Manhattan College
Mechanical Engineering
MECH 405:
Thermal/Fluids
Laboratory
Student Manual
Fall 2023 Semester
Instructor: Dr. S. Peluso
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Table of Contents
Preface............................................................................................................................................ iii
Thermal/Fluids Laboratory Introduction ........................................................................................ 1
Introduction ................................................................................................................................. 1
Laboratory Safety Rules ............................................................................................................. 1
Preparation for Laboratory Classes ................................................................................................. 2
Attendance Policy ....................................................................................................................... 2
Grading Policy ............................................................................................................................ 2
Laboratory Notebook .................................................................................................................. 2
Laboratory Experiment, Presentation, and Report Schedule ...................................................... 3
Sample Reports ........................................................................................................................... 3
Design of Experiments Project ................................................................................................... 4
Recommended References .......................................................................................................... 4
Uncertainty Analysis ....................................................................................................................... 5
Introduction ................................................................................................................................. 5
Uncertainties in Measured Data .................................................................................................. 6
Uncertainties in Reference Data ................................................................................................. 7
Uncertainties in Computed Data ................................................................................................. 8
Plotting Data with Uncertainties ............................................................................................... 11
Uncertainties in a Mean Value .................................................................................................. 11
Uncertainties in Slopes of Lines ............................................................................................... 12
Computation of Uncertainty ..................................................................................................... 14
References ................................................................................................................................. 15
Format for Reports ........................................................................................................................ 16
Experiment #1: Calibration of a Venturi Meter ............................................................................ 17
Introduction ............................................................................................................................... 17
Apparatus .................................................................................................................................. 17
Procedure .................................................................................................................................. 18
For the Pre-Lab Report: ............................................................................................................ 18
For the Final Report: ................................................................................................................. 19
References ................................................................................................................................. 19
Experiment #2: Use of Thermal Conductivity to Identify an Unknown Material ........................ 20
Introduction ............................................................................................................................... 20
Apparatus .................................................................................................................................. 20
Procedure .................................................................................................................................. 21
For the Pre-Lab Report: ............................................................................................................ 22
For the Final Report: ................................................................................................................. 22
References ................................................................................................................................. 23
Experiment #3: Aerodynamics Studies ......................................................................................... 24
Introduction ............................................................................................................................... 24
Apparatus .................................................................................................................................. 24
Data Acquisition ....................................................................................................................... 26
Experiment 3a: Indirect Measurement of Drag ........................................................................ 27
Procedure .................................................................................................................................. 28
For the 3a Pre-Lab Report: ....................................................................................................... 29
Page i
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
For the 3a Final Report: ............................................................................................................ 29
References ................................................................................................................................. 29
Experiment #4: Unsteady Conduction of Solids........................................................................... 30
Introduction ............................................................................................................................... 30
Apparatus .................................................................................................................................. 30
Procedure .................................................................................................................................. 30
For the Pre-Lab Report: ............................................................................................................ 32
For the Final Report: ................................................................................................................. 32
References ................................................................................................................................. 32
Page ii
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Preface
This manual represents a change from what you may be used to in previous laboratory courses.
In an effort to focus on proper methods of experimentation and communication, this manual
reflects a “less is more,” or rather, a “quality over quantity” approach. By reducing the number of
experiments performed, the hope is that more focus will be put on the process of research,
experimentation, and analysis, rather than cranking out as many experiments and reports as
possible.
The student should be referring to the “Green Book” while generating reports and presentations
for this course… and if there is anything you are not sure about, just ask! Use the resources
available to you.
Page iii
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Thermal/Fluids Laboratory Introduction
Introduction
This laboratory course is the last in a sequence of three laboratory courses you will take as a
Mechanical Engineering major. The goal of this course is to teach you how to perform experiments
correctly, to interpret data, and to properly communicate your findings. In the process, you will:
1) Learn how to plan and design an experiment.
2) Use some of the instruments that are commonly used in the laboratory.
3) Learn how to acquire and analyze data.
4) Develop your skill in writing technical reports.
5) Develop your skill in making oral presentations of technical materials.
Consequently, this course will also enhance your understanding of the material you have
learned in MECH 302: Applied Thermodynamics, MECH 318: Fluid Mechanics I, MECH 319:
Fluid Mechanics II, and MECH 325: Heat Transfer, by experimentally verifying many of the
concepts you have learned in these courses. You will also learn how to design, perform, and present
the results of an experiment created by yourselves to investigate specific physical phenomena.
Laboratory Safety Rules
During the semester you will be operating equipment that may be dangerous if not operated
correctly. This includes equipment that generates high pressures, temperatures, velocities, and/or
voltages. It is essential for your safety, and the safety of others, that you understand and follow the
safety rules:
1) You must follow all safety procedures as described by your instructor, teaching assistant, or
laboratory technician.
2) Appropriate dress for the laboratory includes long pants and closed-toe shoes.
3) Wear any required personal protective equipment (PPE) during the experiment.
4) Make sure you know the location of safety equipment and room exits.
5) Remove jewelry, watches, ties, etc. that could become caught in machinery.
6) Long hair should be tied back.
7) All accidents and injuries should be reported to the laboratory instructor at once.
8) The campus emergency number is 7333.
9) Smoking, eating, and drinking in the laboratory are not permitted.
10) At all times behave in a professional and responsible manner.
11) All items (tools, PPE, etc.) should be returned to where you found them.
If a student fails to follow the above regulations and precautions, or otherwise makes himself
or herself a source of danger, he or she will be asked to leave the laboratory and be credited with
an absence. Due to the nature of the experiments, carelessness can lead to the harm of the careless
individual, his or her classmates, as well as the experimental equipment.
Page 1
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Preparation for Laboratory Classes
The experiments are performed in groups, normally with two to three students per group. All
students are expected to be thoroughly prepared for the experiment before the class begins by
reading the manual for the experiment, consulting any reference material, and generating a “prelab report.” This is to ensure that you understand the procedure so that you can perform the
experiment properly and safely, and so that the experiment can be completed in the designated
laboratory period. For each experiment, a Principal Investigator (PI) should be selected. The PI
will be primarily responsible for understanding how the experiment is to be conducted, and how
the data are to be reduced. If, after reading the pre-lab report (see below), the instructor has
questions on the execution of a portion of the experiment or the data analysis, the PI should be
able to explain. Of course, the entire group is responsible for the work, and will share the grade (to
a degree, see below).
Attendance Policy
Since the experiments are all performed in groups, the absence of anybody from the group will
increase the work required by the other members of the group. An absence without sufficient cause
will therefore give the offending student a zero grade for that particular experiment. If the
laboratory instructor considers the student to have sufficient justification for missing the
laboratory, the experiment may be made up by the student at a time assigned by the instructor.
Grading Policy
Since one report is submitted per group, it is incumbent upon each group to work together in
the generation of their reports. Each member of the group will indicate which portions of the report
were his or her responsibility. If the contributions of one member are found to be lacking (or
superior) compared to those of the others, the instructor, at his discretion, may apply a penalty (or
bonus) to that member’s grade.
Laboratory Notebook
Each laboratory group will be responsible for maintaining a laboratory notebook. This notebook
will document their research for each project and will contain the pre-lab reports (see below) for
each experiment, including the raw data. This notebook should contain two sets of interleaved
pages, which allow for carbon copies of each page to be automatically generated. One set of pages
is white, which is permanently bound into the notebook. The other set is yellow, and is perforated
for removal. Copies are made either using carbon paper, or pressure-sensitive carbonless copies.
Two types sold in the Manhattan College Bookstore are made by Roaring Spring; item #77649
(100 sets of pages) and item #77645 (50 sets of pages) are acceptable. However, any notebook that
contains interleaved pages that permit copies to be made is acceptable.
The first page of the notebook should be a table of contents, which indicates the first page of
each new report in the notebook. The notebook should be written in using an indelible pen with
blue or black ink, and using the carbon paper.
The notebook must be brought to every session in which experiments are being conducted. The
notebooks should contain EVERYTHING you need in order to conduct the experiment, and all
data you collect manually should be in the notebook. If you collect data electronically, the
notebook should indicate this, and the names of the file(s) containing the data.
Page 2
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Laboratory Experiment, Presentation, and Report Schedule
Four experiments will be performed this semester. Each experiment will take three weeks to
complete. This section outlines what is expected of the group in during an experimental cycle.
Preparation
In the first week, the groups will be prepped on the objective of the experiment. The groups
will then generate a “pre-lab report” written in the lab notebook and be submitted for review at the
beginning of the second week’s laboratory session. The pre-lab should have a cover sheet and
consist of an introduction, description of the experimental apparatus and procedure, and data
sheets. The purpose of the pre-lab is to ensure that the group understands the experiment they are
to conduct. When someone reads the pre-lab and inspects the data sheets, he/she should be able to
understand what the objective is, what raw data are going to be collected, how they will be
collected, and exactly how the data are to be processed. Outside sources used in the preparation of
the pre-lab (typically essential for the analysis portion) must be referenced. If the report is
adequate, the group will be allowed to conduct the experiment.
Please note that groups have been sent away to redo a pre-lab if it is considered deficient, and
this will reduce the time available to work in the laboratory on an experiment.
Performing the Experiment
Each group will conduct the experiment once the instructor is sure that the group understands
the objective of the experiment. The experiment should be conducted carefully – make sure all
data necessary are collected. Data should be recorded on the data sheet(s) included with the prelab report in blue or black pen. The only exception to this rule is Experiment #4, since the data
will be collected automatically by computer. However, as mentioned above, information on the
data files should be included here. If mistakes are made, they should be marked with a single
strikethrough.
Experimental Oral Presentation
Each group will perform an oral presentation in the third week. Presentations will be critiqued,
both on technical content, and on style. Comments should be considered during the preparation of
the report (see below). It is a good idea to take notes on the comments made, so that nothing is
forgotten.
Experiment Reports
A report is to be written on each experiment. The group as a whole is responsible for the reports.
Each report must be submitted at the beginning of the class period the week following the
presentation (not at some point during the class, otherwise it will be treated as having been handed
in a day late!). The pre-lab should be attached at the end of the report. The format for Reports can
be found in the Green Book, which describes the formats for written materials generated for classes
taught by Mechanical Engineering instructors.
Sample Results
Samples results are provided for each experiment to give each group a general idea of the
outcome of each experiment. However, do not worry if your results do not exactly match the
sample results. Some of the experiments have been purposely modified to produce different
Page 3
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
values. For the presentation and report, compare your results to the sample results and properly
cite the provided authors of the sample results.
Design of Experiments Project
A semester-long independent study project will be conducted by each group. This project will
involve designing the experiment from a statistical perspective, selecting appropriate
instrumentation, designing and building any hardware, performing the experiment, analyzing the
data, and presenting the results in written and oral form.
Design of Experiments Project Report
The group as a whole is responsible for the final report associated with the semester long
experiment assigned to them by the instructor. This report must be submitted on the day of the oral
presentation. The format for the Final Experimental Report can be found in the manual (the Green
Book) describing the formats for written materials generated for classes taught by Mechanical
Engineering instructors.
Design of Experiments Oral Presentation
Each group will make an oral report of their Experimental Project report at the end of the
semester. Questioning following the presentation will be based on the presented material. The
format for the Oral Presentation can be found in the manual (the Green Book) describing the
formats for written materials generated for classes taught by Mechanical Engineering instructors.
Recommended References
Antony, J. Design of Experiments for Engineers and Scientists, Elsevier, 2003
Hugh W. ColemanW. Glenn Steele Jr., Experimentation, Validation, and Uncertainty Analysis for
Engineers, 3rd Edition, Wiley, Hoboken, NJ, 2009
Çengel, Y. A. and Boles, M. A., Thermodynamics: An Engineering Approach, 8th Edition,
McGraw-Hill, New York, 2014.
Doebelin, E. O. Measurements Systems: Application and Design, 5th Edition, McGraw-Hill, New
York, 2004.
Holman, J. P., Experimental Methods for Engineers, 7th Edition, McGraw-Hill, New York, 2001.
Bergman, T. L., Lavine, A. S., Incropera, F. P., and DeWitt, D. P., Introduction to Heat Transfer,
6th Edition, Wiley, Hoboken, NJ, 2011.
Pritchard, P. J. and Mitchell, J. W., Fox and McDonald’s Introduction to Fluid Mechanics, 9th
Edition, Wiley. Hoboken, NJ, 2015.
Wilson, E. B., An Introduction to Scientific Research, Dover, Mineola, NY, 1990.
Page 4
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Uncertainty Analysis
Introduction
Any experiment report should always include an uncertainty analysis. This is a technique by
which the experimenter can:
1) estimate how accurate, and therefore how meaningful, the results of the experiment
are, and
2) determine which measurement or instrument in an experiment leads to the greatest
error in the final result.
Let us first demonstrate how the method works by considering an example of the drag on an object
in a flow, as shown in Fig. 1 below.
V, ρ
F
D
Measured: V, D, F
Reference: ρ
Fig. 1: Typical Measurements
Suppose an experiment is performed to determine the drag coefficient CD of the cylinder of
diameter D in a fluid of density ρ and velocity V. The definition of drag coefficient is
𝐹
𝐶𝐷 =
(1)
1 2
2 𝜌𝑉 𝐴
where A is the object’s frontal area, in this case given by
𝜋𝐷2
(2)
4
The factors F, V and D are measured, and  is looked up or computed from a reference source.
The measured values all have errors, or uncertainties, and the reference data may or may not be
given with an uncertainty. These uncertainties give rise to an error or uncertainty in the computed
value of CD. It is necessary to determine this uncertainty.
𝐴=
Using the terminology of Pritchard,1 CD is the quantity to be computed, and δCD is the
uncertainty we wish to find. The relative uncertainty is then defined as
𝑢𝐶𝐷 ≡ ±
𝛿𝐶𝐷
𝐶𝐷
(3)
A typical result might then be
𝐶𝐷 = 2.5,
𝛿𝐶𝐷 = 0.2,
⇒
𝑢𝐶𝐷 = ±
Page 5
0.2
= ±0.08 = ±8%
2.5
(4)
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
This final result depends on the quality of the measurements of the quantities F, V, and D. In
this example, an intermediate quantity, A, is also computed, which has an uncertainty that depends
on the uncertainty of D. How do we compute the uncertainty in A from that of D? How do we find
the uncertainty in measured quantity D? This involves the technique of Uncertainty Analysis. This
technique is discussed in detail in Pritchard,1 as well as Holman.2 In the following discussion the
notation of Pritchard is used.
Uncertainties in Measured Data
In the example above, F, D and V were measured. Any measurement involves some error, or
uncertainty. This uncertainty depends on the measurement instrument used, as well as the size
(nominal) of the quantity. For example, using a ruler with 0.1" markings to measure the diameter
of the cylinder would have a larger uncertainty than using a micrometer which is marked off in
steps of 0.001". Using the ruler to measure the length of something about 1' long would have a
larger relative (or percentage) uncertainty than when it is used to measure something about 10'
long, even though the absolute uncertainties would be the same. Suppose in this example that the
ruler shown below (Fig. 2) was use to obtain the cylinder diameter.
Smallest Reading: 0.1″
0
1
2
3
4
5
6
Fig. 2: Ruler Markings
The basic concept is: that the uncertainty in any measurement is approximately equal to one
half of the smallest reading of the instrument. The logic behind this statement is that, for example,
the above ruler could with confidence be used to distinguish between objects that are 1", 1.05",
and 1.1", but any further distinction (say between 1.02" and 1.03") would be inaccurate, and hence
unreasonable.
Suppose that the following data for the cylinder were obtained, using the ruler from Fig. 2.
𝐷=4
δD = 0.05
𝛿𝐷
𝑢𝐷 ≡
= 0.013 = 1.3%
𝐷
𝐷 ± 𝛿𝐷 = 4.00 ± 0.05"
(5)
This result means that in all probability, the actual diameter is between 3.95" and 4.05". The
relative uncertainty is thus 1.3%, which is reasonably good. This indicates that for this experiment,
using a ruler with 0.1" gradations is probably accurate enough.
Using a micrometer that could measure to 0.001" would lead to a relative uncertainty in D of
0.013%, which is probably much more accurate than necessary. Using a tape measure that could
only measure to 0.25" would lead to a relative uncertainty in D of 3.1%, which is probably less
Page 6
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
accurate than necessary. These calculations illustrate how the uncertainty analysis can be used as
a guide in selecting the most appropriate measuring instrument.
Note that the value for the diameter is given as 4.00 0.05", not 4 0.05" or 4.000 0.050". This
brings us to two important rules for stating uncertainties and the values to which they belong:3
1) Experimental uncertainties should be rounded to one significant figure, unless the
leading digit in the uncertainty is a 1. In that case the uncertainty should be rounded
to two significant figures.
2) The stated value have the same precision as the uncertainty.
While the above rules should be used when citing results, calculations should use at least one
additional significant figure (for both uncertainties and values), in order to mitigate roundoff
errors.
Also remember that if you are using scientific notation, (6.02±0.06)10–6 is much clearer than
6.0210–6 ± 610–8, although the latter is certainly accurate. We’re going for clarity here!
For this example, assume that some kind of velocity meter was used to measure V, with the
following result:
𝑉 = 60 𝑚𝑝ℎ
𝛿𝑉 = 1 𝑚𝑝ℎ
𝛿𝑉
𝑢𝑉 ≡
= 0.017 = 1.7%
𝑉
𝑉 ± 𝛿𝑉 = 60 ± 1 mph
(6)
and that the drag force measured data were
𝐹 = 2 lbf
𝛿𝐹 = 0.05 lbf
𝛿𝐹
𝑢𝐹 ≡
= 0.025 = 2.5%
𝐹
𝐹 ± 𝛿𝐹 = 2.00 ± 0.05 lbf
(7)
This completes the compilation of measured data for this example.
Uncertainties in Reference Data
Some data, for example, fluid densities, are often not measured but obtained from a reference.
If these reference data are tabulated, both the nominal value and the uncertainty involved are often
provided. Other data may be available from a graph, in which case the uncertainty in reading the
graph must be estimated. Details of this procedure are given in Pritchard.1
In certain cases needed reference data are not available directly, but must be derived from other
data. An example of this is the density of air at nonstandard conditions (for example, air at
atmospheric pressure but 200°F). In this case the ideal gas equation would be used
𝑝
𝜌=
(8)
𝑅𝑇
and uncertainties in p and T would lead to the uncertainty in ρ. The density in this case would be
handled as if it was a computed result, as described below.
Page 7
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Uncertainties in Computed Data
To illustrate how uncertainties in computed data are obtained, consider as an example the
computation of the cylinder frontal area. Using nominal data for D
𝜋𝐷2
= 12.57 in2
4
(Note that it is not good practice to present more than a few significant figures).
𝐴=
(9)
The uncertainty in A is obtained from performing some simple calculus. First, A is a function of D
only
𝐴 = 𝐴(𝐷)
Hence, a change D in D would cause a change A in A in the following way
(10)
𝜕𝐴
(11)
𝛿𝐷
𝜕𝐷
Dividing both sides by the original A, and multiplying and dividing the right hand side by D leads
to
𝛿𝐴 =
𝛿𝐴 1 𝜕𝐴
𝐷 𝜕𝐴 𝛿𝐷
(12)
=
𝛿𝐷 =
𝐴
𝐴 𝜕𝐷
𝐴 𝜕𝐷 𝐷
The left hand side of equation (12) is the relative uncertainty uA, and the last term on the right hand
side is the relative uncertainty uD. This algebra thus leads to a usable result
𝐷 𝜕𝐴
(13)
𝑢
𝐴 𝜕𝐷 𝐷
This expression enables the effect of the uncertainty in the measured diameter D on the computed
uncertainty of area A to be obtained. For the given data
𝑢𝐴 =
𝐷 𝜕𝐴
𝐷
𝐷
(2𝜋
) 𝑢 = 2𝑢𝐷 = 2 × 1.3% = 2.6%
(14)
𝑢𝐷 =
𝐴 𝜕𝐷
𝜋𝐷2 /4
4 𝐷
Examination of this result shows that the relative uncertainty in area A is twice the uncertainty
in diameter D! This is because the area is proportional to the square of the diameter. This will
always be the pattern, so, for example, if a computed result depended on the cube of a measured
quantity, the relative uncertainty of the computed result would be three times that of the measured
quantity. This quality of these calculations means that final results can have a much larger
uncertainty than measured data. Note that this pattern will not be present when other types of
functions are used, e.g. trigonometric, exponential, or logarithmic functions. However, the
technique described here will still be valid.
𝑢𝐴 =
The above example was for a quantity (A) depending on one other (D). Consider now a more
general formulation in which a result R which depends on several measured quantities x1, x2, x3,...
Performing the chain rule derivative, we can find how the uncertainty R depends on the
uncertainties x1, x2, x3,...
𝛿𝑅 1 𝜕𝑅
1 𝜕𝑅
1 𝜕𝑅
(15)
=
𝛿𝑥1 +
𝛿𝑥2 +
𝛿𝑥 + ⋯
𝑅
𝑅 𝜕𝑥1
𝑅 𝜕𝑥2
𝑅 𝜕𝑥3 3
In this equation we divided throughout by R after performing the chain rule. If we multiply and
divide the first term on the right hand side by x1, the second term on the right hand side by x2, and
so on, we finally obtain
Page 8
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
𝑥1 𝜕𝑅
𝑥2 𝜕𝑅
𝑥3 𝜕𝑅
(16)
𝑢𝑥1 +
𝑢 𝑥2 +
𝑢 +⋯
𝑅 𝜕𝑥1
𝑅 𝜕𝑥2
𝑅 𝜕𝑥3 𝑥3
This complicated looking expression enables the relative uncertainty uR in the computed result
to be obtained from the measured relative uncertainties ux1, ux2, ux3 ...
𝑢𝑅 =
This formula is, however, too conservative in the sense that it implies that the uncertainty in the
final result will be due to adding together (in a weighted sense) the uncertainties in all the
measurements. In reality, uncertainties often cancel one another.
For example, the area of a rectangle is given by the product of its two sides. The uncertainty in
the final area will not be too bad, because there is a good chance that one side might be measured
with a positive uncertainty (it might be actually 6", but be read as 6.05"), and another with a
negative uncertainty (it might be actually 4", but read as 3.95"). The computed area would then be
23.90 in2, an accurate result. Uncertainties are just as likely to cancel one another, as in this
example, as they are to accumulate. The above formula always assumes that uncertainties
accumulate, so it is too conservative.
However, it turns out that if, instead of the arithmetic sum in the above equation, we take the
root of the sum of the squares (the RMS), the final uncertainty will be statistically more meaningful
(THIS IS THE ANALYSIS YOU ARE REQUIRED TO USE). Hence the final uncertainty
formula is
𝑢𝑅
2
2
2
𝑥1 𝜕𝑅
𝑥2 𝜕𝑅
𝑥3 𝜕𝑅
√
= (
𝑢 ) +(
𝑢 ) +(
𝑢 ) +⋯
𝑅 𝜕𝑥1 𝑥1
𝑅 𝜕𝑥2 𝑥2
𝑅 𝜕𝑥3 𝑥3
(17)
and if R is merely the addition or subtraction of variables (i.e. x1 + x2 + x3 +….) then this formula
simplifies to
(18)
𝛿𝑅 = √(𝛿𝑥1 )2 + (𝛿𝑥2 )2 + (𝛿𝑥3 )2 + …
These are complicated looking formulae, but they are fairly easy to compute in practice. This is
illustrated by returning to the drag coefficient problem. As you recall
𝐶𝐷 ≡
𝐹
1 2
2 𝜌𝑉 𝐴
(19)
and the measured and computed data are
𝐹 = 2 𝑙𝑏𝑓
𝑢𝐹 = 0.025 = 2.5%
𝜌 = 0.0024 slug⁄ft 3
𝑢𝜌 = 0.0 = 0.0%
𝑉 = 60 mph
𝑢𝑉 = 0.017 = 1.7%
𝐴 = 12.57 in2
𝑢𝐴 = 0.026 = 2.6%
(20)
In this example, we assume that the density is standard atmosphere, from Table A.3 of
Pritchard,1 with no uncertainty. Note that if you were to read data from a chart, you would be
Page 9
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
expected to indicate an uncertainty, based on the information used to acquire a result from the
chart, e.g. uncertainty in temperature to determine the value and uncertainty of the viscosity of a
liquid.
From this data the nominal value of the drag coefficient is
𝐶𝐷 = 2.5
The uncertainty equation (Eqn. 17), rewritten for this example, is
(21)
2
2
2
2
𝐹 𝜕𝐶𝐷
𝜌 𝜕𝐶𝐷
𝑉 𝜕𝐶𝐷
𝐴 𝜕𝐶𝐷
𝑢𝐶𝐷 = √(
𝑢𝐹 ) + (
𝑢𝜌 ) + (
𝑢𝑉 ) + (
𝑢𝐴 )
𝐶𝐷 𝜕𝐹
𝐶𝐷 𝜕𝜌
𝐶𝐷 𝜕𝑉
𝐶𝐷 𝜕𝐴
(22)
This becomes
𝑢𝐶𝐷
2
2
2
𝐹 𝑢𝐹
𝜌 −𝐹𝑢𝜌
𝑉 −2𝐹𝑢𝑉
𝐴 −𝐹𝑢𝐴
) +(
) +(
) +(
)
= √(
1
1
1
𝐶𝐷 𝜌𝑉 2 𝐴
𝐶𝐷 𝜌2 𝑉 2 𝐴
𝐶𝐷 𝜌𝑉 3 𝐴
𝐶𝐷 1 𝜌𝑉 2 𝐴2
2
2
2
2
which in turn simplifies to
𝑢𝐶𝐷 = √(1 × 𝑢𝐹 )2 + (−1 × 𝑢𝜌 )2 + (−2 × 𝑢𝑉 )2 + (−1 × 𝑢𝐴 )2
2
(23)
(24)
Note that the squaring of terms eliminates the signs.
Two things should be noted here. First, the coefficients in front of the uncertainties almost
always simplify to a number. For a linear relation (in this case for CD  F, CD  ρ–1, etc.) the
coefficient will be 1. For a quadratic relation (CD  V–2) the coefficient will be 2, and so on. Second,
the coefficients indicate how strongly a measured uncertainty affects the final result. In this
example, the uncertainty in velocity measurement is doubled, suggesting that, for a well designed
experiment, velocity, as opposed to diameter and force, should be measured carefully. This alone
makes an uncertainty analysis a useful tool.
Using the data of Equation (20):
(25)
𝑢𝐶𝐷 = √0.000625 + 0 + 0.00116 + 0.00068 = 0.049 = 4.9%
This is the final answer for the drag coefficient uncertainty. Note that in this example the third
term in the root, due to the velocity measurement, is most significant. This is yet another reason
the uncertainty analysis is so useful. For a given experiment, it indicates the “weakest link” in the
experiment. In this example, the major source of uncertainty or error is in the velocity
measurement. Replacing, say, the rule with an accurate micrometer for measuring the cylinder
diameter (and hence area) would be a waste of time, but improving the velocity measurement
device would be a good idea. The final result for the drag coefficient is then
(26)
𝐶 𝐷 = 2.5 ± 0.1𝑢𝐶𝐷 = 4.9%
The drag coefficient for this experiment is thus somewhere between 2.4 and 2.6. Note that the
result is rounded to two significant figures. The procedure for performing an uncertainty analysis
is summarized in the following section.
Page 10
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Plotting Data with Uncertainties
If data have uncertainties, they must either be plotted with error bars, or the error should be
indicated somewhere else in a prominent manner (i.e., in the text describing the figure, or in the
figure caption). The second method is typically used when the error bars would be too small to be
seen.
In some cases theoretical data are developed from some experimental inputs. For example, the
predicted drag on a body is calculated from the velocity, which is a measured quantity. In that case,
the theoretical data should consist of two curves: a lower bound and an upper bound based upon
the uncertainty in the theoretical value.
Uncertainties in a Mean Value
Often, a “single” experimental value consists of several individual measurements of the same
value. This combining is often done to insure that random errors, which could severely skew an
experimental value, are minimized. When we take several experimental values, e.g., x1, x2, x3,…,
xN of a single physical quantity, we estimate the value as the average of the individual
measurements:
𝑁
1
𝑥̄ = ∑ 𝑥𝑖
𝑁
(27)
𝑖=1
Now there are two sources of uncertainty in such a measurement. The first source is the uncertainty
in individual measurements, e.g., δx1, δx2, δx3,…, and the second is due to the “scatter” of the
individual values, i.e., how close the values are grouped about the mean. If the individual
measurements have the same uncertainty, we can replace all of the δxi with δx. If they are not
equal, δx may be taken as either the mean of the individual uncertainties, or the maximum of the
uncertainties. The scatter in the values is best represented by the standard deviation in the data:
2
∑𝑁
𝑖=1(𝑥𝑖 − 𝑥̄ )
(28)
𝑁−1
However, the uncertainty in the mean is slightly different from the uncertainty in the data (we will
leave this until our discussion of statistics later in the semester):
𝜎𝑥
𝜎𝑥̄ =
(29)
√𝑁
To get the complete uncertainty in a mean value, we combine the two sources of uncertainty:
𝜎𝑥 = √
(30)
𝛿𝑥 = √(𝛿𝑥𝑖𝑛𝑠𝑡 )2 + (𝜎𝑥̄ )2
We will illustrate this idea through an example: Suppose that we have made multiple drag force
measurements, and that the measured values (in lbf) were:
1.90, 1.93, 1.94, 1.97, 2.00, 2.01, 2.03, 2.08, 2.15
All measurements have an uncertainty (δFinst) of 0.05 lbf. The mean value is equal to the sum of
the values divided by the number of measurements:
Page 11
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
1.90 + 1.93 + 1.94 + 1.97 + 2.00 + 2.01 + 2.03 + 2.08 + 2.15
lbf
9
(31)
F = 2.001 lbf
Note that Excel can find the mean of a set of data using the AVERAGE() function. The standard
deviation of the data can be calculated, either by using Equation (28), or by using the STDEV.S()
function in Excel*. The result is σF = 0.07849 lbf. To find the standard deviation of the mean, we
use Equation (29):
F=
𝜎𝐹̄ =
0.07849
lbf = 0.02616 lbf
(32)
√9
With the above result and the uncertainty in each measurement, we can now calculate the
uncertainty in the mean value:
F = 0.05 2 + 0.02616 2 lbf = 0.05643 lbf
F  F = 2.00  0.06 lbf
(33)
This example can be seen in the Excel workbook “Statistical Analysis,” posted on the class
website. The workbook can also be used as a boilerplate for your own analyses.
Uncertainties in Slopes of Lines
Often times we will need to perform some kind of regression analysis (sometimes called a bestfit line or a trendline). There are two main reasons for such an analysis:
1) We want to prove a linear relationship between physical quantities (check agreement
with some model)
2) The slope or intercept of the regression is (or is related to) some physical quantity of
interest, for example, a thermal conductivity or an elastic modulus.
The method for finding the regression line for a set of data is call least-squares analysis. Any
statistical analysis which generates a best-fit line for a set of data is performing such an analysis.
Without getting into the details at this point (we will later in the semester), we can just say at this
point that if we fit a set of (x, y) data to a straight line of the form
(34)
𝑦 = 𝐴 + 𝐵𝑥
the analysis selects A and B such that sum of the squares of the deviations between the y values of
the actual points and the y values of the line:
𝑁
𝑑 = ∑(𝑦𝑖 − 𝐴 − 𝐵𝑥𝑖 )2
(35)
𝑖−1
is at a minimum.
Now since the line does not match up exactly with the individual data, the scatter in the data
(deviations between the points and the line) is one source of error. The second source of error is
the uncertainties in the individual measurements. As for the uncertainty in a mean discussed above,
both sources of uncertainty must be accounted for when citing values for the slope and intercept
of the best-fit line. The treatment used here is that from Taylor.3
*
This function is available on Excel versions 2007 and 2010. Earlier versions use the STDEV() function. Take care
not to use the STDEV.P() function, which will give an incorrect result.
Page 12
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
If the only source of uncertainty in A and B were the scatter in the data, then the uncertainties
in A and B can be given as:
2
∑𝑁
𝑖=1 𝑥𝑖
𝛿𝐴 = 𝜎𝑦 √
2
𝑁
2
𝑁 ∑𝑁
𝑖=1 𝑥𝑖 − (∑𝑖=1 𝑥𝑖 )
(36)
𝑁
2
𝑁
2
𝑁 ∑𝑁
𝑖=1 𝑥𝑖 − (∑𝑖=1 𝑥𝑖 )
(37)
𝛿𝐵 = 𝜎𝑦 √
where N is the number of data points and σy is given by
2
∑𝑁
𝑖=1(𝑦𝑖 − 𝐴 − 𝐵𝑥𝑖 )
(38)
𝑁−2
Since there are uncertainties for the data point, we need to add to the expression in Equation (38).
We do that by rewriting Equation (38) as
𝜎𝑦 = √
2
∑𝑁
𝑖=1(𝑦𝑖 − 𝐴 − 𝐵𝑥𝑖 )
(39)
+ (𝛿𝑦)2 + (𝐵𝛿𝑥)2
𝑁−2
where δx and δy are the uncertainties in x and y. If the uncertainties are not the same for every
point, one can either use the largest uncertainty or the mean of the uncertainties.
𝜎𝑦 = √
Let us look at this using the example of a constant volume gas thermometer experiment: in a
gas thermometer, the temperature of the gas is a function of pressure:
(40)
𝑇 = 𝐴 + 𝐵𝑃
4
This relationship is a restatement of Charles’ law. Assume that a set of measurements were
collected from a gas thermometer:
Pressure (mm Hg)
Temperature (°C)
65 3
–205
75 3
175
85 3
425
95 3
945
105 3
1275
By plotting the data in Excel and using the trendline feature, one can see that the best fit line for
the data has the equation:
(41)
𝑇 = −263.35 + 3.7100𝑃
Equation (39) can then be applied to the data, resulting in σy = 13.9108. This result can then be
applied to Equations (36) and (37) to find the uncertainties in the intercept and slope of the line:
𝛿𝐴 = 37.9
𝐴 ± 𝛿𝐴 = −260 ± 40
𝛿𝐵 = 0.440
𝐵 ± 𝛿𝐵 = 3.7 ± 0.4
Page 13
(42)
(43)
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
It is interesting to note that the intercept (A) of Equation (40) should correspond to the value of
absolute zero. We see from Equation (42) that when the uncertainty is considered, the experimental
value of A is in agreement with the accepted value of absolute zero (–273.15°C). A plot of the data
and the regression line can be seen in Fig. 3 below.
200
150
100
Temperature (°C)
50
0
-50
-100
-150
-200
-250
-300
0
20
40
60
80
100
120
Pressure (mmHg)
Fig. 3: Plot of Temperature versus Pressure data (diamonds) with uncertainties, showing
regression and the uncertainty in intercept of regression (square). Triangle shows accepted value
of absolute zero for comparison.
This example can also be seen in the Excel workbook “Statistical Analysis,” posted on the class
website. The workbook can be used as a boilerplate for your own analyses.
Computation of Uncertainty
1) Record the data, including the nominal values and the absolute uncertainties.
2) Use the formulas of the experiment to compute the nominal results.
3) Determine the relative uncertainties of all the measured data.
Page 14
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
4) Compute the relative uncertainties of all the computed data using the summation equation.
This is the longest step, and may have to be done in stages (for example, the uncertainty in A
was computed before that of CD in the drag coefficient problem).
5) Discuss the consequences of the analysis (e.g. which instrument lead to the largest
uncertainty in the final result etc).
In your reports you must include an uncertainty analysis. Do not do this for all data, but for one
representative set of data points, namely, those data points for which you did your sample
calculations. It is highly recommended that you use Mathcad for this part of your reports.
References
Pritchard, P. J., Fox and McDonald’s Introduction to Fluid Mechanics, 8th ed., Wiley.
Hoboken, NJ, 2011, Appendix F.
2
Holman, J. P., Experimental Methods for Engineers, 7th ed., McGraw-Hill, New York, 2001,
pp. 63-77.
3
Taylor, J. R., An Introduction to Error Analysis, 2nd ed., University Science Books, Sausalito,
CA, 1997, Chap. 8.
4
Chang, R., Chemistry, 3rd ed., Random House, New York, 1988, Chap. 5.
1
Page 15
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Format for Reports
It is common for different companies, professional societies, editorial boards, etc., to have
different formats for writing technical reports, memoranda, papers, and articles. You probably
have had different teachers over your high school career require different formats for their reports,
and your professors here have no doubt done the same thing. This class will be no different. The
required format for MECH 405 reports is consistent with the format outlined in the Mechanical
Engineering Department’s Guide to Technical Communication, a.k.a. the “Green Book.”
Keep in mind that the format of the report, including cover page layout and format for
references, will be inspected upon receipt; any mistakes will result in the report being returned
without grading for resubmission, with late penalties accruing. This approach may seem extreme,
but this happens in real life, so we want you to be prepared.
The most important thing to remember when writing or presenting is that you are trying to
communicate what you did, how you did it, and what you learned. Therefore, anything you include
in a report or a presentation should be to that end.
Page 16
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Experiment #1: Calibration of a Venturi Meter
Introduction
In this experiment you will calibrate a venturi meter, as shown in Fig. 1.
1
2
Q
A1
A2
Fig. 1: The venturi meter, with flow moving from left to right.
The purpose of a calibration study is to check the output of some measurement device against
that of a known standard, so that the response of the measuring device can be converted into the
desired data. Calibrations are very important to maintain the integrity of experimental data.
Recall from MECH 318: Fluid Mechanics I, as well as from Pritchard,1 that a venturi is a device
used to measure incompressible fluid flow rates in pipes. If the area of flow is reduced in a pipe,
the velocity increases (due to mass conservation) and the pressure decreases (due to the Bernoulli
principle). In the venturi meter the area is reduced gradually to a throat and then it is gradually
increased to the original area of the pipe. The advantage of the venturi meter over similar meters,
such as the orifice meter, is that the loss of total pressure is minimum (i.e. the loss of mechanical
energy due to the presence of the constriction is relatively small).
The volume flow rate through a venturi meter can be calculated, from the measured pressure
drop and ratio of areas, using the conservations of mass, momentum, and energy. You will calibrate
the meter by comparing this theoretical flow rate to the actual measured flow rate.
Apparatus
The venturi meter used in this experiment has a 10° converging section and a 5° diverging
section. The meter has five pressure taps as shown in Fig. 2, each connected to a manometer used
to measure the local pressure in the venturi meter.
The diameters at the points associated with location of five manometers A, B, C, D, and E and
their relative location are provided in Table 1 below. This meter is specifically designed for
laboratory demonstration. Only points A and C (points 1 and 2) are used in calculating flow. The
remainder are provided to illustrate if the manometer heights can be predicted using the area
changes and Bernoulli’s equation.
Page 17
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
E
D
C
B
A
Fig. 2: Photograph of the venturi meter showing the manometer tap locations. Flow is from right
to left. Tap A is at the entrance, tap C is at the throat, tap E is at the exit.
Procedure
1) Examine the equipment. The meter is mounted on a table and connected to a pump inside a
large tank filled with water.
2) Turn on the pump and adjust the valves to control the flow rate. Two valves, one upstream
and one downstream of the venturi meter, are needed to control both the flow rate and pressure
within the meter. Adjust both valves until the liquid levels in all manometers can be read using
the attached scales.
3) Measure the actual flow rate through the meter by collecting the water exiting the meter in a
bucket and measuring the accumulated its weight in a set time period.
4) While the water is collected in the bucket, record the liquid level in manometers A through E.
5) Decrease or increase the flow rate and repeat steps 2) through 4) for an additional four flow
rates. Attempt to measure five different flow rates.
For the Pre-Lab Report:
1) Do not treat this calibration as an academic exercise; assume that the calibration is being
performed for a customer that will be using this meter in their test facility.
Page 18
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
2) In the Introduction, provide a description of the venturi meter, how it works, and how the
measured data (e.g. pressures) and the physical setup (e.g. diameters at the entrance and throat)
are used to determine flow rate. Provide an equation which relates the flow rate to all of the
necessary measured data, and show the derivation of the equation. Include figures and
equations wherever necessary.
3) In the Experimental section, describe the apparatus you will use, and how you will perform the
experiment. DO NOT PROVIDE A NUMBERED SET OF INSTRUCTIONS! Rather, give
your experimental procedure as a narrative.
4) Create a datasheet to record all of the relevant manometer heights, water mass, and times. In
your data sheets, include sufficient space to perform the calibration for at least five flow rates.
For the Final Report:
In addition to the material included in the pre-lab report, the following material should be
provided in the final report:
1) Discuss the data collected. Describe why the data look the way they do. Do the data match
your expectations? Do the results of your calibration depend upon flow rate? Compare your
results to those provided as sample results – be sure to cite them!
2) Include the raw data, sample calculations, and error analyses as appendices.
Table 1: Locations of Pressure Taps and Diameters on Venturi Meter
Tap
A
Location (mm)
0
Diameter (mm)
12.7
B
C
D
E
12.6
27.1
48.0
67.0
10.5
8.3
10.5
12.7
References
Pritchard, P. J., Fox and McDonald’s Introduction to Fluid Mechanics, 8th Edition, Wiley.
Hoboken, NJ, 2011.
1
Page 19
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Experiment #2: Use of Thermal Conductivity to Identify an
Unknown Material
Introduction
In this experiment you will investigate the phenomenon of steady-state heat conduction.
Specifically, you will experimentally determine the unknown thermal conductivity (k) of a metal
cylinder by heating one end and cooling the other end using two methods, compare the results
using the methods, and determine the material.
Electric
Heater
2 in spacing between
each Thermocouple
Rod Diameter, 1 in
Unknown Length 12 in
Reference Length 2 in
0.5 in spacing between each
Thermocouple
Reference Conductivity
kr = 14.4 BTU/ft∙hr∙°F
Water In
Cooling
Plate
Water Out
Fig. 1: Schematic of experimental apparatus
Apparatus
A schematic of the experimental setup is shown in Fig. 1. The unknown cylinder is heated at
the top via an electric heater and is connected at the bottom to a cylinder of known thermal
conductivity. The assembly is mounted on a water-cooled plate. The temperature of the unknown
cylinder is measured at five points along its length by thermocouples spaced two inches apart. The
temperature of the reference cylinder is measured at three points by thermocouples spaced 0.5
inches apart. Thermocouples are also used to measure the temperature in the room, as well as the
water at the cooling plate entrance and exit. A rotameter is used to measure the volumetric flow
rate of the water.
In a typical heat transfer problem, the three modes of heat transfer - conduction, convection,
and radiation - are usually present. In this experiment, however, convection is suppressed by
running the experiment in a vacuum, and heat loss due to radiation is made negligible by
conducting the experiment at relatively low temperatures (the maximum temperature is about
Page 20
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
250°F) and shielding the cylinders with aluminum foil. Therefore, the heat flow through the
cylindrical test column should be very close to one-dimensional from the heat source (the heater)
to the heat sink (the plate).1 A photograph of the apparatus including the vacuum tube and the
aluminum foil can be seen in Fig. 2.
Fig. 2: Photograph of experimental apparatus, showing vacuum tube and radiation shielding
Procedure
SAFETY PRECAUTIONS: Since the glass jar is subjected to a pressure difference, there is the
possibility of the jar breaking. Therefore, eye protection should be worn when working with this
apparatus.
Please note that due to the time necessary to reach steady state, the experiment will already be
running by the time you reach the laboratory. However, it is important to know and describe the
procedure in order to document what was done.
1) Make sure a bucket or some other receptacle is properly placed to catch the cooling water
leaving the plate. During the course of the experiment, this bucket should be checked to make
sure it does not overflow.
2) Switch on the roughing pump to establish a vacuum in the bell jar containing the test column.
3) Close the needle valve on the rotameter, and turn on the water hand valve. Slowly open the
needle valve controlling the flow and set the valve so that the float on the rotameter is near 95
on the scale. This reading corresponds to about 50 cc/min.
Page 21
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
4) Over the course of the startup, make sure that air bubbles in the system do not catch the float
in the rotameter and carry it to the top of the tube. If this happens, tap the rotameter to allow
the bubbles to get past the float and let the float settle back down.
5) Turn on the Variac controlling the electric heater and set it to 45. (CAUTION: DO NOT
EXCEED THIS VALUE OR YOU WILL BURN OUT THE SYSTEM.) Turn on the power to the
electric heater.
6) Turn on the computer and Data Acquisition System. When the computer finishes booting,
launch the file “transient” on the desktop. When the file is loaded, click on the “play” button
to begin telemetry of the system temperatures.
7) Monitor the temperatures until they have reached steady state (stabilized). At steady state,
record all the temperatures. Be sure to record (manually) the cooling water rotameter reading
on the data sheet. A calibration sheet is mounted on the side of the experiment so that the float
height can be converted to a flow rate. A copy of the calibration data has been provided in
Table 1 below.
8) After collecting the data, switch off the heater and vacuum pump, and after five minutes turn
off the cooling water. Dispose of the water caught in the drain bucket.
For the Pre-Lab Report:
1) Do not treat this test as an academic exercise; assume that the test was performed for a client
who wished to determine the conductive heat transfer coefficient for a new metal alloy.
2) In the Introduction, describe conductive heat transfer, and explain how this setup would be
capable of investigating conductive heat transfer decoupled from the other modes (convection
and radiation). Explain what equations would be used to relate heat transfer rate to change in
temperature for conductive heat transfer. Be sure to cite your sources! Be sure to provide
diagrams when introducing equations, in order to help with context.
3) In the Experimental section, describe how the data collected will be used to determine the
thermal conductivity of the unknown material. Remember that you are devising two methods,
so ultimately you should give two equations here which would relate the thermal conductivity
to the data collected. Describe the method of how you will perform the experiment. DO NOT
PROVIDE A NUMBERED SET OF INSTRUCTIONS! Rather, give your experimental
procedure as a narrative.
4) Create a datasheet to record all of the relevant temperatures and the rotameter height. Note that
these data are given in Table 1.
For the Final Report:
In addition to the material included in the pre-lab report, the following material should be
provided in the final report:
1) Discuss your results. Did the temperature gradients in both cylinders suggest pure onedimensional conduction? If not, why? Do the two thermal conductivity values (obtained from
the two methods) agree? If not, why? Compare the measured values of k to published thermal
conductivity values. From this, identify a metal that is most like the material of the unknown
cylinder in terms of heat conduction. Compare your results to those in the sample report – be
sure to cite them!
Page 22
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
2) Inherent in this experiment is the assumption that convection and radiation were negligible.
Could these effects influence the results? Compute the actual heat transfer rate through the bar.
Now, assume the “worst” radiation heat loss possible: that is, the entire cylindrical test column
is at the highest temperature and the surface emissivity is  = 1. Based on these assumptions,
calculate the heat loss due to radiation qrad. Finally calculate the “worst” natural convection
heat loss possible: i.e., the entire cylinder is at the highest temperature and the air is at
atmospheric pressure. Based on these assumptions, calculate the heat loss due to convection
qconv. Compare q, qconv, and qrad, and discuss how legitimate it was to neglect convection and
radiation effects, based on these computed results and the data you collected.
3) Ensure that all of the raw data and all of the analyses appear in appendices. Perform a full error
analysis for these calculations to determine the errors associated with both values found for the
thermal conductivity value (k) of the unknown material. It can be taken that the rotameter has
an accuracy of ±2 cm3/min, all temperatures have an accuracy of ±0.1°C, and that all
dimensions are measured to within ±0.03 in. Also, assume that there is no error associated with
the values for the thermal conductivity for the known material (kr), and the heat capacity of
water (c).
Table 1: Calibration data for rotameter
Float Height
10
20
30
Flowrate (cc/min)
1.64
5.02
10.04
40
50
60
70
80
90
100
110
120
130
15.90
22.04
28.26
35.07
41.49
46.77
53.27
60.08
66.40
72.18
140
150
78.29
85.39
References
1
Bergman, T. L., Lavine, A. S., DeWitt, D. P. and Incropera, F. P., Fundamentals of Heat and
Mass Transfer, 6th Edition, Wiley, Hoboken, NJ, 2011, Chapter 1.
Page 23
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Experiment #3: Aerodynamics Studies
Introduction
In this experiment you will perform an aerodynamic experiment, outlined later in this section.
You will be using a wind tunnel to measure drag and pressures, and convert pitot-static probe data
to flow velocity.
Apparatus
The test articles will be tested in a wind tunnel, shown in Fig. 1. The tunnel is capable of speeds
up to approximately 35 m/s (80 mph) via a variable power fan. The fan power and speed are
controlled via a pair of buttons and a potentiometer on the lower right corner of the control panel.
The test section of the tunnel has a 306 mm by 306 mm cross-section. In this photo, the air travels
from left to right.
Fig. 1: Wind tunnel.
The instrumentation panels for the tunnel are shown in Fig. 2. The tunnel is equipped with two
differential pressure transducers, a 32-channel gage pressure transducer and a three-component
(lift, drag, pitch) force balance. The pressure transducer is capable of measuring gage pressures
(referenced to the ambient pressure in the laboratory). The digital display on the transducer
displays four channels (pressures) at a time. To scroll through the pressures available use the rocker
switch to the right of the display.
The force balance (Fig. 3) is mounted on the side of the tunnel, and the support for the article
is clamped into the balance. The test article is mounted in a hand-tightened collar on the middle of
the balance. The balance is designed to that the hole in the door of the test section is aligned with
the hole in the collar mount. The collar mount is equipped with a protractor so that the orientation
of the test article can be changed. To adjust the angle, loosen the hand screw to the lower right of
Page 24
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Fig. 2: Instrumentation Panels: VDAS, Load balance, 32-Channel pressure transducer,
Differential pressure transducers, main power and fan control.
the collar mount. This allows the article to be rotated freely. To set the angle, retighten the hand
screw. The angle of the protractor is measured using an angle feedback unit, which sends this
information to the VDAS board (see below). In order to protect the load cells during installation
and setup, the force balance should be locked, except when the instrument panel is being zeroed,
or when a measurement is being taken. The balance is locked when the nuts on the two supporting
screws under the protractor are tightened. To loosen them, turn them counterclockwise.
It is important to note that the force balance reads positive lift when the force is applied
downwards. The orientation of the protractor is consistent with this convention, so when the
Fig. 3: Three-Component Force Balance and Angle Feedback Unit mounted on test section
Page 25
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
protractor and angle feedback unit display a positive angle of attack an airfoil will be pointing
leading-edge down.
The tunnel is equipped with two pitot-static probes (capable of measuring velocity)1 at the front
and back of the test section, along the centerline. The forward probe is mounted on a scale, so that
the location of the probes can be noted during testing. The height of the probe can be changed by
loosening and tightening a small thumbscrew at the base of the bracket holding the probe on the
test section. The location of the aft pitot static probe is shown via a digital display. The probe can
be slid up and down to the desired location in the test section, but care should be taken to insure
that the tubes used to measure the pressures do not pop off the probe. Static pressure taps are also
available on the top wall of the test section, where both probes are located. The pressures from the
pitot and static pressures are measured using the differential pressure transducers.
Data Acquisition
This wind tunnel has been equipped with the TecQuipment Versatile Data Acquisition System
(VDAS). This is currently configured to collect data from the differential pressure transducers, the
32-channel transducer, the force balance, the angle feedback unit, and the digital displacement
transducer from the aft pitot probe. In addition, the pressure and temperature of the air in the room
may be recorded manually.
A screencapture of the VDAS program can be seen in Fig. 4. Data are read into the VDAS
system when the
button is pressed. To stop the transmission of data, press the
button. To
log data, press the
button. A dialog box will appear as shown in Fig. 5. Data may be collected
at regular intervals continuously, or over a set time. Collecting data over a finite time can then be
used to determine uncertainties, by calculating the average and standard deviation of the sample.
The data can be output in HTML format, which can be read in Microsoft Excel. To output the data
in HTML format, press the
button. There is a Microsoft Excel macro workbook capable of
calculating averages and standard deviations of data series – ask the instructor for help.
Fig. 4: VDAS (Versatile Data Acquisition System) Program
Page 26
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Fig. 5: Dialog box for data collection.
Indirect Measurement of Drag
In your research, you should develop two methods to indirectly measure drag. One method
should be based on the pressure forces acting directly on the surface of the cylinder. The other
method should investigate the effect of the cylinder on the flow through the tunnel, i.e. a control
volume analysis.
The body you will be investigating is a cylinder, shown in Fig. 6. The cylinder is 300 mm long
and 63.5 mm in diameter. The cylinder has a single pressure tap, connected to a flexible tube
Fig. 6: Cylinder Test Article
Page 27
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
attached to the end of the support. This tap may be used to measure the static pressure on the
surface of the cylinder. The cylinder may be rotated to any orientation along its axis.
Procedure
SAFETY PRECAUTIONS: Be sure to wear eye and ear protection while using the tunnel! In
addition, no one should walk directly in front of or behind the tunnel while it is operating. Doing
so could affect data collected during the experiment, and being caught in the strong flow of air
could cause a student to lose balance, causing injury or damage to equipment.
1) Mount the force balance on the side of the wind tunnel if it is not already mounted. Be sure
that the data cable from the balance to the display panel has been connected, and that all
instruments are connected to the VDAS board.
2) Turn on the wind tunnel main power switch on the instrument panel. Allow fifteen minutes for
the instrumentation to warm up.
3) Install the test article in the force balance apparatus. Make sure that the test article is properly
aligned with respect to the protractor, i.e., make sure the static pressure tap on the cylinder is
at the leading edge of the cylinder and that the protractor on the force balance reads an angle
of zero degrees.
3) Connect any pressure taps necessary for your experiment to the 32-channel transducer. Make
sure the pitot-static probes are connected to the differential pressure transducers. The forward
probe should be connected to DP Cell 1 and the aft probe should be connected to DP Cell 2.
4) Launch the VDAS software and enter the correct atmospheric conditions (temperature and
pressure) based upon the weather station readout in the laboratory.
5) Zero the instrumentation (DP Cells, 32-channel transducer, force balance) by pressing and
holding the black buttons on the panels under the digital displays. Be sure to unscrew the
locking nuts on the force balance before you zero the force balance display.
6) Initiate communications between VDAS and the computer by clicking the
button.
7) Turn on the wind tunnel, and set the speed using the knob potentiometer. If the pressure and
temperature are entered into the VDAS system, the computer program will automatically
calculate the speed based on the dynamic pressure measurement and the air density.
8) Initiate data capture by clicking the
button. When the pop up window appears, select 0.5
second interval, and stop after 20 readings.
9) Perform the first measurement. When not collecting force data, re-tighten the screws to lock
the force balance back in place.
10) Data should be collected using the VDAS program. It is suggested that the data be collected
using VDAS, but that the displays be checked against the VDAS output periodically.
11) Be careful not to loosen the test article when you change the angle.
12) After collecting data, click button to save as an HTML file. This HTML file can then be opened
in Excel so that the post-processor may be used (directions will be provided in the lab).
Page 28
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
For the Pre-Lab Report:
1) Do not treat this test as an academic exercise; assume that the test was performed for a client
who contracted you to confirm that the alternate methods would be sufficient to measure drag.
2) In the Introduction, start by explaining what drag is. Describe what principles will be used to
indirectly measure drag on a body. Specifically, you should show how the measured data you
collect can be used to determine the drag on the cylinder. For the surface pressure
measurement, how will the pressure data be used to determine a force? For the control volume
analysis, what will the control volume look like? Can you set up the volume to minimize or
eliminate any effects you will not be able to measure? Be sure to cite your sources! Be sure to
provide diagrams when introducing equations, in order to help with context. Provide an
estimate of the drag force based upon the velocity at which you intend to conduct your tests.
3) In the Experimental section, describe what data you will collect to determine the drag.
Remember that you are devising two methods. Describe the method of how you will perform
the experiment. DO NOT PROVIDE A NUMBERED SET OF INSTRUCTIONS! Rather, give
your experimental procedure as a narrative.
4) Describe the calculations necessary using the data available to determine the drag forces.
For the Final Report:
In addition to the material included in the pre-lab report, the following material should be
provided in the final report:
1) Discuss your results. Did the measurements agree within experimental error? If not, why? Do
your results match a theoretical value based on reported values of drag coefficient? If not, why?
Compare your results to those in the sample report – be sure to cite them!
2) Ensure that all of the raw data and all of the analyses appear in appendices. Perform a full error
analysis for these calculations to determine the errors associated with both values found for the
drag on the cylinder.
References
Pritchard, P. J., Fox and McDonald’s Introduction to Fluid Mechanics, 8th ed., Wiley.
Hoboken, NJ, 2011 Chapters 4, 6, 9.
1
Page 29
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Experiment #4: Unsteady Conduction of Solids
Introduction
In this study transient (unsteady-state) heat conduction principles will be used to evaluate the
convective heat transfer coefficient for three different shapes. Numerous experimental techniques
have been devised for the measurement of heat transfer phenomena. Both steady-state and
unsteady-state methods are used – you will be using a steady-state method in Experiment #2.
However, an accurate determination of convective heat transfer demands a careful analysis of the
experimental technique involved. In this study, a simple transient procedure will be used to
determine the convective heat transfer coefficient for three different shapes made of different
materials. You will need to determine how to conduct the experiment and how to analyze the data.
If a solid body is suddenly subjected to a change in environment, e.g., being placed in a bath at
a different temperature, it will take some time to reach a new equilibrium. During this time, heat
transfer will occur between the body and its environment. The time necessary for the temperature
at any particular point within an object, initially at a uniform temperature (which is placed in a
large constant temperature bath), to attain some other temperature is dependent upon the geometry
and dimensions of the object and its physical properties. When geometrically and dimensionally
identical test-specimens are subjected to similar conditions, the time necessary to produce a similar
thermal change is dependent only upon the physical properties (i.e., density, heat capacity and
thermal conductivity) of the specimens; the heat transfer within the body is a function of the
thermal conductivity of the body, and the temperature distribution within the body depends upon
the Biot number and the Fourier number. On the other hand, the convective heat transfer coefficient
(nondimensionalized by the Nusselt number) for natural convection (which is the case here) is a
function of the Grashof number (or the Rayleigh number) and the Prandtl number, which only
include fluid properties, and therefore should be the same for bodies of different materials.
Convenient geometries for test specimens are a slab of finite thickness and a cylinder of finite
diameter (where their ends are either impervious to heat transfer or their lengths are such that heat
transfer at the ends is negligible), and a sphere of finite diameter.2
Apparatus
The photograph of the apparatus used in this experiment is shown in Figure 1. The apparatus
consists of a large tank filled with water. The water is circulated by a pump and it is heated by a
heating element. Using the thermostat, the temperature of the tank can be set to a constant
temperature which can be specified on the thermostat’s dial. There are three test shapes:
rectangular slab, circular cylinder, and sphere. The test shapes are made of different materials:
Aluminum 6061, Stainless Steel 304, and Copper 110. A Type K thermocouple is embedded in
the center of each object.
Procedure
1) Make sure that the tank is filled with water up to the holes in the white inner tank. If the water
level is low, fill it up.
2) Ensure that the thermocouple reader is connected to the K-type thermocouple in the water bath
and that a K-type thermocouple extension wire (yellow) is available to connect each test object
to the reader.
Page 30
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
Fig. 1: Photograph of transient conduction apparatus
3) Set the thermostat to 50°C and turn on the circulation switch. Wait until the water temperature
reaches 50°C. Check the water temperature with the immersed thermocouple; it should take
roughly two hours for the water to reach the set temperature from room temperature.
4) Select one group of three objects:
a)
Group 1: Aluminum sphere, cylinder, rectangular slab
b)
Group 2: Copper, aluminum, and stainless-steel rectangular slabs
5) Measure dimensions of all objects using the provided caliper.
6) Pick one of the test objects. Connect its thermocouple to the reader using the extension cable.
Make sure that the reader is set to read the correct thermocouple type, and that the
thermocouple in the specimen initially reads room temperature (check against the weather
station in the room). If the thermocouple’s reading is different from the weather station by
more than ±2°C, notify the laboratory assistant or the instructor.
7) Load the data acquisition program and begin data acquisition while the object is outside of the
tank. Detailed directions on how to load and the use the data acquisition program will be
provided in the lab.
8) Put the object in the tank carefully and record the temperature history until the object reaches
a steady state temperature. The software will record the temperature of the object and water
Page 31
MECH 405 Thermal Fluids Laboratory Manual – Fall 2023
bath once each second. Once the object’s temperature has reached a steady-state value, pause
the data collection and save the temperature traces to the computer.
9) Repeat steps 6 through 8 for the other objects.
10) After finishing recording the temperature histories for all objects, turn off the circulation pump
and heater. Use a towel to dry the objects and the workbench.
For the Pre-Lab Report:
1) Do not treat this study as an academic exercise; assume that the study is being performed for a
customer that wants to confirm the value of this apparatus and method to determine the
convective heat transfer coefficient for a body placed in a free convective environment.
2) In the Introduction, you should explain what free convection is and why it is important, and
provide a description of the unsteady conduction/convection problem, including figures and
equations wherever necessary. Explain how the data you collect will be used to determine the
heat transfer coefficient. In addition, explain how Nusselt number correlations will be used to
estimate the convective heat transfer coefficient. Be sure to cite your sources! Be sure to
provide diagrams when introducing equations, in order to help with context.
3) In the Experimental section, describe the apparatus you will use, and how you will perform the
experiment. DO NOT PROVIDE A NUMBERED SET OF INSTRUCTIONS! Rather, give
your experimental procedure as a narrative.
4) In your data sheets, include sufficient space to record the dimensions for the three objects and
the names of the data files that contain the temperature traces.
For the Final Report:
In addition to the material included in the pre-lab report, the following material should be
provided in the final report:
1) Discuss the trends formed by the data collected. Compare your results to the sample results –
be sure to cite them!
2) Present the convective heat transfer data for the tested objects. Do they match each other?
Should they? Do they match theoretical predictions? If not, why not?
3) Include the raw data and sample calculations as appendices.
References
2
Holman, J. P., Heat Transfer, 7th ed., McGraw Hill. New York, 1990, Chapter 4.
Page 32