AND HEAT FLOW

IN THE

ELECTROSLAG REMELTING PROCESS

MICHEL ALBERT MAULVAULT

Ingenieur Civil

Submitted in

des Mines, Paris 1967

Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

at the

Massachusetts Institute

January

of Technology

1971

Signature of AuthorDepar tyrenSci en e

t of.-4eal1urgyJanuaryY215, 1971

and Materials

Certi fied by ___

T h o co- c Nono ru, ecn wt

Accepted byuepdrLIIIteILay oi mittee on

Graduate StudentsArchives

. sT. c

MAR 5 1971LIEA I

II

Lw aa, it ~ldr

TEMPERATURE

TEMPERATURE AND HEAT FLOW

IN THE

ELECTROSLAG REMELTING PROCESS

by

MICHEL ALBERT MAULVAULT

Submitted to theScience on Januathe requirements

Department of Metallurgy and Materialsry 15, 1971 in partial fulfillment offor the degree of Doctor of Philosophy.

ABSTRACT

An experimental study was made on a laboratoryESR unit which used a direct current power supply. Thetemperature was measured in the electrode, the slagand the ingot for various operating conditions.

Forflow conditilevel. Thedetermined wessentiallyOne-dimensiodimensionalmodel.

ingot.approxipool.programand the

the electrode, approximately adiabatic heatons were found on its surface above the slagimmersion depth of the electrode in the slaghether heat flow in the electrode wasone-dimensional and axial or two-dimensional.nal heat flow was treated analytically. Two-heat flow was treated with a computer thermal

Steady state thermal models were derived for theA first analytical model used the moving fin

mation but did not predict the shape of the metalA second thermal model, treated with a computer

predicted the entire temperature distributionshape and position of the liquid metal pool.

The electrical power input appears to be themain independent variable determining the melting rateof the electrode.

The slag appeared to be at a nearly uniformtemperature except at boundary layers at surfaces asthe ingot, the electrode and the mold. The slag isthe main heat source in the entire ESR process and heatis generated in the slag non-uniformly.

II

Using the results of this study on the laboratoryunit, a heat transfer study was made on industrial ESRunits. The one-dimensional analytical thermal model wasexpected to give valuable estimates of the temperatureprofile in the electrode. Departures from this profiledue to the effect of immersion depth in the slag and thepresence of a parabolic tip are calculated.

On industrial ingots, an investigation with thecomputer thermal model showed that the dimensionless depthof the metal pool (ratio of the depth of the metal pool toingot radius) is proportional to casting speed and varieslinearly with ingot radius. Other operating parametersconsidered were the temperature at the top of the ingot,the heat transfer coefficient between the ingot and thewater, the effective thermal conductivity of the metalpool. Of all operating parameters, casting speed is foundto be most important in determining the shape of the metalpool.

Temperature distribution in the slag of industrialESR units depends on convective and electromagneticstirring and may not be uniform. The slag constitutes themain electrical resistance and the main heat source in theESR process.

Thesis Supervisor: John F. Elliott

Title: Professor of Metallurgy

I

TABLE OF CONTENTS

Section pageNumber number

ABSTRACT ii

LIST OF ILLUSTRATIONS ix

LIST OF TABLES xiii

ACKNOWLEDGEMENTS xiv

NOMENCLATURE xv

I INTRODUCTION 1

II LITERATURE SURVEY 2

III OUTLINE AND PLAN OF WORK 4

IV EXPERIMENTAL APPARATUS 5

V THE ELECTRODE - LABORATORY UNIT 10

A. Experimental Study 101. Temperature measurements 112. Experimental results 11

B. Analysis of the Results 211. Heat transfer in upper part of

electrode 212. Heat transfer in the lower part

of electrode 23C. Thermal Model of the Electrode 25

1. One-dimensional thermal model 272. Two-dimensional thermal model 30

D. Conclusions 34

VI THE INGOT - LABORATORY UNIT 36

A. Experimental Study 371. Temperature measurements in the

ingot 372. Carbon content in the ingot 413. Liquidus and solidus temperatures

of the remelted material in theupper part of the ingot 41

4. The shape of the metal pool 455. Temperature measurements at the

bottom of the ingot 45

W _ 11

Section pageNumber number

B. Analysis of the Experimental Results 48C. Approximate Analytical Thermal Model

for the Ingots 511. Distribution of the analytical

thermal model 512. Heat transfer coefficient between

ingot and water 533. Application of the model to the

entire ingot 57D. Computer Thermal Model for the Ingots 59

1. Distribution of the computerthermal model 61

2. Application of the computerthermal model to the ingots ofthe experimental study 63

E. Conclusion 67

VII THE SLAG - LABORATORY UNIT 71

A. Experimental Study 731. Pyrometric temperature measurements 732. Thermocouple measurements 753. Analysis of the temperature

measurements 774. Voltage measurements in the slag 785. Melting of electrode in the slag 80

B. Mechanism of Heat Generation andHeat Transfer in the Slag 851. Mechanism of heat generation in

s1ag 852. Melting conditions of the

electrode 903. Temperature distribution in the

slag 91C. Heat Balance on the Slag 92D. Conclusion 96

VIII APPLICATION OF THE RESULTS 98

A. Electrode 981. Summary of the results obtained on

the electrode of the laboratoryESR unit 99

2. Available data on the ESR practice 993. Heat generation in industrial ESR

electrodes 1004. Heat flow conditions in ESR

electrodes 1015. Temperature distribution in the

electrode 1036. Conclusion 110

Ii

B. Ingot

C. SlagConclusion

IX SUMMARY AND CONCLUSIONS

1. Electrode - Laboratory Unit2. Ingot - Laboratory Unit3. Slag - Laboratory Unit4. Application of the Results to

Industrial ESR Units

X SUGGESTIONS FOR FURTHER WORK

XI BIBLIOGRAPHY

BIOGRAPHICAL NOTE

SectionNumber

pagenumber

1. Summary of the results on thelaboratory ESR unit

2. Possible applications of movingfin approximation

3. Influence of various parameters onthe temperature distribution insteel ingotsa. Ingot radius and casting speedb. Temperature at the top of the

ingotc. Heat transfer coefficient

between ingot and waterd. Effect of convection in the

liquid metal poole. Importance of the heat

released upon solidification4. Temperature distribution in ESR

ingots.5. Steady state heat flow conditions

in ESR ingots6. Discussion of the work of previous

investigators7. Conclusion

148

151

153

210

I

114

115

115

116117

120

120

123

127

130

130

135139141144

146

v ii

LIST OF APPENDICES

PageNumber

AppendixNumber

I PHYSICAL DATA FOR STEEL

ThermalSpecifiEl ectriDensityHeat of

Conductivityc Heatcal Resistivity

Fusion

II DETERMINATIONELECTRODE

OF THE MELTING SPEED OF THE

III DIFFERENTIAL THERMAL ANALYSES ON STEEL

Electrode SteelIngot Steel

IV TEMPERATURE RISE DUE TO HEAT GENERATIONIN THE UPPER PART OF THE ELECTRODE

Steady State TemperatureTemperature Rise in the Electrode

V HEAT BALANCE ON THE LOWER PART OF THEELECTRODE

VI STEADY STATE HEAT CONDUCTIONA MOVING CYLINDER

VII

VIII

EQUATION FOR

THE EPS COMPUTER PROGRAM

APPLICATION OF TWO-DIMENSIONAL HEAT FLOWMODEL TO THE ELECTRODE OF EXPERIMENT 2

IX THE APPROXIMATION OF THE MOVING FIN ON THEENTIRE INGOT

X HEAT TRANSFERAND WATER

COEFFICIENT BETWEEN INGOT

A. Heat Transfer Coefficient ThroughContraction Gap, hggp

B. Heat Transfer Coefficient ThroughCopper Wall, hmold

C. Heat Transfer CoefficiCopper Wall and Water,

D. Heat Transfer Coefficithe Slag, hslag

ent Betweenhwater

ent Through

161

163

163165

169

169170

172

178

180

182

189

191

191

194

194

194

156

II

Appendix pageNumber number

XI COMPUTER PROGRAM FOR THE THERMAL MODELOF THE ESR INGOTS 196

XII CALIBRATION OF THE INFRARED PYROMETER 205

XIII DIMENSIONLESS ANALYSIS FOR HEAT FLOW ININGOTS 208

i

LIST OF ILLUSTRATIONS

Experimental app(b) Detailed viefeeder

The water-cooledinitial starting

aratus. (a) General view.w of the mold and the flux

copper mold with theproducts

3 Two steel inunit at a po

4 Installationtemperatures

gots castwer of 15

ofin

in the laboratory ESRkw

thermocouplethe electrode

for measuring

The immersed tip of the electrode

6 Temperature recenter line ofexperiments

7 Experiment 1 -center line of

8 Experimentcenter line

corded with thermocoupleelectrode f

temperatureelectrode

- temperatureof electrode

or fouralong

profile along

profile along

9 Experiment 3 - temperature profile alongcenter line of electrode

10 Experiment 4 - temperature profile alongcenter line of electrode

11 Heat balance in the lower part of theelectrode

12 Experiment 5 -center line of

13 Experiment 6 -center line of

temperatureelectrode

temperatureelectrode

profile along

profile along

14 Thermocouple assembly for temperaturemeasurements inside ingot

15 Temperature measurement inside ingot

16 Experimentalfrom outside

temperature profile at 0.6 cmof ingot

FigureNumber

PageNumber

24

38

40

Ii

Figure pageNumber number

17 Experimental temperature profile at 0.6 cmfrom outside of ingot 43

18 Carbon content in ingot at level of the tipof thermocouple 44

19 Carbon content along axis of ingot versusdistance from bottom of ingot 44

20 Sulfur print of longitudinal section ofingot (0.62 percent carbon) 46

21 Temperature measurements inside the bolt 47

22 Temperature in the bolt versus height ofingot 47

23 Temperature distribution in the upper partof the ingot 49

24 Construction of the temperature profile at0.6 cm from outside of ingot 50

25 Approximation of the moving fin on theentire ingot 52

26 Moving fin approximation for temperaturesbelow 1180 0 C 56

27 Heat transfer coefficient between surfaceof ingot and water 58

28 Temperature profile at 0.6 cm from theoutside of the ingot 60

29 Experimental and calculated mushy zone fora power of 15 kw 65

30 Temperature profile at 0.6 cm from outsideof ingot 66

31 Calculated ingot center line and surfacetemperature (results of Appendix XI) 68

32 Calculated isotherms in ingot (results ofAppendix XI) 69

33 Schematic drawing of the temperaturemeasurements in the slag 72

34 The infrared pyrometer used for thetemperature measurements in the slag 74

i

FigureNumber

35 Measurement of the voltage drop acrossthe slag

36 Melting speed of the electrode functionof power input

37 Electrode tips for various meltingconditions

38 Heat balance on slag for a power of 15 kwand a positive electrode mode

39 Temperature profile in ESR electrodescalculated with one-dimensional thermalmodel (equation 1)

40 Temperature profilecalculated with one-model (equation 1)

Temperatureelectrode

in ESR electrodesdimensional thermal

in 37.5 cm diameter steel

42 Temperature in 37.5electrode

43 Temperature distribuESR steel electrode

44 Temperature distributionESR steel electrode

45 Influence of ingot radius andon the dimensionless depth ofpool

46 Effect ofingot on t

47 Influencebetween ththe 1380 0C

48 Position oflow intothermal copool

cm diameter steel

tion in an industrial

in an industrial

casting speedthe metal

he top of therm

sfer coefficientwater, hi, on

temperature at the 1380 0C isothe

of the heat trane ingot and theisotherm

f solidificationthe ingot as funnductivity in th

49 Effect of thermalmetal pool, kl, on

fronttions1 iqui

and heatof thed metal

conductivity in liquidthe 1380 0C isotherm

pagenumber

105

106

108

109

111

112

119

121

122

125

126

W 11

FigureNumber

50 Effect of the heat released uponsolidification on the 1380 0C isotherm

51 Temperature distribution in a 50 cmdiameter ESR steel ingot

52 Temperature distribution in a 50 cmdiameter ESR steel ingot

53 Effect of ingotline temperaturesteel ingot

height on the centerof a 50 cm diameter

Thermal conductivity of solid 0.230.8 percent carbon steels in solid(reference 31)

Specific heat of iron (reference

andstate

20)

Electrical resistivity of 0.23 percentcarbon steel (reference 31)

Schematic drawing of the experimentalarrangement for the determination of thedriving speed of the electrode

Chart giving a tempedriving speed of the

III.1 Differential thermal analsteel

111.2 Differential thermal analelectrode steel

III.3 Differential thermal analsteel

V.1 Radiation between the elesurroundings

V.2 Heat flux by radiation inelectrode

III.1 Grid for data file (MELEC

Grid formodel of

thethe

rature rise and theelectrode

ysis on electrode

ysis on the 1020

ysis on ingot

ctrode and the

or out of the

INPUT)

use of EPS on the thermalESR ingots

Schematic drawing of the experimentalarrangement for calibration of infraredpyrometer

xi i

pagenumber

128

131

132

134

I.1

1.2

1.3

11.1

II.2

157

158

160

162

162

164

166

168

175

177

186

201

207

V

XI.1 1

XII.1 1

W 11

x ii i

LIST OF TABLES

Table PageNumber Number

1 Operating Conditions in the Experimentson the Electrode (Positive Electrode Mode) 13

2 Heat Generation in the Electrode (Resultsof Appendix I) 22

3 Heat Balance on the Lower Part of theElectrode (Results of Appendix V) 26

4 One-Dimensional Thermal Model for theElectrode 28

5 Two-Dimensional Thermal Model for theElectrode 31

6 Approximation of the Moving Fin on theEntire Ingot 54

7 The Thermal Model of the ESR Ingots 62

8 Temperature Measurements in the Slag 76

9 Voltage Measurements in the Slag 81

10 Melting Speed of Electrode Function ofMelting Conditions 82

11 Heat Balance on the Slag for a Power of15 kw and Positive Electrode Mode 95

VI.1 Heat Balance for Heat Conduction Equation 179

VIII.1 Sensitivity of the Numerical Solution toGrid Size and Delta in the Computer ThermalModel Applied to the Electrode ofExperiment 2 188

X.1 Heat Transfer Coefficient Between Ingotand Water (Results of Appendix X) 193

XI.1 Influence of the Number of Points of theGrid and of Delta on the TemperatureDistribution in an Ingot Calculatedwith EPS 203

XIII.1 Quantities Involved in Heat Flow Analysis 209

1i

xiv

ACKNOWLEDGEMENTS

The author wishes to express his thanks to Professor

John F. Elliott for his invaluable help and understanding

throughout the course of this work.

He also wishes to thank his fellow graduate students

for many stimulating discussions and helpful suggestions,

as well as Professor J. Bruce See for reading the thesis

in draft form. Thanks are also due to Mr. James Stack

and Mr. Edward Bradbury for their technical assistance.

Financial support for this study was provided by

the American Iron and Steel Institute.

NOMENCLATURE

Specific heat,C1 , for liquid

cal /g;steel

Heat transfer coeffichj, between the ingotbetween the immersedliquid slag

C s, for solid steel,

ient, cal/cm2/sec/0C;and the water; hsl,

electrode and the

Enthalpy, cal/g; HT, at temperature, T0C

Thermal conductivity, cal/cm/sec/ 0C; ks,solid steel; k1 , for liquid steel

Latent heat of fusion of steel, cal/g

Mass flow rate, g/sec

Radius,radius;

cm;Rm,

RE, electrodemold radius

radius; R,, ingot

Surface area, cm2; SE, cross section elSP, cross section ingot

Heat flux, cal/sec

Time, min

Temperature, 0C; TE, upper partTme, solid end of electrode; ToTs , bulk of slag; Tssl, surfacan ular space between electrodeTt, top of ingot; TW, water

ectrode;

of electrode;surroundings;

e of slag inand mold;

Speed, cm/min; VE, melting speed of electrode;V1, casting speed of ingot

Heat generation, cal/cm 3/sec; WE, in electrode

Distance, cm; Zc, between top of ingot andbottom of metal pool on the center line ofingot; Zs, between top of ingot and bottommetal pool at the surface of the ingot; Zpzc -z s

theof

emissivity,electrode;

dimensionless; c , copper; cE's slag

density, g/cm 3

for

xvi

o Stefan-Boltzmann constant, 1.356 x 10-12cal/cm2/sec/OK 4

I. INTRODUCTION

Knowledge of temperature

heat flow is very important to

Electroslag Remelting process.

in the electrode is also import

gradients and the resultant

adequately understand the

The absolute temperature

ant for the prediction of

possible tran

In the ingot,

the shape of

These local s

on the macro-

segregation i

also importan

properties of

the slag and

There h

and heat flow

determine the

sformations and

the temperature

the

reactions

distri bu

the electrode.

determines

metal pool and local solidification

olidification times

and microstructure

n the ingot. In the

t because it has an

the slag as well as

heat generation in t

ad been little previ

and this study was

importance

times.

have a profound influence

and on the extent of

slag, the temperature is

effect on the refining

on heat transfer through

he slag.

ous research on temperature

undertaken to attempt to

of different modes of heat transfer.

An experimental study

unit with direct current.

the electrode, the ingot an

conditions. Thermal models

with the experimental measu

of the thermal models to in

considered.

was made on a labo

The temperature was

d the slag for vari

have been derived

rements. Possible

dustrial ESR units

ratory ESR

measured in

ous operating

and validated

appl i cations

are also

II. LITERATURE SURVEY

There have been very

distribution and heat flow

studies are referenced in

will be discussed at appro

Thermal models derived to

casting machine and vacuum

mentioned briefly.

Mitchell, Joshi and

with experimental measurem

for the temperature distri

model only applies to the

slag level. The way the el

rather poorly understood(2

transparent crucible to st.

few investigations on temperature

in the ESR process. Major

this section and their content

priate points in the thesis.

treat ingots made by continuous

arc remelting process are

Cameron(l) wrote and validated

ents a computer thermal model

)ution in some electrodes. This

)art of the electrode above the

lectrode melts in the slag was

,3) until Campbell(4) used a

udy the formation of liquid metal

droplets at the electrode tip. The melting of the

electrode is obviously dependent on the temperature in the

slag which has only been studied qualitatively in labora-

tory ESR unit by Campbell(4) and Panin et al(5). This

temperature in the slag depends on the heat generation and

heat transfer in the slag. An investigation on this

subject has been undertaken by Mitchell(6). Related to

the heat transfer in the slag, Roberts(7) studied some

techniques for maximum melting rates and minimum power

consumption.

Ii

-_ -16i - - - . A _. - I Id

The temperature distribution in the ESR ingots

has a direct influence on the solidification in the ingot.

Some solidification patterns have been reported(3,8-ll) and

for the temperature distribution in the entire ingot, Sun

and Pridgeon(12) proposed a computer model to predict pool

shapes using a finite difference technique. Several other

thermal models have been written for ingots, but they are

only valid for ingots produced by processes such as the

continuous casting machine and the vacuum arc remelting

process. For the continuous casting machine Savage(13),

Hills(14,15) and Irving(16) developed analytical solutions

for the heat transfer using integral profile techniques.

Similarly Cliff and Dain(17) made extensive computerized

calculations on steel billets. Various numerical solutions

for the heat transfer have been developed. By neglecting

axial heat flow, Pehlke(18) and Mizikar(19) solved a transient

case. Schroeder and Lippitt(20) considered transient two-

dimensional heat flow. The two-dimensional steady state has

been treated by Adenis, Coats and Ragone(21) and Kroeger(22).

For the ingots produced by the vacuum arc remelting process,

thermal models have been derived both with an approximate

analytical solution(23) and with a computer program(24).

These models are discussed in Section VIII.B.6.

i

4

III. OUTLINE AND PLAN OF WORK

The experimental study for this investigation was

made on a laboratory ESR unit. Details of the experimental

investigation are given first.

A preliminary investigation showed that it was best

to analyze separately temperature and heat flow in the

electrode, the ingot and the slag.

For the electrode and the ingot, the temperature

measurements are described, the heat flow conditions are

analyzed and thermal models are derived. The predicted

and experimental results are compared to assess the

validity of the proposed models.

For the slag ,the experimental study consisted in

temperature and voltage measurements. Possible mechanisms

of heat generation and heat transfer are briefly analyzed.

A heat balance on the slag is made for a single power input.

The last part of this study concerns the application

of the results obtained on the laboratory ESR unit to

industrial scale units.

IV. EXPERIMENTAL APPARATUS

The laboratory ESR unit used for the experimental

study is shown in Figure 1. The water-cooled copper mold

was 9-1/2 in. high and had an inside diameter of 2 in.

(Figure 2). It was mounted on a 1/2 in, thick steel plate.

The water flow for cooling was about 16 1/min. The

electrode was a 1 indiameter rod of a commercial AISI

1020 steel. This electrode was remelted in a prefused

slag with an initial composition of 20 weight percent

calcium oxide and 80 weight percent calcium fluoride.

During the experiments, a slag crust formed around the ingot

and a continuous flow of powder had to be supplied to the

mold. A Syntron vibratory feeder, model F-70,was used to

supply the powder.

Electrode movement was controlled by the driving

system of a former Lepel unitmodel FLZ-100. A Minarik

speed controller, model SH-63, gave a continuous speed

range from 0 to 13.5 cm/min.

The power to the unit was supplied by a D.C. arc

welder Miller Electric, model SR 1000 Bl (three phases,

input power 55 kw, 74 kva, 230/460 volts). The output

power during the initial 2/3 sec after turning on the

unit and the power during operation could be selected

independently.

The powder for the slag was an insulator when solid.

Initial melting of this powder was carried out as shown in

Figure 1: Experimental apparatus.

(a) General view.

(b) Detailed view of the moldthe flux feeder.

(a)

(b)

and

Figure 2.

cooling of

the powder

in the 1 cm

and the sta

and ferric

mixture. T

starting.

reduced to

velocity of

to maintain

A

th

an

h

rt

an

he

Af

a

steel sheet shaped as a

e powder by maintaining

d the mold. A plug of

igh gap between the tip

ing cap screw. A layer

d ferrous oxides served

full power of the arc

ter 2/3 sec, the power

preset value between 7.

he electrode was contr

a constant voltage

cyl

a 3

s tee

of

of

as

weld

was

5 kw

olle

inder prevented

mm gap between

1 wool was placed

the electrode

magnesium turnings

exothermic

er was used for

automatically

and 24 kw. The

d manually in orde

across the slag. The

selected voltage was between 13 and 28 volts.

Figure 3 shows two ingots cast with a power of 15 kw

and two polarities of the electrode. The negative electrode

mode gave the better surface. The remelted material appears

to be sound after the first 5 cm.

Two recording potentiometers were used for various

types of measurements in the experimental study. One was

a Honeywell instrument with an adjustable span 0-1 to 0-51 m

with up to + 50 mV adjustable suppression. The chart

speed could be selected between 15 and 480 in/min. Full

scale deflection took 1/2 sec. The second recording

potentiometer was a Nulline, model 204, with a span of about

10 mV, a chart speed of 6 in./min and a full scale deflection

of about 1/2 sec.

Other special devices will be described in the

appropriate sections.

r

V

electrode

water

copper tubeO.D. 3 in.wall: 0.065 in.

steel shield

- flux

exothermic mixture

steel cap screw

Figure 2: The water.cooled copper mold with the initialstarting products.

(a) Negative electrodemode, castingspeed: 3.4 cm/min

(b) Positive electrode mode,casting speed: 3.9 cm/min

Two steel ingots cast in the laboratory ESR unit at a power of 15 kw.

5 cm

I

Figure 3:

V. THE ELECTRODE - LABORATORY UNIT

Temperature profiles along the center line of the

electrode of the laboratory ESR unit were measured for

various operating conditions. The electrical power was

varied between 7.5 and 22.4 kw, the melting speeds ranged

from 2.2 to 5.4 cm/min and the immersion depth of the

electrode in the slag varied between 0.3 and 1.4 cm. The

recorded temperature profiles in the electrode indicated

a gradual temperature increase from room temperature to

about 50 0C,followed by a sharp temperature rise up to

about 1460 0C near the tip of the electrode.

Necessary heat transfer conditions to give the

above temperature profiles are analyzed. It is shown

that the immersion depth determines whether the heat flow

in the electrode is one-or two-dimensional.

Two thermal models are derived. For one-dimensional

heat flow, the system may be treated analytically, whereas

for the two-dimensional case, a computer program is used.

The experimental and predicted results are compared to

assess the validity of the proposed models.

A. Experimental Study

Temperature measurements were made with a thermocouple

in the electrode of the laboratory unit. The procedure is

summarized below and results are given for four experiments.

I

1. Temperature measurements

Details of the thermocouple installations for

measuring temperatures in the electrode are given i

Figure 4. A Pt-Pt 10% Rh thermocouple protected wi

thin wall alumina tube (2.7 mm 0.D.) was inserted i

3 mm diameter hole drilled at an angle of 25 degree

the axis of the electrode. The initial position of

thermocouple junction

from the tip of the el

connected through a 00

recording potentiomete

speed of 360 in.per ho

line of the thermocoup

speed of the electrode

by the procedure descr

2. Experimental

The operating co

given in Table 1. The

was at the center line, about 50 cm

ectrode. The thermocouple was

C cold junction to the Honeywell

r (Section IV) which had a chart

ur. The microswitch placed on the

le was used to measure the driving

. The melting speed was calculated

ibed in Appendix II.

results

nditions for f

voltages were

our experiments are

measured between the

electrode clamp and the base of the mold.

Experiments 1 and 2 led to the highest and the

lowest melting velocities for a power of 15 kw as shown

in Table 10.

The immersion depth of the electrodes in the slag

was measured using the height of a solid slag layer found

around the cylindrical surface after fast remoyal of th-e

electrodes. This slag layer had an average thickness of

II

a

to

the

electrode

lumina protection2.7 mm;1.7 mm

Installationtemperatures

thermocouple for measuringthe electrode.

Figure

Table 1

Operating Conditions in the Experiments

on the Electrode (Positive Electrode Mode)

experimentnumber 1 2 3 4

power, kw 15 15 7.5 22.4

voltage, volts 20 20 13 28

current, amps 750 750 580 800

melting velocity,(+5%) cm/min 4.3 3.5 2.2 5.4

d cm 0.3 0.9 0.3 0.6

d cm 0.2 0.7 0.0 0.6

.def cm 0.4 1.4 0.3 1.0

about 0.5 mm. An effective

by neglecting the curvature

Figure 5. Experiment 2 corr

immersion.

The recorded temperatu

6 after conversion of the ti

scale. The distance scale i

tip of the thermocouple and

The temperature at the end o

1460 0C, temperature correspo

of steel. This temperature

thermal analysis explained i

steady heat flow existed nea

immersi

of the

esponds

on depth is obtained

tips as shown in

to the deepest

re rises are sh

me scale into a

ndicates the le

the end of the

f the electrode

nding to the so

was measured by

n Appendix III.

r the tips of t

own in Figure

distance

ngth between the

solid electrode.

is taken as

lidus temperature

the differential

Assuming that

he electrodes,

temperature

electrode

series of

Figure 6.

to measur

failure o

The tempe

near the

is about

140 0C for

last case

above 120

profi 1 es

tips

four

The

ements

f the

rature

tip.

400 0C

the h

, the

00 C.

are pl

over the first fo

otted in Figures

experiments.

recorded tempe

below the sol

thermocouples

profiles indi

One diameter a

in experiments

ighest melting

recorded profi

It is assumed

ur

7

These profi

ratur

idus

due t

cate

bove

2 an

rate

le (F

s abo

sothe

atta

harp

he ti

3 (F

(Figu

gure

cm from the

through 10 for the

les are taken from

ve 1460 0 C correspond

rm obtained prior to

ck by the slags.

temperature rises

p, the temperature

igures 8 and 9), and

re 10). In this

10) is vertical

that this was due to some

The thermocouple slipped down from theexperimental error.

electrode

emersed part

slaglevel

immersed part

The values of di, d2for four experiments.

and d are given in Table 1

The immersed tip of the electrode.

deff

Figure 5 :

10 20 30 40 50Distance between tip of thermocouple and reference noint, cm

Figure 6: Temperature recorded with thermocoupleelectrode for four experiments.

along center line

1500

Q)C.

1000

500

-l0

experiment 1

experiment 2

--- experiment 3

experiment 4

The reference point for distance isat 1460 0COperating data listed in Table 1

1460

500

1460

experimental profile

calculated profile from equation(T E = 500 C)

Melting speed: 4.3 cm/minOther operating data listed in Table 1Reference point for distance at 1460 0C

1 2 3 4Distance from reference point, cm

along center line of electrode.

(1)

15001

o'u 1000

Figure 7: Experiment 1 - temperature profile

calculated profile from equation (1) (TE = 50 C)

- - - ,-.~1 ,-.~,l ~4-e,~A orm u - r v- 4-1 t -ma l mA l- " ' X N -- cL p- %. e %;_

1460 (Appendix VIII, TE = 500 C)

- - calculated with computer therma(Appendix VIII, TE = 950 C)

Melting speed: 3.5 cm/minOther operating data listed in

~ \ Reference point for distance at

-b

1 model

Table 1.1460 0 C.

1111

Distance from reference point, cm

Figure 8: Experiment 2 - temperature profile along center line of electrode.

1676

1500

U0

it

4

ai)

10001

500

experimental profile

1500 1460 experimental profile

calculated profile from equation (1)(TE = 500 C)

U -Melting speed: 2.2 cm/min

1000 Other operating data listed in Table 1.Reference point for distance at 1460 0 C.

500

50

0 I 2 3 4Distance from reference point, cm

temperature profile along center line of electrode.

1760

1640

Figure 9: Experiment 3 -

I

1500

cal(TE

Melting speOther operaReference p

culated profile from equation= 50 0 C)

ed: 5.4 cm/minting data listed in Table 1oint for distance at 1460 0C

01 I I I I I I I I I I I I I I -1 I

0 I 2 3 4Distance from reference

along center line of electrode.

C..)0

S-

a-)

E

(1)

1000

500

point, cm

experimentalA60 prof ilIe

F ig u re 10 : Experiment 4 - temperature profile

electrode into the slag when it reached the tip. This

result is excluded from the following analysis.

B. Analysis of the Results

The temperature profiles in Figure 6 indicate a

gradual temperature increase from room temperature (200)

to about 500C, followed by a sharp rise to 1460 0 C. The

temperature of 50 0C is used as a level dividing the

electrode into an upper part and a lower part. The posi-

tion of this level is read from the plots in Figure 6.

The heat transfer conditions in the two parts of the

electrode are analyzed successively below.

1. Heat transfer in upper part of electrode

The temperature rise up to 50 0C is attributed to

the heat generation by Joule effect. This rise corresponds

to a uniform temperature increase of the upper part of the

electrode as analyzed in Appendix IV. The results are

given in Table 2 for the four experiments. Table 2 shows

that if a steady state temperature had been reached, the

difference between the center line and the surface tempera-

tures would have been less than 0.3 0 C. This shows that in

the upper electrodes, the temperature profile in cross

sections was flat. From the duration of the temperature

rise, effective heat transfer coefficients have been

calculated. They account for the heat transfer by convec-

tion due to the motion of the electrode and to the hot slag

which heated the gases around the electrode. The time

Table 2

Heat Generation in the Electrode

(Results of Appendix IV)

experimentnumber 1 2 3 4

power, kw 15 15 7.5 22.4

current, amps 750 750 580 800

heat generation,cal/cm 3 /sec x 03 79 79 47 90

different betweencenter line andsurface tempera-tures, OC 0.3 0.3 0.2 0.3

time to reach500 C, min 8 8 15 7

effective heattransfer coeffi-ci ent,cal /cm2/sec/OCx 104 6.7 6.7 4.5 7.7

steady statetemperature, OC 95 95 86 94

time constant,t, min 14 14 21 12

time for thermo-couple to reachtip electrode (at14600C), min 11.5 12.3 22.5 8.3

temperature inupper part elec-trode when thermo-couple reached tip,Oc 62 63 63 56

constants for the rises are found to be less than 21

minutes and possible steady state temperatures are

evaluated at less than 950C. During the time it took for

the thermocouples to record temperatures between 50 0 C and

1460 0 C, calculations show that the temperature of the

upper part increased from 500C to less than 63 0C. A

temperature of 63 0C was still lower than the steady state

temperature.

2. Heat transfer in the lower part of electrode

The plots of Figures 7 through 10 are assumed to

correspond to steady state temperature profiles along the

center line near the tip of the electrode. The unsteady

temperature rise of the upper part of the electrode above

500C is neglected. The heat flow conditions are analyzed

in Appendix V using a heat balance on the lower part of

the electrode as shown in Figure 11. The term mE (H500C -

H14600C) corresponds to the enthalpy variation of the flow

of material entering at 500C and leaving at 1460 0C. Q,

and Q5 account for the heat fluxes by conduction coming

from the slag. Q2, the conductive heat flux leaving the

lower part of the electrode, is zero since the upper part

was at a uniform temperature. Q3 is the heat generated by

Joule effect. Q4 corresponds to the heat exchange by

conduction in the gas of the annular space and by radia-

tion with the surface of the slag and the inner wall of the

mold.

mE H50 0 C

Q4

emersed electrode

agsl

immersed

level

electrode

in Table 1.

(b) d em + d eff is

temperature of

(c)

the distance between electrode

50 0C read from Figure 6

heat generation

Heat Balance: Q,

Figure 11: Heat b

+ Q2 + Q3 + Q4 + Q5 + mE(H500C-H1 4 6 0 0C) =

alance in the lower part of the electrode.

dem

d ef f

(a) de f is given

tip and the

0

The results of the calculations

for the first three experiments. For

ment, the gradient at the tip could no

the experimental error mentioned in Se

analysis could not be completed.

These results indicate that two

conditions were encountered. In exper

axial heat flux by conduction from the

respectively for 89 percen

variation. It is conclude

controlled the temperature

essentially adiabatic cond

surface. In experiment 2,

tively high (1.4 cm) and t

for 47 percent of the tota

addition, the radial flux,

This is a case involving t

Although no analysis

are given in Tabl

the fourth experi

t be measured due

ction V.2 and the

types of

iments 1

slag, Q

t and 91 percent

d that the axial

profile at the

itions occured o

the immersion d

he axial heat fl

1 enthalpy varia

Q5 , represented

wo-dimensional h

was possible on

heat fl

and 3,

1 , accou

ow

the

nted

of the enthalpy

heat flow

tip and that

n the cylindrical

epth was rela-

ux, Q, accounted

tion. In

about 45 percent.

eat flow.

experiment 4, it

at

con

n)

this experiment

ditions because

and the rather 1

conforms to the one-dimen

of the high melting veloc

ow immersion (1 cm).

C. Thermal Models of the Electrode

Two thermal models are derived

two-dimensional heat flow conditions

section. The temperature rise in the

neglected and a constant temperature,

below for the one and

analyzed in the previous

upper electrode is

TE, is assumed.

appea

heat

(5.4

rs th

flow

cm/mi

si onal

i ty

Heat Balance on the

(Results

experimentnumber

power, kw

current, amps

melting speed,cm/min

d eff, cm

dem, cm

mE(H500C-H1 4 600C)'cal/sec

temperature gradientat tip, OC/cm(a)

Q1, cal/sec

Q3 , cal/sec

Q4 , cal/sec

Q5 , cal/sec(b)

Table 3

Lower Part of the Electrode

of Appendix V)

15

750

4.3

0.4

14.6

-661

1550

588

24

25

24

15

750

7.5

580

2.2

0.3

16.1

-540

670

253

24

23

241 .5

345

830

314

15

14

2

gradients read from Figures 7, 8

obtained by difference according

in Figure 11.

and 9.

to the heat balance

22.4

800

5.4

1.0

6.0

-840

12

II

Calculated temperature profiles using these models

compared with the experimental profiles of Section

1. One-dimensional thermal model

The electrode in experiments

adiabatic condition at the cylindri

V.B.2). The melting conditions are

classical case of ablation(28) of

temperature, TE, melting at a const

a constant temperature, Tme, and a

The temperature profile can be expr

when the physical characteristics a

(Table 4). Using negative values f

given by

1 and 3 had nearly

cal surface (Section

comparable to the

an infinite slab at

ant position (Z = 0)

constant velocity, V

essed analytically

re assumed constant

)r Z, the profile is

T - TET - T E

From this formula

ep CsV E Zs

and the physical constants given in

Appendix I, temperature profiles have

the four experiments. The results are

7 through 10 for comparison with the e

For experiments 1 (Figure 7) and 3 (Fi

agreement above 350 0C is good. The di

than 50 0 C and becomes less than 100C f

above 1100 0 C. For temperatures lower

discrepancy reaches as much as 100 0C.

is thought to exist because the analyt

take into account the variation of the

been calculated for

plotted in Figures

xperimental results.

gure 9), the

screpancy is less

or temperatures

than 3500 C, the

This last discrepancy

ical solution does not

physical properties

are

V.A.

(1)

Ii

,

E'

Heat conduction equation

d (ks i) _ d (PCsVET) = 0

Boundary conditions:

(1) Z = 0

(2) Z -c

T = Tm

T = TE

when ks and PCsVE are not functions

of Z, the solution is expressed as:

T - T epC VET -ET ks Z)

Gradient at Z =

dTdZ z=0

--0 T=TE

E

0 T=Tm

(Z < 0)

0

(Tine - pGsVEm e-e T E) k s

Table 4: One-dimensional Thermal Model for the Electrode.

I

which at low temperatures are very different from those

at high temperatures (Appendix I). This discrepancy is

considered acceptable since the low temperatures are

certainly of lesser importance in the ESR process. For

experiment 2 (Figure 8) the calculated profile under-

estimates the experimental profile by as much as 300 0 C,

at true temperatures of 700 0C. This shows that, as mentioned

in Section V.B.2, a one-dimensional model cannot apply to

this experiment.

The calculated profile of Figure 10 is almost parallel

to the experimental profile between 500 0C and 1200 0C. This

confirms that experiment 4 certainly corresponds to one-

dimensional heat flow conditions, as expected from the

discussion in Section V.B.2.

An effect of a change in the entering temperature of

the material, TE, on the temperature profile is shown by

the following derivation of equation (1).

T -TEdT = [1 - Tine - TE] x dTE

A maximum increase of TE from 50 0 C to 950C as mentioned in

Section V.B.1, would shift up the calculated temperature

profile by about 14 0C at 500 0C. This variation decreases

at higher temperatures. This effect is considered

negligible. It is concluded that the assumption of a

steady state temperature profile made in Section V.B.2 is

valid.

(2)

2. Two-dimensional thermal model

The two-dimensional thermal model consists of the

solution of a steady state heat conduction equation for

appropriate boundary conditions.

The heat conduction equation, derived in Appendix

VI, is expressed by the following second order partial

differential equation:

(ksr 2) + (ksr T - PC V rT) + rWE = 0 (3)

The boundary conditions are listed in Table 5. The

electrode is considered semi-infinite in length with a

constant temperature, Tme, at the tip (boundary condition

1) and a temperature, TE, as Z goes to minus infinity

(boundary condition 4). Boundary conditions 2 and 3

express respectively the heat transfer between the slag and

the surroundings above the slag level as mentioned in

Section V.B.2. Boundary condition 5 takes into account the

axial symmetry.

Numerical solutions to equation (3) are obtained with

a finite difference method of relaxation using the general

purpose computer program EPS.(26). This program is briefly

described in Appendix VII. Its application to experiment 2

is detailed in Appendix VIII. The physical data are those

of Appendix I. The boundary conditions correspond to the

heat flow by conduction and radiation analyzed in Section

V.B.2. A calculated temperature profile along the center

Heat conduction equation

(ksr )

Boundary

+ {(ks rl -

.+ rWE

conditions:

(1) At Z = 0, T = Tme

(2) At r = RE and for

Z > -d ,

pCs V ErT)

=0

-dSe ff

--C (4)

tT = TET=E

(5)

(3)

T=T me

s- r r=R

(3) At r = RE

= hsl(T-Tsl)

and for Z < -deff,

sr r=R q

(4) For Z -- , T = TE

(5) At r = 0, D= 0

Thermal Model for the Electrode.

h sl ,Tsl

Table 5: Two-dimensional

line for an

in Figure 8

upper part of an electrode at 50"

and compared with the experimenta

is plotted

profi 1 e.

Above 900 0 C the discrepancy is less than

discrepancy of 60 0C occurs at about 600 0C

appears to be of little significance and

considered satisfactory.

In Figure 8, a calculated profile i

for a steady state temperature of 95 0 C in

of the electrode (see Table 2). This pro

less than 40 0C of the profile calculated

of 50 0C in the upper part of the electrod

concluded that the assumption of a steady

profile made in Section V.B.2 is satisfac

100 C.

Su

the a

s als

the

file

wi th

e. I

temp

tory.

ch

gr

o p

upp

is

a t

t i

era

A maximum

an error

eement is

lotted

er part

within

emperature

s

ture

0

Two fi

measurements

completely a

experiments

The fi

made with a

Due to exper

could not be

assumed to b

was 0.5 cm.

nal experiments were performed

in the electrode. The results

nalyzed because of the similari

treated in this section.

rst of these two experiments, e

positive electrode mode at a po

imental difficulties, the melti

determined accurately. This v

e 3.9 cm/min. The effective im

The experimental temperature p

with temperature

were not

ty with the

xperiment 5, was

wer of 15 kw.

ng velocity

elocity was

mersion depth

rofile along the

line and the profile calculated with

thermal model using equation (1) are

12. Due to the inaccuracy in the mel

the one-dimen-

plotted in

ting velocity,

center

s i onal

Figure

1460

experimental profile

calculated profile from equation (1)(TE = 500 C)

Power: 15 kwPositive electrode modeMelting speed: 3.9 cm/minImmersion depth: 0.5 cmReference point for distance at 1460

I a ~ - ~f i i

Distance from reference point, cm

Figure 12: Experiment 5 - temperature profile along center line of electrode.

U0

0)

40)

w)

1000

500

0 I II I I I I I I

- - ' - -- 11 -- W- 11 -- N

ftmanff

1706

C

the calculated profile follows the experimental profile

within less than 100 0 C above 700 0C.

The last experiment, experiment 6, was made with a

negative electrode mode at a power of 15 kw. The melting

rate was 3.4 cm/min and the immersion depth 1.1 cm. The

experimental temperature profile along the center line of

the electrode and two profiles calculated with the two

thermal models are plotted in Figure 13. Due to the

relatively high immersion depth of the electrode, only

the profile calculated with the two-dimensional heat flow

model, is in good agreement with the experimental profile.

D. Conclusions

Experimental measurements on the laboratory ESR unit

gave temperature profiles along the center line of the

electrode for various operating conditions. The analysis

of the experimental results showed that heat flow in the

electrode was one-dimensional and axial for low immersion

depths as in experiments 1, 3, 4 and 5 (Figures 7, 9, 10

and 12). For higher immersion depths as in experiments 2

and 6 (Figures 8 and 13), heat flow became two-dimensional

due to heat exchange between the electrode and the slag

through the immersed cylindrical surface.

Two steady state thermal models were derived for the

two types of heat flow conditions. The validity of these

models is shown by the satisfactory agreement found between

the calculated and the experimental temperature profiles

along the center line of the electrodes.

I

r

experimental profile

------- calculated profile from equation (1)(TE = 500C)

calculated profile from computer thermalmodel (Appendix VIII, TE = 500 C,T = 1650 0C)slag

Power: 15 kwNegative electrode modeMelting speed: 3.4 cm/minImmersion depth: 1.1 cmReference point for distance at 1460 C

O)

N -%

N

* a a I a a . I . * ~ a a B I I I L

2 3 4

Distance from reference point, cm

Figure 13: Experiment 6 - temperature profile along center line of electrode.

1000U

0

4IJ

C)a

500H-

0

1658

VI. THE INGOT - LABORATORY UNIT

Experimental measurements were performed on the

laboratory ESR unit. A power of 15 kw was used with a

current of 750 amps and a positive electrode mode. A

casting velocity of 1.1 cm/min was obtained. Thermocouple

measurements were made on a vertical line at 0.6 cm from

the cylindrical surface of the ingots and over the first

5 cm from the top. As explained later, carbon was added

in the upper part of the ingots. The carbon content in

the upper part was found to be almost uniform at about

0.65 percent. At this carbon content, the liquidus and

solidus temperatures were measured by differential thermal

analysis. These temperatures were found to be 1484 0C and

1380 0C respectively. The approximate shape of the solidi-

fication front was determined by sulfur prints. Finally

thermocouple measurements were made in the starting bolt.

From the experimental measurements the temperature

distribution in the entire ingot was derived. The tempera-

ture was found to become steady after the height of the

ingots reached at least 1.8 times their diameter.

Uniform temperature is assumed in cross sections

of the ingots and is derived using the analytical approxi-

mations of the moving fin. An average heat transfer

coefficient between the solid ingot and the water is

evaluated with this model by matching the calculated

I

temperature profile with the experimental profile at 0.6 cm

from the cylindrical surface of the ingot. The thermal

conductivity of the solid slag crust around the ingot is

deduced.

A second thermal model is derived and consists of

the solution of a heat conduction equation for the proper

boundary conditions. This model accounts for the shape

and position of the mushy zone and is treated with a

computer program.

The experimental and predicted results are compared

to assess the validity of the proposed models.

A. Experimental Study

1. Temperature measurements in the ingot

The temperature measurements in the ingot had to be

performed underneath the slag bath which was at a high

temperature (above 1655 0C as shown in Section VII, Table 8)

and was also chemically corrosive. For this reason, a

double tube was used to protect the Pt-6%Rh/Pt-30%Rh thermo-

couple wires. As shown in Figure 14, the inner alumina

tube was surrounded by a graphite tube sealed at one end.

The thermocouple was connected directly to the Honeywell

recording potentiometer described in Section IV. The chart

speed was 360 in/hour. The above thermocouple had an emf

of -0.002V at 250 C and a 0 0C junction was not necessary as

the error was less than 1 0C at all temperatures.

I

Honeywell recordingpotentiometer

alumina insulator

alumina protection

graphite tube

tube, diameter

Section B-B

*Wdiameter

diameter

Section A-A.- graphite

-A

Thermocouple wires:

plug

0.7

Pt-6%Rh/Pt-30%Rh

diameter

(0.011 indiameter)

Figure 14: Thermocouple assemblyinside ingot.

for temperature measurements

LfEC\U

0.3 cm

B

0.8

0.4

cm.

cm

1 cm

2.5Fcm

A

0.8

I

Initially the thermocouple assembly was placed

vertically in the

at 7 mm from

of the electrode,

tube from attack b

tube dissolved in

tube first

(Figure 15c

The

temperature

temperature

liquid slag

conversion

as indicate

on the line

in dire

) and 1

recordi

versus

versus

and th

was as

d in Ap

of the

couple temperature

mold with the axis of the thermocouple

the wall (Figure 15a). During melting

the graphite tube protected the alumina

y the slag (Figure 15b). The graphite

the liquid steel, leaving the alumina

ct contact with the liquid metal

ater with the solid metal (Figure 15d).

ng potentiometer gave plots of

time. These plots were converted to

distance from the interface between the

e liquid metal

follows. The

pendix II, usi

thermocouple

at the end of

each experiment, the distance, Z

junction and the top of the ingo

removal of the thermocouple from

was possible, or by grinding the

. The procedure for this

casting speed was determined

ng the microswitch placed

(Figure 14). The thermo-

casting was noted. After

between the thermocouple

t was measured either by

the alumina tube if this

surface of the ingot until

the junction appeared. No gas bubbles were observed in the

slag during experiments and the ingot did not show any

sign that gas evolution occured from the metal pool

(Figure 3). It was assumed that the distance, Z, also

corresponded to the distance between the thermocouple junc-

tion and the interface liquid slag-liquid metal.

I

0.7cm

mold base

beforeexperiment

(a)

thermo-couple

slag

durin Qexperiment

(b)

duringexperiment

(c)

duringexperiment

(d )

Details of thermocouple are given in Figure 14.

Temperature measurement inside ingot.Fi gure 15 :

Finally it was assumed that the recorded temperature

profiles corresponded to steady state heat flow conditions.

The validity of this last assumption is proved in Section

VI .B.

Two temperature profiles corresponding to a

distance of 0.6 cm from the cylindrical surface of the

ingot are plotted in Figures 16 and 17. These profiles

are in good agreement with each other for temperatures

below 1483 0C. At 1100 0 C the discrepancy is about 250C.

Above 1483 0C, the discrepancy reaches 60 0 C at the top of

the ingot.

2. Carbon content in the ingot

The carbon content was determined in the ingot for

which temperature measurements are given in Figure 16.

The results are plotted in Figure 18 for the level of the

tip of the thermocouple and in Figure 19 along the axis of

the ingot. A maximum carbon content of about 0.8 percent

was found in the immediate vicinity of the thermocouple

(Figure 18). Above 10 cm from the bottom of the ingot,

where the thermocouple measurements were made, the carbon

content was about 0.65 percent. Below 10 cm, the carbon

content was 0.25 percent. This value corresponds to the

carbon content of the steel electrode.

3. Liquidus and solidus temperatures of theremelted material in the upper part of theingot

The two profiles of Figures 16 and 17 do not indicate

the liquidus and solidus temperatures of the steel due to

casting speed 1.1 cm/min

power: 15 kw

positive electrode mode

measured

extrapolated

1 2 3Distance from top of ingot, Z, cm

at 0.6 cm from outside of ingot.

1500

1000

500

Figure 16 : Ex pe r ime ntal I temperature profile

measured

extrapolated

Casting speed: 1.1 cm/minPower: 15 kwPositive electrode mode

0 I 2 3 4 5Distance

Experimentalingot.

from top

temperature profile at

of ingot, Z, cm

0.6 cm from outside of

1500

C-,0

a,S.-

5-a,0~Ea,I-

1000

500

Figure 17:

thermocou

1.0 x measured

extrapolatedprofile

A0---

0.5L-2.5 2 0 1

Distance from axis

Carbon contentthermocouple.

2 2.5

of ingot,

in ingot at level of the tip of

level of tip ofthermocouple

addition oC starts

f Ix

x measurements

--- extrapolated r

5 10 15

Distance from the bottom of ingot, cm

Figure 19: Carbon content along axisfrom bottom of ingot.

Figure 18:

0.5

rofi le

of ingot versus distance

the sharp gradients at high temperatures (about 250 0 C/cm

at 1400 0 C). The plot of Figure 16 shows a change of slope

at about 1485 0C. Using a differential thermal analysis

on a 0.65 percent carbon sample of the first ingot, the

liquidus and solidus temperatures were found to be equal

to 1484 0C and 1380 0C respectively. The details of the

procedure are given in Appendix III.

4. The shape of the metal pool

A 4mm diameter

ngle of 750 to

rom the tip of

iron sulphide

de were melted,

g into the liqu

a longitudinal

n in Figure 20.

line is about 9

bon content near

5. Temperaturei ngot

and 1.5 cm long hole was drilled

the axis of t

the electrode

powder (about

the iron sulr

id metal pool.

section of th

The depth of

mm. A chemic

the interface

measurements

ie electrode at about

The hole was filled

2 g). After 40 cm of

hide went through

A sulfur print was

e ingot. The result

the interface at the

al analysis showed that

was about 0.62 percent.

at the bottom of the

Temperature measurements were performed at the

bottom of a third ingot by inserting a Pt-6% Rh/Pt-30% Rh

thermocouple in the starting bolt as shown in Figure 21.

The recorded temperature profile is plotted in Figure 22

as temperature versus height of the ingot.

This last measurement completed the experimental

study on the ingot.

at

40

up

ele

the

mad

is

cen

the

an a

cm f

with

ctro

sla

e on

give

ter

car

46

- -3

Sulfur print of longitudinal(0.62 percent carbon).

section of ingotFigure 20:

steel cap screw

--- steel base

alumina protection tube 0.3 cmdiameter

Pt-6%Rh/Pt-30%Rh thermocouplewires (0.011 in.diameter)

Figure 21: Temperature

1000

500V-

measurements inside the bolt.

Casting speed: 1.1 cm/minPower: 15 kw

Positive electrode mode

I I I I I I I I t10 15

Height of ingot, cm

Temperature inof ingot.

the bolt versus height

- I

Figure 22:

B. Analysis of the Experimental Results

From Figures 16 and 17, the experimental temperatures

at the top of the ingots were equal to 16500 C and 1590 0 C

respectively.

true values.

cooled as it w

(Figure 15a).

by the thermoc

the tip of the

lef

low

sol

sla

in

fo u

fac

The

par

was

iso

pri

fro

usi

(Fi

t around

ering of

id steel

The t

g and liq

the plots

nd to be

e is assu

validity

The c

t of the

determin

The e

The gr

as onl

The c

oupl e

al umi

xpe

aph

y 3

ool

as

na

the alumina

temperature

interface.

rue tempera

uid metal i

of Figure

rimental values are lower than the

ite protection tube was partially

mm from the inner wall of the mold

ing lowered the temperature recorded

long as graphite was present around

tube. No graphite was found to be

tube in the solid steel. The

occured only above the mushy zone-

t

s

1

about 1690 0 C

med to be

of this

onstructe

ingot is

ed by ass

re at the interface between liquid

obtained by extrapolation as shown

and 17. This temperature is

in the two cases.

at the abo

assumption

d temperatu

shown in Fi

uming that

ve

is

re

gur

the

tempera

proved

di stri b

e 23.

liquid

The entire

tu

in

ut

Th

us

re of

Sect

ion i

is di

and

i

1690 0C

ion VII

n the u

stri but

sol idus

nter-

.A.

pper

ion

therms were parallel to the trace given by the sulfur

nt (Figure 20).

Figure 24 shows the temperature profile at 0.6 cm

m the outside of the ingot. This profile was obtained

ng the experimental temperature measurements above 900 0C

gures 16 and 17) and adding the measurements in the bolt

r - IE

Power: 15 kwCasting speed: 1Positive electrode

Temperaturethe ingot.

distribution in the upper part of

1 cm/minmode

Figure 23:

Power: 15 kwPositive electrode modeCasting speed: 1.1 cm/min

measured

extrapolated

I I I I

Distance from top of ingot, Z, cm

Figure 24: Construction of the temperature profile at0.6 cm from outside of ingot.

1500

1000a>-

4-3

E

500

L7

I I I I I I I I II I I I

(Figure 22). Two interpolations were made: one between

1690 0 C and 1540 0C and the other between 900 0 C and 570 0 C.

The temperature measured in the bolt was assumed to be the

temperature at the bottom of the ingot. In Section VI.D.2,

this assumption is shown to be correct.

The temperature profile in Figure 24 shows that

steady state conditions for heat flow existed in the ingots

once their height was greater than 9 cm. This height

corresponds to 1.8 times the diameter of the ingots.

C. Approximate Analytical Thermal Model for the Ingots

The moving

analytical thermal

the heat transfer

water. The result

tion in the entire

1. Descrip

fin app

model.

coeffi c

s obtai

ingot

tion of

roximation was used to obtain an

This model is used to evaluate

ient between the ingot and the

ned for the temperature distribu-

are also given.

the analytical thermal model

shows th

roximated

radius, R

the solid

at condi

by thos

con'si

ingot.

ions

for

ti ng

The

for heat fl

a semi-inf

of two med

mushy zone

replaced by an interface where heat is generated by solidi-

fication at a rate, pVIL (in cal/cm 2/sec) and where the

temperature, Tin, is intermediate between the liquidus and

solidus temperatures. The heat transfer coefficients

between liquid steel and waterand solid steel and water,

are hI and h2 respectively. The height of the liquid pool

Fi

may

er

po

ingot

cyl i nd

liquid

gure 2

be ap

with a

ol and

in the

te

the

ow

ini

ia,

is

in

--Z.

0

z

h1 , TW

'pVL(cal/cm 2/sec)

h2, Tw

dT 0

T is only a function of Z

Figure 25: Approximation

2 RI

ki, C1

2

pow

ks s

of the moving fin on the entire ingot.

is Z . The heat generation by Joule effect is n1

The temperature at the top of the ingot is Tt'

The mathematical formulation of the model

marized in Table 6. For each medium, i, the heat

equation is expressed as:

eglected.

is sum-

conduction

d2T.

dZ

dT. h.PC Vi -T - 2 R (T - TW) = 0

Boundary conditions 1, 2, 3 and 5 define the values

temperature at the boundaries. Boundary condition 4

correct when the ingot is assumed to be infinite. B

condition 6 shows that the heat flux into the solid

is the sum of the heat flux from the liquid and the

released upon solidification.

Assuming that the physical properties and the

transfer coefficients are constant, an analytical so

can be given for the temperature, T(Z), versus the d

from the top of the ingot. This solution is given i

Appendix IX. In particular, the temperature in the

ingot is expressed as:

T - TW

T in - T W exp

where

[1 PCs I PCs I 2 +8h 2s kS s ks I

2. Heat transfer coefficient between ingot andwater

Heat is transferred from the surface of the ing

into the water in various ways. Conduction occurs thr

(4)

of the

is

oundary

ingot

heat

heat

1 ution

i stanc

n

solid

(5)

(6)

ot

ough a

Table 6

Approximation of the Moving

Heat conduction equation

d2T.

k dZ2

Boundary

P V dT

conditions:

Fin on the Entire Ingot

for medium,

h.-2 1 (T5 - TW) = 0

(a) for medium 1

(b) for medium

(1) Z = -Z

(2) Z = 0

(3)

T T t

T = Tin

T= TinT2 in

dT2

gd =

= 0

(4) Z

(c) at interface between the two media

(5) T1 = T2 Tin

dT(6) -k g |Z 0

dT2

= -ks lz=O

The solution to the heat conduction equation for the above

boundary conditions is given in Appendix

+ pVI L

IX.

I

solid crust of slag (heat transfer coefficient, hsliag)

Conduction and radiation take place through a gap due to the

shrinkage of the ingot and where this shrinkage has not

started, a contact resistance exists (heat transfer coeffi-

cient, h ). Heat is transferred by conduction through the

inner copper wall of the mold (heat transfer coefficient,

hmold) and finally by forced convection into the water flow-

ing in the annular space (heat transfer coefficient, hwater)'

The total heat transfer coefficient, hi, can be expressed as:

= [h + h1 + h 1 + h 1slag hgap mold water(7)

h1 depends on the distance from the top of the ingot.

The thermal conductivity of the slag crust which is

required for the calculation of the coefficient, hslag, was

not found in the literature. This thermal conductivity may

range from 2.5x10~4 to 10-2 cal/cm/sec/ 0C according to values

given by McAdams(30) for similar materials. A value for this

thermal conductivity was obtained indirectly using the moving

fin approximation. This approximation was used on the solid

ingot for temperatures below 11800C as follows. The follow-

ing form of equation (5) was used with the actual water

temperature of 10 0C:

T (0 C)

A is

shown

using

shows

- 10 + 1170 exp(XAZ) (8)

a function of the heat transfer coefficient, h2, as

in equation (6). Temperature profiles were calculated

equation (8) for various values of h2 . Figure 26

that the average value for h2 of 0.0045 cal/cm2 /sec/ C

r

experimental profile

calculated profile f(hI = 0.0045 cal

of Figure 24

rom equation/cm2/sec/OC)

I I I I I I I I I I I I I I I I I3 5 10 15 20

Distance from top of ingot, cm

Moving fin approximation for temperatures

IOOK

500h-

(8)

Power: 15 kw, casting speed: 1.1 cm/mi n

below 1180 0C.F igu r e 2 6:

gave a calculated temperature profile in

the experimental profile

The heat transfer

was also calculated dire

of the calculations are

0.002 cal/cm/sec/OC for

crust led to the proper

0.0045

greater

the hea

the ing

0.2 cal

Irving(

cal/cm 2/sec/ C for

than 3 cm (Figure

t transfer coeffic

ot using a contact

/cm 2/sec/ C. This

16) for contact re

3. Application of

The approximation

Figure 25, was applied to

given in the experimental

ture at the top was 16900

cm/min. The temperature,

as 1432 0C, which is the a

liquidus temperatures for

face which replaced the m

from the top of the ingot

temperature of 1432 0C rea

coefficients h1 and h2 in

of Figure 24.

coefficient between

ctly from equation (7

given in Appendix X.

the thermal conductiv

average heat transfer

distance

27). As

ient was

resistan

value is

sistances

the mode

of the mo

the enti

study (S

C and the

Tin, of

verage of

the ingo

ushy zone

. This

d on Fi

di cated

ingot and water

). The details

A value of

ity of the slag

coefficient of

s from the top of the ingot

indicated in Appendix X,

extrapolated to the top of

ce over the first 2 cm of

based on the work of

between metals.

1 to the entire ingot

ving fin, summarized in

re ingot for the conditions

ection VI.B). The tempera-

casting vel6city 1.1

the mushy zone was taken

the solidus and the

t composition. The inter-

was positioned at 1.6 cm

distance

gure 24.

in Figure

correspo

The heat

25,were

nds to

trans

taken

the

fer

as

0.01 and 0.0045 c

previous section.

Appendix I.

al/cm 2/sec/0C from

The other physica

the ca

1 data

lculation in the

are those of

good agreement with

r

Distance from top of ingot, cm

coefficient between surface of

0

(-)w(A

C\i

EU

U

0.005

--- VWM

direct calculation of hI (Appendix X)

average h calculated with moving finapproximalion (Section VI.C.2)

approximated hi for computer treatment(Appendix XI)

Figure 27: Heat transfer ingot and water.

An effective value of

for the thermal conductivity

calculated temperature profi

factory agreement with the ex

from the outside of the ingot

0.16 cal/cm/sec/0C

in the liquid pool

le was found to be

perimental profile

(Figure 28). The

was used

The

in satis

at 0.6 ci

above value

of the thermal conductivity of the liquid steel accounts

fo r

of

0.0

pre

for

i ng

Sng

0.1

the

(Se

the

convection in metal pool. The thermal

stagnant liquid steel (32,33) would be i

25 to 0.05 cal/cm/sec/ 0C.

The main deficiency of the model is

dict the shape of the metal pool. This

a pure metal and applies when the Biot

ot (Bi = hIRI/k) is less than 0.17(28).

ot considered in this section, the Biot

5 showing that the approximation of the

calculation of the average heat transfe

ction VI.C.2) is satisfactory. Near the

Biot number increases to about 0.33. T

the results obtained on the entire

approximate.

conductivity

n the order of

the inability to

model was derived

number in the

In the solid

number was about

moving fin to

r coefficient

mushy zone,

his shows that

ingot can only be

Computer Thermal Model for the Ingots

A thermal model for the entire ingot,including the

zone of finite dimensions,is described below. This

consists of the solution of the applicable steady

heat conduction equation for given boundary conditions.

mushy

model

state

Power:Posi tivCasting

60

15 kwe electrode modespeed: 1.1 cm/min

calculated with moviapproximation (Sec.k = 0.16 cal/cm/sec

experimental profile(Figure 24)

5 10 15

of ingot,

Figure 28: Temperature profile at 0.6 cm from theoutside of the ingot.

ngVI./ 0C

finC.3)

1500

1000

500

0

Distance from top

M

its complexity, the model is treated by

. An application is made to the ingots

experimental study (Section VI.A).

1. Description of the computer thermal

Due to

program

in the

appl i ca

has the

state heat

s derived

conduction

in Appendix

a computer

considered

model

equation

VI and

3T.(k r ) +

DT.(k r - p C VIrT ) + rW = 0

Subscript i corresponds

mushy zone or to the sol

Wi, corresponds to the J

Wic(cal/cm 3/sec) = resis

In the mushy

upon solidifi

zone, W.

cation.

to

id

oul

the liqu

ingot.

e effect

id metal pool, to the

The heat generation term,

and is expressed as

tivity x (current

is in

This

creased by the

heat released i

density) 2/4.18

(10)

heat released

s expressed at a

distance,

from the

r, from the

top, as:

axis of the ingot and a distance, Z,

W (r,Z) = pVIL/6m(r) (11)

6m (r) is the height of the mushy zone.

Equation (9) was solved for the boundary conditions

listed in Table 7. Boundary condition (1) expresses the

constant and uniform temperature at the top of the ingot.

Boundary condition (2) takes into account the heat exchange

steady

ngots i

orm:

The

ble

fo 1

general

to the

lowing

(9)

0110, -Awww - - - - I I --- - - - - -

Table 7. The Thermalof the ESR

General differential

(k r )D r 1 ~r

Boundary

equation:

- p C VirT )

conditions:

T = T

-k iIr=R = h, (Z) (T-TW)

ModelIngots

= 0

+ rW. = 0

(1)

(2)

(3)

(4)

Z =0

r = R1

Z = Zb

r =0

T t

+ { (k r u

=. 0

Ry

h, (Z),T ,A

between the

function of

that after

ingot and the

distance, Z.

a distance, Zb,

water. The coefficient, hBoundary condition (3) indi

the heat flow through the

of the ingot can be neglected.

takes into account the radial

Because of the complexi

model, a computer program was

applications. This was done w

described in Appendix VII. A

specified the geometry of the

the operating parameters, the

boundary conditions. A grid r

difference method was automati

tion of the shape and positio

Boundary condition (4)

symmetry.

ty of the above thermal

used for the numerical

ith the EPS program briefly

data file was written which

ingot, the physical properties,

differential equation and the

equired for the finite

cally defined after specifica-

n of the mushy zone. Itera-

tions were

mushy zone

temperature

2.

required until the upper and

corresponded to the

s.

Application o

lower limits of the

proper liquidus and solidus

f the computer thermal model to theingots of the experimental study

The above

cast with

(Section

tails of t

ysical dat

:ompute

power

.A).

compu

were t

r thermal model was applied to the

of 15 kw and at a speed of 1.1

The data files coupled to EPS and

tation are given in Appendix XI.

hose of Appendix I and the heat tr

fer coefficient between ingot and water wa

shown in Figure 27.

ans-

s approximated as

is a

cates

bottom

i ngots

cm/min

the de

The ph

I

For a first calculation of the temperature

distribution, a value of 0.16 cal/cm/sec/OC was used for

the thermal conductivity in the liquid metal pool. This

value was calculated with the moving fin approximation

(Section VI.C.3). The predicted shape of the mushy zone

(Figure 29) was similar to the experimental shape. The

predicted position of this mushy zone was about 5 mm below

the experimental position (discrepancy of 20 percent).

A second calculation was made using a value of 0.11

cal/cm/sec/0 C for the thermal conductivity in the liquid

pool. Figure 29 shows that the position of the predicted

mushy zone at 0.6 cm from the outside of the ingot is in

good agreement with the experimental results. If the

sulfur print were to give the shape of an intermediate

isotherm between the liquidus and solidus isotherms the

calculated mushy zone would be about 2 mm too deep at the

center line (8 percent of the total depth). Figure 30

shows that the calculated temperature profile at 0.6 cm

from the outside of the ingot is within less than 200C

of the experimental profile above 9000 C (positive

departure). A maximum discrepancy of 400C is reached at

550 0C.

Appendix XI shows that the solution obtained for

the temperature distribution by the finite difference method

was insensitive to the number of grid points and to the

tolerance parameter.

Power: 15 kwPositive electrode mode

mushy zone - computer modelcal /cm/sec/0C

mushy zone -cal/cm/sec/0

computer model

experimental positionssolidus temperatures (

(k1

(k1

0.11

0.16

of the liauidusFigure 24)

--- trace from sulfur print (Figure 20)

Figure 29: Experimentala power of 15 kw.

and calculated mushy zone for

and

calculated with computer model(k, = 0.11 cal/cm/sec/0 C, Appendix XI)experimental profile (Figure 24)

Power: 15 kwPositive electrode modeCasting speed: 1.1 cm/min

5 10

Distance from top of ingot,

Temperature profile at 0.6 cm from outside

1500

1000

500

Fi gure 30 : of ingot .

The temperature distribution calculated w

11 cal/cm/sec/0 C for the thermal conductivi

d pool, was considered to be in reasonable

the experimental results.

The calculated temperature profiles along

and at the surface of the ingot are plotted

The shape and position of several isotherms

ith a value

ty in the

agreement

the center

in Figure

are drawn

in Figure 32. These isotherms become flatter with

decreasing temperature. This result shows that nearly

flat profiles could be assumed in cross sections of the

ingot for the calculation of the average heat transfer

coefficient between solid ingot and water (Section VI.C.2).

The same result proves the validity of the extension of

the experimental temperature measurements in the bolt to

the bottom of the ingot (Section VI.B).

The calculated value of 0.11 cal/cm/sec/0 C for the

thermal conductivity in the liquid metal pool indicates

that convection occurred in this pool. The thermal

conductivity for stagnant liquid steel(32,33) would be in

the order of 0.025 to 0.05 cal/cm/sec/0 C. This convection

explains the rather uniform carbon content found in the

upper part of the ingot (Section VI.A.2).

Conclusion

An experimental study was performed in the ingot of

the laboratory ESR unit. A power of 15 kw was used with a

current of 750 amps and a positive electrode mode. A

of 0.

1 iqui

with

line

31.

center line temperature

- -- surface temperature

Power: 15 kwPositive electrode modeCasting speed: 1.1 cm/min

5 10

Distance from top of ingot, cm

Figure 31: CalculatedAppendix

ingotI).

center line and surface temperature

1500

1000

500

(resultsX

mushy zone

Power: 15 kw; casting speed: 1.1 cm/min; positive electrodemode

CalculatedAppendix X

isothermsI)

in ingot (results of

14840 C

1380 0C

12000C

10000 C

Or-

SH-

|Oh-

15 -

8000 C

6000 C

400 0C

300 C

Figure 32:

casting velocity of 1.1 cm/min was obtained. The temperature

distribution in the entire ingot was constructed. This

temperature was found to become steady after the ingot

reached a height 1.8 times its diameter.

A first analytical thermal model was derived using

the approximation of the moving fin. This model, applied

to the solid ingot,permitted the evaluation of the thermal

conductivity of the slag crust around the ingot.

A second thermal model was derived. This model

consisted of the solution of the heat conduction equation

applicable to the ingot for the appropriate boundary

conditions. A computer treatment of this model was made

on the ingot of the experimental study. The effective

thermal conductivity in the liquid metal pool was evaluated

at 0.11 cal/cm/sec/ 0C. This value accounts for convection

in the liquid pool. A satisfactory agreement was found

between the predicted and the experimental results. This

agreement proved the validity of the proposed model.

II

VII. THE SLAG - LABORATORY UNIT

A heat transfer study was performed on the slag of

the laboratory ESR unit. Power input varied between 7.5

and 22.4 kw. Temperature measurements were taken as shown

in Figure 33. Bubbles were blown at the end of a graphite

tube immersed in the slag and the temperature was measured

with a pyrometer sighting down the tube. Other thermo-

couple measurements were made underneath the electrode and

in the annular space between the electrode and the mold.

A series of temperature measurements was obtained

for a power of 15 kw and for a positive electrode mode.

These results show that the bulk of the slag was at a

uniform and steady temperature after the ingots reached a

height of about 10 cm.

Sensitive voltage measurements were made by insertion

of a small graphite rod in the electrode steel. Only the

tip of the graphite rod was in contact with the electrode.

At a power of 15 kw,the voltage between the graphite rod

and the bottom of the ingot was found to fluctuate. The

voltage drop at the interface electrode-liquid slag was

observed to be higher for a positive electrode mode than

for a negative electrode mode.

The relationship between melting rate and power is

given for a positive electrode mode. A change of polarity

at 15 kw was found to lower the melting rate and to affect

the shape of the electrode tip.

II

pyrometer

electrode

electrode thermocouple

ingot thermocouple

Schematicmeasuremen

drawing of thets in the slag

temperature

argon

gas ~bubble

ingot

Figure 33:

infrared

The heat generation and heat transfer mechanisms in

the slag are briefly discussed. Finally a heat balance

is made on the slag for a power of 15 kw and a positive

electrode mode.

A. Experimental Study

1. Pyrometric temperature measurements

The instrument for pyrometric temperature measurements

is shown in Figure 34. A 25 cm long graphite tube (0.D.

7 mm, I.D. 3.5 mm) was attached to an infrared radiation

pyrameter provided by Irtronics (model Pacemaker 1000

SP/REL Code W). The attachment was made through a water-

cooled brass clamp designed so that the alignment of the

tube with the pyrometer could be adjusted. A connection

for a gas line was provided. The pyrometer had two

temperature ranges: a low scale from 1400 0 C to 1750 0 C and

a high scale from 1700 0C to 2300 0C. The output of the

pyrometer was read from the amplifier (Figure 34) or

recorded on the Nulline recording potentiometer (Section IV).

The temperature was obtained using calibration curves.

Details of the calibration are given in Appendix XII.

The accuracy of the pyrometric temperature measurements

was estimated at +20 0C. The flow of argon used to form the

bubbles was 0.3 1/min.

Temperature measurements were made at powers of 7.5 kw,

15 kw and 22.4 kw after the ingots were about 10 cm high.

The recorded temperatures were found to fluctuate within

74

L

Figure 34: The infraredmeasurements

pyrometer used for the temperaturein the slag.

of the values

temperature wi

graphite tube

indicated in Ta

th time or with

in the slag was

ble 8. No variation

the immersion depth

observed.

2. Thermocouple measurements

The thermocouples used for the temperature measurements

in the electrode broke in the slags due to chemical attack

(Section V.A.2). The distances travelled by these thermo-

couples below the solid end of the electrode (temperature

of 1460 0 C) before failure and the maximum temperatures

recorded are listed in Table 8. In the same table,

figures are also referenced which indicate that, before

failure, the thermocouples recorded constant temperatures

over a short distance (at least 1 mm).

were taken as the temperatures of the

electrode. For a power of 15 kw, the

were found to vary between 1659 0C and

Other thermocouple measurements

annular space between the electrode an

33). The thermocouple was similar to

ingot temperature (Figure 14). The gr

end of the graphite tube was shorter (

sl

sl

17

we

d

th

ap

0.

These temperatures

ags below the

ag temperatures

06 0C.

re obtained in the

the mold (Figure

at used to measure

hite plug at the

3 cm instead of

2.5 cm) and the thickness of the graphite tube at the end

was about 0.1 cm over 1.5 cm. This thermocouple assembly

remained immersed in the slag to a depth greater than 1 cm

(more than 3 times the diameter of the inner alumina tube).

The graphite was slowly attacked by the slag. The maximum

temperature recorded before the thermocouple brokewas taken

+15 0C.

of the

of the

M - M M M - M11

Table 8

Temperature Measurements in

power, polaritykw electrode

15 +

15 +

15 +

15 +

15 +

15 +

15 +

15 +

15 +

22.4 +

7.5 +

15

meltingspeed,cm/mi n

3.5

4.0

3.9

4.3

4.0

3.9

3 .9*

3.9*

3 .9*

6.3

2.2

3.4

hei ghtof

slag, cm

4.5

2.9

2.5

2.5

5.2

4.1

4.0

0.8

3.0

T, 0 Cpyrometer

+200C

the Slag

T, 0 Cannular

space

1667

1655

1708

1694

1697

1675

1680

2000

1625

T, 0 Cbelow

electrode

1676

1660

1706

1659

1640 to17 60 (b)

1 658

Zt mm

(a)

4.0

4.1

2.0

1.7

figurenumberfor Tbelow

electrode

8

12

7

0.2

estimated value

distance travelled by thermocouple below solin the slag

temperature fluctuations between 16400C andsecond)

d end

760 0 C

of electrode before

(three fluctuations

*

(a)

(b)

failure

per

temperature of the slag for the particul

position. Spot measurements obtained wi

que are listed in Table 8.

3. Analysis of the temperature measureme

pyrometric temperature measurements

thermo-

this

nts

(Secti on

VII .A

space

were

at 15

measu

.1) indi

stayed

at least

Table 8

kw with

rements

C

a

u

space gave sim

The pyrometric

the temperatur

constant and u

at least two t

Table 8

conditions of

positive elect

between 1659 0 C

ate tha

t a val

10 cm.

shows t

a posit

ndernea

ilar re

and th

e in th

niform

imes th

shows v

operati

rode mo

and 17

t the temperature

ue almost constant

high.

hat, for the first

ive electrode mode

th the electrode a

sul ts

ermoco

e bulk

after the

eir diame

ar iations

on. At a

de, the s

060C. At

negative electrode mode,

This measurement underes

due to the rapid failure

the

tima

of

(dis

uple

crepancy

measure

in the

after

annular

the ingots

two experiments

the thermocouple

nd in the annular

of 9 0C and 50C).

ments prove that

of the slag remained

0

te

th

height of the ingots

ter.

of slag temperature

power of 15 kw with

lag temperatures vari

the same power with

nly measurement gave

s the true slag tempe

e thermocouple in the

nearly

reached

with the

58 0C.

ture

lag

(0.2 mm bel

a power of

For a

gave 1625 0C

ow solid end of the electrode, Figure 13). For

22.4 kw, the temperature was about 2000 0C.

power of 7.5 kw, the pyrometric measurement

(Table 8) and the thermocouple measurement below

as the

couple

techni

II

-1 Ow 00600 -- -

the electrode indicated temperature fluctuations between

16400C and 1760 0C (Figure 9). The frequency of the

fluctuation was about 3 per second. The slag was about

8 mm high. The pyrometric measurement was made in the

liqui

is as

bel ow

meta

umed

the e

4.

Th

a graphi

electrod

rod was

V

1

to

le

ol

pool due to the s

have caused the

ctrode (Section V

tage measurements

voltage

e rod (3

over a

ncased i

from the electrode

graphite rod

the electrod

tip, before

the graphite

Honeywell re

reduction th

During

voltage acro

resistances

(Section VII

drop

mm di

istan

an a

across

hallow slag

temperature

II .B.1 ).

in the slag

the

ameter) in

ce of 4.5

lumina tub

except at the

was in contact with

e at a

mel tin

rod a

cordi n

rough

ESR o

ss the

were n

.B.1 ).

i initial dist

, of about 30

d the base of

potentiomete

voltage divi

eration, the

slag, because

egl igi ble

slag was

serted in th

cm (Figure 3

e which insu

tip. The ti

the steel at

nce from the

cm. The vol

the mold was

(Section IV

er (1/1300,

ecorded volt

the electrod

compared

bath. Arcing

fluctuations

measured with

e steel

5). This

lated it

p of the

the axis of

electrode

tage between

read on the

) after

Figure 35).

age was the

e and ingot

to the slag resistance

After 30 cm of electrode melted,

the graphite sensor went i

voltage was instantaneousl

taken at the voltage drop

liquid slag.

nto the sl

y lowered.

across the

ag. The recorded

This voltage drop was

interface electrode-

I

oneywell

ecording

otenti o-eter

alumina insulating tube

graphite rod(3 mm diameter)

5 mm diameter hole

steel electrode

mold base

Figure 35: Measurement of the voltage drop acrossthe slag.

vol tagedivider(1/1300)

30 cm

The results of voltage measurements are given in

Table 9 for two experiments

For a pc

cm/mi n,

20 + 0.6

and the

8.3 V.

was 3.4

within 2

second a

slag was

si t

the

V

vol

For

cm/

0 +

nd

v e

vol

with

tage

a n

mi n,

2.2

the

electrode mo

tage across

a frequency

drop at the

egative elec

the voltage

V with a fr

made at a

de, the mel

the slag fl

of 4 fluct

interface

trode mode,

across the

equency of

power of 15 kw.

ting speed was 3.9

uctuated within

uations per second

electrode-slag was

the melting speed

slag fluctuated

one fluctuation per

voltage drop at the interface electrode-

6.7 V. These results are analyzed in Section VII.B.

5. Melting o

Melting speed

input are shown in

15 kw and 22.4 kw.

and show that, for

speed was proportio

The relationship be

be expressed as:

f electrode in the slag

s of electrodes as functions of

Table 10. The power inputs are

These results are plotted in F

a positive electrode mode, melt

nal to power input within + 10

tween melting speed and power i

power

7.5 kw,

igure 36

ing

percent.

nput can

VE(cm/min, + 10 percent) = 0.26 x P (kw) (12)

Power input appears to be the main independent variable

which determined melting rate.

Typical shapes of electrode tips are given in Figure

37 for powers of 7.5 kw and 22.4 kw (positive electrode

mode) and for a power of 15 kw (two electrode modes). From

this figure, it may be seen that, for a positive electrode

, , TO - , , -- I

Table 9

Voltage Measurements in the Slag

power, polaritykw electrode

averagecurrent

amps

750

mel tingspeed,cm/mi n

3.9

vol tagevariationacrossslag,volts

20+0.6

frequencyof

vol tagefl uctua-ti ons

persecond-

vol tagedrop at

interfaceelectrode-

slag,volts

8.3

750 3.4 20+2.2 6.7

Table 10

Melting Speed of Electrode Function

of Melting Conditions

meltingconditions

powerkw

melting speedcm/min

averagemelting speed

cm/min

medium power,(+) electrode,750 amps, 20V

high power,(+) electrode,800 amps, 28V

low power,(+) electrode,580 amps, 13V

medium power,(-) electrode,750 amps, 20V

3.74.13.83.73.93.9

3.9

22.4

7.5

5.85.46.3

1.82.2

3.4

2.0

* experimental values

linear approximation

Positive electrode mode

7.5 15 22.4

power

power input, kw

Melting speedpower input.

of the electrode

0 2

lowpower

highpower

Figure 36: function

(a) 7.5 kw, positive electrodemode, melting speed: 2.2 cm/min

(b) 22.4 kw, positive electrodemode, melting speed: 5.4 cm/min

(c) 15 kw, positive electrodemelting speed: 3.9 cm/min

mode,

(d) 15 kw, negative electrode mode,melting speed: 3.4 cm/min

Electrode diameter:

Figure 37: Electrode tips for various

1 in.

melting conditions.

mode, a decrease in power makes the electrode tip more flat.

At a power of 15 kw, there was a marked difference in shape

of the electrode tip between the positive and negative

electrode modes.

The difference in shape of the electrode tip for the

two polarities and the results obtained for the voltage

fluctuations (Table 9) indicate that liquid metal

droplets(4,5) may have formed at the tip of the electrode

in two different ways. It was assumed that each fluctuation

in the voltage (Table 9) corresponded to the detachment of

a liquid metal droplet. The weight of these droplets was

calculated and found to be 0.6 g for positive electrode mode

and 2.2 g for negative electrode mode.

The lower melting rates obtained with a negative

electrode mode than with positive electrode mode have also

been observed by several investigators(3,7,39). An explana-

tion of the effect of polarity is given in the next section.

B. Mechanism of Heat Generation and Heat Transferin the Slag

1. Mechanism of heat generation in the slag

The mechanism of heat generation in the ESR process

at high current densities (115 to 160 amps/cm2 in the

laboratory unit) is not very well understood. Previous

investigators(6,9) have attributed heat generation to

resistance heating or arcing. A detailed investigation on

this aspect would have required an extensive experimental

I

study whi

and quali

is given

ch was

tati ve

in this

beyond the scope of this

survey of possible modes

section.

work. A brief

of heat generation

The slag used

initial composition

percent CaO. In the

present: Ca 2+, 02-

manganese, sulfur an

electrode steel. By

added supplementary

The transport

have been measured.

fluoride, the transp

0.6

In

be

Poo

in the laboratory ESR unit had an

of 80 weight

molten stat

and F .

d oxygen

transfer

ions: S2

numbers i

In anoth

ort numbe

6). The conduction

rder for the curren

ischarged at the sl

interfaces. At thi

in th

to g

g-ele

anod

percent

e, the f

Species su

were added

into the s

-, 02-, Fe 2

n CaF 2-CaO

er fluoride

r of F has

e CaF 2 -CaO

o through t

ctrode and

ic interfac

CaF2

ol 1 owi

and 20 weight

ng ions were

ch as iron,

by melting the

lag, these species

+(3+) adM2+(3+)and M

slags do not seem to

like liquid sodium

been found to be

slag may be ionic.

he slag, ions must

slag-liquid metal

es (electrode-slag

for positive electrode mode or liquid metal pool-slag for

negative electrode mode), possible charge transfer

reactions may be

[Fe] + (Fe ) + 2e~ (

[Mn] + (Mn ) + 2e (1

(0 ~) + [0] + 2e~ (

(S~~) [S] + 2e~ (

(FK) + 1/2 F2 (g) + e~ (

I

At the cathodic interfaces (liquid metal pool-slag for

positive electrode mode or electrode-slag for negative

electrode mode), possible reactions would be

(Fe2 +) + 2e~ [Fe] (18)

(Ca 2+) + 2e~ Ca(g) (19)

(Mn2+) + 2e + [Mn] (20)

[0] + 2e~ (0 2-) (21)

[S] + 2e + (S 2-) (22)

These electrochemical reactions originate surface over-

potentials. These overpotentials result from a reversible

potential, a reaction overvoltage, a concentration over-

voltage and a charge transfer overvoltage at the metal-

slag interfaces. From the high voltage drops measured at

the interface electrode-slag (Table 9) any combination of

electrochemical reactions (reactions 13 to 22) appears

possible. These electrochemical reactions may account for

the voltage drops measured at the electrode-slag interface

(Table 9) or at least for part of it.

In the case of reaction 19, metallic calcium may

enter in solution. The activity of metallic calcium may

increase and electronic conduction may interfere with ionic

conduction. In a work on galvanic cells, Wagner showed

that electronic conduction in CaF 2 equilibrated with Th +

ThF 4, was insignificant (activity of calcium below 10-5 )

but that electronic conduction in CaF 2 would interfere in

cells involving metallic Ca with an activity of Ca of 1.

An increased electronic conduction in the ESR slag would

reduce the importance of electrochemical reactions.

88

Gas evolution may also occur at the slag-metal

interfaces. Due to reactions 17 and 19, fluorine and

calcium gas may be evolved at the interfaces. Oxygen

can also be evolved. These gases may form a gas film at

the interfaces. Thus the large potential drops measured

at the interface electrode-slag may also be caused by

resistance through a gas film.

Gas evolution can

a transparent crucible

that arcs formed at the

at small slag depths.

study on an induction u

a water-cooled iron rod

densities above 50 amps

formation of arcs could

arcing can occur at the

voltage drops at interf

voltage drops originate

The present exper

toward temperature measurements

conclusion about the origin of t

at the electrode-slag interface.

ments made over a short distance

a power of 15 kw indicated a con

VII.A. 2). This result seems to

heating was more likely than arc

ment at a power of 7.5 kw, rapid

also be a source of arcing. With

in an ESR unit-, Campbell(4) showed

electrode tip for high voltage

Mitchell(6) conducted an experimental

nit with a fluoride-based slag using

as a working electrode. For current2/cm , and when the iron was anodic,

be observed. Thus on an ESR unit,

electrode tip. Such arcs would give

ace electrode-slag higher than

d by polarization(6).

imental study was aimed essentially

and does not permit a definite

he observed potential drop

The temperature measure-

below the electrode tip at

stant temperature (Section

indicate that resistance

ing. In the single experi-

temperature fluctuations

II

were observed between 1640 0 C and 17600C (Section VII.A.3).

The slag bath also was very shallow. These results may

show the possibility of arcing in such an experiment.

The occurrence of electrochemical reactions and gas

evolution at the slag-metal interfaces and the possibility

of arcing at the electrode tip show that heat is not

generated uniformly in the slag. In particular, intense

heat generation would occur at the electrode tip. In

the experiments made at a power of 15 kw (Table 9),

assuming that the voltage drops were constant at the

electrode-slag interface and that heat was generated by

Joule effect only, the heat generation was proportional

to the voltage drops. About 41 percent of the total heat

was generated at the electrode tip for the positive

electrode mode and 33 percent for the negative electrode

mode.

The slag in the ESR process is the main heat source

because the slag constitutes the main resistance. The

resistance of the steel electrode and of the ingot was

approximately 10~4 0 due to the low resistivity of steel

(Appendix I). Using a voltage of 20 V across the slag

and assuming a constant current of 750 amps, the slag

resistance is found to be approximately 0.0260. The heat

generated in the steel is negligible compared to the heat

generated in the slag.

2. Melting conditions of

The intense heat generati

interface may account for the f

electrode tips observed with po

(Figure 37 a, b and c). A stee

conditions as in ESR but withou

would have a more conic shape a

At a power of 15 kw, the

electrode-slag was found to be

negative electrode mode (Table

the previous section, the heat

tip might be higher at positive

mode. This may account for the

instead of 3.4 cm/min) and for

(Figures 37c and d) at positive

the electrode

on

la

si

l

t

t

vo

hi

9)

ge

at

t to

ti ve

rod

heat

the

1 tag

gher

F

nera

than at negative

the electrode-slag

slightly conic

electrode mode

melting in the sam

generated at the

tip.

e drop at the inte

at positive than

rom the discussion

ted at the electro

electrode

higher melting rate (3.9

the flatter electrode tip

than at negative electrode

mode.

The melting of the

more than heat transfer.

of variable dimensions as

recent study with a trans

of these droplets depends

interfacial tensions, hyd

of the slag, melting rate

particular the interfacia

chemical and electrochemi

electrode-slag interface.

electrode in the slag involves

The liquid metal may form droplets

proved by Campbell(4) in a

parent crucible. The detachment

on such factors as density differences,

rodynamics in the slag, temperature

and possibility of arcing. In

1 tensions also depend on the

cal reactions occurring at the

For example transport of sulfur

from liquid to a slag phase has been shown to alter

I:

H

e

tip ,

rface

at

in

de

instantaneous and non-equilibrium surface tensions by

factors approaching 200(3).

A detailed investigation on the detachment of the

liquid metal droplets from the electrode tip has not been

undertaken. In spite of the numerous parameters involved

in the determination of the melting rate, the melting

rate was found to be proportional to the power input

(within + 10 percent, Section VII.A.5). It is possible

that lower interfacial tensions at the electrode tip may

explain the smaller and more frequent droplets found at

positive than at negative electrode (Section VII.A.5).

3. Temperature distribution in the slag

Although

the tem

uni for r

VII .A.3

magneti

This mi

bulk of

boundar

1 ayers

metal p

annular

perature

after t

). The

c effect

xing app

the sla

y layers

are situ

ool -slag

surface

The conc

temperature in

the assumption

heat is not generated un

in

he i

slag

the

ngot

has

ulk of the s

were about

a low viscos

gives vigorous stirring

ears

g.

whe

ated

i nt

of

Isio

the

nade

l

to make

Temperatu

re heat i

at the e

erf

the

n a

bul

of

le tempe

drops

extract

lectrode

ra

wo

ed

tip

iformly in the slag,

was found to be

cm high (Section

and the electro-

(Section VIII.C).

ture uniform, in the

uld occur in

. These boundary

the liquid

ace, the water-cooled wall and the

slag between electrode and mold.

bout the steady and uniform

k of the slag shows the validity of

a uniform temperature at the top of

the ingot (Section VI.B).

ti

C. Heat Balance on the Slag

An approximate heat

experiments at a power of

balance is made on the slag for

15 kw with a casting speed of

1.1 cm/min and a positive electrode mode (Section VII.A).

The metal is assumed to enter and leave the slag at

the temperature of the slag (1700 0C). The heat absorbed

or dissipated by possible reactions in the slag is ignored.

The heat generated in the slag is assumed to be dissipated

by Joule effect and to correspond to the electrical power

input, P. As indicated in Figure 38, P is taken as the

sum of the heat loss by conduction into the incoming

liquid metal, Qm, into the ingot, Q1 , and into the water,

Qw, and by radiation with the surroundings, QR' in

the annular space between the electrode and the mold.

The total heat generated, P, is equal to 3,600

cal/sec. The electrode having almost adiabatic heat flow

conditions on the cylindrical surface above the slag level

(Section V.B.2), the heat loss, Qm, is the enthalpy varia-

tion of the metal from room temperature to the temperature

of the slag. This heat loss, Qm, is expressed as:

QM (cal/sec) = PSE VE x (HTsl- HTo) (23)

Using tabulated enthalpies for iron(29), Qm is found to

have a value of about 1,000 cal/sec.

The heat flow by conduction into the ingot, Q1, is

calculated using the results obtained with the computer

M I

Positive electroCasting speed:Temperature of sMetal is assumed

leave the slag

(34.8%)

de mode1.1 cm/minlag: 1700 0 Cto enter andat 1700 0C

= Qm + Q + QR + QW

Figure 38: Heat balance on slag for a power of 15 kw anda positive electrode mode.

thermal model of the ingot (Appendix XI). The value of Q

is found to be 1,180 cal/sec.

The heat loss by radiation, QR, is approximated

with:

QR(cal/sec) = T(R -R2 )Ea(T -T ) (24)

The emissivity of the slag is assumed to be 0.7 and the

temperature at the surface of the slag and of the surround-

ings are taken as 18500K and 300 0 K, respectively. QR is

found to be equal to 165 cal/sec.

The heat loss into the water, QW, is obtained by

difference using the heat balance (Figure 38). QW is equal

to 1,255 cal/sec.

The results are summarized in Table 11 and in Figure

38. 34.8 percent of the total heat input is lost by water-

cooling around the slag. A small amount (4.6 percent) is

lost by radiation to the surroundings at the surface of the

slag. 32.8 percent is lost by conduction into the ingot.

The remaining 27.8 percent is entirely used to heat

up the metal to the temperature of the slag. This last

amount is brought into the ingot as enthalpy where it is

lost by water-cooling.

In view of the approximation made in this heat

balance, it is concluded that most of the heat dissipated

in the slag appears to be lost almost equally by conduction

into the water around the slag, the ingot, and the electrode.

II

Table 11

Heat Balance on the Slag for a Power ofand Positive Electrode Mode

heat fluxcal /sec

3,600

1,000

1 ,180

165

1 ,255

percentageof heat input

100

27.8

32.8

4.6

34.8

(a) calculated from(Appendix XI)

ingot computer thermal

(b) obtained by difference with heat balance (Figure 38)

Casting speed 1.1 cm/min.

15 kw

QI(a)

Q )

QW (b)

model

'I

Conclusion

For a power of 15 kw, the temperature in the bulk o

the slag appears to be steady and uniform after the ingo

were about 10 cm high. Temperature drops would occur in

boundary layers at interfaces between the slag and the

surrounding media.

For positive electrode mode, the power appears to

the main independent variable controlling the melting

velocity of the electrode in the slag. The relationship

between melting velocity and power was found to be:

VE(cm/min), + 10%) = 0.26x P (kw)

f

ts

be

(12)

At a power of 15 kw, a change of polarity from

positive to negative electrode mode was found to lower th

melting velocity (3.9 to 3.4 cm/min) and to decrease the

voltage drop at the interface electrode-slag (8.3V to 6.7

With this change of polarity, the voltage fluctuations

across the slag become larger and less frequent, due to a

apparent change in the size of the metal liquid droplets

(0.6 g for positive electrode mode and 2.2 g for negative

electrode mode).

The heat in the ESR process is generated in the

slag. Because of the non-uniform resistance of the slag,

heat is generated non-uniformly. In particular, intense

heat may be generated at the electrode-slag interface due

to the high voltage drops observed (at a power of 15 kw,

8.3 V at positive electrode mode and 6.7 V at negative

e

V

n

'I

).

97

electrode mode for a total of 20 V across the slag).

An approximate heat balance on the slag for a power

of 15 kw, a casting rate of 1.1 cm/min and a positive

electrode mode, showed that little heat is lost by radia-

tion (about 5 percent of total heat input) and that heat

is lost almost equally by conduction into the electrode

to heat up the remelting material, into the ingot and the

water-cooling around the slag.

-Alft-MI I

VIII. APPLICATION OF THE RESULTS

Details of temperature measurements were not

available for industrial ESR units. Thus an investiga-

tion has been made of possible applications of the results

obtained on the laboratory ESR unit to systems on an

industrial scale.

Heat transfer in the electrode, the ingot and the

slag are investigated separately. Various industrial

units are considered, but emphasis is placed on units

producing 50 cm diameter steel ingots with an electrode-

ingot diameter ratio of 0.75. This size appears to be

representative of industrial practice.

A. Electrode

After a brief summary of the results obtained

on the electrode of the laboratory ESR unit (Section V),

available data on industrial ESR practice are given. The

heat generation inside the electrode is shown to have a

negligible effect on the temperature in the electrode.

An investigation is made on the heat exchange between

the electrode and the surroundings above the slag level.

Using the heat flow models derived and validated on the

laboratory ESR electrode, the temperature distribution

is analyzed for industrial electrodes. First the

electrode is assumed to melt with a flat tip and the

immersion depth is neglected. Then the effect of this

immersion depth is investigated and it is shown, that for

high enough immersion depth, the electrode tip cannot

possible be flat. Finally the temperature distribution is

given for electrodes with parabolic tips.

1. Summary of the results obtained on theelectrode of the laboratory ESR unit

In the electrode of the laboratory ESR unit, current

densities of 115 to 160 amps/cm2 were used. The Joule

effect was found to gradually increase the temperature in

the upper part of the electrode. Possible maximum steady

state temperatures of 950C were calculated (Table 2).

The electrode was found to have nearly adiabatic heat flow

conditions for the portion above the slag level. The heat

flow in the electrode was one-dimensional and axial or two-

dimensional depending on the immersion depth. The two-

dimensional heat flow conditions were caused by radial heat

flow from the slag into the immersed cylindrical surface of

the electrode.

2. Available data on ESR practice

In industrial ESR practice, the most common electrode-

ingot diameter ratios appear to be between 0.6 and 0.8(39,42).

Such high electrode-ingot diameter ratiosare selected mainly

to avoid too high a heat loss by radiation from the slag

surface in the annular space between the electrode and the

mold.

100

The casting speed depends on the ingot size and is

selected to obtain a shallow metal pool, i.e., depth of

the metal pool less than or equal to the ingot radius(10).

With these casting speeds, the solidification pattern in

the ingot is more marked in axial direction than in a

radial one. Calculated correct casting speeds are given

in Figure 45 for various steel ingot radii. This figure

is obtained from the computer study in the section on the

industrial ESR ingots (Section VIII.B).

3. Heat generation in industrial ESR electrodes

In industrial ESR electrodes, the current density

appears to be approximately proportional to 1/RE'(3942).

Thus, the importance of the Joule effect as a heat source

decreases with increasing electrode diameter. For

example,in a 37.5 cm diameter steel electrode melting in a

50 cm diameter ingot, a current of about 12,000 amps would

be used(42). The temperature increase above room tempera-

ture may be calculated with equation (IV.4) (Appendix IV).

Using an approximate heat transfer coefficient between

electrode and surroundings of 4.5 x 10~4 cal/cm 2/sec/ C

(lowest value given in Table 2), the temperature increase

is found to be approximately 180C. The differemce between

the center line and the surface temperature would be

about 0.5 0C.

In the rest of this study, Joule effect in the

electrode is neglected.

- OM*MR-_ aoaw - -

101

4. Heat flow conditions in ESR electrodes

In ESR

slag into the

exchange also

between the el

and by conduct

determi nati on

the electrode

level depends

most important

the slag, the

sion depth of

fer coefficien

electrodes, heat conduction occurs from the

immersed portion of the electrode. Heat

occurs above the slag level by radiation

ectrode and the slag surface essentially

ion with the gas around the electrode. The

of the relative amount of heat going into

in the immersed portion and above the slag

on many factors. Among these factors, the

are the emissivities of the electrode and

electrode-ingot diameter ratio, the immer-

the electrode in the slag, the heat trans-

t between electrode and slag in the immersed

portion, the physical properties of the electrode, the

melting speed, and convection of the gas around the

electrode.

On the laboratory ESR unit, calculations showed

that a maximum of 5 percent of the total amount of heat

required to heat up the electrode from 50 0C to 1460 0C was

due to heat coming by radiation from the surface of the

slag (Table 3). The electrode was considered to have

almost adiabatic heat flow conditions on the surface.

For industrial ESR electrodes, the lack of data

does not permit a general conclusion. A specific

example was treated on what might be a typical ESR steel

electrode. The electrode was 37.5 cm in diameter, melted

at a speed of 0.9 cm/min and formed a 50 cm diameter

II

102

ingot (casting speed of 0.5 cm/min). The properties of

the electrode and the slag were assumed to be the same as

for the laboratory unit (emissivities of electrode and

slag, 0.25 and 0.7, respectively). The immersion depth

of the electrode was assumed to be 5 cm. The calculated

heat exchange t

electrode above

similar way to

Heat was found

level. This he

necessary to ma

620 cal/sec com

The same

steel electrode

speed of 0.25 c

temperature gra

effect of radia

rate. The main

hrough the cylindrical surface of the

the slag level was approximated in a

that for the laboratory unit (Appendix V).

to flow into the electrode above the slag

at was about 2 percent of the total heat

intain 1460 0C at the electrode tip (about

pared to about 30,000 cal/sec).

result holds approximately for the same

melting at a speed of 0.45 cm/min (casting

m/min). This is due to the less steep axial

dient in the electrode, which reduces the

tion from the slag and to the lower melting

heat into the electrode has then to

supplied into the immersed portion of the electrode by

conduction from the slag.

The possibility of adiabatic heat flow conditions

can also be shown in a qualitative way. For a given

material and at a given melting rate, when the electrode-

ingot diameter ratio approaches one, the melting speed of

the electrode becomes minimum, the temperature gradients

at the electrode tip given by one-dimensional heat flow

(Section V.C.1, equation 1) would be minimum and heat loss

103

would occur on the side of the electrode. On the contrary,

for decreasing electrode-ingot diameter ratios, the

melting speed of the electrode increases, the vertical

temperature gradient at the tip of the electrode increases

and heat radiation into the electrode from the slag

increases.

In industrial practice on steel, where high electrode-

ingot diameter ratios are used (Section VIII.A.2), almost

adiabatic conditions may exist at the surface of the

electrode as on the laboratory unit. This appears to be

the case on the 37.5 cm diameter steel electrode with an

emissivity of 0.25, melting into 50 cm diameter mold,

considered in this section. Under such conditions, heat

into the electrode is entirely used to heat up the

electrode.

5. Temperature distribution in the electrode

In a first hypothetical case, the electrode is

assumed to melt with a flat tip, without immersion in the

slag and with adiabatic heat flow conditions on the side.

Heat flow is one-dimensional and the temperature profile

in the electrode may be calculated with equation (1),

rewritten below:

T - TE pCsVET - TE k exp( Z) (Z < 0) (26)me E s

This temperature profile is independent of the size of

the electrode.

104

For steel electrodes (data of Appendix I), dimension-

less temperature is calculated versus distance from electrode

tip using equation (26). The results are plotted in Figure

39, for a casting speed of 0.5 cm/min and electrode-ingot

diameter ratios of 0.25, 0.5 and 0.75. Decreasing the

electrode-ingot diameter ratio increases the melting speed

and increases the temperature gradient at the tip of the

electrode.

40

el

ca

Si

which

ectrode

sting s

electrode

mi larly

show the

-ingot d

peed of

is stee

temperature profiles are plotted

effect of the casting speed for

iameter ratio of 0.75. Even for

0.25 cm/min, the gradient at the

p. The dimensionless temperature is

0.3 at 10 cm from the elec

The effect of the i

slag on the temperature di

be studied

V.C.2). T

electrode

0.9 cm/min

for the he

and for th

trode of t

respective

electrode

was calcul

wi

his

mel

(c

at

e s

he

ly,

was

ate

th the two-di

is done for

ting into 50

asting speed:

transfer coef

lag temperatu

laboratory un

Append

taken

like

ix

d

The calculated

trode tip.

mmersion of the electrode in the

stribution in the electrode may

mensional thermal model (Section

a typical 37.5 cm diameter steel

cm diameter ingot at a speed of

0.5 cm/min). The values used

ficient between electrode and slag

re were the same as for the elec-

it (0.04 cal/cm 2/sec/0C, 1650 0 C,

VIII). The melting point of the

as 1460 0 C. The heat flow above the

for the laboratory unit (Appendix V

center line and surface

slag

temperatures are

in

an

the

tip

Figure

lowest

of the

about

M M - M M1

A: electrode-ingot

pC /k

LUJ

LUJ

E

0)

4-

Ln

E

Lin

diameter ratio

= 16.8 sec/cm

Casting speed: 0.5 cm/min

2 5

Distance from tip of electrode,

Figure 39: Temperature pthermal model

rofile in ESR(equation 1).

electrodes calculated with one-dimensional

1 .0

0.5 A0 .75

A=0 .5

A=0.25

r

0 5 10Distance from electrode

Figure 40 : Temperature profilethermal model (equa t

in ESRion 1).

electrodes calculated with one-dimensional

1.0

0.5

LUJ

LI-

I-

F- IF-E

S.-

E4-)

(n

C

E

tio, cm

ratio: 0.75

107

given in

cm respec

file with

is also p

Figures 41 and 42 for immersion depths of 5

tively. On the same figures, the calculated

the one-dimensional heat flow model (equati

lotted. Figure 41 shows that for an immersi

depth of 5 cm, the center 1

not affected by radial heat

is significantly affected i

electrode. For an immersio

radial heat flow increases

a maximum of 100 0C above th

for one-dimensional heat fl

effect of radial heat flow

is due to the low thermal c

and 10

pro-

on 26)

on

ine temperature profile is almost

flow. The surface temperature

n the immersed portion of the

n depth of 10 cm (Figure 42),

the center line temperature by

e temperature profile calculated

ow conditions. This limited

on the center line temperature

onductivity of steel. Figure 42

also shows that the surface temperature remains at the

melting point over about 3 cm from

result indicates that with an imme

electrode cannot melt with a flat

The computer thermal model

V.C.2) may be adapted for electrod

With this model, the temperature w

37.5 cm diameter steel electrodes

diameter molds at speeds of 0.9 an

speeds of 0.5 and 0.25 cm/min). T

assumed to be parabolic over 10 cm

the electrode

rsion depth of

tip.

for electrodes

e tips of any

as investigate

melting into 5

d 0.5 cm/min (

he electrode t

and a total i

This

cm, the

(Section

shape.

d for the

0 cm

casting

ip was

mmersion

depth of 12 cm was taken. Over the 2 cm of ve

electrode surface immersed in the slag, a heat

coefficient of 0.04 cal/cm 2/sec/ C was assumed

rti cal

transfe

between

r

the

108

calculated surface temperature -computer thermal model (Sec. VI.D)

calculated center line temperature-computer thermal model (Sec. VI.D)

calculated profile fromequation (26)

b 5 10 15Distance from electrode tip, cm

Melting speed: 0.9 cm/minElectrode-ingot diameter ratio: 0.75Ingot diameter: 50 cmImmersion depth: 5 cmFlat electrode tipMelting point of electrode: 1460 C

Temperature in 37.5 cm diameter steel electrode.

1500

S1000

500

Figure 41 :

1500

4-.)

(0

-

1 000

500

0

DistaMelting speed: 0.45 cm/minElectrode-ingot diameter raIngot diameter: 50 cmImmersion depth: 10 cmFlat electrode tipMelting point of electrode:

calculated surface temperature -computer thermal model (Sec. VI.D)calculated center line temoerature-computer thermal model (Sec. VI.D)calculated profile fromequation (26)

nce

tio:

10 15from electrode tip, cm

0.75

1460 0C

Temperature in 37.5 cm diameter steel electrode.

109

n

Fi gure 42 :

110

electrode and the slag (val

unit, Appendix VIII). The

The calculated temperature

Figures 43 and 44 indicate

the electrode tip. Along t

profiles are not so steep a

flow conditions (Figure 40)

0.9 cm/

300 0C C

the mel

is abou

tempera

heat fl

the par

el ectro

radi us

ue found for the laboratory

slag temperature was 1650 0C.

distributions, given in

sharp temperature gradients at

he center line, the temperature

s for the one-dimensional heat

For the melting speed of

min (Figure 43), the maximum de

n the center line at 5 cm from

ting speed of 0.45 cm/min (Figu

t 320 0 C at 10 cm from the elect

ture differences between the on

ow conditions are due to radial

abolic tip.

Even for the parabolic tip, the

de rises over a short distance.

from the tip, the temperature i

parture is about

the tip. Similarly

re 44), the departure

rode tip. These

e- and two-dimensional

heat flow through

temperature in the

Above one electrode

less than 200 0C

for a melting speed of 0.9 cm/min and less than 450 0C

a melting speed of 0.45 cm/min.

6. Conclusion

An investigation was made on industrial ESR

electrodes using the results obtained on the electrode

of the laboratory ESR unit. Emphasis was placed on a

typical 37.5 cm diameter steel electrode melting into

cm diameter mold at speeds of 0.45 and 0.9 cm/min.

Joule effect has a negligible effect on the

temperature distribution in large ESR electrodes. The

for

50

111

R = 18.75 cm

cm

15-

200 0 C

slaglevel

10 -4 00 C

600 0 C

800 C

10000 C

1200 0C --0 Melting speed: 0.9 cm/min

0 1460 C Casting speed: 0.5 cm/minIngot radius: 25 cmElectrode-ingot diameter

ratio: 0.75Slag temperature: 16500C

Figure 43: Temperature distribution in an industrial ESRsteel electrode.

Eli

112

R = 18.75 cm

cm

25 - 200 0 c C - -

20

20 00 C -~

20 -

---

0

15 - ----

6000 C -----

slag

10 - 800 0 C level

1000 0 C

1200 0 C

1460 0 C Melting speed: 0.45 cm/min0 Casting speed: 0.25 cm/min

Ingot radius: 25 cmElectrode-ingot diameter

ratio: 0.75Slag temperature: 1650 C

Figure 44: Temperature distribution in an industrial ESRsteel electrode.

113

above steel electrode was found to have almost adiabatic

heat flow conditions on its cylindrical surface. Heat

transfer into the electrode occurs by conduction of heat

from the slag into the immersed portion of the electrode.

This heat flow into the electrode is entirely used to heat

up the electrode.

Temperature profiles were first calculated with the

one-dimensional heat flow model (equation 26), assuming

no radial heat flow. The effect of immersion depth was

studied with a two-dimensional thermal model (Section

V.C.2). On the electrode melting at 0.9 cm/min, and for

a flat tip, immersion depths below 10 cm affect the center

line temperature by less than 100 0 C. For immersion depths

above 10 cm, the electrode tip cannot be flat. Parabolic

shape was assumed over 10 cm and an immersion depth of

12 cm was taken. The discrepancy between the temperature

profiles along the center line calculated with and without

radial heat flow, was less than 320 0C for the two melting

speeds of 0.9 and 0.45 cm/min.

In industrial ESR electrodes ,the temperature

distribution is essentially determined by axial heat flow.

A temperature profile can easily be calculated with the

one-dimensional thermal model (equation 26). Radial heat

flow may also occur mainly in the immersed portion of the

electrode. This causes a positive departure in the

temperature distribution from that calculated with one-

dimensional heat flow.

M - M - IM

114

B. Ingot

After a brief summary of the results obtained on

the ingot of the laboratory ESR unit, the applicability

of the model using the approximation of the moving fin to

the industrial situation is discussed. The computer

thermal model is applied to the steel ingots and the

influence of various parameters on the temperature distri-

bution in the ingot is studied. These parameters are ingot

radius, casting speed, temperature at the top of the

ingot, heat transfer coefficient between the ingot and

the water, convection in the liquid metal pool, and heat

released upon solidification. The entire temperature

distribution is given for typical 50 cm diameter steel

ingots cast at speeds of 0.5 and 0.25 cm/min. The case of

steady state heat transfer is investigated.

Other heat transfer studies on ingots produced by

ESR, continuous casting machine or vacuum arc remelting

process are discussed.

To present the results, the possibility of using

dimensionless numbers was investigated. As indicated in

Appendix XIII, the dimensionless numbers for an ingot of

pure metal are as follows: a dimensionless temperature,

the dimensionless coordinates, the dimensionless tempera-

ture at the top of the ingot, the dimensionless effective

thermal conductivity in the liquid pool, k1/ks, the

dimensionless casting speed, pCs IRI/ks, and the Biot

115

number, hIRI/ks. The Biot number varies with the vertical

coordinate at the surface of the ingot, so that the

similarity between ingots can only be approximate. Because

of the variation of the Biot number with vertical coordinate

and to permit a more clear presentation, most of the results

are given in absolute dimensions.

1. Summary of the results on the laboratory ESR unit

In the laboratory ESR unit, the moving fin approxi-

mation for heat transfer (Section VI.C) was applicable to

the "solid ingot" and gave approximate results on the

entire ingot. The computer thermal model (Section VI.D)

was successfully applied to the entire ingot. For a

casting speed of 1.1 cm/min, steady state heat transfer

existed in the ingot once its height was greater than 1.8

times the diameter.

2. Possible applications of moving fin approximation

using the moving

essentially valid for a pur

numbers (hR/k). According

Biot numbers should be less

restricts the range of appl

materials with low thermal

steel (thermal conductivity

and for a heat transfer coe

the radius would have to be

with higher thermal conduct

is wider.

fin approximation

e material

to Rohsenow

than 1/6.

icability o

conductivit

of about 0

fficient of

less than

ivity, the

and

and

Thi

f th

y.

.075

0.0

2.5

rang

for low Biot

Choi(28), the

s seriously

is model for

For iron or

cal/cm/sec/0 C)

05 cal/cm 2/sec/ C,

cm. For materials

e of applicability

The model

11

116

The model of the moving

modified for ingots of rectangu

main deficiencies of this model

low Biot numbers and its inabil

and position of mushy zones in

no further attempts were made t

3. Influence of varioustemperature

fin can also be slightly

lar cross sections. The

are its limitations to

ity to predict the shape

alloys. For these reasons

o use this model.

parameters on thedistribution in steel ngots

The

was applied

computer

to steel

ermal

ngots

model

of ind

for i

ustri

ngots (Section VI.D)

al sizes, cast

under steady

The sa

were used for

liquidus and

and 1380 0C re

liquid metal

(Section VI.D

convection.

and water was

laboratory ES

of 0.01 cal/c

where no shri

occurs, the h

and the water

state heat flow conditions.

me characteristics as for the laboratory ingo

the larger ingots. In particular, the

solidus temperatures were taken as 1484 0C

spectively. The thermal conductivity in the

pool was taken as 0.11 cal/cm/sec/0 C

.2) except for the study on the effect of

The heat transfer coefficient between ingot

calculated from the values used for the

R ingot (Appendix X). In particular, a value

m 2/sec/ C was used at the top of the ingot

nkage of the ingot occurs. Where shrinkage

eat transfer coefficient between the ingot

depends on the distance from the top of the

ingot and on the ingot

coefficient was approxi

for the laboratory ESR

diameter. This he

mated in a similar

unit (Figure 27).

at transfer

way to that

For example, for

ts

II

117

a 50 cm diameter ingot, an average value of 0.001

cal/cm 2/sec/oC was used for the heat transfer coefficient

in the lower part of the ingot.

The temperature at the top of the ingot was taken

as 1600 0C unless otherwise specified. Joule effect was

neglected. In the laboratory unit, by neglecting Joule

effect, the temperature was lowered by a maximum of about

50C. In large ingots, the current densities are lower

than in the laboratory unit (Section VIII.A.3) and the

effect of neglected Joule effect on temperature would be

even less.

The computer treatment of the thermal model was

similar to the treatment for the small inqot (Appendix XI

a. Ingot radius and casting speed

The computer thermal model was applie

with radii of 2.5 cm, 15 cm and 25 cm. The

were varied between 0.25 and 2 cm/min. The

given in Figure 45 which shows the dimension

the metal pool as a function of ingot radius

speed. The dimensionless depth of the metal

is defined as

d to

casti

resul

less

and

pool

Z p /R1 = (Zc - zs)/R

i ngots

ng speeds

ts are

depth of

casting

(27)

Z is the depth of the 1380 0 C isotherm at the centerc

and Zs the depth of the same isotherm at the surface.

Because of the relatively high heat transfer coeffici

between ingot and water, near the top (0.01 cal/cm 2/s

line,

ent

ec/ 0 C)

).

118

the depth of the metal

all cases (less than 1

Figure 45 shows

relationship exists bet

pool, Zp /R , and ingot

and that dimensionless

proportional to casting

The following relations

pool at

cm) and

that es

ween di

radius,

depth o

speed,

hip can

the surface i

has not been

sentially a li

mensionless de

RI , at fixed

f metal pool,

VI, at fixed

be written:

s very low in

plotted.

near

pth of metal

casting speed

ZP/R 1 , is

ingot radius.

= (0.32 + 0.078 x R1 )V1

where RI and V1Equation

speeds

has me

may as

in the

along

"seal ed

also ci

large -

are given in cm and cm/min, respectively.

(, the depth

itioned tha

;ume a V-sh

last part

:he center

I off" regi

use porosi

ngots if t

Thus casti

28

of

t,

apE

to

1 in

ons

ty.

)oo

n g

parameter controlli

Further work

shows that for increasing

the metal poo

for high casti

. Accumulatio

solidify givin

e of the ingot

of liquid nea

Such a resul

high a casting

speed is an im

g solidificati

on the influen

casting

Il increases. Chalmers(43)

ng speeds, the metal pool

n of solute would occur

g marked segregation

. The shrinkage of

r the center line would

t would easily occur in

speed was used.

portant operating

on in ingots.

ce of various parameters

is made on a

of 0.25 cm/mi

confirmed by

Kroeger(22) o

in Section VI

50 cm diameter steel ingot cast at a speed

n. Various trends are indicated and are also

an extensive computerized study made by

n continuous castings of copper (discussed

II .B.6).

Zp /R1 (28)

Ii

)

Steel ingot

Thermal conductivity in liquid 0metal pool = 0.11 cal/cm/sec/ C

Temperature at top of ingot: 1600 0 C

VI = 1.0 cm/min

X calculated

V, = 2 cm/min-- extrapolated

= 0.5 cm/min

00

-4

-ii

rd

U N

rz:

0.- 4Cl)

E)

- / V = 0.25 cm/min

o 1 1 10 2.5 5 15 25

Ingot radius, cm

Figure 45: Influence of ingot radius and casting speed on the dimensionlessdepth of the metal pool.

XC

z

120

b. Temperature at the top of the ingot

The effect of a change of temperature at the top

of the ingot from 1600 0 C to 1800 0C on the 1380 0C isotherm

is shown on Figure 46. This 200 0 C temperature increase

causes the 1380 0C isotherm to move down with a change in

shape. At the surface, the depth of the isotherm changes

from 0.75 cm to 1.5 cm and at the center line from 14 cm

to 20 cm.

The temperature at the top of the ingot affects

the shape of the metal pool. However this temperature

appears to be difficult to control during ESR operation

and the possible range of temperature may be rather narrow.

Thus this temperature is not a significant operating

parameter for the control of the solidification pattern

as is the casting speed.

c. Heat transfer coefficient between ingot andwater

In the ingots considered previously, the heat

transfer coefficients between ingot and water where no

shrinkage occurred was 0.01 cal/cm 2/sec/ 0C, value found

on the laboratory ingot (Appendix X). This value has been

changed in two computer runs to 0.02 and 0.005 cal/cm 2/sec/0 C

by modification of the heat transfer coefficients through

the slag crust and into the water. The resulting effect

on the 1380 0C isotherm is shown in Figure 47.

For a heat transfer coefficient of 0.01 cal/cm 2/sec/ 0C,

solidification starts at the surface, 0.75 cm from the top

and on the center line, 14 cm below the top. For a heat

121

R, = 25 cm

VI = 0.25cm/min

1380 0 C isotherm,

1380 0 C isotherm,

Tt = 1600 0 C

Tt = 1800 C

Figure 46: Effect of temperature at the top of theingot on the 1380 0C isotherm.

cm

RI = 25 cm

V1 = 0.25cm/mi n

--- - 1 380 0 C

1380 0C

1380 0C

isotherm for hI

isotherm for hl

isotherm for hT

= 0.02 cal/cm 2/sec/ C

= 0.01 cal/cm2/sec/0C

= 0.005 cal/cm 2/sec/ C

Influence of the heatbetween the ingot andthe 1380 0 C isotherm.

transfer coeffi ci entthe water, hi, on

122

16000 C

Figure 47:

123

transfer coefficient of 0.005 cal/cm 2/sec/ C, the 13800 C

isotherm moves down. The ingot solidifies at the surface

about 2 cm from the top and at the center line at 18.5 cm.

For a heat transfer coefficient of 0.02 cal/cm, the 13800 C

isotherm moves up. The ingot solidifies at the surface

practically at the top of the ingot and at about 12 cm

from the top on the center line.

Doubling the heat transfer coefficient between ingot

and water affects the metal pool less than reducing the

same heat transfer coefficient by a factor of 2. This

result indicates that there may be a limiting value of

the heat transfer coefficient between ingot and water above

which the metal pool remains practically unchanged.

d. Effect of convection in the liquid metal pool

Convection in the liquid metal pool causes the

thermal conductivity in the metal pool to become higher

than for a stagnant liquid. The effect of an increasing

effective thermal conductivity was first studied for a small

iron ingot using the moving fin approximation. The ingot

had a diameter of 5 cm and the conditions were similar to

those given in Section VI.C.3. The casting speed was

1 cm/min, the temperature of solidification was 15360C,

and the temperature at the top was kept at 1600 0C. The

thermal conductivity in the liquid pool was varied between

0.1 and 10 cal/cm/sec/ 0C. The distance between the liquid-

solid interface and the top of the ingot (given by equation

IX.13 in Appendix IX) and the total heat coming into the

- - -- -- 44W&Wn "- --

124

ingot were calculated. These results are given in

Figure 48. This figure shows that for a constant

temperature at the top of the ingot, increasing the

thermal conductivity in the liquid pool markedly

displaces the liquid-solid interface downward and signi-

ficantly increases the heat input into the ingot. For

exampleja change in thermal conductivity of the liquid

metal from 0.1 to 1.0 cal/cm/sec/0 C, causes the solidifi-

cation front to move from 0.3 cm to 2 cm from the top of

the ingot and changes the heat into the ingot from 32 to

55 cal/cm 2/sec.

The same effect of an increased thermal conductivity

is shown in Figure 49, for a 50 cm diameter ingot cast at

a speed of 0.25 cm/min. Maintaining a temperature of

1600 0C at the top of the ingot and varying the thermal

conductivity from 0.11 to 1.1 cal/cm/sec/0 C causes the

metal pool to move down and to increase in depth. For

such conditions, the total heat into the ingot is also

increased by about 50 percent.

When the temperature at the top of the ingot is

maintained constant, the main effect of increasing convec-

tion is to introduce more heat into the ingot. The

increased heat flux displaces the metal pool downward.

An alternative method to determine the effect of

convection in the liquid metal pool is based on the

assumption of constant heat input into the ingot.

125

/ 4-4

10 - 100 4f /

+so o/

0-0

40.1 1.04o0

liqui pol0clc/sc)

0OC r .,

5J 50 4

Ingo dimtr:5c

4- 0 0cI) 4->

M -)4-3 4=

M~ 4--

L'~0

0 0

0.1 1.0 10

Effective thermal conductivity in theliquid pool, cal/cm/sec/0 C

Results of calculation with moving fin approximation(Section VI.C) .

Ingot diameter: 5 cmCasting speed: 1 cm/mmnTemperature at the top of the ingot: 1600 C

Heat tranfer coeficient between solid steel and water0.01 cal/cm.2 /sec/ C.Heat transfer coefficient between solid ingot and water0.0045 cal/cm2 /sec/OC.other data from Appendix I.

Figure 48: Position of solidification front and heat flow intothe ingot as functions, Of the thermalconductivity in the liquid metal pool.

126

= 25 cm

0

10

20

30

cm

Figure

1380 0 C isotherm,Tt = 1600 0 C

1380 0 C isotherm,Tt = 1600 0 C

1380 0C isotherm,Tt = 15100C

Effect of thermal conductivitypool. k on the 1380 0C isothe

'

= 0.25cm/mi n

= 0.11 cal/cm/sec/ C,

= 1.1 cal/cm/sec/ C,

= 1.1 cal/cm/sec/0C,

in liquid metalrm.

iV

49:

127

According to the previous discussion, the temperature

the top of the ingot would then decrease with increasi

convection.

steel ingot

heat input,

liquid from

temperature

1510 0C. Fig

slightly fla

unchanged.

An example

cast at a sp

an increase

0.11 to 1.1

at the top o

ure 49 shows

tter and its

was treated on the 50 cm diameter

eed of 0.25 cm/min. At constant

in thermal conductivity of the

cal/cm/sec/0 C decreases the

f the ingot from 1600 0C to

that the 1380 0C isotherm becomes

position remains essentially

Hence at constant heat input into the ingot, the

effect of

the solid

resistance

convection is very limited. Th

ingot constitutes the dominant

e. prac fteha eesduo

is is because

thermal

e. Importance of the heat released uponsolidification

The importance of the heat released upon solidifi-

cation is shown in Figure 50 for the 50 cm diameter steel

ingot cast at a speed of 0.25 cm/min. When the heat

released upon solidification is neglected, the 1380 0C

isotherm moves up at the center line (18 percent higher

than when heat released upon solidification is taken into

account). In this example, the heat released upon

solidification represents about 20 percent of the heat

coming by conduction from the slag into the ingot.

The heat released upon solidification is important

in determining the shape of the metal pool.

128

1600 0 C -- R = 25 cm

0

10

20

V= 0 .25cm cm/min

1380 0 C isotherm when heat releasedupon solidification is taken intoaccount

1380 0 C isotherm when heat releasedupon solidification is neglected

Figure 50: Effect of the heat released upon solidificationon the 1380 0 C isotherm.

129

130~ -1 -2

4. Temperature distribution in ESR ingots

Temperature distributions are given in Figures 51

and 52 for two typical 50 cm diameter steel ingots cast

at speeds of 0.25 and 0.5 cm/min. The physical charac-

teristics are the same as in the previous section. In

particular the temperature at the top of the ingot is

1600 0C, the heat transfer coefficient between the ingot

and the water is 0.01 cal/cm 2/sec/0C where no shrinkage

occurs. Due to this high heat transfer coefficient,

efficient cooling occurs at the surface where there is

no shrinkage. At the center line, the ingot cools

gradually due to the low thermal conductivity of steel.

This causes a distortion of the isotherms at temperatures

near the solidus temperature. This distortion of the

isotherm is much more pronounced at the casting speed of

0.5 cm/min than at 0.25 cm/min. This is due to the

higher melting velocity and to the shrinkage which occurs

at a greater distance from the top of the ingot.

With increasing casting speed, the depth of the

isotherms at elevated temperatures increases (Figures 51

and 52), causing the importance of radial heat flow to

increase compared to axial heat flow.

At low temperatures, the isotherms would become

more flat. For the ingot cast at 0.25 cm/min (Figure 51)

this would occur at temperatures below 6000 C.

5. Steady state heat flow conditions in ESR ingots

All the results in the present study are only valid

131

mushy zone liquidus temperature:solidus temperature:

0

10

20

30

40

50

60

70

cm

Casting speed: 0.25 cm/minIsotherms above 600 0 C

Temperature distributionsteel ingot.

a 50 cm diameter ESR

200 0 C

000 C

800 0C

6000 C

14840C13800 C

Figure 51:

132

KZ2 mushy zone

Temperature distributionESR steel ingot.

1200 0 C1 000 C

- 8000 C

liquidus temperature: 1484 0Csolidus temperature: 1380 0C

Casting speed: 0.5 cm/minIsotherms above 800 0 C

a 50 cm diameterFigure 52:

133

for steady state heat flow conditions. In the laboratory

ingots cast at a speed of 1.1 cm/min, the temperature was

found to be steady above 570 0C after the height of the

ingot reached about 1.8 times its diameter (Figure 24).

For industrial ESR ingots, steady state heat flow

conditions were studied for the two 50 cm diameter steel

ingots of the previous section. The temperature was

calculated for the ingot cast at a speed of 0.25 cm/min

with three different ingot lengths: 75 cm, 100 cm, and

150 cm. Insulation was imposed at the bottom of the ingot

and quasi-steady state heat flow conditions were assumed.

The calculated center line temperature, plotted in Figure

53, shows that, once the height of the ingot reaches about

75 cm, temperatures above 1100 0C are no longer affected.

When the height reaches 100 cm, temperatures above 6000C

become steady. Hence the shape of the metal pool becomes

steady before the ingot is about 1.5 diameter high.

A similar calculation on the 50 cm diameter steel

ingot cast at a speed of 0.5 cm/min showed that temperatures

above 900 0C were no longer affected once the height of the

ingot reached about 75 cm. This result indicates that

the shape of the metal pool becomes steady for a shorter

ingot length at a casting speed of 0.5 cm/min than at

0.25 cm/min. This is due to the effect of the higher

radial heat flow at 0.5 cm/min than at 0.25 cm/min, as

discussed in the previous section.

calculated temperaturefrom computer thermal model (Sec. VI.D)

calculated center line temperature for100 cm high ingotcalculated center line temperature for150 cm high ingotcalculated center line temperature for

ingot

0 1 1 1 I I I I 10 20

Figure 53: Effect of ingot heighdiameter steel ingot.

100Distance from top of ingot, cm

t on the center line temperature of a 50 cm

1500

0

4--)

caE(U

1000

500

135

The minimum ingot length for the shape of the metal

pool to become steady, can also be obtained qualitatively by

considering the shape of the steady state heat flow

isotherms. This is because heat flows perpendicularly to

the isotherms. For the two 50 cm diameter ingots

considered in this section, Figures 51 and 52 indicate

qualitatively that the heat from the metal pool is indeed

mainly extracted through the surface of the ingot over a

height of about 1.5 diameter.

6. Discussion of the work of previous investigators

In this section previous heat transfer studies are

discussed for ingots produced by the ESR process, the

continuous casting machine and the vacuum arc remelting

process. Some of the results are used to extend the

results of this present investigation.

On ESR ingots, the only detailed study is that of

Sun and Pridgeon(12). The temperature and the shape of

the metal pool were determined experimentally for 5.5

in.diameter Hastelloy X ingots. These results served

to validate a computer program which uses a finite

difference technique. Unlike EPS, this program requires

a rectangular grid and the heat released in the mushy zone

was approximated with an increased specific heat. This

last approximation could not permit an accurate

determination of the pool shape for pure metal castings.

Sun and Pridgeon showed that an increased melting rate

increased the effective thermal conductivity in the metal

136

pool by i

the casti

current,

expressed

m(ft/hr)

ncreased stirring. They also related empirically

ng rate, m, to the electrode diameter, RE, the

I, and the amount of slag, SG* The relation is

as:

= 0.6 RE(in,) + 1.57

- 0.18 SG(pounds)

10-3 I(amps)

1 .67 (29)

cation of these results to industrial ESR

nvestigated.

re extensive work has been carried out on

produced in the

process, the hea

in the ESR proce

cooling normally

In t

than

the

mi n

cont

meta

cont

flow

cont

heat

Irvi

Pehl

his last pro

in the ESR

casting spee

in an ESR un

inuous casti

1 pool may b

inuous casti

may become

inuous casti

flow were d

ng(16) using

ke(18) and M

continuous cast

t transfer cond

ss

us

ces

pro

d w

it

ng

e e

ng

pre

ng

eri

an

i zi

ing machine.

i tions

In this

are similar to those

except for a secondary direct water

ed on the continuous casting machine.

s, casting speeds are usually higher

cess. For a 30 cm diameter steel ingot,

ould be typically between 0.5 and 1.2 cm/

and between 10 and 50 cm/min in the

machine(10). Thus the depth of the

xpected to be much greater in the

machine than in ESR, and radial heat

dominant (Section VIII.B.4). For the

machine thermal models neglecting axial

ved by Savage(13), Hills(14,15) and

integral profile technique and by

kar(19) using numerical solutions.

an analytical expression

The

was

appl i

not i

Mo

units

i ngots

Boi chen ko (44) de rivedS im ilIa rly

137

relating the depth of the meta

The liquid metal was assumed t

heat (temperature T m), and the

solidification was assumed to

the height of the metal pool.

was expressed as:

Sp/R

1 pool to various parameters.

o be poured without super-

heat released upon

be extracted radially over

The depth of the metal pool

S pV L'R I4k (T - T )7s m s u (30)

where L'T - T

=L + Cs y m 2su

Tsu is an average temperature at the surface of the ingot.

Equation (30) shows that the dimensionless depth of the

metal pool is approximately proportional to the casting

speed and to the ingot radius. On ESR ingots, the dimen-

sionless depth of the metal pool was found to be proportional

to the casting speed and to vary linearly with the radius

(equation 28).

The above models neglect axial heat flow and are

not really applicable to the ESR ingots in which rather

shallow metal pools tend to be formed (Section VIII.A.2).

Under such conditions, axial heat flow cannot be neglected.

Recently Kroeger(22) made an extensive computer

study on two-dimensional heat flow conditions in continuous

casting machine. The material properties of pure copper

were used and the computer program approximated the heat

released upon solidification with an increased specific heat.

(31)

138

The results were given in terms of dimensionless numbers.

The dimensionless depth of the metal pool was found to be

essentially proportional to the casting speed at all Biot

numbers. This shows that the similar result obtained on

the ESR process for one heat transfer coefficient (Section

VIII.B.3.a) may be expected to hold for any heat transfer

coefficient. The effect of ingot size on the shape of the

metal pool was not studied. Kroeger also found that the

depth of the metal pool goes to an asymptotic value for

increasing heat transfer coefficient near the top of the

ingot. This

(Section VIII

ingot was als

shape of the

in the contin

in the liquid

at fixed heat

sol i difi catio

on the ESR un

confirm the s

ESR process a

result appea

.B.3.c). Th

o found to h

metal pool (

uous casting

metal pool

input into

n front was

it (Section

imilarity of

rs to

e temp

ave a

Sectio

machi

was mu

the in

found

VIII .B

heat

the continuous

hold

erat

1 imi

n VI

ne,

1 tip

got.

to b

.3.d

flow

also on ESR units

ure at the top of the

ted effect on the

II.B.3.b). Finally,

the thermal conductiv

lied by a factor of 1

The effect on the

e almost negligible a

). These results

conditions between t

casting machine. These

results also appear to confirm that the validity of the

results which were obtained in the present study (Section

VIII.B.3), is not restricted to the particular examples

which were selected.

A recent study was made by Eisen and Campagna(24)

on vacuum arc remelted slabs. A computer program was

developed for unsteady state heat flow conditions. Two

it

00

s

he

I'

139

main approximations were made: a constant heat transfer

coefficient between ingot and water was assumed, and the

heat of fusion was approximated using an artificially

increased temperature at the top of the ingot. This

program could give valuable information on the unsteady

state temperature but could not be used for an accurate

determination of the shape of the metal pool.

7. Conclusion

The application to industrial scale of two thermal

models derived for the ingot of the laboratory ESR unit has

been investigated. The model using the moving fin approxi-

mation (Section VI.C) does not have a wide range of

applicability. This model is limited to pure metals

solidifying with almost flat liquid-solid interfaces. The

Biot numbers have to be low (less than about 1/6).

The computer thermal model (Section VI.D) was used

to investigate heat flow conditions in industrial ESR steel

ingots.

The following results were obtained:

1. The dimensionless depth of the metal pool was

found to be proportional to the casting speed and to vary

linearly with ingot radius (equation 28). Casting speed

is the main operating parameter controlling the shape of

the metal pool.

2. The temperature at the top of the ingot, which

is difficult to control, is found to have a limited effect

on the metal pool.

140

3. The importance of the heat transfer coefficient

on solidification front decreases with increasing values

of the coefficient and an asymptotic shape of metal pool

may be reached.

4. Increasing effective thermal conductivity in the

liquid metal pool displaces the solidification front down-

ward at constant temperature at the top of the ingot,

but has a negligible effect on the solidification front

for constant heat input into the ingot.

5. The heat released upon solidification is

important to the determination of the shape of the metal

pool

These results are in agreement with those obt

by Kroeger(22) on the continuous casting machine exc

for the effect of ingot radius which Kroeger did not

apparently consider. This agreement appears to indi

that the results of the present study are not restri

to the particular examples which were selected.

In industrial practice, the dimensionless dep

the metal pool, Z /Ri, may be expected to be related

ingot radius and casting speed in the following way:

Z /R = (a + bR1 )V1

ai ned

:ept

cate

cted

th of

to

(32)

At fixed heat transfer properties through the slag crust

around the ingot and into the water, and at fixed tempera-

ture at the top of the ingot, Z /RI is proportional to

casting speed. The coefficients a and b of equation (32)

II

141

could be determined with a minimum of two experiments with

different ingot radii and the linear dependence of Z /RIwith ingot radius would be found. Since the heat transfer

properties and the temperature at the top of the ingot are

difficult to control, departure from equation (32) may be

noticed but at least estimates of the dimensionless depths

of the metal pool may be obtained.

C. Slag

This section will be

discussion of those factors

influencing heat generation

in an industrial unit. Such

necessary because no data is

in the slag for industrial E

information obtained on the

present laboratory study,is

polation to larger units.

The results obtained

(Section VII) show that the

process occurs almost entire

in the slag is non-uniform a

near the electrode tip. The

limited to a qualitative

which are most important in

and heat transfer in the slag

a limited discussion is

available on heat generation

SR systems. In addition, the

slag as a heat source from the

not sufficient to permit extra-

on

hea

ly

nd

the laboratory

t dissipated in

in the slag. H

intense heat is

temperature

ESR unit

the ESR

eat generation

generated

in the slag was

found to be almost uniform except at boundary layers

between the slag and the surrounding media.

In an industrial ESR unit, heat is also generated

mainly in the slag. According to Salt(41), the resistivity

1]

142

of calcium fluoride based slag varies between 0.5 and 0.2

Q cm for temperatures between 1400 0C and 2000 0C. The

resistivity of steel varies from about 10-5 to 1.2 x 10-4

Q cm for temperatures between 00C and 1500 0C. Thus the

slag constitutes the main electrical resistance in the

system electrode-slag-ingot.

Because of the complex phenomena which occur in the

slag (Section VII.B.1), heat is generated non-uniformly in

the slag. Intense heat generation may occur near the

electrode tip where current density is highest. The

voltage drop at the interface electrode-slag has never been

measured in industrial units. The voltage drop measure-

ments on two experiments in the laboratory unit are not

sufficient to permit an extension of the results from the

small unit to large units. Thus the relative amount of

heat generated at the electrode tip remains unknown.

Heat is extracted from the slag at the electrode

tip, top of ingot, water-cooled mold and slag surface at

the annular space between electrode and mold.

Due to the non-uniform heat generation and

extraction, the degree of uniformity of the temperature

distribution in the slag depends mainly on the thermal

conductivity of the slag. For increasing thermal conduc-

tivity of the slag, the temperature would be increasingly

more uniform.

I]

143

One of the main factors which increases thermal

conductivity in the slag is mixing. Recently a study of

this mixing was made by Campbell(4) on an ESR laboratory

unit using transparent crucibles (maximum diameter of the

crucibles: 7.6 cm). Low melting point metals such as

Pb, Zn, Al and Cu were selected. The slag was the LiCl-

KCl eutectic. In all the experiments, the slag had a

torroidal movement downward away from the electrode tip

and upward at the mold wall. Considering essentially the

electromagnetic effect of the current on the slag,

Campbell showed that the motion should increase with

increased current and with decreased viscosity and electrode

size. The effect of the electrode size was shown in an

experiment where electrode diameter was slightly less than

crucible diameter. In such a case the velocity of the slag

became imperceptible.

In the same experiments, the bulk of the slag had a

reasonably uniform temperature. Two cool boundary layers

were found at the electrode tip and at the top of the ingot.

There was no cool boundary layer at the pyrex or silica

crucible wall because of the absence of water-cooling.

The cool boundary layer at the electrode tip was attributed

to the heat extraction by the electrode which compensated

the intense heat generation. In similar experiments made

with a non-consumable electrode, the slag was found to

become extremely hot near the electrode tip because of the

absence of a significant heat sink.

11

144

In view of Campbell's results, the nearly uniform

temperature found in the small ESR unit (Section VII) may

be attributed to the low viscosity of the slag (about 3

centipoises at 1700 0C(41)) and to the relatively small

size of the electrode compared to the size of the ingot

(ratio of about 1/4 for the cross sections).

Non-uniform temperatures may be observed in the

slag for specific conditions such as a slag of high

viscosity (either because of its structure or a tempera-

ture effect) or alternatively for a large electrode in a

mold.

Conclusion

Heat generation in the ESR process is essentially

in the slag. The heat is generated in the slag non-

uniformly. The degree of uniformity of the temperature in

the slag depends on convective and electromagnetic stirring.

From this problem outlined in this section and in the

section on the slag of the laboratory unit (Section VII),

much more detailed experimental work is required for a more

definitive conclusion about the role of the slag in the ESR

process.

The experimental work could be broadly divided into

two categories. In one, the chemical and electrochemical

reactions could be studied. The interfacial voltage drop

especially at the electrode tip could be measured for

various current densities and slag compositions. This

would give a better understanding of the phenomena occurring

at the electrode tip and of the heat generation. Another

||

145

investigation, could be made on parameters which may affect

convective and electromagnetic stirring (current density,

slag viscosity, immersion depth of electrode, system

geometry). The influence of these parameters on the effec-

tive thermal conductivity of the slag could also be studied.

146

IX. SUMMARY AND CONCLUSIONS

An experimental heat transfer investigation has

been made on the electrode, ingot and slag of a laboratory

steel ESR unit using direct current and various operating

conditions. The various modes of heat transfer were

analyzed and thermal models were derived. Conclusions

are given separately for heat flow characteristics of

the electrode, ingot and slag of the laboratory unit.

An attempt was made to predict temperature distribution

in industrial units.

1. Electrode - Laboratory Unit

Thermocouple measurements along the center line

of the electrode i ndi ca

in the upper electrode

the tip. T

to Joule ef

95 0 C. Heat

above the s

The heat in

the electro

the slag de

essentially

For

he g

fect

flo

lag

to t

de.

term

one

the

radual t

with ca

w condit

level we

he elect

The imm

ined whe

-di mens i

one-dim

ted a gradu

and a sharp

emperature

lculated ma

ions on the

re found to

rode is ent

ersion dept

ther heat f

onal and ax

ensional he

al temperature increase

temperature rise near

increase was attributed

ximum temperatures of

surface of the electrode

be almost adiabatic.

irely used to heat up

h of the electrode in

low in the electrode is

ial or two-dimensional.

at flow conditions,

an analytical model was derived and the temperature was

calculated with the following equation:

147

T E T- exp Vk (Z < 0) (32)Tme E s

Two-dimensional heat flow was treated with a computer

thermal model. Satisfactory agreement was found between

the calculated and experimental temperature profiles along

the center line of the electrode.

2. Ingot - Laboratory Unit

Thermocouple measurements and sulfur prints were

used to determine the temperature distribution in the

ingot of the laboratory unit. An analytical thermal model

was not developed further because of its failure to predict

the shape of the metal pool. A computer thermal model was

derived which consisted of the solution of the heat conduc-

tion equation applicable to the ingot for the appropriate

boundary conditions. This model predicted the shape and

position of the mushy zone and also gave satisfactory

agreement between the calculated and experimental tempera-

ture distributions.

3. Slag - Laboratory Unit

The slag of the laboratory unit was at a nearly

uniform temperature because of convective and electro-

magnetic stirring except for boundary layers at interfaces

between the slag and the surrounding media.

The power was found to be the main independent

variable controlling melting rate. For positive electrode

mode, the melting velocity (within 10 percent) was related

to power input in the following way:

148

VE (cm/min,+ 10 percent) = 0.26 x P (KW) (33)

The slag constitutes the main electrical resistance

and the main heat source in the ESR process. Heat is

generated in the slag non-uniformly and in particular

intense heat may be generated at the electrode-slag inter-

face where high interfacial voltage drop occurs.

The effect of electrode polarity was not

exhaustively examined. A change of polarity from positive

to negative electrode mode appears to lower the melting

rate.

4. Application of the Results toESR Units

Emphasis has been placed on some i

units producing steel ingots.

Typical 37.5 cm diameter steel ele

in 50 cm diameter

electrode 0.25).

temperature almost

almost adiabatic h

surface, as on the

of the electrode i

temperature distri

velocities increas

of the electrode.

along the center 1

dimensional heat f

immersion depth of

tips cause a posit

steel

The e

ingot

lectrode

was selected

Industrial

ndustrial ESR

ctrode melting

(emissivity of

was found to have a

unaffected by Joule effect and to have

eat flow conditions on the cylindrical

laboratory unit. The melting velocity

s the main factor in determining

bution in electrode. Increasing melting

es the temperature gradient near the top

An estimate of the temperature profile

ine can be easily obtained by the one-

low model (equation 32). Increasing

the electrode and non-flat electrode

ive departure of the temperature in the

__ Aogaphii - - " _ -

electrode from

dimensional he

For th

applied to the

influencing th

(i) Di

the temperature calculated by the one-

at flow model.

e ingot, the computer thermal model was

investigation of the main parameters

e shape of the metal pool.

mensionless depth of metal pool was fou

to casting speed and

with ingot radius. Casting speed is an

operating parameter controlling ingot s

(ii) For the possible range of

top of the ingot, the depth of the meta

significantly affected by this temperat

(iii) The influence of the heat

between the ingot and the water on the

decreases with increasing values of the

asymptotic pool shape may be reached.

to vary linearly

essential

olidification.

temperature at the

1 pool is not

ure.

transfer coeffi ci

solidification fro

ent

nt

coefficient and an

(iv)

in the liqui

downward at

(v)

conductivity

d

c

A

Increasing

metal pool

onstant tem

t constant

in the liqu

effec

disp

perat

heat

id me

ti ve

lace

ure

i npu

tal

thermal cond

s the solidif

at the top of

t, increasing

pool decrease

uctivity

'ication front

the ingot.

thermal

s the

temperature at the top of the ingot and makes the metal

slightly flatter but does not affect its position.

In the ESR process, the slag constitutes the

electrical resistance and the main heat source. Due

lack of data, it can only be concluded qualitatively

heat is generated non-uniformly in the slag and that

149

to be proportional

nd

pool

ma i n

to a

that

the

150

degree of uniformity of the temperature depends on

convective and electromagnetic stirring.

I

151

X. SUGGESTIONS FOR FURTHER WORK

The present study did not emphasize the explanation

of the complex phenomena which occur in the ESR slags.

As mentioned in the section on the industrial slags

(Section VIII.C), experimental work could be directed

toward a better understanding of the chemical and electro-

chemical reactions which occur in the slag. This would

involve experiments with various slag compositions and

electrical conditions. This would give a better under-

standing of the phenomena occurring at the electrode tip

and of the heat generation.

Other investigations could be directed toward

determining th

electromagneti

studied are th

slag, the elec

the electrode

Such st

conditions of

In the

the thermal mo

commercially a

on industrial

e

c

e

parameters

stirring.

composition,

that affect convecti

The main parameters

ve and

to be

viscosity and density of the

trical conditions, the immersion dep

and the system geometry.

udies would help explain the melting

the electrode in the slag.

future, the EPS computer program use

dels derived in this study, will bec

vailable. Extensive experimental st

ESR units would permit the evaluatio

th of

d in

ome

udies

n of

the data necessary for the use of the thermal models (for

example, slag temperature, heat transfer coefficients).

These thermal models may then help improve the results

II

152

obtained with ESR units by optimization of the operating

parameters.

153

XI. BIBLIOGRAPHY

1. A. Mitchell, S. JoshiTemperature GradientsUniversity of

and J. Cameron, Electrodein the Electroslag Process,

British Columbia, 1969.

2. M. M. Klyuev

3. D. A.Int.

and Yu. M. Mironov, Stal,

Whittaker, R. G. Ward and G. R. PuSymposium on ESR, Mellon Institute,

4. J. Campbell, Journal of Metals, 22,

6, 480 (1967).

rdy, 2nd3 (1969).

23 (1970).

Panin, 0. B. Borovskii, I.Iodkovskii, Avt. Svarka, 2,

S. Ivakhenko and72 (1964).

6. A. Mitchell, 2nd Int.Institute, 1 (1969).

Symposium on ESR, Mellon

7. R. J. Roberts, 2nd Int. SymposiumInstitute, 2 (1969).

on ESR, Mellon

8. W. HolStahl

zbruber and Eund Eisen, 12

Plockinger638 (1968).

of Kapfenberg,

9. W. Holzgruber, K. Petersen and P.Trans. Int. Vacuum Metallurgy Con

E. Schneference,

i der,499 (1968).

10. W. Holzgruber,Int. Symposium

11. R. P. Devries, 2nd IInstitute, 3 (1969).

12. R. C. Sun and JESR, Mellon Ins

SchneidoferESR, Mellon

nt.

. W. Priti tute,

Symposium

dgeon, 2nd3 (1969).

and M. Kroneis, 2ndInstitute, 1 (1969).

on ESR, Mellon

Int. Symposium

13. J. Savage, J. Iron and Steel Inst., 200,

14. A. W. D.(1965).

41 (1962).

Hills, J. Iron and Steel Inst., 203,

15. A. W. D. Hills, Trans. A.I.M.E., 245, 1471 (1969).

16. W. R. Irving, J. Iron and Steel Inst., 205,

17. K. Cliff and278 (1967).

271 (1967).

R. J. Dain, J. Iron and Steel Inst., 205,

18. R. D. Pehlke, Metals Eng. Quart., 4, 43 (1964).

19. E. A. Mizikar, Trans. A.I.M.E., 239, 1747 (1967

5. V.S.

).

154

20. D. L. Schroeder and D. L. Lippitt,1968 A.I.M.E. Annual Meeting, New

paper given at theYork City.

21. D. J-P. Adenis, K. H. Coatsof Metals, 91, 395 (1963).

and D. V. Ragone, J. Inst.

22. P. G. Kroeger, paper to be published at theHeat Transfer Conference, Paris, 1970.

23. F. W. Wood, U. S.Mines, Investigati

24. W.1,

B. Eisen and849 (1970).

Depart. of Interior,n 7151, 1968.

Campagna, Metallurgical

Bureau of

Trans.,

25. C. C. Tillman,presented, 1970

26. C. C. Tillman,MAC-TR-62,

Jr., Ph.D. Thesis, M.I.T., to be.

Jr., EPS User's Guide, M.I.T.,ESL-R-395 (1969).

27. A.S.M.E.(1954).

Handbook, Metals Properties, McGraw Hill

28. W. M. Rohsenow and H. Y. CMomentum Transfer, Prentice

29. J. F. Elliott, M. Gleiser, J.Thermochemistry for Steelmaki

hoi, Heat, Mass andHall (1961).

Ramakrishna,ng, A.I.S.I. (1963).

30. W. H. McAdams, Heat Transmission, McGraw Hill (1954).

31. A.S.M. Metals Handbook, 19

32. R. Schumann, MetallurgicalWesley (1952).

33. M. M. El-Wakil,Hill (1962).

48 Edition.

Engineering,

Nuclear Power Engineering,

34. R. Hultgren, R. Orr and K. Kelley, Supplement toSelected Values of Thermodynamic Properties ofMetals and Alloys, University of California (1965).

35. K. Kelley,Metallurgy,

36. J. Yavorsky,1970.

Contribution to the DataBureau of Mines, Number

Ph.D. Thesis, M.I.T.,

of Theoretical584 (1960).

to be presented,

37. E. R. Eckert and R.McGraw Hill (1959).

M. Drake, Heat and Mass Transfer,

38. R. B. Bird, W. E. StewartTransport Phenomena, John

and E. N.Wiley and

Lightfoot,Sons (1960).

El II

4th Int.

Addison

McGraw

39. W. Holzgruber, P. MachMetallurgy Conference,

ner,Pitt

and E. Ploeckinger,sburgh (June 1969).

40. C. Wagner, J. Electrochemical Soci ety, 115, 933 (1968).

41. D. J. Salt,Institute,

42. G. K. Bhat,Institute,

1st Int.1, (1967).

1st Int.1, (1967).

Symposium on

Symposium on

ESR, Mellon

ESR, Mellon

43. B. Chalmers, Principlesand Sons (1964).

of Solidification, John Wiley

44. M. C. Boichenko, ConButterworths (1961).

45. H. Silderberg and J.32, 129 (1953).

ti nuous Casting of Steel, London

J. McKetta, Petroleum

155

Vacuum

Refi ner,

156

APPENDIX I

Physical Data for Steel

Some physical data used in this study are given

below for steel as a solid, as a liquid or in a state

partially liquid and partially solid.

A. Thermal Conductivity

The thermal conductivity of 0.23 and 0.8 percent

carbon steels in the solid state is plotted versus

temperature in Figure I.1. An average value of 0.075

cal/cm/sec/0 C is used in the calculations made on the

electrode and on the ingot. This average value proves

to be satisfactory for temperatures above 500 0C.

The thermal conductivity of liquid stagnant steel would

be between 0.025 to 0.05 cal/cm/sec/ 0C(32,33). Because of

the effect of convection, the actual thermal conductivity in

the liquid metal pool at the top of the ingot would be higher

than the above values. Indirect methods are used to

determine this actual value (Section VI.D).

For steel in the mushy zone, the thermal conductivity

is assumed to be 0.075 cal/cm/sec/0 C.

B. Specific Heat

The specific heat of steel is assumed to be the same

as the specific heat of pure iron as shown in Figure 1.2.

In this study, constant values of 0.16 and 0.18 cal/g are

taken for the specific heat of solid and liquid steel

157

0.23

0.8

percent carbon

percent carbon

500 1000

Temperature , 0C

Figure I.1: Thermal conductivity of solid 0.23 andpercent carbon steels in solid state(reference 31).

0.8

0.10

0.05

0

C-,a,(A

EC.,

(0U

C-,

~0

0C-)

(0

E

a,

I-

0.2

0

--

0.18

*-- 0.15

0.10 01

I0

500 1000

1500

Te

mp

era

ture

, 0

C

oF

igure

I . 2:

Sp e c if ich

eat

of

i ron

( R

ef ere nce 2 0)

.

159

respectively. For a steel in the mushy zone, a value of

0.17 cal/g

C. Electrical Resistivity

The electrical resistivity of a 0.23 percent carbon

steel is plotted versus temperature in Figure 1.3.

the calculations, the following linear approximation

used:

resistivity = (20 + 0.08 x T) x 10-6

T is in 0C. This approximation is assumed valid for the

three states.

D. Density

A constant value of 7.86 g/cm3 for the density

used in all the calculations.

E. Heat of Fusion

The latest value of 59 cal/g for the heat of fusion

is assumed to be valid for steel.

is used.

Q cm (I.1)

of pure iron(34)

1000

Temperature, UC

Figure 1.3: Electrical resistivity of 0.23 percent carbon steel(Reference 31).

100x

U)

4-)rH

50

500 1500

____________ ~ i~~§_-~

161

APPENDIX II

Determination of the Melting Speed of the Electrode

The melting speed of the electrode was calculated

from the driving speed of the electrode. This driving

speed was determined using the experimental arrangement

shown in Figure II.1. A scale with pinch clamps spaced

every 5 cm was placed parallel to the driving screw of

the clamp. A microswitch attached to this clamp moved

along the scale striking the pinch clamps and short-

circuiting one of the thermocouples measuring a temperature

rise. The thermocouple was either inserted in the electrode

(Figure 4) or in the ingot (Figure 14). The average

driving speed of the electrode was calculated from the

measured distances between the short-circuits as shown on

the typical recorder chart of Figure 11.2. The melting

velocity of the electrode is expressed as:

V = driving speed x S ESI

-microswitch

microswitch(OFF ON)

pinch cla

scale

162

clamp

driving speed

screw

Figure II.1: Schematic drawing of the experimentalarrangement for the determination of thedriving speed of the electrode.

shortcircuiting of the thermocoupleby the microswitch

-4

Time

Figure 11.2: Chart giving a temperature rise and thedriving speed of the electrode.

163

APPENDIX III

Differential Thermal Analyses on Steel

Two differential thermal analyses were conducted on

the steel of the electrode and of the upper part of one

ingot. They are described below and the results are

reported.

A. Electrode Steel

The AISI 1020 steel used for the electrodes has a

nominal composition(27) in weight per cent of 0.18-0.23 C,

0.3-0.6 Mn, 0.04 P maximum and 0.05 S maximum. However,

a chemical analysis on carbon gave 0.25 percent. The

melting point was determined using the following differ-

ential thermal analysis.

The experimental arrangement is shown in Figure

III.1. Three alumina crucibles were placed in a graphite

crucible, two of which contained samples of the steel rod

(25 g) while the third contained 30 g of OFHC copper. In

each, a Pt-6 percent Rh/Pt-30 percent Rh thermocouple,

encased in an alumina protection sheath, was inserted.

One thermocouple in steel and the thermocouple in copper

were in opposition to one another and were connected to

the Honeywell recording potentiometer described in Section

IV. This could detect a variation in temperature between

the two samples. The third thermocouple, connected to a

Aluminacruci bl es

ID 1.4 cmOD 1.8 cmheight

2.5 cm

Graphitesusceptor

ID 4 cm0D 6 cmheight

12 cm

Thermocouples:

graphitesuscepte

steelsample

copper -

sample

layerof alumina

(1 cm th ick)

Pt-6%Rh/Pt-30%Rh(0.011 in.diameterwires)

Heating unit

Recorder 1 Recorder 2

L Y-j TC3steel

differentialanalysi

thermals

temperature measurementinside steel sample

Figure III.1: Differential thermal analysis on electrodesteel.

164

TC

copper steel

165

a second recording

temperature in the

The samples

an induction unit u

susceptor.

Po

111.2. On

began simu

(point A)

curve show

down to 14

(point F).

To cool

ts of the

heating,

ltaneousl

and ended

s that on

460C (point

potentiometer, measured the

second steel sample.

were heated at a power of 2

sing the graphite crucible

the samplesthe power was

recorded curves are shown

a transformation, assumed

y for both steel samples at

at about 1525 0 C (point B).

absolute

.5 kw with

as a

turned off.

in Figure

to be melting,

about 1460 0 C

The cooling

one steel sample undercooling occurred

E) with recoalescence up to

On the other steel sample, undercooling

1507 0C

wi th

recoalescence probably occurr

the first steel sample (point

to have occurred near the eut

Solidification was completed

the two samples.

The iron-carbon phase

percent carbon steel a liquid

which is found at point B of

temperature of 1494 0C. This

point A. The temperature of

ture of the solid end of the

,ed about 15 seconds before

D). No transformation seems

ectic temperature (1153 0 C).

at about 1472 0 C (point G) for

diagram(29) gives for a 0.25

us temperature of 1525 0C,

the heating curve and a solidus

is 340C higher than found at

1460 0C is used as the tempera-

electrode.

Ingot Steel

A differential thermal analysis was made on the 0.65

percent carbon steel used in the ingots of the experimental

Temperature in steel sample

(D)

(F)

(B)1507 0C

- 1472 0 C

1700

1600

1500

1400

1300

1200

1100

1

Time, min

Differential thermal analysis

5 10

(G)steel hotter

15 2 0 25

(D) Time,I(C) mm

(B)copper hotter

Figure 111.2: Differential thermal analysis on the 1020 electrode steel.

1525 C

14600C

(A)

(E)

heating cooling

2.5 kw

0.5

0

-0.5

-l --. i

167

study (Section VI.I). The experimental arrangement was

almost identical to the arrangement for the electrode steel

(Figure III.1). Only two samples were used, a lOg OFHC

copper sample (1 cm diameter) and a 35 g steel sample

(2 cm diameter). Two thermocouples were inserted in the

steel sample.

The recorded curves are shown in Figure 111.3. On

heating, a transformation assumed to be melting occurred

at 1386 0 C (point A). At 1484 0C (point B) a sharp change

of slope indicated that most of the material was liquid

and another change of slope at 1493 0C (point C) showed that

the material finished melting at that temperature (peri-

tectic temperature). After the power was turned off,

undercooling occurred down to 1471 0C (point D) with

recoaslescence to 1476 0C. A change of slope at 13800C

(point E) was attributed to the end of solidification.

The temperature profiles recorded during ESR

solidification (Figure 16 and Figure 17) did not indicate

undercooling. The temperature of 14840C given by point B

of Figure 111.2 was taken as the liquidus temperature and

the temperature of 1380 0C (point E) as the solidus

temperature.

0 5 10 15 20 25Time, min

(A)

I I I I a I 1 ' 1 1 1

5 10 15

Time, mi

I II I I I I I I -U Tv -- 1 -- T

20 25

n

thermal analysis on ingot steel.

1600

1500

o 1400

(U

a,

Ea)

1300

1200

1100

1000

1.0

0.5

0

0.5

-1 .0

copperhotter

t0steelhotter

Figure III.3: Differential

169

Appendix IV

Temperature Rise Due to Heat Generation

in the Upper Part of the Electrode

In this appendix, detailed

on the steady state temperature

rise in the upper part of the el

is consider

generation

ed to

per un

be an infinite

it volume, WE (

calculations are made

and on the temperature

ectrode. The electrode

cylinder with a heat

cal/cm 3/sec).

A. Steady State Temperature

The steady

case is given by

state heat

the followi

d ( ksr ) + r WE 0di- s dr E

The boundary conditions are:

(1) r = 0, dT 0dr 0

(2) r = R ,-k U|r*R

nduction equation for this

differential equation

(IV.1 )

(IV.2)

(IV.3)= h(T - T )

The first condition expresses the radial symmetry

and the second, the heat flux with the surroundings at a

temperature T .The solution to equation (IV.1) is given by:

T )-WE (R 2 - r2 W ER E+T4k s E r)+ +T (IV.4)

170

The difference of temperature between the center line

and the surface is:

ERE2ATc 4ks (IV.5)cs 4%

The values of the thermal conductivity and of the electrical

resistivity are taken as 0.12 cal/cm/sec/ C and 15 x 10-6 0

cm respectively (values at about 100 0C, as shown in

Appendix I). ATcs is calculated with equation (IV.5) for

the four experiments. The results, reported in Table 2,

show that this difference was less than 0.3 0 C. The tempera-

ture profile in a cross section was nearly flat.

The calculation of the absolute steady state

temperature requires the determination of the effective

heat transfer coefficient h. This is done below.

B. Temperature Rise in the Electrode

The temperature in the electrode is assumed to depend

only on time, t. The unsteady state heat conduction equation

is then given by:

pGRdTPsE dr = REWE - 2h(T - T ) (IV.6)

The initial condition is

T = T9 at t = 0 (1V.7)

Assuming that the various parameters of equation

(IV.6) are constant, the solution can be written as:

T E E _ exp(- E t)] + T

This last expression s

increases exponentially towa

with a time constant t given

hows that the

rd the steady

by:

temperature

state temperature

pC sRET = 2h (IV.9)

The time it took for the electrodes in the four experiments

to reach 500C is given in Table 2. From these values and

using equation (IV.8), the heat transfer coefficients, h,

are calculated and given in Table 2. The calculated steady

state temperatures and the calculated temperatures of the

upper part of the electrodes when the thermocouples

reached the tips are also given.

171

(IV.8)

.:

172

Appendix V

Heat Balance on the Lower Part

of the Electrode

The various terms involved in

the lower part of the electrode acco

are calculated below. The physical

Appendix I and the results are shown

are given in cal/sec.

The enthalpy variation of the

ing at 500 C and leaving at 1460 0C is

the heat

rding to

data are

in Table

balance on

Figure 11

those of

3. The fluxes

flow of material enter-

calculated with:

mE(H500C - H14 600C) = pVESE(H500C - H14 600C) (V.1)

The values of the enthalpies are taken from the tabulated

data of Elliott, Gleiser and Ramakrishna(29) on iron.

Heat flux, Ql, is calculated from the temperature

gradients read at 1460 0C on the plots of Figure 7, 8 and 9.

These gradients are assumed constant at the tip. Q, is

expressed as:

(V.2)1 ~ SE k T=1460 0 C

Heat flux

temperature of

The value

, Q2, has a value of zero because the

the upper part of the electrode is constant.

of the heat generated, Q3 , is given by

_ 1 .2- 4.18 resistivity x iE x SE x (deff + dem

.... (V.3)

173

An average value of 60 x 10-6 Q

the resistivity. This value correspon

temperature of about 500 0 C (Appendix I

current density.

Q4 has two terms: the first one

due to convection of the gas, the seco

to the heat transfer by radiation with

slag and with the inner wall of the mo

evaluated successively below.

In the calculation of Q4c' the v

heat transfer coefficients determined

assumed to be applicable. The heat fl

= 27TR

cm

ids

, Q

nd ,

th

ld.

alu

in

ux

is

to

iE

taken for

an average

is the

4c, is a heat flux

Q4r' correspondse surface of the

They are

es of the effective

Appendix IV are

is approximated by

d em +d

d/ h(T-T 0)dZ ~27Rh(T-T O)avr x d eeff

..... (V.4)The heat flux by radiation, Q4r' is calculated from

the heat flux by radiation per unit area, q,4r, in cal/cm 2/sec

which varies with the distance from the slag. q4r isevaluated as shown in Figure V.1. Due to the indicated

configuration on this figure, only the first order radiant

interchanges between the element of surface dA and the

surroundings are considered. The interchanges with the slag

and the mold are expressed respectively with

q4r dA+A5

-dA - F UT4 + AsFA aT 4

.... (V.5)

- - 410", -

174

4 = -dA FdAA EaT4 + A F acaT

..... (V .6)

Fi is the view factor of surface i with surface j.

at, Ci and T are respectively the absorbtivity, the

emissivity and the temperature of surface i. Equations

(V.5) and (V.6) can be simplified. The reciprocity

theorem gives

dA - FdA+A

Ac ' FAc+dA

A d - FA5 dA

dA - FdA+A

The summation theorem leads to

FdA+A = F = 1 - FA +dAc

Kirchoff's law gives the following equality:

at = C i

The flux

is given by

by radiation per unit area of the electrode

= -GeE[F(T

The view factor, F,

Ts ssl)+

according to

- F)(T 4 - T)]c c

Figure V.1 is expressed as:

F = tan -Tr

R E -Rm)

Z d eff

These calculations have

of the three first experiments

(V.12)

been applied to the condition

. The emissitivities of the

(V.7)

(V.8)

(V.9)

(V.10)

(V.11)

175

Zd -f

Ieff

deff

inner wall ofcopper mold

surface of slag

between the electrode andFigure V.1 : R ad ia t io n the surroundings.

176

steel and of copper are taken as 0.25 and 0.5(

respectively. The emissivity of the slag is a

be 0.7. The temperature of the copper wall is

25 0C. The temperature of the surface of the s

three experiments are assumed to be 1550 0C, 15

1550 0 C respectively. These three temperatures

150 0C less than the temperatures measured in t

shown in Table 8.

The flux, q4r, has been plotted versus d

from the slag level in Figure V.2 for the four

The total flux by radiation, Q4r' is given by

Q4r = 2TrRE

d eff+de

de ffq4r dZ

30)

ssumed to

taken as

lags in the

00 0C and

are about

he slags as

stance,

experiments.

(V.13)

After deduction of the heat flux by conduction, Q4c' the

final heat flux, Q4, is given in Table 3. Its positive sign

shows that heat was actually going into the electrode.

The heat flux, Q5 , corresponding to the heat flowing

into the electrode below the slag level, is obtained by

difference according to the heat balance shown in Figure 11.

ii

1.0 experiment 1

0 experiment 2-4-)

---- experiment 3

S.- c, Operating data given in Table 1.>) V

C\j

1 1a)f

-0.5

Figure V.2: Heat flux by radiation in or out of the electrode.

178

APPENDIX VI

Steady State Heat Conduction Equationfor a Moving Cylinder

An axisymmetrical cyl

in the direction of its axi

small volume fixed in space

(Figure VI.1) leads to the

conduction equation. This

equation has the following

inder is considered moving

s. A heat balance on a

and included in the cylinder

relevant steady state heat

is done in Table VI.l. The

form:

(Kr ) + 9 (Kr - pCVrT) + rW = 0

+ dr

r dedr dz

Elemental

Figure VI.1

volume for heat conduction

Heat Balance

Table VI.1for Heat Conduction

equation

Equation

heat outheat in

- heat out

conduction(radial )

conduction(vertical)

enthalpy

generation

-K }-

-K 9T9 z

rdedz|

rdedr|

pCVTrdedrfz

r 9

z - z

rdedz|

rdedr

pCVTrdedr z+dz

r (Kr T)dedzdrr+dr 9Fr dr

(Kr T' )dodrdzz+dz 9z 9z

(pCVrT)dedrdz

Wrdedrdz

179

z + dz

z

Vol ume:

heat in

Wrdedrdz

180

Appendix VII

The EPS Computer Program

The EPS computer program(26) (Equilibrium Problem

Solver) provided by the project MAC at M.I.T. is designed

to solve two-dimensional boundary value problems for

elliptic systems of second order partial differential

equations in the steady state. It uses a finite

difference method permitting irregular lattices.

Systems of equations, of the following kind, can be

solved:

[- (akl ax + bkl+ Ckl ul )

ekl + fklul) + gklul] =

[k = 1, 2, .. .n]

u1 , u2, ... u n are the dependen

y the independent variables. akl, bkl

either constants or functions of x and

specified.

The boundary conditions for each

must be take the following form:

t v

y,

aniabl

wh 9kw h ic h

boundary

(VI I.1)

es and x and

and hkl are

have to be

sequent, m,

nn (pklm F1 + qklm ul) = rkm [k = 1, .. ., n]

1=1(V II1.2)

pklm' qklm, rkm are constants or functions of position to be

specified. Fk is expressed by

9u 1+ 9 (dk l +

3y

181

n = u I b u1Fk [-(akl + bk + ckl u ]

kl + ekl 9y + 1k u1) ]

3x/Ds and Dy/as are local boundary slopes where s

length.

Details about the method empl

Tillman(25) and Yavorsky(36).

For the solutions of a specif

parameters in the differential equa

conditions have to be defined. An

be provided. A relaxation factor,

parameter, delta, and an upper limi

limit, have to be specified.

This computer program can onl

the M.I.T. compatible time sharing

(VII .3)

is an arc

oyed are given

ic problem, the

tions and in the boundary

appropriate grid has to

omega, a tolerance

t to the number of passes,

y be run pre

system using

sently on

an IBM 7094

computer.

future.

It will be commercially available in the near

182

Appendix VIII

Application of Two-Dimensional Heat Flow Model

to the Electrode of Experiment 2

The applica

(Section V.C.2) to

described below.

tion of the two-dimensional

the electrode of experiment

thermal model

2 is

The physical data were those of Appendix I, and the

operating data are given in Table I. The boundary condi-

tions, given in Table 5, have been applied in the following

way. For the numerical solution, the height of the elec-

trode was fixed to an arbitrary value of 20 cm. The

temperature, TE, was 500C. The flux, q, for boundary

condition 3, corresponding to flux, Q4, calculated in

Appendix V, was approximated with a linear function leading

to a value of 0.6 cal/cm 2/sec at the level of the slag (1.4

cm from the tip) and to a value of zero at the top.

The value of the thermal conductivity of the solid

slag, kslag, was taken as 0.002 cal/cm/sec/0 C as shown in

Appendix X. From the thickness of the slag layer around

the tip found experimentally to be equal to 0.05 cm (Section

V.A.2), the average heat transfer coefficient, h sl in

boundary condition 2 was calculated with

h slag - 0.002 = 0.04 cal/cm 2/sec/OC (VIII.1)eslag 0.05

The temperature of the surrounding slag is assumed to be

1650 0C, which is slightly lower than the temperature of

I'

1676 0 C measured

is proved

below the tip according to Figure 8.

later in this appendix that the values for

temperature of sl

heat flux, Q5, ca

The physica

operating paramet

(equation 3) and

transcribed in a

program as shown

A grid with 84 po

difference method

in the attached c

program, the data

parameters omega,

ag

lcU

1 c

ers

the

lan

in

int

omp

fi

li

and for h sl led to the appropriate

lated in Table 3.

haracteristics of the electrode, the

, the heat conduction equation

above boundary conditions were

guage understandable by the EPS computer

the enclosed data file MELEC INPUT.

s is also defined for the finite

It is drawn in Figure VIII.l. As shown

uter run, after calling for the EPS

le was coupled to EPS. The three

mit and delta were defined and the

solution was obtained after 37 passes. The output is

given in terms of horizontal line, j, vertical coordinate,

Z, center line and surface temperature, and a last quantity

which corresponds to the flux per unit area multiplied by

the radius of the electrode, RE. The total computer time

was about 21 seconds. The center line temperature is

plotted in Figure 8 for comparison with the experimental

results. The flux Q5 of Figure 11, calculated from the

above fluxes given by EPS, is found to be 220 cal/sec.

This value is 8 percent lower than the value calculated

with the heat balance of Table 3. In view of the

uncertainties involved, this result was considered

acceptable.

183

It

the

I I I

print nelec inputW 2216.4

MELEC INPUT 08/03 2216.4

SET YTIP=-1.4, JTIP=-7, HSLLAY=.04, TSL=1650, TMP=1460,TTOP=50$SET N=1, VELEC=.058, RELEC=1.27, IELEC=3, YTR=-1.4, JTR=-7, YTOP=-20, JTOP=-20$DEFINE FX=FIT(X), X=.42*I, GRID=1+Gl(J GEQ JTR)$SET G1(1)=0, G1(O)=1$.DEFINE Y=YY(GRID), YY(")=YTR/JTR*J, YY(2)=YTR+(YTOP-YTR)"(JTOP-JTR)*(J-JTR)$SET K=.075, SPECH=.16$DEFINE A=E=K*FX, F=-7.86*SPECH*VELEC*FX, G=.00172*FX" H=-.103.FX$SET B=C=D=0$APPEND 0,0, IELEC,0, IELECJTOP,0,JTOP, 0,0 TO BORDER$TALLY$IMPOSE 1 ALONG 0,0, 1,0, IELEC,0IMPOSE 2 ALONG IELEC,0,IELEC,-1,IELECJTIPIMPOSE 4 ALONG IELEC,JTIP, IELEC,JTIP-1,IELEC"JTOPIMPOSE 3 ALONG IELECJTOP, 1,JTOP, 0,JTOPIMPOSE 5 ALONG 0,JTOP, 0,-1,0,0SET P(1,1,1)=P(1,1,3.=Q(1,1,4)=Q(1,1,5.=R(1,5)=0,

P(1, 1,2)=P(,1,4)=P(1,1,5)=Q(1,1,1)=Q(1,1,3)=1, R(1,3)=TTOP, R(1,1)=TMP$DEFINE Q(1,1,2)=HSLLAY*FX, R(1,2)=HSLLAY*TSL*FX,

R(1,4)=(.645+.0323*Y)*FX$R 1.166+1.216

r epsW 2217.9.THIS VERSION OF EPS CREATED 4/21/68, PATCHED 12/11/69.

PROCEED:read melec input$FILE MELEC INPUT HAS BEEN OPENED.DEFINITION OF NEW CURVE 'BORDER' HAS BEEN COMPLETED.'BORDER' HAS BEEN CLOSED.POINT TALLY IS 84.END OF FILE ENCOUNTERED. FILE MELEC INPUT HAS BEEN CLOSED.

OF-

PROCEED WITH CONSOLE INPUT:form$SPACE FOR SOLUTION MATRIX HAS BEEN ALLOTTED AND ZEROED.

PROCEED:set omega=1.65, limit=50, delta=.1relax$RELAXATION TERMINATED AFTER 37 PASSES~ MAX SOLN CHANGE: 0.8741863E-01.

PROCEED:do print j,y,u(1,0,j), u(1,3,j), flux(1,3,j) for j=0 step -1 until -20$

0.00000000000-1.OOOOOOOE+00-2.OOOOOOOE+00-3.OOOOOOOE+00-4.0000000E+00-5.0000000E+00-6.OOOOOOOE+00-7.OOOOOOOE+00-8.OOOOOOOE+00-9.OOOOOOOE+00-1.0000000E+01-1.1000000E+01-1.2000000E+01-1.3000000E+01-1.4000000E+01-1.5000000E+01-1.6000000E+01-1.7000000E+01-1.8000000E+01-1.9000000E+01-2.0000000.+01

PROCEED:quit$R 16.016+4.500

0. 00000000000-2.0000000.-Ol-3.9999999E-01-5.9999998E-01-7.9999999E-01-9.9999998E-01-1.2000000E+00-1.4000000E+00-2. 8307692E+00-4.2615384E+00-5. 6923076E+00-7.1230769E+00-8.5538460E+00-9.9846152E+00-1.1415384E+01-1.2846154E+01-1. 4276923E+01-1.5707692E+01-1.7138461E+01-1.8569230.+01-1.9999999E+01

1.4600002E+031.3245949EA 031.2079322E+031. 1036890E+031.0078695E+039.1842393E+028.3460087E+027.5657996E+023.4456639E+022.3247599 E+021.9267285E+021.6665686E+021.4518527E+021. 26328 50E+~21. 0969 511E+029. 5078213E+018.2381103E+017.1603278E+016. 2686344E+015. 5526866E+015.0000005E+01

1.4600002E+031.3855744E+03

1.30749 74. +031. 2291660 . + 031.14889 36E+031. 0624786E+039.6 175042E+028.2766917E+023.4070209E+022.3791556E+021. 9639$ 25E+021. 70089 74E+~'21. 4823523E+ 021. 2903750E+021.1200595 E+029 .6984005E+018. 3925190E+017. 2773438E+016.3441634E+015.5863841E+015. 0000005E+01

-5.5884792E+011.3349480E+011. 7262167E+012. 1219381E+012. 529 5 414 E+012.9699940E+013.4901901E+015.5381242E+006.9997606E-016. 4 311198E-015. 8052825E-015.2049440E-014.6422037E-014.0785787E-013.4823160E-012. 8822989E-012.3077494E-011. 73986 27 E-011.1603878E-015. 7702798E-029. 0315267E-01

186

-20 -20

-7 -1.4

ycm

(3) ' top

0.645

cal/cm

0.0323 x y/sec

= 1650 0 C

x, cmSI I ..

0.42 0.84 1.27

Figure VIII.1: Grid for data file (MELEC INPUT).

187

Changes in the grid size and in delta did not alter

the solution within more than 30C as shown in Table VIII.l.

It was concluded that the computer results converged

toward the proper solution. The value of 20 cm given to the

height of the electrode was also found sufficient for the

temperature in the electrode to be unaffected by the value

of height.

A change in the

from 500 C to 95 0C gave

which has been plotted

temperature of the entering material

a center line temperature profile

in Figure 8.

11

Sensi ti

to Grid

Thermal

number ofpoints in

grid

Table VIII.1

vity of the Numerical Solution

Size and Delta in the Computer

Model Applied to the Electrode

of Experiment 2

84

delta

Z, cm

0.4

1.0

2.8

5.7

1

1208.0

918.2

344.7

192 .2

0.1

1207.9

918.4

344.6

192.7

0.01

1207.9

918.4

344.6

192.6

188

126

0.1

1206.6

915.9

342.3

191.5

189

APPENDIX IX

The Approximation of the Moving Fin

on the Entire Ingot

This appendix describes the solution of the

differential equation for the moving fin approximation.

The appropriate boundary conditions are given in Figure

25 and Table 6.

The solution for each medium, i, using the

reference temperature, T,, is of the form:

T = ai exp(Xi2

where

and

1PC V

7[k

k +

) + S3 exp(x Z)

FpC V1 2 8h

fki 2+- ],)1 i I

PC -V 8h.(k I 2 + ],h1 i I

The

of equation

undary cond

between the

ons lead to th

ntinuous, al,

e

S I

following sy

, a2' 2 and

Tt al exp(-x Z ) + S1 exp(-j Z, )

= 1 +

n a 2 + 2

0 = 2

-kIal I + i i] = -k2 x2a2 + pV1L

(IX. )

(IX.2)(<0)

(>0) (IX.3)

stems

z1

Tin

Ti

(IX.4)

(IX.5)

(IX.6)

(IX.7)

(IX.8)

When the exponential

approximated with

exp(-XZ )

the unknowns

terms of equation

2Z2= 1 - Z + - -2

can be expressed as:

1c4~1 Al-Al

__ 1Ai - Al

[Tin( -k

[Tin A2

x2 ) + ] VzLk1

k 2 - kk1 1 k ]

a2 Tin

is given by one of the two roots of

2 21l+ '2] + Z [

1k1

When the exponential

- 2 Tin

term of equation

further approximated with - A the solution

then simply given as

k, (TinTt)

-k2X2Tin + pV1L

The temperature in the solid ingot is given by:

T - TWTin - T W - exp(AZ)

where A is given by equation

190

(IX.4) can be

(IX.9)

(IX.10)

(IX.11)

(IX.12)

in 't

(IX.4)

= 0

(IX.13)

can be

for ZI

(IX.14)

(IX.15)

(I X. 2) .

191

APPENDIX X

Heat Transfer Coefficient Between Ingot and Water

The heat transfer coefficient between ingot and

water, hi, is calculated directly with equation (7) for

the conditions obtained on the ingots of the experimental

study (Section VI.A). The heat transfer coefficients

h gap, hmold, h water' and hslag, explained in Section

VI.C.2 are successively calculated.

A. Heat Transfer Coefficient Through ContractionGap, hgap

The heat transfer coefficient, hgap, is the sum of

two terms. A first term, h cg, corresponds to the heat

transfer by conduction through the gas in the gap and a

second term, h rg' to the heat transfer by radiation between

the slag crust and the copper wall. The contraction gap,

eg is evaluated with:

e (Z) = cRI (T(Z) - Tco) X.l)

Ac, the expansion coefficient, has a value(31) of

1.2 x 10-5 (31). T(Z) is taken from the temperature profile

at 0.'6 cm from the surface of the ingot given in Figure 24.

T cois the temperature at which contraction started. The

coovalue of Tco is taken as 1350 C.

Assuming that the thermal conductivity of the gas

in the air gap is the thermal conductivity of stagnant air,

-&M -wa,

7

192

k air' h is calculated with:

hcg = kar ai/ (X.2)

At a distance, Z, the value of k air (37) is taken for an

average temperature between T(Z), previously defined,

and the water temperature of 100C. The results for hcg

are given in Table X.1.

The heat transfer coefficient, h rg, is evaluated

from the total heat transferred by radiation, q, from an

element dA of the slag crust at temperature, Ts, to the

corresponding element of the copper wall at temperature,

Tc. Heat flow q is given by(38):

dA a (T - T )q s c (X.3)

s c

es and Ec are the emissivities of slag and copper wall.

The heat transfer coefficient, h rg, is then expressed as:

T - T 4

h = x s c (X.4)rg 1 + 1 Ts -Tcs c

Es is assumed to be 0.7 and E c is taken as 0.5(30). TS is

approximated with the experimental profile T(Z) of Figure

24. Tc is taken as the temperature of the water 283 0 K.

The value of h is given versus distance Z in Table X.1.

hrg is less than 1 percent of the heat transfer coefficient

by conduction,hcg'

Table X.1

Heat Transfer Coefficient Between Ingot

(Results

and Water

of Appendix

Z T(Z)

cm 0_C

1350

1180

900

g

x10 3 cm

5.1

13.5

Kaircal/cm/!sec/0 C

1.5

1.3

hcg 2cal/cm /

sec/0C

0.029

0.0096

hrg 2cal/cm2/sec/OC

0.0017

0.001

hgap

2cal/cm /sec/OC

0.2

0.3

0.01

h +

slag+s1 1

cal/cm /sec/ C

0.0105

0.0105

0.0105

h 1 2

cal/cm /sec/OC

0.010

0.0078

0.0051

0.0105500 25.5 1 .0 0.004 0.0004 0.004 0.0029

194

The heat transfer,

As shown in Table X.1, h

cal/cm 2/sec/ C over the

occurred. This value is

on contact resistances b

casting machine.

h gap' is the sum of hcg and h .

gap was given a value of 0.2

first two cm where no shrinkage

based on the work by Irving(16)

etween metals in the continuous

B. Heat Transfer Coefficient Through Copper Wall, hmold

h mold is expressed as:

hmold = Kcopper/e wall

The thermal conductivity of copper, Kcopper, has

of 0.9 cal/cm/sec/0 C(30) and the wall thickness,

0.35 cm. hmold is in the order of 2.5 cal/cm 2/se

a

(X.5)

value

ewal 1

c/ 0C.

, is

C. Heat Transfer Coefficient Between Copper Wall andWater, hwater

The flow of water i

16 liters/min. This flow

along the copper wall. T

number were calculated.

was evaluated at 0.02 cal

nto the 0.8 cm annular spa

gave a water velocity of

he Graetz modulus and the

Using McAdams data(30),

/cm 2/sec/ C.

ce was

16 cm/sec

Nussel t

hwater

D. Heat Transfer Coefficient Through the Slag, hslag

hslag can be expressed as:

hslag = slag / slag (X.6)

H

195

The thickness of the slag crust is about 0.9 mm. The thermal

conductivity of the slag, Kslag, is unknown as indicated in

Section VI.B.2. A value for Kslag of 2 x 10-3 cal/cm/sec/ 0C

was assumed and h was calculated with equation (7). The

results are given in Table X.1 and plotted in Figure 27.

For distances from the top of the ingot greater than 3 cm,

hI is found to have the average value of 0.0045 cal/cm 2

sec/ 0 C calculated with the moving fin approximation

(Section VI.C.2). It was concluded that the value for Kslag

of 2 x 10-3 cal/cm/sec/ 0 C was satisfactory.

'I

196

APPENDIX XI

Computer Program for the Thermal Model

of the ESR Ingots

The application of the computer thermal model (Section

VI.

is

cal

par

com

are

D) to

detail

cul ate

ameter

Two

puter

shown

The first

the ingots of the

ed below. Proof

d solution to the

delta, is also g

data files were

program described

on the computer

data file, MESR

experimental study

of the insensitivit

grid and to the to

i ven.

written and coupled

in Appendix VII.

print-out enclosed

PART

(

y

le

Section VI.A)

of the

rance

to the

These fi

in this

EPS

1 es

appendix.

contains the information

valid for all ESR ingots, mai nl

tial equation and of the

data file, MESR INS, def

of the experimental stud

of the ingot, the physic

parameters and the bound

The thermal conductivity

as 0.11 cal/cm/sec/ 0 C.

the position of the mush

boundary

s the

Thes

data

cond

the

100

i ne

y.

al

ary

of

The

the

conc

data

data

Appen

tions

iquid

oi nt

form of

itions.

specifi

compri

dix I),

given

metal

grid is

the

The

c to

se th

the

in Ta

pool

defi

y zone as schematically

t

e

0

b

i

n

ifferen-

second

he ingots

geometry

perating

le 7.

s taken

ed from

drawn in

Figure XI.l. The mushy zone is limited by two parabolas.

Z s(cm) = 2.6 - 0.184 x r

Z1 (cm) = 2 - 0.16 x r2

These parabolas correspond respectively to the lower and

( XI.1 )

( XI.2)

print mesr part2W 2142.7

MESR PART2 07/20 2142.8

SET N=1, KING=.075, FING=1.25, FPOOL=1.4, TW=O, B=C=D=0$DEFINE FX=FIT(X), HHG=YMZ-YMPFY=FIT(Y),

GRID=1+Gl(J GEQ JMP)+Gl(J GEQ JMZ)+Gl(J GEQ JL)$SET Gl(0)=0, G1(1)=1$DEFINE Y=YY(GRID), YY(4)=YL+(YING-YL)/(JING-JL)*(J-JL),

YY(2)=YMP+(YMZ-YMP)/(JMZ-JMP)*(J-JMP)"DEFINE ZONE=1+Zl((J GRT JMP-1 )+( J GRT JMP))+

Zl((J GRT JMZ-1 )+( J GRT JMZ))$SET Zl(0)=0, Z1(2)=1$DEFINE Zl(1)=OCTANT GRT 3$DEFINE K=KK(ZONE), FF=FFF(ZONE), H=HH(ZONE)-.007*FX' G=.SET KK(1)=KPOOL, KK(2)=KMUSZ, KK(3.=KING, FFF(1)=FPOOL,

FFF(3)=FING, HH(1)=0, HH(3)=0$DEFINE HH(2)=-464/HHG*VING*FX, F=-FF*VING*FX, A=E=K*FX$APPEND 0,0, 0,JING, IINGJING, IING,0, 0,0 TO BORDER$TALLY$IMPOSEIMPOSEIMPOSEI MPOSESET P(

R(

12,3J4

",1

1,3DEFINE Q(R .866+1.

ALONG 0,JING, 1,JING, IINGJINGALONG IING,JING, IING,1, lIING,0ALONG IING,0, 1,0, 0,0ALONG 0,0, 0,1, 0,JING$,1)=P(1,1,2)=P(1,1,14)=Q(1, 1,3.=1, P(1,1,3)=)=TS, Q(1,1,14)=0$1,1,1)=-F, Q(1,1,2)=HI*FX, R(1,2)=HI*TW*FX$633

000028*FX$FFF(2)=FMUSZ,

R(1,1)=R(1,14)=0,

print mesr insW 2144.9 -

.0

MESR INS 07/20 2144.9

DEFINE YMZ=-.184*(FX POWER 2)+2.6, YMP=-.16*(FX POWER 2)+2, HI=HHI(HT),HT=1+HT1(Y GRT 2)+HT1(Y GRT 5), HHI(2)=-.00183*Y+.0137$

SET HTl(1)=1, HT1(0)=0, HHI(1)=.01, HHI(3)=.0045$.SET KPOOL=.11, IING=3, JMP=5, JMZ=7, JL=12, YL=3, JING=24, YING=25,

TS=1680, VING=.0177, RINGOT=2.5, KMUSZ-.075, FMUSZ=1.34$DEFINE X=XX(I), YY(1)=YMP/JMP*J, YY(3)=YMZ+(YL-YMZ)/(JL-JMZ)*(J-JMZ)$SET XX(0)=0, XX(1)=1, XX(2)=1.9, XX(3)=2.5$R .666+.700.

r epsW 2145.8THIS VERSION OF EPS CREATED 4/21/68, PATCHED 12/11/69.

PROCEED:read mesr ins$FILE MESR INS HAS BEEN OPENED.END OF FILE ENCOUNTERED" FILE MESR INS HAS BEEN CLOSED.

PROCEED WITH CONSOLE INPUT:read mesr part2$FILE MESR PART2 HAS BEEN OPENED.DEFINITION OF NEW CURVE 'BORDER' HAS BEEN COMPLETED.'BORDER' HAS BEEN CLOSED.POINT TALLY IS 100.END OF FILE ENCOUNTERED. FILE MESR PART2 HAS BEEN CLOSED.

PROCEED WITH CONSOLE INPUT:form$SPACE FOR SOLUTION MATRIX HAS BEEN ALLOTTED AND ZEROED.

PROCEED:set omega=1.65, limit=60, delta=1".1relax$RELAXATION TERMINATED AFTER 57 PASSES. MAX SOLN CHANGE: 0.9009841E-01.

PROCEED:set i=0do print j,y,u(1,0,j), u(1,2,j), u(l,3,j), flux(1,3,j) for j=0 step 1 until 12$

0.000000000001.2724706E+021.0000000E+00

-4.0374631E+012.0000000E+00

-3.9089688E+013.0000000E+00

-3.8087109E+014.0000000E+00

-3.7285640E+'15.OOOOOOOE+00

-3.6545070E+'16.0000000E+-0-3.5603211E+017. 0000000E+00-3.4416093E+018.0000000E+00

-3. 2937774E+019.0000000.+00-3.1142627E+011.OOOOOOOE+01

-2.8055514E+011.1000000E+01

-2. 5217712E+011.2000000E+01

-2.1816151E+01

0.00000000000

4.0000000.-01

7.9999999E-01

1.2000000E+*0

1.6000000E+00

2.OOOOOOOE+00

2. 3000000E+00

2.6000000E+00

2.6800000E+00

2. 7600000E+00

2. 8400000E+00

2.9200000E+00

3.0000000.+00

1.6800000E+03

1.6360434E+03

1.5910910E+03

1.5478164E+03

1.5082034E+03

1.4734743E+03

1. 4262771E+'3

1.3681145E+03

1.3490856E+03

1.3305841E+'3

1.3125668E+03

1.2950070E+03

1.2778971E+03

1.6800000E+03

1.6296370E+03

1.5811693E+03

1.5385746E+03

1.5025644E+03

1.4725484E+03

1. 4290207E+-'3

1.3794588E+03

1.332 73 49E+03

1.2887314E+03

1.2471539E+03

1.2079494E+03

1.1712862E+03

1.6800000.+03.

1.6116633E+03.

1.5609652E+03

1.5210820E+03

1.4890349E+03.

1.4636817E+03.

1.4233485E+'3.

1. 3813794E+03

1.3169341E+03

1.2560309E+03.

1.2002822E+03.

1.1487058.+03

1.0997845E+03

PROCEED:do print j,y,u(1,0,j), u(1,2,j), u(1,3,j), flux(1,3,j) for j=14 step 2 until 24$

1.4000000E+01-8.4855084E+00

1.6000000E+01-5.6251553.+001. 8000000E+01

-3.7318774E+002.0000000E+01

-2.4889350E+002.2000000E+01

-1. 71292 00E+002.4000000.+01-2.6579685E+00

6.6666666E+00

1.0333333E+01

1.4000000E+01

1.7666667E+01

2.1333333E+01

2.5000000E+01

8.0850087E+"2

5.3566212E+02

3.5540741E+02

2.3699803E+02

1.6276105E+"2

1.3296144E+02

7.7494699E+02

5.1355218E+02

3.4074323E+02

2.2722446E+02

1.5605289E+02

1.2748240E+02

7.5082160E+02

4.9767005E+02

3.3020938.+02

2.2020217E+02

1.5123183.+02

1.2354438E+02

PROCEED:do print i,x, flux(1,0"i,o)""0), flux(1,i,24) for i=0 step 1 until

0.000000000001.0000000E+"02.OOOOOOOE+003.OOOOOOOE+00

PROCEED:quit$R 32.216+7.400

0.000000000001.0000000.-+001.9000000E+002.5000000E+00

9.5749907E+00 -2.5902716E-015.3946269E+01 -2.8263377E+001.1206843E+-2 -5.1528643E+00.1.2724706.+02 -2.6579685E+00

.0

201

y1

In theiing =

0

0

J

Jmp

jmz

ii

1 .9 2.5cm

ying I. j ing

enclosed computer run: jmp = 5, jmz = 7, jj = 12, jing = 24,3, y1 = 3, ying = 25.

Figure XI.l: Grid for the use of EPS on the thermal model of theESR ingots.

ig

) mushy zone

202

upper limits. The height of the ingot was taken equal to

25 cm. This length is the minimum length for the tempera-

ture in the first 15 cm of the ingot to be unaffected by

the value of length.

The enclosed computer sheets show that the EPS

program was first called and

in. The finite d

relaxations were

parameters omega,

obtained after 57

are given in term

y, of temperature

of the ingot and

physical flux per

ingot. The total

the two data files were read

ifference equations were generated and

initiated after definition of the three

limit and delta. The solution was

passes. The results on the print-out

s of line, j, of the distance on the axis,

along the axis, at 0.6 cm from the surface

on the surface. The calculated flux is the

unit area multiplied by the radius of the

computer time was about 40 seconds.

The sensitivity of the solution to the number of

grid points and to the tolerance parameter is shown in

Table XI.1 for a few selected points. On the 150 point

grid, the solution is the same for the delta of 0.1 and

0.01 except near the bottom of the ingot were the difference

is 10 C. Between the solutions obtained with the 150 point

grid, delta of 0.1 and the 228 point grid, delta of 0.5,

the discrepancy is less than 50C except near the bottom of

the ingot were this discrepancy reaches 13 0 C. Only the

higher temperatures are really of interest. At 1350 0C,

the temperature gradient is at least 150 0C/cm. An error

of 5 0C alters the position of a given temperature by less

than 0.3 mm, which is within the experimental inaccuracy.

203

Table XI.1

Influence of the Number of Points of the Gridand of Delta on the Temperature Distribution

in an Ingot Calculated with EPS

numberof

poi ntsin thegrid

delta

100

0.1

center linetemperature(O C) at dis-tance, Z (cm)from top ofingot

surfacetemperature(OC) at dis-tance, Z (cm)from top ofingot

2.0 1483

2.6

3.0

14.0

1.4

1.1

3.0

1378

1288

365

1474

1391

1110

150

1

1494

1389

1298

359

1468

1412

1119

150

0.1

1496

1391

1300

369

1469

1413

1120

150

0.01

1496

1390

1300

369

1469

1413

1120

228

0.5

1491

1386

1297

360

1465

1409

1120

14.0 340 335 343 344 330

204

The solution obtained

0.1 was considered to

satisfactorily.

with the 100 point grid, delta of

approximate the true solution

205

APPENDIX XII

Calibration of the Infrared Pyrometer

Calibration cu

for the Irtronics in

curves were adjusted

instrument was used

made ove

using a

(Figure

Rh wires

alumina

graphite

with a 4

Th

graphite

r a range of

thermocouple

XII.1). The

(0.01 in.di

tube. This

tube (I.D.

mm high gra

e slag was h

crucible as

rves were supplied

frared pyrometer.

for the exact con

on the ESR unit.

temperatures

and the same

thermocouple

ameter) encas

alumina tube

3.5 mm, O.D.

phite plug.

eated with an

a susceptor

by the manufacturer

These calibration

ditionsin which the

A calibration was

from 14000C t

slag as in th

consisted of

ed in a 3 mm d

was protected

7 mm) closed a

induction

Figure XII

o 166

e ESR

Pt/

i amet

by a

t one

0 0C,

uni

Pt-1

er

end

unit using a

.1). The tube

of the pyrometer and the thermocouple were immersed in the

slag over a distance of about 3 cm. By moving the thermo-

couple up and down, the temperature of the slag was found

to be essentially uniform in the volume where the tempera-

ture was measured.

The calibration of the pyrometer was obtained during

cooling of the slag from 1660 0C to 14000C.

The argon flow rate used to form bubbles was 0.3 1/min.

Doubling this flow rate was found to change the temperature

measurement by about 100 C.

206

The inaccuracy on the temperature measurements was

estimated at about + 200C.

207

pyrometer

Pt/Pt-10% Rh

argon

I'.

I I

thermocouple wires (diam.

alumina tube 0.01 in

(diam. 3 mm)

graphite tubeO.D. 7 mm; I.D. 3.5 mm

7-

0

80% CaF

220% CaO slag

0

inductioncoil

0

0

0

0

graphite susceptorI.D. 2-1/4 in.O.D. 2-3/4 in.height 5-1/2 in.bottom 1/4 in.

Figure XII.l: Schematic drawing of the experimentalarrangement for calibration of infraredpyrometer.

77-

0cm

O0

0

0

0

0

0

0y

208

APPENDIX XIII

Dimensional Analysis for Heat Flow in Ingots

A dimensional heat flow analysis is made for ingots

of pure material.

ham Pi theorem(45

in Table XIII.l.

can be combined w

four. Using as a

VI, the analysis

completely descri

replaced by combi

been rearranged t

Table XIII.2. Th

This analysis is

). The 15 quantit

The number of max

ithout forming a d

base the quantiti

gives 11 dimension

be the problem. S

nation of other gr

o produce 11 more

e seventh group, C

based on

i es

imum

i men

es,

less

ince

oups

comm

s (Tm

i nvol v

quant

si onl e

T m-TW,

group

a gro

, the

on gro

-TW)/V

the Bucking-

ed are listed

ities that

ss group is

ks, RI and

s which

up may be

results have

ups given in

2 represents

the ratio of

which is not

For a

Table XIII.2

an enthalpy term to a kinetic energy term

really relevant to this study.

given material, the last four numbers of

represent the operating dimensionless groups.

209

Table XIII.1

Quantities Involved in Heat Flow Analysis

quantity

1. T-TW

units(absolute)M.L.e.T.

3. Z

4. R

5. Tm-T W

7. ps

8. C1

T

M/L3

M/L3

L2 2T

quantity

9. Cs

10. k

11. ks

12. L

13. h

14. Tt-TW

15. VI

units(absolute)M.L.eT.

L22TL 2/0 2T

ML/3 T

ML/03T

L2 /02

M/3 T

T

Table XIII.2

Dimensionless

1. T-TW/Tm-Tw

2. r/R1

3. Z/R1

4. p /ps

5. C /Cs

6. L/Cs (Tm-TW)

Numbers in Heat Flow Analysis

7. C (Tm-T 2

8. Tt-TW/TM- TW9. psCsVIRI/ks

10. ks/k 1

11. hIRI/ks

I

210

BIOGRAPHICAL NOTE

The author was born in Saint-Etienne, Loire, France

on January 26, 1945.

He attended the "Lycee Pasteur" in Neuilly-sur-Seine,

obtained his "Baccalaureat" in 1962 majoring in

Mathematics and continued with the programs of "Mathe'matiques

Superieures" and "Mathematiques Speciales" at the same

institution.

In 1964, he successfully passed the entrance

examination to the "Ecole des Mines" in Paris from which he

graduated in 1967.

He has since been a candidate for the degree of Doctor

of Philosophy at the Massachusetts Institute of Technology

with a major in Metallurgy and a minor in Industrial

Management.

He is a member of the Society Sigma Xi and is a

student member of the Metallurgical Society of the A.I.M.E.

and of the American Society for Metals.