging jet and h is the height of the liquid metal in the sleeve. The above statement
1 These are academic descendants of the Ohio State University so one should not be supersized from this quality
of work.
127
4.1. INTRODUCTION
leads to
Ucritical ∝
r
g h2
r
(4.1)
The critical velocity on the other hand has to be
(4.2)
Uc ritical = g hL
where hL is the distance of the ladle to the height of the liquid metal in the sleeve. The height
where the hydraulic shock will not exist is
p
hcritical ∼ r hL
(4.3)
ies
ilit
tab
I ns
This analysis suggests that decreasing the ladle height and/or reducing less mass flow rate
(the radius of the jet) result in small critical height. The air entrainment during that time
will be discussed in the book “Basic of Fluid Mechanics” in the Multi–Phase flow chapter. At
this stage, air bubbles are entrapped in the liquid metal which augment the heat transfer. At
present, there is an extremely limited knowledge about the heat transfer during this part
of the process, and of course less about how
to minimize it. However, this analysis suggests
that minimizing the ladle height is one of the
HH
ways to reduce it. The heat transfer from liquid metal to the surroundings is affected by the
velocity and the flow patterns since the mechanism of heat transfer is changed from a domiBubles
nated natural convection to a dominated force
Fig. 4.2 – Filling of the shot sleeve with liquid
convection. In addition, the liquid metal jet
metal.
surface is also affected by heat transfer to some
degree by change in the properties.
4.1.1.0.1 Heat Transferred to the Jet
The estimate on heat transfer requires some information on jet dynamics. There are two
effects that must be addressed; one the average radius and the fluctuation of the radius. As
first approximation, the average jet radius changes due to the velocity change. For laminar
√
flow, (for simplicity assume plug flow) the velocity function is ∼ x where x is the distance
from the ladle. For constant flow rate, neglecting the change of density, the radius will change
√
as r ∼ 1/ 4 x. Note that this relationship is not valid when it is very near the ladle proximity
(r/x ∼ 0). The heat transfer increases as a function of x for these two reasons.
The second effect is jet radius fluctuations. Consider this, the jet leaves the ladle in
a plug flow. Due to air friction, the shear stress changes the velocity profile to parabolic.
1 Very few papers (∼
0) can be found dealing with this aspect.
1 Some elementary estimates of fluid mechanics and heat transfer were made by the author and hopefully will be
added to this book.
128
CHAPTER 4. THE DIE CASTING PROCESS STAGES
For simple assumption of steady state(it is not steady state), the momentum equation which
governs the liquid metal is
assume 0 constant
z}|{
z }| {
 ∂uz
∂uz
∂uz 
1 ∂


r
ρ
+ ur
=µ
∂t
∂r 
r ∂r
∂r
(4.4)
Eq. (4.4) is a simplified equation form for the gas and liquid phases. Thus, there are two equations that needs to be satisfied simultaneously; one for the gas side and one for the liquid side.
Even neglecting several terms for this discussion, it clear that both equations are second order
differential equations which have different boundary conditions. Any second order differential equation requires two different boundary conditions. Requirement to satisfy additional
boundary condition can be achieved. Thus from physical point of view, second order differential equation which needs to satisfy three boundary conditions is not possible, Thus there
must be some wrong either with the governing equation or with the boundary conditions.
In this case, the two governing equations must satisfy five (5) different boundary conditions.
These boundary conditions are as follows: 1) symmetry at r = 0, 2) identical liquid metal and
air velocities at the interface, 3) identical shear stress at the interface, 4) zero velocity at infinity for the air, and 5) zero shear stress for the air at the infinity. These requirements cannot
be satisfied if the interface between the liquid metal and the gas is a straight line.
The heat transfer to the sleeve in the impinging area is significant but at present only
very limited knowledge is available due to complexity.
4.1.2
Plunger Slow Moving Part
4.1.2.1
Fluid Mechanics
Before the plunger starts pushing the liquid metal there should be a quieting time in which
gases have the time to escape. Or in different words, the gases entrapped during the jet enters
into the liquid with the “help” of boundary (no–slip condition) into the liquid pool. Fig. 4.3
depicts the water jet entering into water medium from the air. At the entrance the liquid
presents increase of the height of the water while in the same time visible depression for the
air is shown. The depression is in the periphery of the jet which the air attached to the jet
entering to water medium. This “boundary” layer is breaking down as bubbles and the air
“climbing” out in a stokes flow.
129
4.1. INTRODUCTION
Fig. 4.3 – Jet dragging air into the liquid medium in water analogy. the bubbles are very large as they “given”
the opportunity to expand and escape. In the reflection shows that expansion of the jet. The quieting time
is related to the gravity.
The main point is the estimate for energy dissipation. The dissipation is proportional to
2
µU L. Where the strange velocity, U is averaged kinetic velocity provided by jet. This kinetic
energy is at most the same as potential energy
Shot
of liquid metal in the ladle. The potential enSleeve
ergy in the ladle is H > m g where H is averheat transfer
to the air
aged height see Fig. 4.2.
process 1
p The averaged velocity
y
in the shot sleeve is 2 g H The rate energy
2
dissipation can be estimated as µ U
πRL
R
T
as L is the length of shot sleeve. The shear
heat transfer
stress is assumed to occur equally in the volto the sleeve
Solidificaiton
process 2
layer
ume of liquid metal in the sleeve. This assumpFig. 4.4 – Heat transfer processes in the shot
tion of shear stress grossly under estimates the
sleeve.
dissipation. The actual dissipation is larger due
to the larger velocity gradients.
4.1.2.1.1
Heat Transfer
In this section, the solidification effects are examined. One of the assumptions in the analysis
of the critical slow plunger velocity is that the solidification process does not play an important role (see Fig. 4.4). The typical time for heat to penetrate a typical layer in air/gas phase is
in the order of minutes. Moreover, the density of the air/gas is 3 order magnitude smaller than
liquid metal. Hence, most of the resistance to heat transfer is in the gas phase. Additionally,
it has been shown that the liquid metal surface is continuously replaced by slabs of material
below the surface which is known in scientific literature as the renewal surface theory. Thus,
the main heat transfer mechanism is through the liquid metal to the sleeve. The heat transfer
rate for a very thin solidified layer can be approximated as
Q ∼ klm
∆T
π r L ∼ Ls π r L t ρ
r
(4.5)
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CHAPTER 4. THE DIE CASTING PROCESS STAGES
Where Ls is the latent heat, klm is the thermal conductivity of liquid metal and t is the thickness of the solidification layer. Equation (4.6) results in
t
k ∆T
∝ lm 2
r
Ls r ρ
The value for this die casting process in minutes is in the range of 0.01-0.001 after the thickness reaches to 1-2 [mm]. The relative thickness
further decreases as the inverse of the square
solidified layer increases. If the solidification
is less than one percent of the radius, the speed
will be very small compared to the speed of the
plunger. If the solidification occur as a mushy
zone then the heat transfer is reduced further
and it is even lower than this estimate and
ℓ
R ≪ 1). Therefore, the heat transfer from the
liquid metal surface to the air, as shown in Figure 4.4 (mark as process 1), acts as an insulator
to the liquid metal.
(4.6)
T emperature
y
(Liquid)
Liquid metal
δ (Solid)
`
Shot Sleeve (metal)
Insulation
Fig. 4.5 – Solidification process in the shot sleeve
time estimates. Note the temperature profile
at the steel should close to ln while the solid
part of liquid metal should be closer to a linear. Sorry E. Sparrow for the poor drawing
skill.
The governing equation in the sleeve is
∂T
ρd c p d
= kd
∂t
∂2 T
∂y2
(4.7)
where the subscript d denotes the properties of the sleeve material.
Boundary condition between the sleeve and the air/gas is
∂T
∂n
=0
(4.8)
y=0
Where n represents the perpendicular direction to the die. Boundary conditions between the
liquid metal (solid) and sleeve
∂T
∂T
ksteel
= kAL
(4.9)
∂y y=l
∂y y=l
The governing equation for the liquid metal (solid phase)
2 ∂T
∂ T
ρlm cp lm
= klm
∂t
∂y2
(4.10)
where lm denotes the properties of the liquid metal. The dissipation and the velocity are
neglected due to the change of density and natural convection.
Boundary condition between the phases of the liquid metal is given by
∂(Tl − Ts )
∂T
∂T
− ks
∼ k
(4.11)
vs ρs hsf = kl
∂y y=l+δ
∂y y=l+δ
∂y
y=l+δ
131
4.1. INTRODUCTION
hsf
ρs
vn
k
the heat of solidification
liquid metal density at the solid phase
velocity of the liquid/solid interface
conductivity
Neglecting the natural convection and density change, the governing equation in the
liquid phase is
ρl cp l
∂T
∂2 T
= kl 2
∂t
∂y
(4.12)
The dissipation function can be assumed to be negligible in this case.
There are three different periods in heat transfer;
1. filling the shot sleeve
2. during the quieting time, and
3. during the plunger movement.
In the first period, heat transfer is relatively very large (major solidification). At present, there
is not much known about the fluid mechanics not to say much about the solidification process/heat transfer in fluid mechanics. The second period can be simplified and analyzed as
known initial velocity profile. A simplified assumption can be made considering the fact that
Pr number is very small (large thermal boundary layer compared to fluid mechanics boundary layer). Additionally, it can be assumed that the natural convection effects are marginal. In
the last period, the heat transfer is composed from two zones: 1) behind the jump and 2) ahead
of the jump. The heat transfer ahead of the jump is the same as in the second period; while
the heat transfer behind the jump is like heat transfer into a plug flow for low Pr number. The
heat transfer in such cases has been studied in the past2 .
4.1.3
Runner System
The prevailing or even the exclusive opinion in die casting is that the flow enters into runner
as a sharp interface between the liquid metal and air (gases). In reality, the liquid metal flow
front in the runner has at least two distinct flow reigns. Oppose to the common research belief
which plagues the die casting industry, the flow flows in the runner as a liquid metal with gases
(mostly air) and later the flow changes to only liquid flow. For this discussion, it is assumed
that the flow reaches to runner entrance with only liquid metal. This assumption of course
have a perquisite that a reasonable engineers who did not subscribed to the nonsense spread
by NADCA or the hundred “scientific” papers on this problem, and know how to calculate
the critical plunger velocity using Bar–Meir’s equations. With this critical velocity, the flow
rate in the runner can be estimated. The mass is considered to be at semi steady state as
Urunner =
Ucritical Ashot sleeve
Arunner
2 The reader can refer, for example, to the book “Heat and Mass Transfer” by Eckert and Drake.
(4.13)
132
CHAPTER 4. THE DIE CASTING PROCESS STAGES
The typical velocity in the shot sleeve is in the range of 0.5 to about 3 [m/sec]. Typical area ratio
(shot sleeve to runner) is 10-50 and thus the gas (air) velocity in the runner can reach in hundred
of meters a second. While the flow is not necessarily choked, yet the compressibility affects
the flow always. In the low end, the compressibility has a minimal effect (unlikely). Currently
the knowledge on the effect of compressibility on the mixing is very limited. Nevertheless,
when gas (air) is pushed even with or without sharp transition a shock is created. This shock
moves downstream to mold and increases the pressure and temperature downstream (in front
the transition). The following example is provided to demonstrate the effects to the shock.
Example 4.1: The Shock Speed in Runner
Level: Advance
The plunger diameter is 4[in] and with correct critical velocity of 0.5[m/s]. The runner
cross section area is 0.5[in2 ]. The temperature in the runner head the liquid movement is 27◦ C (before the shock). What is the shock velocity a head of the liquid metal
front? What is the temperature downstream of the shock? If the pressure in runner
upstream the shock is atmospheric what is the pressure in front the mixing zone?
Solution
This situation of moving shock where condition downstream are provided. The velocity down
stream (front) can be calculated using Eq. (4.13).
Urunner =
h m i
0.5 × π 22
∼ 12.56
0 .5
sec
(4.1.a)
This kind of problem deals with shock dynamic because the shock is moving which was solved
by this author (Bar-Meir 2021b, section 6.3). In the traditional books there is no solution for
this kind of problem. The solution this kind problem based on the relationship between the
sides of shock. The movement of the shock is not affected by mixing which will be discussed
below. The summary of the solution is presented. In the moving coordinate the velocity of the
shock is zero (by definition). The velocity after the shock is 12.56[m/sec] and it is denoted Uy′ .
The subscribe y downstream the shock and x upstream the shock. Here, the prime “”’ denotes
the values of the static coordinates. The pressure is not affected by the moving coordinates
hence
(4.1.b)
Px = Px′ Py = Py′
The same can be said for temperature
Tx = Tx′
Ty = Ty′
(4.1.c)
The velocity measured by the observer moving with shock is
Ux = Us − Ux′
(4.1.d)
Where Us is the shock velocity. The “downstream” velocity is
Uy′ = Us − Uy
(4.1.e)
The upstream prime Mach number can be defined as
Mx′ =
Us
Us − Ux
=
− Mx = Msx − Mx
cx
cx
(4.1.f)
133
4.1. INTRODUCTION
End of Ex. 4.1
Additional definition was introduced for the shock upstream Mach number Msx . The downstream prime Mach number can be expressed as
My′ =
Us − Uy
Us
=
− My = Msy − My
cy
cy
(4.1.g)
The relationship between the two new shock Mach numbers is
cy Us
Us
=
−−→ Msx =
cx
cx cy
s
Ty
Msy
Tx
(4.1.h)
This derivation plus considerable manipulation shown in a great details given in (Bar-Meir
2021b) leads to
r
′
(k + 1)Uy +
Msx =
h
i2
′
Uy (1 + k) + 16cx 2
(4.1.i)
4 cx
The speed of sound in front the shock is
cx =
h m i
p
√
k R Tx = 1.4 × 287 × 300 = 347.1
sec
(4.1.j)
Substituting the values into equation provides
Msx =
(1.4 + 1) × 12.5 +
q
(2.4 × 12.5)2 + 16 × 347.12
4 × 347.1
∼ 1.021
(4.1.k)
with the values of upstream parameters the downstream Mach number and the temperature
ratio can be obtained. According (Bar-Meir 2021d, moving shocks section) the temperature
increase by 1.5% while the pressure increase by about 6%. It is suggested to read the section
on shock dynamics in this author book on compressible flow. For those who do not take the
interested in compressible flow it is sufficient to know that shock wave appear a head to the
gas–liquid front.
Ex. 4.1 demonstrated that the liquid metal pushing the gases creates some effects on the
pressure and temperature in the runner. However the most significant part is the mixing processes that occurs in the runner. For example, Andritsos et al (1987) experimentally observed
three types of instabilities which they categorized them as regular 2–D waves are associated
with pressure variations in phase with the wave slope, irregular large–amplitude waves and
atomization of the liquid is associated with pressure variations in phase with the wave height.
Shevtsova et al(2013) studied several effects numerically and suggested the instabilities affected
by the temperature. In a typical numerical simulation carried for the die casting simulation,
the flow is assumed to be with a sharp transition and with boundary condition of “no–slip”.
The no–slip condition means here that the velocity at the wall is zero since the wall does not
move and or the velocity of gas–liquid interface is the same for the gas and liquid. Here the
instabilities due to the temperature are assumed to be insignificant. The focus is on the hydrodynamics and hence the thermal instabilities are ignored for this analysis. Analysis not
presented here shows that the thermal driving force for the instabilities is much smaller as
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CHAPTER 4. THE DIE CASTING PROCESS STAGES
compared to other forces. Additionally, the decoupling seem to be justified because they not
directly related or influence each other.
A sharp transition between the liquid metal and gas (air) with the no–slip condition
is not physically attendable in the same time. This explanation provides an idea of how the
interface appears in reality. Beside the flow in the runner this description explains why similar
phenomena occur.
Example 4.2: Heat Transfer vs Mixing
Level: Advance
Opening the valve in the shower or faucet the transition from cold to hot water never
occur in a sharp transition. Actuality there is a transition period in which temperature increases gradually. There two reasonable explanations one: some of the heat
initially is transferred to heat the pipe material, two: the mixing occurs because the
flow is not a plug flow. For a relatively short pipe (the die casting case) which which
mechanism is more dominate.
Solution
Poor man analysis for the heat transfer relates that the amount of energy in system to the area
area it leave the system. The typical analysis include convection coefficient pipe properties.
The relationship is between heat transfer to mixing the
L∝
D2 h
=D
Dh
(4.2.a)
If the length is only larger than 10 the heat transfer play a significant role. Hence,the mixing
process are the dominate effect.
Assume that the liquid velocity profile in
the runner is a parabolic (similar analysis
can be done for other profiles with exception of plug flow). The liquid metal is
t3
p1
p2 t 2 p3
considered incompressible material for this
t1
analysis. The parabolic profile is the same
in any cross section upstream the interface
for the runner. In the same time, it can
Fig. 4.6 – Interface instability interface of liquid
into gas exhibits new location at three difbe observed that with the exception of the
ferent time. Flow should be vertical and not
plug flow, the material does not move as
horizontal as depicted in this figure.
a unit. The material on any cross section
is separated immediately when it moves to
the “next” cross section. Thus, the “neighbors” always are changing during the flow. For example, the point on centerline has a different velocity than the neighboring points thus move
to different locations. The centerline “meets” the fluid from the downstream cross section.
Yet, the velocity profile remains the same for every cross section up to the mixing zone.
Fig. 4.6 displays the interface at three different times if the interface starts from a sharp
transition. The lines t1 , t2 and t3 represents the actual interface at different times. While the
135
4.1. INTRODUCTION
lines p1 , p2 , and p3 represents the cross sections moving with the averaged velocity at the
same time difference. Notice that t1 and p1 are on the same line and at the same time the cross
section is the sharp transition from liquid (metal) to gas (air). After some time, the interface
moves to t2 line. After the same time difference, the line p2 will be at a different location as
exhibited in Fig. 4.6. If the flow was continuous then the velocity profile should be the same as
at p1 . However, the cross section at p2 is not continuous as shown in the Fig. 4.6. As oppose
to continuous flow, the cross section at p2 is made of the liquid on the inside and the gas
outside. The contradiction is created by the fact of coexist of the gas and liquid at p2 cross
section with initial assumption of identical velocity profiles. If the flow is one dimensional
(continuous) then the parabolic profile cannot coexist with no–slip boundary conditions.
In this context, the mixing zone is defined as the zone where the liquid (metal) and gas
are coexist. This zone starts from p1 or t1 plus ϵ to the point where liquid front reaches.
This zone has a fix cross border (section) on the left while on the right the border moves with
the liquid front velocity. The mixing zone expands by almost double the liquid volume enters
into the zone due to the gas volume absorbed in the same time.
Example 4.3: Mixing Zone Time
Level: Intermediate
In Ex. 4.1 description of a typical die casting machine was given. For that machine
estimate the time mixing zone reaches to the gate if the runner length is 0.4 [m]?
Solution
The liquid velocity profile is affected by the gas (air) present in the runner. This velocity is
affected as the scratching occurs because the boundaries are not solid but gaseous (smaller
shear stress). As first approximation, the parabolic profile is assumed to be the same as in the
initial/entrance point. The speed of the front tip (in the middle the conduit) is approximately
twice the averaged velocity which is about 12.1. Hence the maximum velocity (of the tip) at
the mixing zone is 24.2[m/sec]. The time is
t=
distance
0 .4
=
= 0.016[sec]
U
24.2
(4.3.a)
The actual duration or the length of the mixing zone in the runner is depends on the runner
geometry and the instabilities created for that shape.
The gas portion of the cross section decreases with the direction of the flow. A limiting
case, when the gas (pushed material) is actually the same material as the pushing material (in
this case liquid metal). In that limiting case every cross section has the same flow rate. Even
though the composition of the cross section from the pushed and pushing material changes
every cross section. It can be thought as the pushed materail to be green and pushing material
to be red for example. Outside the mixing zone, the flow is constant but made of only pushing
material at upstream and the only pushed material downstream. Beside the pressure variations, a shock wave moves ahead the interface a upstream the mixing zone. After the shock,
the gas (air) is compressed and the temperature increases slightly (see the example Ex. 4.3).
The pressure at p1 is larger than the pressure at p2 which is larger than pressure p3 . This
136
CHAPTER 4. THE DIE CASTING PROCESS STAGES
variation creats additional small shock waves which increase the instability of the flow.
There are several possibilities or limited cases for the situation. On one hand, the liquid
continues as if the gas does not exist and flows in a parabolic velocity profile. This assumption
imposes conditions on the gas flow via the boundary conditions (through the sides of conduit).
On the other hand, the situation can be assumed that in every cross section the flow rate of
both liquid and gas remain constant as if a single material flows. This assumption imposes
conditions on the velocity profiles. In both cases the no–slip hypothesis is imposed on every
boundary.
The first limit requires to find the location of the front and then compute the velocity
profile of the gas and flow rate. It can be noticed that the liquid cross section area decreases
with the increase of the x coordinate. Naturally, the gas portion of the cross section area, on
the other hand, increases with the flow direction. Notice the maximum velocity is different
from the averaged velocity (for parabolic profile Uave ∼ 2 Umax /3 also note the flow is not
fully parabolic). The liquid averaged velocity at moving coordinates is
Z
1
ULave =
UL dA
(4.14)
A A
where ULave is the liquid averaged velocity. This averaged velocity is defined similarly for
the gas.
The situation suggests that gas “flows” into liquid in the moving coordinates (the averaged velocity). The experimental observations shows that the liquid interface indeed increases
the speed with some gas bubbles in certain ranges. Furthermore, how long the transition zone
has limitedly studied and of course the length is not settled. In this case, the flow is opposed
to the typical two phase flow where the liquid on the inside and the gas is on outside. The
common observation for multiphase flow of gas liquid is that of the reverse (Bar-Meir 2021a,
Multiphase Chapter) the liquid on the outside and gas on the inside. The current situation is
not a real semi–state two phase flow. This reversal occurs because the transitional effects even
though this situation appears as a pseudo steady state. Notice as oppose to “regular” steady
state every cross section here has a different velocity profile.
The personal observation of this author suggests that this mixing zone range is about 2
to 14 times the pipe (conduit) diameter or equivalent and depends on the speed of the plunger
(similar on Re number based on the averaged velocity). It has to be emphasized that this
statement is not conclusive and it is not clear under what exact conditions. Yet, this statement
should be treated better than educated guess. All the numerical works in this die casting
area assumed a sharp interface and are simply wrong (it is physically not possible). In the
assumption of sharp interface, the models missed on the extra energy needed, entrapment,
and location of the liquid front.
Here, the gas flows against the pressure gradient in the moving coordinates with the
average liquid velocity in stationary coordinates. In the gas side, this pressure increase in
turn increases the gas density and hence reduces the gas velocity. The point has to be made
that the pressure in the gas phase increase by two different mechanisms, shock wave and the
pressure “transiting side way” from liquid side. This effect is ignored for time being because
it is relatively secondary. Later, this topic will be discussed in connection to the length of
137
4.1. INTRODUCTION
the mixing zone. The no–slip condition at the interface has to be maintained between the
liquid and gas and gas to the sold wall. The gas (air) acts as a “lubricant” for the flow because
it reduces the resistance at the wall.
Example 4.4: Liquid Acceleration
Level: Intermediate
Show that the liquid metals front during the flow in the runner is going thorough an
acceleration process.
Solution
The simplistic approach is to look at the control volume. In this control volume, the pressure
difference between the liquid and gas is the constant. In other words, the pressure difference
between p1 and the tip cross section is the same at all the time. In reality this pressure difference increases with the time because the mixing zone size increases. The pressure difference
in the direction of the flow is assume to larger than the pressure in perpendicular to the flow.
This assumption is common and realistic in this case.
In this case the driving forces are constant (or increasing) the question now focus on
resistance to the flow. For simplicity it is assumed that wall liquid metal covered by the gas
(air). As approximation, the viscosity of gas is approximated by the air viscosity. Newton’s law
applied to control volume results in
m
}| {
V
z }| {
Wa
∆P 2 d Z
W − τ LZ
W = ρ 2 d LZ
z
(4.4.a)
Notice that the liquid metal mass is over estimated. The shear stress is the standard formula as
τi = µi
dU
dy
(4.4.b)
where i is either liquid metal (to be denoted as L(iquid)) or the gas (to be denoted as G(as)).
It can be noticed that at the limited case where the pushed and pushing material are the same
analysis is simplified and later the deviation can be shown to further enhance the conclusion.
The typical velue of the viscosity of aluminum is about µ = 1.0 − 1.4 MPa · s (Dinsdale and
Quested 2004). The air viscosity is 1.467 × 10−5 Pa · s. For practical purpose the air does not
provide any resistence to the flow and the liquid metal will accelated to the point were liquid
metal could be atomized. It hard to estimate the value the pressure difference because the gas
(air) increases through serveral small shock until it reach the presure of liquid metals (minus
the surface tension effect).
138
CHAPTER 4. THE DIE CASTING PROCESS STAGES
The constant flow rate for the liquid
metal on p1 suggest that the location of the
“old” interface t2 after sometime, t, is elapsed
(separation between the gas to the liquid).
t2 = UL t
2d
y0
y
(4.15)
x0
In parallel after some time p2 will be some distance from the original p1 line. This line of the
cross section p2 depends on the time elapsed
and is denoted here as x ′ and expressed as
x
L
y0G
ξ
Fig. 4.7 – 2–D of Velocity Profile to explain the
nomenclature. Not to scale.
x ′ = Uave t
(4.16)
Note that x is the coordinate while x ′ is the distance. Also note that the averaged velocity is
the based on the velocity in p1 which is different from the averaged velocity at any other point
in mixing zone. To simplify the calculations, the conduit is assumed to be 2–D shape and the
flow is fully developed (not correct but it is approximation). The initial velocity profile as
function (y/d) (at p1 ) is
y
y 2 UL
= Umax 1 −
(4.17)
d
d
Where Umax is the maximum velocity. If the liquid continues to flow in the entire cross
section, the distance t2 is then
y 2 t2 = Umax 1 −
t
(4.18)
d
The point, (p2 ), is located at the new location when the old front (p1 ) moves at the average
velocity in a stationary coordinates after time t is elapsed. The average velocity is 2 Umax /3
for parabolic velocity.
The averaged velocity is based on the velocity when the flow rate at p1 is divided by the
entire cross section area. The intersection of the lines p2 and t2 point(s) is border between
flows into the liquid” refers to the fact
the area of the gas flows into the liquid. The term “flows
that in the moving coordinates the gas flows and the gas is not stationary. The calculations of
the gas flows do not require control volume because liquid profile is predetermined by this
assumption. The intersection of p2 with t2 is at the same time, t, hence
y 2 tC
(4.19)
Uave tC = Umax 1 −
d
The relationship between the Umax and Uave can be quickly estimated by looking at the
integration of velocity profile (q = constant).
Z1
Uave = Umax (1 − η2 )dη
(4.20)
0
139
4.1. INTRODUCTION
In writting Eq. (4.20) several substituions were made which include η = y/d and noticing
that η = 1 when y = d. Also notie that LHS (left hand side) the integraiton was carreid over
distance 0 −→ d which results in Uave = 2/3 Umax . Substituting into Eq. (4.20) reads
XX X
2X
Umax
XX (1 − η2 ) −−−→ η =
= Umax
3
r
1
3
(4.21)
There are several observations can be pointed out from this results Eq. (4.21). The flow rate is
reduced for the parabolic velocity profile. If the flow was plug flow the flow rate was maintained and velocity at p2 was the same as at p1 . The fraction of liquid according to this
p
assumption is constant
1/3 . The flow rate cannot be indentical to the flow rate at p1 or
as if the flow was a plug flow because some of the liquid material exited the conrol system
(between the p1 and p2 ). The amount liquid that leaves the volume between the p1 and p2 is
Z 1/3
qleaving = d
0


Uave
z }|
{


Umax 1 − η2 − 2 Umax  dη =


3
Z 1/3 2
Umax d 2
Umax d
1−η −
dη =
η − η3
3
3
0
−−−→ qleaving =
1/3
0
8 Umax d
81
(4.22)
According to this picture, about 10% of the liquid metal leave the “control volume” (between p1 and p2 ). While this estimate is interesting, the fact suggests that other things occurs.
This scenario breaks after a while due to Kelvin Helmholtz instabilities plus additional issues
such acceleration of the leaving liquid metal material. A considerable amount of material
was further developed here but now will be transferred to section dealing with filling and
evacuation pipes and only abbreviated version will be presented.
To ascertain how much air/gas entrapped due to this instability, consider that some experiments show that for pipe flow air flows back about 5% of the exterior radius (in somewhat
resemble the situation). Hence, as much as 10% of the volume of the air could be entrapped
during process of the filling the runner. While this number is only estimate, the actual value
depends on the liquid metal velocity profile and could increase by as much 10%. In same time
notice that the fact that the air is flowing on periphery reduces the resistance to flow can be as
much as 90% in the relevant area. None of the simulations of flow into the die in the literature
even considered these important facts. The flashing out material has to be about at least 10%
to remove the entrapped gas. This instability also provide a hint to the design of the runner.
The typical runner is trapezoid for which the gas return is larger and larger porosity.
After the initial part, the runner becomes full with liquid metal and the resistance increases but the additional entrapment is eliminated in the runner (in the mold continue). The
liquid pressure downstream increases. These different flow regimes have to be accounted in
the calculations yet all numerical models until now ignore these facts to demonstrate how
140
CHAPTER 4. THE DIE CASTING PROCESS STAGES
useless these models. The flow into cavity in the later part (about 50% of the filling process)
has a different character and there is less “jerking”.
Before going to continue, a point has to be made on how the penetrating streaks are
broken to understand the recommendation at the section end for a different geometry for
the runner. After the streak is created, a new instability created of two streams (liquid metal
and air) with different density and velocity plus and another instability due to the compressibility pushes (due to Bernoulli’s effect) the liquid metal to the wall. The streak is broken and
“grabbed” by the liquid and mixed into the stream. It can be noticed that this streaks are larger
for trapezoid as compared to circular shape. Hence, it is recommended to change the cross
section of the runner from the trapezoid to circular.
Again, the flow in the runner system has to be divided into sections; 1) flow with free
surface 2) filling the cavity when the flow is pressurized (see Figs. 4.8 and 4.9). In the
first section the gravity affects the air entrapment/entrainment. The dominant parameters in this case
are Weber number, We and Reynolds number, Re
and the Mach number, M (it is not the full list
liquid
metal
and real dimensional analysis should be made). This
phenomenon determines how much metal has to be
air
flushed out. Above certain velocity (typical to die
streaks
casting, high Re number) air leaves streaks of air/gas
Fig. 4.8 – Entrance of liquid metal to
slabs behind the “front line” as displayed in Fig. 4.8.
the runner.
These streaks create a low heat transfer zone at the
head of the “jet” and “increases” its velocity. The air
entrapment creates in this case supposed to be flushed out through the vent system in a proper
process design.
4.1.3.0.1
Gravity Limited Effect in Runner System
In the second phase, the flow in the runner system is pressurized. The typical velocity is large
of the range of 10-15 [m/sec]. The typical runner length is in order of 0.1[m]. The velocity
due to gravity is ≈ 2.5[m/sec]. The Fr number assumes the value ∼ 102 for which gravity
play a limited role. The converging nozzle such as the
transition into runner system (which a good die casting engineer should design) tends to reduce the turbulence, if turbulence exists, and can even eliminate it. The
Fig. 4.9 also exhibits entrance pressure lost. In that view,
the liquid metal enters the runner system as a laminar
flow (actually close to a plug flow depending on the entrance design.). For a duct with a typical dimension of
10 [mm] and a mean velocity, U = 10[m/sec], (during the
second stage), for aluminum die casting, the Reynolds
number is:
x
P ressure
Fig. 4.9 – Flow in runner when during pressurizing process.
141
4.1. INTRODUCTION
Re =
Ub
≈ 5 × 107
ν
which is a supercritical flow. However, the flow is laminar flow due to the short time with
mixing processes.
Another look at turbulence issue: The boundary layer is a function of the time (during
the filling period) is of order
δ = 12 ν t
The boundary layer in this case can be estimated as3 the time of the first phase. Anyhow, utilizing the time of 0.01[sec] the viscosity of aluminum in the boundary layer is of the thickness
of 0.25[mm] which indicates that flow is laminar.
4.1.4
The Mold
All the numerical simulations of die (mold) filling were done almost exclusively by assuming
that the flow is turbulent and continuous (no two phase flow what so ever). In fact this author contact numerous people to ask them about what lead them to assume turbulence and
what where their assumptions in doing so. Apparently none even consider this question and
automatically assume that there is turbulence. As in all the models numerical calculating the
critical plunger velocity they deploy wrong boundary conditions but could not explain why.
In the section 4.2.1 a question about the question whether existence of turbulence is
presented and if so what kind of model is appropriate. Thus, the validity of these numerical
models is examined. The liquid metal enters the cavity as a non–continuous flow. According
to some researchers, it is preferred that the flow will be atomized (spray). While there is a
considerable literature about many geometries none available to typical die casting configurations4 . The flow can be atomized as either in laminar or turbulent region. The experiments
by the author and by others, showed that the flow turns into spray in many cases ( See Figures
4.10).
In the section 4.3.1 it was shown that the time for atomization is very fast compared with
any other process (filling time scale and, of course, the conduction heat transfer or solidification time scales). Atomization requires two streams with a significant velocity difference;
stronger surface tension forces against the maintaining stability forces. Numerous experimental studies have shown that better castings are obtained when the injected velocity is
above a certain value. This fact alone is enough to convince researchers that the preferred
flow pattern is a spray flow. Yet, only a very small number of numerical models exist assuming spray flow and are used for die casting (for example, the paper by Bar–Meir (1995d).
Experimental work commonly cited as a “proof” of turbulence was conducted in the mid 60s
3 only during the flow in the runner system, no filling of the cavity
4 One
can just wonder who were the opposition to this research? Perhaps one of the referees as in the Appendix
C for the all clues that have been received.
142
CHAPTER 4. THE DIE CASTING PROCESS STAGES
Fig a. Flow as a jet
Fig b. Flow as a spray
Fig. 4.10 – Typical flow pattern in die casting, jet entering into empty cavity.
(Stuhrke and Wallace 1966) utilizing water analogy5 . The “white” spats they observed in their
experiments are atomization of the water. Because these experiments were poorly conducted
(no similarity to die casting process) the observation/information from these studies is very
limited if any. Yet with this limitation in mind, one can conclude that the spray flow does
exist.
Experiments by Fondse et al (Fondse, Leijdens, and Ooms 1983) show that atomization
is larger in laminar flow compared to a turbulent flow in a certain range. This fact further
creates confusion of what is the critical velocity needed in die casting. Since the experiments
which measure the critical velocity were poorly conducted, no reliable information is available on what is the flow pattern and what is the critical velocity6 .
4.1.5
Intensification Period
The two main concerns in this phase are to extract heat from the die and to solidify the liquid
metal as raptly as possible and to obtain the final shape. Thus, two operational parameters
are important; one the (minimum) time for the intensification and two the pressure of the
intensification (the clamping force). These two operational parameters can improve casting
design to obtain good product. In the literature there are papers dealing with filling time.
Perhapes of the typical misunderstaings can be expressed by the paper by (Adamane, Arnberg,
Fiorese, Timelli, and Bonollo 2015). While Adamane et al saw some of the phenomena in die
casting they not aware that all of their finding was discussed in the past. For example their
show a plot depicting the effect of varying plunger velocity on the tensile properties. This data
point is meaningless without providing precentage filled and other data to find the critical
plunger velocity.
5 The problems in these experiments were, among other things, no simulation of the dimensional numbers such
as Re, Geometry etc. and therefore different differential equations not typical to die casting were “solved.” The
researchers also look at what is known as a “poor design” for disturbances to flow downstream (this is like putting a
screen in the flow.). However, a good design requires smooth contours.
6 Beside other problems such as different flow velocity in different gates which were never really measured, the
pressure in the cavity and quality of the liquid metal entering the cavity (is it in two phase?) were never recorded.
143
4.1. INTRODUCTION
The main resistance to the heat flow is in the die and the cooling liquid (oil or water
based solution). In some parts of the process, the heat is transformed to the cooling liquid via
the boiling mechanism. However, the characteristic of boiling heat transfer time to achieve
a steady state is larger than the whole process and the typical equations (steady state) for the
preferred situation (heat transfer only in the first mode) are not accurate. When there is very
limited understanding of so many aspects of the process, the effects of each process on other
processes are also cluttered.
It is communly assumed that intensification is arbitrary and their is no specific desired
pressure for spesifc mold. The only interst was to find the last place to fill. The assumption
has been exclusive in the die casting industry. Hence many of the numberical simulations
emphasize this point. Since all these simulatons have nothing with reality, their finding are
minnegless.
In the currect version of the book, this
point was connected the physis of the problem.
It has been realized that the intesfication pressure is to compansate for the shrinkage (BarMeir 1995d). Hence, these ellements were connected but even after this breakthrough there
are still several elements that will require a refinement.
Suppose the mold contains only liquid metal and small amount of gases and
it is almost in a static solidification. The
Fig. 4.11 – Thermal expansion of Aluminum Afkentic energy (movements) in liquid after Hidnert 1925.
fects the liquid metal energy. While this
point was not investigate thoroughly, the
estimate shows that it is not signifigatnt (minor point of refinement). The question in this
discussion, what should be the intensification pressure should be overcome shrinkage in this
case. Pushka equation that develped by the auther is not relavent in this case (answer to a
posted question on the web.). The shrinkage of body from above the liquidus point (melting
point) to the room temperature is
∆V = V αV s ∆T
(4.23)
where in this section αV s is the solid volumetric expantion coefficient. Accoring to Blumm
(2000) this coefficient is constant or close it. As a first approximation, it reasonable to assume
that this coefficient a constant. The total volume is made out of three different zone: liquid,
mushy, and solid. Generally speaking the expansion of liquid metal shows some hysteresis
behavior (Hidnert 1925) and the colling path has to be selected. Furthermore, the reference is
old and today work shows more linear behavior. However, some of the latest works put this
author in the awkward position of not beeing able to be more deterministic as these works
are conflicting. Yet, with the above in mind, some averges values can be used.
Kohlstädt et al (2021) provided an excellent example how to spend large amount of efforts and yet getting useless information at the end. In the word “useful information” it means
144
CHAPTER 4. THE DIE CASTING PROCESS STAGES
something tangible and die casters somewhere can use it. Kohlstädt used the turbulence model
in a domain where there is no turbulence like a still liquid metal which violated the first and
second laws of thermodynamics. The reason that this assumption violates the thermo’s laws
is that in a still liquid there is no kinetic energy to fuel and draw from to create turbulence.
This invented energy continue stirring the liquid metal maybe or it is only imaginary stirring.
Yet, in the same time, the liquid metal has a zero velocity at the wall (no slip condition) when
the liquid is forced to move. Additionally, the pressure is not calculated but is assumed (atmospheric) to have erroneous values. As these errors are not enough, there is no results but
only specific case (which is wrong) and there is no path to expend it. This work should enter
to Rube Goldberg machine competition and probably will win the first prize easily.
The main issue in any search is to find the mechanism(s) which affects the issue at hand.
Here are examples of clueless research and reviews of the die casting. Adamane et al (2015)
stated that ”[a]lthough the design of vents and overflows in a die casting is rarely discussed in
the literature, it constitutes an important die parameter that can influence the casting quality.”
Only poor literature review can show no research while google scholar shows about 29,400
results. The solution of the critical vent area has been solved since 1995 (Bar-Meir 1995d). While
one can observe the same poor quality as Adamane’s work appears in literature also such as
(Wang 2007) because no one in world can reproduce this work due to missing important
parameters. This fact of poor research does not mean that there are no breakthroughs work
in the die casting. Bonollo et al (Bonollo, Gramegna, and Timelli 2015) reviews named as
“contradictions and challenges” of the die casting without actually doing any real literature
review. In way, Bonollo et al seems to be under the illusion that the intensification pressure
should be only monitored.
The shrinkage porosity was approached by the “last place to fill” method (Ma, Huang,
Chu, and Cheng 2022). Beside many of the typical errors , Ma et al also believe that solidification occurs in the long dimension while as in the reality it is the opposite. In this method,
is a hope that somehow as the freezing occurs while the liquid metal continue to be fed to
the mold. In this method it is assumed that as the liquid metal solidified and shrinkage the
liquid will be supplied to fill the gap. The problem in this method is that no one can predict
this/these location(s). Furthermore, even if the location(s) was/were known it not possible
to supply gate to that point. In fact, this author observed that if last location is known and
somehow the feeding of that location is done, it will result in moving the last fill location. The
icing on the top of all these problems, even if these locations miraculously solved, the fact remains find the location does not solve the problem. Regardless to the location the shrinkage
porosity will continue as the last location does not prevent it.
The purpose of the intensification is to compensate for the thermal shrinkage (and some
of the other shrinkage). Thus, the thermal expansion needs to be equal to the mechanical contraction due to the pressure (by volumetric Hook’s law). As the calculations of energies should
be part of the analysis of the elastic system at hand based on various distortions of the body
should be included when refinements are investigated. While the body in part of the temperature path goes in the liquid phase, the mushy zone, and solid phase all these contributions:
stretching, shearing, bending, twisting cannot be accounted in a simple way. Thus, others
4.1. INTRODUCTION
145
are invited to improve this section. Additional point to consider is that cooling path in the
liquid phase occur under constant pressure and not under variable pressure. This effect is
ignored because the mathematical complication and lack ability to quickly estimate it. The
dimensional analysis not presented here hints that this change is less significant. For course,
this analysis assumed that the result depends on the state and not the path (typical thermo assumption) which the author cannot completely justifies as there are possible residuals stress
(which should be accounted for). It is hoped that these assumptions will be corrected in the
future. Thus, as a first approximation it can be written that
(4.24)
∆V = V αV L (Tm − TL ) + αV mushy (Tm1 − Tm2 ) + αV s (Tm2 − Te )
where the
T
∆V
V
αV
subscript
e
L
m
m1
m2
mushy
s
temperature
Shrinkage volume
Volume
Expansion coefficient
Room temperature
Liquidus temperature
Injected temperature
Upper mushy zone temperature
Lower mushy zone temperature
Mushy zone
Solid zone
While the temperature change is applied to runner as well, the “transfer” of volume
(material) from or to mold is minimal during the cooling process. The predicted shrinkage is
based on the volume of the mold plus the vent system. The shrinkage and/or contraction are
in a small range, the difference is neglected because the simplicity is paramount here. It can
be noticed that in most cases, the flow, during the solidification, is from the runner system to
the mold and from the mold to the venting system. Material that transferred from the runner
to the mold put this value in a bit in the over estimate range. While the transfer to the vent
system put in the under estimate range. These two transfers, while not totally canceling each
other, are conflicting and reducing the combined effect.
The contraction of the volume due to the pressure based on [Section 1.62 p. 24] (BarMeir 2021a)
∂P
BT = −V
(4.25)
∂V T
The intensification process is to a large degree can be considered as isothermal since the
amount of heat lost in the process is relatively minimal and thus the temperature variation is
small as well. While in theory the increase of the pressure can cause change in temperature
which push liquid metal past the liquidus line. Hence, it is reasonable to assume the process
146
CHAPTER 4. THE DIE CASTING PROCESS STAGES
is isothermal. This point is dwelled because for some materials this assumption is not proper
(if time permited will be visited). The total change is
Z P2
Z V−∆V
∂P
∂V
−
(4.26)
=
B
V
T
P1
V
which can be integrated as
−
P2 − P1
∂V
=
BT
V
V−∆V
= ln
V
V − ∆V
∆V
= ln 1 −
V
V
(4.27)
Rearranging Eq. (4.27) yields
e
P1 −P2
BT
or
∆V = V
∆V
= 1−
V
(4.28)
P1 −P2
1 − e BT
(4.29)
Notice that the assumption of constant bulk modulus is very strong. However, it can be replaced by numerical integration for a specific material. As stated earlier, this point is only a
refining point.
These two changes (mechanical and thermal) of volume should be about equal as
P1 −P2
BT
V 1−e
=@
V αV L (Tm − TL ) +
@
αV mushy (Tm1 − Tm2 ) + αV s (Tm2 − Te )
(4.30)
It should be noted that a more precise way will be to consider the variations of the pressure on
BT . Again, the paramount in this poor man analysis is the simplicity. As the most fundamental
and breakthrough models are normally the poor man analysis, the point here to extract the
range of the ideal intensification pressure.
P1 = P2 + BT ln 1 − αV L (Tm − TL ) +
αV mushy (Tm1 − Tm2 ) + αV s (Tm2 − Te )
(4.31)
In the derivations, some simplifications were made but yet the reasonable results which show
the indication of the intensification required to reduce to zero the thermal shrinkage. It is
interesting no one consider this mechanisms or other explanation to obtain the required intensification pressure. Another point for the solid the linear expansion can be used in case it
is the unknown multiplied by the right coefficient (3).
The relationship between the Young’s modulus or modulus of elasticity and the bulk
modulus for linear and continuous material (no phase change) is
BT =
E
3 (1 − 2 ν)
(4.32)
147
4.2. SPECIAL TOPICS
Symbol
E
ν
Definition
Young’s modulus
Poisson ratio
This relationship is adapted here and a refinement probably is required to deal with
the phase change of various material. For example, the Poisson ratio is a function of the
temperature where here is assumed constant. Hence, Eq. (4.33) can be written in term of
Young’s modulus
P1 = P2 +
E
ln 1 − αV L (Tm − TL ) +
3 (1 − 2 ν)
αV mushy (Tm1 − Tm2 ) + αV s (Tm2 − Te )
Note that Young’s modulus should be taken at the value at liquid phase.
4.1.6
(4.33)
Concluding Remarks For Intensification
Here the relationship between the die, the die cast material and the intensification pressure
was established. The presentation provides the framework on which the improved models
for specific materials can be build. This paper is only preliminary examination of this topic.
Yet, the intensification can be considered as a solved problem and only a refinement is needed.
4.2
4.2.1
Special Topics
Is the Flow in Die Casting Turbulent?
It is commonly assumed that the flow in die casting processes is turbulent in the shot sleeve,
runner system, and during the cavity filling. Further, it also assumed that the k−ϵ model can
reasonably represent the turbulence structure. These assumptions are examined herein. The
flow can be examined in three zones: 1) the shot sleeve, 2) the runner system, and 3) the mold
cavity. Note, even if the turbulence exists in some regions, it doesn’t necessarily mean that all
the flow field is turbulent.
4.2.1.1
Transition from laminar to turbulent
Is the flow in the shot sleeve turbulent as the EKK sale engineers claim? These sale engineers
did not present any evidence or analysis for such claims. For a simple analysis, the initial
part of the shot sleeve filling, the liquid metal goes through a hydraulic jump. The flow after the hydraulic jump is very slow because the increase of the ratio of cross section areas.
148
CHAPTER 4. THE DIE CASTING PROCESS STAGES
Re number 103
For example, casting of the 1[kg] from height
10
of 0.2[m] to a shot sleeve√of 0.1[m] creates a velocity in shot sleeve of 2[m/sec] which re8
sults after the hydraulic jump to be with veloc6
ity about 0.01[m/sec]. The Reynolds number
4
4
for this velocity is ∼ 10 and Froude number of
about 10. After the jump the Froude number is
2
reduced and the flow is turbulent. However, by
0
the time the hydraulic jump vanishes, the flow
0
1
2
3
4
5
6
7
Time [sec]
turns into laminar flow and no change (waviness) in the surface can be observed. It can be
Fig. 4.12 – Transition to turbulent flow in cirnoticed that the time scale for the dissipation
cular pipe for instantaneous flow after Wygnanski (1973) by interpolation. Pink is Lamis about the same scale as the time for the opinar and blue is Turbulent and between is
eration of the next stage.
not determined.
Fig. 4.12 exhibits the transition
to a turbulent flow for instantaneous
starting flow in a circular pipe. The abscissa represents time and the y–axis represents the
Re number at which transition to turbulence occurs. The points on the graphs show the
transition to a turbulence. This figure demonstrates that a large time is required to turn the
flow pattern to turbulent which is measured in several seconds. The figure demonstrates that
the transition does not occur below a certain critical Re number (known as the critical Re
number for steady state). It also shows that a considerable time has elapsed before transition to turbulence occurs even for a relatively large Reynolds number. The geometry in die
casting however is different and therefore it is
expected that the transition occurs at different times. Our present knowledge of this area
is very limited. Yet, a similar transition delay
is expected to occur after the “instantaneous”
start–up which probably will be measured in
seconds. The flow in die casting in many situations is very short (in order of milliseconds)
and therefore it is expected that the transition
to a turbulent flow does not occur.
Very thin
boundary
layer plus
solidification
Transition Almost
zone still flow
Fig. 4.13 – Flow patterns in the shot sleeve.
After the liquid metal is poured, it is normally repose for sometime in a range of 10 seconds. This fact is known in the scientific literature as the quieting time for which the existed
turbulence (if exist) is reduced and after enough time (measured in seconds) is illuminated.
Hence, the turbulence, which was created during the filling process of the shot sleeve, ‘disappear” due to viscous dissipation. The question is, whether the flow in the duration of the slow
plunger velocity turbulent (see Fig. 4.13) can be examined.
Clearly, the flow in the substrate (a head of the wave) is still (almost zero velocity) and
therefore the turbulence does not exist. The Re number behind the wave is above the critical
4.2. SPECIAL TOPICS
149
Re number (which is in the range of 2000–3000). The typical time for the wave to travel to the
end of the shot sleeve is in the range of a ∼ 100 second. At present there are no experiments
conducted on on the flow behind the wave7 . The estimation can be done by looking at what is
known in the literature about the transition to turbulence in instantaneous starting pipe flow.
It has been shown (Wygnanski and Champagne 1973) that the flow changes from laminar flow
to turbulent flow in an abrupt manner for a flow with supercritical Re number.
A typical velocity of the propagating front (transition between laminar to turbulent) is
about the same velocity as the mean velocity of the flow. Hence, it is reasonable to assume
that the turbulence is confined to a small zone in the wave front since the wave is traveling
in a faster velocity than the mean velocity. Note that the thickness of the transition layer is
a monotone increase function of time (traveling distance). The Re number in the shot sleeve
based on the diameter is in a range of ∼ 104 which means that the boundary layer has not
developed much. Therefore, the flow can be assumed as almost a plug flow with the exception
of the front region.
4.2.1.2
A Note on Numerical Simulations
The most common models for the turbulence that are used in the die casting industry for
simulating the flow in cavity is k–ϵ. This model is based on several assumptions:
1. isentropic homogeneous turbulence,
2. constant material properties (or a mild change of the properties),
3. continuous medium (only liquid (or gas), no mixing of the gas, liquid and solid whatsoever), and
4. the dissipation does not play a significant role (transition to laminar flow).
The k–ϵ model is considered reasonable for the cases where these assumptions are not far
from reality. It has been shown, and should be expected, that in cases where assumptions are
far from reality, the k–ϵ model produces erroneous results. Clearly, if one cannot determine
whether the flow is turbulent and in what zone, the assumption of isentropic homogeneous
turbulence is very questionable. Furthermore, if the change to turbulence just occurred, one
cannot expect the turbulence to have sufficient time to become isentropic homogeneous. As
if this is not enough complication, consider the effects of properties variations as a result
of temperature change. Large variations of the properties such as the viscosity have been
observed in many alloys especially in the mushy zone.
While the assumption of the continuous medium is semi reasonable in the shot sleeve
and runner (not really see previous discussion for two phase flow in the runner in the initial
part), it is far from reality in the die cavity. As discussed previously, the flow is atomized and
it is expected to have a large fraction of the air in the liquid metal and conversely some liquid
7 It has to be said that similar situations are found in two phase flow but they are different by the fact the flow
in two phase flow is a sinusoidal (periodic in some respects. Another point since these statements were written (30
years) not much has changed.
150
CHAPTER 4. THE DIE CASTING PROCESS STAGES
metal drops in the air/gas phase. In such cases, the isentropic homogeneous assumption is
very dubious. For these reasons the assumption of k–ϵ model seems unreasonable unless
good experiments can show that the choice of the turbulence model does not matter in the
calculation. Furthermore, the flow is in the transition period from laminar to turbulence. At
present, the author is not aware of any model to describe such turbulence8 .
The question whether the flow in die cavity is turbulent or laminar is secondary. Since
the two phase flow effects have to be considered such as atomization, air/gas entrainment etc.
to describe the real flow in the cavity.
4.2.1.3
Additional note on numerical simulation
The solution of momentum equation for certain situations may lead to unstable solution.
Such case is the case of two jets with different velocity flow into a medium and they are adjoined (see Fig. 4.14). The solution of such flow can show that the velocity field can be an unstable solution for which the flow moderately changes to become like wave flow. However, in
many cases this flow can turn out to be full with vortexes and such. The reason that this happened is the introduction of instabilities. Numerical calculations intrinsically are introducing
instabilities because of truncation of the calculations. In many cases, these truncations results
in over–shooting or under–shooting of the nature instability. In cases where the flow is unstable, a careful study is required to make sure that the solution did not produce an unrealistic solution for larger or smaller than reality introduced instabilities. An excellent example of
such poor understating is a work made in EKK company (Backer and Sant 1997). In that work,
the flow in the shot sleeve was analyzed. The
nature of the flow is two dimensional which
can be seen by all the photos taken by numerous people (staring from the 50s). The presenter of that work explained that they have used
U1
U1
3D calculations because they want to study the
U2
U2
instabilities perpendicular to the flow direction. The numerical “instability” in this case is
larger than real instabilities and therefore, the
Fig. 4.14 – Two streams of fluids into a medium.
numerical results show phenomena does not
exist in reality.
Reverse transition from turbulent flow to laminar flow
After filling the die cavity, during the solidification process and intensification, the attained
turbulence (if exist) is reduced and probably eliminated, i.e. the flow is laminar in a large
8 It is so fanny to see so many ridiculous numerical works(Cleary, Ha, Alguine, and Nguyen 2002). This work is
excellent example how not do research. Simply ignore the physics or in other words, they simply divorced the reality
from their simulation. There experiment actually proves that there is no sharp front as they alleged. See for example
where the research understand that multiphase was and should be considered. (Saeedipour, Schneiderbauer, Pirker,
and Bozorgi 2016).
151
4.2. SPECIAL TOPICS
portion of the solidification process. At present we don’t comprehend when the transition
point/criteria occurs and we must resort to experiments. For example Sibulkin (1962) studie
the transition in pipe from turbulent to laminar. He found that is rapid at the wall and centerline at lower Re number. The decay in main direction starts by fluctuations. It is a hope
that some real good experiments using the similarity technique, outlined in this book, will be
performed. So more knowledge can be gained and hopefully will appear in this book.
4.2.2
Dissipation effect on the temperature rise
The large velocities of the liquid metal (particularly at the runner) theoretically can increase
the liquid metal temperature. To study this phenomenon, compare the of maximum effect of
all the kinetic energy that is transformed into thermal energy.
U2
= cp ∆T
2
(4.34)
This equation leads to the definition of Eckert number
Ec =
U2
cp ∆T
(4.35)
When Ec number is very large it means that the dissipation plays a significant role and conversely when Ec number is small the dissipation effects are minimal. In die casting, Eckert
number, Ec, is very small therefore the thermal dissipation is very small and can be ignored.
4.2.3
Gravity effects
The gravity has a large effect only when the gravity force is large relatively to other forces. A
typical velocity range generated by gravity is the same as for an object falling through the air.
The air effects can be neglected since the air density is very small compared with liquid metal
density. The momentum is the other dominate force in the filling of the cavity. Thus, the ratio
of the momentum force to the gravity force, also known as Froude number, determines if the
gravity effects are important. The Froude number is defined here as
Fr =
U2
ℓg
(4.36)
Where U is the velocity, ℓ is the characteristic length g is the gravity force. For example, the
characteristic pouring length is in order of 0.1[m], in extreme cases the velocity can reach
1.6[m] with characteristic time of 0.1[sec]. The author is not aware of experiments to verify the
flow pattern in such cases (low Pr number due to solidification effect)9 Yet, it is reasonable
to assume that the liquid metal in such a case, flow in laminar regimes even though the Re
number is relatively large (∼ 104 ) because of the short time and the short distance. The Re
9 It be interesting to find such experiments.
152
CHAPTER 4. THE DIE CASTING PROCESS STAGES
number is defined by the flow rate and the thickness of the exiting typical dimension. Note,
the velocity reached its maximum value just before impinging on the sleeve surface.
The gravity has dominate effects on the flow in the shot sleeve since the typical value
of the Froude number in that case (especially during the slow plunger velocity period) is in
the range of one(1). Clearly, any analysis of the flow has to take into consideration the gravity
(see Chapter 8).
4.3
4.3.1
Estimates of the time scales in die casting
Utilizing semi dimensional analysis for characteristic time
The characteristic time scales determine the complexity of the problem. For example, if the
time for heat transfer/solidification process in the die cavity is much larger than the filling
time, then the problem can be broken into three separate cases 1) the fluid mechanics, the
filling process, 2) the heat transfer and solidification, and 3) dissipation (maybe considered
with solidification). Conversely, the real problem in die filling is that we would like for the
heat transfer process to be slower than the filling process, to ensure a proper filling. The same
can be said about the other processes.
filling time
The characteristic time for filling a die cavity is determined by
tf ∼
L
U
(4.37)
Where L denotes the characteristic length of the die and U denotes the average filling velocity,
determined by the pQ2 diagram, in most practical cases this time typically is in order of 5–100
[millisecond]. Note, this time is not the actual filling time but related to it.
Atomization time
The characteristic time for atomization for a low Re number (large viscosity) is given by
ta viscosity =
νℓ
σ
(4.38)
where ν is the kinematic viscosity, σ is the surface tension, and ℓ is the thickness of the gate.
The characteristic time for atomization for large Re number is given by
ta momentum =
ρℓ2 U
σ
(4.39)
The results obtained from these equations are different and the actual atomization time in die
casting has to be between these two values.
Conduction time (die mold)
153
4.3. ESTIMATES OF THE TIME SCALES IN DIE CASTING
The governing equation for the heat transfer for the die reads
ρd cp d
∂Td
= kd
∂t
∂2 Td ∂2 Td ∂2 Td
+
+
∂x2
∂y2
∂z2
(4.40)
L
cooling
liquid
To obtain the characteristic time the governing equations is
dimensionless–ed and present with a group of constants that
determine value of the characteristic time by setting it to
unity. Denoting the following variables as
′
td =
′
yd =
L̃
y
L̃
t
Fig. 4.15 – Schematic of heat transfer
processes in the die.
x
xd =
L̃
′
tc d
′
zd =
z
L̃
θd =
T − TB
TM − TB
(4.41)
the characteristic path of the heat transfer from the die inner surface
to the cooling channels
subscript
B
boiling temperature of cooling liquid
M
liquid metal melting temperature
With these definitions, Eq. (4.40) is transformed to
tc αd
∂θd
= d2
∂t
L̃
∂2 θd
∂x ′ 2
+
∂2 θd
∂y ′ 2
+
∂2 θd
∂z ′ 2
(4.42)
which leads into estimate of the characteristic time as
2
tc d ∼
L̃
αd
(4.43)
Note the characteristic time is not effected by the definition of the θd .
Conduction time in the liquid metal (solid)
The governing heat equation in the solid phase of the liquid metal is the same as equation (4.40)
with changing properties to liquid metal solid phase. The characteristic time for conduction
is derived similarly as done previously by introducing the dimensional parameters
t′ =
t
x
y
z
T − TB
; x ′ = ; y ′ = ; z ′ = ; θs =
tc s
ℓ
ℓ
ℓ
TM − TB
(4.44)
154
CHAPTER 4. THE DIE CASTING PROCESS STAGES
where tc s is the characteristic time for conduction process and, ℓ, denotes the main path of
the heat conduction process die cavity. With these definitions, similarly as was done before
the characteristic time is given by
tc s ∼
ℓ2
αs
(4.45)
Note again that αs has to be taken for the properties of the liquid metal in the solid phase. Also
note that the solidified length, ℓ, changes during the process and discussing the case where
the whole die is solidified is not of interest. Initially the thickness, ℓ = 0 (or very small).
The characteristic time for very thin layers is very small, tc s ∼ 0. As the solidified layer
increases the characteristic time also increases. However, the temperature profile is almost
established (if other processes were to remain in the same conditions). Similar situations can
be found when a semi infinite slab undergoes solidification with ∆T changes as well as results
of increase in the resistance. For the foregoing reasons the characteristic time is very small.
Solidification time
Miller’s approach
Following Eckert’s work, Miller and his student (Horacio and Miller 1997) altered the calculations10 and based the assumption that the conduction heat transfer characteristic time in
die (liquid metal in solid phase) is the same order magnitude as the solidification time. This
assumption leads them to conclude that the main resistance to the solidification is in the interface between the die and mold 11 . Hence they conclude that the solidified front moves
according to the following
ρhsl vn = h∆T
(4.46)
Where here h is the innovative heat transfer coefficient between solid and solid12 and vn is
front velocity. Then the filling time is given by the equation
ts =
ρhsl
ℓ
h ∆T
10 Miller and his student calculate the typical forces required for clamping.
(4.47)
The calculations of Miller has shown an
interesting phenomenon in which small casting (2[kg]) requires a larger force than heavier casting (20[kg])?! Check
it out in their paper, page 43 in NADCA Transaction 1997! If the results extrapolated (not to much) to about 50[kg]
casting, no force will be required for clamping. Furthermore, the force for 20 [kg] casting was calculated to be in the
range of 4000[N]. In reality, this kind of casting will be made on 1000 [ton] machine or more (3 order of magnitude
larger than Miller calculation suggested). The typical required force should be determined by the plunger force and
the machine parts transient characteristics etc. Guess, who sponsored this research and how much it cost!
11 An example how to do poor research. These kind of research works are found abundantly in Dr. Miller and Dr.
J. Brevick from Ohio State University. These works when examined show contractions with the logic and the rest of
the world of established science.
12 This coefficient is commonly used either between solid and liquid, or to represent the resistance between two
solids. It is hoped that Miller and coworkers refer that this coefficient to represent the resistance between the two
solids since it is a minor factor and does not determine the characteristic time.
4.3. ESTIMATES OF THE TIME SCALES IN DIE CASTING
155
where ℓ designates the half die thickness. As a corollary conclusion one can arrive from
this construction is that the filling time is linearly proportional to the die thickness since
ρhsl /h∆T is essentially constant (according to Miller). This interesting conclusion contradicts all the previous research about solidification problem (also known as the Stefan problem). That is if h is zero the time is zero also. The author is not aware of any solidification
problem to show similar results. Of course, Miller has all the experimental evidence to back
it up!
4.3.1.1
Present approach
Heat balance at the liquid-solid interface yields
ρs hsf vn = k
∂(Tl − Ts )
∂n
(4.48)
where n is the direction perpendicular to the surface and ρ has to be taken at the solid phase
see Appendix ??. Additionally note that in many alloys, the density changes during the solidification and is substantial which has a significant effect on the moving of the liquid/solid
∼ kd ∂T/∂n (opposite to Miller)
front. It can be noticed that at the die interface ks ∂T/∂n =
and further it can be assumed that temperature gradient in the liquid side, ∂T/∂n ∼ 0 , is
negligible compared to other fluxes. Hence, the speed of the solid/liquid front moves
vn =
k ∂Ts − Tl k∆TMB
∼
ρs hsl
∂n
ρs hsl L̃
(4.49)
Notice the difference to equation (4.47) The main resistance to the heat transfer from the die
to the mold (cooling liquid) is in the die mold. Hence, the characteristic heat transfer from
the mold is proportional to ∆TMB /L̃13 . The characteristic temperature difference is between
the melting temperature and the boiling temperature. The time scale for the front can be
estimated by
ρs hsl ℓ2 L̃ℓ
ℓ
ts =
(4.50)
=
vs
kd ∆TMB
Note that
the solidification time isn’t a linear function of the die thickness, ℓ, but a function
of ∼ ℓ2 14 .
Dissipation Time
Examples of how dissipation is governing the flow can be found abundantly in nature.
2
∂θl
∂θl
∂θl
∂ θl ∂2 θl ∂2 θl
∂θl
+
+
+u
+v
+w
= αl
+ µΦ
(4.51)
∂t
∂x
∂y
∂z
∂x2
∂y2
∂z2
13 The estimate can be improved by converting the resistances of the die to be represented by die length and the
same for the other resistance into the cooling liquid i.e. Σ1/ho + L̃/k + CD ots + 1/hi .
14 L̃ can be represented by ℓ for example, see more simplified assumption leads to pure = ℓ2 .
156
CHAPTER 4. THE DIE CASTING PROCESS STAGES
Where Φ the dissipation function is defined as
"
2 2 # ∂u 2
∂v
∂u 2
∂v
∂v
Φ=2
+
+
+
+
+
∂x
∂y
∂y
∂x
∂y
2 2
∂v
∂v
2 ∂u ∂v ∂w 2
∂w
∂w
+
+
+
+
+
−
∂y
∂z
∂y
∂z
3 ∂x ∂y
∂z
(4.52)
Since the dissipation characteristic time isn’t commonly studied in “regular” fluid mechanics, we first introduce two classical examples of dissipation problems. First problem deals
with the oscillating manometer and second problem focuses on the “rigid body” brought to a
rest in a thin cylinder.
Fig b. Oscillating manometer
Fig a. Mass, spring
Fig. 4.16 – The oscillating manometer for the example 4.5.
Example 4.5: Manometer Oscillation
Level: Intermediate
A liquid in manometer is disturbed from a rest by a distance of H0 . Assume that
the flow is laminar and neglected secondary flows. Describe H(t) as a function of
time. Defined 3 cases: 1)under damping, 2) critical damping, and 3) overdamping.
Discuss the physical significance of the critical damping. Compute the critical radius
to create the critical damping. For simplicity assume that liquid is incompressible
and the velocity profile is parabolic.
Solution
The conservation of the mechanical energy can be written as
d
dt
rate of increase of kinetic
and potential energy in
system
∆
=∆
total of inflow of
potential energy
total of inflow of kinetic
energy
+∆
+∆
total net rate of
surroundings work on
the system
total of inflow of
potential energy
+∆
+
total work due to
expansion or
compression of fluid
∆
+
total rate mechanical
energy dissipated because
viscosity
(4.5.a)
157
4.3. ESTIMATES OF THE TIME SCALES IN DIE CASTING
continue Ex. 4.5
The control system is the liquid in the manometer. There is no flow in or out of the liquid
of the manometer, and thus, terms that deal with flow in or out are canceled. It is assumed
that the surface at the interface is straight without end effects like surface tension. This system
is unsteady and therefore the velocity profile is function of the time and space. In order to
demonstrate the way the energy dissipation is calculated it is assumed the velocity is function
of the radius and time but separated. This assumption is wrong and cannot be used for real
calculations because the real velocity profile is not separated and can have positive and negative
velocities. It is common to assume that velocity profile is parabolic which is for the case where
steady state is obtained.
This assumption can used as a limiting case and
the velocity profile is
U(r, t) = U(r) =
r 2 U0 (t) 1 −
R
(4.5.b)
where R the radius of the manometer. The velocity at the center is a function of time but independent of the Length. Notice this assumption
is not consistent with the physics as this velocity
profile contradict mass conservation. A discussion on this topic is covered in “Basics of Fluid
Mechanics” on the filling and emptying pipes by
this author. It can be noticed that this equation
pro:eq:velocityH is problematic because it breaks
the assumption of the straight line of the interface.
U0
H(t)


U0 1 − 
R

r 2

R
V = H π R2
Fig. 4.17 – Mass Balance to determine the
relationship between the U0 and the
Height, H.
The relationship between the velocity at the center, U0 to the height, H(t) can be obtained from mass conservation on left side of the manometer (see Figure 4.17) is
ZR
r 2 z dA
}| {
d ρ H π R2
=
ρ
U
1
−
2 π r dr
0
dt
R
0
(4.5.c)
dH
U
= 0
dt
2
(4.5.d)
Eq. (4.5.c) relates H(t) to the center velocity, U0 , and the integration results in
Note that H(t) isn’t a function of the radius, R. This relationship (4.5.d) is based on the definition that U0 is positive for the liquid flowing to right and therefore the height decreases. The
total kinetic energy in the tube is then
ZL ZR
Kk =
0
0
ρ U0 2
2
r 2 2 z dA
}| {
L U0 2 π R2
1−
2 π r dr dℓ =
R
6
(4.5.e)
where L is the total length (from one interface to another) and dℓ is a coordinate running along
the axis of the manometer neglecting the curvature of the “U” shape. It can be noticed that L
158
CHAPTER 4. THE DIE CASTING PROCESS STAGES
continue Ex. 4.5
is constant for incompressible flow. It can be observed that the disturbance of the manometer creates a potential energy which can be measured from a datum at the maximum lower
point. The maximum potential energy is obtained when H is either maximum or minimum.
The maximum kinetic energy is obtained when H is zero. Thus, at maximum height, H0 the
velocity is zero. The total potential of the system is then
right side
left side
z
}|
{ z
}|
{
dV
dV
Z H0 −H
z }| { Z H0 +H
z }| { Kp =
(ρ g ℓ) π R2 dℓ +
(ρ g ℓ) π R2 dℓ = H0 2 + H2 ρ g π R2
0
(4.5.f)
0
The last term to be evaluated is the viscosity dissipation. Based on the assumptions in the
example, the velocity profile is function only of the radius thus the only gradient of the velocity
is in the r direction. Hence,
Ed = µ Φ = L µ
ZR 0
dU
dr
2 z dA
}| {
2 π r dr
(4.5.g)
The velocity derivative can be obtained by using equation (4.5.b) as
!
(4.5.h)
( dU )
z dr
}| {
ZR 2
4 r U0 2 r dr
Ed = µ 2 π L
= 2 π L µ R2 U0 2
R R
R2
0
(4.5.i)
dU
= U0
dr
−2 r
R2
=⇒
dU
dr
2
=
4 r2 U0 2
R4
Substituting Eq. (4.5.h) into equation (4.5.f) reads
2
The work done on system is neglected by surroundings via the pressure at the two interfaces
because the pressure is assumed to be identical. Equation (4.5.a) is transformed, in this case,
into
d
(4.5.j)
(Kk + Kp ) = −Ed
dt
The kinetic energy derivative with respect to time (using equation (4.5.d)) is
d
d Kk
=
dt
dt
L U0 2 π R2
6
!
=
L π R2
4 L π R 2 d H d2 H
U
2 U0 0 =
6
dt
3
dt dt2
(4.5.k)
The potential energy derivative with respect to time is
i
d Kp
d h 2
dH
H0 + H2 ρ g π R2 = 2 H
=
ρ g π R2
dt
dt
dt
(4.5.l)
dH
4 L π R 2 d H d2 H
+2H
ρ g π R2 + 2 π L µ U0 2 = 0
3
dt dt2
dt
(4.5.m)
Substituting equations (4.5.l), (4.5.k) and (??) into equation (4.5.j) results in
Eq. (4.5.m) can be simplified using the identity of Eq. (4.5.d) to read
d2 H
6 µ dH 3 g
+
+
H=0
2L
dt2
ρ R2 dt
(4.5.n)
159
4.3. ESTIMATES OF THE TIME SCALES IN DIE CASTING
End of Ex. 4.5
This equation is similar to the case mass tied to a spring with damping. This equation
is similar to RLC circuita . The common method is to assume that the solution of the form of
A eξ t where the value of A and ξ will be such determined from the equation. When substituting the “guessed” function into result that ξ having two possible solution which are
ξ=
− ρ6Rµ2
±
r
6µ
ρ R2
2
2
−
6g
L
(4.5.o)
Thus, the solution is
H = A eξ1 t + A eξ2 t
H = Ae
ξt
+Ae
ξt
=⇒
ξ1 ̸= ξ2
=⇒
ξ1 = ξ2 = ξ
(4.5.p)
The constant A1 and A2 are to be determined from the initial conditions. The value under
the square root determine the kind of motion. If the value is positive then the system is over–
damped and the liquid height will slowly move the equilibrium point. If the value in square
is zero then the system is referred to as critically damped and height will move rapidly to
the equilibrium point. If the value is the square root is negative then the solution becomes a
combination of sinuous and cosines. In the last case the height will oscillate with decreasing
size of the oscillation. The critical radius is then
Rc =
s
4
6 µ2 L
g ρ2
(4.5.q)
It can be observed that this analysis is only the lower limit since the velocity profile is much
more complex. Thus, the dissipation is much more significant.
a An electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series
or in parallel.
Example 4.6: Rotating Cylinder
Level: Advance
A thin (t/DL1) cylinder full with liquid is rotating. in a velocity, ω The rigid body
is brought to a stop. Assuming no secondary flows (Bernard’s cell, etc.), describe the
flow as a function of time. Utilize the ratio 1Gt/D.
d2 X µ dX
+ 2
+X = 0
dt
dt2
ℓ
(4.6.a)
Discuss the case of rapid damping, and the case of the characteristic damping
Solution
These examples illustrate that the characteristic time of dissipation can be assessed by ∼
µ(du/d“y ′′ )2 thus given by ℓ2 /ν. Note the analogy between ts and tdiss , for which ℓ2
appears in both of them, the characteristic length, ℓ, appears as the typical die thickness.
160
CHAPTER 4. THE DIE CASTING PROCESS STAGES
D
t
Fig. 4.18 – Rigid body brought into rest.
4.3.2
The ratios of various time scales
The ratio of several time ratios can be examined for typical die casting operations. The ratio
of solidification time to the filling time
Ste
tf Lkd ∆TMB
=
∼
ts
Pr Re
Uρs hsl ℓL̃
ρlm
ρs
kd
klm
L
L̃
(4.53)
where
Re
Reynolds number
Ste
Stefan number
cp
Uℓ
νlm
∆TMB
hsl
lm
the discussion is augmented on the importance of Eq. (4.53). The ratio is extremely important
since it actually defines the required filling time.
tf = C
ρlm
ρs
kd
klm
L
Ste
L̃ Pr Re
(4.54)
At the moment, the “constant”, C, is unknown and its value has to come out from experiments. Furthermore, the “constant” is not really a constant and is a very mild function of the
geometry. Note that this equation is also different from all the previously proposed filling
time equations, since it takes into account solidification and filling process15 .
15 In this book, this equation because of its importance is referred to as Eckert–BarMeir’s equation. If you have
good experimental work, your name can be added to this equation.
4.4. SIMILARITY APPLIED TO DIE CAVITY
161
The ratio of liquid metal conduction characteristic time to characteristic filling time is
given by
tc d UL̃2
Uℓ ν L̃2
L̃2
=
= Re Pr
∼
tf
Lα
ν α Lℓ
Lℓ
The solidification characteristic time to conduction characteristic time is given by
! cp lm
ts
ρs hsl ℓL̃αd
ρs
ℓ
1
∼
=
tc kd ∆TMB L̃2
Ste ρd
cp d
L̃
The ratio of the filling time and atomization is
ta viscosity
νℓU
ℓ
≈
= Ca
∼ 6 × 10−8
tf
σL
L
Note that ℓ, in this case, is the thickness of the gate and not of the die cavity.
ta momentum
ρℓ2 U2
ℓ
≈
= We
∼ 0.184
tf
σL
L
(4.55)
(4.56)
(4.57)
(4.58)
which means that if atomization occurs, it will be very fast compared to the filling process.
The ratio of the dissipation time to solidification time is given by
kd
ρlm
tdiss
ℓ2 kd ∆TMB
Ste
ℓ
∼ 100
(4.59)
∼
=
ts
νlm ρs hsl ℓL̃
Pr
klm
ρs
L̃
this equation yields typical values for many situations in the range of 100 indicating that the
solidification process is as fast as the dissipation. It has to be noted that when the solidification
progress, the die thickness decreases. The ratio, ℓ/L̃, reduced as well. As a result, the last
stage of the solidification can be considered as a pure conduction problem as was done by the
“English” group.
4.4
Similarity applied to Die cavity
This section is useful for those who are dealing with research on die casting and or other
casting process.
4.5
Summary
This chapter is under active construction and evident by several breakthroughs that appear
inside. These works was carried on volunteer based and more work in needed.
162
CHAPTER 4. THE DIE CASTING PROCESS STAGES
5
Fundamentals of Pipe Flow
5.1
Introduction
The die casting engineer encounters many aspects of network flow. For example, the liquid metal flows in the runner is a network flow. The flow of the air and other gases out of
the mold through the vent system is also another example network flow. The pQ2 diagram
also requires intimate knowledge of the network flow. However, most die casting engineers/researchers are unfamiliar with fluid mechanics and furthermore have a limited knowledge
and understanding of the network flow. Therefore, this chapter is dedicated to provide a
brief introduction to a flow in a network. It is assumed that the reader does not have extensive background in fluid mechanics. However, it is assumed that the reader is familiar with
the basic concepts such as pressure and force, work, power. More comprehensive coverage
can be found in books dedicated to fluid mechanics and pipe flow (network for pipe). First
a discussion on the relevancy of the data found for other liquids to the die casting process is
presented. Later, a simple flow in a straight pipe/conduit is analyzed. Different components
which can appear in network are discussed. Lastly, connection of the components in series
and parallel are presented.
The results for the flow in a pipe with orifice The results for the flow in a pipe with
orifice.
163
164
CHAPTER 5. FUNDAMENTALS OF PIPE FLOW
1.2
0.1
Loss Coefficient
1.0
0.01
0.001
1e−05
0.50
1.00
0.12
0.06
0.25
0.03
0.00
0.02
0.00
1e−07
0.01
1e−06
0.8
0.6
0.4
0.2
0.0
8.00
0.0001
4.00
Air
Crude Oil
Hydrogen
Mercury
Water
2.00
Head Loss [meter]
1
100
Velocity[m/sec]
1000
10000
Reynolds Number
Fig. 5.1 – The collapsed results as funciton Reynolds number.
5.2
Universality of the loss coefficients
Die casting engineers who are not familiar with fluid mechanics ask whether the loss coefficients obtained for other liquids should/could be used for the liquid metal.
To answer this question, many experiments have been carried out for different liquids
flowing through different components in the last 300 years. An example of such experiments
is a flow of different liquids in a pipe with an orifice (see Fig. 5.1). Different liquids create a
significant head loss for the same velocity. Moreover, the differences for the different liquids
are so significant that the similarity is unclear as shown in ??. As the results of the past geniuses
work, it can be shown that when results are normalized by Reynolds number (Re) instated of
the velocity and when the head loss is replaced by the loss coefficient, U∆H
2 /2 g one obtains that
all the lines are collapsed on to a single line as shown in Fig. 5.1b. This result indicates that the
experimental results obtained for one liquid can be used for another liquid metal provided
the other liquid is a Newtonian liquid1 . Researchers have shown that the liquid metal behaves
as Newtonian liquid if the temperature is above the mushy zone temperature. This example
is not correct only for this specific geometry but is correct for all the cases where the results
are collapsed into a single line. The parameters which control the problem are found when
the results are “collapsed” into a single line. It was found that the resistance to the flow for
many components can be calculated (or extracted from experimental data) by knowing the
Re number and the geometry of the component. In a way you can think about it as a prof of
the dimensional analysis (presented in Chapter 3).
5.3
A simple flow in a straight conduit
A simple and most common component is a straight conduit as shown in Figure 5.2. The simplest conduit is a circular pipe which would be studied here first. The entrance problem and
the unsteady aspects will be discussed later. The parameters that the die casting engineers interested are the liquid metal velocity, the power to drive this velocity, and
1 Newtonian liquid obeys the following stress law τ
=µ
dU
dy
165
5.3. A SIMPLE FLOW IN A STRAIGHT CONDUIT
the pressure difference occur for the required/desired velocity. What determine these
parameters? The velocity is determined by
U2
U1
the pressure difference applied on the pipe
and the resistance to the flow. The relationship between the pressure difference, the
Fig. 5.2 – General simple conduit description.
flow rate and the resistance to the flow is
given by the experimental Eq. (5.1). This
equation is used because it works2 . The pressure difference determined by the geometrical
parameters and the experimental data which expressed by f3 which can be obtained from
Moody’s diagram.
∆P = fρ
L U2
L U2
; ∆H = f
D 2
D 2g
(5.1)
Note, head is energy per unit weight of fluid
(i.e. Force x Length/Weight = Length) and it
has units of length. Thus, the relationship between the Head (loss) and the pressure (loss) is
∆P =
∆H
ρg
(5.2)
Fig. 5.3 – General simple conduit description.
The resistance coefficient for circular conduit
can be defined as
KF = f
L
D
(5.3)
This equation is written for a constant density flow and a constant cross section. The flow
rate is expressed as
Q = UA
(5.4)
The cross sectional area of circular is A = πr2 = πD2 /4, using equation (5.4) and substituting
it into equation (5.1) yields
∆P = fρ
16 L
Q2
π2 D3
(5.5)
The Eq. (5.5) shows that the required pressure difference, ∆P, is a function of 1/D3 which
demonstrates the tremendous effect the diameter has on the flow rate. The length, on the
other hand, has mush less significant effect on the flow rate.
2 Actually there are more reasons but they are out of the scope of this book
3 At this stage, we use different definition than one used in Chapter A. The difference is by a factor of 4. Eventially
we will adapt one system for the book.
166
CHAPTER 5. FUNDAMENTALS OF PIPE FLOW
The power which requires to drive this flow is give by
P = Q ·P
(5.6)
These equations are very important in the understanding the economy of runner design, and will be studied in chapter 11 in more details.
The power in terms of the geometrical parameters and the flow rate is given
P = {⊂
5.3.1
∞̸ L
Q∋
≈∈ D∋
(5.7)
Examples of the calculations
Example 5.1: Resistance in Pipe Flow
Level: Intermediate
Calculate the pressure loss (difference) for a circular cross section pipe for driving aluminum liquid metal at velocity of 10[m/sec] for a pipe length of 0.5 [m] (like a medium
quality runner) with diameter of 5[mm] 10[mm] and 15[mm]
Solution
Example 5.1:
calculate the power required for the above example
Solution
5.4
Typical Components in the Runner and Vent Systems
In the calculations of the runner the die casting engineer encounter beside the straight pipe
which was dealt in the previous section but other kind of components. These components
include the bend, Y–connection and tangential gate, “regular gate”, the extended Y connection
and expansion/contraction (including the abrupt expansion/contraction). In this section a
general discussion on the good design practice for the different component is presented. A
separate chapter is dedicated to the tangential runner due to its complication.
5.4.1
bend
The resistance in the bend is created because a change in the momentum and the flow pattern.
Engineers normally convert the bend to equivalent conduit length. This conversion produces
adequate results in same cases while in other it might introduce larger error. The knowledge
of this accuracy of this conversion is very limited because limited study have been carry out for
the characteristic of flows in die casting. From the limited information the author of this book
gathered it seem that it is reasonable to carry this conversion for the calculations of liquid
167
5.5. PUTTING IT ALL TO TOGETHER
metal flow resistance while in the air/liquid metal mixture it far from adequate. Moreover,
“hole” of our knowledge of the gas flow in vent system are far more large. Nevertheless, for
the engineering purpose at this stage it seem that some of the errors will cancel each other
and the end result will be much better.
bad english, change it please
R
θ
The schematic of a bend commonly used in die casting is shown in Figure ??. The resistance of the bend is a function of several parameters: angle, θ, radius, R and the
geometry before and after the bend. Commonly, the runner is made with the same geometry before and after the bend. Moreover,
we will assume in this discussion that downstream and upstream do not influence that
Cross
Section
flow in the flow. This assumption is valid when
there is no other bend or other change in the
Fig. 5.4 – A sketch of the bend in die casting.
flow nearby. In cases that such a change(s) exists more complicated analysis is required.
In the light of the for going discussion, we left with two parameters that control the
resistance, the angle, θ, and the radius, R As larger the angle is larger the resistance will be.
In the practice today, probably because the way the North American Die Casting Association
teaching, excessive angle can be found through the industry. It is recommended never to
exceed the straight angle (900 ). ?? made from a data taken from several sources. From the
figure it is clear that optimum radius should be around 3.
5.4.2
Y connection
picture of Y connection
The Y–connection represent a split in the runner system. The resistance
5.4.3
Expansion/Contraction
One of the undesirable element is the runner system is a sudden change in the conduit area.
In some instances they are unavoidable. We will discuss how to design and what are the better
design options which available for the engineer.
5.5
Putting it all to Together
There are two main kinds of connections; series and parallel. The resistance in the series
connection has to be added in a fashion similar to electrical resistance i.e. every resistance
has to be added literally to the total resistance. There are many things that contribute to
the resistance besides the regular length, i.e. bends, expansions, contractions etc. All these
connections are of series type.
168
CHAPTER 5. FUNDAMENTALS OF PIPE FLOW
5.5.1
Series Connection
The flow rate in different locations is a function of the temperature. Eckert (Eckert 1989)
demonstrated that the heat transfer is insignificant in the duration of the filling of the cavity,
and therefore the temperature of the liquid metal can be assumed almost constant during the
filling period (which in most cases is much less 100 milliseconds). As such, the solidification
is insignificant (the liquid metal density changes less than 0.1% in the runner); therefore, the
volumetric flow rate can be assumed constant:
(5.8)
Q1 = Q2 = Q3 = Qi
Clearly, the pressure in the points is different and
(5.9)
P1 ̸= P2 ̸= P3 ̸= Π
However the total pressure loss is composed of from all the small pressure loss
(5.10)
P1 − Pend = (P1 − P2 ) + (P2 − P3 ) + · · ·
Every single pressure loss can be written as
Pi−1 − Pi = Ki
U2
2
(5.11)
There is also resistance due to the parallel connection i.e. y connections, y splits and manifolds
etc. First, lets look at the series connection. (see Figure ??). where:
Kbend
L
f
5.5.2
the resistance in the bend
length of the duct (vent),
friction factor, and
The Parallel Connection
An example of the resistance of parallel connection (see Fig. 5.5).
The pressure at point 1 is the same for
three branches, however the total flow rate is
the combination
Qtotal = Q1 + Q2 + Q3
(5.12)
Between the two branches and the loss in the
junction is calculated as
1
2
3
Fig. 5.5 – 3 Parallel Pipes in parallel connections.
6
Runner Design
Under construction please ignore the rest of the chapter.
6.1
Introduction
In this chapter the design and the different relationship between runner segments are studied herein. The first step in runner design is to divide the mold into several logical sections.
The volume of every section has to be calculated. Then the design has to ensure that the
gate velocity and the filling time of every section to be as recommend by experimental results. At this stage there is no known reliable theory/model known to the author to predict
these values. The values are based heavily on semi–reliable data. The Backward Design is
discussed. The reader with knowledge in electrical engineering (electrical circles) will notice
in some similarities. However, hydraulic circuits are more complex and it was discussed the
previous chapter. Part of the expressions are simplified to have analytical expressions. Yet,
in actuality all the terms should be taken into considerations and commercial software such
DiePerfectshould be used.
6.1.1
Backward Design
Suppose that we have n sections with n gates. We know that volume to be delivered at gate i
and is denoted by vi.The gate velocity has to be in a known range. The filling time has to be in
a known function and we recommend to use Eckert/Bar–Meir’s formula. For this discussion
169
170
CHAPTER 6. RUNNER DESIGN
it is assumed that the filling has to be in known range and the flow rate can be calculated by
Vi
ti
(6.1)
Qi
Ugate
(6.2)
Qi =
Thus, gate area for the section
Ai =
Armed with this knowledge, one can start design the runner system.
6.1.2
Connecting runner segments
Design of connected runner segments have to
insure that the flow rate at each segment has
to be close to the designed one. In Fig. 6.1 a
branches i and j are connected to branch κ
at point K. The pressure drop (difference) on
branches i and j has to be the same since the
pressure in the mold cavity is the same for both
segments. The sum of the flow rates for both
branch has to be equal to flow rate in branch
κ
Qκ = Qi + Qj −→ Qj = Qκ − Qi
Vents
Mold
a
b
(6.3)
Biscuit
The flow rate in every branch is related to the
pressure difference by
∆P
Qi =
Ri
i
j
k
Fig. 6.1 – A geometry of runner connection.
(6.4)
Where the subscript i in this case also means any branch e.g. i, j and so on. For example, one
can write for branch j
Oj =
∆P
Rj
(6.5)
Utilizing the mass conservation for point K in which Qkappa = Oi + Oj and the fact that
the pressure difference, ∆P, is the same thus we can write
Qk =
Ri Rj
∆P ∆P
+
= ∆P
Ri
Rj
Ri + Rj
(6.6)
where we can define an equivalent resistance by
R̄ =
Ri Rj
Ri + Rj
(6.7)
171
6.1. INTRODUCTION
Lets further manipulate the equations to get some more important relationships. Using equation (??) and equation (??)
(6.8)
DeltaPi = DeltaPj −→ Oi Ri = Oj Rj
The flow rate in a branch j can be related to flow rate in branch i and corresponding resistances
Qj =
Ri
Qi
Rj
(6.9)
Using Eq. (6.9) and Eq. (6.8) one can obtain
Rj
Oi
=
Qk
Rj + Ri
(6.10)
Solving for the resistance ratio since the flow rate is known
O
Ri
= i −1
Rj
Qk
6.1.3
(6.11)
Resistance
What does the resistance include? How to
achieve resistance ratio in the previous ?? will
be discussed herein further. The total resistance reads
R = Rii + Rθ + Rgeometry +
Rcontraction + Rki + Rexit
(6.12)
θ1
θ2
Fig. 6.2 – y connection.
The contraction resistance, Rcontraction , is the due to the contraction of the gate.
The exit resistance, Rexit , is due to residence of the liquid metal in mold cavity. Or in other
words, the exit resistance is due the lost of energy of immersed jet. The angle resistance, Rθ
is due to the change of direction. The Rki is the resistance due to flow in the branch κ on
branch i. The geometry resistance Rgeometry , is due to who rounded the connection.
∆P
L U2
=f
ρ
HD 2
(6.13)
∆P
L Qi 2
=f
ρ
HD 2 A2
(6.14)
L Qi 2
DeltaP
= (C)f
ρ
HD 3 2
(6.15)
since Ui = Qi /A
172
CHAPTER 6. RUNNER DESIGN
Lets assume further that Li = Lj ,
(C)f
L
HD 3 i
Oi
Qj
= known
fi = fj = f
(6.16)
Qi 2
L Oj 2
= (C)f
2
HD 3 j 2
(6.17)
HD i
=
HDj
Qi
Qj
2
3
(6.18)
Comparison between scrap between (multi-lines) two lines to one line
First find the diameter equivalent to two lines
∆P = (C)f
L Qk 2
L (Qi + Qj )2
= (C)f
3
2 HD k
2
HD 3k
(6.19)
∆P 2
HD i3
f L
(6.20)
(HD 3i + HD 3j
(6.21)
Qi 2 =
Subtitling in to
HDk =
q
3
Now we know the relationship between the hydraulic radius. Let see what is the scrap
difference between them.
put drawing of the trapezoid
let scrap denoted by η
Converting the equation
HDi =
r
3/2
ηi
constL
(6.22)
The ratio of the scrap is
ηi + ηj
(HD i + 2 + HD j2 )
=
ηk
HD k2
(6.23)
and now lets write HDk in term of the two other
(HD 2i + HD 2j )
2/3
HD 3i + HD 3j
(6.24)
In conclusion, it’s just a plain sloppy piece of work.
Referee II, see In the appendix C
7
pQ2 Diagram Calculations
7.1
Introduction
The pQ2 diagram is the most common calgate
mold
culation, if any at all, are used by most die
3
gate
casting engineers. The importance of this
runner
diagram can be demonstrated by the fact
that tens of millions of dollars have been
2
invested by NSF, NADCA, and other major
Shot sleeve
1
institutes here and abroad in the pQ2 diagram research. The pQ2 diagram is one of
Fig. 7.1 – Schematic of typical die casting mathe manifestations of supply and demand thechine for the illustration of the pQ2 diagram
ory which was developed by Alfred Marshall
(1842–1924) in the turn of the century. It
was first introduced to the die casting industry in the late’70s (Davis 1975). In this diagram, an
engineer insures that die casting machine ability can fulfill the die mold design requirements;
the liquid metal is injected at the right velocity range and the filling time is small enough
to prevent premature freezing. One can, with the help of the pQ2 diagram, and by utilizing experimental values for desired filling time and gate velocities improve the quality of the
casting.
In the die casting process (see Fig. 7.1), a liquid metal is poured into the shot sleeve where
it is propelled by the plunger through a runner and the gate into the mold. The runner is a
element which is connecting the shot sleeve to the mold. The gate thickness is very narrow
173
174
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
compared with the averaged mold thickness and the runner thickness to insure that breakage
point of the scrap occurs at that gate location. A solution of increasing the discharge coefficient, CD , (larger conduits) results in a larger scrap. A careful design of the runner and the
gate is required.
First, the “common” pQ2 diagram which sponsored by NADCA’s books and through
their classes is presented by several such as (Zabel 1980a), (Zabel 1980b) and (Zabel 1981). In
fact this nonsense or propaganda was continue to perpetuated by Miller1 (Miller 2017), (Miller
2015). Despite NADCA enormous efforts over 10000 people download various versions of the
papers of this author on the pQ2 check researchgate.net for the simple version of “How to
Calculate the pQ2 Diagram” shows “6,473 Reads”. This number does not include copies of this
book in many web cites and other versions of the articles.
The errors of common model are analyzed. Later, the reformed model is described.
Effects of different variables is studied and questions for students are given in the end of the
chapter.
7.2
The “common” pQ2 diagram
{
The injection phase is (normally) separated into three main stages which are: slow part, fast
part and the intensification (see Fig. 7.2). In the slow part the plunger moves in the critical
velocity to prevent gas (air) entrapment formation and therefore expels maximum air/gas
before the liquid metal enters the cavity. In the fast part the cavity supposed to be filled in such
way to prevent premature freezing and to obtain the right filling pattern. The pQ2 diagram
deals with the second part of the filling phase. The intensification part is to fill the cavity
with additional material to compensate for the
Compansation
shrinkage porosity during the solidification
For
Shinkage
process. The intensification problem solved
Intesification
Plunger
for the first in version 0.4 of this book.
Location
In the pQ2 diagram, the solution is
Starting Filling
determined by finding the intersecting point
The Cavity
Liquid Metal Pressure
of the runner/mold characteristic line with
At the Plunger Tip
Cavity
the pump (die casting machine) characterisFilling
Time
tic line. The intersecting point sometime
Fig. 7.2 – A typical trace on a cold chamber marefereed to as the operational point. The
chine.
machine characteristic line is assumed to be
understood to some degree and it requires
finding experimentally two coefficients. The runner/mold characteristic line requires knowledge on the efficiency/discharge coefficient, CD , thus it is an essential parameter in the calculations. Until now, CD has been evaluated either experimentally, to be assigned to specific
runner, or by the liquid metal properties (CD ∝ ρ) (Cocks 1986) which is de facto the method
{
Liquid Metal Reachers
to The Venting System
1 Many suggested that this author should be thankful to Miller due to Miller’s “work” so much progress can be
made by this author. This view is not supported by the facts as other try as well and very few or any consider any
value to Miller’s work.
7.2. THE “COMMON” PQ2 DIAGRAM
175
used today and refereed herein as the “common” pQ2 diagram2 . Furthermore, CD is assumed
constant regardless to any change in any of the machine/operation parameters during the calculation. The experimental approach is arduous and expensive, requiring the building of the
actual mold for each attempt with average cost of $5,000–$10,000 and is rarely used in the
industry. A short discussion about this issue is presented in the Appendix C comments to
referee 2.
Herein the “common” model (constant CD ) is constructed. The assumptions made in
the construction of the model are as following
1. CD assumed to be constant and depends only the metal. For example, NADCA recommend different values for aluminum, zinc and magnesium alloys.
2. The liquid metal reaches to gate without mixed front.
3. No air/gas is present in the liquid metal. Or alternatively, if it exist then it still can be
represeted by the simple Bernoulli’s eqution.
4. No solidification occurs during the filling.
5. The velcity at the plunger (tip) is significaly smaller than the static pressure (head).
6. Mixing processes in runner are negligible.
7. The pressure in the cavity is constant during the process.
8. The main resistance to the metal flow is in the runner.
9. A linear relationship between the pressure, P1 and flow rate (squared), Q2 which is
based on Bernoulli’s equation.
On a stream line between point 1 and point 3 (see Fig. 7.1) Bernoulli’s equation applied.
∼0 ∼0
h 1 ,3
U3 2 z }| 2{
P1 U1 2 P3
+
= +
+ KF U3
ρ
2
2
ρ
(7.1)
where ???
In the case of zero velocty, the pressure the plunger tip at attened its the maximum
value which is Pmax = P1 .
s
P1
U3 2
3cm
√ P1
(1 + KF ) −−−→ U3 =
=
(7.2)
ρ
2
ρ
2 Another method has been suggested in the literature in which the C is evaluated based on the volume to be
D
filled (Cocks and Wall 1983). The resistance is not depend on the volume to filled but rather on the path the liquid
metal has pass through and the velocity. If the Cocks suggestion should be more serious it should include the filling
time. Thus, this method can be considered as “good” as the “common” method.
176
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
According to the last assumption, the liquid metal pressure at the plunger tip, P1 , can
be written as
"
2 #
Q
P1 = Pmax 1 −
(7.3)
Qmax
Where:
P1
Q
Pmax
Qmax
the pressure at the plunger tip
the flow rate
maximum pressure which can be attained by the die casting machine
in the shot sleeve
maximum flow rate which can be attained in the shot sleeve
In writing Eq. (7.3) it is assumed that the flow regime remains constant (laminar while the
common believe which is turbulent which erroneous). Thus, the stagnation pressure (pressure
without flow) is converted into partially flowing.
The Pmax and Qmax values to be determined
for every set of the die casting machine and the
shot sleeve. The Pmax value can be calculated
using a static force balance. The determination
of Qmax value is done by measuring the velocity of the plunger when the shot sleeve is
empty. The maximum velocity combined with
the shot sleeve cross–sectional area yield the
maximum flow rate,
Pmax
2
1
Q2 =
2 A 3 2 CD 2 P max
ρ
(Qmax )2
Fig. 7.3 – Pmax and Qmax as a function of the
plunger diameter according to “common”
model for plunger diameter.
Qi = A Ui
(7.4)
where i represent any possible subscription e.g. i = max
Thus, the first line can be drawn on pQ2
diagram as it shown by the line denoted as 1 in
Fig. 7.3. The line starts from a higher pressure
Pmax
Qmax
(Pmax ) to a maximum flow rate (squared). A
∼ (D1 )2
∼ (DP )2
P
new combination of the same die casting machine and a different plunger diameter creates
a different line. A smaller plunger diameter has
a larger maximum pressure (Pmax ) and different maximum flow rate as shown by the line
Pmax
denoted as 2.
Qmax
D1
The maximum flow rate is a funcFig. 7.4 – Depiction of Pmax and Qmax as a
tion of the maximum plunger velocity
function of the plunger diameter according
and the plunger diameter (area).
The
to “common” model.
plunger area is a obvious function of the
plunger diameter, A = πD2 /4. However,
7.2. THE “COMMON” PQ2 DIAGRAM
177
the maximum plunger velocity is a far–more complex function. The force that can be extracted from a die casting machine is essentially the same for different plunger diameters. The
change in the resistance as results of changing the plunger’s diameter depends on the conditions of the plunger. The “dry” friction will be same what different due to change plunger
weight, even if the plunger conditions where the same. Yet, some researchers claim that
plunger velocity is almost invariant in regard to the plunger diameter. Nevertheless, this
piece of information has no bearing on the derivation in this model or reformed one, since it
isn’t used.
Example 7.1: caption
Level: Intermediate
Prove that the maximum flow rate, Qmax is reduced and that Qmax ∝ 1/DP 2 (see
Fig. 7.4). if Umax is a constant
Solution
A simplified force balance on the
rode yields (see more details in
section 7.4.2.4 page 192)
Pmax = ∆PB
=
DB
D1
atmospheric
pressure
UP
air flow
in
2
∆PB
DB 2
D1 2
liquid
metal out
the moving part of
the control volume
PB 2
PB 1
(7.5)
where subscript B denotes the
actuator.
friction is neglected
Fig. 7.5 – A general schematic of the control volume of the hydraulic piston with the plunger and part of the liquid metal
Example 7.2: Pressure at Plunger Tip
Level: Basics
What is the pressure at the plunger tip when the pressure at the actuator is 10 [bars]
with diameter of 0.1[m] and with a plunger diameter, D1 , of 0.05[m]?
Solution
Substituting the data into Eq. (7.5) yields
P1 = 10 ×
0 .1
0.05
2
= 4.0[MPa]
(7.2.a)
In the “common” pQ2 diagram CD is defined as
CD =
s
1
= constant
1 + KF
(7.6)
Note, therefore KF is also defined as assumed to be depend on for every metal and constant.
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
178
Utilizing Bernoulli’s equation (for more details see section 7.4 page 182).
U3 = CD
s
2 P1
ρ
(7.7)
The flow rate at the gate can be expressed as
Q3 = A 3 C D
s
2 P1
ρ
(7.8)
The flow rate in different locations is minimally a function of the temperature. The dimensions analysis chapter 3 demonstrated that the heat transfer is insignificant in during of the
filling of the cavity, and therefore the temperature of the liquid metal can be assumed almost
constant during the filling period (which in most cases is much less 100 milliseconds). As such,
the solidification is insignificant. However the more significat part of the analysis is the fact
the initially the liquid metal mix with the gases (air). The liquid metal density changes less
than 0.1% in the runner as compare to about 5–10% with the gases. Yet, the volumetric flow
rate is assumed constant because the the initial part only up 10% of the filling. Therefore the
flow rate is assumed to be constant and written as
Q1 = Q2 = Q3 = Q
(7.9)
Hence, we have two equations (7.3)) and (7.8) with two unknowns (Q and P1 ) for which the
solution is
P1 =
7.3
Pmax
2 CD 2 Pmax A3 2
1−
ρQmax 2
(7.10)
The validity of the “common” diagram
In the construction of the “common” model, two main assumptions were made: one CD is a
constant which depends only on the liquid metal material, and two) many terms in the energy equation (Bernoulli’s equation) can be neglected. Unfortunately, the examination of the
validity of these assumptions was missing in all the previous studies. Here, the question when
the “common” model are valid or perhaps whether the “common’ model valid at all is examined. Some might argue that even if the model is wrong and do not stand on sound scientific
principles, it still could have a value if the model produces reasonable trends. Therefore, this
model should produce reasonable results and trends when varying any parameter in order to
have any value. Part of the examination is done by varying parameters and checking to see
what happen to trends.
179
7.3. THE VALIDITY OF THE “COMMON” DIAGRAM
7.3.1
Is the “Common” Model Valid?
Is the mass balance really satisfied in the “common” model? Lets examine this point. Eq. (7.9)
states that the mass (volume, under constant density) balance is exist.
(7.11)
A1 U1 = A3 U3
So, what is the condition on CD to satisfy this condition? Can CD be a constant as
stated in assumption 1? To study this point
condier the expression for CD . Utilizing
Eq. (7.7) yields
s
2 P1
A1 U1 = A3 CD
(7.12)
ρ
P̄
From the machine characteristic, Eq. (7.3), it
can be shown that
r
Pmax − P1
(7.13)
U1 = Umax
Pmax
A3
Fig. 7.6 – P̄ as A3 to be relocated
Example 7.3: characteristic Equation
Level: Basic
Derive equation 7.13. Start with machine characteristic Eq. (7.3)
Solution
Substituting Eq. (7.13) into Eq. (7.12) yields,
A1 Umax
r
Pmax − P1
= A3 CD
Pmax
s
P1
ρ
(7.3.a)
It can be shown that Eq. (7.3.a) can be transformed into
CD =
√
A1 Umax ρ
q
A3 2 P1 Pmax
(7.3.b)
Pmax −P1
Example 7.4: Ozer Number Function
Level: Intermediate
Find the relationship between CD and Ozer number that satisfy Eq. (7.3.b)
Solution
According
qto the “common” model Umax , and Pmax are independent of the gate area, A3 . The
term A3
P1
Pmax −P1
is not a constant and is a function of A3 (possibility other parameters).
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
180
Example 7.5: caption
√
Find the relationship between A3 P1 Pmax − P1 and A3
Level: Intermediate
Solution
under construction
Example 7.6: What wrong with A3
Level: Basic
A3 what other parameters that CD depend on which do not provide the possibility
CD = constant?
Solution
To maintain the mass balance CD must be a function at least of the gate area, A3 . Since the
“common” pQ2 diagram assumes that CD is a constant it diametrically opposite the mass conservation principle. Moreover, in the “common” model, a major assumption is that the value
of CD depends on the metal, therefore, the mass balance is probably never achieved in many
cases. This violation demonstrates, once for all, that the “common” pQ2 diagram is erroneous.
Example 7.7: caption
Level: Intermediate
Check what happened to the flow rate at two location ( 1) gate 2) plunger tip) when
discharge coefficient is varied CD = 0.4 − 0.9
Solution
7.3.2
Are the Trends Reasonable?
Now second part, are the trends predicted by the “common” model are presumable (correct)?
To examine that, we vary the plunger diameter, (A1 or D1 ) and the gate area, A3 to see if any
violation of the physics laws occurs as results. The comparison between the real trends and
the “common” trends is discussed in the following section.
7.3.2.1
Plunger area/diameter variation
First, the effect of plunger diameter size variation is examined. In section 7.2 it was shown
that Pmax ∝ 1/D1 2 . Eq. (7.10) demonstrates that P1 increases with an increase of Pmax . It
also demonstrates that the value of P̄ never can exceed
P1
ρ Qmax 2
=
(7.14)
Pmax max
2 CD A3
The value Pmax can attained is an infinite value (according to the “common” model)
therefore P1 is infinite as well. The gate velocity, U3 , increases as the plunger diameter de-
181
7.3. THE VALIDITY OF THE “COMMON” DIAGRAM
creases as shown in Figure ??. Armed with this knowledge now, several cases can be examined
if the trends are realistic.
7.3.2.2
Gate area variation
7.3.2.2.1 Energy conservation (power supply machine characteristic)
Let’s assume that mass conservation is fulfilled, and, hence the plunger velocity can apP
proach infinity, U1 → ∞ when D1 → 0
ρQ
2A C
(under constant Qmax ). The hydraulic piston also has to move with the same velocity,
U1 . Yet, according to the machine characteristic the driving pressure, approaches zero
P
(PB1 − PB2 ) → 0. Therefore, the energy supFig.
7.7
–
P
as
a
function
of
P
max .
1
ply to the system is approaching zero. Yet, energy obtained from the system is infinite since
jet is inject in infinite velocity and finite flow rate. This cannot exist in our world or perhaps
one can proof the opposite.
1
3
2
max
2
2
D
max
7.3.2.2.2 Energy conservation (power supply)
Assuming that the mass balance requirement is obtained, the pressure at plunger tip, P1 and
gate velocity, U3 , increase (and can reach infinity,(when P1 → ∞ then U3 → ∞) when the
plunger diameter is reduced. Therefore, the energy supply to the system has to be infinity
(assuming a constant energy dissipation, actually the dissipation increases with plunger diameter in most ranges) However, the energy supply to the system (c.v. only the liquid metal)
system would be PB1 AB1 U1 (finite amount) and the energy the system provide plus would
be infinity (infinite gate velocity) plus dissipation.
7.3.2.2.3 Energy conservation (dissipation problem)
A different way to look at this situation is check what happen to physical quantities. For
example, the resistance to the liquid metal flow increases when the gate velocity velocity is
increased. As smaller the plunger diameter the larger the gate velocity and the larger the
resistance. However, the energy supply to the system has a maximum ability. Hence, this
trend from this respect is unrealistic.
7.3.2.3
Mass conservation (strike)
According to the “common” model, the gate velocity decreases when the plunger diameter
increases. Conversely, the gate velocity increases when the plunger diameter decreases3 . According to equation (7.4) the liquid metal flow rate at the gate increases as well. However,
according to the “common” pQ2 diagram, the plunger can move only in a finite velocity lets
say in the extreme case Umax 4 . Therefore, the flow rate at the plunger tip decreases. Clearly,
3 again Figure
??
4 this is the velocity attained when the shot sleeve is empty
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
182
these diametrically opposing trends cannot coexist. Either the “common” pQ2 diagram wrong
or the mass balance concept is wrong, take your pick.
7.3.2.3.1 Mass conservation (hydraulic pump):
The mass balance also has to exist in hydraulic pump (obviously). If the plunger velocity have
to be infinite to maintain mass balance in the metal side, the mass flow rate at the hydraulic
side of the rode also have to be infinite. However, the pump has maximum capacity for flow
rate. Hence, mass balance can be obtained.
7.3.3
Variations of the Gate area, A3
under construction
7.4
The Correct pQ2 Diagram
The method based on the liquid metal properties is with disagreement with commonly agreed
on in fluid mechanics (Pao 1961, pp. 235-299). It is commonly agreed that CD is a function of
Reynolds number and the geometry of the runner design. The author suggested adopting
an approach where the CD is calculated by utilizing data of flow resistance of various parts
(segments) of the runner. The available data in the literature demonstrates that a typical value
of CD can change as much as 100% or more just by changing the gate area (like valve opening).
Thus, the assumption of a constant CD , which is used in “common” pQ2 calculations5 , is not
valid. Here a systematic derivation of the pQ2 diagram is given. The approach adapted in this
book is that everything (if possible) should be presented in dimensionless form.
7.4.1
The reform model
Eq. (7.3) can be transformed into a dimensionless from as
p
Q̄ = 1 − P̄
(7.15)
Where:
reduced pressure, P1 /Pmax
reduced flow rate, Q1 /Qmax
P̄
Q̄
Eckert (1989) also demonstrated that the gravity effects are negligible (see for more details chapter 3. Assuming steady state (read in the section 7.4.4 on the transition period of the
pQ2 ) and utilizing Bernoulli’s equation between point (1) on plunger tip and point (3) at the
gate area (see Fig. 7.1) yields
P
U 2
P1 U1 2
+
= 3 + 3 + h 1 ,3
ρ
2
ρ
2
where:
5 or as it is suggested by the referee II
(7.16)
7.4. THE CORRECT PQ2 DIAGRAM
U
ρ
h 1 ,3
183
velocity of the liquid metal
the liquid metal density
energy loss between plunger tip and gate exit
subscript
1
plunger tip
2
entrance to runner system
3
gate
It has been shown that the pressure in the cavity can be assumed to be about atmospheric (for air venting or vacuum venting) providing vents are properly designed Bar–Meir
et al 6 . This assumption is not valid when the vents are poorly designed. When they are poorly
designed, the ratio of the vent area to critical vent area determines the build up pressure, P3 ,
which can be calculated as it is done in Bar–Meir et al However, this is not a desirable situation since a considerable gas/air porosity is created and should be avoided. It also has been
shown that the chemical reactions do not play a significant role during the filling of the cavity
and can be neglected (Bar-Meir 1995c).
The resistance in the mold to liquid metal flow depends on the geometry of the part
to be produced. If this resistance is significant, it has to be taken into account calculating
the total resistance in the runner. In many geometries, the liquid metal path in the mold is
short, then the resistance is insignificant compared to the resistance in the runner and can be
ignored. Hence, the pressure at the gate, P3 , can be neglected. Thus, equation (7.16) is reduced
to
P1 U1 2
U 2
+
= 3 + h 1 ,3
ρ
2
2
(7.17)
The energy loss, h1,3 , can be expressed in terms of the gate velocity as
h 1 ,3 = K F
U23
2
(7.18)
where KF is the resistance coefficient, representing a specific runner design and specific gate
area.
Combining equations Eq. (7.9), Eq. (7.17) and Eq. (7.18) and rearranging yields
s
2 P1
U3 = CD
(7.19)
ρ
where
v
u
CD = f(A3 , A1 ) = u
u
t
6 Read a more detailed discussion in Chapter 9
1−
1
2
A3
A1
(7.20)
+ KF
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
184
Converting Eq. (7.19) into a dimensionless form yields
p
Q̄ = 2OzP̄
(7.21)
When the Ozer Number is defined as
Oz =
CD 2 Pmax
ρ
2
Qmax
A3
=
A3
Qmax
2
CD 2
Pmax
ρ
(7.22)
The significance of the Oz number is that this is the ratio of the “effective” maximum energy
of the hydrostatic pressure to the maximum kinetic energy. Note that the Ozer number is not
a parameter that can be calculated a priori since the CD is varying with the operation point.
7 For practical reasons the gate area, A cannot be extremely large. On the other hand,
3
the gate area can be relatively small A3 ∼ 0 in this case Ozer number A3 A3 n where is a
number larger then 2 (n > 2).
Solving equations (7.21) with (7.15) for P̄, and taking only the possible physical solution,
yields
P̄ =
1
1 + 2 Oz
(7.23)
which is the dimensionless form of Eq. (7.10).
7.4.2
Examining the solution
This solution provide a powerful tool to examine various parameters and their effects on the
design. The important factors that every engineer has to find from these calculations are: gate
area, plunger diameters, the machine size, and machine performance etc8 . These issues are
explored further in the following sections.
7.4.2.1
The gate area effects
Gate area affects the reduced pressure, P̄, in two ways: via the Ozer number which include two
terms: one, (A3 /Qmax ) and, two, discharge coefficient CD . The discharge coefficient, CD
is also affected by the gate area affects through two different terms in the definition (equation
7.20), one, (A3 /A1 )2 and two by KF .
perhaps to put discussion pending on the readers response.
7.4.2.1.1 Qmax effect
is almost invariant with respect to the gate area up to small gate area sizes9 . Hence this part
is somewhat clear and no discussion is need.
7 It should be margin-note and so please ignore this footnote.
how Ozer number behaves as a function of the gate area?
Pmax
A3
Oz =
ρQmax 2 1 − A3 2 + K
A1
F
8 The machine size also limited by a second parameter known as the clamping forces to be discussed in chapter 10
9 This is reasonable speculation about this point.
More study is well come
7.4. THE CORRECT PQ2 DIAGRAM
7.4.2.1.2
185
(A3 /A1 )2 effects
Lets look at the definition of CD equation (7.20). For illustration purposes let assume that KF
is not a function of gate area, KF (A3 ) = constant. A small perturbation of the gate area
results in Taylor series,
∆CD
=
=
(7.24)
CD (A3 + ∆A3 ) − CD (A3 )
A3 ∆A3
1
r
+
23 +
2
2
1 − A3 2 + K F A 1 2 1 − A3 2 + K F
A1
A1




3 A3 2
2
A 2
A1 4 1− 3 2 +KF
A1
2
r
+
1−
A1
2

1

A 2
1− 3 2 +KF
A3 2
A1 2
A1
∆A3 2
+ O(∆A3 )3
+ KF
In this case equation (7.10) still hold but CD has to be reevaluated. repeat the example ?? with
KF = 3.3 First calculate the discharge coefficient, CD for various gate area starting from 2.4
10−6 [m2 ] to 3 10−4 [m2 ] using equation 7.20.
This example demonstrate the very limited importance of the inclusion of the term
(A3 /A1 )2 into the calculations.
7.4.2.1.3
KF effects
The change in the gate area increases the resistance to the flow via several contributing factors
which include: the change in the flow cross section, change in the direction of the flow, frictional loss due to flow through the gate length, and the loss due to the abrupt expansion after
the gate. The loss due to the abrupt expansion is a major contributor and its value changes
during the filling process. The liquid metal enters the mold cavity in the initial stage as a
“free jet” and sometime later it turns into an immersed jet which happens in many geometries
within 5%-20% of the filling. The change in the flow pattern is believed to be gradual and is a
function of the mold geometry. A geometry with many changes in the direction of the flow
and/or a narrow mold (relatively thin walls) will have the change to an immersed jet earlier.
Many sources provide information on KF for various parts of the designs of the runner and
gate. Utilizing this information produces the gate velocity as a function of the given geometry.
To study further this point consider a case where KF is a simple function of the gate area.
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
186
When A3 is very large then the effect on KF are
relatively small. Conversely, when A3 → 0
the resistance, KF → ∞. The simplest function, shown in Figure 7.8, that represent such
behavior is
KF
K0
C
KF = C 1 + 2
A3
(7.25)
C1 and C2 are constants and can be cal-
A3
Fig. 7.8 – KF as a function of gate area, A3
culated (approximated) for a specific geometry. The value of C1 determine the
value of the resistance where A3 effect is minimum and C2 determine the range (point) where
A3 plays a significant effect. In practical, it is found that C2 is in the range where gate area
are desired and therefore program such as DiePerfect™ are important to calculated the actual
resistance.
7.4.2.1.4 The combined effects
3
Consequently, a very small area ratio results in a very large resistance, and when A
A1 → 0
therefore the resistance → ∞ resulting in a zero gate velocity (like a closed valve). Conversely,
for a large area ratio, the resistance is insensitive to variations of the gate area and the velocity
is reduced with increase the gate area. Therefore a maximum gate velocity must exist, and
can be found by
dU3
=0
dA3
(7.26)
which can be solved numerically. The solution of equation (7.26) requires full information on
the die casting machine.
A general complicated runner design configuration can be converted into a straight
conduit with trapezoidal cross–section, provided that it was proportionally designed to create
equal gate velocity for different gate locations10 . The trapezoidal shape is commonly used
because of the simplicity, thermal, and for cost reasons.
To illustrate only the effects of the gate area change two examples are presented: one,
a constant pressure is applied to the runner, two, a constant power is applied to the runner.
The resistance to the flow in the shot sleeve is small compared to resistance in the runner,
hence, the resistance in the shot sleeve can be neglected. The die casting machine performance
characteristics are isolated, and the gate area effects on the the gate velocity can be examined.
Typical dimensions of the design are presented in Figure ??. The short conduit of 0.25[m]
represents an excellent runner design and the longest conduit of 1.50[m] represents a very
poor design. The calculations were carried for aluminum alloy with a density of 2385[kg/m3 ]
and a kinematic viscosity of 0.544 × 10−6 [m2 /sec] and runner surface roughness of 0.01 [mm].
For the constant pressure case the liquid metal pressure at the runner entrance is assumed to
10 read about poor design effect on pQ
2 diagram
7.4. THE CORRECT PQ2 DIAGRAM
187
be 1.2[MPa] and for the constant power case the power loss is [1Kw]. filling time that
tmax ⩾ t =
V
√
Qmax 2Oz∗ P̄∗
(7.27)
The gate velocity is exhibited as a function of the ratio of the gate area to the conduit area as
shown in Figure 7.9 for a constant pressure and in Fig. 7.10 for a constant power.
35.00
∆P = 1.2[Mpa]
31.50
28.00
24.50
21.00
V[m/sec]
17.50
14.00
10.50
CD =0.55
a a
1.00m
a
a
` `
a
. . . . . . . . . . 0.75m
..
a
... . . . .. `
.
a
.
..
`
0.50m
`.
a
... `
a....
.
`
..
a
.
a
0.25m
..
a
`.....
.. `
a
..
..
.
..
`
a..
a
`
..
.
..
a
..
..
`
...
a....
`
a
..
`
.
..
a
..
..
`
.. .
a
a....
`
.. .
a
`
.
...
..
a
...
`
.. .
..
a
.
a..
.. .
`
`
a
...
`
..
...
.
...
`
. . ..
...
` `
.. . ..
a...
`
.. . ..
` `
.. . ..
.
..
..
. .. ..
.
..
7.00
3.50
a....
`
. .. .. .
.
..
..
..
..
a
`
0.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
A3
A2
Fig. 7.9 – Gate velocity area ratio.
TEMPT SPACE HOLDER
65.00
power = 1.0[Kw]
58.50
CD =0.55
1.00m
52.00
..........
45.50
`
a
0.75m
0.50m
0.25m
39.00
V[m/sec]
a`
a..`
a.`
aa. .a.
a.`
`
..
a.... ``
`
..
.
.a
`....
`....
a
..
32.50
..
26.00 ...
a..
`
19.50
..a
`...
...a
`.... a
..`
.. . a
..`
...
.
13.00
a a
a
.`
... .. ` a a
.. . .. `
a a
.. . .. `
a
.. . .. ..`
`. ... ` a a a a a
. .. ...
`. ... ` `
... ... ..
... ... ... ....` `
. .... .
. .`. ..
6.50
0.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
A3
A2
Fig. 7.10 – Gate velocities as a function of the
area ratio for constant power.
TEMPT SPACE HOLDER
7.4.2.2
General conclusions from example 7.4.2.1.4
For the constant pressure case the “common”11 assumption yields a constant velocity even for
a zero gate area.
11 As it is written in NADCA’s books
188
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
The solid line in Fig. 7.9 represents the gate velocity calculated based on the common
assumption of constant CD while the other lines are based on calculations which take into
account the runner geometry and the Re number. The results for constant CD represent “averaged” of the other results. The calculations of the velocity based on a constant CD value
are unrealistic. It overestimates the velocity for large gate area and underestimates for the
area ratio below ∼ 80% for the short runner and 35% for the long runner. Figure 7.9 exhibits
that there is a clear maximum gate velocity which depends on the runner design (here represented by the conduit length). The maximum indicates that the preferred situation is to be
on the “right hand side branch” because of shorter filling time. The gate velocity is doubled
for the excellent design compared with the gate velocity obtained from the poor design. This
indicates that the runner design is more important than the specific characteristic of the die
casting machine performance. Operating the die casting machine in the “right hand side” results in smaller requirements on the die casting machine because of a smaller filling time, and
therefore will require a smaller die casting machine.
For the constant power case, the gate velocity as a function of the area ratio is shown in
Figure 7.10. The common assumption of constant CD yields the gate velocity U3 ∝ A1 /A3
shown by the solid line. Again, the common assumption produces unrealistic results, with
the gate velocity approaching infinity as the area ratio approaches zero. Obviously, the results
with a constant CD over estimates the gate velocity for large area ratios and underestimates it
for small area ratios. The other lines describe the calculated gate velocity based on the runner
geometry. As before, a clear maximum can also be observed. For large area ratios, the gate
velocity with an excellent design is almost doubled compared to the values obtained with a
poor design. However, when the area ratio approaches zero, the gate velocity is insensitive to
the runner length and attains a maximum value at almost the same point.
In conclusion, this part has been shown that the use of the “common” pQ2 diagram with
the assumption of a constant CD may lead to very serious errors. Using the pQ2 diagram, the
engineer has to take into account the effects of the variation of the gate area on the discharge
coefficient, CD , value. The two examples given inhere do not represent the characteristics of
the die casting machine. However, more detailed calculations shows that the constant pressure is in control when the plunger is small compared to the other machine dimensions and
when the runner system is very poorly designed. Otherwise, the combination of the pressure
and power limitations results in the characteristics of the die casting machine which has to
be solved.
7.4. THE CORRECT PQ2 DIAGRAM
189
pressure
efficiency
U3
‘‘common’’ model
power
realistic velocity
P
power
Q
A3
U3 as a function of gate area, A3
Fig. a
η
Fig. b General characteristic of a pump.
Fig. 7.11 – Velocity, U3 as a function of the gate area, A3 and the general characteristic of a pump
7.4.2.3
The die casting machine characteristic effects
There are two type of operations of the die casting machine, one) the die machine is operated
directly by hydraulic pump (mostly on the old machines). two) utilizing the non continuous
demand for the power, the power is stored in a container and released when need (mostly on
the newer machines). The container is normally a large tank contain nitrogen and hydraulic
liquid12 . The effects of the tank size and gas/liquid ratio on the pressure and flow rate can
easily be derived.
Meta
The power supply from the tank with can consider almost as a constant pressure but
the line to actuator is with variable resistance which is a function of the liquid velocity.
The resistance can be consider, for a certain range, as a linear function of the velocity
square, “UB 2 ”. Hence, the famous a assumption of the “common” die casting machine
p ∝ Q2 .
Meta End
The characteristic of the various pumps have been studied extensively in the past (Fairbanks
1959). The die casting machine is a pump with some improvements which are patented by
different manufactures. The new configurations, such as double pushing cylinders, change
somewhat the characteristics of the die casting machines. First let discuss some general characteristic of a pump (issues like impeller, speed are out of the scope of this discussion). A pump
12 This similar to operation of water system in a ship, many of the characteristics are the same. Furthermore, the
same differential equations are governing the situation. The typical questions such as the necessarily container size
and the ratio of gas to hydraulic liquid were part of my study in high school (probably the simplified version of the
real case). If demand to this material raised, I will insert it here in the future.
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
190
is mechanical devise that transfers and electrical power (mostly) into “hydraulic” power. A
typical characteristic of a pump are described in Figure ??.
Two similar pumps can be connect in
two way series and parallel. The serious
connection increase mostly the pressure as
shown in ??. The series connection if “normalized” is very close to the original pump.
However, the parallel connection when “normalized” show a better performance.
To study the effects of the die casting
machine performances, the following funcFig. 7.12 – Various die casting machine performances
tions are examined (see Fig. 7.12):
1
0.8
0.6
P
0.4
P= 1 − Q
P= 1 − Q
4
2
P= 1 − Q
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Q
Q̄ =
(7.28a)
1 − P̄
p
1 − P̄
p
4
1 − P̄
Q̄ =
Q̄ =
(7.28b)
(7.28c)
The functions (7.28a), (7.28b) and (7.28c) represent a die casting machine with a poor
performance, the common performance, and a die casting machine with an excellent performance, respectively.
Combining Eq. (7.21) with Eq. (7.28) yields
p
1 − P̄ = 2OzP̄
(7.29)
p
p
1 − P̄ = 2OzP̄
(7.30)
p
p
4
1 − P̄ = 2OzP̄
(7.31)
rearranging equation (7.29) yields
P̄2 − 2(1 + Oz)P̄ + 1 =
0
(7.32)
1 − P̄(1 + 2Oz) =
0
(7.33)
0
(7.34)
2
4OzP̄ + P̄ − 1 =
Solving Eq. (7.32) for P̄, and taking only the possible physical solution, yields
p
P̄ =
1 + Oz − (2 + Oz) Oz
1
P̄ =
1 + 2 Oz
p
1 + 16 Oz2 − 1
P̄ =
8Oz2
(7.35)
(7.36)
(7.37)
7.4. THE CORRECT PQ2 DIAGRAM
1.00
0.90
0.80
0.70
h 0.60
P
0.50
0.40
0.30
0.20
0.10
0.00
191
..
..
..
..
..
h
hh 4
..
..
0.45
..
P = 1−Q
..
..
..
..
..
h
h2
..
..
0.40
..
P = 1−Q
..
..
..
..
..
h
h 0.35
..
..
..
.
.
.
.
.
.
.
.
.
.
P = 1−Q
..
..
..
..
..
..
0.30
..
..
..
..
..
..
.
..
..
0.25
..
..
..
..
..
..
..
..
..
0.20
..
.
..
..
..
..
..
..
..
0.15
..
..
...
..
...
..
...
..
....
0.10
..
..
..
1.00
2.00
3.00
4.00
..
..
..
...
....
.....
. ... ... ... ..
. ... . . ... ... ... ... ..
. . . ... ... ... ... ... . . ... ... ... ... ... . . ..
. ... ... ... ... . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ..
0.00
3.00
6.00
9.00
12.00
15.00
18.00
21.00
24.00
27.00
30.00
Oz
Fig. 7.13 – Reduced pressure, P̄, for various machine performances as a function of the Oz number.
The reduced pressure, P̄, is plotted as a function of the Oz number for the three die
casting machine performances as shown in Figure 7.13.
Figure 7.13 demonstrates that P̄ monotonically decreases with an increase in the Oz
number for all the machine performances. All the three results convert to the same line which
is a plateau after Oz = 20. For large Oz numbers the reduced pressure, P̄, can be considered
to be constant P̄ ≃ 0.025. The gate velocity, in this case, is
s
Pmax
U3 ≃ 0.22CD
(7.38)
ρ
The Ozer number strongly depends on the discharge coefficient, CD , and Pmax . The value
of Qmax is relatively insensitive to the size of the die casting machine. Thus, this equation
is applicable to a well designed runner (large CD ) and/or a large die casting machine (large
Pmax ).
The reduced pressure for a very small value
of the Oz number equals to one, P̄ ≃ 1 or
rode
Pmax = P1 , due to the large resistance in
plunger
hydraulic
piston
the runner (when the resistance in the runner
Fig. 7.14 – Schematic of the plunger and piston
approaches infinity, KF → ∞, then P̄ = 1).
balance forces.
Hence, the gate velocity is determined by the
approximation of
PB 2
D B PB 1
DR
atmospheric
pressure
P1
D1
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
192
U3 ≃ CD
s
2 Pmax
ρ
(7.39)
The difference between the various machine performances is more considerable in the middle
range of the Oz numbers. A better machine performance produces a higher reduced pressure,
P̄. The preferred situation is when the Oz number is large and thus indicates that the machine
performance is less important than the runner design parameters. This observation is further
elucidated in view of Figures 7.9 and 7.10.
7.4.2.4
(why? perhaps to create a question for the students)
Plunger area/diameter effects
The pressure at the plunger tip can be evaluated from a balance forces acts on the hydraulic
piston and plunger as shown in Figure 7.14. The atmospheric pressure that acting on the left
side of the plunger is neglected. Assuming a steady state and neglecting the friction, the forces
balance on the rod yields
D 2π
DB 2 π
D 2π
(PB1 − PB2 ) + R PB2 = 1 P1
4
4
4
(7.40)
In particular, in the stationary case the maximum pressure obtains
DB 2 π
D 2π
D 2π
(PB1 − PB2 )|max + R
PB2 |max = 1
P1 |max
4
4
4
(7.41)
The equation (7.41) is reduced when the rode area is negligible; plus, notice that P1 |max =
Pmax to read
DB 2 π
D 2π
(PB1 − PB2 )|max = 1 Pmax
4
4
(7.42)
7.4. THE CORRECT PQ2 DIAGRAM
1.0 ...
..
..
..
..
0.9
..
..
..
..
0.8
..
..
..
..
0.7
..
..
..
..
..
0.6
..
..
..
..
0.5
..
..
..
..
0.4
..
..
..
..
0.3
..
..
..
..
..
0.2
..
...
.
0.1
η
1.0
2.0
y
. . . . . . . .. . ..
...
...
0.0
0.0
193
3.0
....
. . . . . ..
. ... ... ... ... ..
. ... ... ... ... ... ... ... ... ... ...
4.0
5.0
6.0
7.0
... ... ... ... ... ... ... ... ... ... ... ... ... ..
8.0
9.0
10.0
χ
Fig. 7.15 – Reduced liquid metal pressure at the plunger tip and reduced gate velocity as a function of the
reduced plunger diameter.
Rearranging equation (7.42) yields
DB
D1
2
=
Pmax
=⇒ Pmax = (PB1 − PB2 )|max
(PB1 − PB2 )|max
DB
D1
2
(7.43)
13
The gate velocity relates to the liquid metal pressure at plunger tip according to the
following equation combining equation (7.7) and (??) yields
v
2
s u
u
DB
(P
−
P
)|
u
B1
B2
max
D1
2u
U3 = CD
(7.44)
2
2
ρt
2 CD A3
B
1 + ρ Qmax
(PB1 − PB2 )|max D
D1
13 Note that P |
1 max ̸= [P1 ]max . The difference is that P1 |max represents the maximum pressure of the liquid
metal at plunger tip in the stationary case, where as [P1 ]max represents the value of the maximum pressure of the
liquid metal at the plunger tip that can be achieved when hydraulic pressure within the piston is varied. The former
represents only the die casting machine and the shot sleeve, while the latter represents the combination of the die
casting machine (and shot sleeve) and the runner system.
Equation (7.14) demonstrates that the value of [P1 ]max is independent of Pmax (for large values of Pmax ) under
the assumptions in which this equation was attained (the “common” die casting machine performance, etc). This suggests that a smaller die casting machine can achieve the same job assuming average performance die casing machine.
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
194
Under the assumption that the machine characteristic is P1 ∝ Q̄2 =⇒ P̄ = 1 − Q̄2 ,
Meta
the solution for the intersection point is given by equation ? To study equation (7.44),
let’s define
r
ρ
D1
Qmax
(7.45)
χ=
(PB1 − PB2 )|max CD A3
DB
and the reduced gate velocity
y=
U3 A3
Qmax
Using these definitions, Eq. (7.44) is converted to a simpler form:
s
1
y=
χ2 + 1
With these definitions, and denoting
2 CD A3 2
η = P1
= 2 Oz P̄
ρ Qmax
(7.46)
(7.47)
(7.48)
one can obtain from Eq. (7.35) that (make a question about how to do it?)
η=
1
χ2 + 1
(7.49)
The coefficients of P1 in equation (7.48) and D1 in equation (7.45) are assumed constant
according to the “common” pQ2 diagram. Thus, the plot of y and η as a function of χ
represent the affect of the plunger diameter on the reduced gate velocity and reduced
pressure. The gate velocity and the liquid metal pressure at plunger tip decreases with
an increase in the plunger diameters, as shown in Figure 7.15 according to equations
(7.47) and (7.49).
Meta End
why? should be included in the
end.
A control volume as it is shown in Fig. 7.5 is constructed to study the effect of the plunger
diameter, (which includes the plunger with the rode, hydraulic piston, and shot sleeve, but
which does not include the hydraulic liquid or the liquid metal jet). The control volume is
stationary around the shot sleeve and is moving with the hydraulic piston. Applying the first
law of thermodynamics, when that the atmospheric pressure is assumed negligible and neglecting the dissipation energy, yields
!
!
!
Uin 2
Uout 2
dm
Uc.v. 2
Q̇ + ṁin hin +
= ṁout hout +
+
e+
+ Ẇc.v. (7.50)
2
2
dt c.v.
2
7.4. THE CORRECT PQ2 DIAGRAM
195
In writing equation (7.50), it should be noticed that the only change in the control volume is in the shot sleeve. The heat transfer can be neglected, since the filling process is very
rapid. There is no flow into the control volume (neglecting the air flow into the back side of
the plunger and the change of kinetic energy of the air, why?), and therefore the second term
on the right hand side can be omitted. Applying mass conservation on the control volume for
the liquid metal yields
dm
= −ṁout
dt c.v.
(7.51)
The boundary work on the control volume is done by the left hand side of the plunger and
can be expressed by
(7.52)
Ẇc.v. = −(PB1 − PB2 )AB U1
The mass flow rate out can be related to the gate velocity
(7.53)
ṁout = ρA3 U3
Mass conservation on the liquid metal in the shot sleeve and the runner yields
A1 U1 = A3 =⇒ U1 2 = U23
A3
A1
2
Substituting equations (7.51-7.54) into equation (7.50) yields
"
U2
(PB1 − PB2 )AB U3 A1 = ρA3 U3 (hout − e) + 3
2
(7.54)
1−
A3
A1
2 !#
(7.55)
Rearranging equation (7.55) yields
U2
A
(PB1 − PB2 ) B = (hout − e) + 3
A1 ρ
2
1−
A3
A1
2 !
(7.56)
Solving for U3 yields
v h
i
u
u 2 (PB1 − PB2 ) AB − (hout − e)
A1 ρ
u
U3 = u
2 t
3
1− A
A1
Or in term of the maximum values of the hydraulic piston
v h
i
u
u 2 (PB1 −PB2 )|max AB − (hout − e)
1+2 Oz
A1 ρ
u
U3 = u
2 t
3
1− A
A1
(7.57)
(7.58)
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
196
When the term (hout − e) is neglected (Cp ∼ Cv for liquid metal)
v
u
u (PB1 − PB2 )|max
u2
u
1 + 2Oz U3 = u
2
t
3
1− A
A1
AB
A1 ρ
(7.59)
Normalizing the gate velocity Eq. (7.59) yields
v
U3 A3 u
u
=u
y=
Qmax t
CD
2 3
2
χ [1 + 2 Oz] 1 − A
A1
(7.60)
The expression (7.60) is a very complicated function of A1 . It can be shown that when
the plunger diameter approaches infinity, D1 → ∞ (or when A1 → ∞) then the gate velocity
approaches U3 → 0. Conversely, the gate velocity, U3 → 0, when the plunger diameter,
D1 → 0. This occurs because mostly K → ∞ and CD → 0. Thus, there is at least one plunger
diameter that creates maximum velocity (see figure 7.16). A more detailed study shows that
depending on the physics in the situation, more than one local maximum can occur. With
a small plunger diameter, the gate velocity approaches zero because CD approaches infinity.
For a large plunger diameter, the gate velocity approaches zero because the pressure difference
acting on the runner is approaching zero. The mathematical expression for the maximum gate
velocity takes several pages, and therefore is not shown here. However, for practical purposes,
the maximum velocity can easily (relatively) be calculated by using a computer program such
as DiePerfect™.
7.4.2.5
Machine size effect
The question how large the die casting machine depends on how efficient it is used. To
maximized the utilization of the die casting
machine we must understand under what condition it happens. It is important to realize that
the injection of the liquid metal into the cavity requires power. The power, we can extract
from a machine, depend on the plunger velocity and other parameters. We would like to design a process so that power extraction is maximized.
U3
‘‘common’’ model
realistic velocity
po
ssib
le m
ax
A1
Fig. 7.16 – The gate velocity, U3 as a function of
the plunger area, A1 .
7.4. THE CORRECT PQ2 DIAGRAM
197
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
...
0.00 .. .. ..
0.00
...
...
0.10
..
.
...
...
...
.. .
.
..
..
..
.. .
.
..
..
..
.. .
.
..
..
..
.. .
..
..
..
.. .
..
..
..
..
.. .
.
0.20
0.30
0.40
0.50
h
Q
...
.....
0.60
. .... . .
...
0.70
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
0.80
0.90
1.00
Fig. 7.17 – The reduced power of the die casting machine as a function of the normalized flow rate.
Let’s defined normalized machine size effect
pwrm =
Q̄∆P
≃ Q̄ × P̄
Pmax × Qmax
(7.61)
Every die casting machine has a characteristic curve on the pQ2 diagram as well. Assuming that the die casting machine has the “common” characteristic, P̄ = 1 − Q̄2 , the normalized power can be expressed
pwrm = Q̄(1 − Q̄2 ) = Q̄2 − Q̄3
(7.62)
where pwrm is the machine power normalized by Pmax × Qmax . The maximum power
of this kind of machine is at 2/3 of the normalized flow rate, Q̄, as shown in Figure ??. It is
recommended to design the process so the flow rate occurs at the vicinity of the maximum
of the power. For a range of 1/3 of Q̄ that is from 0.5Q̄ to 0.83Q̄, the average power is 0.1388
Pmax Qmax , as shown in Figure ?? by the shadowed rectangular. One may notice that this
value is above the capability of the die casting machine in two ranges of the flow rate. The
reason that this number is used is because with some improvements of the runner design
the job can be performed on this machine, and there is no need to move the job to a larger
machine14 .
7.4.2.6
Precondition effect (wave formation)
Meta
discussion when Q1 ̸= Q3
14 Assuming that requirements on the clamping force is meet.
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
198
Meta End
7.4.3
Poor design effects
Meta
discussed the changes when different velocities are in different gates. Expanded on the
sudden change to turbulent flow in one of the branches.
Meta End
7.4.4
insert only general remarks until
the paper will submitted for publication
Transient effects
Under construction
To put the discussion about the inertia of the system and compressibility.
the magnitude analysis before intensification effects
insert the notes from the yellow
folder
7.5
Design Process
Now with these pieces of information how one design the process/runner system. A design
engineer in a local company have told me that he can draw very quickly the design for the
mold and start doing the experiments until he gets the products running well. Well, the important part should not be how quickly you get it to try on your machine but rather how
quickly you can produce a good quality product and how cheap ( little scrap as possible and
smaller die casting machine). Money is the most important factor in the production. This
design process is longer than just drawing the runner and it requires some work. However,
getting the production going is much more faster in most cases and cheaper (less design and
undesign scrap and less experiments/starting cost). Hence, for given die geometry, four conditions (actually there are more) need to satisfied
General relationship between
runner hydraulic diameter and
plunger diameter.
∂U3
=0
∂A1
(7.63)
∂U3
=0
∂A3
(7.64)
The clamping force, and satisfy the power requirements.
For these criteria the designer has to check the runner design to see if gate velocity
are around the recommended range. A possible answer has to come from financial considerations, since we are in the business of die casting to make money. Hence, the optimum
diameter is the one which will cost the least H(the minimum cost). ow, then, does the plunger
size determine cost? It has been shown that plunger diameter has a value where maximum
gate velocity is created.
7.6. THE INTENSIFICATION CONSIDERATION
199
A very large diameter requires a very large die casting machine (due to physical size and
the weight of the plunger). So, one has to chose as first approximation the largest plunger on
a smallest die casting machine. Another factor has to be taken into consideration is the scrap
created in the shot sleeve. Obviously, the liquid metal in the sleeve has to be the last place to
solidify. This requires the biscuit to be of at least the same thickness as the runner.
Trunner = Tbiscuit
(7.65)
Therefore, the scrap volume should be
πD1 2
πD1 2
Tbiscuit =⇒
Trunner
4
4
(7.66)
When the scrap in the shot sleeve becomes significant, compared to scrap of the runner
πD1 2
Trunner = L̄Trunner
4
Thus, the plunger diameter has to be in the range of
r
4
D1 =
L̄
π
7.6
(7.67)
(7.68)
The Intensification Consideration
Intensification is a process in which pressure is increased making the liquid metal flows during the solidification process to ensure compensation for the solidification shrinkage of the
liquid metal (up to 20%). The intensification is applied by two methods: one) by applying additional pump, two) by increasing the area of the actuator (the multiplier method, or the prefill
method)15 .
. The first method does not increase the intensification force to “Pmax ” by much. However, the second method, commonly used today in the industry, can increase considerably the
ratio.
To discussed that the plunger diameter should not be use as varying the plunger diameter to determine the gate velocity
put schematic figure of how it is
done from the patent by die casting companies
Meta
Analysis of the forces demonstrates that as first approximation the plunger diameter
does not contribute any additional force toward pushing the liquid metal.
Meta End
A very small plunger diameter creates faster solidification, and therefore the actual
force is reduced. Conversely, a very large plunger diameter creates a very small pressure for
driving the liquid metal.
15 A note for the manufactures, if you would like to have your system described here with its advantages, please
drop me a line and I will discuss with you about the material that I need. I will not charge you any money.
why? to put discussion
discuss the the resistance as a
function of the diameter
CHAPTER 7. PQ2 DIAGRAM CALCULATIONS
200
7.7
Summary
In this chapter it has been shown that the “common” diagram is not valid and produces unrealistic trends therefore has no value what so ever16 . The reformed pQ2 diagram was introduced. The mathematical theory/presentation based on established scientific principles was
introduced. The effects of various important parameters was discussed. The method of designing the die casting process was discussed. The plunger diameter has a value for which the
gate velocity has a maximum. For D1 → 0 gate velocity, U3 → 0 when D1 → ∞ the same
happen U3 → 0. Thus, this maximum gate velocity determines whether an increase in the
plunger diameter will result in an increase in the gate velocity or not. An alternative way has
been proposed to determine the plunger diameter.
16 Beside the historical value
Garber concluded that his model was not able to predict an acceptable value
for critical velocity for fill percentages lower than 50% . . .
Brevick, Ohio
8
Critical Slow Plunger Velocity
8.1
Introduction
This Chapter deals with the first stage of the injection in a cold chamber machine in which
the desire (mostly) is to expel maximum air/gas from the shot sleeve. Porosity is a major
production problem in which air/gas porosity constitutes a large portion. Minimization of
Air Entrainment in the Shot Sleeve (AESS) is a prerequisite for reducing air/gas porosity.
This can be achieved by moving the plunger at a specific speed also known as the critical
slow plunger velocity. It happens that this issue is related to the hydraulic jump, which was
discussed in the previous Chapters B (accidentally? you thought!).
The “common” model, also known as Garber’s model, with its extensions made by Brevick1 , Miller2 , and EKK’s model are presented first here. The basic fundamental errors of
these models are presented. Later, the reformed and “simple” model is described. It followed
by the transient and poor design effects3 . Afterwords, as usual questions are given at the end
of the chapter.
8.2
The “common” models
In this section the “common” models are described. Since the “popular” model also known as
Garber’s model never work (even by its own creator)4 , several other models have appeared.
1 Industrial and Systems Engineering (ISE) Graduate Studies Chair, ISE department at The Ohio State University
2 The chair of ISE Dept.
at OSU
3 It be added in the next addition
4 I wonder if Garber and later Brevick have ever considered that their the models were simply totally false.
201
202
CHAPTER 8. CRITICAL SLOW PLUNGER VELOCITY
These models are described here to have a clearer picture of what was in the pre Bar–Meir’s
model. First, a description of Garber’s model is given later Brevick’s two models along with
Miller’s model5 are described briefly. Lastly, the EKK’s numerical model is described.
8.2.1
Garber’s model
The description in this section is based on
one of the most cited paper in the die casting
v
v
research (Garber 1982). Garber’s model deals
h
v
only with a plug flow in a circular cross–
P
h
P
section. In this section, we “improve” the
model to include any geometry cross section
with any velocity profile6 .
Fig. 8.1 – A schematic of wave formation in stationary coConsider a duct (any cross section)
ordinates
with a liquid at level h2 and a plunger moving
from the left to the right, as shown in Figure
8.1. Assuming a quasi steady flow is established after a very short period of time, a unique
height, h1 , and a unique wave velocity, Vw , for a given constant plunger velocity, Vp are created. The liquid in the substrate ahead of the wave is still, its height, h2 , is determined by the
initial fill. Once the height, h1 , exceeds the height of the shot sleeve, H, there will be splashing. The splashing occurs because no equilibrium can be achieved (see Figure 8.2a). For h1
smaller than H, a reflecting wave from the opposite wall appears resulting in an enhanced air
entrainment (see Figure 8.2b). Thus, the preferred situation is when h1 = H (in circular shape
H = 2R) in which case no splashing or a reflecting wave result.
1
p
w
2
H
1
1
1
2
2
L(t)
Fig a. A schematic of built wave.
Fig b. A schematic of reflecting
wave.
Fig. 8.2 – The left graph depicts the “common” pQ2 version. The right graph depicts Pmax and Qmax as a
function of the plunger diameter according to “common” model.
It is easier to model the wave with
locity, as shown in Figure 8.3. With the
ary, the plunger moves back at a velocity
the right to the left. Dashed line shows
coordinates that move at the wave vemoving coordinate, the wave is station(Vw − Vp ), and the liquid moves from
the stationary control volume.
5 This model was developed at Ohio State University by Miller’s Group in the early 1990’s.
6 This
addition to the original Garber’s paper is derived here. I assumed that in this case, some mathematics will
not hurt the presentation.
203
8.2. THE “COMMON” MODELS
Mass conservation of the liquid in the
control volume reads:
Z
Z
ρVw dA =
ρ(v1 − Vp )dA (8.1)
A2
1
(vw − vp)
v=0
2R
h1 v1
vw
P1
A1
2
P2 h2
L(t)
where v1 is the local velocity. Under quasiFig. 8.3 – A schematic of the wave with moving coordisteady conditions, the corresponding average
nates
velocity equals the plunger velocity:
Z
1
v1 dA = v1 = Vp
(8.2)
A 1 A1
What is justification for equation 8.2? Assuming that heat transfer can be neglected because
of the short process duration7 . Therefore, the liquid metal density (which is a function of
temperature) can be assumed to be constant. Under the above assumptions, equation (8.1) can
be simplified to
Z hi
Vw A(h2 ) = (Vw − Vp )A(h1 ) ; A(hi ) =
dA
(8.3)
0
Where i in this case can take the value of 1 or 2. Thus,
Vw
= f(h12 )
(Vw − Vp )
(8.4)
1)
where f(h12 ) = A(h
A(h2 ) is a dimensionless function. Equation (8.4) can be transformed into
a dimensionless form:
ṽ
(ṽ − 1)
f(h12 )
=⇒ ṽ =
f(h12 ) − 1
(8.5)
f(h12 ) =
(8.6)
where ṽ = VVwp . Show that A(h1 ) = 2πR2 for h1 = 2R Assuming energy is conserved
(the Garber’s model assumption), and under conditions of negligible heat transfer, the energy
conservation equation for the liquid in the control volume (see Figure 8.3) reads:
#
Z Z "
P B γE (Vw − Vp )2
P2 Vw 2
+
(Vw − Vp )dA =
+
Vw dA
(8.7)
ρ
2
ρ
2
A1
A2
where
γ=
1
A1 (Vw − Vp )3
Z
(Vw − v1 )3 dA =
A1
7 see Chapter 3 for a detailed discussion
1
A1 (ṽ − 1)3
Z
3
v
ṽ − 1
dA
Vp
A1
(8.8)
build a question about what happens if the temperature changes
by a few degrees. How much will
it affect equation 8.2 and other parameters?
204
CHAPTER 8. CRITICAL SLOW PLUNGER VELOCITY
under–construction The shape factor, γE , is introduced to account for possible deviations
of the velocity profile at section 1 from a pure plug flow. Note that in die casting, the flow
is pushed by the plunger and can be considered as an inlet flow into a duct. The typical Re
number is 105 , and for this value the entry length is greater than 50m, which is larger than
any shot sleeve by at least two orders of magnitude.
The pressure in the gas phase can be assumed to be constant. The hydrostatic pressure
in the liquid can be represent by Rȳc gρ (Rajaratnam 1965), where Rȳc is the center of the cross
section area. For a constant liquid density equation (8.7) can be rewritten as:
"
#
(Vw − Vp )2
Vw 2
Rȳc1 g + γE
(Vw − Vp )A(h1 ) = Rȳc2 g +
Vw A(h2 )
(8.9)
2
2
Garber (and later Brevick) put this equation plus several geometrical relationships as the solution. Here we continue to obtain an analytical solution. Defining a dimensionless parameter
Fr as
Fr =
Rg
,
Vp 2
(8.10)
Utilizing definition (8.10) and rearranging equation (8.9) yields
2FrE × ȳc1 + γE (ṽ − 1)2 = 2FrE × ȳc2 + ṽ2
Solving equation (8.11) for FrE the latter can be further rearranged to yield:
v
u
u 2(ȳc1 − ȳc2 )
FrE = t (1+γ )f(h
E
12 )
− γE
f(h12 )−1
(8.11)
(8.12)
Given the substrate height, equation (8.12) can be evaluated for the FrE , and the corresponding
plunger velocity ,Vp . which is defined by equation (8.10). This solution will be referred herein
as the “energy solution”.
8.2.2
Brevick’s Model
8.2.2.1
The square shot sleeve
Since Garber’s model never work Brevick and co–workers go on a “fishing expedition” in the
fluid mechanics literature to find equations to describe the wave. They found in Lamb’s book
several equations relating the wave velocity to the wave height for a deep liquid (water)8 . Since
these equations are for a two dimensional case, Brevick and co–workers built it for a squared
shot sleeve. Here are the equations that they used. The “instantaneous” height difference
(∆h = h1 − h2 ) is given as
2
Vp
∆h = h2 √
+ 1 − h2
(8.13)
2 gh2
8 I have checked the reference and I still puzzled by the equations they found?
205
8.2. THE “COMMON” MODELS
This equation (8.13), with little rearranging, obtained a new form
"s
#
p
h1
Vp = 2 gh2
−1
h2
(8.14)
The wave velocity is given by
Vw =
p
" s
gh2 3
∆h
1+
−2
h2
#
(8.15)
Brevick introduces the optimal plunger acceleration concept. “By plotting the height
and position of each incremental wave with time, their model is able to predict the ‘stability’ of
the resulting wave front when the top of the front has traveled the length of the shot sleeve.”9 .
They then performed experiments on this “miracle acceleration 10 .”
8.2.3
Brevick’s circular model
Probably, because it was clear to the authors that the previous model was only good for a
square shot sleeve 11 . They say let reuse Garber’s model for every short time steps and with
different velocity (acceleration).
8.2.4
Miller’s square model
Miller and his student borrowed a two dimensional model under assumption of turbulent
flow. They assumed that the flow is “infinite” turbulence and therefor it is a plug flow12 . Since
the solution was for 2D they naturally build model for a square shot sleeve13 . The mass balance
for square shot sleeve
(8.16)
Vw h2 = (Vw − Vp )h1
Momentum balance on the same control volume yield
"
#
P B (Vw − Vp )2
P2 Vw 2
+
(Vw − Vp )h1 =
+
Vw h2
ρ
2
ρ
2
(8.17)
and the solution of these two equations is
Frmiller
9 What an interesting idea??
Any physics?
1 h1
=
2 h2
h1
+1
h2
(8.18)
10 As to say this is not good enough a fun idea, they also “invented” a new acceleration units “cm/sec–cm”.
11 It is not clear whether they know that this equations are not applicable even for a square shot sleeve.
12 How they come–out with this conclusion?
13 Why are these two groups from the same university and the same department not familiar with each others
work.
206
8.3
CHAPTER 8. CRITICAL SLOW PLUNGER VELOCITY
The validity of the “common” models
8.3.1
Garber’s model
Energy is known to dissipate in a hydraulic jump in which case the equal sign in equation
(8.12) does not apply and the criterion for a nonsplashing operation would read
FrE < Froptimal
(8.19)
A considerable amount of research work has been carried out on this wave, which is known
in the scientific literature as the hydraulic jump. The hydraulic jump phenomenon has been
studied for the past 200 years. Unfortunately, Garber, ( and later other researchers in die
casting – such as Brevick and his students from Ohio State University (Brevick, Armentrout,
and Chu 1994), (Thome and Brevick 1995))14 , ignored the previous research. This is the real
reason that their model never works. Show the relative error created by Garber’s model when
the substrate height h2 is the varying parameter.
8.3.2
Brevick’s models
8.3.2.1
square model
There are two basic mistakes in this model, first) the basic equations are not applicable to the
shot sleeve situation, second) the square geometry is not found in the industry. To illustrate
why the equations Brevick chose are not valid, take the case where 1 > h1 /h2 > 4/9. For
that case Vw is positive and yet the hydraulic jump opposite to reality (h1 < h2 ).
8.3.2.2
Improved Garber’s model
Since Garber’s model is scientific erroneous any derivative that is based on it no better than
its foundation15 .
8.3.3
Miller’s model
The flow in the shot sleeve in not turbulent16 . The flow is a plug flow because entry length
problem17 .
Besides all this, the geometry of the shot sleeve is circular. This mistake is discussed in
the comparison in the discussion section of this chapter.
8.3.4
EKK’s model (numerical model)
This model based on numerical simulations based on the following assumptions: 1) the flow is
turbulent, 2) turbulence was assume to be isentropic homogeneous every where (kϵ model),
14 Even
with these major mistakes NADCA under the leadership of Gary Pribyl and Steve Udvardy continues to
award Mr. Brevick with additional grands to continue this research until now, Why?
15 I wonder how much NADCA paid Brevick for this research?
16 Unless someone can explain and/or prove otherwise.
17 see Chapter 3.
207
8.4. THE REFORMED MODEL
3) un–specified boundary conditions at the free interface (how they solve it with this kind of
condition?), and 4) unclear how they dealt with the “corner point” in which plunger perimeter
in which smart way is required to deal with zero velocity of the sleeve and known velocity of
plunger.
Several other assumptions implicitly are in that work18 such as no heat transfer, a constant pressure in the sleeve etc.
According to their calculation a jet exist somewhere in the flow field. They use the kϵ
model for a field with zero velocity! They claim that they found that the critical velocity to be
the same as in Garber’s model. The researchers have found same results regardless the model
used, turbulent and laminar flow!! One can only wonder if the usage of kϵ model (even for
zero velocity field) was enough to produce these erroneous results or perhaps the problem
lays within the code itself19 .
8.4
The Reformed Model
The hydraulic jump appears in steady–state and unsteady–state situations. The hydraulic
jump also appears when using different cross–sections, such as square, circular, and trapezoidal shapes. The hydraulic jump can be moving or stationary. The “wave” in the shot sleeve
is a moving hydraulic jump in a circular cross–section. For this analysis, it does not matter if
the jump is moving or not. The most important thing to understand is that a large portion of
the energy is lost and that this cannot be neglected. All the fluid mechanics books20 show that
Garber’s formulation is not acceptable and a different approach has to be employed. Today,
the solution is available to die casters in a form of a computer program – DiePerfect™.
8.4.1
The reformed model
In this section the momentum conservation principle is applied on the control volume in
Figure 8.3. For large Re (∼ 105 ) the wall shear stress can be neglected compared to the inertial
terms (the wave is assumed to have a negligible length). The momentum balance reads:
Z h
Z h
i
i
P B + ργM (Vw − Vp )2 dA =
P2 + ρVw 2 dA
(8.20)
A1
where
γM =
1
A1 (Vw − Vp )2
A2
Z
(Vw − v1 )2 dA =
A1
1
A1 (ṽ − 1)2
Z
2
v
ṽ − 1
dA
Vp
A1
(8.21)
Given the velocity profile v1 , the shape factor γM can be obtained in terms of ṽ. The expressions for γM for laminar and turbulent velocity profiles at section 1 easily can be calculated.
Based on the assumptions used in the previous section, equation (8.20) reads:
h
i
h
i
Rȳc1 g + γM (Vw − Vp )2 A(h1 ) = Rȳc2 g + Vw 2 A(h2 )
(8.22)
18 This paper is a good example of poor research related to a poor presentation and text processing.
19 see remark on page 150
20 in the last 100 years
208
CHAPTER 8. CRITICAL SLOW PLUNGER VELOCITY
Rearranging equation (8.22) into a dimensionless form yields:
h
i
f(h12 ) ȳc1 Fr + γM (ṽ − 1)2 = ȳc2 Fr + ṽ2
Combining equations (8.5) and (8.23) yields
h
i2
h
i2
f(h12 )
f(h )
− γM f(h12 ) f(h 12
−1
f(h12 )−1
12 )−1
FrM =
[f(h12 )ȳc1 − ȳc2 ]
(8.23)
(8.24)
where FrM is the Fr number which evolves from the momentum conservation equation.
Equation (8.24) is the analogue of equation (8.12) and will be referred herein as the “Bar–Meir’s
solution”.
TO WORK ON THIS LATER
21 and the “energy solution” can be presented in a simple form. Moreover, these solutions can be applied to any cross section for the transition of the free surface flow to pressurized flow. The discussion here focuses on the circular cross section, since it is the only
one used by diecasters. Solutions for other velocity profiles, such as laminar flow (Poiseuille
paraboloid), are discussed in the Appendix 22 . Note that the Froude number is based on the
plunger velocity and not on the upstream velocity commonly used in the two–dimensional
hydraulic jump.
The experimental data obtained by Garber , and Karni and the transition from the free
surface flow to pressurized flow represented by equations (8.12) and (8.24) for a circular cross
section are presented in Figure ?? for a plug flow. The Miller’s model (two dimensional) of
the hydraulic jump is also presented in Figure ??. This Figure shows clearly that the “Bar–
Meir’s solution” is in agreement with Karni’s experimental results. The agreement between
Garber’s experimental results and the “Bar–Meir’s solution,” with the exception of one point
(at h2 = R), is good.
The experimental results obtained by Karni were taken when the critical velocity was
obtained (liquid reached the pipe crown) while the experimental results from Garber are interpretation (kind of average) of subcritical velocities and supercritical velocities with the
exception of the one point at h2 /R = 1.3 (which is very closed to the “Bar–Meir’s solution”).
Hence, it is reasonable to assume that the accuracy of Karni’s results is better than Garber’s
results. However, these data points have to be taken with some caution23 . Non of the experimental data sets were checked if a steady state was achieved and it is not clear how the
measurements carried out.
It is widely accepted that in the two dimensional hydraulic jump small and large eddies
are created which are responsible for the large energy dissipation (Henderson 1966). Therefore, energy conservation cannot be used to describe the hydraulic jump heights. The same
can be said for the hydraulic jump in different geometries. Of course, the same has to be said
for the circular cross section. Thus, the plunger velocity has to be greater than the one obtained by Garber’s model, which can be observed in Figure ??. The Froude number for the
21 This model was constructed with a cooperation of a another researcher.
22 To appear in the next addition.
23 Results of good experiments performed by serious researchers are welcome.
209
8.5. SUMMARY
Garber’s model is larger than the Froude number obtained in the experimental results. Froude
number inversely proportional to square of the plunger velocity, Fr ∝ 1/Vp 2 and hence the
velocity is smaller. The Garber’s model therefore underestimates the plunger velocity.
8.4.2
Design process
To obtain the critical slow plunger velocity, one has to follow this procedure:
1. Calculate/estimate the weight of the liquid metal.
2. Calculate the volume of the liquid metal (make sure that you use the liquid phase property and not the solid phase).
3. Calculate the percentage of filling in the shot sleeve,
height
.
r
4. Find the Fr number from Figure ??.
5. Use the Fr number found to calculate the plunger velocity by using equation (8.10).
8.5
Summary
In this Chapter we analyzed the flow in the shot sleeve and developed a explicit expression
to calculated the required plunger velocity. It has been shown that Garber’s model is totally
wrong and therefore Brevick’s model is necessarily erroneous as well. The same can be said
to all the other models discussed in this Chapter. The connection between the “wave” and
the hydraulic jump has been explained. The method for calculating the critical slow plunger
velocity has been provided.
8.6
Questions
210
CHAPTER 8. CRITICAL SLOW PLUNGER VELOCITY
9
Venting System Design
The difference between the two is expressed by changing standard
atmospheric ambient conditions to those existing in the vacuum tank.
Miller’s student, p. 102
9.1
Introduction
Proper design of the venting system is one of the requirements for reducing air/gas porosity.
Porosity due to entrainment of gases constitutes a large portion of the total porosity, especially
when the cast walls are very thin (see Figure ??). The main causes of air/gas porosity are
insufficient vent area, lubricant evaporation (reaction processes), incorrect placement of the
vents, and the mixing processes. The present chapter considers the influence of the vent area
(in atmospheric and vacuum venting) on the residual gas (in the die) at the end of the filling
process.
Atmospheric venting, the most widely
used casting method, is one in which the vent
is opened to the atmosphere and is referred
maximum
shrinkage
herein as air venting. Only in extreme cases
porosity
porosity
are other solutions required, such as vacuum
venting, Pore Free Technique (in zinc and
aluminum casting) and squeeze casting. Vacuum is applied to extract air/gas from the
wall thickness
mold before it has the opportunity to mix
with the liquid metal and it is call vacuum Fig. 9.1 – The relative shrinkage porosity as a function of
the casting thickness.
venting. The Pore Free technique is a varia-
211
212
CHAPTER 9. VENTING SYSTEM DESIGN
tion of the vacuum venting in which the oxygen is introduced into the cavity to replace the air and to react with the liquid metal, and
therefore creates a vacuum (Bar-Meir 1995c). Squeeze casting is a different approach in which
the surface tension is increased to reduce the possible mixing processes (smaller Re number
as well). The gases in the shot sleeve and cavity are made mostly of air and therefore the term
“air” is used hereafter. These three “solutions” are cumbersome and create a far more expensive process. In this chapter, a qualitative discussion on when these solutions should be used
and when they are not needed is presented.
Obviously, the best ventilation is achieved when a relatively large vent area is designed.
However, to minimize the secondary machining (such as trimming), to ensure freezing within
the venting system, and to ensure breakage outside the cast mold, vents have to be very narrow. A typical size of vent thicknesses range from 1–2[mm]. These conflicting requirements
on the vent area suggest an optimum area. As usual the “common” approach is described the
errors are presented and the reformed model is described.
9.2
9.2.1
The “common” models
Early (etc.) model
The first model dealing with the extraction of air from the cavity was done by Sachs. In
this model, Sachs developed a model for the gas flow from a die cavity based on the following
assumptions: 1) the gas undergoes an isentropic process in the die cavity, 2) a quasi steady state
exists, 3) the only resistance to the gas flow is at the entrance of the vent, 4) a “maximum mass
flow rate is present”, and 5) the liquid metal has no surface tension, thus the metal pressure
is equal to the gas pressure. Sachs also differentiated between two cases: choked flow and
un–choked flow (but this differentiation did not come into play in his model). Assumption
3 requires that for choked flow the pressure ratio be about two between the cavity and vent
exit.
Almost the same model was repeat by several researchers1 . All these models, with the
exception of Veinik , neglect the friction in the venting system. The vent design in a commercial system includes at least an exit, several ducts, and several abrupt expansions/contractions
in which the resistance coefficient ( 4fL
D see (Shapiro 1953, page 163)) can be evaluated to be
4fL
larger than 3 and a typical value of D is about 7 or more. In this case, the pressure ratio for
the choking condition is at least 3 and the pressure ratio reaches this value only after about
2/3 of the piston stroke is elapsed. It can be shown that when the flow is choked the pressure
in the cavity does not remain constant as assumed in the models but increases exponentially.
9.2.2
Miller’s model
Miller and his student, in the early 90’s, constructed a model to account for the friction in the
venting system. They based their model on the following assumptions:
1 Apparently, no literature survey was required/available/needed at that time.
213
9.3. GENERAL DISCUSSION
1. No heat transfer
2. Isothermal flow (constant temperature) in the entrance to vent (according to the authors in the presentation)
3. Fanno flow in the rest vent
4. Air/gas obeys the ideal gas model
Miller and his student described the calculation procedures for the two case as choked
and unchoked conditions. The calculations for the choked case are standard and can be found
in any book about Fanno flow but with an interesting twist. The conditions in the mold
and the sleeve are calculated according the ambient condition (see the smart quote of this
Chapter)2 . The calculations about unchoked case are very interesting and will be discussed
here in a little more details. The calculations procedure for the unchoked as the following:
• Assume Min number (entrance Mach number to the vent) lower than Min for choked
condition
• Calculate the corresponding star (choked conditions)
temperature ratio for the assume Min number
• Calculate the difference between the calculated
4fL
D
4fL
D ,
the pressure ratio, and the
and the actual
4fL
D .
• Use the difference 4fL
D to calculate the double stars (theoretical exit) conditions based
on the ambient conditions.
• Calculated the conditions in the die based on the double star conditions.
Now the mass flow out is determined by mass conservation.
Of course, these calculations are erroneous. In choked flow, the conditions are determined only and only by up–steam and never by the down steam3 . The calculations for
unchoked flow are mathematical wrong. The assumption made in the first step never was
checked. And mathematically speaking, it is equivalent to just guessing solution. These errors are only fraction of the other other in that model which include among other the following: one) assumption of constant temperature in the die is wrong, two) poor assumption
of the isothermal flow, three) poor measurements etc. On top of that was is the criterion for
required vent area.
9.3
General Discussion
When a incompressible liquid such as water is pushed, the same amount propelled by the
plunger will flow out of the system. However, air is a compressible substance and thus the
2 This model results in negative temperature in the shot sleeve in typical range.
3 How otherwise, can it be? It is like assuming negative temperature in the die cavity during the injection. Is it
realistic?
214
CHAPTER 9. VENTING SYSTEM DESIGN
above statement cannot be applied. The flow rate out depends on the resistance to the flow
plus the piston velocity (piston area as well). There could be three situations 1) the flow rate
out is less than the volume pushed by the piston, 2) the flow rate out is more than the volume
pushed by the piston, or 3) the flow rate out is equal to the volume pushed by the piston. The
last case is called the critical design, and it is associated with the critical area.
Air flows in the venting system can reach very large velocities up to about 350 [m/sec].
The air cannot exceed this velocity without going through a specially configured conduit (converging diverging conduit). This phenomena is known by the name of “choked flow”. This
physical phenomenon is the key to understanding the venting design process. In air venting,
the venting system has to be designed so that air velocity does not reach the speed of sound: in
other words, the flow is not choked. In vacuum venting, the air velocity reaches the speed
of sound almost instantaneously, and the design should be such that it ensures that the air
pressure does not exceed the atmospheric pressure.
Prior models for predicting the optimum vent area did not consider the resistance in
the venting system (pressure ratio of less than 2). The vent design in a commercial system includes at least an exit, several ducts, and several abrupt expansions/contractions in which the
resistance coefficient, 4fL
D , is of the order of 3–7 or more. Thus, the pressure ratio creating
choked flow is at least 3. One of the differences between vacuum venting and atmospheric
venting occurs during the start–up time. For vacuum venting, a choking condition is established almost instantaneously (it depends on the air volume in the venting duct), while in the
atmospheric case the volume of the air has to be reduced to more than half (depending on
the pressure ratio) before the choking condition develops - - and this can happen only when
more than 2/3 or more of the piston stroke is elapsed. Moreover, the flow is not necessarily
choked in atmospheric venting. Once the flow is choked, there is no difference in calculating
the flow between these two cases. It turns out that the mathematics in both cases are similar,
and therefore both cases are presented in the present chapter.
The role of the chemical reactions was shown to be insignificant. The difference in the
gas solubility (mostly hydrogen) in liquid and solid can be shown to be insignificant (Handbook and Fundamentals 1948). For example, the maximum hydrogen release during solidification of a kilogram of aluminum is about 7cm3 at atmospheric temperature and pressure.
This is less than 3% of the volume needed to be displaced, and can be neglected. Some of the
oxygen is depleted during the filling time (Bar-Meir 1995c). The last two effects tend to cancel
each other out, and the net effect is minimal.
The numerical simulations produce unrealistic results and there is no other quantitative tools for finding the vent locations (the last place(s) to be filled) and this issue is still an
open question today. There are, however, qualitative explanations and reasonable guesses
that can push the accuracy of the last place (the liquid metal reaches) estimate to be within the
last 10%–30% of the filling process. This information increases the significance of the understanding of what is the required vent area. Since most of the air has to be vented during the
initial stages of the filling process, in which the vent locations do not play a role.
Air venting is the cheapest method of operation, and it should be used unless acceptable results cannot be obtained using it. Acceptable results are difficult to obtain 1) when the
215
9.4. THE ANALYSIS
resistance to the air flow in the mold is more significant than the resistance in the venting system, and 2) when the mixing processes are augmented by the specific mold geometry. In these
cases, the extraction of the air prior to the filling can reduce the air porosity which require
the use of other techniques.
An additional objective is to provide a tool to “combine” the actual vent area with the
resistance (in the venting system) to the air flow; thus, eliminating the need for calculations of
the gas flow in the vent in order to minimize the numerical calculations. Hu et al. and others
have shown that the air pressure is practically uniform in the system. Hence, this analysis can
also provide the average air pressure that should be used in numerical simulations.
9.4
The Analysis
The model is presented here with a minimal of mathematical details. However, emphasis is
given to all the physical understanding of the phenomena. The interested reader can find
more detailed discussions in several other sources (Bar-Meir 1995d). As before, the integral
approach is employed. All the assumptions which are used in this model are stated so that
they can be examined and discussed at the conclusion of the present chapter. Here is a list of
the assumptions which are used in developing this model:
1. The main resistance to the air flow is assumed to be in the venting system.
2. The air flow in the cylinder is assumed one–dimensional.
3. The air in the cylinder undergoes an isentropic process.
4. The air obeys the ideal gas model, P = ρRT .
5. The geometry of the venting system does not change during the filling process (i.e., the
gap between the plates does not increase during the filling process).
6. The plunger moves at a constant velocity during the filling process, and it is determined
by the pQ2 diagram calculations.
7. The volume of the venting system is negligible compared to the cylinder volume.
8. The venting system can be represented by one long, straight conduit.
9. The resistance to the liquid metal flow, 4fL
D , does not change during the filling process
(due to the change in the Re, or Mach numbers).
10. The flow in the venting system is an adiabatic flow (Fanno flow).
11. The resistance to the flow,
4fL
D ,
is not affected by the change in the vent area.
With the above assumptions, the following model as shown in Figure ?? is proposed.
A plunger pushes the liquid metal, and both of them (now called as the piston) propel the air
through a long, straight conduit.
216
CHAPTER 9. VENTING SYSTEM DESIGN
The mass balance of the air in the
cylinder yields
dm
+ ṁout = 0.
dt
(9.1)
This equation (9.1) is the only equation that
needed to be solved. To solve it, the physical properties of the air need to be related to
the geometry and the process. According to
assumption 4, the air mass can be expressed
as
m=
Fig. 9.2 – A simplified model for the venting system.
PV
RT
(9.2)
The volume of the cylinder under assumption 6 can be written as
V(t)
t
= 1−
V(0)
tmax
Thus, the first term in equation (9.1) is represented by


t
PV(0)
1
−
tmax
dm
d 

=
dt
dt
RT
(9.3)
(9.4)
The filling process occurs within a very short period time [milliseconds], and therefore the heat transfer is insignificant 3. This kind of flow is referred to as Fanno flow4 . The
instantaneous flow rate has to be expressed in terms of the resistance to the flow, 4fL
D , the
pressure ratio, and the characteristics of Fanno flow (Shapiro 1953). Knowledge of Fanno flow
is required for expressing the second term in equation (9.1).
The mass flow rate can be written as
k+1 s
k
Min (t) P0 (0) 2k
f[Min (t)]
(9.5)
ṁout = P0 (0)AMmax
Mmax P0 (t)
RT0 (0)
where
k−1
(Min (t))2
f[Min (t)] = 1 +
2
−(k+1)
2(k−1)
(9.6)
The Mach number at the entrance to the conduit, Min (t), is calculated by Fanno flow characteristics for the venting system resistance, 4fL
D , and the pressure ratio. Mmax is the maximum value of Min (t). In vacuum venting, the entrance Mach number, Min (t), is constant
and equal to Mmax .
4 Fanno flow has been studied extensively, and numerous books describing this flow can be found. Nevertheless,
a brief summary on Fanno flow is provided in Appendix A.
217
9.5. RESULTS AND DISCUSSION
Substituting equations (9.4) and (9.5) into equation (9.1), and rearranging, yields:
k−1
tmax
2k
M̄f(M
)
P̄
k
1
−
in
tc
dP̄
=
P̄;
P̄(0) = 1.
dt̄
1 − t̄
(9.7)
The solution to equation (9.7) can be obtained by numerical integration for P̄ . The residual
mass fraction in the cavity as a function of time is then determined using the “ideal gas” assumption. It is important to point out the significance of the tmax
tc . This parameter represents
the ratio between the filling time and the evacuation time. tc is the time which would be required to evacuate the cylinder for a constant mass flow rate at the maximum Mach number
when the gas temperature and pressure remain at their initial values, under the condition that
the flow is choked, (The pressure difference between the mold cavity and the outside end of
the conduit is large enough to create a choked flow.) and expressed by
m(0)
tc =
AMmax P0 (0)
q
k
RT0 (0)
(9.8)
Critical condition occurs when tc = tmax . In vacuum venting, the volume pushed
by the piston is equal to the flow rate, and ensures that the pressure in the cavity does not
increase (above the atmospheric pressure). In air venting, the critical condition ensures that
the flow is not choked. For this reason, the critical area Ac is defined as the area that makes
the time ratio tmax /tc equal to one. This can be done by looking at equation (9.8), in which
the value of tc can be varied until it is equal to tmax and so the critical area is
m(0)
Ac =
tmax Mmax P0 (0)
q
k
RT0 (0)
(9.9)
Substituting equation
√ (9.2) into equation (9.9), and using the fact that the sound velocity can
be expressed as c = kRT , yields:
Ac =
V(0)
ctmax Mmax
(9.10)
where c is the speed of sound at the initial conditions inside the cylinder (ambient conditions).
The tmax should be expressed by Eckert/Bar–Meir equation.
9.5
Results and Discussion
The results of a numerical evaluation of the equations in the proceeding section are presented
in Figure ??, which exhibits the final pressure when 90% of the stroke has elapsed as a function
of AAc .
Parameters influencing the process are the area ratio, AAc , and the friction parameter,
4fL
D . From other detailed calculations (Bar-Meir 1995d) it was found that the influence of the
parameter 4fL
D on the pressure development in the cylinder is quite small. The influence is
218
CHAPTER 9. VENTING SYSTEM DESIGN
small on the residual air mass in the cylinder, but larger on the Mach number, Mexit . The
effects of the area ratio, AAc , are studied here since it is the dominant parameter.
Note that tc in air venting is slightly different from that in vacuum venting (Bar-Meir,
Eckert, and Goldstein 1996b) by a factor of f(Mmax ). This factor has significance for small
4fL
A
D and small Ac when the Mach number is large, as was shown in other detailed calculations
(Bar-Meir 1995d). The definition chosen here is based on the fact that for a small Mach number
the factor f(Mmax ) can be ignored. In the majority of the cases Mmax is small.
For values of the area ratio greater than 1.2, AAc > 1.2, the pressure increases the volume
flow rate of the air until a quasi steady–state is reached. In air venting, this quasi steady–state
is achieved when the volumetric air flow rate out is equal to the volume pushed by the piston.
The pressure and the mass flow rate are maintained constant after this state is reached. The
pressure in this quasi steady–state is a function of AAc . For small values of AAc there is no
steady–state stage. When AAc is greater than one the pressure is concave upwards, and when
A
Ac is less than one the pressure is concave downwards. These results are in direct contrast to
previous molds by Sachs , Draper , Veinik and Lindsey and Wallace , where models assumed
that the pressure and mass flow rate remain constant and are attained instantaneously for air
venting.
The reference to the stroke completion
(100% of the stroke) is meaningless since 1) no
gas mass is left in the cylinder, thus no pressure can be measured, and 2) the vent can
be blocked partially or totally at the end of
the stroke. Thus, the “completion” (end of
the process) of the filling process is described
when 90% of the stroke is elapsed. Fig. 9.3
Fig. 9.3 – The pressure ratios for air and vacuum
presents the final pressure ratio as a funcventing at 90% of the piston stroke.
tion AAc for 4fL
= 5. The final pressure
D
(really the pressure ratio) depends strongly
on AAc as described in Fig. 9.3. The pressure in the die cavity increases by about 85% of its
initial value when AAc = 1 for air venting. The pressure remains almost constant after AAc
reaches the value of 1.2. This implies that the vent area is sufficiently large when AAc = 1.2
for air venting and when AAc = 1 for vacuum venting. Similar results can be observed when
the residual mass fraction is plotted.
This discussion and these results are perfectly correct in a case where all the assumptions are satisfied. However, the real world is different and the assumptions have to examined
and some of them are:
20.0
18.0
Vacuum venting
. . .. . . . ..
16.0
Air venting
14.0
4fL
=5.0
D
12.0
h
P (t =0.9)
10.0
h
P (t =0)
8.0
..
..
..
..
..
..
..
..
...
.. .
.
6.0
4.0
2.0
0.0
0.0
0.3
0.6
. .. .
.. . .. . ...
. .... .... .... .... .... .... ....
0.9
1.2
1.5
A
Ac
1.8
.... . .. .. .. .. . . . .. . .. .. . ... . . . . .
2.1
2.4
2.7
3.0
1. Assumption 1 is not a restriction to the model, but rather guide in the design. The
engineer has to ensure that the resistance in the mold to air flow (and metal flow) has
to be as small as possible. This guide dictates that engineer designs the path for air (and
the liquid metal) as as short as possible.
2. Assumptions 3, 4, and 10 are very realistic assumptions. For example, the error in using
9.5. RESULTS AND DISCUSSION
219
assumption 4 is less than 0.5%.
3. This model is an indication when assumption 5 is good. In the initial stages (of the
filling process) the pressure is very small and in this case the pressure (force) to open the
plates is small, and therefore the gap is almost zero. As the filling process progresses,
the pressure increases, and therefore the gap is increased. A significant gap requires
very significant pressure which occurs only at the final stages of the filling process and
only when the area ratio is small, AAc < 1. Thus, this assumption is very reasonable.
4. Assumption 6 is associated with assumption 9, but is more sensitive. The change in the
resistance (a change in assumption in 9 creates consequently a change in the plunger
velocity. The plunger reaches the constant velocity very fast, however, this velocity
decrease during the duration of the filling process. The change again depends on the
resistance in the mold. This can be used as a guide by the engineer and enhances the
importance of creating a path with a minimum resistance to the flow.
5. Another guide for the venting system design (in vacuum venting) is assumption 7. The
engineer has to reduce the vent volume so that less gas has to be evacuated. This restriction has to be design carefully keeping in mind that the resistance also has to be
minimized (some what opposite restriction). In air venting, when this assumption is not
valid, a different model describes the situation. However, not fulfilling the assumption
can improve the casting because larger portion of the liquid metal which undergoes
mixing with the air is exhausted to outside the mold.
6. Assumption 8 is one of the bad assumptions in this model. In many cases there is more
than one vent, and the entrance Mach number for different vents could be a different
value. Thus, the suggested method of conversion is not valid, and therefore the value
of the critical area is not exact. A better, more complicated model is required. This
assumption cannot be used as a guide for the design since as better venting can be
achieved (and thus enhancing the quality) without ensuring the same Mach number.
7. Assumption 9 is a partially appropriate assumption. The resistance in venting system
is a function of Re and Mach numbers. Yet, here the resistance, 4fL
D , is calculated based
on the assumption that the Mach number is a constant and equal to Mmax . The error
due to this assumption is large in the initial stages where Re and Mach numbers are
small. As the filling progress progresses, this error is reduced. In vacuum venting the
Mach number reaches the maximum instantly and therefore this assumption is exact.
The entrance Mach number is very small (the flow is even not choke flow) in air venting
when the area ratio, AAc >> 1 is very large and therefore the assumption is poor. However, regardless the accuracy of the model, the design achieves its aim and the trends
of this model are not affected by this error. Moreover, this model can be improved by
taking into consideration the change of the resistance.
8. The change of the vent area does affect the resistance. However, a detailed calculation
can show that as long as the vent area is above half of the typical cross section, the error
220
CHAPTER 9. VENTING SYSTEM DESIGN
is minimal. If the vent area turns out to be below half of the typical vent cross section
a improvement is needed.
9.6
Summary
This analysis (even with the errors) indicates there is a critical vent area below which the ventilation is poor and above which the resistance to air flow is minimal. This critical area depends
on the geometry and the filling time. The critical area also provides a mean to “combine” the
actual vent area with the vent resistance for numerical simulations of the cavity filling, taking
into account the compressibility of the gas flow. Importance of the design also was shown.
9.7
Questions
Under construction
10
Clamping Force Calculations
Under construction
221
222
CHAPTER 10. CLAMPING FORCE CALCULATIONS
It doesn’t matter on what machine the product is produce, the price is the same
Prof. Al Miller, Ohio
11
Analysis of Die Casting Economy
11.1
Introduction
The underlying reason for the existence of the die casting process is so that people can make
money. People will switch to more efficient methods/processes regardless of any claim die
casting engineers make1 . To remain competitive, the die casting engineer must totally abandon the “Detroit attitude,” from which the automotive industry suffered and barely survived
during the 70s. The die casting industry cannot afford such a luxury. This topic is emphasized and dwelt upon herein because the die casting engineer cannot remain stagnate, but
rather must move forward. It is a hope that the saying “We are making a lot of money · · ·
why should we change?” will totally disappear from the die casting engineer’s jargon. As in
the dairy industry, where keeping track of specifics created the “super cow,” keeping track of
all the important information plus using scientific principles will create the “super die casting
economy.” This approach would be true even if a company, for marketing reasons, needed to
offer a wide variety of services to their customers. Which costs the engineer can alter, and
what he/she can do to increase profits, are the focus of this chapter.
First, as usual, a discussion on the “common” models are presented, the validity and
the usefulness is discussed, and finally proper models are unveiled. Also discussion on what
available in the literature is presented. The quote above the chapter was given by Al Miller a
department head of System Engineering at Ohio State University. While the competition to
1 The DDC, a sub set of NADCA operations, is now trying to convince die casting companies to advertise through
them to potential customers. Is the role of the DCC or NADCA to be come the middle man? A reasonable person
will not think so. The role of these organizations should be to promote the die casting industry as whole and not any
particular company/ies. The question must be raised do they do their job?
223
224
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
most ridiculous statements is on going his contributions cannot be contributions.
11.2
The “common” model, Miller’s approach
Miller, R. Allen Miller’s errors NADCA sponsored research at Ohio State University by Miller
which started with idea that the price is effected by the following parameters: 1)weight, 2)alloy
cost, 3)complexity, 4)tolerance, 5)surface roughness, 7)aspect ratio, 8)production quantity, and
9)“secondary” machining. After statistical analysis they have done they come–out with the
following equation
price = 0.485 + 2.20 weight − 0.505 zinc + 0.791 mag + 0.292 details
+ 0.637 tolerance − 0.253 quantity
(11.1)
where mag, zinc, details(<100 dimension), and tolerance are on/off switch. They alleged that
this formula is good for up to ten pounds (about 4.5[kg]). In summary, if you expect to get
equation that does not have much with the actual cost, you got one. This result means no
mater the design is bad or good, large or small, what machine is all the same.
11.3
The Validity of Miller’s Price Model
There is a saying garbage in garbage out. The proponent conclusion from Eq. (11.1) is
that it does not matter how good the design how much scrap the product generates the
price is the same. This is exactly what this author is preaching against. The question must be asked, how Miller and his team calculate the average price of the product
and what statistic they analyzed? Obviously, if they were clueless on how to the calculate the actual price in the first place, how logical their conclusions are? Furthermore,
how Miller and his team determined the product profit if the price have no relationship what
Combined
what so ever to the actual production cost?
cost
The “critical/optimum point” is defined
cost
as the point above which the quality is good
m1
Scrap
(reasonable) below which the quality is unacm2
cost
ceptable. As it turns out, much above and just
Machine
m
3
cost
above the critical point produces an acceptable
m1 > m2 > m3
quality product for many design parameters in
HD
the die casting process. However, the cost is
Fig. 11.1 – Production cost as a function of the
2
considerably higher . The hydraulic diameter
runner hydraulic diameter.
of the runner system is one such example (see
Fig. 11.1).
2 The price of a die casting machine increases almost exponentially with the machine size. For example, small and
used machine of 60 tons is in the 10,000 dollars range while 600 tons can be approach up million dollars (according
to web). These numbers are subject to negation which was not of this inquiry. Thus, finding the smallest die casting
machine to run the job has a critical importance.
225
11.3. THE VALIDITY OF MILLER’S PRICE MODEL
The price of the runner system (scrap) is related to the hydraulic diameter squared,
scrap cost = f HD 2 (a parabola), as shown by the “scrap cost” curve in Figure 11.1. The
machine cost is approximated constant (as a first approximation) up to the point below
which the machine cannot produce an acceptable quality. Further examination, reveals
that the line in the Figure m1 shows a minor decline because a mild reduction of the energy consumption to push the plunger. The engineer should seek to design the runner
diameter just above this point. Dalquist (2004)
collected data from DOE (department of enMetal Preparation 3.0
ergy) about pollution while the research has
questionable parts, it points to the amount of
energy every kilogram of aluminum requires
to get final product. Clearly, the based on the
current averages but provide some indications
to amount of cost of producing 1 kg aluminum.
Clearly Dalquist and et al do not agree with
Miller because as noted the energy is per kilogram (wight) and not per cast as Miller’s claim.3
Casting
3.2
Finishing
1.2
Total
7.9
Table 11.1 – Energy per kilogram of aluminum part
of the die casting process (MJ per kilogram)
“Machine” cost as a function of the runner diameter for several different machines is
shown by the “machine” curves in Fig. 11.1. The larger and newer machine exhibits larger cost.
Hence, expensive (larger) machine are depicted higher than cheaper (smaller) machines. The
combined cost of the scrap and the machine usage can be drawn, and clearly the combined–
cost curve has a minimum point, and is referred to here as the “optimum” point4 . This is a
typical example of how a design parameter (runner hydraulic diameter) effects the cost and
quality.
The components of the production cost now should be dissected and analyzed, and
then a model will be constructed. It has to be realized that there are two kinds of cost components: 1) those which the engineer controls, and 2) those which the engineer does not control.
The uncontrolled components include overhead, secondary operations, marketing, space5 ,
etc. This category should be considered as a constant, since the engineer’s actions/choices
do not affect the cost and therefore do not affect the cost of design decisions. However, the
costs of die casting machine capital and operations, personnel cost, melting cost, and scrap
cost6 are factors which have to be considered, and are discussed in the succeeding sections.
In this analysis it is assumed that the die casting company is here to make a buck, and it is
also assumed that a competitive price wars for a specific project and/or any other personal
reasons influencing decision making are not relevant7 . These decisions and their costs are
formulated in such a way that the engineer will have the needed tools to make appropriate
3 No one can disregard the talent of Miller to get high academic position while exhibit nonsense as a real research.
4 The change in the parts numbers per shot will be discussed in Section 11.7.
5 The room–amount cost for the machine is almost insensitive to the engineer’s choice of the size or brand of the
die casting machine
6 See the discussion on this topic in section 11.7 page 227 for the more detail.
7 such decisions based on the desire to keep a customer for another project are not relevant here. Yet, this information can be used to make intelligent decision in regard to specific customer.
226
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
decisions.
11.4
The combined Cost of the Controlled Components
The engineer has to choose the least expensive machines available, yet which capable to produce a product of acceptable quality. This design is contingent on the number of idle machines
in the shop. The least expensive machine has to chosen. The price for production cost each
machine is determined from the sums of every component. If the customer is in a rush, the
cost should be calculated for the available die casting machine as follows:
X
ωtotal =
ωi
(11.2)
i
where ω is the individual cost component. These component will be discussed in the following sections.
11.5
Die Casting Machine Capital Costs
The capital cost of a die casting machine (like any other industrial equipment) has two components: 1) money cost and 2) depreciation cost. The money cost in many cases is also comprised
of two components: 1) loan cost and 2) desired profit8 . The cost of a loan is interest. The value
of the interest rate is easy to evaluate – just ask a banker. However, the value of the desired
profit is harder to estimate. One possible way to estimate this is by checking how much it
costs to lease a similar machine. Adding these two numbers yields a good estimate of the
money costs. In today’s values, the money cost value is about 12%–25%. Depreciation is a loss
in value of the die casting machine9 . In this analysis, it is assumed (or at least hoped) that the
other die casting machines have other jobs in the queue. If a company for a short time is not
working to full capacity, the analysis will still be valid with minor modifications. However, a
longer duration of being below full capacity requires the company to make surgical solutions.
The cost of the die casting machine depends on the market and not on the value the
accountant has put on the books for that machine. Clearly, if the machine is to be sold/leased,
the value obtained will be according to the market as “average” value. The market value should
be used since the machine can be sold and this money can be invested in other possibilities.
Amortization is estimated in the same manner. The difference between the current value and
the value at one year older is the depreciation value.10 Having these numbers, the capital cost
can be estimated. For example, a one million dollar machine with a 20% money cost and a 5%
depreciation cost equals about $250,000.00 a year. To convert this number to an hourly base
8 This profit is different from the operational profit. For example, if one own a taxi, he should have two kind of
profits: 1) those from owning the taxi and 2) those from operating the taxi. He can rent the taxi and have a profit just
for owning the vehicle. The owner should earn additional income for the eight hours shift. These refereed herein as
operational earnings.
9 The effects of taxes on the depreciation analysis are sometime significant, but to reduce the complexity of the
explanation here, it is ignored.
10 This cost also depends on changes in the condition of the machine.
11.6. OPERATIONAL COST OF THE DIE CASTING MACHINE
227
rate, the number of idle days (on that specific machine) is required, and in many case is about
60 – 65 days. Thus, hourly capital cost of that specific machine is about $34.70.
A change in the capital cost per unit can be estimate via the change in the cycle time be
estimated. The change in the cycle time is determined mostly by the solidification processes,
which are controlled slightly by the runner design. Yet this effect can be diminished by controlling the cooling rate. Hence, the capital price is virtually unaffected once the die casting
machine has been selected for a specific project. Here, the cost per unit can be expressed as
follows:
ωcapital =
capital cost per hour
Nc N p
(11.3)
where ωcapital is the capital cost per unit produced, Nc is the number of cycles per hour,
and Np the number of parts shot.
11.6
Operational Cost of the Die Casting Machine
Operational costs are divided into two main categories: 1) energy cost, and 2) maintenance
cost. The energy cost is less insensitive to the mold/runner design. The maintenance cost is
determined mostly by the amount of time the die casting machine is in operation. This cost
is comprised of the personnel cost of doing the work hydraulic fluid maintenance components (ladle, etc.) and maintenance, etc. which is different for each machine and company.
However, the value of this cost can be consider invariant for a specific machine in regard to
design parameters. The engineer’s duty is to calculate the operation cost for every die casting
machine that is in the company. This can be achieved by keeping records of the maintenance
for each machine and totaling all related costs performed on that machine in the last year.
The energy costs are the costs of moving the die casting machine and its parts and
accessories. The energy needed to move all parts is the electrical energy which can easily be
measured. Today, electrical energy costs are far below one dollar for one [kW]×hour (0.060.07 of a dollar according to NSP prices). Even a large job will require less than 10[kW×hour].
Thus, the total energy cost is in most cases at most $1.00 per hour. The change in the energy is
insensitive to the runner and venting system designs and can vary by only 30% (15 cents for a
very very large job), which is insignificant compared to all other components. The operation
cost can be expressed as
ωoperation =
11.7
operation cost per hour f(machine size, type etc.)
N c Np
(11.4)
Runner Cost (Scrap Cost)
The main purpose of the runner is to deliver the liquid metal from the shot sleeve to the mold,
since the mold cannot be connected directly on (to) the shot sleeve. The requirements of the
runner have conflicting demands. Here is a partial list of the requirements for the runner:
1. As small as possible so it will create less scrap.
228
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
2. Large as possible so that there is less resistance in the runner to the liquid metal flow,
so that the job can be performed on a smaller die casting machine.
3. Small enough so that the plunger will need to propel only a minimum amount of liquid
metal. In a way this is the same as requirement (item 1) above but less important.
Clearly, a large runner volume creates more scrap and is a linear function of the size of the
runner volume, which is
area
Vrunner
z }| { length
πHDT 2 z}|{
LT
=
4
(11.5)
where HD is the characteristic size of the hydraulic diameter, and LT its length (these values
are not the actual values, but they are used to represent the sizes of the runner). From equation
(11.5), it is clear that the diameter has the greater impacts on the scrap cost. The minimum
diameter at which a specific machine can produce good quality depends on the required filling
time, gate velocity, other runner design characteristics, and the characteristics of the specific
machine.
Scrap cost is a linear function of the volume11 . The scrap cost per volume/weight consists of three components: 1) the melting cost, 2) the difference between the buying price and
the selling price (assuming that the scrap can be sold), and 3) the handling cost. The melting
cost includes the cost to raise the metal temperature to the melting point, to melt the metal,
and to hold the metal temperature above the melting point (liquidus point). The melting cost
can be calculated by measuring energy used (crude oil or natural oil in most cases) plus the
maintenance cost of the furnace divided by the amount of metal that has been casted (the parts
and design scrap). The buying price is the price paid for the raw material; the selling price is
the price for selling the scrap. Sometime it is possible to reuse the scrap and to re–melt the
metal. In some instances, the results of reusing the scrap will be a lower grade of metal in the
end product. If reuse is possible, the difference in cost should be substituted by the lost metal
cost, which is the cost of 1) metal that cannot be recycled and 2) metal lost due to the chemical
reactions in the furnace. The handling cost is the cost encountered in selling the metal, and it
includes changing the mechanical or chemical properties of the scrap, transportation, cost of
personnel, storage, etc. Each handling of the metal costs a different amount, and the specifics
can be recorded for the specific metal.
Every job/mold has typical ranges for the filling time and gate velocity. Moreover,
a rough design for the runner system can be produced for the mold. With these pieces
of information in place, one can calculate the gate area. Then the flow rate for the mold
11 Up to a point about which it becomes more sensitive to the volume. The point where handling of the cast has
extra treatment.
229
11.7. RUNNER COST (SCRAP COST)
can be calculated by
0.20
0.18
Vmold
Q=
Agate Ugate
0.16
(11.6)
0.14
0.12
0.10
Additionally, the known design of the runner
with flow rate yields the pressure difference in
the runner, and this yields the power required
for the runner system,
0.08
0.06
0.04
0.02
.
0.00 .. ...
0.00
(11.7)
Pr = Q ∆P
Q∆P
≃ Q×P
Pmax × Qmax
. ..
...
0.10
.
0.20
0.30
0.40
0.50
h
Q
...
.. ..
0.60
.. . ...... . . .
0.70
. ..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
0.80
0.90
1.00
Fig. 11.2 – The reduced power of the die casting machine as a function of the normalized
flow rate.
or in a normalized form,
Pr =
...
...
...
...
..
..
..
.. .
.
..
..
..
..
..
..
.. .
..
..
..
..
..
..
.. .
..
..
..
..
..
..
..
(11.8)
Every die casting machine has a characteristic curve on the pQ2 diagram as well. As2
suming that the die casting machine has the “common” characteristic, P = 1 − Q , the normalized power can be expressed
2
2
Pm = Q(1 − Q ) = Q − Q
3
(11.9)
where Pm is the machine power normalized by Pmax × Qmax . The maximum power of
this kind of machine is at 2/3 of the normalized flow rate, Q, as shown in Fig. 11.2. It is recommended to design the process so the flow rate occurs at the vicinity of the maximum of
the power. For a range of 1/3 of Q that is from 0.5Q to 0.83Q, the average power is 0.1388
Pmax Qmax , as shown in Figure 11.2 by the shadowed rectangular. One may notice that this
value is above the capability of the die casting machine in two ranges of the flow rate. The
reason that this number is used is because with some improvements of the runner design
the job can be performed on this machine, and there is no need to move the job to a larger
machine.
If the machine power turns out to be larger than the required power of the runner,
Pm > Cs Pr , the job can then be performed on the machine; otherwise, a bigger die casting
machine is required. In general, the number of molds castable in a single cycle is given by
Pm
Pm
Np =
=
(11.10)
Cs Pr
Cs Pr
The floor symbol “⌊” being used means that the number is to be rounded down to the nearest
integer. Cs denotes the safety factor coefficient. In the case that Np is less than one, Np > 1,
that specific machine is too small for this specific job. After the number of the parts has been
determined (first approximation) the runner system has to be redesigned so that the required
power needed by the runner can be calculated more precisely. Plugging the new numbers into
Eq. (11.10) yields a better estimation of the number of parts. If the number does not change,
this is the number of parts that can be produced; otherwise, the procedure must be repeated.
230
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
In this analysis, the required clamping forces that the die casting machine can produced
are not taken into consideration. Analysis of the clamping forces determines the number of
possible parts and it is a different criterion which required to satisfied, this will be discussed in
more detail in Chapter 10. The actual number of parts that has to be taken into consideration is
the smaller of the two criteria. Next, the new volume of the runner system has to be calculated.
The cost per cavity is the new volume divided by the number of cavities:
ωscrap =
11.8
Vrunner × (cost per volume)
Np
(11.11)
Start–up and Mold Manufacturing Cost
The cost of manufacturing of a mold is affected slightly by the shape of the runner. The only
exemption to the above statement is the effect of change of the cross section shape and size
on the cost of manufacturing which will be discussed in Chapter 6. A larger part of expense
is the start–up time cost which is composed of 1) rebuilding the mold, 2) lost time (personnel
time, machine time, etc), and 3) lost material. When dealing with calculation of the start–up
time two things have to be taken into account 1) the ratio of the start up cost to the total cost,
and 2) how long it is expected to take to achieve a product of acceptable quality. The cap cost
has to be determined from the total cost per unit, and then multiplied by the total number of
units. This number is the net production cost. The start–up cost cannot (should not) exceed
10%–15% of that number. Presently, it is very hard to determined the number of trials that
will required per mold. This number is related to the complexity of the shape. The more
complex the shape is, the more likely it is that the number of attempted “shots” will increase.
If it is assumed that the engineer is experienced, the only factor that will affect the number
of shots will be the complexity – provided that the job can be performed on the same die
casting machine. The complexity of the shape should present a general idea of the number of
expected attempts, and should be used in calculating the start–up cost,
ωstartUp =
(Cost per attempt) × Na
Nr
(11.12)
where Na is the number of attempts, and Nr is the number of the total parts to be produced.
11.9
Personnel Cost
The cost of personnel is affected by the cycle time plus the number of parts produced per cycle.
With today’s automaxination, the number of operators is decreasing. In some companies, one
operator controls three or more machines. Hence, the personnel cost is:
ωpersonnel =
salary per hour
number of machines × number of cycle
(11.13)
In toady’s market, the operator cost is in the range of $10–$20 per hour. When automaxination is used, the personnel cost is significantly reduced to the point that it is insignificant.
11.10. UNCONTROLLED COMPONENTS
11.10
231
Uncontrolled components
The price to be charged to the customer has to include the uncontrolled components as well.
There are several methods for adding this fragment to the part cost. First, the total cost of
the uncontrolled components has to be calculated. This can be done by adding up the costs
from the previous year and estimated for this year This cost includes salaries that were paid
in the last year plus the legal expenses, rent, and marketing, etc. Dividing the uncontrolled
components of cost has many reasonable options. Here is a selected list according to:
• the number of parts
• the number of parts and their size/weight
• the number of the parts and their complexity
232
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
11.11
Minimizing Cost of Single Operation
In this section several issues related to cost minimization and/or profit increases are discussed. An example of such problem is when an engineer has to make a decision about supplies. Ordering a supply commonly to has to do with two or more conflicting costs. These
two conflicting costs have a minimum of ordering cost associated with an optimum number
of orders. Consider a simple situation where ABC company produces x devices per year, for
example x hard drive frames per year. These x devices require y items supporting components. For example, these hard drive frames are made of y mass of aluminum12 . So, the ABC
company has to order y items supporting components a year. It is assumed that the number
of items ordered and consumed per year is constant13
In a typical situation, these items are ordered several times, the cost is composed of
two components. The first component is the cost per order which includes such things as
the delivery cost, the time to order (verification if it is in stock). The second component is
associated with the cost of keeping the stock beyond one day supply. This category includes
such as the money tied to material, the storage used (cost of space and handling) etc. For
simplicity, it is assumed that the cost per delivery is constant and does not change during
the year. It also assumed that the storage per day cost is also constant. To illustrate this
point, consider two extreme cases of ordering. One possibility is to order everyday and other
possibility is to order once per year. The cost for ordering very day is 36514 times the ordering
cost. In this case, there is no stocking cost. The other extreme is one order a year for which
there is one time order cost with 364 days of stocking cost.
Example 11.1: caption
Level: Intermediate
Under conditions below, calculate the ordering cost for ordering every day of the
year and once a year. The delivery cost of aluminum is $150 per delivery. The cost of
storing a kilogram aluminum is 0.10 dollars. And the cost of a kilogram aluminum is
10 dollars. The daily aluminum consumption is 750 kilogram. The money cost is 0.01
percent a day.
Solution
The ordering cost is $150. Hence, the cost for the first case is composed of the ordering cost (in
this case only delivery cost) only. The stocking cost is vanished because no stocking is involved.
Thus, the total cost is
(11.1.a)
total cost = 365 × 150 = $54, 750
The ordering occurs once a year with 364 days of stocking cost which must be added. The daily
12 Or the weight for the total of the hard drives [in kilograms].
13 If the number is not constant or even seasonal (depend on the season) this model can be expanded by numerical
analysis.
14 It is assumed that the year is 365 days which about 75% of the occurrence. There are years with 356 days.
233
11.11. MINIMIZING COST OF SINGLE OPERATION
End of Ex. 11.1
stocking cost one day consumption is
stocking cost
=
per day
stocked
material
amount
z}|{
750
cost per
kilogram
z}|{
× 10.0 ×0.0001 = $0.75
(11.1.b)
On the first day has 364 portions (days) to be stocked. On the second day has 363 portions (days)
to be stocked. On every sequence day there is one less portions to be stocked. Hence, there is
a series of 364 items which starts with 364 and end at zero. This exactly algebraic series which
can be calculated as
364 + 0
total por- a1 + a364
=
364 =
∗ 364 = 66248
2
2
tions
(11.1.c)
Thus, the total cost for ordering once is
total cost for
= $150 + 66248 × 0.75 = $49836
once ordering
(11.1.d)
For the purpose of this example, the choice of one time order is better. While the cost of one
time ordering looks better in this example, in real life other factors should be considered. In
this case case, the erosion of the company credit which was not a considered here. That fact
should reversed the decision.
Example 11.1 exhibits the effect of the number of orders on the cost of operation. Intuitively, for example, it can be observed that for two (2) orders, as compare to one order, the
cost is reduced by half plus $150 (why?).
The focus of this discussion is to find the optimum number of times to order the items
per yearOptimum items. Not all years are the same (some years are 366 days) but here it will
be assumed that all years are 365 days. In this discussion, months and weeks do not appear
and the dissimilarity is not discussed. The number of items ordered per year is assumed to be
constant for this discussion and is denoted as N. Hence, the number of order items per day is
N/365. The cost per day of store of item is denoted y. The ordering cost is r. The unknown
number of periods is denoted as p.
The number of days per period is
D=
365
p
(11.14)
The period cost of storage is the number parts not used which were kept for the following
days of the period. The number of items remained for the storage for the first day of the
period are
Remainder = (D − 1) ·
N
365
(11.15)
234
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
On the second day, the number of items that remain in storage is
Remainder = (D − 2)
N
365
(11.16)
There are no items to stored on the last day of the period. The number of days that we had to
store is (D − 1) (no need to store at the last day). The cost of the storage per period is
first day
}|
{
second day
z
}|
3rd last day
z }| {
N
N
N
number of
= (D − 1)
+ (D − 2)
+···+ 2
+
365
365
365
items
z
{
day before last
z }| {
N
1
365
last day
z }| {
N
+0
365
(11.17)
Eq. (11.17) can be rearranged (because it is a regular algebraic series) is
D−1
N X
number of
=
D−1−i
365
items
i=1
(11.18)
or
averaged to
stored days
be
}|
{ number of days
z
(D − 1) + 1 z }| {
D (D − 1) N
N
number of
(D − 1)
=
=
2
365
2
365
items
(11.19)
The cost per period is
storage cost D (D − 1) N
=
y
per period
2
365
(11.20)
Or in terms of the number of period, p (see also equation (11.14)) the storage cost is
365 365
−1
storage cost
N
p
p
y
=
per period
2
365
The yearly storage cost is




N
 yearly storage   storage cost 

 =  per period  × p =
2
cost
The total cost yearly ordering cost, C, is


C=


365
−1
p

yearly
yearly
 

+

storage cost
ordering cost
y
(11.21)
(11.22)
(11.23)
235
11.12. INTRODUCTION TO ECONOMICS
In term of the number of periods the total ordering cost is
!
N 365
yearly cost
= rp+
−1 y
2
p
(11.24)
The minimum cost expressed by the expression (11.24) a derivative with respect to p
number of period as
N y 365
1
∂C
= r+
− 2 =0
(11.25)
∂p
2
p
The solution of equation (11.25) is
p=
r
N y 365
2r
(11.26)
These calculations were made under the assumption that the number of periods is a real number. However, the number of periods and several other parameters must be an integer. It can
be argued that the number of orders can be 6.5 on the account that in two years planning to
have 13 orders (13/2 = 6.5). It is more common to have solution with a totally irrational
number which leads in practicality solution that cannot be used. The real solution (in a yearly
integer planning sense) lay either on one of adjoining sides of the continuous solution. It has
to be manually calculated. The calculation can yield a number of periods to be below one.
The actual meaning is that the ordering cost is significant (dominate) so the order must be
continuous. On the other extreme, when the ordering cost is so insignificant to order can
occur several time a day.
Example 11.2: caption
Level: Intermediate
In ABC die casting company has 68000 kg of aluminum a year. The cost of storage
of 1 kg a day is $0.04 and cost of ordering is $130. What is the optimum order period?
Solution
The information provides that
p=
r
68000 × 0.04 × 365
= 3.23443016
2 × 130
(11.2.a)
The solution is either 3 or 4 times when additional consideration has to be taken into account.
11.12
Introduction to Economics
The main goal of any company is to increase the total profits which is denoted as T . The total
profits, T , is a function of (x) the number of the sold items. It is assumed that number of parts
sold is function of the price. For example, if the price of a item is zero the consumer will
take infinite amount of items. On the other hand, if price is very high no item will sold. This
236
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
connection represets the demand line. Later a discussion on the supply line will be presented.
If the stock or inventory is assumed to be zero or at least constant, the cost (C) and the revenue
(R) are a pure function of the number of items sold, x. The aim of this discussion to find the
optimum number of items that will yield the maximum profits.
A new concept is introduced in this discussion, the marginal profits.The marginal profits refers to the point where additional production, or increase of sold items, actually reduces
the profit. This concept requires a new tool the discrete mathematics. In the previous section,
this concept was used in section 11.11 without proper introduction. The discrete mathematics
can be described as treading integer phenomena as a continuous while recognizing that it is
integer. The practical application is the “virtual” maximum or minimum is found assuming
continuous mathematics but later the actual maximum or minimum is found by looking the
two sides (in case one component issue) of the previous found virtual solution.
Thus, the profits can be written regardless whether the number of items is integer or
continuous as
(11.27)
T (x) = R(x) − C(x)
The revenue is a function of items number, x, as well as the price, C(x) is also a function
of x. The typical cost, C, is composed from several components such as the fix cost, linearly
associated with the number of items, and cost that associated with not linear with the number
of items. The first cost is relatively obvious, while the differentiation between the second and
third cost has to be explained. The second (linear) cost is referred to the cost that occur for
every item. For example, the aluminum consumed for every frame produced depends only
the number of items produced (approximately). On the other hand, the cost production of
the mold for the frame does not directly depends on the number of the items. Thus, the mold
cost is inversely proportional (1/x) to the number of items.
Example 11.3: Average Worker
Level: Simple
Engineer on average work 25 hours to produce reasonable quality mold. The cost
per hour of engineer is about 150 dollars per hours. The cost of machining is 400
dollars per incident. The maximum items that can be produced on that specific mold
is 10,000. Draw the mold cost for every item as a function of the item produced.
Discuss also the effect of several cycles of mold productions.
Solution
In the first cycle the cost is inversely proportional the number of item sold or produced. The
cost of first mold production is made from the engineer cost and the production cost. Hence,
Cmold =
25 × 150 + 400
x
(11.3.a)
There are elements of the cost that depends on the run size. These costs can be written as
fixed
z}|{
C(x) = Fc +
cost per item
z}|{
L
more complex situation
x+
z}|{
F(x)
(11.28)
237
11.12. INTRODUCTION TO ECONOMICS
Hence, the cost per item (L is the linear cost per item and is not a function of x.) is
c(x) =
Fc
F(x)
C(x)
=
+L+
x
x
x
(11.29)
Examining the terms in equation (11.29) reveals that some terms vanish as the number
of items increase to infinity x −→ ∞, (1/x) = 1/∞ ∼ 0. Any term that behaves like 1/xn
when n is positive vanishes. Seldom there are terms that that do not vanish. A typical equation
repressing this situation is
√
C(x) = 10, 000 + 45 x + 100 x
(11.30)
when the results are in dollars or any other currency.
The revenue is a strong function of the items sold. The price id determine by the supply
and demand diagram (origin of the pQ2 diagram)Supply and demand diagram. The interesting part of the supply and demand diagram shows that the price decreases as the number of
items increases. This part of the analysis indicates that the profits are a strong function item
parts. These facts shows that the total profits is reduced when the total revenue is increased.
cost
The supply and demand diagram was
proposed by Marco Fanno (the older brother
of Fanno from Fanno flow). Normally the supply and demand diagram is not determined at
the factory rather is determined by the market forces or trends. Sometime this diagram is
suggested by supplier15 Hence the revenue typically appears as
supply
demand
# units
Fig. 11.3 – Supply and Demand diagram.
R(x) = p(x) ∼ P0 − Pr x
(11.31)
Where P0 is initial price of the item small quantity, and a Pc is the reduction of the price
which is a function of the sold items. Pr does not have to be a continues function.
.
11.12.1
Marginal Profits
One way to look at the suggested price, the alternative to supply and demand diagram price,
is examining the marginal profitsAlternative price|seealsoSupply and demand diagram. Suppose that the company discussed here selling a certain number of devices x. Should the company increase the number of sales. This question leads indirectly to the suggested price (by
the supplier/manufacture). The marginal profits is defined as
dT
∆T
= lim
dx x→0 ∆x
(11.32)
15 For example, when you buy item the supplier state a price for quantity between 1-5 items you pay full price but
you buy between 6-10 items you have 5% discount per item. Buying between 10-20 items gives additional 5%.
238
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
It can be noticed that here the virtual continues assumption is invoked. Clearly, the minimum
number, x of items is one. However, for large number of items one can be consider small
enough. In other cases, when the number of item is not so small there is an error. However,
the sequential operation helps to rectify it. In the same vein several other definitions can be
defined. The marginal cost is defined as
dC
∆C
= lim
dx
x→0 ∆x
(11.33)
The marginal price is defined in the same manner.
Example 11.4: Marginal profits
Level: Intermediate
The cost function was determined to be
C(x) = 10, 000 + 3.5 x + 50
√
5
x.
(11.4.a)
Calculate the average cost and marginal cost for this given function at 2000 items.
Solution
The average cost is determined by the dividing the total cost by the number of items
c(x) =
C(x)
10, 000 + 3.5 x + 50
=
x
x
√
5
x
(11.4.b)
The marginal cost is determined by equation (11.33) to be
dC
50 −4/5
= 3.5 +
x
dx
5
(11.4.c)
At 2000 items the averaged cost is $8.61 and the marginal cost is $3.61. Notice the significance
or the reduction in the profits.
The price function is can be estimated from the feel for the market. This can be illustrated by the following example.
Example 11.5: Reduce Price
Level: Basic
Your costumer suggested that if you reduces the price by 50/c he will buy more 200
more in addition to 2000 that already ordered. The current price is $4. Determine the
price function as a function of x, number of units sold. What is the number weekly
units that will maximize the weekly revenue? Calculate the maximum weekly revenue.
Solution
This stage involves some extrapolations of the data which was provided. It is unknown but
can be assumed that relationships are linear and can be extent to a larger range. In make this
assumption, one does not calculate the exact value but rather predicts and trends in which
239
11.13. SUMMARY
great for uncertain situations. It was provided that the number of items
End of Ex. 11.5
4 − p(x)
× 200
0 .5
(11.5.a)
x − 2000
= 11 − 0.0025 x
400
(11.5.b)
x = 2000 +
Equation (11.5.a) can be rearranged as
p(s) = 4 −
The revenue can be expressed as
R(x) = p(x) x = 11 x − 0.0025 x2
(11.5.c)
It can be noticed, not only the profit are reduced (since the difference between production cost
and revenue) but the revue is reduced when the price is increased.
R(x)
11
= 11 − 0.005 x =⇒= xmax =
= 1200
dx
0.005
(11.5.d)
Example 11.6: caption
Level: Basic
Assume that a die casting in your company has maximum output of 50,000 units per
year.
Solution
11.13
Summary
In this chapter the economy of the design and choices of the casting process have been presented. It is advocated that the “averaged” approach commonly used in the die casting industry be abandoned. This chapter is planed to be the “flagship” chapter of this book. Engineer should adopt a more elaborate method, in which more precise calculations are made, to
achieve maximum profit for there company. It is believed that the new method will create the
“super die casting economy.”
240
CHAPTER 11. ANALYSIS OF DIE CASTING ECONOMY
A
Fanno Flow
w
An adiabatic flow with friction is named after
+ Ginno Fanno a Jewish engineer. This model
P
P + P
is the second pipe flow model described here.
T
g (M)
UT ++ TU g (M + M )
U
The main restriction for this model is that heat
w
c.v.
transfer is negligible and can be ignored1 . This
model is applicable to flow processes which
Fig. A.1 – Control volume of the gas flow in a
are very fast compared to heat transfer mechconstant cross section.
anisms with small Eckert number.
This model explains many industrial
flow processes which includes emptying of pressured containers through relatively a short
tube, exhaust system of an internal combustion engine, compressed air systems, etc. As
this model raised from the need to explain the steam flow in turbines. In the context of
this book, this model describes the air or gases escape from the die cavity (mold) even the
runner system. The model is a major part the Fanno model and this appendix was added.
The history about this model is provided in (Bar-Meir 2021b).
flow
direction
No heat transer
A.1
Introduction
Consider a gas flowing through a conduit with a friction (see Figure (A.1)). It is advantageous
to examine the simplest situation and yet without losing the core properties of the process.
1 Even the friction does not convert into heat
241
242
APPENDIX A. FANNO FLOW
Later, more general cases will be examined2 .
A.2
Fanno Model
The mass (continuity equation) balance can be written as
ṁ = ρ A U = constant
(A.1)
,→ ρ1 U1 = ρ2 U2
The energy conservation (under the assumption that this model is adiabatic flow and
the friction is not transformed into thermal energy) reads
T01
U1 2
,→ T1 +
2 cp
= T02
= T2 +
Or in a derivative form
Cp dT + d
U2
2
U2 2
2 cp
=0
(A.2)
(A.3)
Again for simplicity, the perfect gas model is assumed which is
P = ρRT
,→
P2
P1
=
ρ1 T1
ρ2 T2
(A.4)
It is assumed that the flow can be approximated as one–dimensional. The force acting
on the gas is the friction at the wall and the momentum conservation reads
−A dP − τw dAw = ṁ dU
(A.5)
For conduits that are not circular like in die casting it is convenient to define a hydraulic
diameter as
DH =
4 × Cross Section Area
wetted perimeter
(A.6)
Or in other words
A=
πDH 2
4
(A.7)
It is convenient to substitute D for DH and yet it still will be referred to the same name as the
hydraulic diameter. The infinitesimal area that shear stress is acting on is
dAw = π D dx
2 Not ready yet, discussed on the ideal gas model and the entry length issues.
(A.8)
243
A.3. NON–DIMENSIONALIZATION OF THE EQUATIONS
Introducing the Fanning friction factor as a dimensionless friction factor which is some times
referred to as the friction coefficient and reads as the following:
f=
1
2
τw
ρ U2
(A.9)
By utilizing equation (A.1) and substituting equation (A.9) into momentum equation (A.5) yields
A
ṁ
τw
z }| {
z }|
A
{
z}|{
π D2
2
1
−
dP − π D dx f 2 ρ U = A ρ U dU
4
(A.10)
Dividing equation (A.10) by the cross section area, A and rearranging yields
−dP +
4 f dx 1
2
ρ
U
= ρ U dU
2
D
(A.11)
The second law is the last equation to be utilized to determine the flow direction.
(A.12)
s2 ⩾ s1
A.3
Non–Dimensionalization of the Equations
Before solving the above equation a dimensionless process is applied. By utilizing the definition of the sound speed to produce the following identities for perfect gas
2
M =
U
c
2
=
U2
k |{z}
RT
(A.13)
P
ρ
Utilizing the definition of the perfect gas results in
M2 =
ρ U2
kP
(A.14)
Using the identity in equation (A.13) and substituting it into Eq. (A.10) and after some rearrangements yield
−dP +
4 f dx DH
1
2
k P M2
ρU2
z }| {
dU
=
dU = k P M2
U
U
ρU2
(A.15)
By further rearranging equation (A.15) results in
−
dP 4 f dx
−
P
D
k M2
2
= k M2
dU
U
(A.16)
244
APPENDIX A. FANNO FLOW
It is convenient to relate expressions of (dP/P) and dU/U in terms of the Mach number and
substituting it into equation (A.16). Derivative of mass conservation (A.1) results in
dU
U
z }|
{
dρ 1 dU2
+2 2 =0
ρ
U
(A.17)
Differentiating of the equation of state (A.4) and dividing the results by equation of state (A.4)
results
dρ dT
dP
=
+
P
ρ
dT
(A.18)
Derivation of the Mach identity equation (A.13) and dividing by equation (A.13) yields
d(M2 )
d(U2 ) dT
=
−
2
T
M
U2
(A.19)
Dividing the energy equation (A.3) by Cp and by utilizing the definition Mach number yields
2
dT
1 U2
U
1
d
+
=
kR
T
T U2
2
(k − 1)
| {z }
Cp
,→
(k − 1) U2
dT
+
d
T
k
R T} U2
| {z
c2
U2
2
(A.20)
=
dT
k − 1 2 dU2
,→
+
M
=0
T
2
U2
Equations (A.16), (A.17), (A.18), (A.19), and (A.20) need to be solved. These equations are separable so one variable is a function of only single variable (the chosen as the independent
variable). Explicit explanation is provided for only two variables, the rest variables can be
fL
done in a similar fashion. The dimensionless friction, 4D
, is chosen as the independent
4fL
variable since the change in the dimensionless resistance, D , causes the change in the other
variables.
Combining equations (A.18) and (A.20) when eliminating dT/T results
dρ (k − 1) M2 dU2
dP
=
−
P
ρ
2
U2
(A.21)
The term dρ
ρ can be eliminated by utilizing Eq. (A.17) and substituting it into Eq. (A.21) and
rearrangement yields
dP
1 + (k − 1) M2 dU2
=−
P
2
U2
(A.22)
245
A.3. NON–DIMENSIONALIZATION OF THE EQUATIONS
The term dU2 /U2 can be eliminated by using (A.22)
k M2 1 + (k − 1)M2 4 f dx
dP
=−
P
D
2 1 − M2
(A.23)
The second equation for Mach number, M variable is obtained by combining equation (A.19)
and (A.20) by eliminating dT/T . Then dρ/ρ and U are eliminated by utilizing equation (A.17)
and equation (A.21). The only variable that is left is P (or dP/P) which can be eliminated by
utilizing Eq. (A.23) and results in
1 − M2 dM2
4 f dx
(A.24)
=
k−1 2
D
4
1+
M
kM
2
Rearranging equation (A.24) results in
dM2
=
M2
k M2
k−1 2
1+
M
4 f dx
2
2
D
1−M
(A.25)
After similar mathematical manipulation one can get the relationship for the velocity
to read
dU
k M2
4 f dx
=
U
D
2 1 − M2
(A.26)
and the relationship for the temperature is
dT
1 dc
k (k − 1) M4 4 f dx
=
=−
T
2 c
2 (1 − M2 ) D
(A.27)
density is obtained by utilizing equations (A.26) and (A.17) to obtain
dρ
k M2
4 f dx
=−
ρ
D
2 1 − M2
(A.28)
The stagnation pressure is similarly obtained as
dP0
k M2 4 f dx
=−
P0
2
D
The second law reads
ds = Cp ln
dT
T
− R ln
(A.29)
dP
P
(A.30)
The stagnation temperature expresses as T0 = T (1 + (1 − k)/2M2 ). Taking derivative of this
expression when M remains constant yields dT0 = dT 1 + (1 − k)/2 M2 and thus when
these equations are divided they yield
dT
dT0
=
T
T0
(A.31)
246
APPENDIX A. FANNO FLOW
In similar fashion the relationship between the stagnation pressure and the pressure can be
substituted into the entropy equation and result in
dT0
dP0
ds = Cp ln
− R ln
(A.32)
T0
P0
The first law requires that the stagnation temperature remains constant, (dT0 = 0). Therefore
the entropy change is
(k − 1) dP0
ds
=−
Cp
k
P0
(A.33)
Using the equation for stagnation pressure the entropy equation yields
ds
(k − 1) M2 4 f dx
=
Cp
2
D
A.4
(A.34)
The Mechanics and Why the Flow is Choked?
The trends of the properties can be examined by looking in equations (A.23) through (A.33).
For example, from Eq. (A.23) it can be observed that the critical point is when M = 1. When
M < 1 the pressure decreases downstream as can be seen from equation (A.23) because fdx
and M are positive. For the same reasons, in the supersonic branch, M > 1, the pressure
increases downstream. This pressure increase is what makes compressible flow so different
from “conventional” flow. Thus the discussion will be divided into two cases: One, flow above
speed of sound. Two, flow with speed below the speed of sound.
A.4.1
Why the flow is choked?
Here, the explanation is based on the equations developed earlier and there is no known explanation that is based on the physics. First, it has to be recognized that the critical point is
when M = 1. It will be shown that a change in location relative to this point change the trend
and it is singular point by itself. For example, dP(@M = 1) = ∞ and mathematically it is a
singular point (see equation (A.23)). Observing from equation (A.23) that increase or decrease
from subsonic just below one M = (1 − ϵ) to above just above one M = (1 + ϵ) requires a
change in a sign pressure direction. However, the pressure has to be a monotonic function
which means that flow cannot crosses over the point of M = 1. This constrain means that
because the flow cannot “crossover” M = 1 the gas has to reach to this speed, M = 1 at the
last point. This situation is called choked flow.
A.4.2
The Trends
The trends or whether the variables are increasing or decreasing can be observed from looking at the equation developed. For example, the pressure can be examined by looking at
Eq. (A.25). It demonstrates that the Mach number increases downstream when the flow is
subsonic. On the other hand, when the flow is supersonic, the pressure decreases.
247
A.5. THE WORKING EQUATIONS
The summary of the properties changes on the sides of the branch
A.5
Subsonic
Supersonic
Pressure, P
decrease
increase
Mach number, M
increase
decrease
Velocity, U
increase
decrease
Temperature, T
decrease
increase
Density, ρ
decrease
increase
The Working Equations
Integration of equation (A.24) yields
4
D
Z Lmax
L
Fanno FLD–M
k+1
2
1 1 − M2 k + 1
2 M
f dx =
ln
+
2
k M2
2k
1 + k−1
2 M
(A.35)
A representative friction factor is defined as
f̄ =
1
Lmax
Z Lmax
fdx
0
(A.36)
In the isothermal flow model it was shown that friction factor is constant through the process
if the fluid is ideal gas. Here, the Reynolds number defined in ?? is not constant because the
temperature is not constant. The viscosity even for ideal gas is complex function of the temperature (further reading in “Basic of Fluid Mechanics” chapter one, Potto Project). However,
the temperature variation is very limited. Simple improvement can be done by assuming constant constant viscosity (constant friction factor) and find the temperature on the two sides
of the tube to improve the friction factor for the next iteration. The maximum error can be
estimated by looking at the maximum change of the temperature. The temperature can be
reduced by less than 20% for most range of the specific heats ratio. The viscosity change for
this change is for many gases about 10%. For these gases the maximum increase of average
Reynolds number is only 5%. What this change in Reynolds number does to friction factor?
That depend in the range of Reynolds number. For Reynolds number larger than 10,000 the
change in friction factor can be considered negligible. For the other extreme, laminar flow
it can estimated that change of 5% in Reynolds number change about the same amount in
friction factor. With the exception of the jump from a laminar flow to a turbulent flow, the
change is noticeable but very small. In the light of the about discussion the friction factor is
248
APPENDIX A. FANNO FLOW
assumed to constant. By utilizing the mean average theorem equation (A.35) yields
4 f Lmax
D
Resistance Mach Relationship


k+1 2
M
2
1 1−M
k+1


2
=
ln 
+
k−1 2
k
2k
M2
1+
M
2
(A.37)
It is common to replace the f̄ with f which is adopted in this book.
Advance material can be skipped
fL
For a very long pipe the value of the 4D
is large but the value the Mach number is very
small. Hence equation (A.37) can be simplified for that case as following. The term change as
2
(1 − M2 )/M2 ∼ 1/M2 and term in the parentheses after the ln in the parentheses is k+1
2 M .
Thus equation can be written for small Mach number as
1 1
k+1 2
4 f Lmax
k+1
=
ln
M
(A.38)
+
D
k M2
2k
2
Taylor series centered at 1 is
ln(x) = (x − 1) −
(x − 1)2 (x − 1)3 (x − 1)4
+
−
···
2
3
4
This series does not converge very quickly and a mathematical trick is used where defining
new variable η as
η=
(x − 1)
(x + 1)
or
η=
(M2 − 1)
(M2 + 1)
Thus returning to original equation yields
ln(x) = ln
1+η
1−η
The right hand side can be expanded as
1+η
1 1 2 1 4 1 6 1 8
ln
= 2η
+ η + η + η + η +···
1−η
1 3
5
7
9
For small M2 (therefor for η) the resistance can be approximated as
4 f Lmax
1 1
(k + 1) η 1 1 2 1 4 1 6 1 8
=
+
+
η
+
η
+
η
+
η
+
·
·
·
D
k M2
k
1 3
5
7
9
(k + 1)
k+1
+
ln
k
2
(A.39)
249
A.5. THE WORKING EQUATIONS
End Advance material
Equations (A.23), (A.26), (A.27), (A.28), (A.28), and (A.29) can be solved. For example, the
pressure as written in equation (A.22) is represented by 4fL
D , and Mach number. Now equation (A.23) can eliminate term 4fL
D and describe the pressure on the Mach number. Dividing
equation (A.23) in equation (A.25) yields
dP
1 + (k − 1M2
P =−
dM2
2
k
−
1
dM
2 M2 1 +
M2
2
M2
(A.40)
The symbol “*” denotes the state when the flow is choked and Mach number is equal to 1.
Thus, M = 1 when P = P∗ equation (A.40) can be integrated to yield:
Mach–Pressure Ratio
v
u
k+1
u
P
1 u
2
u
=
k−1 2
P∗
Mt
1+
M
2
(A.41)
In the same fashion the variables ratios can be obtained
Temperature Ratio
k+1
T
c2
2
= ∗2 =
∗
2
T
c
1 + k−1
2 M
(A.42)
The density ratio is
Density Ratio
v
u
u 1 + k − 1 M2
ρ
1 u
2
u
=
k+1
ρ∗
Mt
2
(A.43)
Velocity Ratio
v
u
k+1
u
−1
u
U
ρ
2
u
=
= Mt
k−1 2
U∗
ρ∗
1+
M
2
(A.44)
The velocity ratio is
250
APPENDIX A. FANNO FLOW
The stagnation pressure decreases and can be expressed by
k
P0
P0 ∗
2 k−1
(1+ 1−k
2 M )
z}|{
P0
P
P
=
∗
P0
P∗
P∗
|{z}
(A.45)
k
2
) k−1
( k+1
Using the pressure ratio in equation (A.41) and substituting it into equation (A.45) yields
P0
P0 ∗
v
 k
u
k − 1 2 k−1
u 1 + k − 1 M2
M
1+
u
1


2
2
u
=

k+1
k+1
Mt
2
2

(A.46)
And further rearranging equation (A.46) provides
Stagnation Pressure Ratio

P0
1 
=

P0 ∗
M
1+
 k+1
k − 1 2 2 (k−1)
M

2

k+1
2
(A.47)
The integration of equation (A.33) yields
v
u
u
u
u
∗
s−s
2 u
= ln M u
t
Cp
 k+1
k
k+1
k−1 2
2 M2 1 +
M
2
The results of these equations are plotted in Figure A.2



(A.48)
251
A.5. THE WORKING EQUATIONS
102
4f L
D
P/P∗
P0 /P0 ∗
ρ/ρ∗
U/U∗
T/T∗
10
1
10−1
10−2
0
1
2
3
4
5
6
7
8
9
10
M
Fig. A.2 – Various parameters in Fanno flow shown as a function of Mach number.
The Fanno flow is in many cases shockless and therefore a relationship between two
points should be derived. In most times, the “star” values are imaginary values that represent
the value at choking. The real ratio can be obtained by two star ratios as an example
T
T∗
T2
=
T
T1
T∗
M2
(A.49)
M1
A special interest is the equation for the dimensionless friction as following
Z L2
L1
4fL
dx =
D
Z Lmax
L1
4fL
dx −
D
Z Lmax
L2
4fL
dx
D
(A.50)
Hence,
fld Working Equation
4 f Lmax
4fL
=
−
D
D
2
1
4 f Lmax
D
(A.51)
252
APPENDIX A. FANNO FLOW
A.6
Examples of Fanno Flow
Example A.1: Reservoir
Level: Intermediate
Air flows from a reservoir and enters a uniform pipe with a diameter of 0.05 [m]
and length of 10 [m]. The air exits to the atmosphere. The following conditions
prevail at the exit: P2 = 1[bar] temperature T2 = 27◦ C M2 = 0.93 . Assume that the average friction factor
M2 = 0.9
D = 0.05[m]
to be f = 0.004 and that the flow
L = 10[m]
P0 =?
from the reservoir up to the pipe inT0 =?◦C
T2 = 27◦C
P2 = 1[Bar]
let is essentially isentropic. Estimate
the total temperature and total presFig. A.3 – Schematic of Example A.1.
sure in the reservoir under the Fanno
flow model.
Solution
For isentropic, the flow to the pipe inlet, the temperature and the total pressure at the pipe
inlet are the same as those in the reservoir. Thus, finding the star pressure and temperature
at the pipe inlet is the solution. With the Mach number and temperature known at the exit,
the total temperature at the entrance can be obtained by knowing the 4fL
D . For given Mach
number (M = 0.9) the following is obtained.
M
4fL
D
P
P∗
0.90000 0.01451 1.1291
P0
P0 ∗
ρ
ρ∗
U
U∗
T
T∗
1.0089
1.0934
0.9146
1.0327
So, the total temperature at the exit is
T ∗ |2 =
T∗
T
T2 =
2
To “move” to the other side of the tube the
4fL
D 1
=
4fL
D
+
4fL
D 2
=
4fL
D
300
= 290.5[K]
1.0327
is added as
4 × 0.004 × 10
+ 0.01451 ≃ 3.21
0.05
The rest of the parameters can be obtained with the new
polations or by utilizing the attached program.
M
4fL
D
0.35886 3.2100
(A.1.a)
4fL
D
P
P∗
P0
P0 ∗
ρ
ρ∗
3.0140
1.7405
2.5764
(A.1.b)
either from Table (A.1) by inter-
U
U∗
T
T∗
0.38814 1.1699
Note that the subsonic branch is chosen. The stagnation ratios has to be added for M =
0.35886
253
A.6. EXAMPLES OF FANNO FLOW
End of Ex. A.1
ρ
ρ0
T
T0
M
P
P0
A×P
A ∗ × P0
F
F∗
0.91484
1.5922
0.78305
A
A⋆
0.35886 0.97489 0.93840 1.7405
The total pressure P01 can be found from the combination of the ratios as follows:
P
P01
z
}|1
{
P∗
z }| {
P
P0
P∗
= P2
P 2 P∗ 1 P 1
1
1
=1 ×
× 3.014 ×
= 2.91[Bar]
1.12913
0.915
(A.1.c)
T
T01
}|1
{
z
T∗
z }| {
T∗
T
T0
= T2
T 2 T∗ 1 T 1
1
1
=300 ×
× 1.17 ×
≃ 348K = 75◦ C
1.0327
0.975
(A.1.d)
Another academic question/example:
Example A.2: Fanno with Convergent-divergent
A system is composed of a convergentdivergent nozzle followed by a tube with
length of 2.5 [cm] in diameter and 1.0 [m]
long. The system is supplied by a vessel.
The vessel conditions are at 29.65 [Bar],
400 K. With these conditions a pipe inlet Mach number is 3.0. A normal shock
wave occurs in the tube and the flow discharges to the atmosphere, determine:
Level: Intermediate
P0 = 29.65[Bar]
T0 = 400K
Mx =?
D = 0.025[m]
L = 1.0[m]
M1 = 3.0
shock
d-c nozzle
Fig. A.4 – The schematic of Example (A.2).
(a) the mass flow rate through the system;
(b) the temperature at the pipe exit; and
(c) determine the Mach number when a normal shock wave occurs [Mx ].
Take k = 1.4, R = 287 [J/kgK] and f = 0.005.
3 This property is given only for academic purposes.
atmospheric
conditions
There is no Mach meter.
254
APPENDIX A. FANNO FLOW
continue Ex. A.2
Solution
(a) Assuming that the pressure vessel is very much larger than the pipe, therefore the velocity in the vessel can be assumed to be small enough so it can be neglected. Thus, the
stagnation conditions can be approximated for the condition in the tank. It is further
assumed that the flow through the nozzle can be approximated as isentropic. Hence,
T01 = 400K and P01 = 29.65[Par].
The mass flow rate through the system is constant and for simplicity point 1 is chosen
in which,
ṁ = ρ A M c
The density and speed of sound are unknowns and need to be computed. With the
isentropic relationship, the Mach number at point one (1) is known, then the following
can be found either from Table A.1, or the popular Potto–GDC as
3.0000
ρ
ρ0
T
T0
M
A
A⋆
0.35714 0.07623 4.2346
P
P0
A×P
A∗ ×P0
F
F∗
0.02722
0.11528
0.65326
The temperature is
T1
T = 0.357 × 400 = 142.8K
T01 01
T1 =
Using the temperature, the speed of sound can be calculated as
c1 =
√
√
k R T = 1.4 × 287 × 142.8 ≃ 239.54[m/sec]
The pressure at point 1 can be calculated as
P1 =
P1
P = 0.027 × 30 ≃ 0.81[Bar]
P01 01
The density as a function of other properties at point 1 is
ρ1 =
P
RT
=
1
8.1 × 104
kg
≃ 1.97
287 × 142.8
m3
The mass flow rate can be evaluated from equation (A.1)
ṁ = 1.97 ×
π × 0.0252
kg
× 3 × 239.54 = 0.69
4
sec
255
A.6. EXAMPLES OF FANNO FLOW
continue Ex. A.2
(b) First, check whether the flow is shockless by comparing the flow resistance and the
maximum possible resistance. From the Table A.1 or by using the famous Potto–GDC,
is to obtain the following
4fL
D
M
3.0000
ρ
ρ∗
P0
P0 ∗
P
P∗
0.52216 0.21822 4.2346
U
U∗
T
T∗
0.50918 1.9640
0.42857
and the conditions of the tube are
4fL
D
=
4 × 0.005 × 1.0
= 0 .8
0.025
Since 0.8 > 0.52216 the flow is choked and with a shock wave.
The exit pressure determines the location of the shock, if a shock exists, by comparing
“possible” Pexit to PB . Two possibilities are needed to be checked; one, the shock at
the entrance of the tube, and two, shock at the exit and comparing the pressure ratios.
First, the possibility that the shock wave occurs immediately at the entrance for which
the ratio for Mx are (shock wave Table ??)
Mx
My
Ty
Tx
ρy
ρx
3.0000
0.47519
2.6790
3.8571
P0y
P0 x
Py
Px
10.3333
0.32834
After the shock wave the flow is subsonic with “M1 ”= 0.47519. (Fanno flow Table A.1)
4fL
D
M
0.47519 1.2919
P
P∗
P0
P0 ∗
ρ
ρ∗
2.2549
1.3904
1.9640
U
U∗
T
T∗
0.50917 1.1481
The stagnation values for M = 0.47519 are
T
T0
M
ρ
ρ0
A
A⋆
0.47519 0.95679 0.89545 1.3904
P
P0
A×P
A∗ ×P0
F
F∗
0.85676
1.1912
0.65326
The ratio of exit pressure to the chamber total pressure is
1
P2
=
P0
=
=
1
z }| { z }| {
P0y
P2
P∗
P1
P0x
P∗
P1
P0y
P0x
P0
1×
1
× 0.8568 × 0.32834 × 1
2.2549
0.12476
256
APPENDIX A. FANNO FLOW
End of Ex. A.2
The actual pressure ratio 1/29.65 = 0.0338 is smaller than the case in which shock
occurs at the entrance. Thus, the shock is somewhere downstream. One possible way
to find the exit temperature, T2 is by finding the location of the shock. To find the
P2
location of the shock ratio of the pressure ratio, P
is needed. With the location of
1
shock, “claiming” upstream from the exit through shock to the entrance. For example,
calculate the parameters for shock location with known 4fL
D in the “y” side. Then either
by utilizing shock table or the program, to obtain the upstream Mach number.
The procedure for the calculations:
Calculate the entrance Mach number assuming the shock occurs at the exit:
′
1) a) set M2 = 1 assume the flow in the entire tube is supersonic:
′
b) calculated M1
Note this Mach number is the high Value.
Calculate the entrance Mach assuming shock at the entrance.
a) set M2 = 1
2) b) add 4fL
D and calculated M1 ’ for subsonic branch
c) calculated Mx for M1 ’
Note this Mach number is the low Value.
According your root finding algorithm4 calculate or guess the shock location
and then compute as above the new M1 .
a) set M2 = 1
3) b) for the new
4fL
D
and compute the new My ’ for the subsonic branch
c) calculated Mx ’ for the My ’
d) Add the leftover of
4fL
D
and calculated the M1
4) guess new location for the shock according to your finding root procedure and according to the result, repeat previous stage until the solution is obtained.
M1
M2
4fL
D up
4fL
D down
Mx
My
3.0000
1.0000
0.22019
0.57981
1.9899
0.57910
(c) The way of the numerical procedure for solving this problem is by finding
4fL
D up
that will produce M1 = 3. In the process Mx and My must be calculated (see the
chapter on the program with its algorithms.).
A.7
Supersonic Branch
In die casting this branch is not relevant.
4 You can use any method you which, but be-careful second order methods like Newton-Rapson method can be
unstable.
257
A.8. WORKING CONDITIONS
A.8
Working Conditions
0
4fL 1
0
B
4fL 1C
s BBB D 1 CCCA < s BB D 2 CCA
It has to be recognized that there are two
regimes that can occur in Fanno flow model
4fL
T
Larger )
D
one of subsonic flow and the other supersonic
T
flow. Even the flow in the tube starts as a
supersonic in parts of the tube can be transformed into the subsonic branch. A shock
wave can occur and some portions of the tube
s
will be in a subsonic flow pattern.
fL
Fig. A.5 – The effects of increase of 4D
on the
The discussion has to differentiate beFanno line.
tween two ways of feeding the tube: converging nozzle or a converging–diverging
fL
, the entrance Mach number, M1 ,
nozzle. Three parameters, the dimensionless friction, 4D
and the pressure ratio, P2 /P1 are controlling the flow. Only a combination of these two
parameters is truly independent. However, all the three parameters can be varied and they
are discussed separately here.
0
A.8.1
Variations of The Tube Length ( 4fL
D ) Effects
In the analysis of this effect, it should be assumed that back pressure is constant and/or low
as possible as needed to maintain a choked flow. First, the treatment of the two branches are
separated.
A.8.1.1
Fanno Flow Subsonic branch
constant pressure
lines
For converging nozzle feeding, increasing the
T
tube length results in increasing the exit Mach
number (normally denoted herein as M2 ). Once
T
the Mach number reaches maximum (M = 1), no
further increase of the exit Mach number can be
achieved. In this process, the mass flow rate decreases.
s
It is worth noting that entrance Mach number is reduced (as some might explain it to re- Fig. A.6 – The effects of the increase of 4Df L on the
Fanno Line.
duce the flow rate). The entrance temperature increases as can be seen from Figure (A.6). The velocity therefore must decrease because the loss of the enthalpy (stagnation temperature) is
P
“used.” The density decrease because ρ = RT
and when pressure is remains almost constant
the density decreases. Thus, the mass flow rate must decrease. These results are applicable to
the converging nozzle.
In the case of the converging–diverging feeding nozzle, increase of the dimensionless
0
1’’
2’’
1’
1
2’
2
Fanno lines
258
APPENDIX A. FANNO FLOW
friction, 4fL
D , results in a similar flow pattern as in the converging nozzle. Once the flow
becomes choked a different flow pattern emerges.
A.8.1.2
Fanno Flow Supersonic Branch
There are several transitional points that
change the pattern of the flow. Point a is the
M1
a
choking point (for the supersonic branch) in
which the exit Mach number reaches to one.
M2
M =1
M all supersonic
Point b is the maximum possible flow for sub c
flow
personic flow and is not dependent on the
m m_ = onst mixed supersonic
nozzle. The next point, referred here as the
with subsonic
flow with a shock the nozzle
between
critical point c, is the point in which no suis still
choked
M1
personic flow is possible in the tube i.e. the
4fL
shock reaches to the nozzle. There is another
D
point d, in which no supersonic flow is possible in the entire nozzle–tube system. Be- Fig. A.7 – The Mach numbers at entrance and exit of tube
and mass flow rate for Fanno Flow as a function of the
tween these transitional points the effect pa4fL
D .
rameters such as mass flow rate, entrance and
exit Mach number are discussed.
At the starting point the flow is choked in the nozzle, to achieve supersonic flow. The
following ranges that has to be discussed includes (see Figure (A.7)):
_
0
4fL
D
4fL
D
4fL
D
<
4fL
D
<
choking
<
shockless
<
4fL
D
<
<
4fL
D
<
4fL
D
<
chokeless
0→a
4fL
shockless
a→b
chokeless
b→c
4fL
D choking
4fL
D
D
∞
c→∞
The 0-a range, the mass flow rate is constant because the flow is choked at the nozzle.
The entrance Mach number, M1 is constant because it is a function of the nozzle design only.
The exit Mach number, M2 decreases (remember this flow is on the supersonic branch) and
fL
starts ( 4D
= 0) as M2 = M1 . At the end of the range a, M2 = 1. In the range of a − b the
flow is all supersonic.
In the next range a − b The flow is double choked and make the adjustment for the
flow rate at different choking points by changing the shock location. The mass flow rate
continues to be constant. The entrance Mach continues to be constant and exit Mach number
is constant.
fL
The total maximum available for supersonic flow b − b ′ , 4D
, is only a theomax
retical length in which the supersonic flow can occur if nozzle is provided with a larger Mach
A.8. WORKING CONDITIONS
259
number (a change to the nozzle area ratio which also reduces the mass flow rate). In the range
b − c, it is a more practical point.
In semi supersonic flow b − c (in which no supersonic is available in the tube but only
in the nozzle) the flow is still double choked and the mass flow rate is constant. Notice that
exit Mach number, M2 is still one. However, the entrance Mach number, M1 , reduces with
fL
the increase of 4D
.
It is worth noticing that in the a − c the mass flow rate nozzle entrance velocity and
the exit velocity remains constant!5
In the last range c − ∞ the end is really the pressure limit or the break of the model
and the isothermal model is more appropriate to describe the flow. In this range, the flow
rate decreases since (ṁ ∝ M1 )6 .
To summarize the above discussion, Figures (A.7) exhibits the development of M1 , M2
fL
mass flow rate as a function of 4D
. Somewhat different then the subsonic branch the mass
flow rate is constant even if the flow in the tube is completely subsonic. This situation is
because of the “double” choked condition in the nozzle. The exit Mach M2 is a continuous
fL
monotonic function that decreases with 4D
. The entrance Mach M1 is a non continuous
function with a jump at the point when shock occurs at the entrance “moves” into the nozzle.
Figure A.8 exhibits the M1 as a function of M2 . The Figure was calculated by utilizing
fL
fL
the data from Figure (A.2) by obtaining the 4D
for M2 and subtracting the given 4D
max
and finding the corresponding M1 .
The Figure (A.9) exhibits the entrance Mach number as a function of the M2 . Obviously
there can be two extreme possibilities for the subsonic exit branch. Subsonic velocity occurs
for supersonic entrance velocity, one, when the shock wave occurs at the tube exit and two, at
fL
fL
the tube entrance. In Figure (A.9) only for 4D
= 0.1 and 4D
= 0.4 two extremes are shown.
4fL
For D = 0.2 shown with only shock at the exit only. Obviously, and as can be observed,
fL
creates larger differences between exit Mach number for the different shock
the larger 4D
fL
larger M1 must occurs even for shock at the entrance.
locations. The larger 4D
4fL
For a given D , below the maximum critical length, the supersonic entrance flow
has three different regimes which depends on the back pressure. One, shockless flow, tow,
shock at the entrance, and three, shock at the exit. Below, the maximum critical length is
mathematically
4fL
1 1+k
k+1
>− +
ln
D
k
2k
k−1
fL
For cases of 4D
above the maximum critical length no supersonic flow can be over the whole
tube and at some point a shock will occur and the flow becomes subsonic flow7 .
5 On a personal note, this situation is rather strange to explain. On one hand, the resistance increases and on the
other hand, the exit Mach number remains constant and equal to one. Does anyone have an explanation for this
strange behavior suitable for non–engineers or engineers without background in fluid mechanics?
6 Note that ρ increases with decreases of M but this effect is less significant.
1
1
7 See more on the discussion about changing the length of the tube.
260
APPENDIX A. FANNO FLOW
Fanno Flow
M1 as function of M2 for the sub sonic brench
1
4fL = 0.1

D
= 1.0
= 10.0
= 100.0
0.9
0.8
0.7
M1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
M2
0.6
0.7
Wed Oct 20 10:59:30 2004
Fig. A.8 – M1 as a function M2 for various
4fL
D .
0.8
0.9
1
261
A.8. WORKING CONDITIONS
Fanno Flow
M1 as a function of M2 for the subsonic brench
5
4fL = 0.1

D
= 0.2
= 0.4
= 0.1 shock
= 0.4
4.5
4
3.5
M1
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
M2
1.2
1.4
1.6
1.8
Tue Jan 4 11:26:19 2005
Fig. A.9 – M1 as a function M2 for different
4fL
D
for supersonic entrance velocity.
2
262
A.8.2
APPENDIX A. FANNO FLOW
The Pressure Ratio, P2 / P1 , effects
In this section the studied parameter is the variation of the back pressure and thus, the pressure ratio ( P2 / P1 ) variations. For very low pressure ratio the flow can be assumed as incompressible with exit Mach number smaller than < 0.3. As the pressure ratio increases (smaller
back pressure, P2 ), the exit and entrance Mach numbers increase. According to Fanno model
fL
the value of 4D
is constant (friction factor, f, is independent of the parameters such as, Mach
number, Reynolds number et cetera) thus the flow remains on the same Fanno line. For cases
where the supply come from a reservoir with a constant pressure, the entrance pressure decreases as well because of the increase in the entrance Mach number (velocity).
Again a differentiation of the feeding is important to point out. If the feeding nozzle is
converging than the flow will be only subsonic. If the nozzle is “converging–diverging” than
in some part supersonic flow is possible. At first the converging nozzle is presented and later
the converging-diverging nozzle is explained.
∆P
P1
a shock in
the nozzle
P2
fully subsoinic
flow
P2
P1
critical point a
critical point c
critical point b
critical point d
4f L
D
Fig. A.10 – The pressure distribution as a function of
4fL
D
for a short
4fL
D .
263
A.8. WORKING CONDITIONS
A.8.2.1
Choking explanation for pressure variation/reduction
Decreasing the pressure ratio or in actuality the back pressure, results in increase of the entrance and the exit velocity until a maximum is reached for the exit velocity. The maximum
velocity is when exit Mach number equals one. The Mach number, as it was shown in Chapter (??), can increases only if the area increase. In our model the tube area is postulated as a
constant therefore the velocity cannot increase any further. However, for the flow to be continuous the pressure must decrease and for that the velocity must increase. Something must
break since there are conflicting demands and it result in a “jump” in the flow. This jump is
referred to as a choked flow. Any additional reduction in the back pressure will not change
the situation in the tube. The only change will be at tube surroundings which are irrelevant
to this discussion.
If the feeding nozzle is a “converging–diverging” then it has to be differentiated between
fL
two cases; One case is where the 4D
is short or equal to the critical length. The critical length
4fL
is the maximum D max that associate with entrance Mach number.
P1
P2
∆P
a shock in
the nozzle
4f L
D max

∆ 

fully subsoinic
flow
4 f L 


D

{
P2
P1
M1 = ∞ ∆  4fL 


D

critical point a
critical point b
critical point c
4f L
D
Fig. A.11 – The pressure distribution as a function of
4fL
D
for a long
4fL
D .

264
APPENDIX A. FANNO FLOW
2.0
1.8
1.6
M
1.4
shock location at:
1.2
75%
50%
5%
1.0
0.8
0.6
0.4
0.0
0.05
0.1
0.15
0.2
0.25
0.3
4f L
D
Fig. A.12 – The effects of pressure variations on Mach number profile as a function of
fL
resistance 4D
= 0.3 for Fanno Flow.
A.8.2.2
4fL
D
when the total
Short 4 f L/ D
Figure A.11 shows different pressure profiles for different back pressures. Before the flow
reaches critical point a (in the Figure A.11) the flow is subsonic. Up to this stage the nozzle
feeding the tube increases the mass flow rate (with decreasing back pressure). Pressure between point a and point b the shock is in the nozzle. In this range and further reduction of
the pressure the mass flow rate is constant no matter how low the back pressure is reduced.
Once the back pressure is less than point b the supersonic reaches to the tube. Note however
that exit Mach number, M2 < 1 and is not 1. A back pressure that is at the critical point c
results in a shock wave that is at the exit. When the back pressure is below point c, the tube
is “clean” of any shock8 . The back pressure below point c has some adjustment as it occurs
with exceptions of point d.
A.8.2.3
Long
4fL
D
fL
fL
In the case of 4D
reduction of the back pressure results in the same process as
> 4D
max
4fL
explained in the short D up to point c. However, point c in this case is different from point
fL
fL
. In this point the exit Mach number is equal to
c at the case of short tube 4D
< 4D
max
1 and the flow is double shock. Further reduction of the back pressure at this stage will not
“move” the shock wave downstream the nozzle. At point c or location of the shock wave, is
fL
a function entrance Mach number, M1 and the “extra” 4D
. There is no analytical solution
8 It is common misconception that the back pressure has to be at point d.
265
A.8. WORKING CONDITIONS
4.5
4.0
3.5
P2
3.0
P1
shock location at:
5%
50%
75%
2.5
2.0
1.5
1.0
0.0
0.05
0.1
0.15
0.2
0.25
0.3
4f L
D
Fig. A.13 – Pressure ratios as a function of
4fL
D
when the total
4fL
D
= 0.3.
for the location of this point c. The procedure is (will be) presented in later stage.
A.8.3
Entrance Mach number, M1 , effects
In this discussion, the effect of changing the
throat area on the nozzle efficiency is neglected. In reality these effects have significance and needs to be accounted for some instances. This dissection deals only with the
flow when it reaches the supersonic branch
reached otherwise the flow is subsonic with
regular effects. It is assumed that in this disP2
cussion that the pressure ratio P
is large
1
fL
enough to create a choked flow and 4D
is
small enough to allow it to occur.
fL CA
B 4D
fL
D max1
0
4
1
fL
D retreat
4
M = 1 or less
Mx M
y
shock
M =1
Fig. A.14 – Schematic of a “long” tube in supersonic branch.
The entrance Mach number, M1 is a function of the ratio of the nozzle’s throat area
to the nozzle exit area and its efficiency. This effect is the third parameter discussed here.
Practically, the nozzle area ratio is changed by changing the throat area.
fL
As was shown before, there are two different maximums for 4D
; first is the total max4fL
imum D of the supersonic which depends only on the specific heat, k, and second the maximum depends on the entrance Mach number, M1 . This analysis deals with the case where
4fL
4fL
D is shorter than total D max . Obviously, in this situation, the critical point is where
266
APPENDIX A. FANNO FLOW
fL CA
B 4D
1
0
M
1
=8
M
M =1
1
=5
1
0
fL
D retreat
4
fL
D max
4
Fig. A.15 – The extra tube length as a function of the shock location,
4fL
D
supersonic branch.
fL
is equal to 4D
as a result in the entrance Mach number.
max
The process of decreasing the converging–diverging nozzle’s throat increases the entrance9 Mach number. If the tube contains no supersonic flow then reducing the nozzle throat
area wouldn’t increase the entrance Mach number.
This part deals with cases where some part of the tube is under supersonic regime and
there is a shock as a transition to subsonic branch. Decreasing the nozzle throat area moves
the shock location downstream. The “payment” for increase in the supersonic length is by
reducing the mass flow. Further, decrease of the throat area results in flushing the shock out
of the tube. By doing so, the throat area decreases. The mass flow rate is proportionally linear
to the throat area and therefore the mass flow rate reduces. The process of decreasing the
throat area also results in increasing the pressure drop of the nozzle (larger resistance in the
nozzle10 )11 .
fL
fL
In the case of large tube 4D
> 4D
the exit Mach number increases with the
max
decrease of the throat area. Once the exit Mach number reaches one no further increases is
possible. However, the location of the shock wave approaches to the theoretical location if
entrance Mach, M1 = ∞.
4fL
D
The Maximum Location of the Shock
9 The word “entrance” referred to the tube and not to the nozzle. The reference to the tube is because it is the
focus of the study here.
10 Strange? Frictionless nozzle has a larger resistance when the throat area decreases.
11 It is one of the strange phenomenon that in one way increasing the resistance (changing the throat area) decreases
fL
the flow rate while in a different way (increasing the 4D
) does not affect the flow rate.
267
A.8. WORKING CONDITIONS
M
1 max
1
fL
D
4
4f L
D max∞
Fig. A.16 – The maximum entrance Mach number, M1 to the tube as a function of
4fL
D
supersonic branch.
The main point in this discussion however,is to find the furthest shock location downfL
stream. Figure (A.15) shows the possible ∆ 4D
as a function of retreat of the location of
the shock wave from the maximum location. When the entrance Mach number is infinity,
M1 = ∞, if the shock location is at the maximum length, then shock at Mx = 1 results in
My = 1.
The proposed procedure is based on Figure A.15.
fL
and subtract the actual extra
i) Calculate the extra 4D
side (at the max length).
fL
ii) Calculate the extra 4D
and subtract the actual extra
side (at the entrance).
4fL
D
4fL
D
assuming shock at the left
assuming shock at the right
iii) According to the positive or negative utilizes your root finding procedure.
From numerical point of view, the Mach number equal infinity when left side assumes
result in infinity length of possible extra (the whole flow in the tube is subsonic). To overcome
this numerical problem, it is suggested to start the calculation from ϵ distance from the right
hand side.
Let denote
4fL
4 f¯ L
4fL
∆
=
−
(A.52)
D
D actual
D sup
268
APPENDIX A. FANNO FLOW
fL
fL
Note that 4D
is smaller than 4D
. The requirement that has to be satisfied is
sup
max∞
4fL
that denote D retreat as difference between the maximum possible of length in which the
supersonic flow is achieved and the actual length in which the flow is supersonic see Figure
A.16. The retreating length is expressed as subsonic but
4fL
D
=
retreat
4fL
D
−
max∞
4fL
D
(A.53)
sup
Figure A.16 shows the entrance Mach number, M1 reduces after the maximum length
is exceeded.
Example A.3: Large FLD SS
Level: Intermediate
Calculate the shock location for entrance Mach number M1 = 8 and for
assume that k = 1.4 (Mexit = 1).
4fL
D
= 0.9
Solution
The solution is obtained by an iterative process. The maximum
0.821508116. Hence,
4fL
D
exceed the maximum length
maximum for M1 = 8 is
4fL
D
4fL
D
4fL
D max
for k = 1.4 is
for this entrance
Mach number. The
= 0.76820, thus the extra tube is ∆
4fL
D
= 0.9 − 0.76820 =
= 0.76820 (flow is choked and no
0.1318. The left side is when the shock occurs at
fL
additional 4D
). Hence, the value of left side is −0.1318. The right side is when the shock is
fL
at the entrance at which the extra 4D
is calculated for Mx and My is
4fL
D
Ty
Tx
My
8.0000
0.39289
13.3867
M
4fL
D
P
P∗
P0
P0 ∗
0.39289
2.4417
2.7461
1.6136
P0y
P0 x
Py
Px
y
Mx
x
5.5652
74.5000
0.00849
With (M1 ) ′
∗
2.3591
U
U∗
T
T∗
0.42390
1.1641
fL
The extra ∆ 4D
is 2.442 − 0.1318 = 2.3102 Now the solution is somewhere between the
negative of left side to the positive of the right side12 .
In a summary of the actions is done by the following algorithm:
fL
(a) check if the 4D
exceeds the maximum
ingly continue.
(b) Guess
4fL
D up
=
4fL
D
−
4fL
D max
for the supersonic flow. Accord-
4fL
D max
(c) Calculate the Mach number corresponding to the current guess of
4fL
D up ,
(d) Calculate the associate Mach number, Mx with the Mach number, My calculated
previously,
269
A.8. WORKING CONDITIONS
(e) Calculate
4fL
D
End of Ex. A.3
for supersonic branch for the Mx
(f) Calculate the “new and improved”
(g) Compute the “new
4fL
D down
(h) Check the new and improved
stop or return to stage (b).
Shock location are:
=
4fL
D up
4fL
4fL
D − D up
4fL
D down
against the old one. If it is satisfactory
M1
M2
4fL
D up
4fL
D down
Mx
My
8.0000
1.0000
0.57068
0.32932
1.6706
0.64830
The iteration summary is also shown below
i
4fL
D up
4fL
D down
Mx
My
4fL
D
0
0.67426
0.22574
1.3838
0.74664
0.90000
1
0.62170
0.27830
1.5286
0.69119
0.90000
2
0.59506
0.30494
1.6021
0.66779
0.90000
3
0.58217
0.31783
1.6382
0.65728
0.90000
4
0.57605
0.32395
1.6554
0.65246
0.90000
5
0.57318
0.32682
1.6635
0.65023
0.90000
6
0.57184
0.32816
1.6673
0.64920
0.90000
7
0.57122
0.32878
1.6691
0.64872
0.90000
8
0.57093
0.32907
1.6699
0.64850
0.90000
9
0.57079
0.32921
1.6703
0.64839
0.90000
10
0.57073
0.32927
1.6705
0.64834
0.90000
11
0.57070
0.32930
1.6706
0.64832
0.90000
12
0.57069
0.32931
1.6706
0.64831
0.90000
13
0.57068
0.32932
1.6706
0.64831
0.90000
14
0.57068
0.32932
1.6706
0.64830
0.90000
15
0.57068
0.32932
1.6706
0.64830
0.90000
16
0.57068
0.32932
1.6706
0.64830
0.90000
17
0.57068
0.32932
1.6706
0.64830
0.90000
This procedure rapidly converted to the solution.
270
A.9
APPENDIX A. FANNO FLOW
The Practical Questions and Examples of Subsonic branch
The Fanno is applicable also when the flow isn’t choke13 . In this case, several questions appear
for the subsonic branch. This is the area shown in Figure (A.7) in beginning for between points
0 and a. This kind of questions made of pair given information to find the conditions of the
flow, as oppose to only one piece of information given in choked flow. There many combinations that can appear in this situation but there are several more physical and practical that
will be discussed here.
A.10
The Table for Fanno Flow
Table A.1 – Fanno Flow Standard basic Table k=1.4
M
4fL
D
P
P∗
P0
P0 ∗
ρ
ρ∗
U
U∗
T
T∗
0.03
787.08
36.5116
19.3005
30.4318
0.03286 1.1998
0.04
440.35
27.3817
14.4815
22.8254
0.04381 1.1996
0.05
280.02
21.9034
11.5914
18.2620
0.05476 1.1994
0.06
193.03
18.2508
9.6659
15.2200
0.06570 1.1991
0.07
140.66
15.6416
8.2915
13.0474
0.07664 1.1988
0.08
106.72
13.6843
7.2616
11.4182
0.08758 1.1985
0.09
83.4961
12.1618
6.4613
10.1512
0.09851 1.1981
0.10
66.9216
10.9435
5.8218
9.1378
0.10944 1.1976
0.20
14.5333
5.4554
2.9635
4.5826
0.21822 1.1905
0.25
8.4834
4.3546
2.4027
3.6742
0.27217 1.1852
0.30
5.2993
3.6191
2.0351
3.0702
0.32572 1.1788
0.35
3.4525
3.0922
1.7780
2.6400
0.37879 1.1713
0.40
2.3085
2.6958
1.5901
2.3184
0.43133 1.1628
0.45
1.5664
2.3865
1.4487
2.0693
0.48326 1.1533
0.50
1.0691
2.1381
1.3398
1.8708
0.53452 1.1429
12 What if the right side is also negative? The flow is chocked and shock must occur in the nozzle before entering
the tube. Or in a very long tube the whole flow will be subsonic.
13 These questions were raised from many who didn’t find any book that discuss these practical aspects and send
the questions to this author.
271
A.10. THE TABLE FOR FANNO FLOW
Table A.1 – Fanno Flow Standard basic Table (continue)
M
4fL
D
P
P∗
P0
P0 ∗
ρ
ρ∗
U
U∗
T
T∗
0.55
0.72805
1.9341
1.2549
1.7092
0.58506 1.1315
0.60
0.49082
1.7634
1.1882
1.5753
0.63481 1.1194
0.65
0.32459
1.6183
1.1356
1.4626
0.68374 1.1065
0.70
0.20814
1.4935
1.0944
1.3665
0.73179 1.0929
0.75
0.12728
1.3848
1.0624
1.2838
0.77894 1.0787
0.80
0.07229
1.2893
1.0382
1.2119
0.82514 1.0638
0.85
0.03633
1.2047
1.0207
1.1489
0.87037 1.0485
0.90
0.01451
1.1291
1.0089
1.0934
0.91460 1.0327
0.95
0.00328
1.061
1.002
1.044
0.95781 1.017
1.00
0 .0
1.00000
1.000
1.000
1.00
2.00
0.30500
0.40825
1.688
0.61237 1.633
0.66667
3.00
0.52216
0.21822
4.235
0.50918 1.964
0.42857
4.00
0.63306
0.13363 10.72
0.46771 2.138
0.28571
5.00
0.69380
0.089443 25.00
0.44721 2.236
0.20000
6.00
0.72988
0.063758 53.18
0.43568 2.295
0.14634
7.00
0.75280
0.047619 1.0E + 2
0.42857 2.333
0.11111
8.00
0.76819
0.036860 1.9E + 2
0.42390 2.359
0.086957
9.00
0.77899
0.029348 3.3E + 2
0.42066 2.377
0.069767
10.00
0.78683
0.023905 5.4E + 2
0.41833 2.390
0.057143
20.00
0.81265
0.00609
1.5E + 4
0.41079 2.434
0.014815
25.00
0.81582
0.00390
4.6E + 4
0.40988 2.440
0.00952
30.00
0.81755
0.00271
1.1E + 5
0.40938 2.443
0.00663
35.00
0.81860
0.00200
2.5E + 5
0.40908 2.445
0.00488
40.00
0.81928
0.00153
4.8E + 5
0.40889 2.446
0.00374
45.00
0.81975
0.00121
8.6E + 5
0.40875 2.446
0.00296
1.000
272
APPENDIX A. FANNO FLOW
Table A.1 – Fanno Flow Standard basic Table (continue)
M
A.11
4fL
D
P
P∗
P0
P0 ∗
ρ
ρ∗
U
U∗
T
T∗
50.00
0.82008
0.000979 1.5E + 6
0.40866 2.447
0.00240
55.00
0.82033
0.000809 2.3E + 6
0.40859 2.447
0.00198
60.00
0.82052
0.000680 3.6E + 6
0.40853 2.448
0.00166
65.00
0.82066
0.000579 5.4E + 6
0.40849 2.448
0.00142
70.00
0.82078
0.000500 7.8E + 6
0.40846 2.448
0.00122
Appendix – Reynolds Number Effects
Almost Constant
Zone
Constant
Zone
Linear Representation
Zone
Small Error
Due to Linear
Assumption
Fig. A.17 – “Moody” diagram on the name of Moody who netscaped H. Rouse’s work to claim as his own. In
this section, the turbulent area is divided into 3 zones, constant, semi–constant, and a linear After S Beck
and R. Collins.
The friction factor in equation (A.24) was assumed constant. In Chapter ?? it was shown
that the Reynolds number remains constant for ideal gas fluid. However, in Fanno flow the
temperature does not remain constant. Hence, as it was discussed before, the Reynolds number is increasing. Thus, the friction decreases with the exception of the switch in the flow pat-
273
A.11. APPENDIX – REYNOLDS NUMBER EFFECTS
tern (laminar to turbulent flow). For relatively large relative roughness larger ϵ/D > 0.004
of 0.4% the friction factor is constant. For smoother pipe ϵ/D < 0.001 and Reynolds number
between 10,000 to a million the friction factor vary between 0.007 to 0.003 with is about factor of two. Thus, the error of 4fL
D is limited by a factor of two (2). For this range, the friction
factor can be estimated as a linear function of the log10 (Re). The error in this assumption is
probably small of the assumption that involve in fanno flow model construction. Hence,
(A.54)
f = A log10 (Re) + B
Where the constant A and B are function of the relative roughness. For most practical
purposes the slope coefficient A can be further assumed constant. The slope coefficient
A = −0.998125. Thus, to carry this calculation relationship between the viscosity and the
temperature has to be established. If the viscosity expanded as Taylor or Maclaren series then
µ
A T
= A0 + 1 + · · ·
µ1
T0
(A.55)
Where µ1 is the viscosity at the entrance temperature T1 .
Thus, Reynolds number is
Re =
DρU
A0 +
A1 T
T0
(A.56)
+···
Substituting equation (A.56) into equation (A.54) yield
f = A log10
DρU
A0 +
A 1 T2
T1
+···
!
+B
(A.57)
Left hand side of equation (A.24) is a function of the Mach number since it contains the temperature. If the temperature functionality will not vary similarly to the case of constant friction factor then the temperature can be expressed using equation (A.42).




constant
z }| {



4 
DρU




+
B
(A.58)

A log10 

2



D
1 + k−1
M
1
2
A0 + A1
+···
2
1 + k−1
2 M2
Equation (A.58) is only estimate of the functionally however, this estimate is almost as good
as the assumptions of Fanno flow. Equation (A.51) can be improved by using equation (A.58)




constant
z }| {


 1 1 − M2 k + 1
k+1
2
DρU
4 Lmax 




2 M
+
B
∼
+
ln
A log10 


2 
2

 k M2
D 
2k
1 + k−1
1 + k−1
2 M
2 M
A0 + A1
1 + k−1
2
(A.59)
274
APPENDIX A. FANNO FLOW
In the most complicate case where the flow pattern is change from laminar flow to turbulent
flow the whole Fanno flow model is questionable and will produce poor results.
In summary, in the literature there are three approaches to this issue of non constant
friction factor. The friction potential is recommended by a researcher in Germany and it
is complicated. The second method substituting this physical approach with numerical iteration. In the numerical iteration method, the expression of the various relationships are
inserted into governing differential equations. The numerical methods does not allow flexibility and is very complicated. The methods described here can be expended (if really really
needed) and it will be done in very few iteration as it was shown in the Isothermal Chapter.
B
Flow in Open Channels
B.1
Introduction
Fl
ow
The open channel is a branch of the fluid mechanics which deals with an open surface such
as a river. The topic presented in this author’s
book (Bar-Meir 2021a). This topic is a breach
separated from multiphase flow because it has
a unique characters set of problems. The hydraulic jump phenomenon is one of the feature
of the open channel flow and it will be present
in this chapter.
∆x
y
d
g
Fig. B.1 – Equilibrium of Forces in an open channel.
The discussion on hydraulic jump is on the focus of this section but several definitions
and concepts must be presented. The flow or the hydraulic jump in die casting is done in a
semi circular shape (shot sleeve). This shape is harder to analysis and first a discussion on two
dimensional shape is presented.
The flow in open channel flow in steady state is balanced by between the gravity forces
and mostly by the friction at the channel bed. As one might expect, the friction factor for open
channel flow has similar behavior to to one of the pipe flow with transition from laminar flow
to the turbulent at about Re ∼ 103 . Nevertheless, the open channel flow has several respects
the cross section are variable, the surface is at almost constant pressure and the gravity force
275
276
APPENDIX B. FLOW IN OPEN CHANNELS
are important.
The flow of a liquid in a channel can be characterized by the specific energy that is
associated with it. This specific energy is comprised of two components: the hydrostatic
pressure and the liquid velocity1 .
B.2
What is Open Channel Flow?
B.2.1
Introduction
Open channel flow is a branch of multi phase flow. Traditionally, open channel flow is considered as a direct branch of fluid machines because it was studied much earlier. However,
one can view the open channel flow as (almost) horizontal two phase flow with extremely
large ratio of gas flow to liquid flow. In that case, the flow is stratified flow (as can be observed from the two phase flow regime map). Furthermore, the gas phase can be assumed
almost unchanged, and therefore, the liquid upper surface can be assumed to be under constant pressure.
The open channel flow and the pipe flow move liquids from one place to another.
Yet, the main different between these two flows
is that, in pipe flow, the shape of the pipe determines the flow cross section shape while
vents
in open channel flow the shape of the flow is
die
P
ouring
determined by the flow. The secondary difgate
hole
ference is that in pipe flow the pressure derunner
termined from the flow while in open chancold
chamber
nel flow, the pressure is determined from the
liquid metal
gas phase (through the free surface). In plain
Fig. B.2 – Open Channel flow in die casting.
English, in the pipe flow, the resistance in
the pipe determines the pressure down stream
while for open channel flow, the surroundings pressure (atmosphere) at channel interface
determines the pressure in the flow down stream. In the limiting case, the pressure remains
constant. This limiting case is what will be discussed mostly in this chapter.
The open channel flow occurs in nature as can be observed in river flow (and many
water running systems). Open channel flow occurs in many man made situations like sewer
systems and many water supply systems. While the open channel flow was traditionally dealt
with mostly water (or water base) as the substance it also can be applied to many kind of substances, oil, methanol, liquid metal etc. It also can appear in situations that one does not expect
it. For example, in die casting process (see Fig. B.2), where a liquid metal is injected into the
cavity, creates open channel flow a situation which determines major operating parameters.
1 The velocity is an average velocity
277
B.2. WHAT IS OPEN CHANNEL FLOW?
Another word on the classification of
partial
this flow. Open channel flows are bound by the
F low
in
boundaries on lower part and top is exposed to
pipe
the atmosphere or other gaseous medium (see
Fig. B.3). According to this definition, the flow
in the pipe in Fig. B.3 also will be considered
to be open flow yet some will consider it to
Fig. B.3 – What is open channel flow? Some limbe two phase flow. For any kind flow with a
itations on the definition.
free surface, the flow boundary is can be deformed in contrast to solid boundary (almost).
The conditions at boundary for true open channel are different from the multi–phase which
the shear stress is zero and the pressure is atmospheric. The flow in pipe sometimes referred
as a stratified flow. In this chapter only true open channel is discussed. If one is particular
about the definition, the flow in rivers and other channel is not a open channel flow according
to this definition. However, the effect is not that significant and hence it is considered to be
open channel flow.
All the equations and principles developed
1
earlier still can be applied to new situations. In ad2
dition to the flow that was dealt before, the open
channel flow and in particular the issue of the top
Fig. B.4 – Change of the height of the botboundary is focus here. As opposed to the flow in
tom has two possibilities 1 and 2.
closed conduit, the boundary has to be determine
and cross area is depend on the flow. This new
complexity is one of the main topics in the chapter. The change of the boundary also affect
the kind of flow in open channel flow. As oppose to the close conduit flow, the open channel
flow is strongly affected by the gravity. Additional difference, the waves can be generated on
the free surface regardless to the movement of the liquid.
U
B.2.2
Open Channel “Intuition’
As in compressible flow, the open channel flow, one has to gain new intuition. Supposed
that flow exposed to a change of the height of the channel bottom as shown in Fig. B.4. The
change can be also negative, in other words the bottom located in lower position. Assume
the flow is a two dimensional case (other limitations such as surface tension are insignificant.). What the height of the liquid will be after the obstacle? There is two possibilities, one,
278
APPENDIX B. FLOW IN OPEN CHANNELS
the liquid level increases, and two, the liquid level
decreases.
A
To consider what direction the height takes,
one has to get information from the familiar. Instinctively as the situation described in A is something that most readers (if not all) familiar with.
B
When looking at the situation from a rotated coordinate system it is clear that free surface height
Fig. B.5 – Flow on an include plane to
increases. In this case the height increase unchanges the bottom direction. Figure
A shows the actual flow and B shows
boundedly (without a limit). when the change
the same flow in a rotated coordinate
is limited the height is limited. In the figure,
system.
the change is shown as a gradual transition from
one height to another. This change is only for
illustration and this change in most cases not correct. A more refined analysis is required for
the change describe the change.
B.2.3
Energy Line
As usual engineers do, first build and defined a reference situation which is
used later as a base for further analysis can be carried out.
That is, a flow
with an angle inclination is assumed to be free of the three dimensional
effects. It further assumed that a steady state is
achieved. The transition length is not part of
the discussion here. It is further assumed that
L
A
the velocity profile in any cross section is the
h
B
same. In other words, the flow or the velocity
y
θ
profile in “A” is the same as in “B”. That is, the
x
initial condition does not affect the flow at this
Fig. B.6 – Uniform flow on include plane assume
point. This situation in nature can be closed to
no change from section A to section B
reality and in a laboratory the flow can be even
closer.
The x and the y are defined in the Fig. B.6 and y = 0 is at the bottom. It
is assumed that no–slip condition exist at the bottom (y = 0). The velocity (as it
will be shown) reaches its maximum at the interface (at the conclusion of the analysis).
279
B.2. WHAT IS OPEN CHANNEL FLOW?
The flow is uniform, hence the velocity is
in the x direction only. The control volume is shown in Fig. B.6 and Fig. B.7 from
the front. Assuming that the resistance to
b
the flow at the edge can be considered uniformed. The force balance in the x direction has only the liquid weight and shear
Fig. B.7 – Control volume from the front.
stress. It can be noticed that as stated, the
velocity in and out canceled out and the
pressure on both surfaces is the same. At this stage, it is assumed that the shear stresses at
the wall are the same as the bottom shear stresses. Under these assumptions, the balance (see
B.8) reads
volume
z }| {
ρ g sin θ b L
Ah =

bottom
walls

z}|{
z }| {


τ0  L
Ab + 2hL
A
mg h
ρ g sin θ h
τ0 =
b+2h
τw
(B.1)
P
τw
θ
m g sin θ
(B.2)
Fig. B.8 – Force balance in the flow direction
open channel. Notice that in this case τw =
τb = τ0 .
In general as shown in Fig. B.9, any cross section can have a similar expression for the averaged shear stress. Yet, the only limitation is
that the same cross section remains the same in
the channel. The cross area defined in illustration as A (cross section) and the P the wetted
edge (perimeter).
ρ g sin θ A
τb
b
The averaged shear stress is than
τ0 =
L
U
(B.3)
The shear stresses in the general case is more
uniform as compared to the rectangle case.
A
P
Fig. B.9 – Constant cross section in general. Orange is the perimeter, P , and the area, A, is
the cross section.
The averaged shear stress for the rectangle, which was obtained earlier, can be used to
obtain an expression for two dimensional flow. In that case, Eq. (B.2) reduces to
∼ ρ g sin θ h
τ0 =
b
(B.4)
The shear stress in the rectangle case changes with the height of the liquid. Con-
280
APPENDIX B. FLOW IN OPEN CHANNELS
sider the control volume shown in Fig. B.10 in which forces are similar to previous. For the same reasons as before, the net momentum flux is zero as well as the
net pressure difference on both side. The net
shear stress (force) balance reads
L
τ = ρ g sin θ (h − y)
(B.5)
h τ
The shear stress is a linear function of y and its
maximum is at y = 0 and zero at the surface (as
expected). As it can be recalled, the shear stress
is linearly related to the velocity derivative for
a laminar flow with respect to y.
h τ
τ y
θ
θ
Fig. B.10 – Small control volume to ascertain
shear stress.
τ
ρ g sin θ (h − y)
dU
= =
dy
µ
µ
After the substitution, a very simple ordinary
differential equation is defined for the velocity.
Eq. (B.6) can be integrated to yield
ρ g sin θ
y2
U=
hy−
+C
(B.7)
µ
2
with the no-slip boundary condition of U(y =
0) = 0 then C = 0 and/or no shear stress at
the interface.
U=
ρ g sin θ
µ
(B.6)
τ (y)
U
H0
dU
dy
h
τ0
θ
y
x
Fig. B.11 – to explain the transition from Shear
stress to velocity function.
y2
hy−
2
(B.8)
Note, Eq. (B.8) is correct only in the case where no slip is appeared (not always!). All the
relevant equations are actually plotted on Fig. B.11. The solution was for rectangular shape
and only for laminar flow the assumption of the shear stress). The flow rate per width can be
derived for this velocity profile
Zh
q=
0
ρ g sin θ
µ
y2
ρ g sin θ h3
hy−
dy =
2
3µ
(B.9)
In real application, the flow is not laminar even for relatively small Reynolds numbers. For extremely small Reynolds number (and high viscosity) there is a good agreement
between the theory and the experiments. Please note that there is Reynolds below which no
2-dimensional flow can exist. The change is that information passes from a layer to another
later. The shear stress (viscosity) can be viewed as a transfer of momentum like a transfer of
heat or mass across layers. Another view of the thickness of the liquid essentially depends on
281
B.3. ENERGY CONSERVATION
the shear stress at the wall. Larger forces (shear stress) at the bottom can carry more weight.
Alternatively, the velocity is reduced because the shear force at the bottom overcome it and
to compensate for larger resistance by the liquid height has to increase.
Some empirical equations describes the shear stress such as Chézy coefficient (?) as
τ0 =
f ρ U2
8
(B.10)
With this shear stress, the flow rate can be obtained. A better coefficient is Manning coefficient. Regardless to specific (it not turbulence book) the reader should be aware of the topic.
B.3
Energy conservation
The energy is conserved as long there is no energy loss (by definition) significant. Hence, the
energy equation has to be developed. The energy at every cross section has to include the kinetic and potential, as they changed from a cross section to a cross section. Bernoulli equation
per unit volume of fluid moving along a streamline, ρ U2 /2 + P + ρ g h and is constant. Or
it can be written for dividing by g which the energy per unit weight of fluid (as ρ g is weight).
Ew =
P
U2
+
+H
2g ρg
(B.11)
where H is liquid height from arbitrary point (not the bottom of the channel). This equation
(B.11) is exact for on the same stream line. In order to generalize this equation two assumptions
have to be made. One, the acceleration perpendicular to the flow is insignificant thus the
∼ ρ g (h − y).
pressure is basically the hydrostatic pressure, almost the actual pressure P =
Two, the sum of the height and the pressure can be written as
H
H+
z }| { P
P
ρ g (h − y)
H
= H0 + y +
= H0 + y + H
= h + H0
ρg
ρg
ρH
g
H
(B.12)
where H0 is height from arbitrary datum to channel bottom shown in Fig. B.11. Eq. (B.11) can
be written as
Ew =
U2
+ h + H0
2g
(B.13)
The energy Ew plot as a function of horizontal line is referred to as the energy grade
line. For any kind of the open channel which was discussed here, the energy line decreases
with the horizontal (in the flow direction). The reason for the decrease is because the pressure
remains the same and the liquid height is the same while the elevation (H) is lower with the
downstream progression. The head loss is defined as
Head Loss Rec
H2
Ew2 − Ew1
H
z }| {
z }|1 {
!
!
2
U2
U1 2
=
+ h2 +H02 −
+ h1 +H01
2
2
(B.14)
282
APPENDIX B. FLOW IN OPEN CHANNELS
While technically this equation is not appropriate for the rapid acceleration still for a quick
result, Eq. (B.12) can be used for a quick calculation.
For this uniform flow, the pressure remains the same on a stream line and the velocity
as well, while the potential energy decreases downstream. The energy loss is actually the
change in elevation or in another view, the rate loss is the slop of the channel bottom. When
the change in bottom are relatively small, the loss is negligible and energy (head) Eq. (B.14)
reads
(B.15)
H02 + H2 = H01 + H1 −−→ H2 = H01 − H02 + H1
An energy specific variable is defined as
Specific Energy Rec
H = h+
U2
2g
(B.16)
which presents the energy for unit width.
Using the mass flow rate per unit width q = U h hence the energy
Specific Energy Rec q
q2
2 g h2
(B.17)
h
h
H = h+
b1
b2
ow
Fl
,
se
ea
cr
n
i
te
ra
q
H
(a) Height lines for open channel as a function of the
energy for various constant flow rate lines.
C
p2 p1
∆H
H
(b) Energy line with the effects of elevation change
showing the two possibilities.
Fig. B.12 – The energy lines in general and specific case The reduction of in H result in two possibilities
depending if the flow in the “high” speed or the “low” speed. Points b1 and b2 are denoted the point on
subcritical flow and points p1 and p2 supercritical flow see fig. (b).
This quantify remains the same (constant) for uniform flow at steady state. This equation,
Eq. (B.15), can be used to evaluate the height of the channel for a given flow rate, q. The value
of H is defined from the slope of the channel which determines the velocity of the liquid. That
is, for a given sloop the energy (velocity) is determined by it. For a specific energy and fixed
flow rate there is a height that correspond to these data. Eq. (B.17) is a cubic equation which
283
B.3. ENERGY CONSERVATION
means that there are three possible solutions. This equation can be solved analytically and the
solution of quadratic equation is given in (Bar-Meir 2021b). Yet the expressions are very long
and thus not presented here. For a small range of H, there is only one real negative solution
and two imaginary solutions. For larger values of H, the solution has two positive roots and
one negative root. The negative root is rejected as it is not physically possible (no negative
height). The solution (actually the governing equation) becomes a parabola (two roots) since
one of the root was rejected. In other words, the cubic equation is reduced to a quadratic
equation. The solution is plotted on the diagram Fig. B.12 (part a). These two roots represent
the two different regimes for flow. Similar to compressible flow, one) branch with the smaller
height, h thus with larger velocity and two) branch with the larger height thus smaller velocity.
Fig. B.4 exhibits a situation of the flow in a open channel for which the bottom is elevated. The value of ∆H0 = H02 − H01 is positive. That is according to Eq. (B.15) the value
of H decreases (note the order in the equation). The flow rate is a constant (the flow rate
did not change for the different height). The H reduced is exhibited in Fig. B.12b. There
is two possibilities either the “high” speed and “low” speed. For the “high speed” (the lower
branch), the height increase and therefore the velocity reduces. The opposite occurs on the
“low” speed branch. Again it is similar to compressible flow. The points where the “high” and
“low” heights are the same is refers as the critical height. The speed that correspond to this
height is the critical speed. At this stage the upper branch can be referred to subcritical flow
and the lower branch is referred as the supercritical critical flow.
The reverse situation occurs when the bottom elevation is lowered. In this case, the ∆H
is negative, and thus the new H is larger. For flow that is in the supercritical branch, the velocity increases while on the subcritical branch the velocity describes. As oppose earlier case,
step up (obstacle), there is a critical height above which the flow upstream become affected.
In this case there is no such a limiting case (at least not obvious).
The critical point can be found by taking the derivative of Eq. (B.17) with respect to h
and equating to zero.
0=
−A
2 q2
+1
A2 gh3
(B.18)
Critical Height Rec
q 2 = g hc 3
(B.19)
The notation of subscript c is to indicate that it refers to the critical height. Using this value
for hc , the critical specific energy is obtained by substituting the value in Eq. (B.17) to get
Hc = h c +
g hc 3
3
= hc
2
2
2 g hc
(B.20)
Note this value is correct (only? maybe) to the rectangular shape.
Equation Eq. (B.20) demonstrates that the critical energy linearly depends on the critical
height with as a slope of 2/3. This line is shown in Fig. B.12a. Flow that is above this line is
subcritical and flow below this line is supercritical. As in compressible flow, is possible to
284
APPENDIX B. FLOW IN OPEN CHANNELS
move from one branch to another? In other words, it possible to move from supercritical
to subcritical flow or from supercritical to subcritical? The answer is yes from subcritical to
subcritical but requires a step up change (or similar) and it will remains subcritical for a short
distance (unless there is a steeper slop). It can be noticed that Eq. (B.18) can be rearranged by
substituting q = h U into Eq. (B.18)
h
3
(U h)2 = g h3 −−→ U2 h2 = g h
to be
(B.21)
Froude Definition Rec
Fr =
U2
gh
(B.22)
Froude number for critical flow is Fr = 1. This definition of Froude number is at the critical
condition which equals to one. This situation is similar to situation that occurs at at compressible flow for Mach = 1. Supercritical flows occur for Froude numbers greater than one
while subcritical flows occur at Froude number < 1. The difference in the behavior of the
flow for different regimes is important in analyzing the flow.
The open channel flow has mostly hyperbolic character which is the downstream flow
does not affect the flow upstream. In Fig. B.4 the bottom was raised and the subcritical the
height was lower (and opposite for the supercritical branch). The larger change in bottom
height, the larger the effect is. When the change in the bottom reach to the point that the
liquid reached the critical condition. Any increase of further creates a local dam situation.
In other words, the flow upstream has to increase. The flow rate does not change because
the dam and it remains as before. The nature fixes the situation by changing the height of the
liquid (upstream) approaching the step (the bottom raised to about the critical point). The flow
in this case over the step must be at critical condition. That is, the reason the word mostly
was used in the beginning of this paragraph. As long as the raise is below the critical point
no effect upstream occurs. How far upstream the effect taking place? At this stage, without
doing analysis it cannot be answered precisely. However, a rough estimate can be made. The
distance should be in a magnitude of such that the channel bottom raised as the critical step
(the critical step is the amount needed to get the flow to be at the critical conditions).
A flow approaches a step that goes up
and down as shown in Fig. B.13. Assuming
that the flow is such that the height of the
subcritical
step forces a critical condition at the step.
critical
flow
supercritical
flow
The flow after the step becomes supercritiflow
cal but with the same specific energy. On
s
the diagram Fig. B.11b the liquid goes from
Fig. B.13 – Transition from subcritical to superpoint bi through point c to point pi . The
critical. The curves are not to scale.
symbol i denotes the corresponding point
that is, i = 1 or i = 2. For smooth
transition and gradual enough no significant energy loss and hence energy remains constant
285
B.3. ENERGY CONSERVATION
along the path. Rephrasing the statement: the energy in b and p is the same. Physically, the
flow at the end of the step accelerates and there no sufficient mechanism to elevate the liquid
level. This situation is similar to a nozzle in compressible flow. This situation is not difficult
to achieve by making the step higher even than necessary. In that case, the flow upstream
will be higher. The flow changes to subcritical shortly after the conversion to supercritical
downstream after step for the same slope or smaller.
To summarized the transition from subcritical to supercritical flow, the Smooth erected/created. Up to certain step height the flow return to its original heigh and velocity. After
the critical height, the liquid height is recessed but with the same energy.
Example B.1: Increasing the Step
Level: Intermediate
The step height (s) as shown in Fig. B.13. Assume that the step can be raised slowly
from zero without creating any energy losses. Quantitatively describe the height of
the flow downstream the step. The initial height of the subcritical flow is ξ0 . At what
stage the critical condition start to occur?
Solution
The height downstream is constant until the critical condition is attained. At the critical condition, the downstream regimes change to supercritical flow. After this stage and continue, the
flow at the step flow is critical. However, Hc increases because overcome the obstacle to keep
the same flow rate. The Fr number is one (the flow is sat the critical conditions). Hence,
Fr = 1 =
U2
gh
(B.1.a)
Multiplying by the height, h2 and dividing by h2 right hand side provides
1=
q2
U2 h2
−−→ 1 =
3
gh
g h3
(B.1.b)
Notice that q = h U and the second part of equation Eq. (B.1.b) could be written. Thus h3 is
a function of flow rate, q, which is constant. Hence, the height about the step is constant and
the same argument the velocity is constant at (if h and q are constant U must be constant. The
increase about the critical point cause increase of H. As it was pointed out increase in H push
the flow point to the left on Fig. B.11b.
h2
critical
s
Fig. B.14 – Downstream flow height as a function of the step height.
286
APPENDIX B. FLOW IN OPEN CHANNELS
Example B.2: Given Step what Upstream Height
Level: Advance
The flow rate in a wide channel is 10[m2 /sec] (notice the units are m2 and not m3
because it is flow rate per width). Before the insertion of the step, the water level
was 2.5[m]. A step with a height of 0.5[m] is inserted. What is water height above
the step? Assume no energy loss occurs. Is the flow immediately downstream the
step subcritical or supercritical? Estimate the height immediately upstream of the
step? What is the velocity immediately upstream the step? Estimate the water height
immediately downstream the step? If the step is 1.2[m] what will be the velocity at
the step? Estimate the height of the water just upstream the step.
Solution
hc =
s
3
q2
−−→ hc ∼
g
r
2
3 10
∼ 2.17[m]
9.81
H[m]
The critical height can be obtained from
Eq. (B.18) as
(B.2.a)
The critical velocity is, Uc = q/hc and hence,
Uc = 10/2.17 = 4.61[m/sec]. The flow at upstream is subcritical because 2.5[m] > 2.17[m].
At the critical conditions H = 1.5 × 2.17 ∼
3.25[m]. The velocity before the step was inserted is U = q/h = 10/2.5 = 4m/sec. The
specific energy remains constant and according
to Eq. (B.16) can be calculated as
10
9
8
7
6
5
4
3
2
1
0
(2.17[m], 3.25[m])
q = 10[m2/sec]
0
1
2
3
4
5
6
7
8
h[m]
Fig. B.15 – ]
Energy Diagram q=10[m2 ] for Example
Ex. B.2. It exhibits the critical coordinate in
red.
H = h + U2 /2 g = 2.5 +
42
= 4.25[m]
2 × 9 .8
(B.2.b)
The new value of H1 = H2 − 0.5 = 3.75 (green dash line). The solution of equation between
hc < h < ∞ is govern by
3.75 − h −
102
= 0.
2 g h2
(B.2.c)
Equation Eq. (B.2.c) can be solved by several methods which include numerical, analytical,
graphical. Here, the emphasize is on the conceptual understanding. Hence, from the graph
the value is obtained h ∼ 3.4[m]. The velocity upstream the step is the same velocity. The
downstream the velocity remain the same as upstream and the same as the height (no change
because no dam effect).
If the height of the step increases than velocity at step is the critical velocity and the
upstream H is 3.25 + 1.2 = 4.45. The corresponding height is h = 4.2[m] with the velocity
of 10/4.45 = 2.25[m/sec]. The water will raise upstream the step about 4.2 − 2.5 = 1.7[m]
much more than the step itself.
287
B.3. ENERGY CONSERVATION
Example B.3: Max Step
Level: Intermediate
An open channel flow with velocity of 1.5 [m/sec] with height of 2.0[m]. A step of 0.1
[m] is introduced to the flow. Calculate the velocity and height over the step. What is
the maximum before the dam’s effect appears.
Solution
The energy diagram of the to be computed and
drawn. The flow rate is
q = U h = 1.5 × 2.0 = 3.0[m2 /sec] (B.3.a)
hc =
s
3
H[m]
The critical values are obtained as
q2
−−→ hc ∼
g
r
2
3 3 .0
∼ 0.97[m]
9.81
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
available
(0.97[m], 1.46[m])
q = 3[m2/sec]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
h[m]
(B.3.b)
Fig. B.16 – Energy line for flow rate 1.5.
The critical velocity is then
Uc = q/h −−→ Uc ∼ 3.0/0.97 ∼ 3.09[m/sec]
(B.3.c)
The energy at the critical condition is
Hc = h +
U2
3.092
= 0.97 +
∼ 1.46
2g
2 × 9.81
(B.23)
The maximum is at the point critical point. The current situation is on subcritical branch. The
difference between ∆H is possible available. At H at 2[m] (U = 1.5[m]) is
H = 2+
1 .5 2
2 × 9.81
− 1.46 = 0.65
This value can also be observed from the diagram in thick Magenta.
(B.3.d)
288
B.3.1
APPENDIX B. FLOW IN OPEN CHANNELS
Some Design Considerations
When engineers designing channels one of the
question that the engineer has to look at the
optimal flow rate. Obviously, if one examine diagram Fig. B.11b it can be observed for
given H there can be many heights of liquid
in channel. Discussion on how change height
or velocity is left to later part it is only state
that it partially related to sloop. Assuming that
it is possible, what the flow rate for different
height. It was hinted that on the flow rate in diagram Fig. B.11a that there is a maximum. For
given H Eq. (B.16) provides that
subcritical
flow
hc
q
qmax
supercritical
flow
H
(h)
Fig. B.17 – Flow rate as a function of the energy
or the height.
Velocity–Energy
p
U = 2 g (H − h)
(B.24)
Flow Rate–Energy R
p
q = h 2 g (H − h)
(B.25)
The flow rate can be written
It can be notice that liquid (water) height can be only between zero (0) and H. Obviously,
height can not be below zero. The height can not be higher than H.
dq p
gh
= 2 g (H − h) − p
=0
dh
2 g (H − h)
(B.26)
or
2g
A (H − h) = g
A h −−→ h =
2H
3
(B.27)
The maximum flow rate occurs at the critical conditions. Thus, design should be such that
flow will be at condition close to the critical conditions.
The following two figures show the flow rate for different specific energy, H.
289
B.3. ENERGY CONSERVATION
Flow Rate as a function of h
H=
q [m2/sec]
2.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
h[m]
Fig. B.18 – Flow Rate as Function of height, h, for various, H in the range of .1 to 1
Flow Rate as a function of h
60
H=
1
2
3
4
5
6
7
8
9
10
q [m2/sec]
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
h[m]
Fig. B.19 – Flow Rate as Function of height, h, for various, H in the range of 1.0 to 10.0
Alternatively, all the graphs can be summarized into dimensionless equation as
290
APPENDIX B. FLOW IN OPEN CHANNELS
Dimensionless Flow Rate Height
r
h
q
h
p
1−
=
H
H 2gH H
(B.28)
It can be noticed in the case, the height ratio is really single value function (only one value
between zero and one) for given flow rate. The maximum occurs at h/H = 2/3 and the
maximum value is 4/27. The meaning of the last statement is if the calculations show that
if the value of left hand side of Eq. (B.28) greater than 4/27 the flow is chocked. The value
of H has to be adjusted so that the value of the dimensionless quantity is equal to 4/27. As
approximate value height ratio can assumed (small perturbation analysis) to be
h
H
∼
q
p
H 2gH
(B.29)
The accuracy is greater for supercritical flow. Nevertheless, it acceptable for first approximation.
B.3.2
Expansion and Contraction
Up to his point, the discussion was limited
T OP V IEW
to the same cross section mostly rectangular with only a change in the bottom height
(step or hump). At this stage, a limit exploration on what happen when the cross
Sudden Contration
Sudden Expansion
section is changed by changing the width.
The change can be either expansion or conb1
b2
traction. The change can be symmetrical
Gradual Expansion
Gradual Contration
or non–symmetrical. This discussion mostly
Fig. B.20 – expansion and contraction top view
limited to symmetrical (or close to it to
in gradual and abrupt.
avoid non–symmetrical issues and other complications). Fig. B.20 depicts four possible
situations: gradual and abrupt and for these two also contraction and expansion. Due to the
complications with the energy losses, the abrupt changes are out of the scope of this book.
Also as in the step change, the acceleration effects and 3-dimensional effects are neglected.
The first issue that stare at this topic is the flow rate. In the regular rectangular cross
section the flow rate q and Q are constant. In the present situation, only the total flow rate, Q
is constant, The flow rate per width is at the cross section 1 is q1 = Q/b1 and same for cross 2
which is q2 = Q/b2 . It turned out there are four possible regimes that have to be considered:
contraction/expansion and subcritical/supercritical. The emphasis will be on the subcritical
flow as it more common. The choked flow will be briefly considered for this version. The
heavy lifting for chocking flow will be in future versions. The heavy lifting for chocking flow
will be in future versions.
291
B.3. ENERGY CONSERVATION
B.3.2.1
Subcritical Regime; Contraction
The first case is of flow that undergoes contraction arriving with subcritical flow (Fr < 1).
For contraction b1 > b2 hence q2 > q1 (b1 q1 = b2 q2 ). If the 3-dimensional effects are
ignored then the energy is conserved and can be expressed as
(B.30)
H = H1 = H2
The energy assumed to be constant through out the channel. Hence, it is reasonable to examine the flow in constant specific energy.
q
qmax
∆q
H=3
q2
q1
(h)
Fig. B.21 – Flow in contraction subcritical flow. The specific energy in diagram is H = 3
The flow at section 1 with q1 increase the flow rate per width to section 2. At section 2 now
the flow rate is given and with specific energy H all the parameters can be found. For example
in
H
q
∆q
4
3
2
h
Fig. B.22 – Flow in contraction subcritical energy Diagram exhibiting three flow rates to demonstrate ∆q
effect.
What happen when the flow rate at section 2 is greater than the maximum flow rate.
The flow rate is chocked and maximum flow rate is at section 2 is the maximum possible and
energy has to change as it was discussed just before Eq. (B.29). The flow downstream with
a change area is similar to the flow with a step. Yet there some differences, quantities that
remains constant in each case are different (see the question at the end of the chapter).
292
APPENDIX B. FLOW IN OPEN CHANNELS
Example B.4: Simple Contraction
Level: Intermediate
The water enters to a wide side of contracted section at averaged velocity of
0.3[m/sec] and the water height is 2.0[m]. The ratio of the wide to narrow cross section is 3.0 for this rectangular shape channel. What is the height and velocity at the
exit of the contracted section?
Solution
The specific energy is
H = h+
U1 2
0.32
= 1.5 +
∼ 1.52[m]
2g
2 × 9.81
(B.4.a)
The flow rate is
(B.4.b)
q1 = U h = 0.3 × 1.5 = 0.45[m2 /sec]
√
Froude number is Fr = .3/ 9.81 × 1.5 = 0.078 so the flow is subcritical. Thus procedure
outlined earlier can be used. The flow rate at section 2 is
q2 =
q1 b1
= 0.45 ∗ 3.0 = 1.325[m2 /sec]
b2
(B.4.c)
which means that Eq. (B.25) can be plotted for this situation. The equation can be solved analytically or graphically. Here the graphical solution show that h2 ∼ 1.48 and h1 = 1.517 if the
flow rate at 1 was the same as at 2.
1.475
q
1.525
1.45
1.55
1.5
1.35
0.45
h
1.0
1.6
Fig. B.23 – The Flow Rate for Contraction Exercise. Ex. B.4. Note that h1 is not the actual height
but rather height if the flow rate was same based on section 2.
Notice, the solution can be obtained analytically and numerically in many methods. With
knowledge of the height, the velocity can be calculated as
U2 =
q2
1.325
=
= 0.8953[m/sec]
h2
1.48
(B.4.d)
293
B.3. ENERGY CONSERVATION
B.3.2.2
Supercritical Regime; Contraction
In this cases the supercritical flow approaches a contraction is dealt in a similar logic to the
previous case. The flow rate increases and thus the “left” side of the graph is controlling the
phenomenon. As oppose to the previous case the increase in the flow rate actually reduce the
velocity was it will shown in the following Ex. B.5. The
Example B.5: Simple Contraction Supercritical
Level: Intermediate
A flow enters a channel with a contraction with ratio of b1 /b2 = 1.5 with velocity
of 7[m/sec]. The height at section 1 is 0.8[m]. What is the velocity and height at the
exit?
Solution
The specific energy that appear at section 1 is
H = h1 +
72
U2
= 0 .8 +
∼ 3.3[m]
2g
2 × 9.81
(B.5.a)
The flow rate at section 2 is q2 = q1 b1 /b2 . The flow rate at section 1 is
q1 = h1 U1 = 0.8 × 7 = 5.6[m2 /sec]
(B.31)
The flow rate per width at section 2 is
(B.32)
q2 = 5.6 × 1.5 = 8.4[m/sec]
with this information a diagram can be drawn as
8.4
q
8.3
8.2
8.1
8.4
2
1
8.0
1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37
h
Fig. B.24 – Flow in Contraction Supercritical Energy Diagram.
The diagram shows that at q2 = 8.4 at h ∼ 1.363 which is displayed on the zoom part of
Fig. B.24. Notice that un–zoom part of the diagram is displayed on the bottom right corner.
The velocity is
8.4
q
U2 = 2 =
∼ 6.163[m/sec]
(B.5.b)
h2
1.363
294
APPENDIX B. FLOW IN OPEN CHANNELS
B.3.3
Summery
√
Fr = U/ g h has good representation
when the flow is for wide rectangular
√
channel. If Fr = 1 that is U = g h,
flow is critical. If Fr < 1 that is. U <
√
g h, flow is sub-critical (some refer to it
as tranquil flow). If Fr > 1, flow is supercritical( and some refer to it as torrential
flow). Adding a new point to the discussion while not expanding it. The terms
(H1 − H2 )/L = So and loss hL /L =
Sw = Sf are commonly used when energy is lost and large scale calculation are
needed.
U1 2
2g
1
2
E nergy
W ate
r
Line
S urf a
ce
h1
hL
U2 2
2g
h2
C hannel
bottom
H1
H2
Dotum
L
Fig. B.25 – The energy line and liquid surface line with
the energy lost. As can be observed the flow is subcritical.
Example B.6: Sub and Supper Heights
Level: Advance
For a rectangular channel the height of flow was observed to have value, h1 = 2[m].
A hump or step was inserted and downstream the flow was observed to height of
h2 = 0.8[m]. Assume that there was no energy lost by inserting the step and the flow
at after insertion of the step is supercritical (this should a conclusion). What is the
critical height? What is the specific energy? what is the critical velocity? (this part left
as a challenge).
Solution
H = Hc = H1 = H2
Hence,
U 2
U1 2
= h2 + 2
2g
2g
(B.6.b)
q2
q2
= h2 +
2
2
g
y2 2
2 g h1
(B.6.c)
H = h1 +
or utilizing the flow rate
h1 +
Using critical relationship for rectangular, hc 3 =
hc 3
2
(B.6.a)
q2
g
provides
1
1
−
= h2 − h 1
2
2
h1
h2
s
2
2
3 2 h 1 h2
hc =
h1 + h2
(B.33)
(B.6.d)
295
B.4. HYDRAULIC JUMP
End of Ex. B.6
In this case, h1 = 2[m] and h2 = 0.8[m] thus
hc =
s
3
2 2 × 0 .8 2
∼ 0.9706[m]
2 + 0.8
The specific energy
hc 3
2 h1 2
(B.6.f)
2 h 1 2 h2 2
2 h1 2 (h1 + h2 )
(B.6.g)
H = h1 +
H = h1 +
H=
H=
h1 (h1 + h2 ) + h2 2
h 1 + h2
(B.6.h)
h 2 + h2 2 + 2 h 1 h2 − h1 h2
h1 2 + h 2 2 + h1 h2
= 1
=
h1 + h2
h 1 + h2
(h1 + h2 )2 − h1 h2
h h
= (h1 + h2 ) − 1 2
h1 + h 2
h1 + h2
H = (2 + 0.8) +
B.4
(B.6.e)
0 .8 × 2
∼ 3.37[m]
2.8
(B.6.i)
(B.6.j)
Hydraulic Jump
One of the most common phenomenon which
most people observed every day, is the hydraulic jump. When pouring water (either
P
from your faucet or otherwise) into the sink,
h2
F low
P
Direction
h1
there is a hydraulic jump2 . One can no1
2
tice that water hits the sink and a thin waFig. B.26 – Schematic of hydraulic jump.
ter layer spreads in all angles. At some
point, the thin layer suddenly changes to
thicker layer. This change is the hydraulic jump. Generally, there are several classifications
of hydraulic jump such as stationary, moving. Additionally the jump also classified by the
geometry such radial or two dimensional (there are more). The hydraulic jump also classified
as uniform density (or material) or mixing or chemical interaction also involve. No mater
how complicated the situation considered in most times, it is assumed the jump occurs at
very narrow width. The flow changes from supercritical to subcritical flow. The hydraulic
jump is depicted in Fig. B.26.
The mixing processes at the surface and additionally the mixing inside the jump are
2 This example apparently is used by many to demonstrate hydraulic jump not an original example by this author.
This example in suitable for modern world for which the assumption the reader uses a sink. A more general use is
pouring water to a glass (for drinking).
296
APPENDIX B. FLOW IN OPEN CHANNELS
very substantial hence the energy is not conserved. Some of the energy dissipates in these
eddy in the turbulent processes. Only some of the energy converted into heat and some for
other forms of energy like sound energy (slightly compressed moist air) and other energies.
The amount of lost energy is unknown and hence cannot be used to solve the problem. Another quantity is needed to solve the problem. Under the assumption that jump occurs at
very narrow space, the shear stress can be assumed to negligible (similar assumption to shock
wave). The same argument can be made for the upper surface. Furthermore, at the upper
surface, the air is a light gas (relatively) and hence the shear force is small3 . There are several
methods to analyze this situation. A simple control volume is used for this analysis. The assumption taken here are: the height of the flow is uniform (2D assumption) on both sides of
the jump, (initially) the rectangular cross section is assumed, a plug flow (or averaged velocity
is assumed). It is further assumed that the jump occurs at short distance and hence the shear
stress at the bottom and air are negligible.
The mass conservation of the control volume shows that control volume itself is not
moving and there is only one stream in and one stream out.
Z
Z
ρ U(y)dA =
ρ U(y)dA
(B.34)
A1
A2
If plug flow is assumed (or averaged velocity)
Ur1 h1 = U2 h2
(B.35)
The streamlines are assumed to be parallel, hence the pressure made mostly from the
hydrostatic. The pressure at stations 1 and 2 the average hydrostatic pressure is given by Pa =
g h/2. The momentum conservation can be expressed as
g h1 2 − h2 2
2
2
U2 h2 − U1 h1 =
(B.36)
2
Substituting Eq. (B.35) into Eq. (B.36) provides
!
2
g h1 2 − h2 2
g (h1 − h2 ) (h1 + h2 )
2 h1
U1
− h1 =
=
h2
2
2
(B.37)
which can be reduced to
U1 2
gXX
h 1 XX
XhX
(h1 −XhX
2 ) = (h1 −
2 ) (h1 + h2 )
X
X
h2
2
(B.38)
Which can be expressed as
Hydraulic Jump U1
U1 2 =
3 Why the term light gas is used?
g h2
(h1 + h2 )
2 h1
Because, the force is related to mass.
(B.39)
297
B.4. HYDRAULIC JUMP
Under the symmetry argument Eq. (B.39) can be written for the other velocity. In other words,
the points (1) and (2) can be interchanged and there is nothing significant about their locations.
Hydraulic Jump U2
U2 2 =
g h1
(h1 + h2 )
2 h2
(B.40)
If the momentum is conserved that indirectly imply that some energy is lost (elementary physics which shows no energy lost only when U1 = U2 . This situation is similar to
collision of the two balls from the equations point of view.). The total head (energy per unit
weight) change in the transition is
!
U2 2
U1 2
h L = h2 +
− h1 +
(B.41)
2g
2g
which can be rearranged as
hL = h 2 − h 1 +
U2 2 U1 2
−
2g
2g
(B.42)
Utilizing Eq. (B.39) and Eq. (B.40) Eq. (B.42) can be written as
hL = h2 − h1 +
1 g
A h2 (h1 + h2 ) − 1 g
A h1 (h1 + h2 )
2g
2
h
2
g
1
A
A 2 h2
(B.43)
which can be also written as
hL =
4 h 2 2 h1 − 4 h1 2 h2
h (h + h2 ) h1 (h1 + h2 )
+ 2 1
−
4 h 1 h2
4 h1
4 h2
(B.44)
which can be also written as
hL =
4 h2 2 h1 − 4 h1 2 h2 h2 2 (h1 + h2 ) h1 2 (h1 + h2 )
+
−
4 h1 h2
4 h 1 h2
4 h 2 h1
(B.45)
The numerator is simply a quadric equation (a − b)3 (Notice that coefficient 4 changes to 3
for both terms and the last two terms are in the third power.) and Eq. (B.45) can be written as
Hydraulic Energy Loss
hL =
(h1 − h2 )3
4 h2 h1
(B.46)
The conclusion from Eq. (B.46) is that hL < 0 must be negative (thermo second law). The
above statement means that h1 < h2 . The jump must be from a shallow flow to a deep flow;
In plain English, the energy loss is a strong function of the hydrostatic sides heights. Note
that while the hydraulic jump goes from subcritical to subcritical, it does not mean that it a
proof that subcritical is the preferred flow.
298
B.4.1
APPENDIX B. FLOW IN OPEN CHANNELS
Poor Man Dimensional Analysis
The topic of open channel started long before the dimensional analysis become popular (about
150 years difference). Thus, the serious usage of the dimensional analysis started to appear only
after world war two. Here a simple dimensional analysis is offered. The Froude number is
defined as
Fr1 2 =
U1 2
g h1
(B.47)
Dividing Eq. (B.39) by g h1 and utilizing the definition r = h2 /h1 to reads
h
U1 2
= 2
g h1
2 h1
h2
1+
−−→ Fr1 = 2 r (1 − r)
h1
(B.48)
Similar equation can be written for Fr2 . It is common to solve for r as a function of Fr. There
are two solutions for the equation of
r2 + r − 2 Fr1 2 = 0
(B.49)
The positive solution (no negative height possible)
Heights Ratio Froude
p
h
−1 + 1 + 8 Fr1 2
r= 2 =
h1
2
(B.50)
Based on arguments of symmetry, the reverse equation can be written as
1
h
−1 +
= 2 =
r
h1
p
1 + 8 Fr2 2
2
(B.51)
For completeness, the reverse relationship is
Fr for h1/h2
r
r(1 + r)
Fr1 =
2
(B.52)
The energy loss in the H as
H1 − H2
(r − 1)3
=
h1
4r
The power loss is ρ g q(H1 − H2 )
(B.53)
299
B.4. HYDRAULIC JUMP
Example B.7: Simple Hydraulic Jump
Level: Basic
Flow in channel has hydraulic jump from height of 0.4[m] to 0.8[m]. What are the
upstream and downstream velocities, the volumetric flow rate and the rate of energy
loss at the jump?
Solution
height ratio is r = 2 The upstream Froude number is
Fr1 =
r
r(1 + r)
2
r
2(1 + 3) √
= 3
2
(B.7.a)
Since Froude was calculated the velocity can be obtained according to Eq. (B.47) as
U1 = Fr1 2 g h1 = 3 × 9.81 × 0.4 ∼ 11.8[m/sec]
(B.7.b)
The velocity on the other side can be ascertained from r height ratio as
U2 = U1 /2 = 11.8/2 ∼ 5.89[m/sec]
(B.7.c)
q = h1 ∗ U1 = 0.4 × 11.8 ∼ 4.72[m2 /sec]
(B.7.d)
The flow rate is
The energy lost in the hydraulic jump is
EL =
13
(r − 1)3
=
= 1/8
4r
8
(B.7.e)
Example B.8: Hydrostatic Pressure
Level: Basic
A hydraulic jump occurs in rectangular channel with upstream velocity of U1 =
1.2[m/sec] and h1 = .1[m]. The density of the liquid (water) is about 1000[kg/m3 ]
Calculate the difference in the hydrostatic pressure in both sides.
Solution
The pressure on both sides of the jump is based on the height of the water (liquid). In this case,
first the height has to be calculated on the downstream side. The upstream Froude number is
Fr1 2 =
1+
h 2 = h1
q
1 + 8 Fr1 2
2
The pressure on upstream is
P1 =
U1 2
1 .2 2
=
= 1.46
g h1
9.81 × 0.1
= 0 .1 ×
1+
√
1 + 8 × 1.46
∼= 0.23[m]
2
0.2
× 9.81 × 1000 ∼ 981[N/m2 ]
2
The force per width is also very small since the height is very small. F1 = P h1 .
(B.8.a)
(B.8.b)
(B.8.c)
300
B.4.2
APPENDIX B. FLOW IN OPEN CHANNELS
Velocity Profile
The flow regime (code name for the velocity profile) is important factor in the flow. Up this
point it was assumed that there is some kind overage representing the velocity. Intuitively,
it can be observed that the velocity has effect on the flow. In this section while still dealing
with rectangular shape the velocity profile is arbitrary ( shot sleeve will be discussed in the
main chapter on critical plunger velocity). In this section the velocity profile is assumed to be
known and there is no attempt to solve for it. The question in focus, given a profile what is the
change in the momentum equation and energy equation. It suggested to isolate the velocity
profile from the calculations and to make it as a coefficient.
In order to carry these calculations, the mass conservation has to be solved. The velocity
profile can be any kind of function. Assuming that flow is stationary and two dimensional,
the mass flow is given by
Z h1
ṁ
ρ U(y)dy
(B.54)
0
In this stage, the complication of the air entrainment and similar effects are neglected. Thus,
a good approximation is to assume that the density is constant. Hence, equation can be read
Z h1
ṁ
=q=
U(y)dy
(B.55)
ρ
0
The averaged velocity is U1 = q/h1 .
The momentum conservation required that
Z h1 h
Z h2 h
i
i
P1 + ρ U1 2 dy =
P2 + ρ U2 2 dy
0
0
(B.56)
The pressure, under the assumptions used in this discussion is actually linearly related to the
pressure. Again, the hydrostatic assumption is employed. Thus, the first part of the integral
(for any height) is
Zh
0
h
ξ2
ρ g h2
ρ g (h − ξ) dξ = ρ g h ξ −
=
2 0
2
The part above is almost trivial. Next part is more complicated as
Zh
Zh 2
Zh
U2 (ξ) 2
U (ξ)
2
ρ U2 (ξ)dξ =
ρ
U
dξ
=
ρ
U
dξ
2
2
0
0
0
U
U
(B.57)
(B.58)
Notice that averaged velocity can be pulled out the integral. The combination of the integral
is basically a function of velocity profile and is defined as gamma momentum
Gamma Momentum
Z
1 h U2 (ξ)
dξ
γ=
h 0 U2
(B.59)
301
B.5. CROSS SECTION AREA
The gamma function is a dimensionless function. Notice, that for energy gamma function is
defined a bit different. Using the definition Eq. (B.59) Eq. (B.56) to read
Zh h
i
2
2
∼ g h + γρ q
P + ρ U(y)2 dy =
2
h
0
(B.60)
Now γ can be calculated from various velocity profiles. For example, consider the plug
flow which preferred to uniform velocity.
Example B.9: Gamma for Plug Flow
Level: Intermediate
Calculate profile factor γ for two profiles: plug, laminar flow.
Solution
The calculations are straight forward for plug flow U(y) = q/h = constant. For the plug
flow
Z
q 2
dy = 1
h
(B.9.a)
h
y2
ρ g sin θ h y2 y3
hy−
dy =
−
2
µh
2
6 0
(B.61)
γ=
h
q
2
1
h
h
0
For the laminar velocity it was shown earlier that velocity profile is parabola. The averaged
velocity is
U=
1
h
Zh
0
ρ g sin θ
µ
which result in averaged velocity
U=
A h2
ρ g sin θ h2
≡
3µ
3
(B.62)
The profile factor can be calculated as
γ=
3
A h2
Z
2 Z h 2
y2
A hy−
dy
Z
2
0
(B.63)
which can be rewritten, if ξ = y/h and dξ = dy/h, as
γ = 9h
B.5
Cross Section Area
B.5.1
Introduction
2
Z1 9h
ξ2
dξ =
ξ−
2
30
0
(B.64)
Before considering different cross sections, so far the discussion was focus on the rectangular
shape. At this stage, no discussion was offered on the best ratio of rectangular sides. Now here
it is postulated that reducing of the wetted area can reduced the resistance. To some degree,
it is a valid but it is not universally correct. Regardless to the accuracy of the idea, it will be
302
APPENDIX B. FLOW IN OPEN CHANNELS
examined here. The rectangular cross section has width of b and two sides with height of h.
The wetted perimeter length is
P = b+2h =
A
+2h
h
(B.65)
where P denotes the wetted perimeter. The minimum wetted perimeter will at the derivative
equal zero.
dP
A
hb
b
= − 2 +2 = −A +2 = − +2
2
dh
h
h
hA
(B.66)
Which is h = b/2. This analysis suggests that the closer to the optimal channel is when the
liquid height is designed for width is double height. This design will minimizes the resistance
area and hopefully reduces the construction cost.
One of the concept when discussing non–circular shape is the hydraulic radius which
represents a similarity to circular conduit. In the context of the open channel flow it is defined
as
Hydraulic Radius
RH =
A
P
(B.67)
The analysis of the optimal rectangular suggest that shapes that are closer to circle are
more optimal. For example, the trapezoidal cross section can be used as an example. The
closest trapezoidal shape to the circle is the shape that all the three sides are equable which is
a half of hexagon. Another example is triangular channel (see the next example).
Example B.10: Optimal Triangle for OC
Level: Basic
A triangular open channel depicted in Fig. B.27 the angle θ is half of the total angle.
For given amount of area find the optimal angle.
Solution
303
B.6. ENERGY FOR NON–RECTANGULAR CROSS–SECTION
End of Ex. B.10
The area of the triangle is
A=2
h h tan θ
2
(B.10.a)
From geometry the wetted perimeter is
P=
2
2
√
A sec θ
√
tan θ
(B.10.b)
α
p
1 dP
√
= sec (θ) tan (θ)−
A dθ
sec (θ)
3
(B.10.c)
3
2 tan (θ) 2
α
h
Fig. B.27 – Optimal angle for triangular
cross section.
Equating Eq. (B.10.c) to zero yields
p
sec (θ) tan (θ) =
sec (θ)3
(B.10.d)
3
2 tan (θ) 2
After some manipulations θ = 45◦ .
B.6
Energy For Non–Rectangular Cross–Section
In this section a discussion on the energy line
for non–rectangular is offered. The critical
conditions can be found by generalizing the
energy equation. Eq. (B.16) defines the specific
energy for rectangular shape. In that equation
the averages velocity was used and it will be
modified to be more general. Notice that q was
replaced by Q to denote that there is no possibility to have a flow rate per width. Notice the
plug flow is returned for simplification and γ
can be used when velocity profile is accounted.
The velocity is replaced by U = Q/A to be
H = h+
b dh h
sc
c
hc
Uc 2
2
c
H
Fig. B.28 – Specific energy lines for non–
rectangular channel.
U2
Q2
= h+
2g
2 g A2
(B.68)
The head loss is
H
Ew2 − Ew1 =
H
z }|2 {
z }|1 {
!
!
2
Q2
Q1 2
+ h2 +H02 −
+ h1 +H01
2 A2
2 A2
(B.69)
304
APPENDIX B. FLOW IN OPEN CHANNELS
While geometry of the cross section was not provided yet, the specific energy is a function of
flow depth H = f(h). If the cross section geometry is provided and for known flow rate, Q,
the specific energy can be calculated. The derivative of Eq. (B.68) yield
dH
2 Q2
dA
= 1−
(B.70)
dh
2 g A3 dt
As opposed the rectangular case, another term was added. It can be noticed that ratio of
dA/dh can be only positive. The area cannot decrease with the increase of the height at most
it can be zero if the width is zero. At the surface, the differential infinitesimal element is the
width times the change of the height, b dh. Thus,
bH
dh
dA
H
=
= b(h) ̸= constant
H
dh
dh
H
(B.71)
The value of b, in this case, refers to the value at the free surface width at the cross section.
Hence Eq. (B.70) reads now
dH
2 Q2 b
= 1− A
dh
A2 g A3
Or
(B.72)
Critical Conditions General
Q2
A3
=
g
b
(B.73)
Or Eq. (B.73) can be rearranged as
U2
z}|{
Q2
gA
=
b
A2
(B.74)
It is common to define the hydraulic diameter as
Hydraulic diameter, hD
hD =
A
b
(B.75)
With this definition, Eq. (B.75), Eq. (B.74) becomes
U2 = g hD −→ U =
p
g hD
(B.76)
Thus similar to the rectangular case, using the definition of hydraulic diameter Fr at the critical condition is one.
Critical Fr number NR
Uc
Frc = √
=1
(B.77)
g hD
305
B.6. ENERGY FOR NON–RECTANGULAR CROSS–SECTION
The maximum flow rate can be obtained when H is constant. In other words, finding the
maximum flow for a fix specific energy is done similarly as before. Eq. (B.68) can be written
as
q
p
p
Q = 2 g A2 (H − h) = 2 g A H − h
(B.78)
The derivative of Eq. (B.78) is
dQ p
= 2g
dh
A
dA p
H−h− √
=0
dh
2 H−h
(B.79)
Equating Eq. (B.79) to zero (to get the maximum) and using the value of dA/dh = b provides
b
p
H−h =
2
√
A
−→ H − h =
H−h
A
2b
(B.80)
This results is the critical condition substitute into Eq. (B.78) and can be written as
Critical Flow Rate
Q2 =
2 g A3
g A3
Q2
A3
=
−→
=
2b
b
g
b(h)
(B.81)
It should be noted that b is unknown but if it is obtained or known, there is a critical and
maximum flow rate at that location. Eq. (B.81) is not linear equation because b is not a constant,
the line representing this phenomenon not necessarily a straight line for all geometry. For
instance b = a h2 is parabolic is more common that one expect and in that case it not a
linear equation. In general it can be written as
Q2 b
Q2 b
Q2
= 1 −→
= 1 −→ hc 3 =
3
3
gA
g(b hc )
g b2
(B.82)
For the rectangular shape the specific energy is
U2c
2g
(B.83)
Q2
2 g A2
(B.84)
Q2
2 g h2c
(B.85)
H = hc +
in general the specific energy is
Hc = h c +
Substituting Eq. (B.82) into Eq. (B.84)
Hc = h c +
306
B.6.1
APPENDIX B. FLOW IN OPEN CHANNELS
Triangle Channel
h tan θ
One the common shape of open channel is the
triangle or the trapezoid.
Q2 b
g A3
h
θ
(B.86)
=1
Fig. B.29 – Open channel flow in an isosceles triangular shape.
Defining m = tan θ and assuming that there is
hc relating b = 2 m tan θ, same for the area,
A = m hc 2 one gets
Q2 (2 m hc )
=1
g(m h2c )3
(B.87)
2 Q2
=1
g · m2 h5c
(B.88)
After simplification Eq. (B.86)
Changing the subject of a Eq. (B.88) it becomes
hc =
s
5
2 Q2
g m2
(B.89)
Specific energy at the critical condition is
Hc = h c +
Q2
2 g A2
(B.90)
According to Eq. (B.86) Q2 /g can be replaced by A3 /b and thus Eq. (B.90) becomes
A3
A
b
Hc = h c +
−→ Hc = hc +
2b
2 A2
(B.91)
Again using the value form the geometry i.e. A = m hc 2 /2 and b = 2 m tan θ to be
Hc = h c +
m h2c
hc
5 hc
−→ Hc = hc +
−→ Hc =
2 2 m hc
4
4
(B.92)
Froude number for triangular channel will be
√
2U
√ c =1
g hc
(B.93)
B.6. ENERGY FOR NON–RECTANGULAR CROSS–SECTION
B.6.1.1
307
Section Factor Z
The generalize the treatment and to have general equations
that can be used in a general way
p
(as possible) the following is offered. The expression A/b is a function of the depth h for a
given channel geometry. It is convenient to define
r
A
Z=A
(B.94)
b
It also can be defined for the critical conditions
Zc = Ac
r
Ac
b
(B.95)
Squaring both sides results of Eq. (B.95) results in
Zc 2 =
Ac 3
b
(B.96)
For critical conditions Eq. (B.73) is valid and can be used with Eq. (B.96)
Z2c =
√
Qc 2
−→ Qc = Zc g
g
(B.97)
308
APPENDIX B. FLOW IN OPEN CHANNELS
C
What The Establishment’s Scientists Say
What a Chutzpah? to say samething like that!
anonymous
In this section exhibits the establishment “experts” reaction the position that the “common”
pQ2 diagram is improper. Their comments are responses to the author’s paper: “The mathematical theory of the pQ2 diagram” (similar to Chapter 7)1 . The paper was submitted to
Journal of Manufacturing Science and Engineering.
This part is for the Associate Technical Editor Dr. R. E. Smelser.
I am sure that you are proud of the referees that you have chosen and that you do
not have any objection whatsoever with publishing this information. Please send a
copy of this appendix to the referees. I will be glad to hear from them.
This concludes comments to the Editor.
I believe that you, the reader should judge if the mathematical theory of the pQ2 diagram is correct or whether the “experts” position is reasonable. For the reader unfamiliar with
the journal review process, the associate editor sent the paper to “readers” (referees) which are
anonymous to the authors. They comment on the paper and according to these experts the
paper acceptance is determined. I have chosen the unusual step to publish their comments
because I believe that other motivations are involved in their responses. Coupled with the
1 The exact paper can be obtain free of charge from Minnesota Suppercomputing Institute,
http://www2.msi.umn.edu/publications.html report number 99/40 “The mathematical theory of the pQ2 diagram” or by writing to the Supercomputing Institute, University of Minnesota, 1200 Washington Avenue South,
Minneapolis, MN 55415-1227
309
310
APPENDIX C. WHAT THE ESTABLISHMENT’S SCIENTISTS SAY
response to the publication of a summary of this paper in the Die Casting Engineer, bring me
to think that the best way to remove the information blockage is to open it to the public.
Here, the referees can react to this rebuttal and stay anonymous via correspondence
with the associated editor. If the referee/s choose to respond to the rebuttal, their comments
will appear in the future additions. I will help them as much as I can to show their opinions. I
am sure that they are proud of their criticism and are standing behind it 100%. Furthermore, I
am absolutely, positively sure that they are so proud of their criticism they glad that it appears
in publication such as this book.
C.1
Summary of Referee positions
The critics attack the article in three different ways. All the referees try to deny publication
based on grammar!! The first referee didn’t show any English mistakes (though he alleged that
he did). The second referee had some hand written notes on the preprint (two different hand
writing?) but it is not the grammar but the content of the article (the fact that the “common”
pQ2 diagram is wrong) is the problem.
Here is an original segment from the submitted paper:
The design process is considered an art for the 8–billion–dollar die casting industry.
The pQ2 diagram is the most common calculation, if any that all, are used by most
die casting engineers. The importance of this diagram can be demonstrated by the fact
that tens of millions of dollars have been invested by NADCA, NSF, and other major
institutes here and abroad in pQ2 diagram research.
In order to correct “grammar”, the referee change to:
The pQ2 diagram is the most common calculation used by die casting engineers to
determine the relationship between the die casting machine and gating design parameters, and the resulting metal flow rate.
It seems, the referee would not like some facts to be written/known.
Summary of the referees positions:
Referee 1 Well, the paper was published before (NADCA die casting engineer) and the errors
in the “common” pQ2 are only in extreme cases. Furthermore, it actually supports the
“common” model.
Referee 2 Very angrily!! How dare the authors say that the “common” model is wrong. When
in fact, according to him, it is very useful.
Referee 3 The bizzarro approach! Changed the meaning of what the authors wrote (see the
“ovaled boxed” comment for example). This produced a new type logic which is almost
absurd. Namely, the discharge coefficient, CD , is constant for a runner or can only vary
with time. The third possibility, which is the topic of the paper, the fact that CD cannot
be assigned a runner system but have to calculated for every set of runner and die
casting machine can not exist possibility, and therefore the whole paper is irrelevant.
C.2. REFEREE 1 (FROM HAND WRITTEN NOTES)
311
Genick Bar–Meir’s answer:
Let me say what a smart man once said before: I don’t
need 2000 scientists to tell me that I am wrong. What
I need is one scientist to show what is wrong in my
theory.
Please read my rebuttal to the points the referees made. The referees version are kept as close
as possible to the original. I put some corrections in a square bracket [] to clarify the referees
point.
Referee comments appear in roman font like this sentence,
and rebuttals appear in a courier font as this sentence.
C.2
Referee 1 (from hand written notes)
1. Some awkward grammar – See highlighted portions
Where?
2. Similarity of the submitted manuscript to the attached Die Casting Engineer Trade
journal article (May/June 1998) is Striking.
The article in Die Casting Engineer is a summary of the present article. It
is mentioned there that it is a summary of the present article. There is
nothing secret about it. This article points out that the “common” model is
totally wrong. This is of central importance to die casting engineers. The
publication of this information cannot be delayed until the review process is
finished.
3. It is not clear to the reader why the “constant pressure” and “constant power”
situations were specifically chosen to demonstrate the author’s point. Which situation
is most like that found [likely found] in a die casting machine? Does the “constant
pressure” correspond closely to older style machines when intensifyer [intensifier]
bottle pressure was applied to the injection system unthrottled? Does the “constant
power” situations assume a newer machines, such as Buher Sc, that generates the
pressure required to achieve a desired gate velocity? Some explanation of the logic of
selecting these two situations would be helpful in the manuscript.
As was stated in the article, these situations were chosen because they are
building blocks but more importantly to demonstrate that the “common”
model is totally wrong! If it is wrong for two basic cases it should be
absolutely wrong in any combinations of the two cases. Nevertheless, an
additional explanation is given in Chapter 7.
312
APPENDIX C. WHAT THE ESTABLISHMENT’S SCIENTISTS SAY
4. The author’s approach is useful? Gives perspective to a commonly used process
engineering method (pQ2 ) in die casting. Some of the runner lengths chosen (1 meter)
might be consider exceptional in die casting – yet the author uses this to show how
much in error an “average” value for CD be. The author might also note that the
North American Die Casting Association and many practitioners use a A3 /A2 ratio
of ≈ .65 as a design target for gating. The author" analysis reinforces this value as a
good target, and that straying far from it may results in poor design part filling
problems (Fig. 5)
The reviewer refers to several points which are important to address. All
the four sizes show large errors (we do not need to take 1[m] to demonstrate
that). The one size, the referee referred to as exceptional (1 meter), is not
the actual length but the represented length (read the article again). Poor
design can be represented by a large length. This situation can be found
throughout the die casting industry due to the “common” model which does
not consider runner design. My office is full with runner designs with
represent 1 meter length such as one which got NADCA’s design award2 .
In regards to the area ratio, please compare with the other referee who claim
A2 /A2 = 0.8 - 0.95. I am not sure which of you really represent NADCA’s
position (I didn’t find any of NADCA’s publication in regards to this point).
I do not agree with both referees. This value has to be calculated and
cannot be speculated as the referees asserted. Please find an explanation to
this point in the paper or in even better in Chapter 7.
C.3
Referee 2
There are several major concerns I have about this paper. The [most] major one [of these] is
that [it] is unclear what the paper is attempting to accomplish. Is the paper trying to suggest
a new way of designing the rigging for a die casting, or is it trying to add an improvement to
the conventional pQ2 solution, or is it trying to somehow suggest a ‘mathematical basis for
the pQ2 diagram’?
The paper shows that 1) the “conventional pQ2 solution” is totally wrong, 2) the
mathematical analytical solution for the pQ2 provides an excellent tool for
studying the effect of various parameters.
The other major concern is the poor organization of the ideas which the authors [are] trying
to present. For instance, it is unclear how specific results presented in the results section
where obtained ([for instance] how were the results in Figures 5 and 6 calculated?).
I do not understand how the organization of the paper relates to the fact that the
referee does not understand how Fig 5 and 6 were calculated. The organization of
the paper does not have anything do with his understanding the concepts
2 to the best of my understanding
C.3. REFEREE 2
313
presented. In regard to the understanding of how Figure 5 and 6 were obtained,
the referee should referred to an elementary fluid mechanics text book and
combined it with the explanation presented in the paper.
Several specific comments are written on the manuscript itself; most of these were areas
where the reviewer was unclear on what the authors meant or areas where further
discussion was necessary. One issue that is particularly irksome is the authors tendency in
sections 1 and 2 to wonder [wander] off with “editorials” and other unsupported comments
which have no place in a technical article.
Please show me even one unsupported comment!!
Other comments/concerns include• what does the title have to do with the paper? The paper does not define what a pQ2
diagram is and the results don’t really tie in with the use of such diagrams.
The paper presents the exact analytical solution for the pQ2 diagram. The
results tie in very well with the correct pQ2 diagram. Unfortunately, the
“common” model is incorrect and so the results cannot be tied in with it.
√
• p.4 The relationship Q ∝ P is a result of the application of Bernoulli’s equation
system like that shown in Fig 1. What is the rational or basis behind equation 1; e.g.
Q ∝ (1 − P)n with n =1, 1/2, and 1/4?
Here I must thank the referee for his comment! If the referee had serious
problem understanding this point, I probably should have considered adding
a discussion about this point, such as in Chapter 7.
• p.5 The relationship between equation 1(a) to 1(c) and a die casting machine as “poor”,
“common”, and “excellent” performance is not clear and needs to be developed, or at
least defined.
see previous comment
• It is well known that CD for a die casting machine and die is not a constant. In fact it
is common practice to experimentally determine CD for use on dies with ‘similar’
gating layouts in the future. But because most dies have numerous gates branching off
of numerous runners, to determine all of the friction factors as a function of Reynolds
number would be quite difficult and virtually untractable for design purposes.
Generally die casting engineers find conventional pQ2 approach works quite well for
design purposes.
This “several points’ comment give me the opportunity to discusses the
following points:
314
APPENDIX C. WHAT THE ESTABLISHMENT’S SCIENTISTS SAY
⋆
I would kindly ask the referee, to please provide the names of any
companies whom “experimentally determine CD .” Perhaps they do it
down under (Australia) where the “regular” physics laws do not apply
(please, forgive me about being the cynical about this point. I cannot
react to this any other way.). Please, show me a company that uses the
“common” pQ2 diagram and it works.
⋆
Due to the computer revolution, today it is possible to do the
calculations of the CD for a specific design with a specific flow rate (die
casting machine). In fact, this is exactly what this paper all about.
Moreover, today there is a program that already does these kind of
calculations, called DiePerfect™.
⋆
Here the referee introduce a new idea of the “family” – the improved
constant CD . In essence, the idea of “family” is improve constant CD in
which one assigned value to a specific group of runners. Since this idea
violate the basic physics laws and the produces the opposite to realty
trends it must be abandoned. Actually, the idea of “family” is rather
bizarre, because a change in the design can lead to a significant change
in the value of CD . Furthermore, the “family” concept can lead to a
poor design (read about this in the section “poor design effects” of this
book). How one can decided which design is part of what “family”?
Even if there were no mistakes, the author’s method (calculating the
CD ) is of course cheaper and faster than the referee’s suggestion about
“family” of runner design. In summary, this idea a very bad idea.
⋆
What is CD =constant? The referee refers to the case where CD is
constant for specific runner design but which is not the case in reality.
The CD does not depend only on the runner, but on the combination of
the runner system with the die casting machine via the Re number.
Thus, a specific runner design cannot have CD “assigned” to it. The CD
has to be calculated for any combination runner system with die
casting machine.
⋆
I would like to find any case where the “common” pQ2 diagram does
work. Please read the proofs in Chapter 7 showing why it cannot work.
• Discussion and results A great deal of discussion focuses on the regions where A3 /A2
0.1; yet in typical die casting dies A3 /A2 0.8 to 0.95.
Please read the comments to the previous referee
In conclusion, it’s just a plain sloppy piece of work
I hope that referee does not mind that I will use it as the chapter quote.
(the Authors even have one of the references to their own publications sited incorrectly!).
C.4. REFEREE 3
315
Perhaps, the referee should learn that magazines change names and, that the
name appears in the reference is the magazine name at the time of writing the
paper.
C.4
Referee 3
The following comments are not arranged in any particular order.
General: The text has a number of errors in grammar, usage and spelling that need to be
addressed before publication.
p 6 1st paragraph - The firsts sentence says that the flow rate is a function of temperature,
yet the rest of the paragraph says that it isn’t.
The rest of the paragraph say the flow rate is a weak function of the temperature
and that it explains why. I hope that everyone agrees with me that it is common
to state a common assumption and explain why in that particular case it is not
important. I wish that more people would do just that. First, it would eliminate
many mistakes that are synonymous with research in die casting, because it forces
the “smart” researchers to check the major assumptions they make. Second, it
makes clear to the reader why the assumption was made.
p 6 - after Eq 2 - Should indicate immediately that the subscript[s] refer to the sections in
Figure 1.
I will consider this, Yet, I am not sure this is a good idea.
p 6 - after equation 2 - There is a major assumption made here that should not pass without
some comment[s]3 “Assuming steady state ” - This assumption goes to the heart of this
approach to the filling calculation and establishes the bounds of its applicability. The
authors should discuss this point.
Well, I totally disagree with the referee on this point. The major question in die
casting is how to ensure the right range of filling time and gate velocity. This
paper’s main concern is how to calculate the CD and determine if the CD be
“assigned” to a specific runner. The unsteady state is only a side effect and has
very limited importance due to AESS. Of course the flow is not continuous/steady
state and is affected by many parameters such as the piston weight, etc, all of
which are related to the transition point and not to the pQ2 diagram per se. The
unsteady state exists only in the initial and final stages of the injection. As a
general rule, having a well designed pQ2 diagram will produce a significant
improvement in the process design. It should be noted that a full paper discussing
the unsteady state is being prepared for publication at the present time.
In general the organization of the paper is somewhat weak - the introduction especially
does not very well set the technical context for the pQ2 method and show how the present
work fits into it.
3 Is the referee looking for one or several explanations?
316
APPENDIX C. WHAT THE ESTABLISHMENT’S SCIENTISTS SAY
The present work does not fit into past work! It shows that the past work is
wrong, and builds the mathematical theory behind the pQ2 diagram.
The last paragraph of the intro is confused [confusing]. The idea introduced in the last
sentence on page 2 is that the CD should vary somehow during the calculation, and
subsequently variation with Reynolds number is discussed, but the intervening material
about geometry effects is inconsistent with a discussion of things that can vary during the
calculation. The last two sentences do not fit together well either - “the assumption of
constant CD is not valid” - okay, but is that what you are going to talk about, or are you
going to talk about “particularly the effects of the gate area”?
Firstly, CD should not vary during the calculations it is a constant for a specific
set of runner system and die casting machine. Secondly, once any parameter is
changed, such as gate area, CD has to be recalculated. Now the referee’s statement
CD should vary, isn’t right and therefore some of the following discussion is wrong.
Now about the fitting question. What do referee means by “fit together?” Do the
paper has to be composed in a rhyming verse? Anyhow, geometrical effects are
part of Reynolds number (review fluid mechanics). Hence, the effects of the gate
area shows that CD varies as well and has to be recalculated. So what is
inconsistent? How do these sentences not fit together?
On p 8, after Eq 10 - I think that it would be a good idea to indicate immediately that these
equations are plotted in Figure 3, rather than waiting to the next section before referring to
Fig 3.
Also, making the Oz-axis on this graph logarithmic would help greatly in showing the
differences in the three pump characteristics.
Mentioning the figure could be good idea but I don’t agree with you about the
log scale, I do not see any benefits.
On p. 10 after Eq 11 - The solution of Eq 11 requires full information on the die casting
machine - According to this model, the machine characterized by Pmax, Qmax and the
exponent in Eq 1. The wording of this sentence, however, might be indicating that there is
some information to be had on the machine other that these three parameters. I do not
think that that is what the authors intend, but this is confusing.
This is exactly what the authors intended. The model does not confined to a
specific exponent or function, but rather gives limiting cases. Every die casting
machine can vary between the two extreme functions, as discussed in the paper.
Hence, more information is needed for each individual die casting machine.
p 12 - I tend to disagree with the premises of the discussion following Eq 12. I think that
Qmax depends more strongly on the machine size than does Pmax. In general, P max is the
intensification pressure that one wants to achieve during solidification, and this should not
change much with the machine size, whereas the clamping force, the product of this
pressure and platten are, goes up. On the other hand, when one has larger area to make
larger casting, one wants to increase the volumetric flow rate of metal so that flow rate of
metal so that fill times will not go up with the machine size. Commonly, the shot sleeve is
C.4. REFEREE 3
317
larger, while the maximum piston velocity does not change much.
Here the referee is confusing so many different concepts that it will take a while
to explain it properly. Please find here a attempt to explain it briefly. The
intensification pressure has nothing to do with the pQ2 diagram. The pQ2 does
not have much to do with the solidification process. It is designed not to have
much with the solidification. The intensification pressure is much larger than
Pmax . I give up!! It would take a long discussion to teach you the fundamentals of
the pQ2 diagram and the die casting process. You confuse so many things that it
impossible to unravel it all for you in a short paragraph. Please read Chapter 7 or
even better read the whole book.
Also, following Eq 13, the authors should indicate what they mean by “middle range” of the
Oz numbers. It is not clear from Fig 3 how close one needs to get to Oz=0 for the three
curves to converge again.
The mathematical equations are given in the paper. They are very simple that
you can use hand calculator to find how much close you need to go to Oz = 0 for
your choice of error. A discussion on such issue is below the level from an
academic paper.
.
Besides being illustrative of the results, part of the value of an example calculation comes
from it making possible duplication of the results elsewhere. In order to support this, the
authors need to include the relationships that used for CD in these calculations.
The literature is full of such information. If the referee opens any basic fluid
mechanics text book then he can find information about it.
The discussion on p 14 of Fig 5 needs a little more consideration. There is a maximum in this
curve, but the author’s criterion of being on the “right hand branch” is said to be shorter fill
time, which is not a criterion for choosing a location on this curve at all. The fill time is
monotone decreasing with increasing A3 at constant A2, since the flow is the product of
Vmax and A3. According to this criterion, no calculation is needed - the preferred
configuration is no gate whatsoever. Clearly, choosing an operating point requires
introduction of other criteria, including those that the authors mention in the intro. And the
end of the page 14 discussion that the smaller filling time from using a large gate (or a smaller
runner!!??) will lead to a smaller machine just does not follow at all. The machine size is
determined by the part size and the required intensification pressure, not by any of this.
Once again the referee is confusing many issues; let me interpret again what is
the pQ2 diagram is all about. The pQ2 diagram is for having an operational
point at the right gate velocity and the right filling time. For any given A2 , there
are two possible solutions on the right hand side and one on the left hand side
with the same gate velocity. However, the right hand side has smaller filling time.
And again, the referee confusing another issue. Like in many engineering
situations, we have here a situation in which more than one criterion is needed.
The clamping force is one of the criteria that determines what machine should be
318
APPENDIX C. WHAT THE ESTABLISHMENT’S SCIENTISTS SAY
chosen. The other parameter is the pQ2 diagram.
It seems that they authors have obscured some elementary results by doing their
calculations.4
For example, the last sentence of the middle paragraph on p 15 illustrates that as CD
reaches its limiting value of 1, the discharge velocity reaches its maximum. This is not
something we should be publishing in 1998.
CD ? There is no mention of the alleged fact of “ CD reaches its limiting value of
1.” There is no discussion in the whole article about “ CD reaching its maximum
(CD = 1)”. Perhaps the referee was mistakenly commenting on different articles
(NADCA’s book or an other die casting book) which he has confused with this
article.
Regarding the concluding paragraph on p 15:
1. The use of the word “constant” is not consistent throughout this paper. Do they mean
constant across geometry or constant across Reynolds number, or both.
To the readers: The referee means across geometry as different geometry
and across Reynolds number as different Re number5 . I really do not
understand the difference between the two cases. Aren’t actually these cases
the same? A change in geometry leads to a change in Reynolds number
number. Anyhow, the referee did not consider a completely different
possibility. Constant CD means that CD is assigned to a specific runner
system, or like the “common” model in which all the runners in the world
have the same value.
2. Assuming that they mean constant across geometry, then obviously, using a fixed
value for all runner/gate systems will sometimes lead to large errors. They did not
need to do a lot calculation to determine this.
And yet this method is the most used method in the industry(some even
will say the exclusive method).
3. Conversely, if they mean constant across Reynolds number, i.e. CD can vary through
the run as the velocity varies, then they have not made their case very well. Since they
have assumed steady state and the P3 does not enter into the calculation, then the only
reason that mention for the velocity to vary during the fill would be because Kf varies
as a function of the fill fraction. They have not developed this argument sufficiently.
4 If it is so elementary how can it be obscured.
I have broken–out this paragraph for purposes of illustration.
5 if the interpretation is not correct I would like to learn what it really mean.
C.4. REFEREE 3
319
Let me stress again the main point of the article. CD varies for different
runners and/or die casting machines. It is postulated that the velocity does
not vary during run. A discussion about P3 is an entirely different issue
related to the good venting design for which P3 remains constant.
4. If the examples given in the paper do not represent the characteristics of a typical die
casting machine, why to present them at all? Why are the “more detailed calculations”
not presented, instead of the trivial results that are shown?
These examples demonstrate that the “common” method is erroneous and
that the “authors‘” method should be adopted or other methods based on
scientific principles. I believe that this is a very good reason.
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APPENDIX C. WHAT THE ESTABLISHMENT’S SCIENTISTS SAY
D
My Relationship with Die Casting
Establishment
I cannot believe the situation that I am in. The hostility I am receiving from the
establishment is unbelievable, as individual who has spent the last 12 years in research to
improve the die casting. At first I was expecting to receive a welcome to the club. Later
when my illusions disappeared, I realized that it revolves around money along with avoiding
embarrassment to the establishment due to exposing of the truths and the errors the
establishment has sponsored. I believe that the establishment does not want people to know
that they had invested in research which produces erroneous models and continues to do so,
even though they know these research works/models are scientifically rubbish. They don’t
want people to know about their misuse of money.
When I started my research, I naturally called what was then SDCE. My calls were never
returned. A short time later SDCE developed into what is now called NADCA. I had hoped
that this new creation would prove better. Approximately two years ago I wrote a letter to
Steve Udvardy, director of research and education for NADCA ( a letter I never submitted).
Now I have decided that it is time to send the letter and to make it open to the public. I have
a long correspondence with Paul Bralower, former director of communication for NADCA,
which describes my battle to publish important information. An open letter to Mr. Baran,
Director of Marketing for NADCA, is also attached. Please read these letters. They reveal a
lot of information about many aspects of NADCA’s operations. I have submitted five (5)
articles to this conference (20th in Cleveland) and only one was accepted (only 20%
acceptance compared to ∼ 70% to any body else). Read about it here. During my battle to
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APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
“insert” science in die casting, many curious things have taken place and I wonder: are they
coincidental? Read about these and please let me know what you think.
Open Letter to Mr. Udvardy
Steve Udvardy NADCA,
9701 West Higgins Road, Suite 880
Rosemont, IL 60018-4721
January 26, 1998
Subject: Questionable ethics
Dear Mr. Udvardy:
I am writing to express my concerns about possible improprieties in the way that
NADCA awards research grants. As a NADCA member, I believe that these
possible improprieties could result in making the die casting industry less
competitive than the plastics and other related industries. If you want to enhance
the competitiveness of the die casting industry, you ought to support die casting
industry ethics and answer the questions that are raised herein.
Many of the research awards raise serious questions and concerns about the ethics
of the process and cast very serious shadows on the integrity those involved in the
process. In the following paragraphs I will spell out some of the things I have
found. I suggest to you and all those concerned about the die casting industry
that you/they should help to clarify these questions, and eliminate other problems
if they exist in order to increase the die casting industry’s profits and
competitiveness with other industries. I also wonder why NADCA demonstrates
no desire to participate in the important achievements I have made.
On September 26, 1996, I informed NADCA that Garber’s model on the critical
slow plunger velocity is unfounded, and, therefore so, is all the other research
based on Garber’s model (done by Dr. Brevick from Ohio State University). To
my great surprise I learned from the March/April 1997 issue of Die Casting
Engineer that NADCA has once again awarded Dr. Brevick with a grant to
continue his research in this area. Also, a year after you stated that a report on
the results from Brevick’s could be obtained from NADCA, no one that I know of
has been able to find or obtain this report. I and many others have tried to get
this report, but in vain. It leaves me wondering whether someone does not want
others to know about this research. I will pay $50 to the first person who will
furnish me with this report. I also learned (in NADCA’s December 22, 1997
publication) that once more NADCA awarded Dr. Brevick with another grant to
do research on this same topic for another budget year (1998). Are Dr. Brevick’s
results really that impressive? Has he changed his model? What is the current
model? Why have we not heard about it?
I also learned in the same issue of Die Casting Engineer that Dr. Brevick and his
colleagues have been awarded another grant on top of the others to do research on
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the topic entitled “Development and Evaluation of the Sensor System.” In the
September/October 1997 issue, we learned that Mr. Gary Pribyl, chairman of the
NADCA Process Technology Task Group, is part of the research team. This
Mr. Pribyl is the chairman of the very committee which funded the research. Of
course, I am sure, this could not be. I just would like to hear your explanation. Is
it legitimate/ethical to have a man on the committee awarding the chairman a
grant?
Working on the same research project with this Mr. Pribyl was Dr. Brevick who
also received a grant mentioned above. Is there a connection between the fact that
Gary Pribyl cooperated with Dr. B. Brevick on the sensor project and you
deciding to renew Dr. Brevick’s grant on the critical slow plunger velocity project?
I would like to learn what the reasoning for continuing to fund Dr. Brevick after
you had learned that his research was problematic.
Additionally, I learned that Mr. Steve Udvardy was given a large amount of
money to study distance communications. I am sure that Mr. Udvardy can
enhance NADCA’s ability in distance learning and that this is why he was
awarded this grant. I am also sure that Mr. Udvardy has all the credentials needed
for such research. One can only wonder why his presentation was not added to the
NADCA proceedings. One may also wonder why there is a need to do such
research when so much research has already been done in this area by the world’s
foremost educational experts. Maybe it is because distance communication works
differently for NADCA. Is there a connection between Mr. Steve Udvardy being
awarded this grant and his holding a position as NADCA’s research director? I
would like to learn the reasons you vouchsafe this money to Mr. Udvardy! I also
would like to know if Mr. Udvardy’s duties as director of education include
knowledge and research in this area. If so, why is there a need to pay Mr.
Udvardy additional monies to do the work that he was hired for in the first place?
We were informed by Mr. Walkington on the behalf of NADCA in the Nov–Dec
1996 issue that around March or April 1997, we would have the software on the
critical slow plunger velocity. Is there a connection between this software’s
apparent delayed appearance and the fact that the research in Ohio has produced
totally incorrect and off–base results? I am sure that there are reasons preventing
NADCA from completing and publishing this software; I would just like to know
what they are. I am also sure that the date this article came out (Nov/Dec 1996)
was only coincidentally immediately after I sent you my paper and proposal on
the shot sleeve (September 1996). What do you think?
Likewise, I learned that Mr. Walkington, one of the governors of NADCA, also
received a grant. Is there a connection between this grant being awarded to
Mr. Walkington and his position? What about the connection between his
receiving the grant and his former position as the director of NADCA research? I
am sure that grant was awarded based on merit only. However, I have serious
concerns about his research. I am sure that these concerns are unfounded, but I
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APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
would like to know what Mr. Walkington’s credentials are in this area of research.
The three most important areas in die casting are the critical slow plunger
velocity, the pQ2 diagram, and the runner system design. The research sponsored
by NADCA on the critical slow plunger velocity is absolutely unfounded because
it violates the basic physics laws. The implementation of the pQ2 diagram is also
absolutely unsound because again, it violates the basic physics laws. One of the
absurdities of the previous model is the idea that plunger diameter has to decrease
in order to increase the gate velocity. This conclusion (of the previous model)
violates several physics laws. As a direct consequence, the design of the runner
system (as published in NADCA literature) is, at best, extremely wasteful.
As you also know, NADCA, NSF, the Department of Energy, and others
sponsoring research in these areas exceed the tens of millions, and yet produce
erroneous results. I am the one who discovered the correct procedure in both
areas. It has been my continuous attempt to make NADCA part of these
achievements. Yet, you still have not responded to my repeated requests for a
grant. Is there a reason that it has taken you 1 12 years to give me a negative
answer? Is there a connection between any of the above information and how long
it has taken you?
Please see the impressive partial–list of the things that I have achieved. I am the
one who found Garber’s model to be totally and absolutely wrong. I am also the
one responsible for finding the pQ2 diagram implementation to be wrong. I am
the one who is responsible for finding the correct pQ2 diagram implementation. I
am the one who developed the critical area concept. I am the one who developed
the economical runner design concept. In my years of research in the area of die
casting I have not come across any research that was sponsored by NADCA which
was correct and/or which produced useful results!! Is there any correlation
between the fact that all the important discoveries (that I am aware of) have been
discovered not in–but outside of NADCA? I would like to hear about anything in
my area of expertise supported by NADCA which is useful and correct? Is there a
connection between the foregoing issues and the fact that so many of the die
casting engineers I have met do not believe in science?
More recently, I have learned that your secretary/assistant, Tricia Margel, has
now been awarded one of your grants and is doing research on pollution. I am sure
the grant was given based on qualification and merit only. I would like to know
what Ms. Margel’s credentials in the pollution research area are? Has she done
any research on pollution before? If she has done research in that area, where was
it published? Why wasn’t her research work published? If it was published, where
can I obtain a copy of the research? Is this topic part of Ms. Margel’s duties at her
job? If so, isn’t this a double payment? Or perhaps, was this an extra separated
payment? Where can I obtain the financial report on how the money was spent?
Together we must promote die casting knowledge. I am doing my utmost to
increase the competitiveness of the die casting industry with our arch rivals: the
325
plastics industry, the composite material industry, and other industries. I am
calling on everyone to join me to advance the knowledge of the die casting process.
Thank you for your consideration.
Sincerely,
Genick Bar–Meir, Ph.D.
cc: NADCA Board of Governors
NADCA members
Anyone who care about die casting industry
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APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
Correspondence with Paul Bralower
Paul Bralower is the former director of communications at NADCA. I have tried to publish
articles about critical show shot sleeve and the pQ2 diagram through NADCA magazine.
Here is an example of my battle to publish the article regarding pQ2 . You judge whether
NADCA has been enthusiastic about publishing this kind of information. Even after Mr.
Bralower said that he would publish it I had to continue my struggle.
He agreed to publish the article but · · ·
At first I sent a letter to Mr. Bralower (Aug 21, 1997):
Paul M. Bralower
NADCA, Editor
9701 West Higgins Road, Suite 880
Rosemont, IL 60018-4721
Dear Mr. Bralower:
Please find enclosed two (2) copies of the paper “The mathematical theory of the
pQ2 diagram” submitted by myself for your review. This paper is intended to be
considered for publication in Die Casting Engineer.
For your convenience I include a disk DOS format with Microsoft WORD for
window format (pq2.wid) of the paper, postscript/pict files of the figures (figures 1
and 2). If there is any thing that I can do to help please do not hesitate let me
know.
Thank you for your interest in our work.
Respectfully submitted,
Dr. Genick Bar–Meir
cc: Larry Winkler
a short die casting list
Documents,
Disk
He did not responded to this letter, so I sent him an additional one on December 6, 1997.
encl:
Paul M. Bralower
NADCA, Editor
9701 West Higgins Road, Suite 880
Rosemont, IL 60018-4721
Dear Mr. Bralower:
I have not received your reply to my certified mail to you dated August 20, 1997
in which I enclosed the paper ”The mathematical theory of the pQ2 diagram”
authored by myself for your consideration (a cc was also sent to Larry Winkler
from Hartzell). Please consider publishing my paper in the earliest possible issue.
I believe that this paper is of extreme importance to the die casting field.
327
I understand that you have been very busy with the last exhibition and congress.
However, I think that this paper deserves a prompt hearing.
I do not agree with your statement in your December 6, 1996 letter to me stating
that ”This paper is highly technical-too technical without a less-technical
background explanation for our general readers · · · . I do not believe that
discounting your readers is helpful. I have met some of your readers and have
found them to be very intelligent, and furthermore they really care about the die
casting industry. I believe that they can judge for themselves. Nevertheless, I have
yielded to your demand and have eliminated many of the mathematical
derivations from this paper to satisfy your desire to have a ”simple” presentation.
This paper, however, still contains the essentials to be understood clearly.
Please note that I will withdraw the paper if I do not receive a reply stating your
intentions by January 1, 1998, in writing. I do believe this paper will change the
way pQ2 diagram calculations are made. The pQ2 diagram, as you know, is the
central part of the calculations and design thus the paper itself is of same
importance.
I hope that you really do see the importance of advancing knowledge in the die
casting industry, and, hope that you will cooperate with those who have made the
major progress in this area.
Thank you for your consideration.
Sincerely,
Dr. Genick Bar-Meir
cc: Boxter, McClimtic, Scott, Wilson, Holland, Behler, Dupre, and some other
NADCA members
ps: You probably know by now that Garber’s model is totally and absolutely
wrong including all the other investigations that where based on it, even if
they were sponsored by NADCA. (All the researchers agreed with me in the
last congress)
Well that letter got him going and he managed to get me a letter in which he claim that he
sent me his revisions. Well, read about it in my next letter dated January 7, 1998.
Paul M. Bralower,
NADCA, Editor
9701 West Higgins Road, Suite 880
Rosemont, IL 60018-4721
Dear Mr. Bralower:
Thank you for your fax dated December 29, 1997 in which you alleged that you
sent me your revisions to my paper “The mathematical theory of the pQ2
diagram.” I never receive any such thing!! All the parties that got this information
and myself find this paper to of extreme importance.
I did not revise my paper according to your comments on this paper, again, since I
did not receive any. I decided to revised the paper since I did not received any
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APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
reply from you for more than 4 months. I revised according to your comments on
my previous paper on the critical slow plunger velocity. As I stated in my letter
dated December 6, 1997, I sent you the revised version as I send to all the cc list.
I re–sent you the same version on December 29, 1997. Please note that this is the
last time I will send you the same paper since I believe that you will claim again
that you do not receive any of my submittal. In case that you claim again that
you did not receive the paper you can get a copy from anyone who is on the cc
list. Please be aware that I changed the title of the paper (December, 6, 1997
version) to be ”How to calculate the pQ2 diagram correctly”.
I would appreciate if you respond to my e-Mail by January 14, 1998. Please
consider this paper withdrawn if I will not hear from you by the mentioned date in
writing (email is fine) whether the paper is accepted.
I hope that you really do see the importance of advancing knowledge in the die
casting industry, and, hope that you will cooperate with those who have made the
major progress in this area.
Sincerely,
Dr. Genick Bar-Meir
ps: You surely know by now that Garber’s model is totally and absolutely wrong
including all the other investigations that where based on it
He responded to this letter and changed his attitude · · · I thought.
January 9, 1998.
Dear Mr. Bar-Meir:
Thank you for your recent article submission and this follow-up e-mail. I am now in
possession of your article "How to calculate the pQ2 diagram correctly." It is the version
dated Jan. 2, 1998. I have read it and am prepared to recommend it for publication in Die
Casting Engineer. I did not receive any earlier submissions of this article, I was confusing it
with the earlier article that I returned to you. My apologies. However I am very pleased at
the way you have approached this article. It appears to provide valuable information in an
objective manner, which is all we have ever asked for. As is my policy for highly technical
material, I am requesting technical personnel on the NADCA staff to review the paper as
well. I certainly think this paper has a much better chance of approval, and as I said, I will
recommend it. I will let you know of our decision in 2-3 weeks. Please do not withdraw
it–give us a little more time to review it! I would like to publish it and I think technical
reviewer will agree this time.
Sincerely,
Paul Bralower
Well I waited for a while and then I sent Mr. Bralower a letter dated Feb 2, 1998.
Dear Mr. Bralower,
Apparently, you do not have the time to look over my paper as you promise. Even
a negative reply will demonstrate that you have some courtesy. But apparently
329
the paper is not important as your experts told you and I am only a small
bothering cockroach.
Please see this paper withdrawn!!!!
I am sorry that we do not agree that an open discussion on technical issues should
be done in your magazine. You or your technical experts do not have to agree
with my research. I believe that you have to let your readers to judge. I am sure
that there is no other reasons to your decision. I am absolutely sure that you do
not take into your consideration the fact that NADCA will have to stop teaching
SEVERAL COURSES which are wrong according to this research.
Thank you for your precious time!!
Dr. Bar-Meir
Please note that this letter and the rest of the correspondence with you in this
matter will be circulated in the die casting industry. I am sure that you stand by
your decision and you would like other to see this correspondence even if they are
NADCA members.
Here is the letter I received in return a letter from Paul Bralower Feb 5, 1998.
Dear Mr. Bar-Meir:
I’ll have you know that you have inconvenienced me and others on our staff today with your
untoward, unnecessary correspondence. If you had a working telephone or fax this e-mail
would not be necessary. As it is I must reply to your letter and take it to someone else’s office
and have them e-mail it to you right away.
I tried to telephone you last week on Thurs. 1/29 with the news that we have agreed to
publish your article, “How to Calculate the pQ2 diagram correctly.” I wanted to ask you to
send the entire paper, with graphics and equations, on a disk. Because of the current status
of our e-mail system, I would advise you not to e-mail it. Send it on any of the following:
Syquest, Omega ZIP or Omega JAZ. Use Microsoft Office 97, Word 6.0 or Word Perfect 6.0.
The problem is I couldn’t reach you by phone. I tried sending you a fax several times Thurs.
and last Friday. There was no response. We tried a couple of different numbers that we had
for you. Having no response, I took the fax and mailed it to you as a letter on Monday 2/2. I
sent Priority 2-day Mail to your attention at Innovative Filters, 1107 16th Ave. S.E.,
Minneapolis, Minn, 55414. You should have received it today at latest if this address is correct
for you, which it should be since it was on your manuscript.
Now, while I’m bending over backwards to inform you of your acceptance, you have the
nerve to withdraw the paper and threaten to spread negative gossip about me in the
industry! I know you couldn’t have known I was trying to contact you, but I must inform
you that I can’t extend any further courtesies to you. As your paper has been accepted, I
expect that you will cancel your withdrawl and send me the paper on disk immediately for
publication. If not, please do not submit any further articles.
My response to Paul M. Bralower.
Feb 9, 1998
Dear Mr. Bralower:
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APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
Thank you for accepting the paper ”How to calculate the pQ2 diagram
correctly”. I strongly believe that this paper will enhance the understanding of
your readers on this central topic. Therefore, it will help them to make wiser
decisions in this area, and thus increase their productivity. I would be happy to
see the paper published in Die Casting Engineer.
As you know I am zealous for the die casting industry. I am doing my utmost to
promote the knowledge and profitability of the die casting industry. I do not
apologize for doing so. The history of our correspondence makes it look as if you
refuse to publish important information about the critical slow plunger velocity.
The history shows that you lost this paper when I first sent it to you in August,
and also lost it when I resubmitted it in early December. This, and the fact that I
had not heard from you by February 1, 1998, and other information, prompted me
to send the email I sent. I am sure that if you were in my shoes you would have
done the same. My purpose was not to insults anyone. My only aim is to promote
the die casting industry to the best of my ability. I believe that those who do not
agree with promoting knowledge in die casting should not be involved in die
casting. I strongly believe that the editor of NADCA magazine (Die Casting
Engineer) should be interested in articles to promote knowledge. So, if you find
that my article is a contribution to this knowledge, the article should be published.
I do not take personal insult and I will be glad to allow you to publish this paper
in Die Casting Engineer. I believe that the magazine is an appropriate place for
this article. To achieve this publication, I will help you in any way I can. The
paper was written using LATEX, and the graphics are in postscript files. Shortly, I
will send you a disc containing all the files. I will also convert the file to Word 6.0.
I am afraid that conversion will require retyping of all the equations. As you
know, WORD produces low quality setup and requires some time. Would you
prefer to have the graphic files to be in TIFF format? or another format? I have
enhanced the calculations resolution and please be advised that I have changed
slightly the graphics and text.
Thank you for your assistance.
Sincerely,
Dr. Genick Bar-Meir
Is the battle over?
Well, I had thought in that stage that the paper would finally be published as the editor had
promised. Please continue to read to see how the saga continues.
4/24/98
Dear Paul Bralower:
To my great surprise you did not publish my article as you promised. You also did
not answer my previous letter. I am sure that you have a good reasons for not
331
doing so. I just would like to know what it is. Again, would you be publishing the
article in the next issue? any other issue? published at all? In case that you
intend to publish the article, can I receive a preprint so I can proof-read the
article prior to the publication?
Thank you for your consideration and assistance!!
Genick
Then I got a surprise: the person dealing with me was changed. Why? (maybe you, the
reader, can guess what the reason is). I cannot imagine if the letter was an offer to buy me
out. I just wonder why he was concerned about me not submitting proposals (or this matter
of submitting for publication). He always returned a prompt response to my proposals, yah
sure. Could he possibly have suddenly found my research to be so important. Please read his
letter, and you can decide for yourself.
Here is Mr. Steve Udvardy response on Fri, 24 Apr 1998
Genick,
I have left voice mail for you. I wish to speak with you about what appears to be
non-submittal of your proposal I instructed you to forward to CMC for the 1999 call.
I can and should also respond to the questions you are posinjg to Paul.
I can be reached by phone at 219.288.7552.
Thank you,
Steve Udvardy
Since the deadline for that proposal had passed long before, I wondered if there was any
point in submitting any proposal. Or perhaps there were exceptions to be made in my case?
No, it couldn’t be; I am sure that he was following the exact procedure. So, I then sent Mr.
Udvardy the following letter.
April 28, 1998
Dear Mr. Udvardy:
Thank you very much for your prompt response on the behalf of Paul Bralower.
As you know, I am trying to publish the article on the pQ2 diagram. I am sure
that you are aware that this issue is central to die casting engineers. A better
design and a significant reduction of cost would result from implementation of the
proper pQ2 diagram calculations.
As a person who has dedicated the last 12 years of his life to improve the die
casting industry, and as one who has tied his life to the success of the die casting
industry, I strongly believe that this article should be published. And what better
place to publish it than “Die Casting Engineer”?
I have pleaded with everyone to help me publish this article. I hope that you will
agree with me that this article should be published. If you would like, I can
explain further why I think that this article is important.
I am very glad that there are companies who are adopting this technology. I just
wish that the whole industry would do the same.
Again, thank you for your kind letter.
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APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
Genick
ps: I will be in my office Tuesday between 9-11 am central time (612) 378–2940
I am sure that Mr. Udvardy did not receive the comments of/from the referees (see
Appendix C). And if he did, I am sure that they did not do have any effect on him
whatsoever. Why should it have any effect on him? Anyhow, I just think that he was very
busy with other things so he did not have enough time to respond to my letter. So I had to
send him another letter.
5/15/98
Dear Mr. Udvardy:
I am astonished that you do not find time to answer my letter dated Sunday,
April, 26 1998 (please see below copy of that letter). I am writing you to let you
that there is a serious danger in continue to teach the commonly used pQ2
diagram. As you probably know (if you do not know, please check out IFI’s web
site www.dieperfect.com), the commonly used pQ2 diagram as it appears in
NADCA’s books violates the first and the second laws of thermodynamics, besides
numerous other common sense things. If NADCA teaches this material, NADCA
could be liable for very large sums of money to the students who have taken these
courses. As a NADCA member, I strongly recommend that these classes be
suspended until the instructors learn the correct procedures. I, as a NADCA
member, will not like to see NADCA knowingly teaching the wrong material and
moreover being sued for doing so.
I feel that it is strange that NADCA did not publish the information about the
critical slow plunger velocity and the pQ2 diagram and how to do them correctly.
I am sure that NADCA members will benefit from such knowledge. I also find it
beyond bizarre that NADCA does not want to cooperate with those who made the
most progress in the understanding die casting process. But if NADCA teaching
the wrong models might ends up being suicidal and I would like to change that if I
can.
Thank you for your attention, time, and understanding!
Sincerely, Dr. Genick Bar-Meir
ps: Here is my previous letter.
Now I got a response. What a different tone. Note the formality (Dr Bar-Meir as oppose to
Genick).
May 19, 1998
Dear Dr Bar-Meir,
333
Yes, I am here. I was on vacation and tried to contact you by phone before I left for vacation.
During business travel, I was sorry to not be able to call during the time period you
indicated.
As Paul may have mentioned, we have approved and will be publishing your article on
calculating PQ2. The best fit for this is an upcoming issue dedicated to process control.
Please rest assured that it will show up in this appropriate issue of DCE magazine.
Since there has been communications from you to Paul and myself and some of the issues
are subsequently presented to our Executive Vice President, Dan Twarog, kindly direct all
future communications to him. This will assist in keeping him tied in the loop and assist in
getting responses back to you. His e-mail address is Twarog@diecasting.org.
Thank you,
Steve Udvardy
Why does Mr. Udvardy not want to communicate with me and want me to write to
Executive Vice President? Why did they change the title of the article and omit the word
“correctly”. I also wonder about the location in the end of the magazine.
I have submitted other proposals to NADCA, but really never received a reply. Maybe it isn’t
expected to be replied to? Or perhaps it just got/was lost?
Open letter to Leo Baran
In this section an open letter to Leo Baran is presented. Mr. Baran gave a presentation in
Minneapolis on May 12, 1999, on “Future Trends and Current Projects” to “sell” NADCA to
its members. At the conclusion of his presentation, I asked him why if the situation is so
rosy as he presented, that so many companies are going bankrupt and sold. I proceeded to
ask him why NADCA is teaching so many erroneous models. He gave me Mr. Steve
Udvardy’s business card and told me that he has no knowledge of this and that since he
cannot judge it, he cannot discuss it. Was he prepared for my questions or was this merely a
spontaneous reaction?
Dear Mr. Baran,
Do you carry Steve Udvardy’s business card all the time? Why? Why do you not
think it important to discuss why so many die casting companies go bankrupt and
are sold? Is it not important for us to discuss why there are so many financial
problems in the die casting industry? Don’t you want to make die casting
companies more profitable? And if someone tells you that the research sponsored
by NADCA is rubbish, aren’t you going to check it? Discuss it with others in
NADCA? Don’t you care whether NADCA teaches wrong things? Or is it that
you just don’t give a damn?
I am sure that it is important for you. You claimed that it is important for you in
the presentation. So, perhaps you care to write an explanation in the next
NADCA magazine. I would love to read it.
Sincerely,
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APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
Genick Bar–Meir
Is it all coincidental?
I had convinced Larry Winkler in mid 1997 (when he was still working for Hartzell), to ask
Mr. Udvardy why NADCA continued support for the wrong models (teaching the
erroneous Garber’s model and fueling massive grants to Ohio State University). He went to
NADCA and talked to Mr. Udvardy about this. After he came back, he explained that they
told him that I didn’t approach NADCA in the right way. (what is that?) His enthusiasm then
evaporated, and he continues to say that, because NADCA likes evolution and not
revolution, they cannot support any of my revolutionary ideas. He suggested that I needed
to learn to behave before NADCA would ever cooperate with me. I was surprised and
shaken. “What happened, Larry?” I asked him. But I really didn’t get any type of real
response. Later (end of 1997) I learned he had received NADCA’s design award. You, the
reader, can conclude what happened; I am just supplying you with the facts.
Several manufacturers of die casting machines, Buler, HPM, Prince, and UBE presented
their products in Minneapolis in April 1999. When I asked them why they do not adapt the
new technologies, with the exception of the Buler, the response was complete silence. And
just Buler said that they were interested; however, they never later called. Perhaps, they lost
my phone number. A representative from one of the other companies even told me
something on the order of “Yeah, we know that the Garber and Brevick models are totally
wrong, but we do not care; just go away–you are bothering us!”.
I have news for you guys: the new knowledge is here to stay and if you want to make
the die casting industry prosper, you should adopt the new technologies. You should
make the die casting industry prosper so that you will prosper as well; please do not look at
the short terms as important.
The next issue of the Die Casting Engineer (May/Jun 1999 issue) was dedicated to machine
products. Whether this was coincidental, you be the judge.
I submitted a proposal to NADCA (November 5, 1996) about Garber/Brevick work (to which
I never received a reply). Two things have happened since: I made the proposal(in the
proposal I demonstrate that Brevick’s work from Ohio is wrong) 1) publishing of the article
by Bill Walkington in NADCA magazine about the “wonderful research” in Ohio State
University and the software to come. 2)a “scientific” article by EKK. During that time EKK
also advertised how good their software was for shot sleeve calculations. Have you seen any
EKK advertisements on the great success of shot sleeve calculations lately?
Here is another interesting coincidence, After 1996, I sent a proposal to NADCA, the cover
page of DCE showing the beta version of software for calculating the critical slow plunger
velocity. Yet, no software has ever been published. Why? Is it accidental that the author of
the article in the same issue was Bill Walkington.
And after all this commotion I was surprised to learn in the (May/June 1999) issue of DCE
magazine that one of the Brevick group had received a prize (see picture below if I get
NADCA permission). I am sure that Brevick’s group has made so much progress in the last
335
year that this is why the award was given. I just want to learn what these accomplishments
are.
For a long time NADCA described the class on the pQ2 diagram as a “A close mathematical
description.” After I sent the paper and told them about how the pQ2 diagram is erroneous,
they change the description. Well it is good, yet they have to say that in the past material was
wrong and now they are teaching something else. or are they?
I have submitted five (5) papers to the conference (20th in Cleveland) and four (4) have been
rejected on the grounds well, you can read the letter yourself:
Here is the letter from Mr. Robb.
17 Feb 1999
The International Technical Council (ITC) met on January 20th to review all submitted
abstracts. It was at that time that they downselected the abstracts to form the core of each of
the 12 sessions. The Call for Papers for the 1999 Congress and Exposition produced 140
possible abstracts from which to choose from, of this number aproximately 90 abstracts
were selected to be reviewed as final papers. I did recieve all 5 abstracts and distribute them
to the appropriate Congress Chairmen. The one abstract listed in your acceptance letter is
in fact the one for which we would like to review the final paper. The Congress Chairmen
will be reviewing the final papers and we will be corresponding with all authors as to any
changes revisions which are felt to be appropriate.
The Congress Chairmen are industry experts and it is there sole discretion as to which
papers are solicited based on abstract topic and fit to a particular session.
It is unfortunate that we cannot accept all abstracts or papers which are submitted. Entering
an abstract does not constitue an automatic acceptance of the abstract/or final paper.
Thank you for your inquiry, and we look forward to reviewing your final paper.
Regards,
Dennis J. Robb
NADCA
I must have submitted the worst kind of papers otherwise. How can you explain that only
20% of my papers (1 out of 5) accepted. Note that the other researchers’ ratio of acceptance
on their papers is 65%, which means that other papers are three times better than mine.
Please find here the abstracts and decide if you’d like to hear such topics or not. Guess which
the topic NADCA chose, in what session and on what day (third day).
A Nobel Tangential Runner Design
The tangential gate element is commonly used in runner designs. A novel approach to this
runner design has been developed to achieve better control over the needed performance.
The new approach is based on scientific principles in which the interrelationship between
the metal properties and the geometrical parameters is described.
put the picture of Brevick, Udvardy and price guy
336
APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
Vacuum Tank Design Requirements
Gas/air porosity constitutes a large part of the total porosity. To reduce the porosity due to
the gas/air entrainment, vacuum can be applied to remove the residual air in the die. In
some cases the application of vacuum results in a high quality casting while in other cases
the results are not satisfactory. One of the keys to the success is the design of the vacuum
system, especially the vacuum tank. The present study deals with what are the design
requirements on the vacuum system. Design criteria are presented to achieve an effective
vacuum system.
How Cutting Edge technologies can improve your Process Design
approach
A proper design of the die casting process can reduce the lead time significantly. In this
paper a discussion on how to achieve a better casting and a shorter lead time utilizing these
cutting edge technologies is presented. A particular emphasis is given on the use of the
simplified calculations approach.
On the effect of runner design on the reduction of air entrainment:
Two Chamber Analysis
Reduction of air entrapment reduces the product rejection rate and always is a major
concern by die casting engineers. The effects of runner design on the air entrapment have
been disregarded in the past. In present study, effects of the runner design characteristics are
studied. Guidelines are presented on how to improve the runner design so that less air/gas
are entrapped.
Experimental study of flow into die cavity: Geometry and Pressure
effects
The flow pattern in the mold during the initial part of the injection is one of the parameters
which determines the success of the casting. This issue has been studied experimentally.
Several surprising conclusions can be drawn from the experiments. These results and
conclusions are presented and can be used by the design engineers in their daily practice to
achieve better casting.
Afterward
At the 1997 NADCA conference I had a long conversation with Mr. Warner Baxter. He told
me that I had ruffled a lot of feathers in NADCA. He suggested that if I wanted to get real
results, I should be politically active. He told me how bad the situation had been in the past
and how much NADCA had improved. But here is something I cannot understand: isn’t
there anyone who cares about the die casting industry and who wants it to flourish? If you
337
do care, please join me. I actually have found some individuals who do care and are
supporting my efforts to increase scientific knowledge in die casting. Presently, however,
they are a minority. I hope that as Linux is liberating the world from Microsoft, so too we
can liberate and bring prosperity to the die casting industry.
After better than a year since my first (and unsent) letter to Steve Udvardy, I feel that there
are things that I would like to add to the above letter. After my correspondence with Paul
Bralower, I had to continue to press them to publish the article about the pQ2 . This process
is also described in the preceding section. You, the reader, must be the judge of what is really
happening. Additionally, open questions/discussion topics to the whole die casting
community are added.
What happened to the Brevick’s research? Is there still no report? And does this type of
research continue to be funded?
Can anyone explain to me how NADCA operates?
Is NADCA, the organization, more important than the die casting industry?
338
APPENDIX D. MY RELATIONSHIP WITH DIE CASTING ESTABLISHMENT
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Index of Subjects
pQ2 , 126
pQ2 diagram, 11
1D Control Volume, 36
Control Volume Surface, 24
converging–diverging nozzle, 67
Courant number, 100
Critical design, 6
Critical Froude number, 284
Critical Gate velocity, 10
critical height, 283
Critical plunger diameter, 2
Critical plunger velocity, 2, 6, 11
Cubic equation, 282
Cubic to Parabola, 283
absolute viscosity, 32
Actual cost, 6
adiabatic nozzle, 69
Adiabatic process, 27
Air Entrapment, 18
Air entrapment, 6
Air streaks
Runner, 140
Archimedes number, 100
Atwood number, 100
Avi number, 97
Dean number, 100
Deborah number, 100
Deformable control volume, 24, 34
Detroit’s attitude, 8
Die Casting
Cost, 1
Guess work, 2
Typical run, 4
Die casting economy, 223
Dimensional analysis, 12
Basic units, 85
Typical parameters, 99
Discrete mathematics, 236
Doehler–Jarvis, 1
Drag coefficient, 100
BEP break even point, 3
Biscuit, 6
Bond number, 100
Break Even Point, 4
Linear relationship, 4
Brinkman number, 100
Buckingham’s pi Theory, 84
Bulk modulus, 102
Capillary number, 104
Capillary numbers, 100
Cauchy number, 100
Cavitation number, 100
Chézy coefficient, 281
Control Volume, 24
Control volume
example, 66
Eckert number, 100
Ekman number, 101
Energy conservation, 24
Enthalpy, 27
per mass, 27
347
348
Entropy, 27
Equation of state, 28
Erroneous models, 2
Euler equations, 40
Euler number, 101, 105
Example of poor research
Davis and pQ2 , 11
Wallace, 10
External forces, 38
fanno
second law, 243
Fanno flow
choking, 246
fanno flow, 241, 4fL
D 245
average friction factor, 247
entrance Mach number calculations,
256
entropy, 246
shockless, 255
star condition, 249
Fanno flow trends, 246
Filling the shot sleeve, 126
First Law of Thermodynamics, 25
Fluid, 31
What is, 31
Fluid Mechanics, 31
Fr number, 284
Froude number, 101
rotating, 106
Galileo number, 101, 106
Grashof number, 101
Heat transfer
jet, 127
hydraulic diameter, 304
Hydraulic jump, 11, 126, 295
Hydraulic Radius, 302
Ideal gas, 28
Instability, 126
Interfacial instability, 40
INDEX OF SUBJECTS
Internal Energy
per mass, 25
Internal energy, 25
Isotropic viscosity, 46
Kinematic boundary condition, 50
Kinetic energy, 25
Kolmogorov time, 102
Laplace Constant, 101
Laplace number, 106
Least expensive machine, 226
Leibniz integral rule, 37
Lift coefficient, 101
Mach number, 67, 101
Manning coefficient, 281
Marangoni number, 101
Marginal profits, 236
Mass conservation, 24, 33
Material replacement
Cross section, 134
Minimum Gate velocity, 10
Minimum profits, 4
Mixing zone
Runner, 135
Mold design, 1
Momentum Conservation, 37
Momentum conservation, 45
Momentum equation
Accelerated system, 39
Morton number, 101
Moving boundary, 50
Moving surface
Free surface, 51
Moving surface, constant of integration,
51
NADCA
Indoctrination, 3
Newtonian fluid, 33
No–slip condition, 49
Non–deformable control volume, 34
349
INDEX OF SUBJECTS
Normal Shock
Solution, 76
Numerical simulations, 14
Nusselt number, 97
Nusselt’s dimensionless technique, 89
Ohnesorge number, 107
OMC, 1
Open Channel
perimeter, 279
Open channel
Bernoulli’s equation, 281
Classification, 277
Intuition, 277
Open channel flow Chocking, 293
Ozer number, 101
Poiseuille flow, 58
Pore free technique, 12
Porosity, 18
pQ2 diagram, 2
Prandtl number, 102
Production
cost, 2
Production Cost, 3
Production Cost link to Design, 6
Reactions, see also Pore free technique
Reason people work in die casting, 3
Relative Profit Change, 3
Reversible process, 27
Reynolds number, 102, 103
Reynolds Transport Theorem, 37
Rossby number, 102
Runner
design, 2
Flow reigns of multiphase flows, 131
Multiphase flow, 136
Runner system, 6
Scrap, 3
cost, 3
Designed, 5
leakage, 4
minimum, 3
reduction, 3
Undesigned, 5
Scrap-Cost Diagram, 2
Second law of thermodynamics, 26
Shear number, 102
Shear Stress, 31
Shear stress, 31
Shelby, 1
shock wave, 71
star velocity, 78
trivial solution, 75
Shot sleeve, 11
Shrinkage Porosity, 18
Similitude, 85
Slip condition range, 50
Sloop, Given, 282
Solidification layer, 126
Specific energy, 282
Specific Heat
pressure, 28
volume, 28
Specific heats ratio, k, 28
speed of sound
star, 68
speed of sound
ideal gas, 65
St. Paul Metalcraft, 1
stagnation state, 67
Star conditions, 77
State property, 25
Stokes number, 102
Strouhal number, 102
Subcritical flow, 293
subcritical flow, 282
Supercritical flow, 293
supercritical flow, 282
Supply and demand method, 11
System
Definition, 24
system energy, 25
350
Taguchi’s method, 3
Taylor number, 102
Thermodynamical pressure, 47
Tool Products, 1
Transition to continuous, 37
Universal gas constant, 28
Velocity profile, 300
Vent system, 6
Vent system design, 2
INDEX OF SUBJECTS
viscosity
Murray, 81
von Karman vortex street, 103
Weber number, 102, 104
Why Die Casting?, 223
Work
Definition, 24
Young modulus, 102
Index of Authors
Faura, F, 7
Favi, Claudio, 7
Fondse, H, 142
Fourier Jean B. J., 84
Froude, William, 84
Fu, Li Bing, 11
Fu, Penghuai, 12
(Manning, 281
Altuncu, Ekrem, 11
Andritsos, N, 133
Backer, G, 17
Baek, U. H., 11
Baicheng, Liu, 13
Bar–Meir, Genick, 141
Bochvar, A. A., 10
Brauner, Neima, 8
Brevick, 12
Brinkman, CR, 16
Buckingham, 85
Gaponenko, YA, 133
Garber, Lester W, 11
Gašpár, Štefan, 9
Gebauer–Teichmannb, A, 14
Germani, Michele, 7
Goekec, S, 14
Gojdan, Dominik, 9
Go´ mez, P, 7
Griffith, 281
Griffiths, J. R., 81
Guerra, Marco Antonio Pego, 11
Chandrasekaran, 5
Coranič, Tomáš, 9
Dai, Wei, 11
Davis, A. J., 11
Dinsdale, 137
Dou, Kun, 3, 14
Doğan, A, 11
Draper, Alan, 10
Dupláková, Darina, 9
Han, J. W., 11
Hanratty, TJ, 133
Hatala, Michal, 9
Herna´ ndez, J, 7
Husár, Jozef, 9
E. R. G. Eckert, 84
Eckert, E. R. ,George, 12
El-Mehalawi, Mohamed, 6
Erdil, B, 11
Euler, Leonahard, 104
Ipek, Osman, 8
Jacot, A, 14
Jacot, Alain, 3
Ji, Shouxun, 8
Jiang, Haiyan, 12
Fan, ZY, 14
Fanno, Marco, 237
Kallien, Lothar H., 15
351
352
Kan, Mehmet, 8
Karni, Y, 7, 10
Kenar, O, 11
Kim, C. M., 16
Knapcikova, Lucia, 9
Knapčíková, Lucia, 9
Kohlstädta, S, 14
Konopka, Z, 9
Koru, Murat, 8
Lazaro-Nebreda, Jaime, 3
Lee, B. D., 11
Leibniz, 37
Leijdens, H, 142
Li, Tian, 8
Liu, Guangyu, 8
Liu, Jihua, 6
Lordan, E, 14
Lordan, Ewan, 3
Lo´ pez, J, 7
Luo, Alan A, 12
Lagiewka, M, 9
Maier, Ralph David, 10
Majerník, Ján, 9
Mandolini, Marco, 7
Martinelli, 84
Martinho, RP, 13
Maxwell, 84
Miller, R. A, 6
Miller, R. A., 15
Minaie, B, 14
Mobley, C. E., 15
Murray, M. T., 81
NADCA, 2
Board of Director, 7
Nadolski, M, 9
Nepomnyashchy, Alexander, 133
Neto, Belmira, 3
Newton, 84
Niu, Zhichao, 8
Notkin, E. M., 10
INDEX OF AUTHORS
Nusselt, Ernst Kraft Wilhelm, 59, 84, 85
Ooms, Gysbert, 142
Osborne, M. A., 15
Pan, XZ, 9
Patel, Jayesh, 3
Paško, Ján, 9
Peng, Liming, 12
Pigot, 281
Pinto, AG, 13
Pinto, HA, 13
Poiseuille, Jean Louis, 58
Porter, WD, 16
Purgert, RM, 16
Quested, 137
Radchenko, Svetlana, 9
Rajini, N, 9
Rajkumar, I, 9
Rayleigh, 84
Reed, George, 2
Reiner, M., 102
Reynolds, Osborne, 37, 85
Riabouchunsky, 85
Sachdev, Anil K, 12
Sachs, Bruno, 10
Sadchikova, N. M., 10
Sant, Frank, 17
Sant, Frank J., 16
Schmidt, 84
Shevtsova, Valentina, 133
Shoumei, Xiong, 13
Silva, FJG, 13
Smeaton, John, 84
Spektorova, S. I., 10
Stelson, KA, 14
Strouhal, Vincenz, 115
Stuhrke, W. F., 10, 141
Tzileroglou, Chrysoula, 3
353
INDEX OF AUTHORS
Vaschy, Aiméem, 84
Veinik, Albert, 10
Veinik, Albert Iozefovich, 10
Vernon-Harcourt, 281
Viswanathan, S, 16
Voller, VR, 14
Von Karman, 103
Vynnyckya, M, 14
Wallace, J. F., 10, 141
Wang, Shihao, 3
Wang, Yun, 3
Weishan, Zhang, 13
Yu, Yandong, 12
Zhai, Chunquan, 12
Zhang, Wei, 11
Zhang, Yijie, 3
Zhang, YJ, 14
Zhou, Xiaorong, 3
Zyska, A, 9