A STUDY OF HEAT TRANSFER AND FLUID FLOW
IN THE ELECTROSLAG REFINING PROCESS
by
MANOJ KUMAR CHOUDHARY
(Chemical Engineering)
B. Tech. Hons.
Indian Institute of Technology, Kharagpur
(1974)
(Chemical Engineering)
M.S.
State University of New York at Buffalo
(1976)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1980
O
Manoj Kumar Choudhary 1980
The author hereby grants to M.I.T. permission to reproduce and
to distribute copies of this thesis document in whole or in part.
Signature of Author
Certified by
II
,,
-
I·
Accepted by
_
Department of Materials Science
and Engineerq, May -,f 00
.
I
'Y
Julia.h Seekely,
Thesis Supervisor)
Y
'I
ARCHIVES
MASSACHUSETTS INSTITUTE
OF TECH!OLOGY
JUL 2 1980
LIBRARES
Regis M. Pelloux
Chairman, Departmental Committee
on Graduate Students
2-.
A STUDY OF HEAT TRANSFER AND FLUID FLOW
IN THE ELECTROSLAG REFINING PROCESS
by
MANOJ KUMAR CHOUDHARY
Submitted to the Department of Materials Science
and Engineering on May 2, 1980
in partial fulfillment of the requirements
for the degree of Doctor of Science
ABSTRACT
A mathematical model has been formulated to describe
the electromagnetic field, fluid flow, heat transfer and
solidification phenomena in electroslag refining systems.
The formulation is based on the simultaneous statement of Maxwell's equations written for the MHD approximation,
the equations for turbulent fluid flow in the slag as caused
by both electromagnetic and natural convection forces (due
to temperature gradients) and the differential thermal
energy balance equations with allowances made for the spatial
distribution of heat generation rate in the slag, for the
moving interfaces, for the transport of heat by metal droplets
falling through the slag and for the release of latent
heat in the mushy zone. The effective viscosity and the
effective thermal conductivity in the slag are calculated
by using a two equation model for turbulence. The equations
are first stated in vector notations and then simplified
for an axi-symmetric cylindrical coordinate system. An
outline of the computational approach is also included.
The theoretically predicted pool profiles and
temperature fields are found to be in reasonable agreement
with experimental measurements reported in literature for
a laboratory scale system. The predictive capability of the
model makes it possible to relate the heat generation pattern,
the temperature and the velocity fields, the casting rate
and the pool profiles to the operating power and current, to
the amount of slag used and to the geometry of the system.
Thesis Supervisor:
Title:
Dr. Julian Szekely
Professor of Materials Engineering
3.
TABLE OF CONTENTS
Chapter
II
III
Page
ABSTRACT
2
TABLE OF CONTENTS
3
LIST OF FIGURES
7
LIST OF TABLES
12
ACKNOWLEDGEMENTS
13
INTRODUCTION
14
LITERATURE SURVEY
17
for ESR
2.1
Mathematical Models
2.2
Turbulent Recirculating Flows
in Metallurgical Systems
FORMULATION OF MATHEMATICAL MODEL
18
23
27
3.1
Process Description
27
3.2
Summary of Basic Processes
28
3.3
Assumptions Made in Model
30
3.4
Statement of Mathematical Tasks
34
3.5
Governing Equations for Flow and
Heat Transfer Phenomena in the
Electroslag Refining Process
35
3.5.1
Maxwell's equations
35
3.5.2
Fluid flow equations
39
3.5.3
Heat transfer equations
45
3.5.3A
Heat transfer in slag
45
3.5.3B
Heat transfer in other
portions of ESR
47
3.5.4
Droplet Behavior
50
3.5.4A
Droplet radius
50
3.5.4B
Droplet motion in the
slag
51
3.5.4C
Rate of heat removal
by droplets from the
slag
55
4.
Chapter
Page
3.6
3.7
3.8
IV
The Boundary Conditions
57
3.6.1
Boundary conditions for
the magnetic field equation
57
3.6.2
Boundary conditions for the
fluid flow equations
60
3.6.3
Boundary conditions for
temperature
64
General Nature of Solutions
69
3.7.1
The nature of stirring
69
3.7.2
Relationship between velocity
and current
72
3.7.3
Heat input vs. energy for
stirring
73
3.7.4
Dimensionless form of governing equations
75
3.7.4A
Dimensionless form of
magnetic field equation
75
3.7.4B
Dimensionless form of
flow equations
76
3.7.4C
Dimensionless form of
the heat transfer equation
77
Concluding Remarks
79
Nomenclature
82
NUMERICAL SOLUTION OF GOVERNING EQUATIONS
91
4.1
91
4.2
Summary of Governing Equations and
Boundary Conditions
4.1.1
Equations for magnetic field
91
4.1.2
Equations for fluid flow and
heat transfer
92
4.1.3
Boundary conditions
93
Derivation of the Finite-Difference
Equations
4.2.1
4.2.2
4.3
Expressions for interior
nodes
Expressions for boundary
nodes
Wall Function Approach
93
98
103
107
5.
CHAPTER
4.4
V
VI
Solution Procedure
Page
Page
114
4.4.1
Flowsheet for computation
115
4.4.2
Introduction to computer
program
116
4.4.3
Stability and convergence
problems
116
Nomenclature
125
COMPUTED RESULTS AND DISCUSSION
129
5.1
Description of the System Chosen
for Computation
129
5.2
Physical Properties and Parameters
Used in Computation
132
5.3
Computational Details
140
5.4
Results and Discussions
142
5.4.1
Computed results on electromagnetic parameters
142
5.4.2
Computed results on Fluid flow
and heat transfer
153
CONCLUSIONS AND SUGGESTIONS FOR FURTHER
WORK
184
6.1
Conclusions
184
6.2
Suggestions for Further Work
189
6.2.1
Suggestions for short term
plans
190
6.2.2
Suggestions for long term
plans
190
6.
Chapter
Page
APPENDICES
A
A BRIEF NOTE ON PHASOR NOTATION
192
B
BOUNDARY CONDITIONS ON VORTICITY
196
C
CALCULATION OF RADIATION VIEW FACTORS
201
D
USE OF TEMPERATURE DEPENDENT ELECTRICAL
CONDUCTIVITY IN THE SLAG
206
E
THE COMPUTER PROGRAM
208
E.1
List of Fortran Symbols
208
E.2
Program Listing
217
REFERENCES
273
BIOGRAPHICAL NOTE
280
7.
List of Figures
Page
Number
3.1
Schematic sketch of the electroslag
29
refining process
3.2
Physical concept of the process
33
model
3.3
The effect of current on the maximum
74
value of the linear velocity in the slag
4.1
Boundary conditions for the magnetic
95
field intensity
4.2
Boundary conditions for fluid flow
96
variables
4.3
Boundary conditions for temperature
97
4.4
A portion of the finite-difference
99
grid
4.5
Illustration for calculating first
108
order derivatives at boundaries
4.6
Domains for wall function method
110
4.7
Illustration for wall function approach
111
.
Page
4.8
Simplified flow diagram for the
117
computational scheme
5.1
Computed ratio of effective and atomic
139
thermal conductivities in both slag and
metal pools for operation with 1.7 kA
(rms) and for an idealized metal pool
shape and size.
5.2
Details on the grid configuration
141
5.3
Computed magnetic field intensity for
143
operation with 1.7 kA (rms)
5.4
Computed current density vectors in
145
slag for operation with 1.7 kA (rms)
5.5
Computed radial distribution of vol-
146
umetric heat generation rate in slag
for operation with 1.7 kA.
5.6
The effect of electrode penetration
148
depth on heat generation rate in slag.
5.7
The effect of the amount of slag used
149
on the heat generation rate in slag.
5.8
The effect of fill ratio on current distribution in the slag.
151
9.
Page
5.9
Computed and measured axial temper-
155
ature profiles for ingot 15 (rms current = 1.7 kA) at r = 2.5 cm.
5.10
Computed temperature distribution at
157
the slag-metal interface for ingot 15.
5.11
Results from Fig. 5.10 expressed in an
159
alternative way.
5.12
Computed liquidus and solidus isotherms
160
for ingot 15.
5.13
Computed and measured axial temperature
161
profiles for ingot 17.
5.14
Computed liquidus and solidus isotherms
163
for ingot 17.
5.15
Computed isotherms in slag for ingot
164
15.
5.16
The effect of electrode penetration
167
depth on temperature distribution in
the slag.
5.17
Computed streamline pattern in slag
for ingot 15.
169
10.
Page
5.18
Computed velocity vectors in slag
170
for operation with 1.7 kA and for
a fill ratio of 0.325.
5.19
Computed velocity vectors in slag
173
for operation with 1.7 kA and for a
fill ratio of 0.09.
5.20
Radial distribution of buoyancy/elec-
174
tromagnetic contributions to vorticity,
2 cm below the electrode.
5.21
Computed distribution of turbulence
175
kinetic energy in the slag for ingot 15.
5.22
Computed contours of the ratio effective
177
thermal conductivity/atomic thermal conductivity in the slag for ingot 15.
5.23
Computed metal pool depths for different
178
currents.
5.24
Variation of maximum pool depth with
179
current.
5.25
Variation of casting rate with power.
181
A.1
Graphical representation of a phasor.
193
11.
Page
B.1
Evaluation of wall vorticity.
198
C.1
Schematic representation of the
202
system for calculating view factors.
12.
List of Tables
Page
Number
3.1
Summary of governing equations in
70
vector notation
3.2
Summary of dimensionless variables
80
4.1
Summary of governing differential
94
equations in cylindrical coordinate
system
4.2
Description of the computer program
119
5.1
Composition of electrodes used in
130
Mellberg's experiments
5.2
Remelting parameters in Mellberg's
132
experiments
5.3
Physical property values used
134
5.4
Numerical values of parameters used in
136
computation
5.5
Values for droplet parameters
182
13.
ACKNOWLEDGEMENTS
I express my gratitude to Professor Julian Szekely
for his invaluable guidance, assistance and encouragement
during the course of this work.
He has helped make my
graduate study at MIT academically fruitful as well as
an enjoyable experience.
I am grateful to Professors Thomas B. King, John
F. Elliott, and Thomas W. Eagar for their interest in my
research program.
I am indebted to my father and to other members
of my family in India for their constant encouragement.
Special gratitude to my wife, Saraswati, for too many
contributions to mention.
I thank my friends for their advice and good wishes.
One of them, Karin, deserves special thanks for her typing
of the manuscript.
The financial support
gratefully acknowledged.
for this work from NSF is
14.
CHAPTER I
INTRODUCTION
In recent years there has been a growing interest
in the development of mathematical models for the representation of heat transfer and fluid flow phenomena in the
electroslag refining process.
While the earlier models
concentrated on the calculations of pool profiles and temperature fields in the ingot, a more fundamental approach
was taken in recently reported models where allowance has
been made for the thermally and electromagnetically driven
flow in the system.
However, these latter models were
primarily of theoretical interest because the shape and size
of the molten metal pool had to be specified and because
heat transfer in the mushy zone and in the ingot was
neglected.
The work to be described in this thesis represents
an attempt towards developing a predictive model for flow
and thermal characteristics of the ESR process.
The model
developed in this work seeks mathematical representations
for the electromagnetic field, for the turbulent recirculating flow in the slag (due to both electromagnetic and
natural convection forces) and for heat transfer with
phase change.
It therefore involves the simultaneous
statement of Maxwell's equations, equations for turbulent
motion and the differential thermal energy balance equations.
The model is then used to make predictions for a laboratory
15.
scale system reported in literature.
The predictive
capability of the model is utilized to investigate the
interdependence of principal operating parameters.
Regarding the organization of this thesis, it
is divided into six chapters in the following manner:
In Chapter 2,a literature survey is presented,
which reviews mathematical models on ESR and on turbulent
recirculating flow in metallurgical systems.
The formulation of the mathematical model is given
in Chapter 3.
After discussing the basic processes involved
and the assumptions made, the governing differential equations
are first written in vectorial forms so that some general
conclusions can be drawn regarding the behavior of ESR
systems.
Then they are presented in the cylindrical
coordinate system with axial symmetry.
Boundary conditions
are discussed and some dimensionless parameters are derived.
Numerical procedure used to solve the governing
differential equations is outlined in Chapter 4.
Computed results on current distribution, heat
generation pattern, velocity and temperature fields and
pool profiles are discussed in Chapter 5.
these
Wherever possible,
results are compared with experimental measurements
available in literature.
Concluding remarks and some suggestions for further
work in modelling of ESR process are made in Chapter 6.
16.
Appendix A contains a brief note on phasor notation
used for AC operation.
Derivations of vorticity boundary
conditions are discussed in Appendix B.
Calculation of
radiation view factors is outlined in Appendix C.
The use
of a temperature dependent electrical conductivity for slag
is discussed in Appendix D and a listing of the computer
program is presented in Appendix E.
17.
CHAPTER II
LITERATURE SURVEY
The past decade has seen a rapid increase in the
application of electroslag process in this country and in
other industrialized nations.
During the same period
numerous papers dealing with both physical and mathematical
modelling of ESR have been published.
These models have
resulted from a need to have a better understanding of the
relationships among key process parameters so as to be able
to devise effective strategies for controlling structure
and composition of remelted ingots.
Mathematical models for the ESR process can be
classified into two groups --
(1) those dealing with physical
phenomena such as heat transfer, fluid flow and solidification (i.e. thermal and fluid flow models)
1-22
22 and (2) those
dealing with chemical and electrochemical reactions (i.e.
2 3-26
chemical models) 23-26.
The review presented here is
restricted to thermal and fluid flow models for this is the
category into which the present work falls.
From the point of view of mathematical modelling,
the electroslag refining process represents a complex
group of problems involving turbulent recirculating flow
driven by electromagnetic and buoyancy forces, heat and
mass transfer phenomena and phase change (both melting and
solidification) with free boundaries.
Because of the way
ESR model described in this thesis has evolved, it appears
18.
best to divide this chapter into two sections.
The first
section reviews mathematical models for ESR and the second
section presents an overview of literature on the mathematical modelling of turbulent recirculating flows in metallurgical systems.
2.1
Mathematical Models for ESR
Some of the earliest modelling work on ESR dealt
with temperature distributions in the electrode.
These
involved the solution of one dimensional 2,3 or two dimensional 1,7 heat conduction problems with experimentally
established boundary conditions.
While these models were
helpful in visualizing the relative magnitudes of various
heat transfer mechanisms (i.e. conduction, convection and
radiation)
so far as the electrode was concerned, they
could not provide insight into the local or the overall
heat transfer rate between the electrode and the slag.
The heat transfer coefficient between the electrode and the
slag was, instead, used as an adjustable parameter to
interpret measured temperature distributions in the electrode.
Most of the early models on ESR 4-12,18 have concentrated on the representation of thermal field in the ingot.
While these models may differ in the form (e.g. transient
vs. quasi steady state) or in the type of boundary conditions chosen (e.g. specified flux vs. specified temperature
at the slag-metal interface) or in the way the release of
19.
latent heat is accounted for (e.g. adjustment of specific
heat in the mushy zone vs. use of solidification models),
the unifying themes behind these models are:
1) transport processes taking place in the slag
are ignored with the
boundary condition at the slag-metal
interface being considered adjustable.
2) casting rate is used as an input to the model.
and 3) an effective thermal conductivity is used to
account for convection in the metal pool.
These models then reduce to a set of heat conduction equations (with movement of slag-metal interface being
accounted for) for the metal pool, for the mushy zone and
for the solid ingot with appropriate boundary conditions.
Some of these earlier models have been reviewed by Mitchell
et al.
9
and by Ballantyne and Mitchell
presented by Sun and Pridgeon
Paton et al. 5,10
4
12
.
The models
, Carvajal and Geiger 88,
Ballantyne and Mitchell 12 and Jeanfils
18
et al. 18 are transient in nature.
While all these authors
used the transient models to study the development of isotherms from the initial stages of remelting up to the
attainment of quasi-steady state, Jeanfils et al. 18
also utilized their model to investigate the response of
the system (as characterized by change in pool depth and
mushy zone thickness) to specified changes (e.g. ramp,
sinusoidal) in the melt rate.
20.
11
Elliott and Maulvault 11 noted that the thermal
conditions in an ESR system became reproducible in time
after the ingot had grown to sufficient length (e.g. about
2.5 times the radius of the ingot when casting steels
and other metals with similar conductivities) and developed
a quasi steady state model for calculating thermal field
in the ingot.
This reduced the dimensionality of the
problem without seriously affecting the scope of the model.
Furthermore, the numerical scheme chosen by Elliott and
Maulvault 11
11 allowed for arbitrary grid configurations.
This in turn enabled them to concentrate the nodes in
critical areas without excessive grid requirements.
Apart from being successful in the interpretation
of experimental measurements on pool depth and on local
solidification time in the mushy zone, these models 4-12,18
illustrated the influence of casting rate and the effective
thermal conductivity on thermal fields in the ingot.
Further-
more many of these papers provided measurements which were
needed for model validation.
The inability of these models to generate predictive
relationships among key process parameters such as power
input, geometry, slag depth on one hand and melting rate,
pool depth, width of the mushy zone etc. on the other hand
stems from ignoring transport processes in the slag.
The
calculation of thermal fields in the slag necessitates the
21.
solution of electromagnetic field equations in order to
obtain the local rate of heat generation in the slag.
Furthermore there is vigorous convection in slag.
The
driving force for flow is provided by both electromagnetic
and buoyancy forces and the flow, in general, is turbulent.
The use of a spatially independent effective conductivity
will not provide a realistic representation of convection in
the slag.
Thus a predictive model for the ESR process will have
to seek additional mathematical representations for the electromagnetic force field, turbulent fluid flow field and for
convective heat transfer in the slag.
This more fundamental approach has been taken in
models published by Dilawari andSzekely 13,14,15, Kreyenberg
and Schwerdtfeger 16 and Inoue and Iwasaki 21.
A review of
these recent models as well as a survey on measurements of
temperature and electric potential reported in the literature
have been made by Kawakami and Goto 17.
The basic approach taken by the three groups of
investigators is to first calculate the current paths for
an assumed electrode melting tip shape (flat in references
13-16 and conical in reference 21) and then to solve fluid
flow and convective heat transfer equations.
Among these,
Kreyenberg and Schwerdtfeger 16
16 considered fluid flow and
heat transfer in the slag phase only while the other two
groups examined the behavior of both metal and slag phases.
22.
16
The approach of Kreyenberg and Schwerdtfeger 16 necessitated an assumed temperature distribution at the
slag-metal interface.
This assumed boundary condition was
found to have a strong effect on calculated flow and temperature fields in the slag.
The formulation presented by
Dilawari and Szekely 15
15 was particularly comprehensive since
it allowed for the turbulent nature of flow (in both liquid
pools) and accounted for heat transfer between the slag and
the falling metal droplets. Although the models put forward
15
21
by Dilawari and Szekely 5 and Inoue and Iwasaki
were of
fundamental interest, their practical use was limited
because the shape and the size of the molten metal pool
had to be specified and heat transfer in the mushy zone
and in the solid ingot was neglected.
These models,
therefore, could not be addressed to the metallurgically
important question of how to relate the shape and the size
of the metal pool and the mushy zone to the operating parameters.
Furthermore, the arbitrarily assumed pool shape
precluded a meaningful comparison of experimentally measured
temperature profiles below the slag-metal interface with
predictions based on the model.
The work described in this thesis represents a
significant step in our continuing efforts to develop a
predictive model for ESR operations.
The nature of the model,
its scope and limitations are detailed in subsequent chapters.
23.
Some applications of the model described in this thesis
have already been published 19,20,22
Before closing this section it should be pointed
out that some studies have been made 27,28 to calculate
segregation in ESR by solving interdendritic flow and
temperature fields in simulated ESR ingots.
The latter
paper, in fact, investigated the suppression of macrosegregation by rotating the ingot.
In both these papers,
no account was taken of motion in the metal pool above
the liquidus isotherm and electromagnetic effects were
either avoided or ignored.
Recently, however, Mehrabian
29
and Ridder9 have extended their model to account for
motion in the metal pool
(laminar motion caused by natural
convection) and have elaborated on the important influence
of fluid motion inthe metal pool on solute redistribution
in ingots.
It will be a worthwhile exercise to combine
some aspects of the model to be described in this thesis
with models for calculating segregation so as to minimize
uncertainties in specifying various boundary conditions
in the latter models.
2.2
Turbulent Recirculating Flows in Metallurgical Systems
Some of the recent models of ESR (13-16, 19-22)
have involved the solution of turbulent recirculating flow.
These models have benefited greatly from experiences gained
with modelling of such flows in connection with various
24.
metallurgical processes such as continuous casting (Szekely
and Yadoya 30
30), argon stirring (Szekely et al. 34
34), deoxidation in the ASEA-SKF furnace (Szekely and Nakanishi 31 ),
induction stirring and melting (Szekely and Chang 32,
Tarapore and Evans
33
) etc.
It is to be noted that the
computation of flow profiles in the latter two cases involved
the simultaneous solution of Maxwell's and the Navier-Stokes
equations.
A growing interest in ladle metallurgy 37 opera-
tions to achieve bath homogenization (with respect to temperature and composition), deoxidation, degassing, inclusion
removal, desulfurization etc. is likely to enhance the
application of transport fundamentals in these systems.
A detailed review of mathematical and experimental tools
available for the study of transport phenomena in agitated
ladle systems has been presented by Szekely 38.
The basic approach in the papers mentioned above
has been to model the eddy transport terms through the use
of an eddy viscosity (i.e. Boussinesq's proposal) which
in turn is computed by solving additional equations.
Unlike boundary layer flows or simple one dimensional flows
where the spatial dependence of eddy viscosity can be
given by an algebraic expression
(e.g. mixing length hypo-
thesis) or by a one equation model (which uses a differential
equation for k, the turbulence kinetic energy and some
suitable algebraic expression for 1, the length scale of
25.
turbulence), turbulent recirculating flows, in general,
require a two equation model (which uses two differential
equations - one for k and another for some appropriate
parameter of turbulence).
Launder and Spalding
36 have
reviewed various mathematical models of turbulence.
Except
Szekely and Yadoya 30
30 who used a one equation model, the
rest of the papers listed above used a two equation model
(Spalding's k-W model, where W is a statistical characteristic of turbulence).
The computational algorithm employed
in these papers used the Stream function-vorticity technique
as detailed by Gosman et al. 39
39
Experimental proof for
the predictions reported in these papers was in terms of
tracer dispersion rates or in terms of surface velocities
(for both laboratory and industrial systems) and thus was not
very direct.
Szekely et al. 34
34 measured velocity and
turbulence kinetic energy in the water model of an argon
stirred ladle.
This study however was not conclusive
because of the uncertainties regarding boundary conditions
atthe gas-liquid interface and because of inherent experimental
inaccuracies involved in determining low velocities using a
hot film anemometer.
35
In a recent study Szekely et al. 35 have reported
accurate measurements (using a laser doppler anemometer) of
time averaged and fluctuating velocities in a system in
which recirculating motion was created by a moving belt.
26.
They also refined the mathematical model by incorporating
wall functions to represent momentum transfer in the
vicinity of solid surfaces.
This refinement is all the more
necessary because the transport processes taking place in
the vicinity of bounding surfaces are usually of more
A good agreement between measurements
35
and predictions is found in this paper.
practical interest.
In conclusion, it may be stated that while a great
deal more work needs to be done to characterize gas agitated
systems, the mathematical treatment of turbulent recirculating flows in single phases and for axial symmetry is
relatively well developed and is supported by measurements
made in both laboratory and plant scale systems.
27.
CHAPTER III
FORMULATION OF MATHEMATICAL MODEL
In this chapter, a mathematical model is developed
to describe flow and heat transfer phenomena in ESR systems.
A brief description of the electroslag refining process is
first presented so as to provide a clear perspective on the
processes and components involved in the system.
3.1
Process Description
A detailed technical description of the ESR process
is available in literature
40 41
, .
sketch of a typical ESR system.
Figure 3.1 shows a
As seen here a solid
consumable electrode of the primary metal which may be
cast or wrought or be composed of scrap is made one pole
of a high current source
plate is the other pole.
(AC or DC) and a water cooled base
A slag bath contained in the water
cooled mold acts as ohmic resistance and the Joule heating produced in it melts the electrode tip.
The metal droplets
fall through the slag and collect in a pool on the base
plate to solidify.
The electrode is fed into the slag
bath and the liquid metal solidifies progressively
forming an ingot which now acts as the secondary electrode.
An important feature of the process is that a
slag skin is formed at the inner surface of the mold,
which provides an electrical insulation separating the
mold from the molten slag, the metal pool and the solid-
28.
fying ingot.
The most usual slag compositions fall
within
the system CaF 2 + CaO + A203 and fulfill the basic
requirements imposed by electrical and thermal conductivity, high temperature stability and phase behavior.
Refining takes place because of reactions between
the metal and the slag in three stages:
i)
during formation of a droplet on the electrode
ii)
as the droplet falls through the slag, and
iii)
at the slag-molten metal
tip
pool interface.
By suitable choice of slags, chemical and electrochemical reactions can either be encouraged or inhibited.
3.2
Summary of Basic Processes
The basic processes taking place during electroslag
refining can be summarized as follows:
1)
media.
Passage of electric current through conducting
This gives rise to spatially distributed joule
heating in the slag.
The interaction between current and
the induced magnetic field results in Lorentz forces which
cause circulation in slag and in metal pools.
2)
Heat transfer, melting and solidification.
Convective heat transfer takes place in the slag and in
the metal pool.
Metal droplets extract heat from the slag
and get superheated.
molten metal pool.
This superheat is released in the
Heat transport in other portions of
29.
I'-W
4
W
0
o
0
C-)
LU
-J
LU
3.1
-J
0
Sceai
wa:
al-
J
J
0
0
0
0
a.
Z
a.
oo
<4
o
0
0
.4
-J
w
¢
Z
_
N
w
>'
U
LL
0
I
E
U
CD
)
z
b
nE
o
..
0C
z
Lu
1-j
a.
0
LU
-j
G,)
(I,
4
sketch of the electroslag refining process.
30.
an ESR unit is characterized by conduction with account
being taken of the movement of various interfaces. The electrode
tip melts and solidification takes place in the
mushy zone.
Heat is removed through the mold by cooling
water and there is radiative exchange of thermal energy
between the free surface of the slag, the outer surface
of the electrode and the inner surface of the mold.
3)
Recirculation.
There is recirculating motion
in the slag and in the metal pool due to the combined effect of
electromagnetic
(Lorentz)and buoyancy
gradients) driving forces.
(due to thermal
The fluid motion is, in
general, turbulent.
4)
Chemical and electrochemical reactions.
This aspect of the ESR operation is not considered in
the present work.
The implications of ignoring chemical
and electrochemical effects while modelling the thermal
character of ESR are detailed in the next section.
3.3
Assumptions Made in Model
The physical concept of the process model is
sketched in Fig.
3.2 which shows the coordinate system
and the assumptions made regarding the geometry of the
system.
The assumptions are as follows:
1)
Cylindrical symmetry.
31.
2)
Slag-electrode and slag-metal boundaries are
represented by horizontal surfaces.
planar
The assumption of a
electrode melting tip is thought to be reasonable
for large scale systems 42
However, we have retained
this assumption even for the small scale system considered
in this work.
Other key assumptions made in the model are as
follows:
wall.
3)
Quasi-steady state.
4)
The slag-metal interface is modelled as a rigid
This is thought to be reasonable in view of the
42
modelling work reported by Campbell 2
5)
Fluid flow equations are solved for the
slag phase only.
Motion in the metal pool is accounted
for by using an effective thermal conductivity.
an attempt is made to deduce this parameter from
However,
the cal-
culated flow field in the slag phase.
6)
In most of the calculations electrical con-
ductivity of the slag is assumed uniform.
However, in
some calculations the temperature dependence of electrical
conductivity of slag is approximately accounted for.
7)
The effect of metal droplets on the motion
of the slag is neglected.
8)
Effects associated with chemical and electro-
chemical reactions are not considered.
There are two
32.
aspects to these effects.
The first is the influence of
these processes on the nature of heat release in the slag
and on motion of the slag.
The second is the refining of
metal as it is melted, passed through the slag, and
collected in a pool at the top of the ingot.
Since the
present work is concerned with the flow and the thermal
characteristics of ESR, the second aspect (i.e. refining)
is not important here.
It should be recognized, however,
that even though the amount of matter involved in exchange
reactions is quite small as compared to the total amount
of metal being transferred from the electrode to the ingot,
the thermal effects arising from concentration polarization
and the enthalpy involved in various exchange reactions may
influence the net heat supply rate in the regions of the
slag near its interfaces withthe electrode and the metal
pool.
These, in turn, will affect the melting rate of the
electrode and the depth of molten metal pool.
9)
The interaction between the electromagnetic
force field and
the turbulent fluctuations is neglected in
absence of satisfactory methods for treating it.
10)
An insulating slag skin is assumed to form
on the interior surface of the mold.
Mathematical statements of these assumptions
together with assumptions made in formulating the boundary
conditions will be presented at appropriate places.
33.
ELECTRODE
z O
MOLD
SLAG POOL
*z=Z
Z 21I
z=Z2
z=Z 3
METAL POOL
z=Z 4 r)
MUSHY ZONE
z: Z
5
INGOT
Z=Z 6
3.2
Physical concept of the process model.
(r)
34.
3.4
Statement of the Mathematical Tasks
In context of the assumptions outlined above,
the model to be developed in this work seeks mathematical
representation for the following physical phenomena:
(1)
Electromagnetic field
This is represented by
the magnetohydrodynamic form
of Maxwell's equations written for different portions of
an ESR system and interconnected through boundary conditions.
Solution of these equations gives spatial distributions
of current densities, Joule heat and Lorentz
(2)
forces.
Recirculating flow in slag
This is represented by the turbulent Navier Stokes
equations with due allowance for body forces
magnetic and natural convection).
(electro-
Turbulent viscosity
is computed by solving two additional differential
equations.
(3)
Heat transfer and phase change
The mathematical statement of heat transfer penomena in the system is given by convective heat transport
equations.
The convection terms in these equations account
for heat transfer due to turbulent recirculating flow in
the slag and heat transfer due to movement of various
interfaces.
In the slag, allowance has to be made for
Joule heat generation and heat extraction by metal droplets
and in the mushy zone account has to be taken of the release
of latent heat.
35.
3.5
Governing Equations for Flow and Heat Transfer
Phenomena in the Electroslag Refining Process
Equations mentioned in the previous section are
now presented.
First the vectorial forms of these equations
are given in order that some general conclusions can be
obtained and then the equations are stated in the cylindrical
coordinate system with axial symmetry.
3.5.1
Maxwell's Equations
Upon applying the MHD approximation, Maxwell's
equations take the following form 43
43
aB
(Faraday's Law)
V x E =
-
(Ampere's Law)
-
-
(3.1)
a~t
V x H =
J
(3.2)
V - B=
(3.3)
V
(3.4)
J =
Here,
E
is the electric field, Volt/m
B
is the magnetic flux density, Weber/m 2
Teslas)
H
is the magnetic field intensity, Amp/m
J
is the current density, Amp/m2
(or
36.
t
is time, s
Furthermore, we have
J =
(E + V x B)
(3.5)
B =
0
(3.6)
and
Where
is the electrical conductivity in
/Ohm-m,
%0
is the magnetic permeability of free space in Henry/m
and V is the velocity of medium in m/s.
In brief, the meaning of these equations is as
follows:
Eq.
(3.1) relates the change in
density to
the
induced
emf.
the magnetic flux
Eq. (3.2) is Ampere's
circuital law which relates the induced magnetic field
intensity to current in the circuit.
Eqs.
(3.3) and (3.4)
represent the continuity of the magnetic lines of force
and conservation of current respectively.
Eqs. (3.1) through (3.6) can be combined 43
43 to
give:
H
2
-
nV H +
at
(3.7)
x (V x H)
-...-
1
where n =
is called the magnetic diffusity.
-
Ul-0
Terms arising from the spatial dependence of
neglected.
Eq.
(3.7),
along with Eq.
(3.4),
have been
contains all
the information about H included in Maxwell's equations.
37.
By using dimensional relation, the ratio of the terms on
the rh-s of Eq.
(3.7) is:
= magnetic convection
IQx ( x
In V 2 HI
0.
magnetic diffusion
0 /L.
H
H /L2
= 0 (Re m )
(3.8)
where Re m = V LGI
is called magnetic Reynolds number.
m 0
00
L and V are characteristic length and velocity respectively.
In Eq. (3.8) the symbol
(Q) stands for the order
of magnitude of a physical quantity Q.
<< 1 and hence the convection term can
in general, Re
be neglected
m
13
For ESR systems,
.
The magnetic field equation then reduces
to,
aH
~H
2
at
(3.9)
-=
nV2 H
The electromagnetic body force (in N/m 3 ) is given by:
J x B =
Fbe
0
J
(3.10)
x H
In cylindrical coordinate system with axial symmetry
(H = H = r
Z
=
~H0
l0
0t
0),
Eq.
3
(3.9) can be written as
-~
a
Lrr
or~~
r
1-
(rH
j
2
~ 2H
~z2
(3.11)
38.
In order to account for AC operation, phasor notation
(explained in Appendix A) is used.
43,44
In this notation
He = HeJWt
J
= J ejw t
r
r
/%
and
Jz
zA
Jzet
z
(3.12a,b,c)
Where He, J
J are the complex amplitudes of H, J and
r
z
Jr'
Jz respectively,
is the angular frequency and j is Z
The momentary physical values of He, Jr and Jz are the
real parts of the complex functions given above.
In phasor notation, Eq.
(3.11) takes the following
form:
2%
a
A
jo 0wH6
a
w
a2
Dr (rH3)
-
Dr
+
Dz 2(313)
(3.13)
which has to be solved for the real and imaginary parts.
After solving Eq.
tions, Eq.
(3.13) with appropriate boundary condi-
(3.2) can be used to calculate current densities
as follows:
He
A
and
J =
r
_-
^
1
z
A
(rHe)
Jz=
r
r
(3.14a,b)
39.
Using Eq.
(3.10) and averaging 4344 over the period
gives the following relationships for the time averaged
components of electromagnetic body force:
r= _
and
Fz =
z
2
Jz
o Re(H
e(HJ)
^
^
1
(3.15a,b)
0 Re(HJr)
Re( 0 J
2-I
where Re stands for the real part and the overhead bar
Similarly the time averaged
denotes the complex conjugate.
heat generation rate per unit volume is given by:
A
x
^
rJrJr+
1Re
Qj =
z
(3.16)
The electrical power input to the system is computed by
using:
W = 2
Q.(r,z) r dr dz
(3.16a)
zr
3.5.2
Fluid Flow Equations
Turbulent motion in the system is represented by
the time-smoothed equations of continuity and motion (i.e.
Navier-Stokes equations) written below in vectorial
from 45:
V
V = 0
(3.17)
40.
p(V*V)
V
=
inertial force
-
V
-
pressure
force
VT
F
+
+
viscous and
Reynolds
forces
[b
body
force
(3.18)
Here
p
is the (average) density of the fluid
V
is the velocity vector
P
is the pressure
T
is the stress tensor, which includes both
viscous and Reynolds stresses
Fb
(pvi'v.')
is the body force (per unit volume) vector
which incorporates both electromagnetic and
buoyancy driving forces and is given by
F
~b
= JxB + p[l-3(T-T)]g
0
(3.19)
Here
is the coefficient of volume expansion
g
is the acceleration due to gravity
T
is the temperature at a given location
in the fluid
T
0
is a reference temperature
The overhead bar in the above equations represents timesmoothed parameters.
An assumption inherent in writing
equations (3.17) and (3.18) is that the density variations
due to temperature gradients are of importance only in
41.
producing buoyancy forces and density p is supposed to
be evaluated at the reference temperature T.
0
Following Boussinesq 45 , turbulent or Reynolds
stresses can be computed using the same relationships
which exist for viscous stresses in a Newtonian fluid
but by replacing molecular viscosity of the fluid with a
scalar turbulent viscosity.
As mentioned in the previous
chapter, turbulent viscosity is computed by using a suitable
model of turbulence.
In the present work a two equation
model of turbulence, called the
k -
model
46
is used.
Here k is the turbulence kinetic energy per unit mass
and
is the dissipation rate of turbulence energy.
As pointed out by Launder and Spalding 46 , a wide variety
of flows may be adequately represented by this model without
adjustments to model parameters in the near wall regions.
Also a comparison of the predictions of various models, with
each other and with experiments has shown 4 6 the k -
£
model
to be surpassed only by more complex "Reynolds - Stress"
models.
for
It should be noted, furthermore, that the equation
contains fewer terms, the exact form of the equation
can be derived relatively easily and that
as an unknown in the equation for k.
HIt
appears directly
The model postulates:
CdP k2/£
(turbulent viscosity)
Here C d is a dissipation constant.
(3.20)
42.
Distributions of k and
in the flow field are represented
by transport equations for scalar quantities.
form, these can be represented as
-- V I ~~~~eff
p(V)
eff VX
V
convective transport
In vectorial
6,46
+
viscous and
S
(3.21a,b)
source
turbulent
diffusive transport
Here
represents k or
number for transport of
,
,
is the effective Prandtl
ef is the effective viscosity
and is the sum of molecular viscosity ()
viscosity (t)
and S
of generation of
and turbulent
represents the net rate (volumetric)
.
Equation of motion and the transport equations for
k and
will now be given for an axisymmetric cylindrical
coordinate system.
Upon introducing the vorticity,
r
- z
z
;r
(3.22)
and the stream function,
v
r
z
=
1
Pr
z
Pr r
,
v
=
1
~(3.23
~tb~
a,b)
43.
the equation of motion [Eq.
(3.18)] can be written as the
vorticity transport equation given below
(
-
r20 a (~ a)
-)J
r3(
-
)
3
3~(eff
-r
(3.19),
Using Eqns.
+
39
effr )
ar
Dr
r2
)
= 0
(3.24)
J---
(3.15a,b) and (3.14a,b), the last term
in the above equation can be shown to take the following
form:
r2
ar
z
j
=_
+
roRe(H0Jr)
(3.25)
r=2pg(T)J
buoyancy
electromagnetic
contribution
contribution
In addition the following relationship exists between
+
9*
aI
3zX pr
z
1 r= '
+
bT
pr
r
J
Transport equations for k and
0
=
,
and
(3.26)
in the axisymmetric
cylindrical coordinate system, are given below:
Transport Equation for k
k~eff k
- ~9z)
D
k
'3
k3~~
3
-3r
3±j
[k
3kz±I
zo k r1 Tz)
~z~ ~~ ~ r - ~
where
Sk = G - D
-
=r
r effk
r_~fT
=
r Sk
r
.a.rJ
~
(3.27)
(3.28)
44.
2
-
g = 2
t
- aJ
l
;V
2
fV
r
aJ
t+
+
2
rJ
_V__
+ 12az r +
2
(3.29a)
ar
and D = p
(3.29b)
Transport Equation for
a
im
or sr) ;z
a~
_ai_
'z
r
-
1 eff
r
a-
3z
_aLr
-
z
r
eff
a
rS
r(3.30)
where
S
Se
£2
CC2 p -k
=C 1 £G
G
As seen from Eqn.
(3.31)
(3.28), the source S k of the turbulence
kinetic energy is made up of two terms G and D.
The
generation term, G, represents kinetic energy exchange
between the mean flow and the turbulence.
The dissipation
term, D, represents the rate at which viscous stresses
perform deformation work against the fluctuating strain
rate.
The origin and the form of these terms are discussed
47
by Tennekes and Lumley
The source of
and a negative term.
terms in Eqn.
£,
48
and by Hinze 48.
S
£
is also made up of a positive
As in the previous case for S k , the
(3.31) represent interaction of turbulence
with the mean flow and the self interaction of turbulence.
The parameters
k
£'
1'
C,C 2 originate because of the
assumptions made in representing diffusive action of
45.
turbulence by means of a gradient law
Pt k
_k - )
ak 9y
(e.g. p u'k' =
in modelling source terms.
and
The
significance of these parameters and their estimation are
discussed in reference 36.
3.5.3
Heat transfer equations
Equations for heat transfer phenomena taking
place in different portions of an ESR system are given
below.
3.5.3A
Heat transfer in slag
The vectorial form of the convective heat transfer
45
equation is written as 45.
p Cp(VVT) = V-K ff VT
p
eff
+
-
convective
laminar and
transport
turbulent diffusive
ST
T~+
(3.32)
(3.32)
source
transport
Here
C
p
T
is the specific heat of slag
ST
is the net volumetric heat generation rate
is the time-smoothed temperature
in the slag
Kef f is the effective thermal conductivity in the
slag.
46.
ST, the source term in Eqn.
(3.32) consists of two terms
as shown below:
T
where Q
Qd X
jQ
(3.33)
is the volumetric rate of Joule heat generation
given by Eqn.
extracted
(3.16) and Qd is the rate at which heat is
(per unit volume)
metal droplets.
subsequently.
from the slag by the falling
An expression for Qd will be derived
X
in Eqn.
(3.33) is defined as follows:
X= 1
when r< R
X= 0
when r> R
(3.34a,b)
where Re is the radius of the electrode.
Conditions
(3.34a,b) reflect the fact that droplets remove heat from
the central column of the slag which has a radius equal to
that of the electrode.
The effective thermal conductivity, Kef f is given
by
Keff
eff K
+
molecular
K ~~t
(3.35)
turbulent
conductivity
conductivity
After it' the turbulent viscosity, has been calculated using
the k-e model, K t can be evaluated by using,
C
t
t K
1
(3.36)
47.
The convective
where at is the turbulence Prandtlnumber.
transport term in Eqn.
(3.32) accounts for the fluid velocity
as well as the rise of the slag.
In the axisymmetric cylindrical coordinate system
and using Eqns.
(3.23a,b), Eqn.
rpC V UT + C |
p
p c z
ar
where V
3.5.3B
Ta3_a(
r
eff r
i
-z
r
T
(3.32) can be written as:
+
3z Keff r
- + rST
(337)
is the casting rate.
c
Heat transfer in other portions of ESR
The temperature distribution in the electrode,
the molten metal pool, the mushy zone and the solid ingot
can be expressed by the following general equation:
(V.VT) = VKiVT + STi
~
1- ,
p.C
1 P. -2.
where,
i = e (electrode),
m (mushy zone),
and
(3.38a,b,c,d)
(metal pool),
s (solid ingot)
accounts for the rise of the ingot surface
V
(i.e. casting rate) and the downward movement of the
electrode. V
~1
has only the axial (i.e. z-direction)
component.
For the coordinate being used, Eqn (3.38) can
be written as:
48.
rp
Pi v i
z = a
i r
+
(K i r zT) + r Si
(3.39 a,b,c,d)
For the electrode, we have
(3.40)
t + Vc
V=V
e
where Vt is the speed of travel of electrode
ST,e = 0
(3.41)
For the metal pool
VZ = V c
ST,
=
(3.42)
0
(3.43)
= effective thermal conductivity in
the metal pool
= (1 +A ) Km 9
(3.44)
where KmZ is the atomic thermal conductivity of molten
metal.
The evaluation of A will be discussed in Chapter
V.
For the mushy zone
V
m
= V
(3.45)
c
af
T,m
Vc m
-
(3.46)
49.
where X is the latent heat of fusion and f
S
of solids in the mushy zone.
is the fraction
ST m represents the rate of
T,m
heat release (per unit volume) due to solidification.
For simplicity, a linear relationship will be
*
assumed between f
S
=
and T
fZ- m,m-
, i.e.
TT
(3.47)
Tst
where Tm
and T
are the liquidus and solidus temperZ,m
s,m
atures of the metal.
Then Eq.
(3.36)
T
VcPmkX
ST,m
-
can be written as
T Z,m - T sza
(3.48)
For the ingot we have,
V
s
=V
c
(3.49)
STs= 0
(3.50)
It is to be noted that Joule heating has been ingnored
everywhere except in the slag.
This is reasonable because
electrical conductivity of the metal is very large.
The melting rate of the electrode is calculated
by making a heat balance at the slag-electrode interface;
*
The use of more complex relationships
models) can be easily accommodated.
(e.g. solidification
50.
Vme
me
e y Ze)) /
e
e
-qs se
(e
)
(3.51)
where
qse
is heat flux from the slag to the
electrode surface
K
e
X
e
~Tz
is heat flux conducted into the electrode
e
is the latent heat for melting of electrode.
qse in Eq. (3.51) is evaluated either by using wall flux
relation to be described later or by using
-q
se =- =KTZ
3.5.4
sz
(3.52)
Droplet Behavior
In this section, expressions are developed to
calculate heat transport by metal droplets falling through
the slag.
The treatment given here follows that of
Dilawari and Szekely
3.5.4A
.
Droplet radius, rd
For large electrodes, it has been suggested by
Campbell 42 that metal droplets are formed at discrete
locations on the tip of the electrode.
The droplet
51.
radius rd is, therefore, assumed independent of Re.
Then
from dimensional arguments and by using experimental
results, Campbell has given the following relationship,
1/2
rd
rd=
J
(2.0
2gA0 4y ~
(g6
A
(3.53a)
where y is the interfacial tension between liquid slag
and liquid metal,Ap is the difference in density between
the two liquids ang g is the acceleration due to gravity.
For small electrodes, the following relation is
deduced by equating gravitational and surface tension
forces:
1/3
1.5y Re]
rd = [lYA](3.53b)
3.5.4 B
Droplet motion in the slag
Considering the slag to be stagnant (it is shown
later that the slag velocity is substantially lower than
the average falling velocity of the drop) and assuming
the droplet to be a rigid sphere the equation describing
the droplet motion takes the following form 49:
4 /3
3
rd3 (d
+
P
du
2 ) du
=
4/3
3
rd3Apg
-
CD
7
2 U2
rdP 2
(3.54)
where
d
p
is the density of metal droplet
is the density of slag
52.
is the drag coefficient
and
U
is the velocity
Ap
is the difference in densities of the drop
and the slag
Eq.
d -
(i.e.
P
).
(3.54) can be written as follows:
dU
CD
U2 - a 2
where
a 2 = 8/3
rd
Pd + 0.5p
dt
pp grd
P CD
On integration, Eq.
A
(3.56)
2 A/ t
2
e fA
- 1
A = [ AP/(pd + 0-5P)
and
B = 3/8
P
Pd +
(3.57)
t + 1
where
CD
rd
(3.55)
(3.55) yields:
e
U =
3/8
-
P
0
It follows from Eq.
] g
.5p
(3.58a,b)
(3.57) that the terminal
velocity of the droplet is given by:
ut
Ut
Thus Eq.
IT
(3.59)
iB
(3.57) can be written as
U =
t
Ct
eCt - 1
e
+ 1
(3.60)
53.
where
C = 2A/Ut
(3.61)
If L 1 is the distance from the electrode tip
to the slag-metal interface and
is the residence time
of the droplet, then
1
L1 =
U dt
(3.62)
0
Substituting Eq. (3.60) in the above equation gives:
(1 + eC)
4 eC
C
2
e
=
CL1 /Ut
t
(3.63)
from which, the following expression can be deduced,
T
where
=
m
1/C in [ (m-l) +
2e
2AL /U 2
12AL
/Ut
m -2m
]
(3.64)
(3.65)
In the equations given above, CD, the drag
coefficient is not known and thus U t is unknown. Following
15
Dilawariand Szekely
, U t is estimated by using a correlation proposed by Hu and Kintner 50
50 who studied the steady
motion of single drops of various organic liquids falling
through stationary water.
motion
can
They concluded that the droplet
be represented in terms of two variables
defined below:
Y = C
We Pd 0.15
X =(RedP
d
) + 0.75
+ 0.75
X = (Red,/pdO 15 )
(3.66)
(3.67)
(3.67)
54.
where
p
Ut2d
We
=
d
=
Weber number
Y
P¥Y 34 P
a physical property group gV A P
Pd
=
droplet Reynolds number = UtddP
= ~~~~~~~~1diameter
of a droplet
dd
diameter of a droplet
(3.54) one can derive,
From Eq.
Ap gdd
4/3
CD
(3.68)
2
P U
Using definitions of C D and We, Eq.
(3.66) can
be written as;
gd
Y
=
p215
P0
P
4/3
Y
(3.69)
a
Thus Y depends on physical properties of the system only.
50
Hu and Kintner
proposed the following relationships
between Y and X:
= (O. 75Y) 0.784
X
=
(22.22Y) 0.422
for
2 < Y< 70
for
Y > 70
(3.70a,b)
55.
Once X is calculated using one of the above equations,
U t can be calculated as follows:
Re
d
=
(X - 0.75)
0.1 5
d
which gives
Redp
Ut
(3.71)
dd
3.5.4C
Rate of heat removal by droplets from the slag
By assuming that heat transfer from the slag to
a droplet can be characterized by a single heat transfer
coefficient, h and an average temperature of the slag
between the electrode tip and the slag-metal interface
(TB), one can write the following equation for heat
balance on a single drop:
4/3 rr d
3
id
CP,d
dTd
d
dt
=
h (TB - T d )
With the initial condition, t = 0, T
=
Tm
4
rd
2
(3.72)
where, T
is the temperature of the drop and Tme is the melting
temperature of the electrode.
The heat transfer coefficient, h can be estimated
using the average velocity, Uav ( =L1/T ) of the droplet
11
in the slag, from suitable correlations 51
In the
present work a correlation proposed by Spelles 52
52 is used.
This correlation is given below:
56.
hd
-
( d
= 0.8
K
where
1/3
1/2
, K, Cp, i
UP/
)
av
(C /K)
P
(3.73)
are respectively the density, the
thermal conductivity, the specific heat and the viscosity
of the slag.
Eq. (3.72) can be integrated to give the following
expression for the final temperature, Tf of a droplet:
Tf
=
B - (
f TB
B - TTme
B
me ) eS
(3.74)
where
3h
(3.75)
rd Cp,dP d
The rate at which heat is removed from the slag by the
falling droplets is given by:
R 2
Qs
Re
v
Vme
e Cp,d ( Tf
as defined by Eq.
Where V,
Qd
the electrode.
-
Tme )
(3.76)
(3.51) is the melting rate of
appearing in Eq (3.33) is obtained
from:
Qd
Qs
2
=
TR e
(3.77)
L1
i.e. droplets are assumed to remove heat uniformly from
the volume of slag forming a central column of radius Re
and height L 1 .
57.
The Boundary Conditions
3.6
In this section expressions are developed for the
dependent variables (or their gradients) on the bounding
surfaces sketched in Fig. 3.2.
3.6.1
Boundary conditions for the magnetic field equation
Boundary conditions for Eq.
(3.13) have to express
the following physical constraints 43:
(1)
the continuity of the tangential component
of the electric field across the phase boundaries, i.e.
nx [
1
2]
-
= 0
(3.78)
where n is the unit vector normal to the boundary separating
(3.78) is obtained by applying the
Eq.
media 1 and 2.
integral form of Eq. (3.1) across the phase boundary.
(2)
the statement of Ampere's Law [obtained
by applying Stoke's theorem to Eq.(3.2)],
H
Eq.
d
=
i.e.
(3.79)
J ' dS
(3.79) states that the line integral of H around
the path enclosing an area through which a current is
passing is equal to the current.
Constraints 1 and 2 are the statements of physical
laws derivable from Maxwell's equations.
In addition, the
following assumptions have to be made:
(3)
axial symmetry gives
Ha = 0
at r = 0
(3.80)
58.
J
z
=
(4)
at the free slag surface (z = Z 1 , Re< r< R )
(5)
at the upper boundary of electrode (z = 0,
0
A
J
O < r<- R ee )
(6)
0O
r
at the lower boundary of ingot (z=Z 6 '
0< r< R )
m
J
0.
r
Mathematical statements for these assumptions
in terms of the magnetic field intensity, H6
are given
below:
< r < R e , (upper boundary ofelec-
(1) at z = 0,
trode)
H
=
0
(3.81)
A
(J r = 0)
(2)
0
at r = R e ,
z -<
_
(surface of electrode
1
above slag)
H0
=
I 0 /(2R
(3.82)
e)
(Statement of Ampere's Law)
where I
is the maximum value of the total current.
(3)
at r = Re
Z
<
z <
Z2
(vertical surface of
electrode immersed in slag)
1
=
z e
1lA
O
z
of
Ss
A(Continuity
E)
(Continuity of Ez )
(3.83)
59.
(4)
at r = R,
< z < Z6
Z
(slag-mold interface)
IO
(3.84)
He
m
(Ampere's Law)
(5)
at z = Z 2
,
(slag- electrode
0 < r < R
interface)
1
a jr
=
e
1J
r
S
a
(Continuity of Er)
which gives,
aH
1
a ~ z
1
He
1
az
e
(6)
at z = Z
(3.85)
sQ
Re < r
Rm
(free surface of
slag)
I0
He =
(from
(7)
(3.86)
^
_1
Jz = r
at z = Z 3,
A
r ( rH
0 < r
R
) = 0 )
(slag-metal interface)
^
1
He
a
UZ
1
SQ
__e
daz
(continuity of Er)
(3.87)
60.
(8)
at z =Z
6,
0 < r < R
(lower boundary
of ingot)
0
~z
=
0
(3.88)
A
(Jr
0)
It is to be noted that boundary conditions are
stated in terms of H
from which real and imaginary parts
can be separated.
3.6.2
Boundary conditions for the fluid flow equations
In a physical sense the boundary conditions for
the fluid flow equations have to express the following:
(1)
Symmetry about the centerline
(2)
The "no-slip" condition for the velocity
at the solid boundaries (i.e. zero velocity at the stationary solid boundaries).
As discussed
in section 3.3,
the slag-metal interface is assumed to be a rigid interface
(3) At the free surface of the slag, the fluxes
of momentum and turbulence quantities
(k and
) are assumed
to be zero.
Since we use the vorticity transport equation, the
above conditions have to be stated in terms of vorticity
( ) and stream function (4).
Furthermore, boundary
conditions have to be stated for k and
£
as well.
The
expressions for boundary conditions in terms of ~ and 4
61.
are derived in texts on computational fluid dynamics 39,40
and a brief summary of these derivations are given in
In this chapter, only the final form of the
Appendix B.
Also the wall function treatment
expressions are given.
for the turbulence quantities is discussed in the next
With reference to Fig. 3.2 the boundary condi-
chapter.
tions for the flow equations are as follows:
at r = 0,
(1)
=k
r
k_ = 0
Dr
=~~~~~~
(i)=r
r0
and
Z2 < z < Z3
[
a8
-~2
2
2 + ~1 1 -'0
r2 2
]/(r 22
r1 2
(3.89a,b)
where suffixes 0, 1, 2 denote the points on the axis of
symmetry and the adjacent grid nodes in the r-direction
respectively.
at z = Z1 ,
(2)
ri=
r
Re < r < Rm
Dk
9z
-
= 0 follows from VzZ = 0 and
aV
(
D~
Dz
= 0
0
(free slag surface)
=
i = 0 follows
(3.90)
from
DV
+
reff
* Derivation
3 z
z) = 0 and from the definition of C (Eq.3.22).
in Appendix Bff
given
3r
Derivation given in Appendix B
62.
(3)
at z = Z 2
0 < r < Re
(slag-electrode inter-
face)
(3.91a)
**
k=E
(3.91b)
= 0
and
3 (
(i)
0
=
Pr2 (z
-
1
*
1)
0 -
z0 )
(3.91c)
1
2
0
where suffixes 0 and 1 refer to a grid node on the boundary
and to the adjacent node in z - direction, respectively.
(4)
0 < r < Rm
at z = Z 3 ,
(slag-metal interface)
= 0
(3.92a)
**
k =
(3.92b)
= 0
and
*
3 (
r0
pr
0
(z
-
W1)
- z
0
)2
2
r
(3.92c)
1
where suffixes have the same meaning as in the case of
Eq.
(3.91c).
*
derivation given in Appendix B
**
alternate and more realistic formulation through the use of
wall functions is discussed in next chapter.
63.
(5)
at r = R e
(vertical surface
Z1 < z < Z2
of electrode immersed in slag)
=
(3.93a)
**
k =
(3.93b)
= 0
and
1)
-
(
3
1
r 1
-
0
P
(r - r
) r 0 r1
*
pgP
+
4
(rB - r0 )
(T
-T1 )
(3.93c)
Re eff 1
where suffixes 0 and 1 refer to a grid point on the boundary
and to the adjacent node in r - direction, respectively.
(6)
at r = R,
Z1 <
Z3
(slag-mold interface)
(3.94a)
**
k =
(3.94b)
= 0
and
r 0
) 3
P
+
( 0 -
)
(r1 -r0 ) r 0 r 1
_ 2 (ir )1
-
*
pg3
(r1 -r0)
1 0
(TO - T 1)
(3.94c)
4 RmIeff, 1
derivation discussed in Apendix B
**
alternate and more realistic formulation through the use of
wall functions is duscussed in next chapter.
64.
where suffixes have the same meaning as in the case of
Eq.
(3.93c).
3.6.3
Boundary conditions for temperature
These boundary conditions have to express the
following physical constraints:
(1)
symmetry about the centerline
(2)
continuity of heat fluxes at all the external
surfaces and at the slag-metal interface.
(3)
the electrode tip is at the liquidus temper-
ature.
Again, with reference to Fig. 3.2, boundary
conditions for the temperature equations can be written
as follows:
(1)
at r = 0,
Z6
0 < z
3T
at = 0
(2)
at z = 0,
electrode) and at z=Z6,
3T
gz-
(3)
above slag)
(3.95)
0 < r < R
-- - e
0<r<R
(lower boundary of ingot)
-- in
0
at r = R e ,
(upper boundary of
(3.96)
0 < z <
Z1
(surface of electrode
65.
-K
T
~e
=
h
c
[ T(R
e,
z)
T
+ e 6 [ T(R
-
F
T
m em
a
e
e, z)
I
- eFes(Z) T
a v
4
]
(3.97)
II
where
T
a
is the temperature of the gas
6
is the Stefan-Boltzmann constant
e 'es 'em
are the emissivities of the dry electrode surface, the free surface of the
slag and the inner surface of the mold
respectively.
T
m
is the temperature of the inside surface
of the mold above the slag
T
s,av
is the average temperature of the free
surface of the slag
F
es
is the view factor between the electrode
element and the slag surface
F
em
is the view factor between the electrode
element and the mold wall.
It is understood here that the temperatures are in the
absolute scale.
The first term in Eq. (3.97) represents
the convective exchange between the electrode surface
and the ambient gas whereas the second term represents
66.
the radiative exchange between the electrode surface, the
free slag surface and the inner surface of the mold.
In
order to fascilitate the calculation of view factors, the
slag surface is represented by a single temperature,
T
av which is calculated from the following equation:
T
s,av
=
2Rm T(r,Zl)r dr
2
2
(R
R )
m
e
(4)
at z = Z 1,
Re < r < Rm
K
3Tt
[T
(3.98)
(free surface of
slag)
Keff
sk
= 6s
s
4 '
4'
- F T
s,av
m sm m
F
T
4
]
e se e,av
(3.99)
where
T
e,av
is the average temperature of the dry
electrode
F
sm
is the view factor between the slag
surface and the mold wall
and
F
se
is the view factor between the free
surface of the slag and the dry surface
of the electrode.
As seen from Eq.
(3.99), the convective heat loss from
the slag surface to the ambient gas has been neglected.
Also, in order to simplify calculations the slag surface
and the electrode surface have been represented by their
average temperatures.
67.
View factors appearing in Eqns.
(3.97) and (3.99)
are calculated using the techniques discussed by Leunberger
54
The calculation procedure is outlined in
and Person 5.
Appendix C.
(5)
at r = Re
<
Z
< Z2
(vertical surface
of electrode immersed in slag)
*
(3.100)
re
(6)
sz
at z = Z 2
(slag-electrode
0 < r < Re
interface)
T = T
(7)
(3.101)
m,e
0 < r -< R m
at z = Z
(slag-metal inter-
face)
*
-
Kff
eff
~zSk
Q
+
where Qs is defined by Eq.
Eq.
X =
e
-K
T
(3.102)
z
(3.76) and X is defined by
(3.34a,b); i.e. the heat extracted by droplets from
the slag is given uniformly to the liquid metal over an
2
area -iR
e
*
Alternate expressions for these fluxes can be given by
wall function approach discussed in the next chapter.
68.
(8)
at r = Rm,
Z
< z < Z3
(slag-mold interface)
Since a solidified slag layer is assumed to exist
at the inside wall of the mold, the temperature condition
can be specified as
where T
s
T = T
s
(3.103a)
is the melting temperature of the slag.
Alternatively one may write
*
-K
T
-K rs
h'Ws -
(T
W
)
(3.103b)
where h W s is the overall heat transfer coefficient
which describes the heat transfer from the molten slag/
slag skin interface to the cooling water in the mold.
TW is the average temperature of the cooling water.
(9)
at r = R
- K
m
=
hWi
Z
<
3-
< Z
-6
(T - T)
(3.104)
where i stands for interfaces between various media
(i.e. metal pool, mushy zone, solid ingot)and the mold.
hw
i
represents overall heat transfer coefficient at
these interfaces.
*
Alternate expressions for these fluxes can be given by
wall function approach discussed in the next chapter.
69.
3.7
General Nature of Solutions
The mathematical statement of the model is now
complete.
Before presenting the detailed results, it is
worthwhile to discuss the general nature of solutions.
The governing equations
(in vectorial form) for the
system are grouped together in Table 3.1.
3.7.1
The nature of stirring
Using Eqns.
(3.10),(3.2) and (3.6),
the electro-
magnetic body force Fbe can be written as follows:
Fbe
= J xB
j
=
=1
-1
(VXB)XB
x
0 B
B 2 ) + 1 (B.V) B
V(2
I
II
x (i.e. curl) on the r.h.s. of Eq.
Upon operating with
(3.105), the first term vanishes.
V [-
(3.105)
B )
=
0
i.e.
(3.106)
while in general,
x
(B-V)B
0(3.107)
It follows therefore that the term I (called the magnetic
pressure), cannot do any work on a circulating fluid and
that the second term (II) is responsible for doing work.
70.
Table 3.1
Governing equations for fluid flow and heat
transfer in the ESR process
1.
Maxwell's equations
V x E
2
~t ==~nnVH
VxH = J
OR
V -J = 0
V*B = 0
V
0
J
Constitutive equation
B =
O
Ohm's law
J
2.
E
Equations for fluid flow (in slag)
Equation of continuity
VV = 0
Equation of motion
p(VV)V =-Vp-
V*
+
-Re(JxB)
Transport equation for k and
p(V.7V)
= V
where
= k or
eff1
+ S
- p(1-(-T )
71.
3.
Thermal energy equations
Heat transfer in slag
pCp(VVT)
p~~
V-Kff VT +
eff
-2Re(J-J)- QX
a
Heat transfer in other portions of ESR
PiCpi(V.VT) = VKiVT + STi
where
i = e (electrode),
(metal pool),
m (mushy zone), s (solid ingot)
and V
accounts for movement of various interfaces.
72.
The inertial force in the equation of motion
[Eq.
(3.18)]
can be written as
1-
_
-I
p(V.V)V = pl (VxV)xV
+ pV(2
(3.108)
Introducing the definition of vorticity,
= V xV
(3.109)
we can rewrite Eq. (3.108) as follows:
p(~xV) = -pV( )
+
(V.V)V
(3.110)
vortex force
From Eqs.
(3.105) and (3.110), the analogy between Fbe and
the vortex force is evident.
3.7.2
Relationship between velocity and current
Let us assume that the buoyancy driving force is
small compared to the electromagnetic driving force
(i.e.
we operate with high current and with a small fill ratio)
then Eqs. (3.18) and (3.19) can be combined to give:
p(V.V)V = -VP - V.
+ J xB
(3.111)
where Pt represents the sum of static pressure and gravitational force.
If the inertial forces dominate (i.e. at high
Reynolds number) then as a first approximation we may neglect the first two terms on the r.h.s. of Eq.(3.111).
order of magnitude of the remaining terms are:
The
73.
P
2
L
O JX B
where J
JO x POOL
(3.112a,b)
is a characteristic current density of the system.
From Eqs.
(3.112a) and (3.112b) we can write
2
PVO
'L
(3.113)
i 21L
00
i.e.
V
0
jO
L
Op
which indicates that the characteristic velocity is proportional to the current.
Figure 3.3, taken from Dilawari
and Szekely 15
15 shows the effect of current on the velocity
in the slag phase.
As seen here, under the isothermal
conditions (i.e. in the absence of buoyancy
driving
forces) the relationship between the velocity and the
current is linear.
3.7.3
Heat input vs. energy for stirring
From dimensional arguments it may be shown that:
energy input for stirring
energy input for heating
- (J B).V
0
j2/
_
0
J0 P0 LV0°
2
J0
~~~~~(3.115a)
(3.115b)
74.
E~~~~
LO)
..4
IC)
-H
0
1
0=
C4)
~C4t
Li)~~
~ ~~L
-
g~~~~~_H
H
r .
10 "'C
A
~~-
I--~~~~~~f
0
o4
ci)
4-4
.NQN
O
0 r
0
_
10~~
~
0~~~E
O
-1
I
II
I
o
el
N
co
..
(S/
O) Ail0913A
0
~~~~~~~
,..{H
-J
".
4C)
_
75.
(0
*
1/2 L 2 a J
0
(3.115c)
For illustrative purposes, let us assume L = 0.5 m,
= 3.OxlO3 kg/m3, J =40 kA/m 2
=250 (ohm- )
3
L 2 aJ
P
=
-
0
13xl03
= 6.5 xQ10
3.7.4
1
(1.25x0l -6)3
/2 x.25x250x4xlO4
1
<< 1
Dimensionless form of governing equations
Let us now make the vector form of the governing
equations dimensionless.
This will generate parameters
characterizing the behavior of the system.
3.7.4A
Dimensionless form of magnetic field equation
Let us introduce the following dimensionless variables:
V
= LV
H
= H/H 0
H 0 = I0/L
(3.116)
then the magnetic field equation (Eq. (3.9))
can be
written, in the phasor notation, as:
^*
jaH = V
*2^*
H
(3.117)
where
_
Using eq. (3.114)
__
76.
2
a = ~dpL
L2.C
0
-
=
(1/w)
characteristic time for magnetic diffusion
characteristic period for electric current
(3.118)
Dimensionless form of flow equations
3.7.4B
In addition to dimensionless variables already
defined in Eq.
(3.116) let us introduce
=
k
= k/V 0
T
*
V/V o
V
p
2
*
£
= (T- T
=
£L/V0
0
)/pV 0
3
)/(T
Q,s
Z,m - TZ, )
l
(3.119)
then the equation of motion
*
(P - P
= JL/H 0
-
J
*
*
2
_
=
(V .V )V
= -
*
*
* *
*
P
-
T
(Eq.
1
O
(3.18))
can be written as
+ N 1 2Re(J
g
(T -y)J
*
^**
B ) +N 2 -N 3
(3.120)
where
*
T
2
= T/pV 0
= (viscous + Reynolds force)/(inertial
-
force)
(3.121)
77.
N1 =
(3.122)
o0H 0 /pV0 = (Lorentz force)/(inertial force)
N2 = gL/V
NN3 =
(Gravitational force)/(inertial force)
=
g(T
2
0
0
T ,s )
(T,m
(3.123)
(3.124)
(Buoyancy force)/(inertial force)
=
Y =0
Z,
(3.125)
Tm-Ts
'Qm iQs
similarly, the transport equations for k and
[Eqs.
(3.21a,b)]
can be written as
*** *
V V k =
*
1
Re
-
where
V
* (~ ef' *
*(fVeff
V k
Ref =
pVoL
1
.*
H
11eff
Sk
**
+ Sk
(3.126)
(Reynolds no.)
Effective viscosity
Molecular viscosity
t
1.
= SkL/pVo
(3.127a,b,c)
and
*
V * *V * £ * =
=
1
Ref
*
-
J
'~eff
a
V
*,
I
+
£
(3.128)
where
where
S
SE == SL2/pV4
S L2
3.7.4C
Dimensionless form of the heat transfer equation
(3.129)
Using dimensionless variables defined in Eqs.
and (3.119),
[Eq.
(3.116)
the convective heat transfer equation for slag
(3.32)] can be written as,
78.
*
(V
V
~
*
1
*
T )=
~
Pe- V
*
*
~~~
-
QdX
*
Keff V T
eff
*
1
+ NT[- Re(J
~~
T
*
*
)
(3.130)
/Jo ]
where
= Effective thermal conductivity
=Molecular thermal conductivity
Kt
K eff
+
K
=
it
Pr
-
+ 1
1-I
=
where
Pr
Pr
(eff
- 1) + 1
(3.131)
= Prandtl no.
=
K
Jo2/a
NT
P CpV (T
m
- T z,s)/L
Heat generation by Joule effect (3.132)
Heat transport by convection
PC V L
Pe
-
ppo=
Peclet no. = Ref Pr
K
Heat transport by convection
Heat transport by conduction
By using the approach outlined here on Eq. (3.38), we can
similarly define Peclet numbers for other portions of ESR.
For example, below the slag-metal interface, we can
define
79.
PZCp,lVc,0 L
Pe I
=
(3.133)
Kl
where
Vc, 0
is a reference casting rate
Kg
is effective thermal conductivity in
and
metal pool.
Similarly, a group of other dimensionless parameters
can be derived by considering the boundary conditions
(e.g. Biot number at the ingot-mold interface).
The dimensionless parameters developed here
and their physical interpretations are summarized
in Table 3.2.
3.8
Concluding Remarks
A mathematical model has been formulated to
describe the current distribution, fluid flow and heat
transfer phenomena in the electroslag refining process.
The model involves simultaneous statement of Maxwell's
equations, equations for turbulent motion and convective
heat transfer equations.
The limitations of the model,
in the form of assumptions made, are listed in section
3.3.
On the positive side, the model accounts for some
of the features of the ESR process which are considered
80.
t)
N
.,-
0i
(3)
0
>1
4
-I O
0
0d
0 H
4.-I
14-4
a)
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U
0)
w
0
,.1:
4
4 ¢)
a. H
O
C'~
;)
O
c
-H
-4
u4
-4
0
'4-4
·,0
~
,
ud
0
~
4-4
h
4
u4
40
a, 44
¢')
0 4U
O
O
-3
a) 0a,
0
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0¢)
u4
4
"O
O
4.
O C*
-4
.IU
0
5-4
H
O
NJ
4P
U
~3
£~00oa,
Q>*
)
EH
G
CdH
v-N,
45-4
a,
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a,
04a,
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O
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H
40
H
-H
4~
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5.4.
U a,
04
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h
402
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ro
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ty)
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0)
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l
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0<
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o
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EH
z
>
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(Ir~
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04
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54
P4
O
0
OJ¢
·H
q.H: .,U
'4-44
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rdo
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>
M
::
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>i4
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0
.,--I
-n
>
G
O
0
4-
0
.r4
U
O
0 .--I
a,
q4
.,Cd
-H
H
0
4-4
0
Cd
a)
0
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0a4
.Jo
4
04
04
Uo
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a,
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.
4J
W
81.
crucial from the point of view of flow and heat transfer
in the system.
Thus, allowance has been made for both
electromagnetic and buoyancy driving forces, for the
spatial distribution of Joule heat release, for heat
transport by metal droplets and for the release of
latent heat in the mushy zone, etc.
Boundary conditions for the governing differential
equations have been stated in section 3.6.
Finally, brief
discussions have been given on the general nature of the
solution and the dimensionless parameters for the system.
82.
Nomenclature
B
Magnetic flux density
Constants in k-
model
C-d
Dissipation rate constant
CpCp,e,eCp
p,
Specific heat of slag, electrode,
C
,C
,C
p,m Cp,s'C p,d
molten metal, mushy zone, solid
ingot and metal droplet
CD
Drag coefficient
dd
Diameter
D
Dissipation term for turbulence
of a droplet
kinetic energy
E
f
Electric field
s
Fraction of solids in the mushy
zone
Body force (per unit volume) vector
Fbe
Electromagnetic
(Lorentz) body
force vector
F r ,F z
Radial and axial components of
body force
F es ,F em
Radiation view factors between the
electrode element and slag,
83.
Nomenclature
(cont'd)
electrode element and mold
Radiation view factors between
F' ,F'
se
sm
the slag surface and the mold,
slag surface and the dry
surface of electrode
g
Acceleration due to gravity
G
Generation term for turbulence
kinetic energy
Heat transfer coefficient between
h
the slag and a droplet
Overall heat transfer coefficient
hW
,ts
at the slag-mold interface
W,
iW~L~
heat transfer coefficients
,~~~Overall
i
for the regions defined in
equation
HH 0 1 H 0 1 H 0
(3.104)
Magnetic field intensity, its
component in the
-direction,
complex amplitude of H,
reference value
I0
Amplitude of total current,also
reference value for current
84.
Nomenclature (cont'd)
j
J,J r ,Jz,J
,J z
r
Current density components in
r- and z- direction and their
complex amplitudes
conjugate
of
J
AComplex
J
Complex conjugate of J
Jo
Reference current density
k
Kinetic energy (per unit mass)
of turbulence
K,Ke'KZiKmKs
Thermal conductivity of slag,
electrode, molten metal (effective value), mushy zone and ingot
Thermal conductivity (atomic)
of molten metal
Keff
Effective thermal conductivity
(sum of molecular and turbulent
contributions) in slag
Kt
Turbulent thermal conductivity
in slag
L1,L
Depth of slag below the electrode,
characteristic length scale
85.
Nomenclature (cont'd)
Normal vector
n
Ratio of (Lorentz force/ inertial
N
1
force)
N.2
Ratio of
Eq.
(3.122)
(Gravity
force/
inertial force) Eq.
NT
(3.123)
Heat generation rate / Heat
transport by convection,
Eq. (3.132)
Time- smoothed pressure, its
reference value
Pe
Peclet no. for slag
Peg
Peclet no. for metal pool
Pd
A physical property group
for droplets
Pr
Prandtl no. for slag
Qj
Rate of Joule heating (per unit
volume)
Rate of heat extraction
(per
unit volume) from slag by metal
droplets
86.
Nomenclature (cont'd)
Rate of heat extraction from slag
by metal droplets
r
Radial coordinate
rd
Radius of a metal droplet
R
Radius of the electrode
R
e
m
Radius of the mold
Re
m
Magnetic Reynolds number
Ref
Reynolds number for flow
Red
Droplet Reynolds number
ST
Source term for temperature
equation
SkS
Source terms for k,
t
Time
T,T
Temperature in the slag (time
smoothed), elsewhere
T
a
Temperature of gas above the
slag surface
TB
Bulk temperature of the slag in
the region O< r< Re
87.
Nomenclature (cont'd)
Final temperature attained by a
droplet
Temperature of a droplet
T
m
Liquidus temperature of the metal
Melting temperature of the slag
T m,e
Melting temperature of electrode
T Sm
Solidus temperature of the metal
Tw
Temperature of the cooling water
T s,av
Average temperature of the slag
surface
T e,av
Average temperature of the dry
electrode surface
Velocity of a metal droplet
U
U.
Average velocity of a droplet
av
Utt
Terminal velocity of a droplet
Vc VcO
Casting rate, its reference value
Vt
Rate of travel of electrode
Y,V
0
Time-smoothed velocity vector,
its reference value
88.
Nomenclature (cont'd)
Radial and axial components of
Vr,Vz
r z
velocity vector
W
Input power
We
Weber no.
X
Variable defined in Eq.
Y
Variable defined in Eq. (3.66)
z
Axial coordinate
Zl,Z 2 ,Z
3 ,Z 6
(3.67)
Position of free slag surface,
melting tip of electrode, slagmetal interface, lower boundary
of ingot
GREEK SYMBOLS
Characteristic time for magnetic
diffusion/ Characteristic period
for electric current
Coefficient of thermal expansion
of slag
Y
Interfacial tension between liquid
metal and liquid slag
6
Stefan - Boltzmann constant
Nomenclature (cont'd)
89
GREEK SYMBOLS
Dissipation rate of turbulence
£
energy
eI
s
m
Emissivity of electrode, slag,
mold
&-
component of the vorticity
vector
Magnetic diffusivity
x
Latent heat of fusion of metal
A
Defined by Eq.
I-'
Viscosity of slag
(3.44)
Magnetic permeability of free space
Effective viscosity of slag
neff
Stream function
PIPe'P 'PmrPs'Pd
Density of slag, electrode, metal
pool, mushy zone, solid ingot and
droplet
'5e
m
Electrical
conductivity of slag,
elctrode and metal
90.
Nomenclature (cont'd)
GREEK SYMBOLS
a ka
t
Tr
Prandtl number for k, E
Turbulence Prandtl number
Sum of viscous and Reynolds
stresses
Residence time of a droplet
X
w
Angular frequency of current
x
Defined by Eqs.
(3.34a,b)
Superscript
*
Dimensionless quantities
91.
CHAPTER IV
NUMERICAL SOLUTION OF GOVERNING EQUATIONS
This chapter presents an outline of the numerical
technique used for solving the differential equations
developed in the previous chapter.
After discussing the
derivation of finite difference equations for the dependent
variables,a brief treatment on the use of wall functions for
representing heat and momentum transfer in the near wall
regions is presented.
The computational scheme and the
computer program are discussed at the end of the chapter.
4.1
4.1.1
Summary of Governing Equations and Boundary Conditions
Equations for magnetic field
A
A
Let H0R and H8I denote the real and imaginary parts
/%~~~~0
of H
respectively, then Eq.
(3.13) can be decomposed to
give:
r(rH
-rr
)I
+
R =
_
1wH e
-
(4.1)
3z 2
and
^~~~
arOHIJ +
3r
3r r ei
2
qz2
I
It should be noted that,
I
H
i
(H2OR
=e
+He
01
1/2
O1
wH
0e
(4.2)
92.
and that the phase angle
H = tan
HiI/H RJ
H
Similarly from Eq.
(4.3a,b)
(3.14a,b) one can write;
MHeR
JrR =Re(J)
=
az
JrI = Im(Jr) =
;z
(4.4a,b,c)
IJrI =[ J ^ zR
2
zI
izI
J
1/2
and
J
= Re(J )
JzR
=
Im(J )
Tm(J ) =
L z2I+
R
5 2
4.1.2
-
-
(rH
)
(r
-
)
-~-(rHa1)
2f J/2
(4.5a,b,c)
Equations for fluid flow and heat transfer
Eqs. (3.24) , (3.27) , (3.30) ,
(3.37),
(3.39a,b,c,d)
constitute the mathematical statement of the fluid flow and
heat transfer phenomena in the system.
These equations can
be represented by the following general elliptic partial
differential equation:
a
r
z
+ a rC( + a )
qb
;r
3Dr q-z)z
bqr
I-b
r
[Z
(c)-
'3~z
-
93.
r r br
J-
(c&P)rJ
rrS
=
(4.6)
0
stands for the dependent variables
where
Values of the coefficients al,,a,, b,
term S
4.1.3
(,,k,,T).
and the source
are listed in table 4.1.
Boundary conditions
Boundary conditions for the magnetic field intensity,
H 8 are stated in Fig. 4.1.
flow variables
Boundary conditions for fluid
~
,k and
(i.e. -,
r
) are summarized in Fig.
4.2 and finally, boundary conditions for temperature are
given in Fig. 4.3.
In these figures the symbols e,sZ,Z,m,
s stand for electode, slag, metal pool, mushy zone, and
solid ingot respectively.
4.2
Derivation of the Finite- Difference Equations
In this section, transformation of differential equations
listed in section 4.1 to finite- difference forms is presented.
This transformation involves the following steps:
1)
The domain of integration is represented by a
two dimensional
2)
(r,z) array of points called a grid.
A set of algebraic equations is derived, from
the differential equations and the boundary conditions,
which connect the values of the dependent variables at
the grid nodes
(points of intersection of grid lines)
94.
.
i=
..
~
_
-
_
Ii
I
I
I
_
Cd4
A
I
H
I
co
1
_vi
(N
rd
m
$4
M
rc
+
(N
M
JIlk
CJ2
m
< "n
m
Ca0
a
;4
$4
I
xI
I
I
I
I >0I
I I
I
I0
I p>
I
I
I
xI
hL)
Ur
I
4-
44
o
I
I
0
I
I
I
I
~~~~~~~~~~~~~~~~~I
I
II !
I
~~~~~~~~~~~~~~~~~~~~I
HI!
C)
0
I i I >Jol 1
Io
0
0
-H
~~~I
I UZ~~I
I cI I
I
· Ol
.
I
I
c I
I
m
U)
-V1
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I
I
I
I
I
I
I
I
I
I
-Q
_
H
i i
H
I
1
I! '
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Nq~
-Q
C)
C1)
CN
a4G
-.1
-H1
1
C)
,^
H
0
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I
H
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04I!
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CNM
o4
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I I I .
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ar-;:
> £
Eo
4.
0
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C)
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Q)
>
C4 V
4-4 0OC
I>)I
II >
0
1
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> '1>1>
4
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-E-
0a
0
O
0
Q~
a
g14
-i
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m
I
i
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J©
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'I
Io
Q
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II
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III
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I f I
I £ I
Q
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I
a
0
o
Q
95.
Eq.
Eq.
(3.84)
z
z
Z
4.1
Boundary conditions for the magnetic field intensity.
96.
Eq. (3.90)
Z1
z1
__
___
__
_
__
_
-I
_
Eq.(3.93a,b,c)
Z2
z2
Eq. (3.91a,b,c)
Eq. (3.94a,b,c)
Eq. (3.89a,
b)
z3
_
__
Eq.
Fig. 4.2
I___
__
__
__
(3.92a,b,c)
Boundary conditions for fluid flow variables.
97.
Z1
Z2
Eq. (3.103a)
Eq. (3.9
z3
z 4 (r
Eq. (3.104)
z5(r
z6
-
Fig. 4.3
Boundary conditions for tmeaue
98.
with each other and with other quantities.
4.2.1
Expressions for interior nodes
Fig. 4.4 shows a part of the orthogonal grid; with
a typical node P and the surrounding nodes E,NE,N,NW,W,SW,
SE.
The neighboring nodes need not be equally spaced.
The finite -difference
equations are derived by using the
technique described by Gosman et al 39.
A brief outline
of this technique is presented with respect to the general
elliptic equation
[Eq.
(4.6)].
The derivation of finite
difference equations for the magnetic field intensity
A
(HeR,HeI) is analogous and only the final results will be
given.
Let us proceed by integrating Eq.
(4.6) over the
area enclosed by the small rectangle, shown by the dotted
lines in Fig. 4.4, which encloses the point P.
The sides
of this rectangle lie midway between the neighbouring
grid lines.
The double integral to be evaluated is
rn z e
f
+ a4 a(2¢--)
r
r-+
3zr
-
-
Dr
dr dz
)
Dz
rs z w
convection terms
z
in
r
((c
C
z~
D
(c
b)l
____br
±
Dz
+ b r 4
Dr
C
r z
s w
A,"
-9' .9"
'"c""'
%..A...xr
t
I.
Ad
__
.I L-
I
-J.| - _/
'Y"".
I
)]
-[
dr dz
99.
I
LiJ
,
Z
]
w
(fr
I
>
L.
ci)
C-
I
©
"l
L_
I
I
I
cl
z
I
fI
_
U)
I
1
C
t'-
0
.J)
I
.")
AN
i~~~~~~
Z
(I)
N
L~~~
4.4
Aprino
the finite-difference grid.
100.
rn ze
j
rS
r
drdz = 0
(4.7)
z
w
source term-/
Following the details given by Gosman et al 39 and after
some algebraic manipulation, we obtain the following
successive substitution formula for
which is valid for
any interior point P in the integration field:
{A
j=¢,Ew
j=N,S,E,W
{
+ c
~
. +
i
(b
'
'
+ b
+ S
}
~p
cf (b j + b p)Bj
¢,P ,J
¢,P
}
(4.8)
where
A=
(A
+ A l j )/V
B. = B /{v (b,
j
j {P
j
li
+ b
(49)
P
~~~~~~~(4.9)
(4.10)
(4.10)
¢,P
with
as [
AE
(SE
W=
N
AS
=
S
'8
$N)+I$SE
S-$NE-'N
8
N(NW WN WSW
8
($NE+$E-tNW-W)+ INE+E-$NW-'JW
8
S)+I$NW 'N
INE
SW ~S
J
W SE E)ISW+
[(JSW W - SE-EI
(4.11a,b,c,d)
101.
AlE
A
1W
4 alp(rN
(4.12a)
- rS )
1a
=
4-1~,P (rN - r)
r5
(4.12b)
A1N
=
0
(4.12c)
A1 s
=
0
(4.12d)
b,E
+
b,P
8
b ,W
+
rN -r
zE
b ,p
- (rE + rp)
- ZP
rN -r
(rW + rp)
BW
8
bi,N
+
zP -z
bp
- zW
E W(r
rN - rp
8
BS
b ,s
+
b ,P
(4.13b)
W
z
BN
(4.13a)
+rp)
(4.13c)
zE - z W
E W(r+ r)
(4.13d)
N
8
and
1
Vp
=4 (zE-
zW) (rN - rs) rp
(4.14)
The form of the convection coefficients Ajs, as given by
Eqs.
(4.11a,b,c,d) arises because of the "upwind differencing"
used in representing the convection terms in Eq.
Similarly Eqs.
(4.7).
(4.1) and (4.2) can be integrated
over the area enclosed by the rectangle sw,se,ne,nw
in
Fig. 4.4 and the two resulting algebraic equations can be
102.
solved to give the following expressions for HR
and H I
at any interior node P,
r.
{ D1 -
I
HeR,P
D. HeR
j=N,S,E,W
p
jI
r.
HeI,iJ }
p
(4.15)
{D1[ rp i
j=N,S,E,W
p
=
He
I,j
- D2 -
D.
r
] HGR,j }
p
(4.16)
where
D1 =
D
=
2
S1
1
S1 +
1
(4.17)
S2
2
S2
S2
+
(4.18)
S2
1
2
Z
S1 -
D
(4.19)
]
j=N,S,E,W
S 2 = 0.5
0
(4.20)
P
with
1
2rp
D =
N
rN + rP
(rN - r)(r
N -
rS )
rS) (rN -
rS )
2rp
D
=
2
rS rP
1
DE =
(zE -
(rp
-
1
zP )
(zEB - zWW )
103.
1
DW =
4.2.2
(Zp
1
zW
-
(zE
)
-
zW
(4.21a,b,c,d)
Expressions for boundary nodes
The successive substitution formulae given by
Eqs. (4.15),
(4.16) and (4.8) are valid for interior nodes.
At the boundary nodes, the substitution relationships have
to be derived using the boundary conditions shown in Fig.
4.1, 4.2 and 4.3.
These boundary conditions are of the
following general types:
=
(1)
1
i.e. the value of the variable at the boundary is a specified
constant.
For example, w.r.t. Fig. 4.1,
He = 0
(at
r=0,
0<z
Z6 )
I
A
H0
He
1
( at r= R, 0 <-z < Z
2R0
e
)
etc.
etc.
similarly w.r.t. Fig. 4.2
=k = e= 0
at all rigid boundaries
and w.r.t. Fig. 4.3
TT = Tm
=
T T
(2)
e
at z = Z 2,
s
at r = R
atSR
f(_i) =
O< r< R
-
-
e
Z1
-3
1 -<z<Z
3
104.
i.e. a functional relationship is specified among the
dependent variables.
The functional relationship f is
such that it is possible to solve explicitly for a variable
ijat the boundary node in terms of variables at the same
and adjacent nodes.
For example, w.r.t. Fig. 4.2
3(~
0 - _- 1) ) 2
pr2(Z
0
rI ~I0
-II
2
E
r
K1
at z= Z,0<r<R
- n
where suffixes 0 and 1 refer to a grid node on the boundary
and to the adjacent node in the z- direction, respectively.
(3)
where
a8 is
p
+ qF(~) = 0
the gradient, normal to the boundary surface.
These boundary conditions contain statements for the flux,
normal to the bounding surfaces.
For example, the symmetry
boundary conditions, i.e.
9k-=
Dr
9E
-=
~r
9T =- 0
~r
(p=l, q=0)
fall in this class.
Other examples are (w.r.t. Fig. 4.3)
K -
at r = R
+ h~(T T T )i0
=0
+h
Z < z < Z6
~ P=Ki q=hw
Thi i
F(T)=T-TW
105.
T
eff ~z
-6s
asK
s
T
414
£-sF
T
m sm
Sav
-
F T 4
e se eav
= 0
Re < r< R
at z=Z 1 ,
(4)
P a
+
(4)l3n 1
P2 n
-72 n1
u
i.e. continuity of fluxes (or electric field) across an
interface between media 1 and 2.
For example, w.r.t. Fig.
4.1
13H3
a1
a___ ee=~z
l
z sl
at z= Z ,
2
< r< R
e
and
-K
eff
-jI
zl
+
TR2
X=
-K
k
l
e
at z=Z 3 ,
0< r< R
Boundary conditions of type 1 are specified, once
and for all, at the beginning of the computation.
The rest
of the boundary conditions need updating after every iteration.
Boundary conditions (3) and (4) involve calculating first
order derivatives in a direction normal to the boundary.
The calculation is illustrated below:
With reference to Fig. 4.5, let us suppose that AA'
represents a boundary surface and that we wish to evaluate
at node 0. Let 1 and 2 be adjacent nodes in the direction
n 2.
Let us denote
normal to AA' and that 01 = n and 02
normal to A
and that 01 = n and 02 = n 2
Let us denote
gn
106.
the values of
at nodes 0,1, and 2 by
respectively.
0, (1' and ~2
Assuming a quadratic profile for
we can
write
(4.22)
+ an + bn2
=
Then,
1
+
f=0 + an
(4.23)
bn2
and
2 = (
From Eqs.
(4.24
+ an2 + bn2
(4.23) and (4.24) we get,
n2 (
n2 (1
a
dn 0
~0
at=
For the case when
n(2 (
-
-0)
0
1
)
0O
(4.25)
(e.g. symmetry, free surface
boundary conditions for k,e)
we get,
2
2
1-
2
n
0
2
2 2
n2
(4.26)
2
1
To illustrate let us consider the boundary condition for the
energy equation at the free slag surface, i.e.
Keff
z
= 6IT
s1
By using Eq.
sav
sav
-
F'
T -E
m sm m
F
T
e se e,av
--I
(4.25) we get,
(3.99)
107.
_T
n2 (T
9T
2
1 - T)
z 0
(n
n
2
T
-n
2
_ 1
Putting this in Eq.
T
n~~~~~2
T
=
0
n2
n
2
_
1
1
2
2
-n2)T
2
1
0
(4. 27)
nl)nln2
(3.99) gives,
1-
2
t
1
T
1
T
n2 6£
_
2
n2-n
2
(n
1 2
(n2
2
- n)nn
-
2T
n2 (T - T)
-
1
n
S
+n
1
2
s,av
FT
Tn
m sm m
F'
4
e
se e,av
-
(4.28)
Thus at the end of an iteration, temperature at a node
lying on the free surface of the slag can be updated using
the above equation.
4.3
Wall Function Approach
One of the assumptions inherent in deriving the
successive substitution formulae [Eq.
(4.8)] is that the
transport properties of the fluid vary so little between
grid points that its value at a point such as e in Fig.
4.4 can be given by the arithmetic mean of the values
at P and E.
b
b,e
Thus,
b,
+ b
ypU
q,E
2
(4.29)
It is known however that steep gradients of transport
properties occur near the walls which bound a turbulent
stream.
In the immediate vicinity of the wall, the fluid
108.
A
/
/
/
/
/
/
/
/
id
-
fl
/I
/
/
/
/
0
.
-
--
- -W-
1
2
/
/
/
/
/
/
A
Mr
4.5
L·
-
w-
·
I IIII
_~.
/
Illustration for calculating first order derivatives
at boundaries.
109.
is in laminar motion and the effective viscosity and
thermal conductivity are very much lower than they are
at even a short distance from the wall.
The behaviour
of the near wall region can be modelled either by using
the low Reynolds number modelling method or by using the
46
wall function method
. The latter method has been used
It economizes computer time and storage and
more widely.
allows the introduction of additional empirical information
in special cases, as when the wall is rough.
A brief treatment of wall function method for
turbulence quantities
(k and
the slag is given here.
) and for heat transfer in
Fig. 4.6 shows the regions in the
slag where wall function method has to be used.
These
regions which lie in the vicinity of various rigid walls
are indicated by inclined lines and numbers 1, 2, 3, 4.
Let us now illustrate the use of wall function approach
A portion of the grid in this
wrt region 1 in Fig. 4.6.
region is shown in Fig. 4.7.
Let us assume that region 1 represents a constant
shear layer with the value of shear stress being equal to
T W.
Let n represent distance from the wall, in the direc-
tion normal to the wall.
Velocity distribution for this
region can be written in terms of a "logarithmic law" 55
V +
+
-
i
( En
E n
)
(4.30)
110.
4Ke
-F
Slag
I
1_
'-O
4.6
Domains for wall function method.
4I-
111.
n:
W
SW, -
S
Nw
N
P
se
!,
.---
e
Ne
SEm
E
6'.1
4.7
0
Illustration for wall function approach.
112.
where
K
=
von Karman's constant
=
0.4
E
=
a function of wall roughness, approximately
equal to 9 for a smooth wall
V+ =
V/VT = dimensionless velocity parallel to
wall (z-direction for Fig. 4.7)
V
=
friction velocity = TW/(PCdl/ 4 k 1/ 2
=n+
nV~
=V-,
nT
-
dimensionless distance from the wall.
For Eq.
C1/4 pkl/2 n/Cd
(4.30) to be valid, n
larger than unity.
For turbulent flow in a pipe, Pun and
Spalding 55 suggest that n
tions,
TW
tionship
should be much
> 11.5.
Under these condi-
can be evaluated by using the following rela-
55
=
K
Cd/
kp 1
V
2 /Zn
[Ep/
k
1 2
/4V]
(4.31)
The source term for the turbulence kinetic energy,
Sk can be written as
Sk
=
G - D
=
T
_v
W
n
(3.28)
-
d
d
at
~It
(4.32a)
113.
= TWn
= W
p 2k 2
V
~
- Cd k(4.32b)
n'
'rn
W
Near a wall, length scale of turbulence is porportional
to distance from the wall and one can write 55
C 3/4 k 3/2/(
C=
p
d
(4.33)
/(K
1)
Let us now discuss wall function for the transfer
of heat.
By using the analogy between heat and momentum
transfer, it can be shown 56 that the local Nusselt number
Nu
can be given by a relationship of the following type :
Rex
1/2Cf
Nu
where
~~~x
t {1 +
71/2 Cf
(4.34)
(/t
1) a}
Re X
is
the local Reynolds number
aF
z
is
the Prandtl number of the fluid
at
is the turbulence Prandtl number
Cf'
is the coefficient of skin friction defined
as
and
f
'a'
l/2 Cf' =
w/pV 2
is a parameter which makes an allowance for
the transfer of heat through the viscous
sublayer and depends on the ratio as/at'
Eq. (4.34) is valid in the absence of viscous dissipation
or any other source of heat and for a constant wall temperature.
114.
The final expression for the local flux of heat through
the wall shown in Fig.
4.7 can be written as 55
T /PV 2
w
P
q
_s__
_
_
PCpVp(T
with
P
=
_
at
-T N )
9 (/%
t
- 1) (
+
(4.36)
PvTW/pVp2
'}
zct)1/4
(4.37)
The differential equation (4.6) is integrated over the area enclosed by the rectangle sw, se, Ne,
Nw in Fig. 4.7.
In the case of turbulence kinetic energy,
diffusion through the wall is set equal to zero and source
term is given by Eq.
energy,
p
P
(4.32b).
Dissipation rate of turbulence
is calculated by using Eq.
(4.33).
In the case
of energy equation, heat flux through the wall is replaced
by Eq.
4.4
(4.36).
Solution Procedure
The governing differential equations and their
boundary conditions have now been converted into a set of
algebraic equations.
These equations will now have to be
solved by an iterative technique.
The solution procedure
used in this work is the Gauss-Seidel method and uses
successive substitution as compared to the "block-methods"
which use matrix inversion techniques.
Among the point
methods, the Gauss-Seidel method is known to yield rapid
convergence and is efficient from the view point of
115.
computer storage
39
.
In the following subsections, the
computational flowsheet will be provided together with a
description of the computer program.
4.4.1
Flowsheet for computation
The equations for magnetic field intensity [Eqs.
(4.1),
(4.2)] are first solved to calculate the electromag-
netic driving forces and the spatial distribution of the
Joule heating rate.
This involves using Eqs. (4.15) and
(4.16) to update the values of
H R
and HaI at the interior
nodes and to use algebraic equivalent of boundary conditions to update the values at the boundary nodes.
After
convergence is obtained, the spatial distribution of
current densities and the volumetric Joule heating rate
are calculated using Eqs. (3.14 a,b) and (3.16).
Next,
the dependent variables for flow and heat transfer
k,
-,
r
,
, and T are taken up in the order indicated here.
The procedure adopted can be summarized as follows:
(1) Each cycle of the iterative procedure is made
up of IV subcycles where IV is the number of dependent
variables.
(2)
In each sub-cycle the domain of integration is
scanned row by row and a single variable is updated.
If the node being considered is an interior node, Eq. (4.8)
is used; otherwise, the appropriate substitutional formula
for the boundary node is used.
116.
(3)
When all of the sub-cycles have been completed,
a new iteration cycle is commenced.
(4)
The procedure is repeated until the changes
in the values of the variables between successive iterations are less than a small specified quantity.
A simplified conceptual flow chart of the computational scheme is shown in Fig. 4.8.
4.4.2
Introduction to computer program
In this subsection a brief description of the
computer program is given.
The program was developed by
following the outlines of the basic computer program
published by Gosman et al
39
.
However, extensive modifi-
cations and additions had to be made to tailor it for
the present purposes.
The program is subdivided into
a number of subroutines, each one designed to perform
a specific task.
The listing of the computer program is
given in Appendix E.
The main features of this program
are outlined in Table 4.2 which gives the names and the
functions of various subroutines.
4.4.3
Stability and convergence problems
The search for a solution to the mathematical
problem posed in Chapter III is based on the premise that
the model describes the physics of the system adequately
and that the set of differential equations together with
117.
\0
_~~
SU-~tjP P
7
I ~TA
.1
COMPUTE & STORE GRID PTS.
AND GEOMETRICAL FACTORS
I
I
UP iNT;AL VALUES AND
S,
FIXED BOUNDARY CONiDITIONS
_2-
-
J
I
I-
.
COMPUTE VELOCITY
VECTORS IN SLAG
0
--
--
~~~~~~~~~~~
CALCULATE SHEAR STRESS
B NUSSELT NO. AT W'ALL
No
-o
CONVERGENCE
SATISFIED?
~CRITERION
2
COMPUTE TURS. KINETIC ENERGY
AT iNTERIOR NODES IN SLAG
f
I
~
~~
S
COMPUTE DISSIPATION RATE
OF TURB. ENERGY AT INTERIOR
NODES IN SLAG
CALCULATE CURRENT DIST.
AND JOULE HEAT RATE
r
CALCULATE RADIATION
VIEW FACTORS
COMFUTE TEMERXA7URE AT
ALL INTERIOR NODES
I
I
PERFORM ONE ITERATION
ON BOUNOARY VALUES
_
3
l
;_
~-
Ill
CALCULATE NEW VALUES FOR
TEMPERATURE DEPENDENT
PHYSICAL PROPERTIES
.
__
CALCULATE MELTING
RATE OF ELECTROCE
I
,_,~
4.8
I
CO-MPUTE STREAM FUNCTION
AT INTERIOR NODES N SLAG
COMPUTE MAGNETIC FIELD
INTENSITY AT INTERIOR
AND BOUNDARY NODES
z
rW
COMPUTE VORTICITY AT
!NTERIOR NODES IN SLAG
Simplified flow diagram for the computational scheme
118.
i
z
4.8 Continued
2
3
119.
Table 4.2
Description of the Computer Program
Name
MAIN
Function
Starts the computations and controls the
iteration procedure for flow and heat transfer
variables.
BLOCK DATA
Supplies reference values for physical properties, values for operating parameters and
dimensions of the system as well as control
indices for the program.
CORD
Calculates coordinates of the grid nodes as
well as Dj of Eqs.
Eq.
INIT
(4.13),
(4.14) and Bj' of
(4.6).
Provides initial values for the dependent
variables and computes those prescribed boundary conditions for which no iteration is
required.
FIELD
Solves for magnetic field intensity.
Computes
the distribution of current density and Joule
heat and total power usage.
VF DROP
Calculates radiation view factors and droplet
parameters.
120.
EQN
Performs one cycle of iteration on the complete
(both interior and boundary nodes) set of successive substitution equations by calling
various subroutines.
Also updates physical
property values at the end of each iteration.
VORITY
Computes vorticity at the interior nodes in
slag.
STRFUN
Computes stream function at the interior nodes
in slag.
TURVAR
Computes turbulence kinetic energy and the
dissipation rate of turbulence energy at the
interior nodes in slag.
TEMPR
Computes temperature at all the interior
nodes.
WALL
Computes shear stress and Nusselt no. at
rigid boundaries.
BOUND
Computes dependent variables at those boundary
nodes where iteration is required.
CONVEC
Calculates A' of Eq. (4.6) and makes modificafor incorporating
Jtions
wall functions.
tions for incorporating wall functions.
SORCE
Calculates
source term, S
of Eq.
~p
(4.6)
121.
VELDIS
Calculates velocity distribution in the slag.
VISCOS
Calculates effective viscosity in the slag.
pROP
Allows the use of temperature dependent thermophysical properties.
Calculates the liquidus
and the solidus isotherms and the melting
rate of the electrode.
ADF
Calculates first order derivative by the
three point quadratic approximation.
PRINT
Prints out calculated results.
122.
their boundary conditions constitute a complete and a
well-posed problem.
Even when these conditions are ful-
filled it requires a great deal of
numerical experimen-
tation to ensure that the solution procedure converges
to the "correct" solution.
Another important aspect of
the solution procedure is the speed of convergence, since
the computation time has to be realistically limited.
A discussion on the factors which may influence the convergence, accuracy and the economy of the procedure is
39
given by Gosman et al
The computer program described earlier has evolved
in stages and has involved considerable numerical experimentation.
Following the practice in literature, the
method of under-relaxation is used to reduce the chances
of instability.
If
(N-1)
is the value of the variable
calculated in the (N-l)th iteration and
(N)
is the
value which would be computed in the Nth iteration, then
the value which is actually used in iteration N is
computed from:
=
where
aUR
aUR
(N)
+
(1
UR
-
aUR)
UR
(N-1) ~~~~~~~(4.38)
(4.38)
is called the under-relaxation parameter and
is a number between 0 and 1.
However, the rate of conver-
gence of an iterative solution procedure can sometimes be
123.
improved by over-relaxation
39
.
Mathematically, over-
relaxation can be represented by an equation analogous
to the above equation, i.e.,
=
where
OR
OR
(N)
+
(1 -
OR)
(N-(4.39)
' the over-relaxation parameter, lies between
1 and 2.
During numerical experimentation, the magnetic
field equations were found to be very well behaved and
over-relaxation was found to greatly enhance the rate
of convergence.
For the laboratory scale system discussed
in the next chapter,
for
Hei
6
aOR = 1.5 for
K8R and
OR = 1.2
were found to give the optimum convergence
rate.
In the case of other dependent variables
k,
,
T),
(-,
I,
r
over-relaxation for the stream function and under-
relaxation for the rest of the variables was found to be
the best practice.
The attempt to over-relax the vorticity
equation lead to divergence.
To observe the convergence rate, two different
criteria for convergence were employed.
In one following
39
Gosman et al 39, the maximum fractional change of
in the
field was dictated to be less than a prescribed value,
i.e.;
124.
[(
f
[(N)
-
Usually
e
)/( (N)]max
3mx<
(N-1))/
£1
(4.40)
(4.40)
has been set in the range of 0.001 to 0.005.
It sometimes happens that when the value of a
variable at a particular node is much smaller than the
values at surrounding nodes, fluctuations, in the small
value will occur which are unacceptable by the above criterion, even though the rest of the field has converged.
In this case an alternative criterion used by some other
authors 32,3
was employed.
[I(N)
_(N-1
This is given as
)t
<
F_12
where
X
(4.41}
means summation over all the interior nodes.
£2 has been set in the range
0.001 to 0.005.
The numerical solution was carried out on IBM 370
at MIT.
Calculated results and discussions are given in
the next chapter.
125.
Nomenclature
Coefficients in the general
a,,ai,,b,,cc
elliptic equation (4.6)
Coefficients in the general
A ,B
3 3
substitution formula, Eq. (4.8)
AEAW,ANAS
Coefficients in the convection
terms of the general substitution
formula, given by Eqs.(4.lla,b,c,d)
A1E AlW,A1N AlS
Coefficients accounting for the
movement of interfaces, given by
Eqs.
BE BW,BNB
S
(4.12a,b,c,d)
Coefficients in the diffusion
terms of the general substitution
formula, given by Eqs.(4.13a,b,c,d)
Dissipation rate constant
Cd
Coefficient of skin friction
D1,D
2
Terms in substitution formula
for magnetic field intensity
DN DS,DEDW
Coefficients defined by Eqs.(4.21
a,b,c,d)
E
A function of wall roughness appearing
in logarithmic law Eq.
(4.30)
126.
Nomenclature
A
(cont'd)
He
He, 1%
Complex amplitude of magnetic
field intensity in
- direction,
its magnitude
HeR,HeI
r, IJr I
Real and imaginary parts of He
Complex amplitude of current
density in r- direction, its
magnitude
JrR' rI
Real and imaginary parts of Jr
r
A
JzI IZI
Complex amplitude of current
density in z- direction,
its
magnitude
JzR'JzI
Real and imaginary parts of Jz
z
k
Kinetic energy (per unit mass)
of turbulence
Nu
x
nn+
n,n
Local Nusselt number
Dimensional, dimensionless distance
normal to wall
p
Re
Defined by Eq. (4.37).
x
Local Reynolds number
127.
Nomenclature
Radial coordinate
r
S1,S
(cont'd)
2
Defined by Eqs.(4.19),(4.20)
Source term in the general
S
elliptic equation
(4.6)
T
Temperature
V+,V
Dimensionless, dimensional velocity parallel to wall
VT
Friction velocity
z
Axial coordinate
GREEK SYMBOLS
aUR'aOR
Parameters for under and over
relaxation
Phase angle of H
6116 2163f64
Domains for the wall function
approach
Dissipation rate of turbulence
£:
energy
E1
2
Convergence parameters defined
by Eqs. (4 .4 0),(4.41)
128.
Nomenclature (cont'd)
General notation for dependent
variables
Vorticity
'p
Stream function
Prandtl number, turbulence
Prandtl number
T
Shear stress at wall
K
von Karman's constant
129.
CHAPTER V
COMPUTED RESULTS AND DISCUSSION
The model developed in Chapter III is now used
to make predictions on the flow and thermal characteristics in an ESR system.
Computed results presented in
this chapter show typical temperature and velocity distributions and pool profiles as well as the interdependence
of key process parameters, with the power input, fill
ratio, amount of slag used and the position of the electrode as the independent variables and the casting rate,
pool depth, velocity and temperature fields as the dependent variables.
Wherever possible these predictions will
be compared with experimental measurements available in
57
the literature 57
5.1
Description of the System Chosen for Computation
The experimental results, to which the predictions
will be compared, are those reported by Mellberg 5 7 who
studied the electroslag remelting of ball bearing steels
in a laboratory scale system, using electrodes of 0.057 m
diameter and a stationary water cooled copper mold of 0.1 m
internal diameter.
Remelting was done with alternating
current and the mold was electrically insulated from the
base plate.
The electrode composition is given in Table
5.1 and the remelting parameters are given in Table 5.2.
130.
CN
To
0
H
O
tp
,
0
H,
H
rQ4
rH
0
cl)
0
U)
U)
0
o
44
0
ro
o
r~~~~~t-N
u
0)
G
O
U
ro
0
m
H
*4-
0
o
C~
0
-U)
0)
f
O
-kr'
H
U0
I-
Qf-
0
EP
tO
.4
U
O
0
ro
O
V]
0
o
-r..
3:
,,
£
o\O
131.
q il a
0~~~
4
0
00
~~4
0
0
A
U]
C
x
-W
m
u
H
-4
a)
Q4
54
a)
C)
-Q
a)
N
.H
4
o'p
04
-)
o
H
H
4CJ2
U)
~-
c
~
C'
~4
c-)
Cd
~4)
l
u
4.)
0~~~~~~~C
H
132.
5.2 Physical Properties and Parameters Used in Computation
The computer program described in Chapter IV and included
in the thesis as AppendixE allows for the temperature dependence of physical properties (,C
As described in Chapter
,P,K) appearing in the model.
P
, in order tokeep the equation for
magnetic field intensity decoupled from the flow and heat
transfer equations, the model uses average values for the
electrical conductivities of slag and metal.
However, in some
of the calculations an approximate allowance has been made for
the temperature dependence of electrical conductivity ofslag.
In the absence of specific information on the temperature
dependence of properties for the system being modeled, it was
decided to perform calculations with constant values for the
properties.
Physical property values used in the computation are
listed in Table 5.3.
The liquidus temperature, the density
and the viscosity of the slag were estimated from data reported
in reference 40 and its electrical conductivity was estimated
from compilations made by Hajduk and E
Gammal553.
In the
absence of better information, values for the specific heat,
the thermal conductivity and the coefficient of volume expansion
of the slag were taken the same as those used by Dilawari and
Szekely
15
. The liquidus and the solidus temperature of the metal
were given by Mellberg5 7
Values for the other properties
from
taken
Dilawari and Szekely 15were
and from
from Elliott
Elliott
were taken from
Dilawari
and
133.
11
and Maulvault 11.
These values apply for pure iron or
for intermediate carbon steels.
The value for atomic
thermal conductivity of molten metal was taken to be half
of that for solid metal.
The dimensions of the system and the values for
other parameters appearing in the model are shown in
Table 5.4.
£
The values for the constants C 1 , C21 C d ' ok,
of the k -
and Spalding
model are those recommended by Launder
46
.
These recommendations were made on the
basis of extensive examination of free turbulent flows.
However, one must be aware that these are not universal
constants and may change in different situations.
The
value for the convective heat transfer coefficient
between the electrode surface and the ambient was suggested
2
to be 17.2 W/(m2K)
by
endrykowski et al 33
From experi-
mental measurements, Maulvault 77 calculated this to be in
the range
A value of 25 W/m 2 K was
19 - 32 W/m2 K.
chosen for the present case.
Values for heat transfer
coefficients below the slag-metal interface were deduced
from suggestions made by Elliott and Maulvault 11
11 and
Ballantyne and Mitchell
12
.
As seen in Table 5.4, below
the slag-metal interface, heat removal by cooling water
is represented by three heat transfer coefficients.
Instead of using discontinuous values for heat transfer
coefficient, as has been done here, it will certainly be
134.
Table 5.3
K
Physical Property Values Used
thermal conductivity of electrode,
e
31.39
KsZ
thermal conductivity of slag, 10.46
Km
thermal conductivity of mushy zone, 31.39
K
thermal conductivity of solid ingot,
m
s
KmZ
m
~~~~~~~~~~~~~
31.39
thermal conductivity of molten metal,15.48
HeP
Pe
~
density of electrode, 7.2 x 103
PSQ ~density
PPm
P
of slag, 2.85 x 10
Cpts ~
Cp',s
7.2 x 10 3
specific heat of electrode, 502
specific heat of slag,
837
Cp jCpmCp sspecific heat of liquid metal, mushy
zone,
solid ingot,
753
TZ m
liquidus temperature of metal, 1723K
TSm
solidus temperature of metal, 1523 K
T
liquidus temperature of slag, 1650 K
s
3
density of liquid metal, mushy zone,
solid ingot,
Cpe
m
kgK
135.
coefficient of cubical expansion of slag,
x10-4
X
a
-1
latent heat of fusion of metal, 247 kJ/kg
e
m
electrical conductivity of electrode,
metal, 7.14 x 10
5
mho/m
electrical conductivity of slag,
2.50 x 102
mho/m
e
Fs
emissivity of electrode surface, 0.4
emissivity of free slag surface, 0.6
£
Y
interfacial tension between molten slag
and molten metal, 0.9 N/m
F-I
viscosity of slag, 0.01 kg/m-s
136.
Table 5.4
Numerical Values of Parameters Used In Computation
Re
electrode radius, 0.0285 m
R
mold inside radius, 0.05 m
Z1
free slag surface, 0.30 m
Z2
melting tip of electrode, 0.32 m
Z3
slag-metal interface, 0.37 m
Z6
lower boundary of the ingot, 0.73 m
C1
constant in k -C
model, 1.44
C2
constant in k -
model, 1.92
Cd
dissipation rate constant, 0.09
Prandtlnumber for k, 1.0
aE
IA 0
h
c
Prandtlnumber for
, 1.3
permeability, 1.26
1.26 x
magnetic magnetic
permeability,
x 10
10 -6Henry/m
Henry/in
heat transfer coeffieicnt between the
electrode and the gas, 25.1
W
h
w ,1
heat transfer coeffieicnt at the molten
metal-mold interface, 272
m2K
K
137.
hw,2
heat transfer coefficient at the mushy
zone-mold interface, 272
hw,3
heat transfer coefficient at the ingot-
W
m2K
mold interface, 188
Xw
I0
~
angular frequency of current, 377 radians/s
total current (rms value), 1.4 - 2.5 kA
138.
more realistic to use heat transfer coefficient as a
function of position if such information is available.
The evaluation of the parameter A which appears
in Eq.
(3.44) remains to be discussed.
(3.44),
( 1 +
As seen from Eq.
A ) represents the ratio of effective and
atomic thermal conductivities in the metal pool.
In the
present work an attempt has been made to link this parameter
to operating conditions by evaluating it from the computed
flow field in the slag.
Calculations were carried out for
turbulent fluid flow and heat transfer in both the slag
and the metal pool for the conditions when the shape of
the metal pool was assumed cylindrical and its size predetermined.
This approach is analogous to the one taken
by Dilawari and Szekely
15
.
A typical result on the compu-
ted ratios of effective and atomic thermal conductivities
in both the slag and the metal pools is shown in Fig.
5.1.
This figure represents computation for an operating
current of 1.7 KA and for an assumed metal pool depth of
0.03 m.
Calculations like these indicated that the average
ratio of the effective and atomic thermal conductivities in
the metal pool was about one third of the corresponding
ratio for the slag pool, i.e.
K~
~
~
( = 1 + A
mZ ~KmQ~3
1 ( eff)
KK avg
(5.1)
139.
E
I-li
C0-
D
0LI
-J
U)
n
at:
LiL
IO
a:
U_
H
aw
Q_
iW
an
0
I
2
3
4
5
RADIUS, cm
5.1
Computed ratio of effective and atomic thermal conductivities in both slag and metal pools for operation with
1.7 kA (rms) and for an idealized metal pool shape and
size.
140.
where, as mentioned before, K
and K
denote the
effective and the atomic thermal conductivities of the
metal pool whereas Keff and K denote the corresponding
quantities for the slag.
An attempt such as
Eq.
(5.1)
appears crude at best and it is entirely possible that
the value of the coefficient in this equation would differ
from the value of 1/3 used in the present instance for
different geometries and for much different current levels.
Furthermore, according to the scheme chosen here, we only
account for the mixing or dispersive action of turbulence
in the metal pool, and the convective term in the equation
for heat transport is still unaccounted for.
These criti-
cisms can not be taken care of unless the model is extended
to calculate fluid flow in the metal pool.
In the mean
time, however, it does appear more reasonable to calculate
the effective thermal conductivity in the metal pool from
the model itself (albeit in an approximate manner) than
selecting it to give a better agreement with the experimental measurements.
For the results presented here,
calculated values for A fell in the range
5.3
0.7 - 3.4.
Computational Details
The numerical scheme
outlined in Chapter IV was used
to solve the governing equations.
Fig. 5.2 shows the
grid which is typical of those used in the calculations
presented in this chapter.
As shown in this figure,
141.
r(U)
,
IJ
.I
le8 i%)
I
tI
I I 10J"i
Z(1)
Z I-
-X(I), II
Z2-
-
X (2), I2
-X(3), IA
z3-
m
-X(4),I3
I
-X(5), I4
IN = 51, JN = 12
J1 = 7
I
= 16,
I3 = 27,
-X(i),I5
I2 = 19,
I4 = 31, I5 = 41
X(1) = 0.0285 m
X(2) = 0.3 m
X(3) = 0.361 m
X(4) = 0.37 m
X(5) = 0.38 m
X(6) = 0.43 m
-
Z6-
-
-X(7),
IN
X(7) = 0.73 m
R
R
Fig. 5.2
e
m
Details on the grid configuration.
IA = 24-
= 0.0285 m
= 0.05 m
142.
there are 51 I-lines
( rdirection ).
( z-direction ) and 12 J-lines
While the spacing between the J-lines
was kept the same in all the computations reported here,
spacing between the I-lines was adjusted for various cases
to concentrate nodes in the critical areas.
The computa-
tions were carried out using the IBM 370 digital computer
at MIT; the time interval involved in the computation
was in the range of 100-200 seconds.
5.4
Results and Discussions
A selection of computed results on the electro-
magnetic aspect of the process is first presented and
then results for the flow and thermal aspects are given.
In the following discussions ingots 15 and 17 refer to
operations with 1.7 kA and 1.55 kA (both rms values)
respectively.
5.4.1
Computed results on electromagnetic parameters
Fig. 5.3 shows the radial distribution of the
magnetic field intensity in the slag and in the metal
calculated at different axial positions and for an operating current of 1.7 kA (rms).
Curves 1 and 2 refer to
calculations for the slag at vertical positions, 0.016 m
and 0.044 m below the electrode respectively.
Curves
3 and 4 refer to calculations for the metal at positions,
0.045 m and 0.22 m below the slag-metal interface
143.
8
77
-5
co
ILl
w
0
.'
XCb
-J
ILd
lL
A 4
0ZF
zw
0 <3
2
I-F_
Q.
il
2
0
1
2
3
4
5
RADIUS ,cm
5.3
Computed magnetic field intensity for operation with
1.7 kA (rms).
1
1.6 cm below electrode
2
4.4 cm below electrode
3
4.5 cm below the slag-metal interface
4
22
cm below the slag-metal interface
144.
respectively.
As seen in this figure, the radial distri-
bution of the magnetic field intensity in the metal is
linear and the distributions at the two locations are
identical.
This implies that the current in the radial
direction is negligible and that the axial component of
the current is distributed uniformly across the cross
section (with a density of 216 kA/m2).
In the slag, as
can be inferred from curves 1 and 2, the radial component
of current is finite and becomes small as the slag-metal
interface is approached.
This is readily seen in Fig. 5.4
which shows the current density vectors (length of a vector
represents the rms value of current density) in the slag.
The current path diverges in the vicinity of the electrodeslag interface
but it becomes almost parallel as the slag-
metal interface approaches.
Fig. 5.5 shows the radial distribution of volumetric Joule heat generation rate in the slag for the same
two axial positions as in Fig. 5.3 (i.e. 0.016 m and
0.044 m below the electrode).
In the vicinity of the
slag-electrode interface, heat generation rate is quite
high and the distribution is non-uniform.
As is to be
expected from the discussions given in connection with
previous figures, the radial distribution of heat generation
rate becomes uniform when the distance from the slagelectrode interface increases.
The heat generation rates
__
_
145.
_
_
_
wW
lU
283. kA/m
n
V
2
I
ELECTRODE
w
9
0
N\
If
LUJ
x
I
0
(n
I
-J
t
V
(I)
m
LJ
LU
Li
I iI
Ld
C
m
LI
0
_
I
I
_
I
I
_
2
I
_
I
_
I
3
I_
4
_
_I
6
7
I__
5
RADIUS , cm
5.4
Computed current density vectors
with 1.7 kA (rms).
in slag for operation
146.
3
E
,% 2
0o
x
iW
LUI
LUj
_J
DI
0
0
1
2
3
4
5
RADIUS, cm
5.5
Computed radial distribution of volumetric heat generation rate in slag for operation with 1.7 kA.
1
2
1.6 cm below electrode
4.4 cm below electrode
147.
are completely uniform in the ingot
(65.45 kW/m ) and
in the upper portions of the electrode
(621.6 kW/m3).
Fig. 5.6 shows the effect of electrode penetration
depth in the slag.
Here the solid lines represent
calculations already shown in Fig. 5.5 whereas the broken
lines denote calculations where the electrode protrudes
0.01 m into the slag
previous case).
The total amount of slag used (1.5 kg)
and the power input
cases.
(as compared to 0.02 m for the
(73 kW) were the same in both the
Curves 1 and 2 in this figure have the same
meanings as in Fig. 5.5
As seen here, the case with a
lower electrode penetration depth has a lower volumetric
heat generation rate in the bulk portion of the slag.
This is to be expected since the volume of slag below
the electrode is larger in this case.
The different heat
generation patterns in the two cases will lead to different
temperature distributions in the slag.
This will be
examined subsequently.
Fig. 5.7 shows the effect of the amount of slag
used on the heat generation rate in the slag.
Once again
the solid lines represent the results already described
in Fig. 5.5 whereas the broken lines represent calculations
for a higher amount of slag
former case).
(1.9 kg vs 1.5 kg for the
The electrode penetration depth
(0.02 m)
and the input power (73 kW) were the same in both the
148.
3
...
E
I
O
X 2
LLJ
:]
D0
0
1
2
3
4
5
RADIUS, cm
5.6
The effect of electrode penetration depth on heat
generation rate in slag.
--1
2
depth of penetration of electrode 2 cm,power 73 kW
depth of penetration of electrode 1 cm,power 73 kW
1.6 cm below electrode
4.4 cm below electrode
149.
3
E
32
'N
I
1o
X
:
LLW I
-_
D
0
0
1
2
3
4
5
RADIUS, cm
5.7
The effect of the amount of slag used on the heat
generation rate in slag.
amount of slag 1.5 kg
---amount of slag 1.9 kg
1
2
1.6 cm below electrode
4.4 cm below electrode
150.
cases.
As seen here, the larger volume of slag in the
second case gives rise to a lower volumetric heat generation
rate in the slag.
Based on a uniform current distribution
over the entire cross section of slag, the volumetric
heat generation rate is
1.32 x 105 kW/m 3
with 1.9 kg of slag as compared to
the other case.
for
operation
1.87 x 105 kW/m 3 for
For both the lower electrode penetration
depth and the larger amount of slag used, the resistance
of the system was found to increase in accordance with
2
the operating experience 6
62
Fig. 5.8 shows the effect of fill ratio
(cross
sectional area of electrode/c.s-area of mold) on current
distribution in the slag.
The solid lines denote calcu-
lations for an electrode radius of 0.0285 m (Re1 ) whereas
the broken lines represent calculations for an electrode
radius of 0.015 m (Re2 ).
The mold radius in both the
cases is 0.05 m and the total current in each case is
1.7 kA.
As in the case of previous figures, 1 and 2
denote vertical positions 0.016mand 0.044 m below the
electrode-slag interface.
In the vicinity of the elec-
trode and directly below it, there is an appreciable
difference in the current density in the two cases with the
lower fill ratio case having a much larger current density.
The difference narrows as the radius increases.
From the
discussions given in section 3.7.2, we expect that a larger
151.
c'J
E
is
N-
LU
-J
l,
en
E
H
t
LLU
0
H
z
W
0
0
I
2
3
4
5
RADIUS, cm
5.8
The effect of fill ratio on current distribution in the
slag.
1 1.6 cm below electrode
--fill ratio = 0.09
fill ratio
=
0.325
2
4.4 cm below electrode
152.
current density will give rise to increased velocity
in the slag.
This effect of the fill ratio will be
examined subsequently.
As the slag-metal interface is
approached, the current density tends to be uniform and
the difference in the two cases becomes small.
Another
effect of the reduced fill ratio is that the "effective
cross section" for the passage of the current decreases,
thereby increasing the electrical resistance of the
system and the heat generation rate.
Thus the total
power input for the case with 0.015 m electrode radius
(fill ratio = 0.09) is calculated to be 94 kW as compared
to 73 kW for the case with 0.0285 m electrode radius
(fill ratio = 0.32).
The experimental observation of
this effect is reported in the literature 61,
62.
Before closing this subsection, it should be noted
that the computed current distribution and the heat
generation pattern, and consequently fluid flow and
temperature distrubution, will depend on the assumed shape
of the electrode tip and on the assumptions made regarding
the boundary conditions
(e.g. insulating slag skin on the
inside wall of the mold , the presence of a solidified
slag crust on the submerged vertical wall of the electrode).
As mentioned in Chapter III, the model developed in this
work assumes a flat melting tip for the electrode and an
insulating slag skin on the interior surface of the mold.
The latter assumption is in accord with observations made
153.
in literature 40
The assumption of a flat melting
tip is made for computational convenience and ideally
the shape of the electrode tip should be calculated using
the mathematical model itself.
However, this refinement
to the model can only be accomplished if a more realistic
understanding is developed of the melting process.
This
will require both experimental and analytical work.
The effect of a solidified slag crust on the
submerged, curved surface of the electrode will be analogous to the effect of a reduced fill ratio since the
effective c.s.area for current will decrease.
Then,
from discussions given in connection with Fig. 5.8, for
the same total current, power requirement will be higher
for this case and the current distribution will be less
uniform.
From the preceding discussions, the presence
of a solidified slag crust on the electrode will be
expected to increase the velocity in the slag.
These
observations are confirmed by calculations reported by
Dilawari and Szekely 14,59
5.4.2
Computed results on fluid flow and heat transfer
Results will now be presented for the temperature
and velocity distributions in the system.
Computed re-
sults on the temperature distribution will be compared
with the limited measurements available.
154.
It is noted that most of the calculations were
carried out using a constant electrical conductivity of
the slag.
In some calculations, however, an allowance
has been made for the temperature dependence of the
electrical conductivity of the slag.
The modifications
required to the governing equations in order to accomplish
this are discussed in Appendix D.
Fig. 5.9 shows a comparison between the experimentally measured axial temperature profile at a radial position r = 0.025 m and those predicted from the model.
The
discrete data points denote measurements, the broken line
denotes predictions for the condition where the experimentally measured casting rate was used as an input to the
model.
The two solid lines denote predictions for the
condition when the casting rate was computed from the
model; curve I refers to calculations using a uniform
electrical conductivity in the slag while curve II refers
to calculations using
a temperature dependent electrical
conductivity in the slag.
As discussed in Appendix D,
temperature dependence of the electrical conductivity used
in the calculations was deduced from experimental measurements reported by Mitchell and Cameron 60
in both the cases is kept the same (73 kW).
Power input
It is seen
that measurements correspond to a steeper axial temperature
gradient in the vicinity of slag-metal interface than that
155.
^f%
0
0
aL-
w
-2
0
2,
4
6
8
DEPTH FROM SLAG-METAL INTERFACE, cm
5.9
Computed and measured axial temperature profiles for
ingot 15
-
(rms current= 1.7 kA) at r = 2.5 cm;
I
casting rate (Vc) calculated from the model
c
uniform electrical conductivity in slag
II
temperature dependent electrical conductivity in
slag
--- experimentally measured value of V
(0.74 m/hr)
input
as
an
to
the
model
cused
used as an input to the model
C
measured values from lMellberg 57
156.
exhibited by the calculated results.
This, as pointed
out by Melberg, may partly be due to inaccuracies involved
in experimentally defining the zero position.
In general,
all three curves shown in Fig. 5.9 provide a reasonable
representation of the data points below the slag-metal
interface.
Curve II provides a somewhat better agreement
with measurements primarily because in this case, calculated
casting rate
(0.7 m/hr)
in the metal pool
and effective thermal conductivity
(27 W/mK) are lower than the corresponding
values for case I (casting rate = 0.76 m/hr, effective
thermal conductivity in metal pool = 30 W/mK).
One important derived quantity in these calculations is the radial temperature distribution at the slagmetal interface.
Since there is not much difference in
the three cases cited in Fig. 5.9, only the distribution
for case I is discussed.
This is shown in Fig. 5.10.
As seen here there is quite an appreciable radial variation in the temperature
(about 225°C in the present case).
It is noted that several authors, when modelling pool
profiles and the temperature fields in the ingot, used the
temperature at the slag-metal interface as an arbitrarily
adjustable boundary condition.
The choice of this boundary
condition may be critical, because this represents the
coupling between the heat source in the slag and the heat
transfer processes that take place within the ingot.
157.
170
W
0
w
O
Ww
z
t-
160
Ld
wC3
en
-J
I)
0
150'
U
H_
140(
0
I
2
3
4
5
RADIUS, cm
5.10
Computed temperature distribution at the slag-metal
interface for ingot 15.
158.
Fig. 5.11 shows an alternative way of expressing the
radial distribution of temperature at the slag-metal
T
-T
interface.
Here, Zn( T max
) is shown plotted against
T
-T
min
~max
rA~
n(~-) where T
and T
are the maximum (1675°C) and
RM
max
min
m
the minimum (1450°C) temperatures respectively and Rm
is the radius of the mold.
From this figure, it can
be shown that
T
- T
max
r
1.375
=____(-)
T
-T
max
min
(5.2)
R
m
This type of relationship has been used by other authors
to specify temperature distribution at the top of the
ingot.
Fig. 5.12 shows the computed solidus and liquidus
isotherms for the three cases that were represented in
the previously given Fig. 5.9.
Also shown, for the sake
of comparison is a sulfur print reported by Mellberg,
for corresponding operating conditions.
It is seen that
the sulfur print falls between the solidus and the liquidus
curves, as one would expect.
As before, curve II which
represents computation with a temperature dependent
electrical conductivity in the slag gives a closer agreement
with the experimental measurements.
Fig. 5.13 shows a comparison between the experimentally measured and the theoretically predicted temperature
profiles, for an ingot produced with a 1.55 kA current, at
159.
3.0
2.5
C
2.0
E
I-
1.5
Ia
'E
I0
E
Hx
1.0
c
I
0.5
0
0.5
1.0
1.5
2.0
-In (r/Rm)
5.11 Results from Fig. 5.10 expressed in an
alternative way.
160.
E0.,E
w-
I
_J
O
I
0
Lb.
LU
-
a.
w
a
o
I
2
3
4
5
RADIUS, cn
5.12
Computed liquidus and solidus isotherms for ingot 15.
-0-sulfur
print from Mellberg 57 rest of legends
-0- sulfur print from Mellberg
, rest of legends5.9.
same as in Fig. 5.9.
161.
hUUI
180
0
0
C)
T-
IGO
H
140
Ld
I--
H:
120
100
-2
0
2
4
6
8
DEPTH FROM SLAG-METAL INTERFACE,cm
5.13
Computed and measured axial temperature profiles for
ingot 17.
-
--*
casting rate (V ) calculated from the model
c
experimentally measured value of V (0.62 m/hr)
used as an input to the model
c
measured values from Mellberg
162.
the mid-radius position.
Here again the solid line repre-
sents the predictions for a condition, when the casting
rate was computed, while the broken line corresponds to
predictions, where the experimentally determined casting
rate was used as an input for the model.
It is seen once
again that there is reasonably good agreement between the
measurements and the predictions below the slag-metal interface.
The computed liquidus and solidus isotherms for these
two cases are shown in Fig. 5.14.
pose of
Also given for the pur-
comparison are some of the measured locations
of the isotherms.
It is seen that the solidus and the
liquidus lines predicted by the two techniques are comparable, and that there is a somewhat better agreement between
measurements and predictions for the case in which the
casting rate is calculated from the model (0.64 m/hr vs.
measured value of 0.62 m/hr).
Fig. 5.15 shows the computed isotherms in the
slag for ingot 15.
The solid lines represent calculations
with a uniform electrical conductivity of the slag while
the broken lines represent calculations with a temperature
dependent electrical conductivity. As seen here, the latter
case gives a lower maximum temperature (1787
C vs. 1832 °C
for the former case) with a little more uniform distribution.
This is as expected since the electrical conductivity of the
slag increases with temperature.
It is seen that the
163.
U
2
E
c.)
a
3W
L
C)
Q:
lc
LI
4 Fz
_1
5
6
W
E
!l
_J
CO
-J
Cf)I
E
0
CC
r
F7 0.
.
W
CX
B
9
10
RADIUS, cm
5.14
Computed liquidus and solidus isotherms for ingot 17.
Legends same as in Fig. 5.13.
164.
E
W
LI
C)
C,
0
UJ
en
CD,
.. I
w
w
L
m
Cr
IL
1-r
a
Li
0
I
2
3
4
5
RADIUS, cm
5.15.
Computed isotherms in slag for ingot 15.
uniform electrical conductivity in slag
temperature dependent electrical conductivity
in slag
165.
hottest region of the system is in the central portion,
close to the electrode, but not in the immediate vicinity
of the electrode.
It has to be stressed that at the surface of
the electrode the temperature in the slag must equal the
melting point of the electrode material, hence the positive
axial temperature gradient in the immediate vicinity of
the electrode.
This positive axial temperature gradient
is also consistent with the physical requirement that
thermal energy has to be transferred from the slag to the
electrode.
It should be noted that the temperature profiles
depicted in Fig. 5.15 result from the combined effect
of Joule heating and the convective fluid flow field in
the slag phase.
If convection had been neglected, one
would have obtained unrealistically high temperatures
in the vicinity of the electrode surface.
An important
effect of this convection, readily seen in Fig. 5.15,
is the quite uniform temperature field in the central core
of the slag; indeed, most of the temperature gradients
appear to be confined to the vicinity of the wall.
This knowledge of the temperature field is of
course quite important, if we wish to represent chemical
reactions between the droplets and the slag, that may
be temperature dependent.
166.
Fig. 5.16 shows the effect of electrode penetration depth in the slag on the isotherms in the slag.
The isotherms shown here are computed for an electrode
penetration depth of 0.01 m.
The amount of slag used
(1.5 kg), the input power (73 kW) and the value used for
the effective thermal conductivity in the metal pool
(30 W/mK) are the same as in the case of Fig. 5.15.
Comparing Figures 5.15 and 5.16 reveals that the temperatures in the bulk portion of the slag below the electrode
are somewhat reduced when a lower electrode penetration
depth is used (maximum temperature 1807
for the other case).
C vs. 1832 °C
This is to be expected from the
discussions given in connection with Fig. 5.6 where it
was shown that a lower penetration depth resulted in a
lower volumetric heat generation rate in the bulk portion
of the slag.
Also the casting rate is now reduced (0.61 m/hr
vs. 0.76 m/hr for the other case).
As is to be expected
from the discussion given in connection with Fig. 5.9, the
agreement with measured temperatures below the slag-metal
interface was somewhat better.
However, the temperature
profiles in the ingot are not shown here, for the sake of
brevity.
Although the immersion depth of the electrode
appears to have an important effect on the casting rate
and hence on the position of liquidus and
solidus iso-
therms, its effect on the temperature and velocity in
167.
O
E0
2
w
0
LicD
e)
3O
-5
I)
ww
4 cc
L0M
iw
0
6
0
1
2
5
4
5
RADIUS, cm
5.16
The effect of electrode penetration depth on temperature distribution in the slag.
168.
the slag is less significant from the global viewpoint.
Also, it should be noted that in practice, the immersion
depth of the electrode is determined by the process itself.
However , this aspect of the ESR process is yet to be
modelled.
In the next few figures, calculated results on
the flow and turbulent mixing in the slag are discussed.
Fig. 5.17 shows the calculated stream lines in the slag
for ingot 15.
Computed velocity vectors in the slag are
presented in Fig. 5.18.
It follows from the discussions
given in Chapter III that there are two principal forces
that drive fluid motion in the slag -- the electromagnetic
force field which in the present case would tend to generate
an anti-clockwise circulation pattern and the buoyancy
force field, which would tend to generate a clockwise
circulation pattern in the bulk of the slag.
It is seen
in Figs. 5.17 and 5.18 that for the case considered,
buoyancy forces tend to dominate in the bulk of the slag,
producing a clockwise circulation pattern.
The anti-
clockwise circulation pattern in the annular space between
the electrode and the mold results from the combined
effects of buoyancy and electromagnetic forces (if buoyancy
forces alone acted in the region one would obtain two
circulating loops, as caused by a hotter fluid being located between the two cold surfaces, viz the electrode and
169
E
U
ULU
0
IIL
F0A
LU
MD
0
1
2
3
4
5
RADIUS, cm
5.17
Computed streamline pattern in slag for ingot 15.
170.
-OI
I-
U
ct--I
UU·IIII-
ELECTRODE
\
---
2 cri/sec
/
--
E
-~ ----
I
j/7
/ lo
---
/
I/
I
P
/
i
_J
1)
¢D
v.)
L
a~~~~~'~
1 I
I t
/
-*
I
-
LCCCL
t
t
*\,~ V-"
-,
,
*
4
fm w-~~~~~~~~~~~~~--- --I~~
0
i
-
V--
0
5- W
CL
/
Hj
.0
-
-
w
I
i
6
-
!
I
I
2
3
4
RADIUS
5.18
9.
C)
0
I
- \
[LI
0
7
57
cm
Computed velocity vectors in slag for operation with
1.7 kA and for a fill ratio of 0.325.
171.
the mold wall).
The absolute magnitude of the velocities
calculated for the system, viz 0 - 5 cm/s are comparable
to computed values reported in earlier papers describing
fluid flow in ESR process 13-15
The circulation is an important characteristic
of ESR systems and it would be desirable to define the
conditions under which either the electromagnetic or the
buoyancy forces dominate the flow field.
By dimensional
arguments it may be shown that the following group may
be used to define the nature of the circulation:
- 2
-_~
4TA Lp
e
where A
e
and A
m
Ae
(1 - Am)
(5.3)
gAT
are the cross sectional areas of the
electrode and the mold respectively, L is the depth of
the slag below the electrode, and I
is the rms value
of the total current.
The higher the value of
the stronger is the
domination of the electromagnetic forces.
Thus the
relative importance of electromagnetic stirring is
increased by increasing the current, decreasing the fill
ratio and decreasing the slag depth below the electrode.
These findings seem reasonable on physical grounds,
because a diverging current field will produce a stronger
J x B force.
Extreme examples of this have been found in
electroslag welding, using wire electrodes.
172.
Figure 5.19 shows the velocity field computed
for an identical current to Fig. 5.18 but for a greatly
reduced fill ratio
vious case).
(0.09 as opposed to 0.32 in the pre-
It is clearly seen that under these conditions
the flow is now dominated by electromagnetic forces.
Also as expected from discussions given in connection with
Fig. 5.8, the magnitude of the velocity increases when
the fill ratio is decreased.
Fig. 5.20 shows the radial distribution of the
ratio of buoyancy/electromagnetic contributions to vorticity, some 0.02 m below the electrode, computed for
the cases that have been given in Figs. 518 and 5.19.
It is seen that buoyancy forces dominate in the
vicinity of the walls because of the very steep radial
temperature gradients.
However, the curve drawn with the
broken line, depicting the behavior of the system shown
in Fig. 5.19 clearly indicates that the electromagnetic
forces dominate in the central core for that case.
Fig. 5.21 shows the computed values for the
turbulence kinetic energy in the slag for ingot 15.
The
maximum value for this parameter occurs in the annular
space between the electrode and the mold and this value
A
A,
1
173.
U
E
2 o
w
0
L<
0
-j
Un
L.
E
4 O
W,
L;
0
5
M
L
M
LA
0
6
.4
7,
0
1
2
3
4
5
RADIUS, cm
5.19
Computed velocity vectors in
slag for operation with
1.7 kA and for a fill ratio of 0.09.
174.
-
o Ca
%I
_
0
1
2
3
.4
5
RADIUS , cm
5.20
Radial distribution of buoyancy/electromagnetic contributions to vorticity, 2 cm below the electrode.
1.7 kA, Ae/Am = 0.325,
1.7 kA, Ae/Am = 0.09
~ (E
.(5
3)1=
1.7
175.
F%
0
%
0
IL
C)
(9
_J
U)
4 a
li
U.
LL
0
5
L_
FLd
6
7
i
0
2
3
4
RADIUS, cm
5.21
Computed distribution of turbulence kinetic energy
in
the slag for ingot 15.
176.
is about 25% of the maximum kinetic energy of the mean
motion.
Fig. 5.22 shows a plot of the ratio:
effective
thermal conductivity/atomic thermal conductivity in the
slag for ingot 15.
It is seen that the strong convective
field does indeed produce turbulent conditions, corresponding to an appreciable enhancement of the effective
thermal conductivity.
As in the case of Fig. 5.21, the
regions of the maximum effective conductivity correspond
to the zones where the circulation has its maximum intensity -- a behavior that is consistent with physical
reasoning.
The role of the current in determining the shape
and the depth of the metal pool is examined in Figs. 5.23
and 5.24.
Fig.
5.23 shows computed pool profiles for
different current levels.
The curves, drawn with the
solid line, show that the higher the current, the deeper
the pool profile.
This finding has been established
experimentally a long time ago; however, this is the
first time that these experimental findings have been
predicated from first principles.
The curve drawn with
the broken line (2A) depicts the computed pool profile
177.
E
LL
Ocr2
C)
CD
3 <
CO
I)
(9
IiJ
LO
4
L
2:
5 0FLiJ
0
6
7
0
1
2
3
4
5
RADIUS, cm
5.22
Computed contours of the ratio effective thermal
conductivity/atomic thermal conductivity in
slag
for ingot 15.
the
178.
E
U
w
0
I-
W
z
-J
,O
IE
wi5
0
co
nCU)
W
IL
0
I
2
3
4
5
RADIUS, cm
5.23
Computed metal pool depths for different currents.
1. 1.55 kA, 2. 1.7 kA, 3. 2.0 kA, 4. 2.5 kA
In all these cases amount of slag = 1.5 kg.
2A. Same power as 2
(73 kW),
amount of slag = 1.9 kg.
179.
ir
8
E
04-
I'- 6
Ld
--
0
0
X;
-o
ZZ
As
1.5
2.0
CURRENT (rnms)
5.24
2.5
kA
Variation of maximum pool depth with current.
180.
for identical conditions of power to that given by
(2)
but for a greater amount of slag present (1.9 kg for the
case 2A vs.
1.5 for the case 2).
Heat loss from the
slag to the mold wall for the case 2A was 30% higher
than that for the case 2.
pool depth.
Fig.
This resulted in a smaller
5.24 shows a plot of the computed
maximum pool depth against the current used.
Over the
range examined the relationship is found to be linear.
Fig.
5.25 shows a comparison between the theoretically
predicted casting rates, as a function of power and the
values reported experimentally by Mellberg
57
seen that the agreement is not unreasonable.
.
It is
The pre-
dicted relationship between the casting rate and the
power is linear.
ting experience
This again is consistent with opera62
62
Values for the parameters associated with a
metal droplet are shown in Table 5.5.
The superheat
given this table is for operation with 1.7 kA (i.e.
ingot 15).
As seen here, the residence time of the
droplet in the slag is very small.
city of a droplet
(0.28 m/s)
The average velo-
is substantially larger
than the velocity of slag and therefore the assumption
of a quiescent slag made in calculating droplet para-
181.
1.50, Y_
__
_____
__
_I_
_ ____IC_
__
__· _
I
_____
C
__
_ ___
_
_
1.25
i.00
k-
0: 0.75
~m
E
'1
w
to
0.50
C)
o VALUES FROM MELLBERG
0.25
0
4
40
I
60
-L
I
80
I1I
I·
I00
I
-----------
·
120
Variation of casting rate with power
* measured values from Mellberg
I
140
POWER, kW
5.25
3
.
-·
--
1
160
182.
Table 5.5
Values for Droplet Parameters
Parameter
Values
diameter, dd
1.3 cm
residence time, T
0.18 s
average velocity, Uav
28 cm/s
superheat,
(T
- Tm,
e ).
m _e
.
72
.C
...
183.
meters
(Section 3.5.4B)
is reasonable.
This concludes the discussions of the computed
results.
The results presented in this chapter were
selected to illustrate the predictive capability of
the model developed in this work, and to investigate
the interdependence of key process parameters.
cluding remarks are made in the next chapter.
Con-
184.
CHAPTER VI
CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK
In this chapter concluding remarks are made and
some suggestions are made for further work in mathematical
modelling of ESR process.
6.1
Conclusions
In the work reported in this thesis a mathematical
model has been developed to describe electromagnetic field,
fluid flow, heat transfer and solidification phenomena in
ESR systems.
The model involved the simultaneous statement
of Maxwell's equations, equations for turbulent flow and
the differential thermal energy balance equations.
These
equations were first written in vectorial forms so that
some general conclusions could be drawn regarding the
behavior of ESR systems.
Then the equations were presented
in the cylindrical coordinate system with axial symmetry.
The limitations of the model are inherent in the assumptions
made in developing the model.
While these assumptions are
detailed in Section 3.3, some of the principal shortcomings
of the model in its present form are as follows:
(1)
only.
Fluid flow equations are solved in the slag
Motion in the metal pool is accounted for by using
an effective thermal conductivity.
However, as discussed
in section 5.2, an attempt is made to deduce this parameter
in the model itself.
185.
(2)
The model assumes a predetermined shape
(flat)
for the melting tip of the electrode and a known electrode
penetration depth in the slag.
Ideally these parameters
should be calculated using the mathematical model.
However,
this refinement can be made only if a more realistic understanding is developed of the melting process.
(3)
A constant value is used for the electrical
conductivity of the slag.
In some of the calculations
presented in Chapter 5, an approximate allowance has been
made for the temperature dependence of this parameter.
(4)
Effects associated with chemical and electro-
chemical reactions are not considered.
On the positive side, the model accomplishes
the following:
(1)
It allows for turbulent flow in slag as
caused by both electromagnetic and natural convection forces.
(2)
It accounts for the spatial distribution of
heat generation rate,
the transport of heat
by the metal
droplets falling through the slag and for the movements
of the ingot and the electrode.
(3)
By integrating transport processes taking
place in different portions of an ESR system, the model
allows predictive relationships to be developed among key
process parameters.
The governing differential equations developed in
Chapter
III
were
solved
using
a
numerical
186.
technique outlined in Chapter IV to provide current distribution, velocity and temperature profiles for a laboratory
scale system used by Mellberg
57
.
In general the theoreti-
cal predictions for the temperature profiles and for the
pool profiles were found to be in reasonable agreement with
the experimental measurements, thereby indicating experimental support for the model.
The predictive capacity which is inherent in this
model enables one to develop theoretical relationships
predicting the interdependence of the key process parameters.
For example,
it has been possible to relate the heat gener-
ation pattern, the temperature and the velocity fields, the
melting rate and the pool profiles to the operating power
and current and to the geometry of the system.
The principal findings may be summarized as
follows:
(1)
The temperature at the slag-metal interface
was found to be strongly spatially dependent.
As seen
from the work reported by Kreyenberg and Schwerdtfeger 16
the computed temperature and velocity distributions in
the slag are strongly affected by the assumed temperature
distribution at this interface and therefore it should be
computed by the model rather than being specified arbitrarily.
(2)
It was found that for a fixed geometry, the
higher the current, the deeper was the pool profile;
this
187.
mode of operation also resulted in a faster casting rate.
Over the range of parameters used in the computation, the
relationship between maximum pool depth and the current
and the relationship between casting rate and power were
found to be linear.
(3)
A deeper slag bath results in a larger heat
loss through the mold wall.
Thus for the same input
power, a larger amount of slag gives rise to a lower
pool depth.
(4)
The depth of penetration of the electrode was
found to have some effect on the slag temperature; more
specifically the lower the penetration, the lower the slag
temperature in the bulk and hence the lower the casting
rate.
However as mentioned before, in practice the
electrode penetration is a factor that is determined by
the process itself and this aspect of the problem is
yet to be modelled.
(5)
The model enables us to distinguish between
the role buoyancy forces and electromagnetic forces play
in driving the flow in the slag phase.
It was found that
the higher the current and the smaller the fill ratio,
the more important was the role of electromagnetic
forces.
It has to be stressed, however, that in general,
both these forces could play an important role in determining the flow field.
188.
(6)
The temperature distribution in the slag was
found to be quite uniform with strong gradients in the
vicinity of solid walls.
(7)
The use of a temperature dependent electrical
conductivity for slag gave a reduced maximum temperature
in the slag and a somewhat more uniform temperature distribution.
This resulted in a lower amount of heat being
transported to the electrode and thus in a reduced casting
rate.
(8)
The ratio of effective/atomic thermal conduc-
tivity in the metal pool, computed according to suggestion
outlined in section 5.2 was found to lie in the range of
1.7 to 4.5.
This is consistent with values deduced from
experimental works 66
In closing, it is worthwhile to comment briefly
on the principal differences and similarities between
the present work and the earlier work reported by Dilawari
and Szekely 13-15
1315.
As mentioned in Chapter II, in the
earlier work, the electromagnetic field, the fluid flow
field and the temperature field were represented for an
ESR system for a predetermined and idealized pool shape
and size.
Heat transfer phenomena in the mushy zone and
in the ingot were ignored.
While many of the transport
equations and the boundary conditions used in the present
work are identical to those used in the earlier work, the
advances made in the present work are as follows:
189.
(1)
The shape and the size of the molten metal
pool is no longer specified but calculated by solving heat
transfer equations in the mushy zone and in the ingot.
The present work allows for the incorporation of solidification models to represent the release of latent heat in
the mushy zone.
(2)
The model allows for the first time for a
comparison to be made between the measured and the predicted
pool profiles and temperature fields in the ingot.
(3) A more up-to-date model is used to represent
the turbulent viscosity and the turbulent thermal conductivity in the slag.
The model allows for special treat-
ment of flow and heat transfer phenomena in the near wall
regions.
6.2
Suggestions for Further Work
From the point of view of mathematical modelling,
ESR process represents a fascinating, if complex, group
of problems.
The model described in this work was an
attempt at grouping together the salient features of
current distribution, fluid flow and heat transfer phenomena so that the model could be used to investigate the
interdependence of key process parameters.
A few sugges-
tions are given here for some further work in the area
of modelling of ESR process.
The suggestions given here
are limited to mathematical modelling but it should be
190.
recognized that physical modelling is a vast and extremely
important area and that there is a need for a strong
interaction between the two.
6.2.1
Suggestions for short term plans
These include the following:
(1)
In the work reported in this thesis, limited
measurements on a laboratory scale system were used.
It
would be desirable to have some measurements, specially
on temperature distribution in the slag, available for
a large scale system so as to be able to test the predictions of the model for this system.
(2)
The model, in its present form, uses an
effective thermal conductivity to account for convective
heat transfer in the metal pool.
The model can be extended
to calculate flow in the metal pool.
To achieve this it
will be advantageous to employ a numerical scheme
11
of the type used by Elliott and Maulvault 11 below the
slag-metal interface.
(3)
An attempt can be made to use the information
on velocity and temperature distribution generated by the
model in developing a kinetic model.
6.2.2
Suggestions for long term plans
These include the following:
(1) There is a strong need for both experimental
and analytical work in order to develop a realistic under-
191.
standing of melting phenomena at the slag-electrode
interface.
(2)
The model developed in this work can be
extended to calculate fluid flow in the mushy zone so
that it can be used to calculate macrosegregation in
the ingot.
(3)
Some preliminary calculations can be made
for multiple electrode configurations.
The modelling of
multiple electrode system represents a major additional
difficulty, because under these conditions axial symmetry
is no longer observed.
(4) An approach similar to the one described
in this thesis can be used to model the electroslag
casting process.
192.
APPENDIX A
A BRIEF NOTE ON PHASOR NOTATION
In dealing with sinusoidally time-varying quantities
it is convenient to use phasor approach.
A phasor is a
complex number which can be represneted graphically as shown
in Fig. A 1.
The length of the line is equal to the magni-
tude of the complex number and the angle that the line makes
with the positive real axis is the angle of the complex
number.
For example when the functional form is cosinusoid-
al, i.e. A cos
(wt + P), the magnitude of the phasor is
equal to the magnitude A of the cosinusoidal function and
the angle of the phasor is equal to the phase angle
the function for
is
A cos4
t = 0.
of
The real part of the phasor
which is the value of the function at
t = 0.
As seen from Fig. A.1, if the phasor is rotated about the
origin in the counter-clockwise direction at the rate of
w rad/s, the time variation of its projection on the real
axis describes the time variation of the cosinusoidal
function.
Using the terminology adopted in Chapters III
and IV, a cosinusoidal function
f
can be represented, in
phasor notation, as
where
f
=
Re
A
=
JAI
IAI =
[A e
e
]
j
magnitude of the phasor at
(A.1)
(A.2)
t = 0.
193.
.
_ ..
.
i----
/S
I
I
I
(a.
1
EAL
0
V,
.
A-
.
I
Gahia
REAL PART
II
representation of a phasor.
194.
Similar considerations hold when dealing with sinusoidally
time-varying vectors.
To illustrate the application of phasor concept,
let us derive the expression for time average electromagnetic body force component
F
[i.e. Eq. (3.15b)].
z
(3.10), the instantaneous value of this quantity
From Eq.
can be written, in phasor notation, as
jWt
"'
F'
Re(H
=
z
e
-0
0 IHeI
=
=
i
cos(Wt +
I0 HeIJrI
^
= 2e0
-
t) x Re(J e
~~~~~~~~A r
^
He
1) x IJrl
[cos(Wt +
Jrl
jc~tt
(A.3)
cos(wt +
1) cos(Wt +
[cos(2wt +
1 +
2)
2) ]
2)
I
+
cos
(
-
2) ]
(A.4)
II
The time-average value can be found as follows:
T
F
where
T
=
F '
dt
(A.5)
is the period.
The time-average value of the first term in Eq.
(A.4) is equal to zero, thus
195.
^ IIJr' ^
cos(Pe 1 2]
I~ HaH
2 VO
0 Re [ IH9 |r'
Z=
!2
Fz=
2
0
J
IHE I e
Re[
*
]
^ Je
IJrl
^
1
=2
where
i
gl
_1
0
2]
(3.15b)
Re( r )
is the complex conjugate of
Jr.
Following a similar approach, expressions can be
derived for other time-averaged parameters such as
Fr
[Eq.
(3.15a)], Q
[Eq.
(36)
] , etc.
196.
APPENDIX B
BOUNDARY CONDITIONS ON VORTICITY
In this appendix, expressions will be derived for
vorticity at the axis of symmetry and at the walls.
B.1
Vorticity at the Axis of Symmetry 39
Form Eq.
Vv
-
(3.23b)
1
a2
pr
9r
in the text,
It follows that in order for V
z
to be finite at
r = 0,
must tend to zero at the same rate as r near the axis.
It follows that in the immediate vicinity of the axis,
the
-
r
distribution is parabolic.
Furthermore,
because of symmetry the second term in the
~
-
r
expansion
should be the fourth power one; thus:
where
40
ar 2
W0=
+ br 4
(B.1)
is the stream function at the axis and a and b
are constants for a fixed z.
Using the definition of vorticity
the symmetry condition
V
r
= 0
at the
[Eq.
(3.22)],
r = 0
and Eq.
(B.1) we can write
8br
P
\k
.
G
197.
Thus
r = 0,
at
= 0
i
on the other hand is finite.
-
r
If 1 and 2 denote the nodes which are once and twice removed
from the node 0 on the symmetry axis, we can evaluate b
from
2
1
-
0
=
-0
-
2
= ar1
4
+ br1
ar2 2 + br2
4
(B.3a,b)
2'Po2
Then Eq. (B.2) gives;
(i)
(i)
B.2.
~
=
~r
[
0 -
2 +1
-20/r
2
2
2
8 [022+120+ We
0]/(r
- rl2)
P
r2
r!22
p r~~~~
1
(3.89b)
2
Vorticity at a Wall 53
Let us illustrate the calculation of wall vorticity
w-r-t slag-electrode interface.
As seen in Fig. B.1, 0 is
a node on the wall and 1 is the adjacent node in the zUsing Taylor series expansion, the value of
direction.
stream function at node 1,
1
1 2q
'P
= 'P
1A
+
can be expressed as
2
13
34
3 +
(
(B.4)
where
A
From Eq.
=
1
(B.5)
- z0
(3.23a)
3z2o
Tz
0
=-PrV1
0
(no slip condition)
(B.6)
198.
w
a
0
Cl
Q
u w
,,,
.
_L I
I--
-
-
,
w,
f:
A-i
.H
u
f,,
.H
q-4
0
A-i
ff
H
44
H
,,
N
199.
2
and
From Eq.
9V
av7
- r
rI
=
z2
(B.7)
a.z o
(3.22)
-r
V-
0
3/r
z
1 22
(B.8)
Thus from Eq. (B. 8) we can write
2
o
9z
-pr I1
=
(B.9)
-pr
(B.10)
3
and
Kz3
O
Using Eqs.
(B.6),
-
$l
=
=
I
(B.9) and (B.10) in Eq. (B.4) gives
1
[2
-
~O-pr
If as shown in Fig. B.1,
~
1
+
il
A ]
2
A
(B.ll)
is taken as varying linearly
with z in the vicinity of the wall,
Da
A
Solving for
()
r0
=
(l
-
o)
(B.12)
0 from Eq. (B.11) then gives,
3(90 - 1)
2
pr (z1 -z0 ) 2
-
1
(3.91c)
1
200.
Following an identical approach, expression for
vorticity at the slag-metal interface can be derived.
For the vertical surfaces,
and
r = Rm ,
Z
(i.e. r = Re
Z
<
z < Z2
< z < Z3), a similar approach is used
but instead of assuming a linearly varying vorticity, the
vorticity transport equation is used to give an expression
for the normal gradient of vorticity at the wall.
201.
APPENDIX C
CALCULATION OF RADIATION VIEW FACTORS
In this appendix, radiation view factors appearing
in Eqs.
(3.97) and (3.99) are calculated using the compil-
54
ation made by Leunberger and Person 54 for finite coaxial
cylinders.
Fig. C.1 shows a schematic representation of
the system for calculating view factors.
Symbols s, e, m
and t represent free surface of the slag, outer surface
of the electrode, inner surface of the mold and top (open)
surface respectively.
View factors are calculated with the aid of "view
factor algebra" which relies on the reciprocity rule and
on the fact that radiation is conserved.
According to
Leunberg and Person 54
F
1
=
(z)
es
-
[ cos
1 A2
2Tr
cos
x
x {
A1
-1 A2 Re
R
cos
-
A1 Rm
A33
eA 2e - 4R 2R2
R
3
e m
-1 R
e}
]
(C.l)
Rm
where
A
A
1
2
A
x
x
2
=
x
=
x2
=
x2
=
L
=L
-z
2
R
m
-
Rm 22
2 +
R 2
R 2
R 2
z
R
2
+
+
e
e
(C.2,3,4,5)
(C.2,3,4,5)
202.
17
C.1
Schematic representation of the system for calculating
view factors.
203.
The view factor
(C.1) with
Fet(z)
can be evaluated by using Eq.
Then
x = z.
Fem(Z)
em =
F
1 - F
em
(z) can be calculated using,
(z)
~es
(C.6)
Ft(z)
et
--
Let us now consider the evaluation of
F'
and F'
sm
se
First we note that,
F'
where
'
+F' + F' +
ms
mm
F'
mt
me
=
1
(C.7)
indicates that the view factors are w-r-t-the
entire surface.
F'
=
ms
F'
(C.8)
mt
1
Thus,
ms
2
mm
- F'
me
)
(C.9)
Again, from reference 54,
F'
Re
R
=
me
1
{
-1 A5
A
-- cos
4
m
1
1 ~~1
[ A
A6
-Re
7
cos
R
-
A
A5
-
R
mn
A4
R
+
sin 1
A
-
e
R
A ]}
2
m
(C.10)
where
A
A
A
4
5
6
A7
=
L
+ R
=
L
=
2R R
=
/(L22
+
2
-R
- R
m
2
m
+ R
e
2
e
(C.11,12,13, 14)
m e
A7
Re )2
4R 2R2
=/ (L2 + Rme m
204.
and
R
F'
R
1 _
=
e +
1{ 2
Rm
-
A l l-
A0
L
[ A 8 sin- A 9
2R
m
Rm
i-a +
sin
e tan
(A8
-
1)]}
(C.15 )
where
2/R 2 -2
m
e
All
L
/4R2
=
A
+ L2
L
A 2
Al +
A9
10
A2
Ai-i + 1
R
1-
=
2
(C. 16,17,18,
19)
2 ( e)
R
m
Thus after calculating F'
from Eq. (C.10)and F'
me
Eq. (C.15), we can calculate F'Ms from Eq. (C.9).
ms
Then, from reciprocity rule,
2R L
F'
R 2
R 2
Rm -R e
sm
similarly noting that
F'
F
-m
St
=
1 -
from
(C20 )
ms
54
L
R 2_
m
e
R
R (F
+ 2F'
mm
m
me
e
(C.21)
1)]
205.
and that
F'
se
+ F' +F'
sm
st
= 1
we can calculate
F'
se
= 1 - F'
sm
-
F'
st
(C.22)
206.
APPENDIX D
USE OF TEMPERATURE DEPENDENT ELECTRICAL CONDUCTIVITY IN
THE SLAG
The magnetohydrodynamic form of Maxwell's equation
(for low magnetic Reynolds no.)
UH
at
_
can be written as:
[Vx (H)]
a
-
(D.1)
In cylindrical coordinate and for axial symmetry (Hr = H =
z
a
0), Eq.(D.l) can be written as follows:
D0
a
;H0
2
°
lH e
a 11
[
] + -[- - -(rH)]
a
z
~r
r r
-
at
z
(D.2)
Expanding the terms in Eq.(D.2) gives:
2
MHe
0
aH
3t
He
-
ana
[--
9z
9z
;r
( - -(rH))]
r
r
-
1 a
+ -
az
2 +
1
r
DZna
(rH )
-]
r
0
r
(D.3)
For the calculations reported in the text, the
second term on the rh.s.ofEq. (D.3) has been neglected.
However, in some of the calculations the temperature dependence of
, appearing on the
accounted for.
-h-s of Eq.(D.3) has been
The basic approach in these latter calcu--
lations involved:
207.
(1)
Solution of field equations for a specified
temperature distribution in slag
(2)
Values for electromagnetic force and Joule
heat rate generated in step 1 was then used to solve flow
and temperature equations.
(3)
Temperature field generated in step 2 was
used to calculate the new distribution of
in slag.
Steps 1 to 3 were repeated until the temperature
fields in slag calculated in two consecutive iterations
agreed within a specified limit.
It is to be noted that
each of steps 1 and 2 is iterative in addition to the
overall process being iterative.
The temperature dependence of electrical conductivity was deduced from data published by Mitchell and Cameron 60
on 70% CaF 2 ,
30% A203 slags.
Their data in the range
1500 °C - 1700 °C, can be represented by the following
equation:
Zna
where
-=9888
T
+ 10.467
is in ohm - 1 m-1 and T is in
K.
208.
APPENDIX E
THE COMPUTER PROGRAM
The computer program used for the solution of the
governing differential equations is presented in this
Appendix.
A brief introduction to the program and the
functions of various subroutines have already been given
in Chapter IV.
The computer program given here incorpor-
ates wall function approach for turbulence quantities and
for temperature.
However, the computed results reported
in the thesis did not utilize this feature.
E.1
List of Fortran Symbols
Meaning
Symbol
A(I ,J,K)
The dependent variables for
flow and temperature equations
and V z , VrI
t'
eff' K,
p,
Cp.
B (I,J,K)
Variables for electromagnetic
field.
AE, AW, AN, AS
Aj' of Eq. (4.8)
ANAME (6, 12)
An alphameric array containing
the names of variables for
flow and temperature equations.
209.
ASYMBL
An alphameric array containing
the names of residuals as defined
by Eq.
AKEFF
(4.40).
Effective thermal conductivity
in metal pool.
APP
al
of Eq (4.11 a,b,c,d)
aP,p
BNAME (6,10)
An alphameric array containing
the names of variables for electromagnetic field.
BE (I), BW(I), BN(J), BS(J)
Bj' of Eq.
(4.8)
BPP
b~p of Eq.
BBE, BBW, BBN, BBS
The group of terms c j (b,j
(4.8)
b~ p)Bj' in Eq.
BETA
+
(4.8).
Coefficient of volume expansion
(3).
C1, C2, C3, CD
Constants in the turbulence
model.
CC,CP
The convergence criteria for flow
and temperature equations, magnetic
field equations.
CU
Total current (maximum value), kA
210.
CAPPA
Von Karman's constant
D
S in Eq.
Dl,
D2,
DN, DS
(K)
(3.75)
Parameters in Eqs.
(4.15) and
(4.16).
E
The wall roughness factor, E
in Eq.
ES, EE, EM
Emissivities of slag surface
(E )
mold
FES, FEM, FSM, FSE
(4.30)
of electrode
(
(
)
and of
m)
Radiation view factors Fes
F'
Fem , F' s
se
sm'
em
FR
Geometric factors to account
for the fact that control volumes
for integration in near wall
regions are different from those
in the interior of the domain.
GAMA
Interfacial tension between
molten metal and slag ().
HC
Heat transfer coefficient between
electrode and gas
HW
Overall heat transfer coefficients
at interfaces defined by Eq.
(3.104).
211.
HCD
Re
e 2VmePeCd
me e P,d in Eq.
(3.76)
Heat of fusion
HL
Index for constant - z grid
lines
The number of differential
IE
I1, I2, I3,
Ji
IN
I4, I5,
IA,
equations
(for flow and heat
transfer)
to be solved.
Indices defined in Fig. 5.2
The total number of constant
z grid lines
IP
The number of successive
iterations for which print-out
is to be produced
IV
The number of variables whose
values are to be printed out.
J
Index for constant-r lines.
JN
The total number of constant-r
lines.
K
The index which denotes the
dependent variable in question.
212.
The maximum number of iterations
MIT
for magnetic field equations
The number of iterations between
MPRINT
print-out cycles for magnetic
field variables.
xTrDr l
Dissipation rate of turbulence
energy
NF
Stream function
NK
Turbulence energy
NM
Turbulent viscosity
NMt
Effective viscosity
NRC
for flow
tr.
Density
s
NSI
Specific heat
NT
Temperature
NTC
Thermal conductivity
NVI
Velocity in z-direction
NV2
Velocity in rdirection
NW
Vortiity
/r)
213.
Real part of magnetic field
NHR N
I
intensity
Imaginary part of magnetic
NHI
field intensity
NH
Magnetic field intensity
NJRR
Real part of current density
in rdirection
NJRI
I
Indices for
electromagnetic
field variables
Imaginary part of current density in rdirection
NJR
Current density in rdirection
NJZR
Real part of current density
in z-direction
NJZI
Imaginary part of current density
in z-direction
NJ Z
NJJ
Current density in z-direction
J
NMAX
Joule heat (kCal/m3s)
The maximum number of iterations
for flow and heat transfer
equations.
214.
NPRINT
The number of iterations
between print out cycles for
flow and heat tr. variables
p
Magnetic permeability,
PR(K)
The Prandtl or Schmidt numbers.
QS
QS of Eq (3.76)
R(J)
r
RE,
M
Radius of electrode
0
(Re ), radius
of mold (Rm)
RES
The maximum residual (1 N/ N-1)
for all of the
p
u
max
C equations.
ROREF (5)
Reference densities for various
media.
RP(K)
Relaxation parameters for flow
and temperature equations.
RS
The residual at a particular node.
RSDU (K)
The maximum value of RS for each
Q
SB
equation.
1012 x Stefan-Boltzmann constant
(kCal/m sK 4 )
215.
S(4)
Electrical conductivities of
electrode, slag, molten metal,
ingot.
SN1, SN2, SN3, SN4
Local heat transfer coefficients
in regions
defined in Fig. 4.6.
SOURCE
Sfp,P
p of Eq.
SPREF (5)
Reference specific heats
TLM, TSM, TLS, TA, TM, TW
T ,m' Tsm
TF, TB
Tf, T B in Eq.
TCREF (5)
Reference thermal conductivities.
TAUW1, TAUW2, TAUW3, TAUW4
Shear stress values in regions
(4.8).
TQ,s, Ta' Tm' T w
(3.74).
defined in Fig. 4.6.
TAU
Residence time of a droplet.
UP
Velocity parallel to a wall.
VOLT
Voltage.
VE
Melting rate of electrode.
VC
Casting rate.
W
Relaxation parameters for
magnetic field equations.
216.
X(7)
Defined in Fig. 5.2.
xi (I)
z-coordinate of the grid nodes.
X2 (J)
r-coordinate of the grid nodes.
YL(J)
Position of liquidus isotherm.
YS(J)
Position of solidus isotherm.
ZMUREF
Reference value for viscosity.
217.
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BIOGRAPHICAL NOTE
The author was born in Bihar, India on June 26,
1952.
After finishing high school in Calcutta he attended
Indian Institute of Technology, Kharagpur, from 1969 to
1974 and obtained the degree of Bachelor of Technology
(Honours) in Chemical Engineering.
During both high
school and undergraduate years he received numerous awards
for achievements in academics and extra-curricular activities.
On completion of his undergraduate studies in 1974,
Department of Chemical Engineering awarded him Professor
S.K. Nandi Gold Medal for being the "best all rounder" Chemical Engineering student in the class of 1974, the Institute
Silver Medal for securing first rank in B. Tech. examinations
and an award for the Best Undergraduate Thesis.
In June 1974,
In September 1974,
the author married Saraswati Jha.
he joined State University of New York
at Buffalo and obtained the degree of Master of Science
in Chemical Engineering in June 1976.
His thesis work was
on the effect of flow maldistribution on the performance of
catalytic packed bed reactors.
He joined the Department of
Materials Science and Engineering at MIT in September 1976.
In 1978, the department awarded him Falih N. Darmara
Materials Achievement Award "in recognition of outstanding
academic performance, excellent research work and extra
curricular activities".
The same year he received the
281.
Annual Student Paper Award of the Eastern Iron and Steel
Section
f AIME for a paper "Optimization of Burden Size
for a Blast Furnace".
As a research associate
(Summers of 1976, 1977,
1978) at both SUNY at Buffalo and at MIT, the author has
worked on a number of projects, both experimental and theoretical.
He has worked on a broad range of problems as
parts of various consulting assignments.
Some of the
assignments have involved heat transfer and fluid flow
calculations in an electric furnace smelter, heat transfer
calculations for recuperators using impinging jets, material
and energy balance analysis for coal gasification and direct
reduction processes, etc.
He has also published papers in
the areas of chemical reaction engineering, gas-solid
reaction, and modelling of electroslag refining process.
The author is a student member of AIME and ASM.
He is also a member of Sigma Xi.