Magnetic Convection 9781860947124, 9781860945786

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Magnetic

CONVECTION

Magnetic^

CONVECTION

by

Hiroyuki Ozoe

Kyushu University, Japan

//S^i

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

MAGNETIC CONVECTION Copyright © 2005 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 1-86094-578-3

Printed in Singapore by Mainland Press

Preface This book focuses on the convection of liquid metal caused by a Lorentz force and the convection of fluids such as air and water due to a magnetizing force. The convection of liquid metal due to the Lorentz force has been studied extensively, and here its study was motivated by the desire to clarify the effect of a magnetic field on the convection of molten Si in a crucible of the Czochralski crystal-growing system. This is because the characteristics of convection of molten silicon have been considered to be responsible for the solidifying processes at the melting temperature and for the final crystal composition and dopant distribution. On the other hand, study of the convection of non-electro-conducting fluid due to the magnetizing force appears to have started since the report by Braithwaite et al. in 1991 on the experimental measurement of the heat transfer rates of the gadolinium nitrate solution for the configuration of Rayleigh-Benard type natural convection in the bore space of a super-conducting magnet. The driving force for this convection is the magnetizing (or magnetization or magnetic) force, which is proportional to the magnetic susceptibility of the fluid and approximately proportional to the gradient of the square of magnetic induction. This force has become widely applicable with the development of the super-conducting magnet, and the motivation for this study is due to its vast potential for application, ranging from quasi-non-gravitational or enhanced gravitational acceleration in the bore space of a super-conducting magnet to micro-scale magnetic effects in atomic level processes. The title of this book, "magnetic convection," may not be a familiar or widely employed term, but it is employed herein to mean the convection caused either by a Lorentz force in an electro-conducting fluid or by a magnetizing (or magnetization or magnetic) force in a non-electro-conducting fluid. The origin of this term lies with Professor Shigeo Asai, Nagoya University, who led and chaired a research project, "The new developments in the electro-magnetic-processing", sponsored by the Ministry of Education, Japan from 1999 to 2002. There appears to be no earlier term that covers the convection of fluid caused by magnetic field, either by a Lorentz force or by a magnetizing force. Most of the latter part of this book represents results acquired by my research group in the above research project. The preparation of this book was proposed by Imperial College Press. Its style is a summary of our research results. Even though the above funded research project has now ended, we have continued to develop our research in line with the contents of this book. Future research results of our group will be listed on our home page.

v

vi

Magnetic Convection

http://trans2.cm.kyushu-u.ac.jp/, http://www.hiroyuki-ozoe.net. Various new findings on the phenomena caused by a strong magnetic field will emerge, and some of them may be employed for developing new industries. I believe this new wave of applications of a strong magnetic field will develop further, and this book may provide an introduction to the current state of this new art.

Hiroyuki Ozoe June 14, 2003 in Fukuoka

Contents Preface

v

Chapter 1 Application of a Magnetic Field for Materials Processing

1

Chapter 2 Natural Convection of Liquid Metal without a Magnetic Field 2.1 Two-Dimensional Computation of Oscillatory Natural Convection of Low-Prandtl-Number Fluid Heated from Below 2.2 Experimental Heat Transfer Rates of Rayleigh-Benard Oscillatory Natural Convection of Liquid Gallium Heated from Below 2.3 Three-Dimensional Computation of Natural Convection in a Shallow Rectangular Region Heated from Below 2.4 Closing Remarks

3

Chapter 3 Two-Dimensional Numerical Analyses for Natural Convection of Liquid Metal in a Magnetic Field 3.1 Two-Dimensional Numerical Computations in a Vertical Magnetic Field 3.2 Two-Dimensional Analysis for a Lateral Magnetic Field Chapter 4 Three-Dimensional Natural Convection of Liquid Metal in a Cubical Enclosure with a Magnetic Field 4.1 The Effect of the Direction of the External Magnetic Field 4.2 Experimental Results for Molten Gallium under an External Magnetic Field in the X, Y or Z Direction 4.3 Summary Appendix A Appendix B Chapter 5 Enhanced Heat Transfer Rate of Natural Convection Due to a Magnetic Field 5.1 Three-Dimensional Numerical Analyses 5.2 Further Experimental Measurement for the F-Directional Magnetic Field Appendix

vii

3 7 10 13

15 15 20

25 25 30 35 35 37

41 41 48 50

viii

Magnetic Convection

Chapter 6 Natural Convection of Liquid Metal in a Cube with the Seebeck Effect Under a Magnetic Field 6.1 Introduction 6.2 Mathematical Model 6.3 Computed Results 6.4 Discussion 6.5 Conclusions

51 51 51 53 56 57

Chapter 7 Flow Visualization of Oscillatory Czochralski Convection 7.1 Introduction 7.2 Experiments 7.3 Conclusions

59 59 60 65

Chapter 8 Liquid Encapsulated Czochralski Bulk Flow with Flow Visualization 8.1 Introduction 8.2 Mathematical Model Equations 8.3 Experimental Set-up and Conditions for Flow Visualization 8.4 Results and Discussion 8.5 Conclusions

67 67 68 69 71 76

Chapter 9 Effect of Radiation Cooling from a Free Surface in Czochralski Melt Flow 9.1 Introduction 9.2 The System and Model Equations 9.3 Computational Schemes 9.4 Computed Results 9.5 Conclusions

79 79 80 82 83 89

Chapter 10 Effect of an Axial Magnetic Field on the Melt Convection of Czochralski Crystal Growth 10.1 Introduction 10.2 Computed Results for a Static Crucible 10.3 Computed Results for a Rotating Crucible 10.4 Summary of the Results in an Axial Magnetic Field

91 92 94 97 98

Contents

ix

Chapter 11 Effect of a Transverse Magnetic Field on the Melt Convection of Czochralski Crystal Growth 11.1 Introduction 11.2 Model System 11.3 Computational Schemes 11.4 Computed Results 11.5 Experimental Measurement of Temperature 11.6 Elliptic Temperature Profile in a Transverse Magnetic Field 11.7 Conclusions

99 99 100 103 104 106 107 111

Chapter 12 Effect of a Cusp-Shaped Magnetic Field on the Melt Convection of Czochralski Crystal Growth 12.1 Introduction 12.2 Numerical Analysis 12.3 Results and Discussion 12.4 Conclusions

113 113 114 118 122

Chapter 13 Effect of a Rotating Magnetic Field on the Melt Convection of Czochralski Crystal Growth 13.1 Introduction 13.2 Model System 13.3 Computational Schemes 13.4 Computed Results 13.5 Conclusions

125 125 126 127 128 130

Chapter 14 Continuous Steel-Casting Systems with Various Magnetic Fields 14.1 Introduction 14.2 Model Systems 14.3 Computed Results 14.4 Conclusions

131 131 132 135 138

Chapter 15 Convection Induced by a Cusp-Shaped Magnetic Field for Air in a Cube Heated from Above and Cooled from Below 15.1 Introduction 15.2 Derivation of Magnetizing Force Model for Paramagnetic Gas 15.3 Computed Results

139 139 140 143

x

Magnetic Convection 15.4 Experimental Flow Visualization 15.5 Conclusions

Chapter 16.1 16.2 16.3 16.4 16.5

16 Rayleigh-Benard Convection of Air in a Magnetic Field Introduction Experiments Numerical Analyses Discussion Conclusions

149 150 151 151 152 155 159 160

Chapter 17 Effects of Various Parameters on the Convection of Air in a Cubic Enclosure Under a Magnetic Field 17.1 Effect of the Direction of the Magnetic Field 17.2 Effect of ^and Ra 17.3 Effect of the Number of Coils

163 163 168 172

Chapter 18 Transient Characteristics of Convection and Diffusion of Oxygen Gas in an Open Vertical Cylinder Under Magnetizing Forces 18.1 Experiments 18.2 Derivation of Model Equation 18.3 Computed Result 18.4 Conclusions

175 175 176 179 182

Chapter 19.1 19.2 19.3 19.4 19.5

183 183 184 185 189 191

19 Rayleigh-Benard Convection of Diamagnetic Fluid Introduction Derivation of Model Equation Model Systems and Computed Result Discussion Conclusions

Chapter 20 Magnetothermal Wind Tunnel 20.1 Introduction 20.2 Numerical Methods 20.3 Results and Discussion 20.4 Conclusions

193 193 195 197 202

Contents

xi

Appendix

205

Nomenclature

207

Acknowledgements

213

Index

219

About the author

223

CHAPTER 1

Application of a Magnetic Field for Material Processing The convection of liquid metal under the influence of a magnetic field has been studied extensively. Early research results are described in a book by Chandrasekhar [1]. Pioneering studies on the effect of a magnetic field on the convection of electro-conducting materials were basically carried out in the fields of the geophysics and cosmology and were related to convection of the Earth's mantle or of gas in a space [2]. In industry, on the other hand, the application of a magnetic field in the convection of liquid metal appears to be a recent development. Applications include convection control for the molten silicon in the crucible of the Czochralski crystal-growing process, convection of molten steel in a continuous steel-casting process, and liquid metal cooling in a nuclear reactor. The present book describes the convection of liquid metal affected by a Lorentz force in the first 14 chapters, and the convection of weakly magnetic materials due to magnetic (or magnetizing) force in a strong magnetic field in the later chapters. Chapter 2 deals with the convection of liquid metal in a shallow layer that is heated from below and cooled from above, i.e., Rayleigh-Benard convection and its oscillatory characteristics. In chapter 3, the magnetic field is applied in the natural convection of liquid metal either vertically or horizontally, and a two-dimensional flow field is studied. In chapter 4, a three-dimensional numerical model system is studied for liquid metal in a cubic enclosure heated and cooled from opposing vertical side-walls without and with a magnetic field. The application of a magnetic field to liquid metal usually results in the suppression of convection due to a Lorentz force. However, the experimental results suggested an enhancing effect, and this effect is studied in chapter 5. In chapter 6, the Seebeck effect is considered for natural convection in a cubic enclosure with its possible application in a dendrite solidification system. These numerical approaches are applied to a Czochralski crystal-growing system without and with a magnetic field. Chapters 7 and 8 present flow visualization for silicon oil in a Czochralski crucible without a magnetic field, while chapter 9 considers further the cooling effect from the free surface. Next, the

1

2

Magnetic Convection

application in a Czochralski system of either a vertical (chapter 10), transversal (chapter 11) or cusp-shaped (chapter 12) magnetic field is considered, and a rotating magnetic field is studied for a vertical cylindrical melt layer in chapter 13. Lastly, inasmuch as the magnetic field acts on liquid metal due to a Lorentz force, the continuous steel-casting system is studied with various directional magnetic fields in chapter 14. From 1986 or after, super-conducting materials at high temperature (about 90 K or so) had been invented and super-conducting magnets became more widely available for laboratory use. In 1991, de Rango et al. [3] reported levitation of nonferrous materials, while Braithewaite et al. [4] reported the enhancement or cancellation of gravitational convection due to a magnetic field for a solution of gadolinium nitrate in a shallow layer heated from below and cooled from above. The later chapters of this book deal with the effects of magnetizing force on air and water. Simple mathematical model equations are derived in a similar way to the usual Boussinesq approximation for both temperature variation or concentration variation. In chapter 15, the stably stratified air in a cube heated from above and cooled from below is disturbed by the magnetic field of four pole magnets located outside the cube. Chapter 16 considers a shallow layer of air heated from below and cooled from above, while chapter 17 examines the effects of various parameters. Chapter 18 describes the convection and diffusion of oxygen gas in a vertical glass tube located in a super-conducting magnet, and examines the effect of concentration of oxygen gas in nitrogen gas due to a strong magnetic field on the convection. Chapter 19 turns to the Rayleigh-Benard convection of water rather than air, and chapter 20 studies numerically the magnetothermal wind tunnel reported by Uetake

etal.[5]. References 1. Chandrasekhar S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Dover, (1961). 2. Saltzmann B., The General Circulation as a Problem in Thermal Convection: A Collection of Classical and Modern Theoretical Papers, Dept. of Meteorology, Massachusetts Institute of Technology (1958). 3. de Rango P., Lee M., Lejay P., Sulpice, Tournier R., Ingold M., Germi P. and Pernet M., Nature, 349 (1991), 770. 4. Braithwaite D., Beaugnon E. and Tournier R., Nature, 354-14 (1991), 134-136. 5. Uetake H. et al., J. Appl. Physics, 85-8 (1999), 5735-5737.

CHAPTER 2

Natural Convection of Liquid Metal Without a Magnetic Field

2.1 Two-Dimensional Computation of Oscillatory Natural Convection of LowPrandtl-Number Fluid Heated from Below [1] Oscillatory natural convection of low-Pxandtl-number fluid is considered to be responsible for the formation of striae in crystal rods [2]. In order to produce striafree crystal rods, this oscillatory convection should be suppressed. Fundamental study on oscillatory convection is expected to clarify the mechanism of its occurrence. Theoretical study on natural convection of the low-Prandtl-number fluids has been reported in many papers, including those by Clever and Busse on the Benard convection [3, 4]. They clarified interesting characteristics in their series of reports on the Prandtl number, oscillatory characteristics, and higher-order oscillation including traveling waves. However, the appearance and disappearance of roll cells may not be described by stability analyses. We try to clarify oscillatory characteristics by numerical analyses, so that more practical industrial problems can be more easily approached. Experiments on natural convection of low-Prandtl-number fluids were carried out by Fultz et al. [5], Verhoeven [6], and Harp and Hurle [7]. The heat transfer rate of Benard natural convection was measured by Rossby [8] for mercury. In the present section, finite difference numerical results are described for multiple roll cells, including the multiplication in the number of roll cells and the initiation of oscillatory characteristics at a specific Rayleigh number. For the Rayleigh-Benard natural convection in an infinitely shallow layer heated from below and cooled from above, the flow mode is known to shift from a rectangular two-dimensional roll cell to a chaotic turbulent mode with an increase in the Rayleigh number [9]. In the present work, we seek the initiation of oscillatory convection for the fluid at low Prandtl number. If we assume the two-dimensional flow mode, the basic model equations consist of the following vorticity Q and temperature 0 after non-dimensionalization:

3

4

Magnetic Convection d£2/dr+ GrV2(Ud£2/dX + VdQ/dY) = d2i2/dX2 + d2i2/dY2 - Gi1/2dT/dX dT/dr+ Gr1/2(Ud T/3X+ Vd T/3Y+ V) = (d2 T/dX2 + d2 T/dY2)/Pi Q = dV/dX- BU/dY = -92W/dX2- d2WldY2, T=0-(Y-

0.5)

(1) (2)

(3), (4)

X = x/h, Y = ylh, U = uhl(vGrm), V=vhl(vGrm), r=vt/h2, * = (O-Oo)/(Q,-0e), V=y/(vGT1/2), Vx=v/a, Gr = gfl 6h - 6c)h V The temperature difference T from a conduction state was employed as shown by eq. (4). The dimensionless variables were also defined above. The boundary conditions for the present system as shown in Fig. 2.1 are as follows. The aspect ration of the domain is set at 4 as described below, h is the vertical height of the enclosure.

&XX=Q,A, W=£?=d0/dX=O, at7=0, W = 0, Q=-dU/dY, = + 0.5 or T= 0.

The basic equations were approximated by the finite difference equations with standard second-order central difference approximation and solved by the alternating direction implicit method. The first computation was started from a static state with random variables in the temperature fluctuation T, but most of the subsequent computations were started from former transient states. The primary purpose of the present work was to simulate oscillatory ntural .

*

COLD «> = -o.5 f=o

YJ

°; £2 = 0 ;

n = -i-4

f

9

i

3x-° •

n

~°

i 3x"°

1i

1 32f *=

o.5

T =O

n =

HOT

Figure 2.1[1]The schematics of the system Figure 2.2 [1]Transient responses of th Figure 2.1 [1] The schematics of the system and the boundary conditions. A = 4 for most of the cases studied.

Figure 22 [1] Transient responses of the average Nusselt number for various Rayleigh numbers atA = 4 (201x51) for Pr = 0.01.

Natural Convection of Liquid Metal Without a Magnetic Field

5

convection in a shallow fluid layer. Vox A = 4, oscillatory response occurred at Ra = 6000 for Pr = 0.01 with 201x51 equal grid divisions. Oscillation may be expected at even lower Rayleigh numbers for a shallower layer, but we employed^ = 4 herein. Figure 2.2 shows transient response curves of the average Nusselt number at various Rayleigh numbers. The response curve for Ra = 8000 starting from Ra = 7000 is quite different from the other curves. Figure 2.3 shows a series of instantaneous contour maps of stream function from r = 0.020 to x = 0.046, corresponding to the present transient time. For comparison, roll cells at T =0.0 are shown at the top of the figure. This series of pictures shows a multiplication process of roll cells from four to six. Two new rolls are born near the two end boundaries and grow to almost equal size at r = 0.046. During this transition there are 17-18 oscillations, as seen in Fig. 2.2, which suggests a lengthy time for bearing new roll cells. It is interesting that the oscillation amplitude attained finally at Ra = 8000 is much smaller than that at Ra = 7000. This appears to suggest that this multiplication of roll cells is induced by large-amplitude oscillation, and that the amplitude becomes much less for a system of multiplying roll cells. In other words, the multiplication of roll cells is induced to stabilize the oscillatory system to one of smaller amplitude. Further computations were carried out for Pr = 0.1 and 0.001, although their response graphs are not listed herein. The critical oscillatory Rayleigh numbers are 2.2xlO5 for Pr = 0.1, 6000 for Pr = 0.01, and 5000 for Pr = 0.001. At Pr = 0.1, the oscillation does not readily start as the Rayleigh number is raised, but once it does begin, the oscillatory amplitude increases greatly. On the other hand, at Pr = 0.001, the oscillation starts at as low as Ra = 5000, and the oscillatory amplitude increases only slightly with increase in the Rayleigh number. The magnitude of the Nusselt number is also different. The oscillation starts at Nu = 4.1 for Pr = 0.1 and Ra = 2.2xlO5, at Nu = 1.3 for Pr = 0.01 and Ra = 6000, and at Nu a 1.004 for Pr = 0.001 and Ra = 5000. The computational grid divisions have been known to affect the result. The effect of grid size was then studied with coarser grids for the same system. Figure 2.4 shows transient responses of the average Nusselt number computed with three different grid sizes, namely, 121x31, 161x41, and 201x51 at Ra = 1.2xlO4, Pr = 0.01 and A = 4. For any other Rayleigh numbers (Ra = 8xlO3, 104, 1.5xlO4) larger amplitudes were obtained for a larger number of grids. If we were to employ even larger grid numbers, we might obtain larger-amplitude oscillations even at lower Rayleigh numbers. Thus, the oscillatory amplitude of the Nusselt number is plotted versus the Rayleigh number for the results obtained with 201x51, 161x41, and

6

Magnetic Convection

Figure 2 3 [1] A series of instantaneous contour maps of stream functions after a step change in the Rayleigh number from 7000 to 8000 a t ^ = 4, Pr = 0.01 with 201x51 grid.

„ , . ,, „ , ., Figure 2.4 r[1] Transient responses of the X, > •_ J •*. •!. average Nusselt numbers computed with three ,._ .. . - . A different grid numbers for the same system at A = j^ ™, 4 and Pr = 0.01.

Figure 2.5 [1] Extrapolation of the critical 6 .,, „ , . , , . . . c .t . „ oscillatory Rayleigh number m the lnfinitesimally ' ' .. , . ., ., ,. small oscillatory amplitude in the average Nusselt number aM = 4 and Pr = 0.01.

Natural Convection of Liquid Metal Without a Magnetic Field

7

121x31 grids in Fig. 2.5. All results are for A = 4 and Pr = 0.01. These mesh sizes correspond to 1/50, 1/40, or 1/30 the height of the fluid layer. For the 1/50 mesh size, the amplitude of the average Nusselt number converges to zero at Rayleigh number a 4500 for convection of four rolls. For convection of six rolls, it converges at Ra a 7800. For the 1/40 mesh size, it converges to the Rayleigh number a 9800. For the 1/30 mesh size, the oscillation appears to start at Ra s 1.4xlO4. The critical oscillatory Rayleigh number Raoc apparently depends on the grid sizes employed. These Raoc values obtained for different grid sizes are further plotted versus the square of the grid size. Then, these three points are employed to extrapolate to the infinitesimal magnitude of the grid size. The results obtained with the 1/50 grid size give six roll cells at higher Rayleigh number and four at lower Rayleigh number. This leads to Raoc = 4000 or less. For Pr = 10"3, Raoc appears to be about 5000 obtained with AX = 1/50. With another graph (omitted herein) we can see that Raoc approaches 2000 or less as AX2—»0. In summary, these results suggest that the oscillatory natural convection occurs very close to the critical Rayleigh number 1708 for Pr = 10"2 and 10~3. The results presented herein may give the impression that any finite difference computation is unreliable. For the oscillatory convection of liquid metal, this may be true, and more accurate prediction may become possible with a more powerful computers. However, we should recognize that the computed results also depend on the model employed (a two-dimensional in the above result) and the finite grid sizes etc. Subsequent computational results would depend on these factors but still are expected to give an insight into convection.

2.2 Experimental Heat Transfer Rates of Rayleigh-Benard Oscillatory Natural Convection of Liquid Gallium Heated from Below [10] Rayleigh-Benard natural convection in a horizontal layer of fluid between two solid surfaces was studied extensively after Rayleigh [11], by Jeffreys [12], Pellow and Southwell [13] and others for ordinary fluids and has been studied theoretically by Busse and Clever [3, 4], but experimental measurements for low-Prandtl-number (Pr) fluids are rather limited. Rossby [8] reported the time-averaged Nusselt number (Nu) for mercury (Pr = 0.025) for Rayleigh number (Ra) up to 4xlO5. The present section describes the results of an experimental investigation of the oscillatory natural convection in a bounded horizontal layer of gallium (Pr = 0.023) heated from below.

8

Magnetic Convection

Figure 2.6 shows the overall experimental set-up. The convective layer is 100x100x10 mm3. Electric power to the heating wire under the lower copper plate was supplied through a regulated electric DC power supply (Kikusui Elec. Model PAD) with a watt meter for the nichrome wire, which had a resistance of 33 Q. The upper copper plate is the bottom of a 180x130x10 mm3 cooling-water jacket. The temperature difference between the upper and lower plates was measured with Ktype Chromel-Alumel thermocouples glued on surface of the copper plates and connected to a Yokogawa HR 2300 strip-chart recorder. The part of the apparatus inside the hatched lines of Fig. 2.6 was placed in a small room maintained at a uniform temperature approximately equal to the average of the plate temperatures. Since gallium is corrosive to copper, the plates were initially coated with a thin sprayed layer of Teflon. However, the thermal resistance of this layer proved to be both unknown and significant. Hence, it was replaced with gold plating. The details of the convection chamber are illustrated in Fig. 2.7. Gallium is easily and quickly oxidized in air to form a viscous film. To avoid this difficulty the air in the experimental enclosure was replaced with nitrogen before supplying the gallium through a vinyl tube.

I 1

LJ __ij

1

'—TT—'

xij

®~!—rKWWH- ; ' I t7ySi-J~' \(

GP-IB

f

\\

9

\^-/

ih°l

Figure 2.6 [10] Schematics of the experimental system: I. temperature controller, 2. temperatureregulated air heater, 3. constant temperature water bath, 4. experimental convection layer, 5. ammeter, 6. voltmeter, 7. constant voltage and ampere power supply, 8. personal computer for data acquisition, 9. stripchart recorder, 10. cold junction, II. constant-temperature room, 12. thermocouple.

Figure 2.7 [10] Details of the convection layer: 1. cooling copper plate, 2. layer of gold plating, 3. rubber sheet, 4. plexiglass sidewall, 5. rubber sheet, 6. layer of gold plating, 7. heated copper plate, 8. nichrome wire heater

Natural Convection of Liquid Metal Without a Magnetic Field

9

The most critical aspect of the experimental measurements was the determination of the heat flux through the gallium. Heat losses were estimated as suggested by Ozoe and Churchill [14] by inverting the apparatus and thereby reducing the heat flux through the gallium to that due to pure conduction. This procedure indicated that the heat flux through the gallium was 0.8859 times that supplied to the heater. Thus, the heat losses were 0.1141 times the total input. This result is judged to be sufficiently accurate. More detailed description of this is in chapter 4. The experimental data were correlated in terms of Ra = gph\0h-6c)l(av) v) and Nu = {Qne,IA)l{k{6h-Bc)lh}. The relevant physical properties of gallium are listed in the Appendix A of chapter 4 [15]. In evaluating these properties, the absolute temperature 6 was taken to be the average of those of the hot and cold plates. Figure 2.8 shows the voltage fluctuations of the thermocouple output. The equivalent scale of the fluctuations in temperature is indicated as an inset. These fluctuations in temperature result from the fluctuations in velocity, which are characteristic of low-Pr fluids. The peak frequency of the oscillation increases with the net heat flux. This slow oscillation with a frequency of several minutes would represent the transient change of flow mode, which will be clarified by numerical analyses in future. The ranges of the fluctuations in Nu and Ra, represented by the X-marks, and the time-averaged values, represented by the solid triangles are shown in Fig. 2.9. The time-averaged values of Rossby [8] for mercury (Pr = 0.025) are included for comparison. The present values extrapolate more satisfactorily to the critical Rayleigh number of 1708 than do those of Rossby, which suggests that they may be slightly more accurate. The oscillations of liquid metal are a consequence of reduced viscous damping for a low-Prandtl-number fluid.

r L.

j

Figure 2.8 [10] Transient temperature difference between the heated and cooled walls at Qnet = 293(W).

Figure 2.9 [10] Summary of the experimental results with the oscillation range.

10

Magnetic Convection

2.3 Three-Dimensional Computation of Natural Convection in a Shallow Rectangular Region Heated from Below [16] Earlier works on oscillatory natural convection of low-Prandtl-number fluids were mostly for two-dimensional systems. The present work is concerned with a transient three-dimensional system. The computational region is shown in Fig. 2.10 for top and side views of the coordinate system. The width and length are 5 times the height. The grid numbers are 51x51x11. Because of its dragless and adiabatic vertical boundaries, this system is expected to simulate an infinitely wide shallow layer. The equations are for transient conservation in an incompressible fluid except for the Boussinesq approximation in the buoyancy term. These equations may be written in non-dimensional form as follows: ^ dX dU

UdU +

VdU +

+

^ +^ =0 dY dZ

WdU +

-BP =

(1)

W

2 32U B2U) n(d U + Pr — - + +

{dX2 dY2 dZ2 J ( 82V d2V d2V\ + + + = + Pr - + + dt dX BY dZ dY [dX2 dY2 dZ2)

dt dV

dw dt

+

dX UdV

Udw vdw +

dY VdV

+

dZ WdV

wdw

-ep

(2)

dX -dP

2

n(e

+ Pr

w

+

d2w

+

e2w)

dX dY BZ dZ \dX2 dY2 dZ2 J 2 2 2 dT UdT V3T WdT (d T B T d T\ TJ7 — + + + +— - + \+W. dt dX dY dZ [dX2 dY2 dZ2 J

(3)

„ „ ^, ,„,

+ RaPrJ (4) (5)

The dimensionless variables are defined as X = x/x0,

Y = y/y0, Z = z/zQ, U = U/U0, V=V/V0,

W = w/w0, v=t/t0,

P=p/p0,

&={6-dQ)l{0h-0c), xo = h, uo=a/xo=a/h, to = xo2/ct = h2/a, 2 2 Pa = /M0 = palh , 0O = (dh+ 6C)I2, ®c = 0.5-Z, T=®-&C=

•

! V

V

•:'••

v

0.04-

\

\

"

|_

n

Top and side views of the

000

Fi

0

•'

/

'"; /

J

-j 'l

//

002

TT~

\\»

/

200

"j 400 "600

800

1000

Nondimensional Time [-]

I

8 u r e 2U [161 Transient responses of the average velocity components in the X, Y and Z directions for Pr = 0.01 and Ra = 2000.

12

Magnetic Convection

at r = 220, they became parallel roll cells with their axes in a horizontal and diagonal direction of the square region. After this, even more complicated modes of flow occurred, but at r = 400, 560 and 680, a similar diagonal roll-cell pattern appeared again. In these instances, the average W velocity component appears to have attained its peak value. At x = 600 or after, the U values decreased in magnitude, suggesting an orientation of roll cells in the X direction. The graphs at

Figure 2.12 [16] A series of top views of instantaneous velocity vectors in the Z = 0.25 plane for Pr = 0.01 and Ra = 2000.

Figure 2.13 [16] Aseries of top views of instantaneous isotherms in the Z = 0.25 plane for Pr = 0.01 and Ra = 2000.

Natural Convection of Liquid Metal Without a Magnetic Field

13

r = 720 to 800 apparently represent dominant Y-velocity components and a strong increase in the K-velocity curve. At or after T = 940, the diagonal roll cells again appeared and attained a similar structure to that before. Because of the very long computing time (2 months), the computations were terminated at this point. Figure 2.13 shows corresponding instantaneous isotherms in the Z = 0.25 plane. At T = 180, multiple, separate, cylindrical cells appeared. However, at r = 220, parallel roll cells in the diagonal direction of the square region become dominant. Over this entire period of time, the average Nusselt number was about 1.001 or less, and hence is not plotted.

2.4 Closing Remark As mentioned in section 2.1, convergence of the numerical computations for oscillatory natural convection in low-Prandtl-number fluids has not been attained with complete success even for two-dimensional systems. Some of the details of the present results for a three-dimensional system would also be expected to be dependent on gridsize. However, on the basis of the previous studies, the basic characteristics of the oscillations would be expected to be independent of gridsize. The computations are limited to a single aspect ratio, Pr = 0.01 and Ra = 2000. More extensive results may be expected with the present computational scheme as larger and faster computer facilities become available in the near future.

References 1. Ozoe H. and Hara T., Numerical Heat Transfer, Part A, 27 (1995), 307-317. 2. Hurle D. T. J., Phil. Mag., 13 (1966), 305-310. 3. Clever R. M. and Busse F. H., J. Fluid Mech., 102 (1981), 61-74. 4. Clever R. M. and Busse F. H., Phys. Fluids, Sen A, 2-3 (1990), 334-339. 5. Fultz D., Nakagawa Y, and Frenzen P., Phys Review, 94-6 (1954), 1471-1472. 6. Verhoeven J. D., Phys. Fluids, 12-9 (1969), 1733-1740. 7. Harp E. J. and D. Hurle T. J., Phil. Mag., 17 (1968), 1033-1038. 8. Rossby H. T., J. Fluid Mech., 36 (1969), 309-335. 9. Krishnamurti R., J. Fluid Mech, 42 (1970), 295-320. 10. Yamanaka Y, et al., Chemical Engineering Journal, 71 (1998), 201-205. 11. Rayleigh L., Phil. Mag., 32 (1916), 529-546. 12. Jeffreys H., Phil. Mag., 2 (1926), 833-844.

14

Magnetic Convection

13. Pellow A. and Southwell R. V, Proc. Roy. Soc. London, 176 A (1940), 312-343. 14. Ozoe H. and Churchill S. W., AIChE Symp. Sen, Heat Transfer, 69-131 (1973), 126-133. 15. Okada K. and Ozoe H., /. Heat Transfer, 114 (1992), 107-114. 16. Nakano A. et al., Chemical Engineering Journal, 71 (1998), 175-182.

CHAPTER 3

Two-Dimensional Numerical Analyses for Natural Convection of Liquid Metal in a Magnetic Field 3.1 Two-Dimensional Numerical Computations in a Vertical Magnetic Field [1] Electro-conducting fluid such as liquid metal and plasma receives a Lorentz force under a magnetic field when the fluid moves. This is also true when the fluid convects in a gravitational field under non-uniform thermal boundary conditions. The convective rates of heat transfer and mass transfer would be affected by both the gravitational and the magnetic fields. This kind of situation is realized in the process of manufacturing crystals of such materials as silicon and gallium arsenide. It is important to clarify the detailed characteristics of transport phenomena in such processes so that well-controlled, good-quality products can be developed with improved design in the manufacturing and operational processes. The natural convection of liquid metal has been studied by, for example, Hunt and Wilks [2]. They studied the convection of a fluid at a low Prandtl number around a semi-infinite vertical plate under a strong cross magnetic field. They reported an increase in the rate of heat transfer near a leading edge as the Prandtl number decreased. Nagase et al. [3] measured the rate of heat transfer for various liquid metals over a heated vertical plate and plotted the Nusselt number against 2Ha2/Grx1/2. The plot of the data itself was widely scattered, reflecting the difficulty of the experiment and the complexity of the phenomena. They also reported complicated perturbations of temperature at 0.1 tesla rather than at zero. For liquid metals in a rectangular box under an external magnetic field, Tabeling [4] studied the natural convection with a linear theory. He discussed the relationship between the flow patterns of the Benard cells and the direction of the magnetic field. Raptis and Vlahos [5] studied the combined effect of the magnetic and the gravitational natural convection in porous media, which is related to geographical phenomena. The effect of various parameters was discussed in terms of the boundary layer approximation. All these works have limitations, owing to either boundary layer approximation and/or linear theory. The work by Weiss [7] is more closely related

15

16

Magnetic Convection

to the present work. He carried out a linear and a numerical analysis for the natural convection of fluid under both magnetic and gravitational fields. He studied the perturbed convection near a critical state, i.e., when Ra was about 6500 or less. His work is, however, limited to slip boundaries which are the case in the cosmos, and is further limited to a Prandtl number equal to unity, for which magnetic force is hardly effective. In this section, the natural convection of the fluid at Pr = 0.054 was numerically calculated under both magnetic and gravitational fields for a two-dimensional flow with rigid boundaries. The mathematical model for natural convection of the fluid consists of the balance equations for mass, energy and momentum. For the convection in a magnetic field, the external force term includes a Lorentz force as well as a buoyant force. According to textbooks on electromagnetics [8, 9], the following relationships should be solved to determine the magnetic field: rotE + dB/dt = 0, D = eE,

rotn-dD/dt

B = MmH,

= ],

divT>=pe,

divB = 0

J - p e u + a e ( l + uxB).

(1), (2), (3), (4) (5), (6), (7)

Equations (1) to (4) are the Maxwell equations for an electro-magnetic field, and Eqs. (5) and (6) relate the electric field to the magnetic field. Equation (7) is Ohm's law. However, these equations can be simplified under a number of assumptions. First, the displacement current dD/dt = 0, because the propagation of electromagnetic waves is not considered. The fluid is assumed to be electrically neutral and the convective term pe u is neglected. These equations are finally reduced to the following equation of induction for a magnetic field B [8]:

dB/dt = rot[u x B ) - rotfyae \ot B//^ }.

(8)

The momentum equation, including a Lorentz force in the magnetic field, is given as p Du/Dt = -VP + pg - B x rot\B/jum )+ /*V2u .

(9)

The energy and the mass balance equations are as follows: pCpDOIDt = AV20,

V•u =0 .

(10), (11)

17

Two-Dimensional Numerical Analyses for Natural Convection of Liquid Metal

These equations are simplified to two-dimensional ones, and a Boussinesq approximation is introduced by assuming that the physical properties other than the density in the buoyant term are constant. The two momentum equations are crossdifferentiated and subtracted from each other to give a vorticity equation. After dedimensionalization, the following equations are obtained:

^ 7 = V2r ' §-[B*§

™V2fi> V-B = 0(12),(13),(14),(15)

+B

= ?T

S)

^

= _ P r ^ + PrV^ + Ha^Ra-»prpr m L

r =

^ l _ ^ BX dY

Dx

dY

JL&_^

}*dX{dX

=

_vV,

U-^-, dY

dY

+B

JLi^JlA

y

dY^dX

dY J

.

(16), (17), (18), (19) V ' l '

V—&-. dX

The dimensionless variables are defined as follows: X = x/x0, Y = yfy0, U = u/u0, V=v/v0, r=t/t0, T = (0-6>0)/(6k - 6>c), Bx = bjbo, By = by/bo, xo=yo = [g/3(8h - 0c)/(av)]-1/3, u0 = vo = a/x0, dQ=(0h+6c)/2, t0 = xo2/a, Ha = (oelpv)mb0t, Prm = vja, Pr = via, L = £/x0 = Ra1/3, 3 Ra = g/3(6h-ec)£ l(av). In a primary state of computation, the magnetic field Bx and By could have been solved successfully, but they failed to satisfy another requirement, Eq. (15). This is because Bx and By were not solved to satisfy Eq. (15). Weiss [7] employed a vector potential, A , in the magnetic field, and we followed his approach. B = VXA,

= Pr m V 2 A

DA/DT

(20), (21)

The induction equations for a magnetic field, Eqs. (13) and (14), can be derived by taking derivatives of Eq. (21) by Y and X, respectively. For a two-dimensional field, Bz = 0 and d/dZ = 0 hold, and then we get the following equations to satisfy the Maxwell relationship automatically: Bx -

^

dY

By 5

V . B 3

^

dX

)

dX

+

^

dY

=^

-

dXdY

^

S

0

dYdX

.

(22), (23),

(24)

18

Magnetic Convection

The Z component of the magnetic vector potential is abbreviated as A, and the next equation to be solved is:

dr

dX

dY

m[dX2

(25)

dY2)-

The solution scheme is as follows: 1) Temperature from Eq. (12). 2) The vorticity and the stream function from Eqs. (16) and (17). 3) The velocity components from Eqs. (18) and (19). 4) The magnetic vector potential from Eq. (25). 5) Two components of the magnetic induction from Eqs. (22) and (23). The finite difference approximation and the computational scheme are the same as that reported by Ozoe et al. [11] and will not be repeated herein. At the beginning, the flow field is solved for a non-magnetic condition. This provides an initial condition for the computation of the magnetic field. The transient computation was continued for a convergence. For a fluid at a low Prandtl number, the thermal conductivity is large and the temperature profile becomes linear, which results in inaccurate values of the temperature gradients. When Pr = 0.054, the variable for temperature was changed from T to

Jn>*, ^ = ip/«, L = I/XO = R*V3,

W=wfu0,

27

r=t/t0,

T=(e-eo)/(eh-ec),

xo = [gP(8h-8cy(av)]-1/3, to=xo2/a, uo=a/xo, d0 = ( 6h + 6C )/2, xpe0 = ab0, Q= ZJ(coc0-2), Pr = via, Ha = [oJij\mb0t, Ra = gP(6h - 0c)£3/(av). Numerical computation for three-dimensional turbulent natural convection in a cubical enclosure for air is presented in ref. [12]. Following this scheme, the laminar natural convection is here computed for a low-Prandtl-number fluid. The computation supplied an initial convection state. The boundary conditions for this system are given as follows for all variables on all six walls of a cubical enclosure. J = 0 . 5 a t X = 0 , T = -0.5 at X = L, 3T/dY= Oat Y= 0 and L, dT/dZ = 0 at Z = 0 and L. U=V=W=Q=0 at Z = 0 a n d Z , F=0andZ,, Z=OandZ,. dWJdX= Wy=Wz = Q at X= 0 andL, Wx = dWy/dY= Vz = 0 at 7 = 0 and£, Wx=Wy = dWJdZ = 0 at Z = 0 andL. Qy=-bWldX, P2=dV/dX, at X = 0 a n d L QX = Q, Qy = Q, Qz=-dUldY, at 7 = 0 andL Qx = dWldY, £2x = -dV/dZ, Qy = dU/dZ, A = 0, at Z = 0andL.

dWJdX=0atX=0andL, L, dWJdY = 0 at Y = 0 andL, dWJdZ = 0 atZ = 0 andL. The boundary conditions for vector potential are given by Hirasaki and Heliums [13]. The vorticity boundary conditions are from non-slip velocity conditions on the walls. The electrical insulation on the walls gives the boundary conditions for scalar potential for an electric field. The cubical enclosure was divided into unequal grids, with 21 grids in the X direction, 11 in the Y direction and 15 in the Z direction. The computational scheme is the ADI method and is similar to that described in ref. [14]. The initial condition for this computation is a static state and is isothermal at an average temperature in all internal fluid regions. Computations were carried out for Pr = 0.054. However, computation is expected to be equally possible for smaller Prandtl number fluids. The convection at Ra = 107 and Pr = 0.054 converged smoothly. The initial condition is the convective state obtained for a non-magnetic field described above. Figure 4.2 shows the response curves of the Y component of a vector potential after a step addition of a magnetic field. Curves (l)-(3) are for the X, Y and Z directions of an external magnetic field, respectively. They are all at Ra = 107, Ha =

28

Magnetic Convection

100 and Pr = 0.054. Curve (2) is not at a completely converged state because extremely small time steps were required to achieve numerical stability. Table 4.1 summarizes the computed results. The average Nusselt numbers are only slightly different from each other under this weak magnetic field. However, the tendency is apparent. The magnetic field in the X direction gave the smallest value of the average Nusselt number on the vertical hot wall. The magnetic field in the Y direction gave the largest. The central value of the Y component of the vector potential also gave the same order of decrease in their magnitude. This indicates that the heat transfer is convection-dominated under these conditions even for the fluid at this low Prandtl number of 0.054. It is commonly accepted that the external magnetic field suppresses the flow most effectively when the magnetic field is imposed perpendicular to the direction of flow of an electroconducting fluid. When the external magnetic field is in the Y direction, the magnetic field is perpendicular to both the vertical boundary layer flow along the hot or cold plates and the horizontal flow along the ceiling and the bottom adiabatic plates, namely, major circulating convection. The authors expected to see the most significant suppression for this external magnetic field in the Y direction. The results were just the opposite. The magnitude of the Lorentz force term was therefore computed term by term. In the momentum equation in the Z direction (and also in the Y component of the vorticity equation), the terms QWJdY-W represent the Lorentz T i- I 0 0 1

500 '

1000 '

(t)X Direction of Magnetic Field (2) Y Direction of Magnetic Field (3)2 Direction of Magnetic Field

~

~

S -10-

1

Table 4.1 [1] Summary of the computed results for Pr = 0.054. ^

'

Ra

107

-

I ^

./C~~~ -20'

77\ 1

(1)

" i

Ha

107 100 X

9.655 -13.682

100 Y 10.445

-15.523

107 100 Z 10.024

-14.805

10

6

0 -

10 6 ~

100

10

X

° ,nn

I

5.7371

-8.876

4.4577

-7.792

X

20

1fl 6

I

^(center)

10.524 -15.708

10

Figure 4.2 [1,24] Transient responses of the central value of the K component of the vector potential at Ra = 107, Pr = 0.054 and Ha = 100: (1) external magnetic field in the X direction; (2) external magnetic field in the Y direction; (3) external magnetic field in the Z direction.

Nu

0 -

7

I

B

Y

I

2

-9168 ~4-670 o ~>S(\a i n n o I I

Three-Dimensional Natural Convection of Liquid Metal in a Cubical

29

force when the external magnetic field is in the X direction only. The terms -d WJdX-W become the Lorentz force when the external magnetic field is in the Y direction only. The magnitude of these terms is shown in Fig. 4.3. When the magnetic field in the X direction is given, the dWJdYX&tm is small but the -Wterm is large along the vertical walls, and the resulting magnitude of the Lorentz force is large along the hot and cold vertical walls. When the magnetic field acts in the Y direction, both the -d WJdX term and the —W term have a large magnitude along the vertical heated boundaries but with opposite signs, so that they cancel each other out to produce a Lorentz force with a small magnitude in the momentum equation in the Z direction. This appears to be why the magnetic field in the Y direction had an unexpectedly small suppressive effect. Table 4.1 also includes the converged solutions for various Hartmann numbers under an external magnetic field in the X direction for Ra = 106 and Pr = 0.054. The average Nusselt number decreases greatly when the Hartmann number is raised stepwise, and a conduction-dominated state was attained at Ha = 500, though not shown. The isotherms at Ra = 106 and Pr = 0.054 are shown in Fig. 4.4. The isotherms at the cross section of Y - 0.5 represent the effect of the magnetic

(a)

(d)

(b)

(e)

(c)

(f)

Figure 4.3 [1] Graphical presentation of the magnitude of the Lorentz force at Y = 0.5 for two cases of an external magnetic field in the X direction only or in the Y direction only: (a)-(c) are for the magnetic field in theXdirection; (d)-(f) are for the magnetic field in the Ydirection, (a) dWJBY-W; (b) dVJdY;

(c) -W; (d) -dVJdX-W; (e) -dVJdX; (f) -W.

30

Magnetic Convection

(a)

(b)

(c)

(d) 6

Figure 4.4 [1] Computed isotherms in the Z-directional magnetic field at Ra = 10 and Pr = 0.054: (a), (b) are for Ha = 0 ; (c), (d) are for Ha = 300. (a) and (c) are at 7 = 0.5 ; (b) and (d) are at Z = 0.5.

field in the X direction. The thermal boundary layer thickness spreads into the core region as seen at the cross section at Z = 0.5. In summary, the ascending and descending boundary layer type flows along the vertical heated and cooled walls were mostly suppressed by the horizontal external magnetic field perpendicular to the heated vertical wall (X direction). However, the horizontal magnetic field parallel to the vertical heated wall (Y direction) was least effective in suppressing the circulation flow. The effect of the vertical external magnetic field (Z direction) was intermediate between these two extremes.

4.2 Experimental Results for Molten Gallium under an External Magnetic Field in the X, Y or Z Direction [15] The experimental schematic is shown in Fig. 4.5. Experimental fluid was poured into a cubic enclosure (3), which was placed between two polar faces of an electromagnet (2). One vertical wall was heated uniformly through a constant power regulator and a watt meter. The opposing vertical wall was cooled with water from a constant temperature bath. Temperature differences between the hot and cold walls were measured by use of thermocouples and a strip-chart recorder. The electromagnet was placed in a small room of 900 x 800 x 1235 mm3 maintained at a constant temperature approximately equal to the average of the hot and cold wall temperatures. A thermocouple hole was drilled through the copper plate forming the cooled side wall up to the geometrical center through which one 0.32 mm0 and three 0.05 mm0 copper-constantan thermocouples were inserted. Other junctions of thermocouples were fixed on the liquidside surface of the heated copper plate with instant glue (Aron Alpher) at the center, at 10 mm up and at 10 mm below midheight as shown in Fig. 4.6. The copper plate was coated with Teflon spray coating to

31

Three-Dimensional Natural Convection of Liquid Metal in a Cubical

avoid direct contact with molten gallium. If this was not done, Ga extensively dissolved the copper plate, whose surface became rough. The copper walls are therefore electrically adiabatic. The heated and cooled walls were clamped firmly with bolts. This experimental enclosure was clamped into another Plexiglas chamber that was fixed between the poles of the electromagnet. This enabled the experimental enclosure to be placed perpendicular or parallel to the pole faces of the electromagnet. A small laboratory jack was employed to position the enclosure centrally between the magnetic pole faces. Inside this chamber, fiberglass thermal insulation was firmly packed to minimize the influence of the variation in the room temperature where the experiment was conducted. The uniformity of the magnetic field in an experimental space was confirmed by the reports supplied by the manufacturer of the electromagnet. The actual internal length of the enclosure is 15 mm from the center, and uniform strength of the magnetic field can be expected for a gap of 100 mm between the two polar end planes employed in this experiment. Thermocouple outputs were measured with a strip-chart recorder (Yokogawa 3056-31). Other temperatures were monitored and recorded on a data logger (Takeda TR2731). These outputs were calibrated to temperature via a curve prepared before the experiment. The electromagnet was made by Electromagnet Inc.

1 to D.C.power source

E| I in constant ""i temperature bath X T " T ~ n

A8 *"

i

I E f i7

| [T

}

^

If

I

> 3

Y

to D.C,power source]

I

from constant [temperature bath

Figure 4.5 [15] Schematics of the experimental setup: (1) Strip-chart recorder, (2) Electromagnet, (3) Convection chamber.

E

~7

!/

io I

I 1 0 ,-!

n'

j

/>

/

li

M

i

U

A6av

^

H

Figure 4.6 [15] Temperature differences between the hot and cold walls measured at three points. Integrated average temperature difference A0av was employed as a reference conductive rate to get an average Nusselt number.

32

Magnetic Convection

(CMD-10). Its one end pole can be moved between 0 and 100 mm from the other end plate. The strength of the magnetic field can be varied up to 0.4 Tesla for pole faces 100 mm apart. The strength of a magnetic field was monitored with a gauss meter whose sensor was attached to the pole face during the experiment. Liquid gallium was selected as the experimental fluid because its melting temperature is 302.93 K and it is easy to handle. Mercury is more common but is documented to be toxic in both liquid and vapor form. Although Ga is not known to have such negative effects, it was handled carefully. The physical properties of Ga are summarized in Appendix A. Molten Ga was found to be oxidized easily and quickly when exposed to air, and then its free surface becomes covered by a thin but rather viscous film layer. To avoid oxidation, the experimental convection enclosure was air-sealed; oxygen-free air was introduced inside and then Ga was injected with a needle-less plastic syringe. Displaced gas was evacuated carefully through a 1 mm0 hole at a corner of the convection chamber, which was then sealed. The experimental room was thermally controlled at the average temperature between the hot and cold walls of the convection enclosure to minimize the thermal loss. However, we can still not deem the heat loss as negligible. An accurate estimate of the heat loss is the most difficult part of the experiment in natural convection because total heating amount is very small. The estimation procedure followed was originally invented by Ozoe and Churchill [16]. First, an experiment was carried out with air (known thermal conductivity) heated from above and cooled from below. The temperature differences between the hot and cold plates were measured for various rates of electrical heating Q,olai. In the working graph, the computed rate of heat transfer by pure conduction in air and the measured heat input are both plotted versus the temperature on the hot wall. The difference between them is considered to be the heat loss, Qioss, as follows: Qioss = Qtotai -Akair{dh - ec)iL

(6)(6)

This heat loss was approximated by a linear equation through a least-square method as eq. (7). This heat loss to the surroundings is expected to be the same even when convection occurs inside the experimental enclosure. Net heat input to the experimental fluid is given by eq. (8). Qloss = - 8 . 6 8 5 + 0.028310 A ,

Qne, = Q,otal - Qloss

(7), (8)

Three-Dimensional Natural Convection of Liquid Metal in a Cubical

33

The net heat flux can be estimated by the above method for the electrically heated system. The rate of heat transfer is represented by a dimensionless Nusselt number, which is defined as a ratio: the total heat flux divided by the conduction heat flux. The system heated from a side wall by an electric resistance dissipates a uniform heat flux, and the vertical heated wall has a temperature distribution. The opposing cold wall is kept at a uniform temperature for this system. The temperature differences between the hot and cold walls were measured at three points for the 30 mm high hot wall as described above. The reference conduction heat flux can be given as follows under such temperature differences. The hot wall temperature distribution was approximated by a second-order equation (9). The coefficients a, b and c are given by the temperatures at three locations. The average conductive heat flux is given by eq. (10). 6h = az2 + bz+ c

(9)

(10) ^^QnetlQcond

(11)

The average Nusselt numbers under various heating rates and magnetic forces were computed from the net heat flux divided by this equivalent conduction heat flux. The average temperature difference is given as shown in Fig. 4.6, and the average Nusselt number is given by eq. (11) above. The following Nusselt numbers were computed by this empirical procedure. The cubic enclosure was heated uniformly from one vertical wall and cooled isothermally from an opposing wall. The other four walls were thermally insulated. All boundaries were electrically insulated. The natural convection heat transfer rates were measured without a magnetic field before the magnetic experiment. The average Nusselt numbers are plotted versus Gr*Pr2 in Fig. 4.7. This abscissa has previously been employed for the liquid metal heat transfer. In the same graph, three-dimensional numerical analyses reported by Viskanta et al. [17] for three different Prandtl numbers and by Ozoe and Okada [1] are also plotted for comparison. The correlation equation derived by Churchill [18] for a rectangular enclosure is indicated by a rigid line for comparison. The other experimental results for free convection found by Takahashi et al. [8] for Pr = 0.016 to 0.022 and by Julian and Akins [19] for Pr = 0.022 are also plotted, although they are for a boundary layer type flow along a single plate with a uniform heat flux. The average Nusselt numbers computed by Viskanta et al. are in excellent

34

Magnetic Convection

agreement with the present experimental rate of natural convection heat transfer in a cube for almost the same Prandtl number in the range Gr*Pr2 = 102 to 104. For a much higher range of the abscissa, the data from various sources show poorer agreement for the modified Grashof number 107 to 1010. The dotted and hatched lines are for boundary layer type flow and give higher average Nusselt numbers. The heat transfer rates in the Y- or Z-directional magnetic field at various modified Grashof numbers are plotted in Fig. 4.8. The data group for the Xdirectional magnetic field almost coincides with that for the Z-directional one and therefore is omitted. The black circles show the rates computed by Ozoe and Okada [1] in the three-dimensional numerical analyses, and they agreed well with the experimental result. The fact that magnetic suppression by the y-directional field was smallest was proved by the much larger rates, represented by white triangles. The black triangles show the rates obtained by three-dimensional computations, which also agree well with the experimental ones. Churchill and Usagi [20] proposed to derive a compact empirical equation for such data as seen herein, using two limiting cases and selecting a single parameter n tofitthe data. Let (Nu B -l)/(Nu 0 -l) be represented by | and Ha/Gr1/3 by rj. The present system has as limiting characteristics § —» 1 as rj —» 0, and § —» 0 as rj —» o°, Then we can presume the following form to satisfy these conditions: {1/(1-1)}" = l" + { ( ^ / / 7 ) T .

20|

' • i|

•'

r-i-rpm,

u|

o pr=0.01 Viskanta, Calc. A Pr=0.02 Viskanta, Calc. 10'

I^IIM!

,--''...-'A

'

.,-'..,•-

°^

> ^

bn^^ ^^fi^

il . • • 102

—

Gr'Pr

2

\

10 5

10 6

I....I 107

Figure 4.7 [15] Comparison of the average Nusselt number for liquid metal without a magnetic field, Viskanta is from [17]; Churchill is from [18]; Takahashi et al. is from [8]; Julian and Akins is from [19]; Ozoe and Okada is from [1].

.

X *}Mag * \* / \

^r\

• Z-Mag

^ ,

' ' ' | " " |

" A ' A\ \ \ *

'

" 3

V

.-•'''..---'kJ^ •••'

1 . 0 | ' ' ' Wl«^!B'"l

,-''h-''''° -'"o-- - "

• Pr=0.054 Ozoe, Calc. , - - ' ..--'' _s»*- V i Pr=0.025 Present Exp.,-' .•-•" ^ ^ ' O

z4

(12)

*

\ \ .

.\

ot " *\ ^n •• ^RL \

O.ol i i , I,ml i ,.hnil^fc—liml "T-^-IMHI 1 1 10 10 '0 ° 1/3

Ha/Gr

Figure 4.8 [15, 24] Comparison of the present experimental results with the theoretical results of Ozoe and Okada [1]; lines are those correlated by Churchill and Usagi plot [20], eqs. (13)-(15).

Three-Dimensional Natural Convection of Liquid Metal in a Cubical

35

When (Nu0 - NuB)/(Nu0 - 1) = 1-g is plotted versus Ha/Gr1/3 = rj on logarithmic coordinates, the gradient gives the unknown index y and any data point gives the cross-sectional point r]A. The exponent n is determined to fit the data well on the graph. This type of equation includes two extreme cases of variables of rj at 0 and infinity as well as the intermediate range in such a simple form. In this way, the unknown parameters were determined for the X-directional data group to give eq. (13), for the 7-directional one to give eq. (14), and for the Z-directional one to give eq. (15). These equations are indicated by rigid lines in the graph that fit the data groups excellently. (NuB - l)/(Nu0 - 1) = 1 - [1 + (0.57Gr1/3/Ha)319]-1/176

(13)

(NuB - l)/(Nu0 - 1) = 1 - [1 + (4.19Gr1/3/Ha)207]-1/L45

(14)

(NuB - l)/(Nu0 - 1) = 1 - [1 + (0.52Gr1/3/Ha)2'72r1/L44

(15)

4.3 Summary The effect of an external magnetic field was studied for the natural convection of liquid Ga in a cubic box heated from one vertical wall and cooled from an opposing wall. The rate of heat transfer was measured and correlated as a function of Ha/Gr-^ for the direction of a magnetic field acting either perpendicular to the vertical hot wall (X direction) or parallel and horizontal to the hot wall (Y direction) or vertical (Z direction). The X- or Z-directional magnetic field was almost 10 times as effective in suppressing the rate of heat transfer as that in the Y direction. The strong suppressive effects in the X- and Z-directional magnetic fields proved the previous prediction from three-dimensional numerical analyses by Ozoe and Okada. The rates of heat transfer were correlated with Churchill and Usagi correlation.

Appendix A [15] Physical properties of gallium (a) Density. K. E. Spells [21] reported measured values, which were correlated by a least-square approximation as follows: for 303.65 K s f l s 375.15 K,

36

Magnetic Convection

p = ao + ai6 [kg/m3], a o = 633O [kg/m3], ax = -0.7717 [kg/(m3K)]. (Al) (b) Viscosity. Measured values by K. E. Spells [21] are correlated by a least-square approximation as follows: for 303.65 K s ^ 323.15 K, H = b0 + bx6 +b20l [kg/(m-s)], b0 = 1.207 x 10~2 [kg/(m-s)] b] = -5.754 x 10"5 [kg/(m-s-K)], b2 = 7.981 x 10~8 [kg/(m-s-K2)]. (A2) (c) Specific Heat. According to the Metals Handbook II [34], Cp = 397.6 [J/(kg-K)] from 285.65 K to 473.15 K.

(A3)

(d) Thermal conductivity. According to the Metals Handbook II [22], A = 28.68,34.04, or 38.31 [W/(m-K)] at 350.15 K.

(A4)

Since these values differ substantially from one another, the thermal conductivity of Ga was measured directly in our convection enclosure. The measurement of thermal conductivity should be carried out for a shallow fluid layer. However, the thermal conductivity of liquid Ga is about 1300 times larger than that of air and 42 times that of Plexiglas, and we presumed the heat flux could be measured accurately by the method described in the text. The thermal conductivity of Ga can be described as follows: kGl = -7.448 +0.12566* [W/(m-K)] for 306.15 K«s 6> P

334

120

0.848

101.8

24.7

1.247

238

156

1.005

156.7

25.9

1.638

138

191

1.319

252

26.4

B

2.535

58

126

2.042

257

28.0

B

3.315

34

116

2.670

310

27.5

Gr/Re

2

Thermocouple v graph No.

Picture .T No.

Fig. 7.2

Fig. 7.3

Flow m o d e A : a cold p l u m e from t h e edge of the rotating cylinder merges with a cold p l u m e descending periodically from the bottom center of a rotating rod; Flow m o d e B : a cold p l u m e descends continuously from the edge of the rotating cylinder and hot p l u m e ascends toward the bottom center of the rotating cylinder periodically.

b)

r av : time-averaged temperature of thermocouple output 30 mm below the center of the rotating cylinder.

Flow Visualization of Oscillatory Czochralski Convection

65

descends from the periphery and the hot plume periodically ascends toward the bottom center of the rotating cold cylinder. These characteristics are expected to exist even for lower values of the Prandtl number of the fluid. In Czochralski crystal pulling, the height of the fluid level decreases gradually and the Grashof number decreases quite rapidly as the third power of the liquid level. When Gr/Re2 decreases below this critical point (Gr/Re2)c, then the flow direction of bulk liquid under the rotating cylinder changes from inward to outward, and various serious changes may occur in the crystal cylinder. In a practical operation, pulling up should be terminated just before this critical point is reached. These characteristics appear to be well visualized and quantitatively represented in terms of the periodic time of temperature fluctuation.

7.3 Conclusions The flow of Czochralski convection was visualized with a liquid crystal technique using silicone oil at room temperature. Periodic change in the convection mode was found to exist for wide ranges of the dimensionless parameter Gr/Re2, which agrees with the report by Munakata and Tanasawa [14]. When natural convection is dominant, i.e., Gr/Re2 is large, the cold plume from the periphery of the rotating cold cylinder (model for a crystal) was found to merge toward the central cold plume descending from the bottom center of the cold cylinder. When inertial convection becomes dominant, i.e., Gr/Re2 becomes smaller »i»ij*(»>>yf*ii'
, Z) = (0.615//, 0, Z) with Z = 0.05//, 0.5// and 0.95//. Almost regular oscillation resulted from the rotation of the crystal rod. The amplitude of the temperature oscillation is much larger than that of Fig. 11.2. Figure 11.5 (a)-(h) shows isothermal lines near the melt surface (Z = 0.05//) for eight consecutive times in Fig. 11.4. The temperature distribution rotates in a circumferential direction with the rotation of the crystal rod, in agreement with one cycle of Fig. 11.4.

(a)

5 o.46O Z=0.05H f j E 1=0 95H h • 0 J . 0 L ~ . . . . : o 2000 4000

I • 6000

A '\ I 8000

r 1 10000

Figure 11.2 [2, 20] Transient responses of the local temperature at Z = 0.Q5H, 0.5H and 0.95H. Temperature at (R, S ide view of isotherms along the line A-B. Reproduced from Modelling of Transport Phenomena in Crystal Growth, Ed. J. S. Szmyd and K. Suzuki, WIT Press 2000 ISBN 1-85312735-3

Effect of a Transverse Magnetic Field on the Melt Convection

105

The dimensional equivalence of the above result is as follows. When the melt height h is 4 cm and the rotation rate of the crystal rod is 20 rpm, the oscillation period is about 860 s and the amplitude of temperature oscillation at Z = 0.05// is about 0.75 degree Celsius, in contrast to the maximum temperature difference of 11.3°C. During this period, the crystal rod rotates about 287 times.

o, 0.400 - • o

•

H

• . O.3751

0

a

Z==0.05H

h

:

•••••••—•• /^—\j,jn ,

z=o,95H i

i

,

i

10

.

.

.

.•

i

20

i

i

i

,

i

30

i

i

i

i

i

L

40x10

i

.

.I

Time [-J Figure 11.4 [2, 20] Transient responses of the local temperature with rotation of a crystal rod at (R, }

3

I I

500 Gauss,,

r

.

J ^-0.0375 m I

•

I Q|

ft

V ^

>»^T^>v

/

/ f SY|A /

\

IJ_

j

NX >w \

\CRYJ3TAL//

\ \

\\ 1 1 .

SY

'

Jj

CRUClBLt^J^^y^cu * •*

(b)

SX'

Figure 12.1 [1, 2, 26] Schematic diagram of silicon melt under a cusp-shaped magnetic field, (a) Side view, (b) Plane view, x Point A (/• = 0.8tth,z = 0.733/;, #= tt), x Point B ( r = 0.5A, z = 0.98/J, $= rt), x Point O ( r = Oh, z = 0.98/)). Reproduced from Modelling of Transport Phenomena in Crystal Growth, Ed. J. S. Szmyd and K. Suzuki, WIT Press, 2000, ISBN 1-85312735-3

116

Magnetic Convection

equation with the assumption of axial symmetry, where r is a vector from a point to either of the coils which are equally spaced above and below the horizontal plane of the crystal-melt interface and the free surface, and which carry equal current / in opposite directions.

jdB = ptm/4nfidsxr/r3).

(10)

The dimensions of the growth system with two coils are shown in Fig. 12.1. At the center of the bottom wall of the crucible, the vertical component of the magnetic density is B o = 1000 gauss, which was defined as the reference magnetic induction, while the horizontal component of the magnetic induction at the crucible-free surface intersection is 500 gauss. Axial symmetry about the crucible axis was not presumed, so that fully three-dimensional results can be expected. A staggered-grid system was employed for the mass balance to be satisfied in each grid cell. Timeaveraged Nusselt numbers at a converged state gave 1.06, 1.41 and 1.49 for grid numbers of 303, 403 and 503. Thus, 503 was employed in this work. A non-uniform grid was employed in the r—z plane, while a uniform grid was set along the circumferential direction. The QUICK scheme [23] and SIMPLE algorithm [24] were employed. Electric potential and current density corrections were solved by the same method as pressure and velocity correction of the SIMPLE algorithm. Difficulty in solving for velocity components at a radial center was circumvented by the technique of Ozoe and Toh [25], which gave a fully three-dimensional cylindrical coordinate system. Table 12.1 [22] Dimensional and dimensionless parameters with properties of silicon melt.

h = 0.0375 [m], d/2 = 0.0375 [m], cvcr=\0 [rpm], &cu = - 3 [rpm], $, = 1715 [K], 6$,= 1705 [K], 6>m=1683 [K], a=2.13xlO" 5 [/n 2 /$], p= 2530 [kg/m3], cre=l.29xl06[n-mY\ Cp =1000 [J/(kg-K)], ju =7x10"" [kgf(m-s)], 4 v=2.11x\0'1[m2/s\, 9=1.4xl0" [^T 1 ], A = 54[W/(m-K)], Z) = 5xlO"8 [m2/s]. y Effective heat transfer coefficient at a free surface, W/(m2-K): 4.0 Effective oxygen mass transfer coefficient at a free surface, m/s: 2.0 Ha =161, Pr = 0.013, Ra = 3.92xlO 5 , Recu = -1596, Re cr =1329, Sc = 5.5. r0 = 0.0005124 [m], u0 = 0.04157 [m/s], t0 = 0.0123 [s]. Note: Computed results of this chapter are applicable only for the system of above parameters, although convection equations are written in dimensionless variables.

Effect of a Cusp-Shaped Magnetic Field on the Melt Convection

117

uimciisiumetis umc [-j

Figure 12.2 (a) [2, 26] Transient responses of volume-averaged velocity components and temperature at point A. (a) Radial velocity, (b) Circumferential velocity, (c) Vertical velocity, (d) Temperature, (e) Period from r = 47480 to r= 48455, (f) Period from t = 81463 to r = 82602.

Reproduced from Modelling of Trasport Phenomena in Crystal Growth, Ed. J. 735-3

Figure 12.2 (b) [1] Oxygen concentration as a function of dimensionless time, (a) Oxygen concentration at point A (r = 0.813h, z = 0.733h, = rr), (b) Average oxygen concentration,, (c) Oxygen concentration at point O (r = Oh, z = 0.98h), (d) Oxygen concentration at point B (r = 0.5h, z = 0.98h, = ?i), (e) Period from r = 47480 to r = 48846, (f) Period from r = 81463 to r = 82602.

118

Magnetic Convection

12.3 Results and Discussion Figure 12.2 (a) shows transient responses of volume-averaged velocity components and temperature at point A (r = 0.813/?, z = 0.733/?) shown in Fig. 12.1. Almost standing oscillation was obtained at Ha = 161. However, at Ha = 0, velocity components and temperature lose their periodicity after x= 50894. Figure 12.2 (b) shows the transient responses of the oxygen concentration averaged over the entire domain and the oxygen concentrations at specific points under a cusp-shaped magnetic field (Ha = 161). The average concentration converges to a certain value, while local concentration shows periodic oscillation as

(ay ^s^kkijij

cu

,

,

,^^^7/^k

(b)^ \Jatz = h

UatSX-SX'

Tatz = /z

TatSX-SX'

Figure 12.3 [2, 26] Instantaneous contours of velocity U at z = h, velocity U in the vertical crosssection SX-SX', T at z = h and T in a vertical cross-section SX-SX' and Ha = 0. (a) r= 81463, (b) t= 81789. Reproduced from Modelling of Transport Phenomena in Crystal Growth, Ed. J. S. Szmyd and K. Suzuki, WIT Press, 2000, ISBN 1-85312735-3

Top view

Side view

Figure 12.4 [1] Contours of oxygen at the top of silicon melt (z = h) and in the vertical cross-section SX-SX' at Ha = 0 and r = 81463.

119

Effect of a Cusp-Shaped Magnetic Field on the Melt Convection

a function of time at Ha = 161. Oxygen concentration at point B (r = 0.5/z, z = 0.98//, = Ti) is lower than that at point A (r = 0.813/J, z = 0.733ft, = 7i), since point B is near the free surface. Oxygen concentrations at points B and O (r = 0, z = 0.98ft) indicate that the radial distribution of oxygen in a grown crystal rod is inhomogeneous. The local concentrations at points A and B show periodic oscillation with relatively high frequency. The average concentration at Ha = 0 increases immediately after the magnetic field is removed. The concentrations at points A, B and O also increase and lose their periodicity. Therefore, a cusp-shaped magnetic field decreases the average oxygen concentration in the melt. Figure 12.3 shows a series of instantaneous contours of velocity U at Ha = 0 and z - h, velocity U in a vertical cross-section SX-SX', T at z = h and T in the vertical cross-section SX-SX'. The temperature shows a non-periodic distribution as a function of time and severely unsteady phenomena. Figure 12.4 shows oxygen distributions without a magnetic field (Ha = 0) at r = 81463 of (f) of Fig. 12.2(b). They are the top view at z = h and the side view in the vertical plane of SX-SX'. Oxygen contours are non-periodic and unsteadiness is very severe in comparison with those of Ha = 161 shown below, although series graphs are omitted herein. Figure 12.5 shows the corresponding features at Ha = 161, namely, instantaneous contours of velocity vectors U from a rotating coordinate, temperature contour T, current density vectors J and Lorentz force vectors F in a horizontal cross-section at z = 0.75ft at the instant %= 47480. Circumferential velocity is much higher near the free surface than at lower levels of a crucible. At the same time, velocity near the bottom wall is a relatively solid rotation with the crucible in comparison with that near the free surface. Near the free surface, both velocity and

U u

T

JJ

F

Figure 12.5 [2, 26] Instantaneous contours of velocity U from a rotating coordinate, temperature T, current density J and Lorentz force F at each horizontal plane at z = 0.75h, r = 47480 and Ha = 161. Reproduced from Modelling of Transport Phenomena in Crystal Growth, Ed. J. S. Szmyd and K. Suzuki, WIT Press, 2000, ISBN 1-85312735-3

120

Magnetic Convection

temperature show an elliptic pattern. The circumferential component of current density is relatively large near the side wall and the bottom wall of the crucible. A large current J in a circumferential direction along the crucible wall and the radial magnetic field B will yield a strong Lorentz force F = JxB in a vertical direction near the upper crucible wall. The circumferential component of a Lorentz force will also be large near the bottom wall of a crucible, since the vector product of the vertical magnetic field B and the centripetal current density J is in a circumferential direction near the bottom wall of the crucible. Figure 12.6 shows -V%, UxB, J and F distributions in the vertical crosssection SX-SX' at the instant T= 47480. Both the gradient of the electric potential !fand the vector product UxB contribute to the current density J and the Lorentz force F. Near the bottom wall, current density J in a circumferential direction and the vertical magnetic field B produce a strong centripetal Lorentz force as seen in Fig. 12.6 (d). Figure 12.7 shows a series of instantaneous contours of temperature at z = h. The elliptic temperature distribution T at the top of the melt rotates about n radian (180 degree) in the same direction as the crucible, while the crucible rotates 1.68;r

(cL

(b)

(a)

mmmm\[\[\\\

(d)

Figure 12.6 [2, 26] Instantaneous distributions at r = 47480 and Ha = 161. (a) -V% in a vertical cross-section SX-SX', (b) UxB in a vertical cross-section SX-SX', (c) J = (- VWe + UxB) in SX-SX', (multiplied ten times over those in (a) and (b)), (d) Lorentz force F = JxB at SX-SX'. Reproduced from Modelling of Transport Phenomena in Crystal Growth, Ed. J. S. Szmyd and K. Suzuki, WIT Press, 2000, ISBN 1-85312735-3

(a)

(b)

(c)

(d)

Figure 12.7 [2, 26] Instantaneous contours of T at z = h and Ha = 161. (a) r = 47480, (b) r = 47805, (c) r = 48130, (d) r = 48455. Reproduced from Modelling of Transport Phenomena in Crystal Growth, Ed. J. S. Szmyd and K. Suzuki, WIT Press, 2000, ISBN 1-85312735-3

121

Effect of a Cusp-Shaped Magnetic Field on the Melt Convection

radian (302 degree). The approximately two peaks of temperature oscillation at Ha = 161 in Fig. 12.2 (a) appear to correspond to one cycle of rotation of this elliptic temperature contour. The above computed cases are for a crucible of 7.5 cm in diameter with a maximum temperature difference in the melt of 42K, a reference magnetic field of 1000 Gauss, and rotation speeds of 10 rpm for the crystal and - 3 rpm for the crucible. This gave a regular oscillation in the velocity and temperature with a period 16.8 s, which corresponds to a characteristic dimensionless oscillation frequency of 0.000721 at one point of the melt. Figure 12.8 shows a series of instantaneous oxygen concentration profiles at the top of the melt (z = h) in the top row and those in a vertical cross-section in the second row, at Ha =161 and at four equally spaced times during the interval denoted by (e) in Fig. 12.2 (b). The elliptic oxygen distribution rotates about TT radian in the direction of crucible rotation, while the crucible rotates 1.68 /r radian. In the vertical cross-section denoted by SX-SX', during the same time interval, oxygen spouts emerge from the side wall of the crucible at r = 47480, as indicated by arrows. These spouts of oxygen develop until r = 47805, and then disappear at r = 47967 (not shown). These results suggest that oxygen distribution at the crystal-melt interface is modified by a cusp-shaped magnetic field. Figure 12.9 shows instantaneous radial distributions of oxygen concentration and the time-averaged distribution (bold line) at the crystal-melt interface for Ha = 0. The instantaneous oxygen concentration at

\

| CRYSTAL |

(a)

J

I

J

(b)

,

I ,,^,_,, j.

(c)

I

1

(d)

Figure 12.8 [1] Instantaneous oxygen concentration at the top of silicon melt (z = h) and in the vertical cross section SX-SX'at Ha =161. (a) r = 47480, (b) r = 47805, (c) r = 48130, (d) r = 48455.

122

Magnetic Convection

the crystal-melt interface was averaged in a circumferential direction. The oxygen concentration changes dramatically as a function of time. The time-averaged value is in the order of 1.2-1.5xlO18 atoms/cm3. Figure 12.10 shows the corresponding results for Ha = 161. Under the present cusp-shaped magnetic field, azimuthally averaged oxygen concentration profiles were almost the same for three consecutive times (a) to (c). The time-averaged bold curve is in the order of 0.4-0.7x1018 atoms/cm3. The local maxima of the oxygen concentration near the periphery of the crystal rod arise from the transient outgoing flow as seen in Fig. 12.8. These numerical results suggest that a cusp-shaped magnetic field suppresses the random fluctuation of the oxygen concentration at the crystal-melt interface.

12.4 Conclusion Transient three-dimensional numerical computations were successfully carried out for the silicon melt in a Czochralski configuration crucible under a cusp-shaped magnetic field. Numerical conditions are Ra = 3.92 X 105, Pr = 0.013, Re cu = -1596, Recr= 1329 and Ha = 0 and Ha = 161. The elliptic isotherms and ordered secondary vortices were found to rotate in the same circumferential direction as the crucible rotation. When the magnetic field was removed, the convection became very 2

|

1

|l.5 -

So, -

1

/

r 0), while in the left half they are generated in the opposite direction. Diagram (d) shows instantaneous Lorentz force vectors F = JxB in a stationary coordinate at T = 130 for this case. Lorentz forces are generated only near the container wall, since (J) z is large near the wall as seen in diagram (c), and they occur in the rotational direction of the magnetic field. The melt flow in the circumferential direction as shown in Fig. 13.2 (b) is produced by these forces.

(a)

(b)

Figure 13.2 [1] (a) Vectors of a rotating magnetic field B with Remag = 1183 in a stationary coordinate at T= 130[—]. (b) Converged velocity vectors U under a rotating magnetic field for Pr = 0.013, Re^i = 0, RelmB= 1183 and Ha = 21.5 (type B). y

"'" y

y

"

\

"

'

(a)

"'

"

"k "

'"(b) (b)

^

"' " °k " ' (c) "" " °k " ^(d) '(d) Figure 13.3 [1] Converged results, i.e., (a) contour lines of (UxB)z, (b) contour lines of (E)z, (c) contour lines of (J)z and (d) instantaneous Lorentz force vectors F in a stationary coordinate at T= 130. (Type B, rotating magnetic field at Pr = 0.013, Recyi = 0 and Remag= 1183, Ha = 21.5.)

130

Magnetic Convection

13.5 Conclusions Two-dimensional numerical calculations were carried out to study convectional mode and electrical field in silicon melts in a cylindrical container which is sufficiently long in the axial direction, under no magnetic field and rotating magnetic fields. Under no magnetic field, the melt rotated like a rigid body (type A). A Lorentz force in a circumferential direction was not generated when the rotational rate of the magnetic field and that of the cylindrical container were the same (type E). Except in this case, melt in the cylindrical container rotated in the rotational direction of the magnetic field, driven by the Lorentz force generated in a circumferential direction. Application of a rotating magnetic field to the Czochralski crystal growing system may become an important technology for controlling melt convection, oxygen transfer, dopant concentration, etc. without crucible and crystal rotations.

References 1. Akamatsu M. et al, Numerical Heat Transfer A, 42 (2002), 33-54. 2. lino E., Takano K., Kimura M. and Yamagishi H., Mater. Sci. Eng. B, SolidState Mater. Adv. Technol, (Switzerland), B36-1-3 (1996), 142-145. 3. Walker J.S. and Williams M.G., J. Crystal Growth, 137-1-2 (1994), 32-36. 4. Watanabe M., Eguchi M. and Hibiya T., Jpn. J. Appl. Phys., 2, 38-1A-B (1999). 5. Dold P. and Benz K.W., Prog. Crystal Growth Charact. Mater., 38-1-4 (1999), 7-38. 6. Ma N. et al, J. Crystal Growth, 230-1-2 (2001), 118-124. 7. Ozoe H. and Toh K., Numerical Heat Transfer, Part B, 33-3 (1998), 355-365. 8. Nakamura S. and Hibiya T., Int. J. Thermophys., 13 (1992), 1061-1084. 9. Kakimoto K., Eguchi M. and Ozoe H., J. Crystal Growth, 180 (1997), 442-449.

CHAPTER 14

Continuous Steel-Casting Systems with Various Magnetic Fields [10]

14.1 Introduction The continuous steel-casting process has recently been employed successfully in the steel industry. In this process, however, the convection of molten steel in the mold is considered to affect the quality of the solidified product. The Prandtl number of molten steel is low, and the convection is therefore oscillatory. This oscillatory flow would impede uniform solidification and the formation of a uniform product. Application of a magnetic field to calm the convection appears to be indispensable to control the quality of the product. There are many reports on the convection in the mold. However, the geometrical layout and arrangement of the magnetic field differ from one steel company to another. This is undoubtedly related to the fact that such details are patented, but it also suggests that there are many possible ways to suitably arrange the magnetic fields, and that the arrangement of the magnetic field may not seriously affect the quality of the product. On the other hand, the rectangular shape of the mold looks to be similar in most steel companies. The present report therefore considers molds of square cross-section in order more easily to study the effect of magnetic field. The following are recent reports on the continuous steel-casting process with the application of a magnetic field. Kobayashi et al. [1] reported on a national project on the application of electromagnetic force to continuous casting of steel, which found that a magnetic field decreased the number of clusters of alumina particles. Toh and Takeuchi [2] in a plenary lecture stated that convection in the mold determines the quality of the solidified product and that the application of a magnetic field works for braking, stirring and casting for better products. A magnetic field suppresses the channeling of melt, and magnetic stirring is employed for convection control. They also stated that detailed study on the convection has been carried out with LES (Large Eddy Simulation) and a two-fluid model, and even the micro-scale behavior has been studied over the solidifying surface. Takatani [3]

131

132

Magnetic convection

also stated in the plenary lecture that a magnetic field may be either alternating or static. He stated that an alternating magnetic field has been employed for stirring and a static one for braking of convection, although he warns that too strong suppression induces defects in heat supply to the meniscus of the solidifying field. Morishita et al. [4] reported the effect of magnetic braking with gold particles as tracers finding that penetration depth of jet flow in the mold from the meniscus decreases from 1500 to 900 mm. Inclusion was also found to have decreased to 50% or less. Kubota et al. [5] also reported the effect of a traveling magnetic field to suppress the inclusion of mold powder in the case of high-speed casting. They also reported an empirical equation showing that the suppression is proportional to the 4th order power of magnetic induction. Mochida et al. [6] reported the use of superconducting magnets for flow control of molten steel, finding an extensive decrease of inclusion of alumina particles even in high-speed casting of 3 m/min. Mochida et al. [7] also reported that the use of a superconducting magnet decreased the depth of oscillation hooks due to the calming of the melt surface. Suzuki et al. [8] reported a decrease in the number of pin-holes and clusters due to the magnetic stirring effect. These are reviewed by Tozawa et al. [9]. The present chapter compares the effects of various magnetic fields on the melt convection in molds of square cross-section. A laminar model was also presumed.

14.2 Model Systems Three different coil locations were considered as shown in Fig. 14.1 (a), (b) and (c). Coils are located outside of the mold to generate a magnetic field in approximately either the X, Y or Z direction. In the X- or 7-directional magnetic fields, two electric coils were located outside the vertical walls with the electric current in the same direction in each coil. The dimensional equivalent size is as follows. The width of the rectangular mold is £ = 0.064 m and the diameter of the electric coils is 0.1 m. The molten steel is presumed to come down from the tundish through a rectangular pipe of O.UxO.U and flow out obliquely into the mold. The molten steel is at high temperature. Four surrounding boundaries of the mold are kept at cold temperature, while the top and bottom boundaries are presumed to be thermally adiabatic. From the bottom boundaries, the molten steel was assumed to flow out with uniform equivalent volume flow rate as that from the top inlet. The three-dimensional model equations consist of the equation of continuity, momentum equations, the energy equation, Ohm's law, and the continuity equation

133

Continuous Steel-Casting Systems with Various Magnetic Fields

of electric current. The magnetic induction was computed from Biot-Savart's law as follows. The dimensionless equations are as follows: V-U = 0 2

(1) 2

Z)U//)r = - V / ' + V u/Re + (jxB)Ha /Re + (0,

1 / H o tfluidinlet I" >

\~

7

j/

I

I/I o.u \

\ !•;: \

;l-te;=-^'t

/]

^ - 9 4 ^ ° 8 !J\

8 ;COi5~~^^\ ] I Coil /

adiabatfc

• u*

/

Hot fluid ou!]ct

0, 7 f ( G r / R e )

77£>r= V277(Re • Pr),

J = E + UxB = - V ! P + U x B

V-J = O,

(3), (4)

B = -JRxdS/(4nR3).

(5), (6)

The initial and boundary conditions for computation are as follows. At r = 0 , U = 7T = J = 0

At A; 7=0, i

u = o,r=-o.5, J = O

At the top inlet, At the top surface, At the bottom boundary,

W=l,T=0.S, 0 = 0 , dTldZ = 0, J = 0 W=\l\00, 8T/8Z = 0, dJJdZ = 0

The variables are defined as follows: b0 = nJlt, B = b/bQ, e = electric field, e0 = uobo, E = e/e^ = - V tFe , Gr = gP(0h~0c) flv2, Ha = (o/[i)ll2b0l, i = electric current in a coil, j = electric current vectors, j0 = auobo, J = j / / 0 , 1 = width of the mold, p = pressure, Po = pwjl, P = p/po, Pr = Prandtl number = via, R = Pointing vector, Re = Reynolds number = WiJ.lv, Rein = Reynolds number = WiJJv, / = time, to=xo/uo, r = t/t0, T= (B- 0Q)I {9h - 6C), S = velocity vector = (u, v, w), u0 = wim U = u/uQ = (U, V, W), woul = wJ\QQ, y/e = electric scalar potential, i//e0 = ab0, *F = y/eiVeo-

The above dimensionless equations were approximated with finite difference equations. The pressure was computed by the HSMAC method. The inertial terms were approximated with the third-order upwind scheme. The dimensionless time increment is 2x10~4. The rectangular enclosure was divided into 503 grids with nonuniform staggered mesh. The nozzle part was divided into sections of 0.01 in dimensionless length. Table 14.1 [10] Computed results for Umax. Umax is 1.233 at Ha = 0.

^ ^ - ^ - ^ ^ ^ X-Mag Y-Mag Z-Mag I

at Ha = 100 L232 L241 1.223

|

at Ha = 200 L220 L217 1.184

[

at Ha = 500 1.131 1.101 0.917

Continuous Steel-Casting Systems with Various Magnetic Fields

135

14.3 Computed Results The computational conditions are as follows: Re = 104, Rein = 103, Gr = 107 and Pr = 0.025. The magnetic field is in either the X, Y or Z direction (not completely uniform). The computed values of maximum dimensionless velocity Umax are listed in Table 14.1. Figure 14.2 shows magnetic induction vectors in the mid-plane under an (a) X-, (b) Y- or (c) Z-directional magnetic field. These vectors are mostly either in the X, Y or Z directions. The transient computations converged smoothly to suggest the stable numerical computation. Figure 14.3 shows a reference result without a magnetic field at Ha = 0 and Pr = 0.025, Re = 104, Gr = 107 and at z= 10. Velocity vectors are shown at left and the

Figure 14.3 [10] Velocity vectors and isothermal contours at reference state, Pr = 0.025, Gr = 107 and Re = 104. Upper row shows Y= 0.5, and lower row shows Z= 1.5.

136

Magnetic convection

75

5

25

v*v

v^v

w

Figure 14.4 [10] Vectorial representation of Lorentz force vectors at Z = 1.5 at Ha = 200, Pr = 0.025, Re = 104, Gr = 107 and r = 10 in the (a) X-directional, (b) 7-directional, (c) Z-directional magnetic field.

5

5

5

5

\'V

\"V

\~S

Figure 14.5 [10] Computed vectors in the (a) X-9 (b) Y-, (c) Z-directional magnetic field at Ha = 500, Pr - 0.025, Re = 104, Gr = 107 and r= 10. Upper row shows at Y= 0.5, and lower row shows Z= 1.5.

137

Continuous Steel-Casting Systems with Various Magnetic Fields

isothermal contours at right. Cross-sections at Y= 0.5 are shown at top, and those at Z = 1.5 are at bottom. Figure 14.4 shows the Lorentz force vectors F = J x B in the X-, Y- or Zdirectional magnetic field at Ha = 200. Comparison suggests the Z-directional magnetic field provides much larger magnetic suppression than those in the X or Y direction. Figure 14.5 shows velocity vectors in the plane of Y = 0.5 in the (a) X-, (b) Y-,

r

•••'-•;•••

\

!

-

r 0

1

i: O

.'

•

I

' -

'

tj

0 25

I

0.5

i

0.75

._'.._'

. ^

^

|

• 1

-

|

- 0.5

-

•

','.

•

^'

•

. '

ll° W r i i

1

0

~T~|i II

0.25

0.S

0.75

!

'

'

"'."''

•>•

- 1

-

.

- 0.5

"

H° 11

:

'

S

••

-

~

••'• i : 1

.

-

.

1

- 0.5

•; i j . \r

°

-'s

~]~|i II

i

1

\

'?'""''

-•; ' 7 ) 7 1 •

' ^

•

.--•''''.'•'••• I

'.', p=p0, p = po and x = Zo, there will be no convection and eq. (3) becomes

(4)

0 = -Vpo+£^Ls/b2+Pog.

When there is a temperature distribution in a fluid, convection will occur due to the difference in the magnetic susceptibility and density in the buoyancy term. Pressure p can be represented as the sum of p0 at the isothermal state and p', the pressure difference due to the perturbed state, as follows: P=Po+p'Subtracting eq. (4) from (3) gives

(5)

Convection Induced by a Cusp-Shaped Magnetic Fieldfor Air in a Cube 141

P^p'+^M-^W+ip-p^.

(6)

' m

Since mass magnetic susceptibility of oxygen gas x is inversely proportional to the absolute temperature 9, following Currie's law [8] we get (7)

X=CIO, where C is a constant. By a Taylor expansion around a static state,

(8), (9) where J3 = 1/0 is used for an ideal gas, p= pRQ

dM.Ax 39

89

+ pdJL = _ph+i^\ 30

HHA

\

e2)

= -2ph.

(10)

Then, we get

(11) With the Boussinesq approximation that physical properties other than the density in the buoyancy terms are constant at 9= 6J>, then ^

= - L v ^ v 0 V 2 » - M ( ^ ^ * 2 - ^ ^ o ) ( 0 . 0 , - g ) r . (12)

The equation of continuity, the energy equation and Biot-Savart's law for magnetic induction complete the model equations as follows: V-u = 0,

D9/Dt=dV29,

h=

~~^\—3~"

(13), (14), (15)

These can be non-dimensionalized by the method of Heliums and Churchill [14] as follows:

142

Magnetic Convection V • U = 0,

DT/Dr= V 2 r

(°YI

£>U/Dr = -V/> + PrV2U + RaPr7' -yVB2 + 0

,

(16), (17)

B = — - c f R x f S (18), (19) An J R3 .

v ) . The following dimensionless variables were employed:

B = b/b0, P=p'lpo, S=s//, U = u/u0, r=t/t0, bo= fxmil£, po= po(a/(.f, to=£2/a, uo=a/t, Ra = afl(ft-&) £3/(av), Pr^vYa, r= ZbYtfoKyiJ). These are the model equations for the present problem of an ideal gas (paramagnetic fluid) in magnetic and gravitational fields. Initial and boundary conditions are as follows.: atr 2 -

2

(2)f f

Silveston's experiment

yf 1

(„$ ' $ '*

'

2000

*

T -$h^~~~~

J^L--

(5)

t

4000 Ra number

Figure 16.3 [1] Summary of experimental results. (1) zb = +66 mm and (2) zb = +66 mm and (3) Zi = +66mmand (4) (5) zb = -66 mm and

j ^ 6000 800010000

Solid line bz(dbjdz) bz(dbjdz) bz(dbjdz) bz{8bjdz) bz(dbjdz)

is from Silveston 's experiment [6]. = 138 T2/m = 49.7 T2/m = 5.52 T2/m = 0 T2/m = -138 T2/m.

156

Magnetic convection

Fig. 16.4. The locations of the cylindrical enclosure correspond to 95, 55, 25, -25, -55, -95 mm in the bore of 100 mm in diameter. The magnetic coil diameter was presumed to be 180 mm. Diagram (a) shows a vectorial representation of magnetic induction B due to the passage of electricity through the magnetic coil. Magnetic induction was computed by Biot-Savart's law below. A super-conducting magnet consists of a coil of several thousand turns, but in the present computations a coil of one turn was presumed for simplicity. Diagram (b) shows a vectorial representation of magnetizing force V 5 2 . The magnetizing force is symmetrical in terms of the coil. Following our recent works [7, 8], dimensionless model equations for the gravitational and magnetizing force fields are given as follows. They are an equation of continuity (4), an energy equation (5), momentum equations (6) that include a magnetizing force term, and Biot-Savart's law (7). V-U = 0,

DT/DT

= V2T

r

2

(4), (5)

2

roY

= -vT + PrV U-Ra-Pr-:r yVB + 0

L lu.

B-U^f

An J

R3

(6)

(7)


)/(2/fr )} vb 2 +pg{ 9- % )( 0, 0, 1 f.

(8)

This leads to the following non-dimensionalized system equations, including equations of continuity, energy and Biot-Savart's law: V-U = 0,

DT/DT=V2T

2

(9), (10)

2

= -W> +PrV U-rRa-PrT-VS /2 + Ra-Pr-r-(0, 0, 1 )T (11)

DU/DT

B = —cfcSxfe)/^ 3 .

'

(12)

The following dimensionless variables were employed.

(R,Z)=(r,z)/h, U=u/u0, W=w/u0, r=tlt0, P=p'/p0, B = b/b0, T = (6 - 60)l{6h - 6C), uo = a/h, to = h2/a, bo = fimi/h, Po^pctlh2, Ra = g^9h-9c)h3/(av), Pr = v/a, y=%b02/(^gh). The initial conditions are as follows:

U=W=T=0

at

rdRa -7.0xl0 6 -2.9xlO 6 -3.5xlO 6 -7.0xl0 6

Ram -4121 1708 2061 4121

Nu ave . 1.000 1.0008 1.063 1.674

A6 [K] 10 10 10 10

k U , e r [T] 0.923 0.594 0.653 0.923

Rayleigh-Benard Convection of Diamagnetic Fluid

187

of the average Nusselt number. For the system without gravitational acceleration, we can see the effect of magnetizing force alone. Transient computation converged smoothly. For the system at g = 0, y becomes infinity and Ra becomes zero, but the system can be defined with finite values of ^.a. Table 19.1 shows the converged values of the average Nusselt number. If the temperature difference between the hot and cold walls is 10 K, the magnetic induction at the center of the enclosure is 0.923 T for )Ra = -7.0xl0 6 and 0.653 T for yRa = -3.5x10 6 . Figure 19.2 shows computed isothermal contours and velocity vectors for position 1 and 2 at Pr = 6 without gravitational acceleration. At position 1, the magnetizing force acts in the positive Z direction (upward), and the water layer in a cylinder heated from below and cooled from above approaches the conduction state. In (i), the velocity vectors are drawn on a scale of 100 times larger than those in (ii) for easy visualization. The actual absolute velocity is almost zero, and a quasi-conduction state is attained. On the other hand, at position 2, the magnetizing force acts downward in the Z direction, and the magnetizing force acts to give larger Nusselt number with the increase in | yRa |. According to the derivation of the model equation, this can be explained in another way as follows. Water near a hot wall has lower density and is less repelled by the magnetic field than that near a cold wall. Therefore, at position 1, water receives less accelerating force with the increase of | ^Ra |, but at position 2, water is unstable and convection results if | ^Ra | exceeds the critical value.

(i)

Hot

'

--^=^^:";"!::;^^:r=r:^--

(ii)

Hot Figure 19.2 [19] Computed isothermals (left) and velocity vectors (right) in a cross-section of a vertical shallow cylindrical enclosure without gravity acceleration but with a magnetic field for Pr = 6: (i)Positionl, ?Ra=-7.0*10*, Nu= 1.000 (ii)Position2, ?Ra= -7.0>