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Heat transfer and convection in liquid metal Harrison, Christine Elizabeth 1984

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HEAT TRANSFER AND CONVECTION IN LIQUID METAL By CHRISTINE ELIZABETH HARRISON B.Sc,  The University of B r i t i s h Columbia, 1981  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES (Department of M e t a l l u r g i c a l Engineering)  We accept this thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1984 © CHRISTINE ELIZABETH HARRISON, 1984  In p r e s e n t i n g  this  requirements  f o r an  of  British  it  freely  available  understood  that  financial  shall  for reference  and  study.  I  f o r extensive copying of be  her  copying or shall  g r a n t e d by  not  be  METALLURGICAL ENGINEERING  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3  Date  DE-6  (3/81)  A p r i l 26  .  1984  of  Columbia  make  further this  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  the  representatives. publication  the  University  Library  h i s or  gain  the  the  s c h o l a r l y p u r p o s e s may by  f u l f i l m e n t of  I agree that  permission  department or  for  in partial  advanced degree a t  Columbia,  agree that for  thesis  written  ii  ABSTRACT  This  investigation  was  undertaken  to  examine  the  heat  flow  c h a r a c t e r i s t i c s of a l i q u i d metal system i n which f l u i d flow i s present due to buoyancy forces. has  Previous investigations of heat flow i n l i q u i d s  been confined to transparent materials,  which have very d i f f e r e n t  flow c h a r a c t e r i s t i c s compared to l i q u i d metals.  Measurements were made on l i q u i d t i n contained i n a thin square cavity which had  a temperature  difference  imposed across the c e l l  produce  convection.  The  flow  natural  heat  across  the  cell  to was  calculated from the measured temperature difference across the cold end plate and the thermal conductivity heat flow and effective  the measured temperature  thermal conductivity  sizes were studied. found  of the plate.  to have a  Using the calculated  difference across the melt  of the melt was  calculated.  Two  the cell  The thermal conductivity of the cold end plate was  s i g n i f i c a n t e f f e c t on  the heat  transfer  through  the  cell.  Radioactive tracers were used to observe the flow pattern i n the melt and to measure the flow v e l o c i t y as a function of the difference  across  radioactive Sn^^  3  the  cell.  The  technique  involved  temperature  insertion  of  into the melt, then quenching the sample after a given  iii length of time.  The sample was  then autoradiographed to determine the  path of the tracer after i n s e r t i o n into the melt. be  very  fast  for the  three-dimensional  smaller of  flow  the  two  characteristics.  laminar, two-dimensional  flow.  The flow was found to  cell  sizes which  The  larger  A c o r r e l a t i o n was  exhibited  cell  produced  observed between the  time per cycle and the temperature difference across the large c e l l .  The  study  also  includes  a  f i n i t e - d i f f e r e n c e model which  developed  to provide further  insight into the thermal and  behaviour  of  model examines the e f f e c t of  the melt.  temperatures  The  along the ends and bottom of the c e l l  and v e l o c i t y f i e l d s  and  i s used  on the  ends of the c e l l  flow  nonuniform temperature  to compare the response of l i q u i d t i n  and l i q u i d s t e e l to i d e n t i c a l temperature differences. model indicate that  fluid  was  either a temperature  drop  Results from the  along the hot and  cold  or the presence of a linear gradient along the bottom  of the c e l l would decrease the maximum f l u i d v e l o c i t y In the c e l l .  The  present  thermal conductivity liquid  metal  conductivity.  can  investigation due be  as  shows  that  the  enhancement  of  the  to the presence of natural convection i n the high  However the degree  as  ten  times  the  stagnant  of enhancement i s influenced  thermal resistance at the boundaries.  thermal by the  iv  TABLE OF CONTENTS  1.0 INTRODUCTION 1.1 F l u i d Flow  1  1.1.1 Natural convection  2  1.1.2 Forced convection  7  1.2 Convection In Industrial Processes  10  1.3 Examination Of Previous Investigations  12  1.4 Heat Transfer Analysis  16  2.0 EXPERIMENTAL DESIGN 2.1 Design Of Experimental Cells  22  2.1.1 C e l l I  24  2.1.2 C e l l II  26  2.1.3 C e l l I I I  27  2.2 F l u i d Flow Measurement  29  2.3 Temperature Measurement  31  3.0 EXPERIMENTAL RESULTS 3.1 Results From C e l l I  34  3.1.1 Calculation of e f f e c t i v e thermal conductivity  34  3.1.2 Discussion of results from c e l l I  37  3.2 Results From C e l l II  38  3.3 Results From C e l l I I I  40  3.3.1 Results of quench tests  42  3.3.2 Temperature measurement  49  3.0 EXPERIMENTAL RESULTS(cont'd) 3.3 Results From C e l l III (Continued) 3.3.3 Calculation of effective  thermal conductivity  57  4.0 MATHEMATICAL MODEL 4.1 Governing Equations  60  4.1.1 Dimensionless variables  63  4.1.2 Stream function and v o r t i c i t y  65  4.2 I n i t i a l And Boundary Conditions  67  4.3 F i n i t e Difference Equations  68  4.4 Results Of Computer Runs  75  5.0 SUMMARY REMARKS  97  REFERENCE LIST  100  APPENDIX I . L i s t of Symbols  102  APPENDIX I I . Chart for Argon Flow Rate Conversion  104  APPENDIX I I I . Computer Program  105  vi  LIST OF TABLES PAGE  TABLE I  Comparison of F l u i d Properties  TABLE II  Properties of Liquid T i n  5 23  TABLE III Comparison of Thermal Resistances  39  TABLE IV  58  Calculation of E f f e c t i v e Thermal Conductivity  )  vii  LIST OF FIGURES PAGE FIGURE 1.1  F l u i d flow progression after start of c a s t i n g .  FIGURE 1.2  Flow patterns induced by electromagnetic s t i r r i n g .  FIGURE 1.3  E f f e c t of input stream on f l u i d flow i n l i q u i d pool (a) Flow patterns induced by d i f f e r e n t types of input streams (b) Example of how input stream induces flow below the mould.  1  1 6  17  18  FIGURE 1.4  F l u i d flow at s o l i d - l i q u i d  interface.  11  FIGURE 1.5  Convection currents i n a s o l i d i f y i n g ingot  FIGURE 1.6  Temperature isotherms f o r (a) purely conductive heat flow and (b) convective heat flow ( T > Tj^).  13  V e r t i c a l (x) v e l o c i t y p l o t t e d versus h o r i z o n t a l position across mid-plane of cavity (x=0.5) according to (a) the a n a l y t i c a l model of Batchelor and (b) the numerical model of Stewart which shows v e l o c i t y p r o f i l e s f o r Pr= 0.0127 and various Grashof numbers.  15  Correlation observed by Stewart between the time to complete one cycle around c e l l and the temperature d i f f e r e n c e across the c e l l f o r average melt temperatures of 237°C, 260°C and 305°C .  17  Radiographs from experiments of Stewart showing (a) two -dimensional laminar flow and (b) three-dimensional vortex f l o w .  18  FIGURE 2.1  Design of experimental c e l l I  25  FIGURE 2.2  Design of experimental c e l l III  28  FIGURE 2.3  Location of thermocouples i n experimental c e l l s a) c e l l I, b) c e l l II and c) c e l l I I I ( thermocouple location i s marked by T ) .  32  1 9  20  11  8  2  FIGURE 1.7  FIGURE 1.8  8  FIGURE 1.9  8  viii  FIGURE 3.1  FIGURE 3.2  FIGURE 3.3  FIGURE 3.4  FIGURE 3.5  FIGURE 3.5  FIGURE 3.5  FIGURE 3.6  FIGURE 3.7  Graph showing results from c e l l I with top of c e l l covered and uncovered. Slopes were calculated using l i n e a r regression of data points. The c o r r e l a t i o n c o e f f i c i e n t i s given i n brackets.  35  Quenched samples from c e l l II (a) Tracer i s well mixed (AT=6°C, time prior to quench = 30 sec), (b) Sample showing vortex motion (AT=8°C, time prior to quench = 30 sec), (c) Sample exhibiting turbulent flow (AT=5°C, time prior to quench = 25 sec).  41  Examples of inadequate quenchs from c e l l I I I (a) Sample quenched by rapidly f i l l i n g inside of furnace with water and (b) Sample quenched using water jets aimed at side walls.  43  Successful quench sample acheived using composite wall design.  44  cell  Quenched samples from c e l l I I I (a) AT=7.6 °C, time p r i o r to quench = 20 sec (b) £T=7.2 °C, time prior to quench = 21 sec.  45  (Continued) Quenched samples from c e l l I I I (c) AT = 1.4 °C, time prior to quench = 22 sec, (d) AT=4.4 °C , time prior to quench = 11 sec.  46  (Continued) Quenched samples from c e l l 2.4 °C, time prior to quench = 18 sec.  47  I I I (e) AT =  Relationship between time required per cycle and the temperature difference across the melt (Temperatures i n melt were measured using a thermocouple probe).  48  Graph of temperature differences across cold end versus temperature difference across melt f o r (a) experiment A which used a heater and argon cooling to produce the temperature difference across melt. Slopes indicated on graph were calculated using l i n e a r regression of data points.  50  ix FIGURE 3.7  (Continued) Graph of temperature differences across cold end versus temperature difference across melt f o r b) experiment B which only used argon cooling to produce the temperature difference across melt. Slopes indicated on graph were calculated using l i n e a r regression of data points.  51  Graphs showing behaviour of temperature differences across (a) the cold end and (b) the melt versus argon flow rate for experiment A.  53  Graphs showing behaviour of temperature differences across (a) the cold end and (b) the melt versus argon flow rate for experiment B.  54  Graphs of temperature at centre of outside face of cold end versus argon flow rate for (a) experiment A and (b) experiment B.  55  FIGURE 4.1  System of reference for mathematical model.  61  FIGURE 4.2  Grid system used i n mathematical model.  70  FIGURE 4.3  Isotherm d i s t r i b u t i o n for Pr=0.0127 and (a) Gr=1.0X10\ (b) Gr=1.0X10 and (c) Gr=1.0X10 .  77  Streamline plot for Pr=0.0127 and (a) Gr^L.OXlO *, (b) Gr=1.0X10 and (c) Gr=1.0X10 .  78  Nusselt number versus v e r t i c a l (x) position along cold wall for Pr=0.0127 and Gr=1.0X10\ 1.0X10 and 1.0X10 .  79  Normalized v e l o c i t y f i e l d for Pr=0.0127 and (a) Gr=10 (maximum v e l o c i t y =487.44) and (b) Gr=10 (maximum velocity = 1185.09). V e l o c i t i e s given are dimensionless v e l o c i t i e s .  80  Isotherm plots with Pr=0.0127 and Gr=1.0X10 . Solid l i n e indicates p r o f i l e s for isothermal ends and dashed l i n e indicates those for (a) a 5% drop along ends and (b) a 10% drop along ends.  81  FIGURE 3.8  FIGURE 3.9  FIGURE 3.10  5  FIGURE 4.4  1  5  FIGURE 4.5  6  6  5  FIGURE 4.6  6  5  6  FIGURE 4.7  5  FIGURE 4.8  FIGURE 4.9  FIGURE 4.10  Isotherm plots with Pr=0.0127 and Gr=1.0X10 . Solid l i n e indicates p r o f i l e s for isothermal ends and dashed l i n e indicates those for (a) a 5% drop along ends and (b) a 10% drop along ends.  82  Streamline plot with Pr=0.0127 and Gr=1.0X10 with (a) isothermal ends and (b) 10% drop along both ends (with coldest temperature at top).  83  Isotherm plot with l i n e a r temperature gradient along bottom (Isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0Xl0 and (b) Gr=1.0X10 .  85  Streamline plots with linear temperature gradient along bottom (isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0X10 and (b) Gr=1.0X10 .  86  Isotherm plots with l i n e a r temperature gradient along bottom and 10% temperature drop along ends with Pr= 0.0127 and (a) Gr=1.0X10 and (b) Gr=1.0X10 .  87  Isotherm d i s t r i b u t i o n f o r a temperature difference of 0.01 °C across c e l l for ( a ) l i q u i d t i n (Pr=0.0127 and Gr =3.6X10 ) and ( b ) l i q u i d s t e e l (Pr=0.11 and Gr=6.0X10 ).  89  Isotherm d i s t r i b u t i o n for a temperature difference of 0.5 °C across c e l l f o r (a) l i q u i d t i n (Pr=0.0127 and Gr=1.8X 10 ) and (b) l i q u i d s t e e l (Pr=0.0127 and Gr=3.0X10 ).  90  Streamline d i s t r i b u t i o n f o r a temperature difference of 0.01 °C across c e l l f o r ( a ) l i q u i d t i n (Pr=0.0127 and Gr =3.6X10**) and ( b ) l i q u i d s t e e l (Pr=0.11 and Gr=6.0X10 ).  91  Streamline d i s t r i b u t i o n for a temperature difference of 0.5 °C across c e l l f o r ( a ) l i q u i d t i n (Pr=0.0127 and Gr =1.8X10 ) and ( b ) l i q u i d s t e e l (Pr=0.11 and Gr=3.0X10 ).  92  V e r t i c a l (x) v e l o c i t y versus horizontal (y) position at x=0.5 for a temperature difference of 0.01 °C across c e l l i n (a) l i q u i d t i n (Pr=0.0127 and Gr=3.6X10 *) and (b) l i q u i d steel (Pr=0.11 and Gr=6.0X10 ).  93  6  6  5  FIGURE 4.11  6  5  FIGURE 4.12  6  5  FIGURE 4.13  6  k  FIGURE 4.14  3  6  5  FIGURE 4.15  3  FIGURE 4.16  6  FIGURE 4.17  5  1  3  xi FIGURE 4.18  V e r t i c a l (x) v e l o c i t y versus horizontal (y) position at x=0.5 for a temperature difference of 0.5 °C across c e l l i n (a) l i q u i d t i n (Pr=0.0127 and Gr=1.8X10 ) and (b) l i q u i d steel (Pr=0.11 and Gr=3.0X10 ).  94  Plot of l o c a l Nusselt number versus position along cold wall for t i n and s t e e l at temperature differences of 0.01 °C and 0.5 °C.  95  6  5  FIGURE 4.19  xii  ACKNOWLEDGEMENTS  I would l i k e to thank my supervisor, Dr. Fred Weinberg for h i s patience and guidance throughout this investigation.  I would also l i k e  to thank the technical staff at the Department of Metallurgy for their valuable assistance during the course of this work.  Financial  support  was provided  by the Alcan Fellowship  grant from the National Science and Engineering Council.  and a  1  1.INTRODUCTION  An of  the  analysis  temperature  problem has analysis data  of many m e t a l l u r g i c a l distribution  been approached  to  estimate  related  to  difficult fluid  since  flow  in  thermal  profiles.  conditions  flow  through  liquid.  a  Such  knowledge  t r a n s f e r mathematical  To  do  and  this  thermal  liquid Is  involves  I n r e c e n t years  I f the system c o n t a i n s l i q u i d  heat the  system.  by u s i n g a heat  boundary  materials involved.  i n the  processes  analysis properties  model  requires of  the  the t a s k becomes more  i s markedly  the  this  case  influenced  for  by  solidification  p r o c e s s e s i n l a r g e i n g o t s , continuous c a s t i n g and c r y s t a l growth as w e l l as  heat  loss  determinations i n l a d l e s  information i s available flow  present.  used which arbitrary  f o r heat  In practise,  consists  an  and  tundishes.  transfer  in a liquid  e s t i m a t e d heat  metal w i t h  transfer  fluid  coefficient  of the atomic  thermal c o n d u c t i v i t y m u l t i p l i e d  present  investigation  was  undertaken  t r a n s f e r through a l i q u i d metal w i t h known f l u i d  by  is an  to determine  the heat  f l o w i n the m e l t .  FLUID FLOW  Fluid or  quantitative  number.  The  1.1  No  be  flow i n a l i q u i d  induced  by  mechanical  metal can r e s u l t or  electrical  from n a t u r a l c o n v e c t i o n  means  to  produce  forced  2 c o n v e c t i o n i n the  Natural  1.1.1  Convection  Natural gradients of  liquid.  convection  which  give  rise  is  caused  to d e n s i t y  by  changes  Boiling  in  1  produced  solvent  Figure  during  Convection  from  are  differences  solidification  as  this  most  cause  affect  is  geometry  the  fluid  difficult  and  flow.  solute  patterns  physical The  solidification continually clear,  can where  be  very the  involved,  size  changing w i t h t i m e .  c o n s i s t i n g of a p a r t i a l l y  complicated  geometry i s  to  required.  investigate  and In  into  when  change  effect  of  refers  the  the  melt.  solute  Such  and  density  appreciably  the  to  system  the  i n the f l u i d .  which shape  with  is of  shape  of  liquid  s o l i d and l i q u i d r e g i o n . and  a  simpler  the  In many cases  especially the  greatly  true pool  a d d i t i o n the boundary i t s e l f  directly  always  define.  properties  geometry  almost  density.  which  to c l e a r l y  are  segregates  in  boundaries and the presence of any b a r r i e r boundaries  The  Thermal c o n v e c t i o n can  pronounced  different  complex f l o w  The  compositional  the f l u i d .  Compositional differences  1.1.  significantly cause  or  low temperature d i f f e r e n c e s as shown by Cole and  time and are t h e r e f o r e  too  in  the buoyancy f o r c e produces the f l u i d m o t i o n .  be p r e s e n t even at very  the  thermal  is  for is not  The system i s steady-state  3  .1  F l u i d flow progression after start of c a s t i n g .  1  4  Many of  the  studies  on  heat  transfer  and  natural convection  have been undertaken because of the interest i n heat transfer i n nuclear reactors,  solar c o l l e c t o r s and other i n d u s t r i a l systems.  the geometries studied have attempted encountered  i n these applications.  Consequently  to duplicate the pipes and ducts These  geometries  often  cannot  be  d i r e c t l y related to those encountered i n l i q u i d pools i n s o l i d i f i c a t i o n or other applications where large volumes of l i q u i d metal are contained in  a  vessel,  which  makes  i t difficult  to  apply  the  heat  transfer  studies  been made using transparent  results.  In materials  addition,  such as water,  these f l u i d s Table I.  most  oils  d i f f e r markedly  and  have gases.  The physical properties  of  from those of l i q u i d metals, as shown i n  The d i s p a r i t y i s p a r t i c u l a r l y pronounced  i n the values of the  density and thermal conductivity.  To physical parameters  understand properties that  it  are  Grashof number and  the  significance  is  of  useful  importance  to  of  the  look  to natural  the Prandtl number.  at  differences the  in  the  dimensionless  convection, namely  the  The Grashof number, Gr i s the  r a t i o of the buoyancy force times the i n e r t i a force over the shear force squared and i s defined as follows:  =  B g L  3  v  2  AT  (1.1.1)  TABLE I  FLUID  COMPARISON OF FLUID PROPERTIES  THERMAL  PRANDTL  GRASHOF  CONDUCTIVITY  COEFFICIENT  NUMBER  NUMBER  cal  OF EXPANSION 1 ( 1 ]  1.02X10-^  0.013  3.6X10  6  10.62  1.15X10" *»  0.024  5.8X10  6  2  6.95  2.0X10" *•  0.11  6.0X10  1  2.37  -  3  1.00  I^XIO- *  THERMAL  VISCOSITY SPECIFIC HEAT  (cp)  (  cal  \  ( 1  DENSITY  gm 1  ~ 1 a ' 1cm cm-sec- C l  J  \  1.88  0.054  8.0X10"  2.39  0.038  3.9X10"  6.5  0.12  7.0X10"  4.5  0.259  2.0X10"  Water  1.38  1.0  1.4X10"  Alr(50°C)  0.019  0.25  2.11X10"  NHjCl  1.30  0.776  1.12X10"  1.013  1.86X10-  2.5X10 *  0.239  2.36X10"^  1.54  1.98X10" *»  2.5X10 1.9X10"  1.9X10  0.45  4.92X10"  1.06  7.6X10- *  1.7X10 * 6.1X10"  Liquid  tin lead  "  8  steel Al  1 5  Oil  #1  21  Oil  #2  21  8  8  1  3  6.953  2  2  11  3  11  0.0011  -  0.058 10.0  1  -  1.3X10  1  6  -  0.225  5  5  9.0  2.8X10 ' 1  5  1  1  3  6 where 8 = C o e f f i c i e n t of thermal expansion g = Acceleration due to gravity L = C h a r a c t e r i s t i c length of the system AT = Temperature difference v = Kinematic v i s c o s i t y  The  Prandtl  number, Pr i s the r a t i o  of the momentum to the thermal  d i f f u s i v i t y and i s defined as:  Pr=J±-£2.  (1.1.2)  k  where u = v i s c o s i t y Cp = S p e c i f i c heat k = Thermal conductivity  Table  I  shows  transparent  the  difference  and m e t a l l i c  i n the Prandtl  materials.  Also  given  number  between the  i n Table  I i s the  Grashof number which has been calculated for a one degree temperature difference and an arbitrary c h a r a c t e r i s t i c length.  The the  heat  Grashof number and the Prandtl number are very important to transfer  c h a r a c t e r i s t i c s of  d i s t r i b u t i o n i s related to the Rayleigh these numbers. function  In natural  of the Grashof  convection  the system.  The  temperature  number which i s the product of the Nusselt  number and the Prandtl  number i s usually a number.  The  Nusselt  7  number, Nu i s a dimensionless parameter which i s defined as:  Nu = —  (1.1.3)  k  where h i s the heat transfer c o e f f i c i e n t .  1.1.2 Forced  Convection  Forced  convection  refers  to f l u i d  motion which  external forces rather than by buoyancy forces.  i s caused by  Many examples of forced  convection can be found i n metallurgical systems.  Figure 1.2 shows the  range of flow patterns that can be produced by electromagnetic used i n the continuous  casting of s t e e l .  stirrers  The exact extent and v e l o c i t y  of the f l u i d flow i s not clear at present.  In represents  continuous  casting  a considerable source  the  input  stream  of  molten  of momentum to the l i q u i d pool.  metal The  flow pattern produced and the extent of i t s effect i s determined by the manner i n which the stream enters 1.3.  Below  the mould  the l i q u i d pool, as shown i n Figure  as the momentum  from  the input  stream i s  dissipated, i t Is unclear to what degree the flow i s caused by the input stream or by natural convection.  The flow region represents a s i t u a t i o n  where there i s combined forced and free convection.  FIGURE 1 . 2  Flow patterns induced by electromagnetic  stirring.  FIGURE 1.3  E f f e c t of input stream on f l u i d flow i n l i q u i d pool (a) Flow patterns induced by d i f f e r e n t types of input streams (b) Example of how input stream Induces flow below the m o u l d . 2 0  20  10 1.2 CONVECTION IN INDUSTRIAL PROCESSES  The interest i n convective flow i s best understood by looking at some of the applications where such flow i s present.  The behaviour of  convecting l i q u i d metal i s of p a r t i c u l a r interest i n the s o l i d i f i c a t i o n of  castings.  The convecting f l u i d  i s a major determinant of the f i n a l  cast structure due to i t s effect on heat and mass transfer. shows the pattern of f l u i d liquid in  the  interface. bottom  flow believed  to be present at the  Natural convection produces  end  of  large  steel  Figure 1.4  counter  ingots which  is  solid-  current flow  thought  to  be  responsible for the segregation pattern observed i n these castings (see Figure 1.5).  In  continuous  casting,  complex than that i n ingots. that  mixing  below  the  the  the  i n the  liquid  pool i s more  Work with radioactive t r a c e r s  i s most pronounced mould  flow  behaviour  i n and of  near  the  the mould but  fluid  is  not  2  has shown i n regions  clear.  In  mathematical models of the s o l i d i f i c a t i o n p r o f i l e i n continuous casting, the  liquid  account  pool  i s assumed  to be  completely  for the enhancement i n the heat  the l i q u i d , the l i q u i d  3  conductivity.  1  everywhere.  To  transfer due to convection i n  i s assumed to act as a conducting s o l i d with an  e f f e c t i v e thermal conductivity, k g f f i n the l i t e r a t u r e ' * '  mixed  5  The value of k ^^ has been taken g  to be seven to ten times the stagnant thermal  FIGURE 1.5  Convection currents i n a s o l i d i f y i n g  ingot.  12 Convection uniformity quality.  of  is  the  also  crystal  Experiments  of  concern  in  composition  crystal  is directly  have shown that f l u i d  6  growth  where  related  flow due  to  the  product  to convection i s  faster than the growth of the advancing interface therefore i t cannot be ignored.  1.3 EXAMINATION OF PREVIOUS INVESTIGATIONS  A good review of recent work on natural convection can be found i n reference 7. theory  Before discussing previous investigations some of the  concerning  natural convection w i l l  analysis of natural convection i s based equation and  the Navier-Stokes  two-dimensional  v are  term The  The  energy  equation for a  (1.3.1)  the x- and y-components of v e l o c i t y ,  respectively.  + ox  input.  equations.  energy  ST = pCp — ot  -k(  first  Theoretical  on the solution of the  5%  2  The  introduced.  system i s as follows:  o !  where u and  be  2  2  i n the  second  9T oT ) + pCp (u — + v — ) oy ox dy  equation  term  is  represents  the  the  convective  heat  from  transport  conductive  contribution.  Assuming no heat generation these two quantities are equal to the heat accumulation which i s the term on the right-hand side of the equation. The  result  isotherms  of  the  convective  for convective heat  transfer by conduction.  terms transfer  in  this  are  not  equation straight  is as  that  the  for heat  Instead they are bent as shown i n Figure  1.6.  13  FIGURE 1.6  Temperature isotherms f o r (a) purely conductive flow and (b) convective heat flow ( T > T ^ . 8  2  heat  14 Since the x- and y-components of v e l o c i t y appear i n the energy equation the v e l o c i t y f i e l d must be known to solve for the temperature field.  For  forced  convection  independant of the thermal  the  field.  velocity  field  can  be  solved  However for natural convection, the  temperature appears i n the equation  for momentum transport i n the  x-  direction(note - the x-direction i s taken to be v e r t i c a l for this system of reference):  du du du — + u — +v dt dx dy  This  coupling  analysis  of  the  thermal  ldP -gB(T-T )  d2u + v(  0  pdx  and  velocity  d2v +  dx  fields  ) dy  2  makes  (1.3.2)  2  theoretical  difficult.  A n a l y t i c a l solutions for these equations have been developed Batchelor ' 9  cavity.  1 0  f o r s t e a d y - s t a t e n a t u r a l c o n v e c t i o n i n a rectangular  The v e l o c i t y p r o f i l e predicted by the solution of Batchelor i s  shown i n Figure 1.7(a). developed  by W i l k e s , 1 1  reported by Stewart  More recently, numerical Vahl D a v i s  Stewart,  boundary with  the  1 2  and  others ' ' 8  of  13  1  maximum  in  the  velocity  profile  .  shifts  increasing Rayleigh number whereas the  Batchelor  1 5  Results  breaks  down at  Grashof  According to the numerical results  constant according to the solution of Batchelor. model  solutions have been  are shown i n Figure 1.7(b) for a range of  numbers with a fixed Prandtl number. of  by  toward  profile  the  remains  Since the a n a l y t i c a l  large Rayleigh  numbers  numerical  15  O.OI5  ud Oi Ro  0.010 -  0.005-  (a)  - 3M0  - MOO  - t400  - tooo  - 1800 tu o - too  (b)  FIGURE 1.7  0.1  02 0.3 DISTANCE, Y  0.4  >  03  V e r t i c a l (x) v e l o c i t y plotted versus horizontal p o s i t i o n across mid-plane of cavity (x=0.5) according to (a) the a n a l y t i c a l model of Batchelor and (b) the numerical model of Stewart which shows v e l o c i t y p r o f i l e s f o r Pr= 0.0127 and various Grashof numbers.  16 models are more useful f o r the modelling of metal systems because the Rayleigh number for many l i q u i d  metal systems i s greater than 10**.  In addition to the numerical model, Stewart conducted one of the few  studies  which used  liquid  metal f o r the experiments.  This work  established a c o r r e l a t i o n between the time per cycle and the temperature difference  across a thin  relationship was v a l i d Figure 1.9(a).  square cavity, as shown i n Figure 1.8.  The  for the type of two-dimensional flow shown i n  For large temperature differences across the c e l l , the  flow pattern showed a vortex type motion (see Figure 1.9(b)) exhibiting three-dimensional rather than two-dimensional c h a r a c t e r i s t i c s . three-dimensional  flow patterns  exceeded a certain value.  were produced  i f the c e l l  For such flow i t i s d i f f i c u l t  time per cycle and therefore d i f f i c u l t  Similar thickness  to define the  to relate i t to the temperature  difference across the c e l l .  1.4 HEAT TRANSFER ANALYSIS  The work by Stewart established that the temperature difference across a rectangular cavity determines the f l u i d v e l o c i t y . cell,  Using such a  the temperature difference can be adjusted to study how the heat  transfer across the c e l l i s affected by the amount of f l u i d flow.  17  FIGURE 1.8  Correlation observed by Stewart between the time to complete one cycle around c e l l and the temperature difference across the c e l l for average melt temperatures of 237 °C, 2 6 0 ^ and 3 0 5 ^ . 8  18  FIGURE 1.9  Radiographs from experiments of Stewart showing (a) twodimensional laminar flow and (b) three-dimensional vortex f l o w . 8  19 For this investigation, i t i s necessary to characterize flow across the c e l l .  the heat  The idea of an e f f e c t i v e thermal conductivity  adopted to describe the heat transfer across the c e l l .  was  Assuming that no  heat i s lost from the top, bottom or side walls then  4 i j cold  4 melt  =  n  where q  n  v  (1.4.1) '  -, i s the rate of heat flow out of the cold end of the c e l l and cold  q , „is the r a t e raelt  of heat flow across the melt.  n  thermal conductivity,  k ^ e  Assigning an e f f e c t i v e &  6  to the melt then  q = k ,-VT ^melt e f f melt  v  (1.4.2) '  q , . = k ,.71 , , cold cold cold  v  (1.4.3) '  Since  H  then  k  TJVT  cold  u = k , VT . cold e f f melt  (1.4.4)  C  If we assume that the heat flow i s primarily one-dimensional then,  k  , cold n  '''cold cold  = k  eff  ^melt melt  (1.4.5) '  v  20  where d  , , and d , ^ are the widths of the c o l d cold melt  respectively  and  J  across the cold 1.4.5  T  - . and cold  end and  T  are the temperature  melt  the melt, respectively.  differences  Rearranging equation  yields  _ eff cold cold k ,* d - " cold melt k  T  (1.4.6a)  d  melt  n  = k ... C . eff  where C i s a constant. of  end and the melt, '  T  ., versus cold  T  T  _ melt  melt  can be used to determine a value f o r k , . eff c  can be defined for any  g  relationship  (1.4.6b) '  Therefore the slope of the tangent to the curve  Using this technique a value for k ^^ difference  v  across the melt.  Once k . i s correlated eff CJ  between the temperature difference  flow v e l o c i t y could be used to relate k ^^ g  to  T  temperature  , t h e n the melt  across the c e l l and the  to flow v e l o c i t y .  eff In the standard heat transfer notation the factor  could be melt  thought of as  n a v  g , the average heat transfer c o e f f i c i e n t for the melt.  The average Nusselt number would then be,  „ Nu  where k  g  avg  ,, = (h  v ^melt ^eff ^melt ^eff ) . = , . , = avg' k d k k Sn melt Sn Sn  i s the stagnant thermal conductivity  of t i n .  . ,v (1.4.7)  / n  To summarize, this study looked at heat transfer across metal that was flowing due to natural convection. difference  across a thin  square  cavity  liquid  Using the temperature  to produce varying degrees of  natural convection, the temperature differences across the cold end and the melt were measured to calculate a value for the e f f e c t i v e conductivity  of the melt.  This value when compared  thermal  to the stagnant  thermal conductivity indicated the magnitude of the enhancement i n heat transfer  due  to  the  presence  of  convection.  experimental work a numerical model was developed the thermal and f l u i d flow behaviour of the c e l l .  In addition to the to give insight  into  22  2.0 EXPERIMENTAL DESIGN  To study the problem  of heat transfer with varying degrees of  convective flow, the geometry of the flow c e l l was kept as simple as possible. enclosure  The c e l l which  would  contained produce  the l i q u i d laminar  metal  in a  two-dimensional  thin flow  square due to  natural convection. The temperature of the ends of the c e l l were varied to was  change the convective flow v e l o c i t y i n the melt. measured  by the flow  techniques described  The flow produced  i n section  2.2.  The  temperature differences were measured across the melt and the cold end. These melt.  values could  then be used to determine the heat flow across the  The l i q u i d metal used i n these experiments was 99.999% pure t i n .  Table II gives the properties of l i q u i d t i n at various temperatures.  2.1 DESIGN OF EXPERIMENTAL CELLS  Three c e l l systems were developed over the course of this study. Certain design features were common to a l l three c e l l s . of  the l i q u i d  ensure  pieces  c e l l was 0.32 cm which should have beeen small enough to  two-dimensional  measurements  required  therefore  materials.  The thickness  flow, according to Stewart. that  the bottom  heat  flow  and side  occurred only walls  The heat  transfer  through  the end  were made of insulating  TABLE I I PROPERTIES OF LIQUID TIN  TEMPERATURE  VISCOSITY  SPECIFIC HEAT  THERMAL  DENSITY  CONDUCTIVITY  COEFFICIENT OF THERMAL EXPANSION  (°c)  (c ) P  cal f ) gm- C  cal  ' fcm-sec-o C1 1  U  1  gm ( } cm  r  V  3  237  2.02  0.0541  0.0798  6.9698  1.0215  260  1.88  0.0543  0.0806  6.9538  1.0239  305  1.68  0.0546  0.0809  6.9217  1.0287  24  2.1.1 C e l l I  To  use the c o r r e l a t i o n  between  time  per c y c l e  and the  temperature difference across the melt determined by Stewart, the c e l l geometry which  and dimensions  he used were adopted  f o r this design,  i s shown i n Figure 2.1. The bottom and side walls were made from  0.32 cm glass sheet. The  which  low  thermal  The cold end piece was made of stainless  conductivity  significant  gradient  across  differences  across the melt.  of  the stainless  the end piece  even  This a b i l i t y  would  steel.  produce  a  f o r low temperature  was considered desireable  since the heat transfer behaviour for low temperature differences across the  melt was of p a r t i c u l a r  flow v e l o c i t y  interest  occurs i n this range.  because  the most rapid change i n  The cold end was cooled by argon  gas jets located i n the assembly attached to the end (see F i g 2.1). The hot  end of the c e l l  consisted of a copper block with a T-shaped  section that had a hole to allow for the heating assembly. consisted sheath.  of chromel heating wire which was wrapped around  cross-  The heater a ceramic  Power was supplied by a variac.  Before assembling the c e l l the inside surfaces were coated with c o l l o i d a l graphite to prevent l i q u i d metal from attacking the c e l l walls and held  to seal any gaps between components. together by bolts  through the bottom  The walls of the c e l l were piece and by the s p e c i a l l y  designed end pieces, as shown i n Figure 2.1. The top of the c e l l was open to allow f o r the thermocouple wires coming out of the melt.  The  Pi  FIGURE 2 . 1  Design of experimental c e l l I  26  mould was suspended the  cell.  i n the furnace by a hangar attached to the ends of  The furnace  temperature  was  controlled  by a  Honeywell  temperature controller (model 5500101).  2.1.2  C e l l II  Since this c e l l was to be used for quench experiments the walls were made of 0.48 cm stainless s t e e l instead of glass which would crack due to thermal shock i f quenched. were  the same as f o r c e l l  Although the o v e r a l l c e l l dimensions  I, new end pieces  were designed  due to  problems with the end pieces i n c e l l I. The cold end was made of copper instead  of stainless  steel.  tube soldered to the back face. cooling  instead  This  copper piece had a U-shaped copper  The tube was used to provide the Ar gas  of the jets used  in cell  I.  soldered between the tube and the end piece.  Two thermocouples were A thermocouple probe was  used to measure temperatures near the hot and cold ends of the melt once the  c e l l was operating.  The bottom piece of the c e l l was made of teflon to minimize heat flow  along  the bottom.  The c e l l  was bolted  clamps were used along the side edges.  through  the bottom and  27  2.1.3. CELL I I I  Results cell  shown  accommodate  from c e l l s  i n Figure  I and II lead to the design of the larger  2.2.  the new c e l l .  A This  different cell  bolts through the bottom of the c e l l . the  furnace was  necessary to  design eliminated  the need f o r  Instead, clamps were used along  bottom and sides of the c e l l , as shown i n the figure.  special  assembly  at the cold  end which  incorporated  There was a  the clamps and  argon jets and provided a shield around the end to prevent argon from flowing into the furnace.  To provide more r e l i a b l e attachment of the thermocouples to the outside face of the cold end, the cold end piece was designed with three threaded holes which allowed the thermocouples to be held i n place by screws.  However the inside thermocouples were s t i l l spotwelded i n place  since the use of screws would d i s t o r t the interface between the melt and the  end piece.  Due to the number of thermocouples, a thermocouple  switch was used to monitor the output of the thermocouples.  It was found that composite c e l l walls were best for quenching. The  cell  wall  consisted  of an inner wall, made from 0.08 cm aluminum  sheet and an outer wall, made from 0.016 cm stainless s t e e l sheet, which were seperated by 0.016 cm t e f l o n spacers at the edges. air  The resulting  gap between the walls provided an insulating layer i n the w a l l .  To  Argon Inlet Tube  D e s i g n of e x p e r i m e n t a l  cell III.  29  quench the melt, water was forced into this a i r gap from a thin-walled stainless  steel  insertion  into  tube  (o.d.«1.2 cm) with  the gap.  Since only  the end flattened  to allow  the aluminum sheet separated the  cooling water from the melt, the molten metal could be quenched i n less than two seconds using this technique.  2.2 FLUID FLOW MEASUREMENT  Two melt. piece  methods were used  to measure  the flow v e l o c i t i e s  in  the  The f i r s t method consisted of monitoring the a c t i v i t y of a small of radioactive  consisted  copper added  of adding radioactive  to the melt.  tin (Sn  1 1 3  The second procedure  ) to the t i n melt and then  quenching the melt to e s t a b l i s h the path of the radioactive t i n i n the melt.  The  general procedure for the copper p a r t i c l e experiment i s as  follows:  i ) The temperature of the melt was monitored by a thermocouple probe.  When the c e l l  radioactive  copper  reached thermal equilibrium,  (Cu  6i+  ),  less  than  2 mm  a piece of  i n diameter was  inserted into the melt.  i i ) Lead bricks were arranged so that the a c t i v i t y of the copper  30  p a r t i c l e could be monitored from the top or side of the c e l l by a fast-rate s c i n t i l l a t i o n counter.  The position of the copper  p a r t i c l e could be determined by estimating the absorption due to the  tin.  The expected p e r i o d i c i t y  i n the count rate would be  d i r e c t l y related to the period of the f l u i d  flow i n the cavity.  The  f o r the experiments  general procedure which  using the radioactive t i n ( S n  1 1 3  ) as tracer was:  i ) Before inserting the S n end  of the melt  Temperatures  were  was followed  was  into the melt, the temperature at  1 1 3  measured  taken  using  a  at approximately  thermocouple  probe.  the middle  of the  inside face of each end.  i i ) The radioactive t i n ( S n the  1 1 3  ) was added near the cold end of  c e l l and after a given time the c e l l was quenched.  i i i ) After quenching, the s o l i d the  block of t i n was removed from  c e l l and placed on a sheet of X-ray f i l m .  Another sheet of  f i l m was placed on top and a glass sheet put on top to ensure good  contact  exposed  between the f i l m  and the sample.  The f i l m was  f o r two to three days  then developed.  The resulting  dark areas on the f i l m indicated the location of the S n thus showed the extent of f l u i d moment the radioactive sample was quenched.  1 1 3  and  flow i n the melt between the  t i n was added  and the moment when the  31 2.3 TEMPERATURE MEASUREMENT  Chrome-alumel  thermocouples  measurements i n this  study.  were  used  f o r the temperature  For c e l l  I, 0.1 mm uncoated thermocouple  wire inside ceramic sheaths was used.  For c e l l s II and I I I , 0.025 cm  wire was used since the f i n e r wire was easier to embrittle during the spot-welding and would often break near the weld.  In addition, coated  wire was used instead of the ceramic sheaths since the larger ceramic sheaths might obstruct the l i q u i d metal from complete contact with the end piece.  The Figure  locations  2.3.  thermocouple since  of the thermocouples  The use of spot-welded  i n each c e l l  thermocouples  probe  was preferred  the location  of the probe  could  be d i f f i c u l t  to reproduce  therefore  i t would  f o r the heat  measurement.  However, f o r the quench  used  i t would  because  have  been  are shown i n instead  transfer  not be defined  of a  measurements precisely and  i t s location  for each  tests, a thermocouple probe was  neccessary  to replace  spot-welded  thermocouples after each sample was removed from the c e l l .  The cold junctions of the thermocouples i n c e l l s I and II were maintained  i n an ice-water bath.  connected d i r e c t l y to  The junctions  to the thermocouple switch.  in cell  A thermometer  I I I were attached  the thermocouple switch was used to determine the temperature of the  32  (a)  (b)  (c)  FIGURE 2.3  Location of thermocouples i n experimental c e l l s a) c e l l I, b) c e l l II and c) c e l l III ( thermocouple location i s marked by T ) .  33  cold junction.  The chart  output from the thermocouples was  recorder  adjusted  to  be  (model no. either  one  194). or  recorded on a Honeywell  The range of the chart two  millivolts  full-scale  recorder  was  deflection,  depending on the magnitude of the temperature differences between the thermocouples.  34  3.0 EXPERIMENTAL RESULTS  3.1 RESULTS FROM FIRST EXPERIMENTAL CELL  Several runs were made using the f i r s t c e l l with the top of the c e l l open and with i t p a r t i a l l y covered. differences  S u f f i c i e n t l y large temperature  were produced without the use of the heater.  Figure 3.1  shows the results plotted as A T , the temperature difference across the gg  stainless s t e e l end versus AT melt. both across  ^ , the temperature difference across the  The experiments with the top of the c e l l covered and uncovered showed  linear  relationships  the cold end and that  between  the temperature  across the melt.  difference  The constant slope i n  F i g u r e 3.1 indicates that kgff> the e f f e c t i v e thermal conductivity, was constant.  3.1.1 Calculation of the E f f e c t i v e Thermal Conductivity  The value of k f f can be c a l c u l a t e d e  graph of AT  versus AT .1- d  from m, the slope of the  . u s i n g equation 1.4.6(a). m »x* t •  According to  equation 1.4.6(a), ^eff ^cold m =^ . -j cold melt  , iv (3.1.1)  / 0  Temperature  FIGURE 3.1  Difference  Across Cell  (°C)  Graph showing r e s u l t s from c e l l I with top of c e l l covered and uncovered. Slopes were calculated using l i n e a r regression of data points. The c o r r e l a t i o n c o e f f i c i e n t i s given i n brackets.  36  Therefore,  , eff k  =  m  h  e  , ^cold * ^melt • — d — ; > cold  ,„ , „ ( - - )  x  (  3  1  v  2  For c e l l I: k  cold  =  T  r  m  a  l  conductivity of stainless s t e e l = 0.044 cal/cm-sec-°C  ^melt  =  D  *-  S t a n c e  across melt = 5.0 cm  ^cold  =  D  ^-  S t a n c e  across cold end = 1.25 cm  (3.1.3)  From figure 3.1:  m (uncovered) = 2.069  (3.1.4a)  m (covered)  (3.1.4b)  Calculating the values f o r k  g f f  = 1.705  using equation 3.1.2 y i e l d s :  k ££ (uncovered) = 0.3641 cal/cm-sec-°C = 4.5 X k Sn e  c  k  eff  (  c o v e r e d  )  =  0.3003 cal/cm-sec-°C s 3.8 X k  0  Sn  k  g n  (stagnant)  =0.08 cal/cm-sec-°C  (3.1.5)  37 3.1.2  Discussion of Results from C e l l I  There are s e v e r a l reasons why k ^ ^ would be constant. have  been  convection.  constant  because  i t was  independent  of  the degree  of  This condition would be expected when the enhancement i n  heat transfer due to convective flow had reached i t s l i m i t . the degree  I t may  of convection produced,  the largest temperature  Looking at differences  employed across the melt were big enough to ensure that the maximum flow was  achieved, according to Figure 1.8.  Therefore i t i s possible that  the heat transfer l i m i t had been reached. the  lowest  temperature  differences  Also according to Figure 1.8,  that were used would only  flows that were marginally slower than the maximum flow. r e a s o n k ^ ^ remained  constant was probably because  g  produce  Therefore the the change i n  convective flow was too small to show an e f f e c t .  The four  times  lower  than  magnitude  of the e f f e c t i v e  the stagnant the seven  thermal  thermal conductivity to t e n times  represents  convection.  the maximum  of t i n .  enhancement  mathematical modelling of continuous casting. number  conductivity  enhancement  i s about  This factor i s  assumed  i n the  I t i s possible that this acheivable  with  natural  However there may be another reason why k ^^ i s low. g  The heat transfer through the c e l l i s not completely determined by  the c h a r a c t e r i s t i c s  of the molten  bath.  The rate  of heat  flow  38 through  the  cell  is  affected  boundaries  as w e l l  as  resistance  of  hot  liquid  t i n , as  the  the  by  the  thermal  thermal r e s i s t a n c e  end  of  the  cell  shown i n T a b l e I I I .  of the melt.  (Cu)  The  resistances  was  lower  of  The  than  the  thermal that  thermal r e s i s t a n c e of the  for cold  end of the c e l l  ( s t a i n l e s s s t e e l ) i s l e s s than the thermal r e s i s t a n c e of  stagnent l i q u i d  t i n but i s g r e a t e r than t h a t p r e s e n t e d by the c o n v e c t i n g  melt. to  From t h i s a n a l y s i s the heat t r a n s f e r a c r o s s the c e l l would  appear  be l i m i t e d by the s t a i n l e s s s t e e l end p i e c e r a t h e r than the behaviour  of  the  melt.  constant.  To  This avoid  reason this  problem  copper f o r both ends of the  3.2  could  also  the  explain  why  next  two  cell  were  conducted  k ff e  remained  designs  employed  cell.  RESULTS FROM CELL I I  The cell  which  next  series  duplicated  of the  experiments dimensions  of  different  materials for i t s construction.  this  attempted  cell  copper would  particle  in  difference state  method.  T h i s method was  p r o v i d e i n f o r m a t i o n about  conditions change  to measure the degree  rather  than  convective was  inferring flow  The  first first  be a  second  but  used  experiments  with  of c o n v e c t i v e f l o w u s i n g the preferred  the f l o w from  could  the  cell  since  the  the f l o w at the temperature  adjusted, providing  behaviour.  the  with  a quench.  observed look at  the  when  technique  measurement As w e l l ,  the  transient  the  temperature and  steady  39  TABLE I I I COMPARISON OF THERMAL RESISTANCES  MATERIAL  THERMAL  LENGTH  THERMAL RESISTANCE*  CONDUCTIVITY L k  L  R = k A  cal (  )  n gm-sec-°C  (cm)  sec- °C ( 1 I ) cal  Copper  0.928  1.27  1.37  Stainless  0.044  1.27  28.97  0.080  5.08  63.50  5.08  16.93  steel Liquid t i n , stagnant Liquid t i n ,  ** 0.30  convecting  For comparison purposes the area, A has been taken = 1 cm Approximate value from equation 3.1.5  40  When the counting rates  from  the copper  particle  experiments  f a i l e d to produce the expected p e r i o d i c i t y the quench technique was used to  look at the flow pattern i n the c e l l .  The radiographs from these  experiments showed nearly uniform greying across the sample indicating that the tracer was well mixed by the time the quench was finished (see Figure 3.2(a)).  Some samples  showed evidence of vortex-type flow (see  Figure 3.2(b)) which would explain why the copper p a r t i c l e experiments failed  to show any p e r i o d i c i t y .  Other  samples  appeared  to be very  turbulent, as shown i n Figure 3.2(c), but i t i s not certain whether the turbulence was present prior to the quench or whether i t was caused by the  quench.  These  results  imply much higher v e l o c i t i e s  than those  expected from the results of Stewart.  3.3 RESULTS FROM CELL I I I  Since  the previous c e l l  size  did not provide the range of  convective flow that was expected i t was decided to conduct experiments on  a  larger  convective  cell  flow  and to establish  and the temperature  quenching the c e l l .  the relationship difference  across  between the the c e l l  by  (c)  Quenched samples from c e l l II (a) Tracer i s well mixed (AT=»6°C, time prior to quench = 30 sec), (b) Sample showing vortex motion ( A T ^ t , time p r i o r to quench = 30 sec), (c) Sample exhibiting turbulent flow (AT-5°C, time prior to quench = 25 sec).  42  3.3.1 Results of Quench Tests  The  first  concern  adequacy of the quench.  of the experiments on the new c e l l was the The stainless s t e e l walls of the previous c e l l  were believed to have had too low thermal conductivity to provide a fast quench  and  i t was  suspected  initiation  of the quench.  tests with  cell  that  some flow  To v e r i f y  had occurred  after the  this suspicion the f i r s t quench  I I I were performed using stainless  s t e e l walls which  were made of two pieces of 0.16 cm stainless s t e e l sheet glued together. The  sampled was  furnace with liquid  from  quenched by rapidly  cooling water.  filling  up the Inside can of the  The quench was very slow and the cooler  the sides would f a l l  pattern shown i n Figure 3.3(a).  to the bottom and produce the flow When quenched i n the manner used f o r  c e l l II by aiming jets of water at the walls the quench was much faster but the flow was s t i l l like  pattern shown  quench  obtained  distorted during the quench producing the fork-  i n Figure 3.3(b).  using  the composite  Figure 3.4 shows a successful cell  wall  design  described i n  section 2.1.3.  The  radiographs  of the quenched samples used to determine the  time per cycle are shown i n Figure 3.5.  The resulting graph of the time  per cycle versus the temperature difference across the melt i s given i n Figure 3.6 and i s very similar to that observed by Stewart.  FIGURE 3.3  Examples of inadequate quenchs from c e l l I I I (a) Sample quenched by rapidly f i l l i n g inside of furnace with water and (b) Sample quenched using water jets aimed at side walls.  44  FIGURE 3.5  (Continued) Quenched samples from c e l l I I I (c) AT-1.4 °C time prior to quench = 22 sec, (d) AT-4.4 °C , time prior to quench = 11 sec.  47  FIGURE 3.5  (Continued) Quenched samples from c e l l III (e) AT=2.4 °C time prior to quench • 18 sec.  48  100  2 4 6 8 10 12 Temperature Difference Across Melt (°C)  FIGURE  3.6  R e l a t i o n s h i p between time r e q u i r e d per c y c l e and the temperature d i f f e r e n c e a c r o s s the melt (Temperatures i n melt were measured u s i n g a thermocouple p r o b e ) .  49 3.3.2  Temperature Measurements  Having established predictable  that the flow was  manner, detailed  temperature  laminar and behaved i n a  measurements were  The measured temperatures were accurate to + 0.3 difference across the melt was  °C.  The  performed. temperature  measured from the mid-point of the hot  face to the mid-point of the cold face.  Temperatures were measured at  three points along the outside face and at three points along the inside face of the cold  end, as shown i n Figure 2.3(c).  These  temperatures  were used to calculate the temperature difference across the cold Rather  than using  the average  temperature  of the outside  and  end.  inside  faces of the cold end to calculate a single temperature difference for the cold end, a temperature difference was thermocouples.  Temperature  calculated for each pair of  differences were calculated for the bottom  and middle sections of the cold end but not for the top section due to problems with the thermocouple at the top of the inside face.  Two  experimental conditions were employed.  For experiment  A,  both the heater i n the hot end and argon gas cooling of the cold end were used  to create the temperature  difference  across the melt.  For  experiment B, only argon gas cooling was used to produce the temperature difference.  Figure  across the cold  3.7  experiment  the plot  of the temperature  difference  end versus the temperature difference across the melt  for the two experiments. for  gives  A,  as  There are two d i s t i n c t regimes i n the results shown  i n Figure  3.7(a).  Initially,  the  50  FIGURE 3.7  Graph of temperature d i f f e r e n c e s a c r o s s c o l d end v e r s u s temperature d i f f e r e n c e a c r o s s melt f o r (a) experiment A which used a h e a t e r and argon c o o l i n g to produce the temperature d i f f e r e n c e a c r o s s m e l t . Slopes i n d i c a t e d on graph were c a l c u l a t e d u s i n g l i n e a r r e g r e s s i o n of d a t a points•  51  0 2 4 6 8 10 Temperature Difference Across Melt ( ° C ) (b)  FIGURE 3.7  (Continued) Graph of temperature differences across cold end versus temperature difference across melt f o r b) experiment B which only used argon cooling to produce the temperature d i f f e r e n c e a c r o s s m e l t . Slopes i n d i c a t e d on graph were c a l c u l a t e d u s i n g linear regression of data points.  52 temperature d i f f e r e n c e the  temperature  difference  transistion  into  between the  two  showed  a  Figure  3.7(b).  associated  across  temperature  with  large  the  flowmeter.  the  that  into  the  argon flow  number  there  contains  linear  linear rates.  experiment A .  across  the  increasing across  end nor  experiment  rate.  behaviour it  B  neither the  At one  point,  d i d not  where  appears  argon  across  the  Figure  shows  the  3.10  to  be  occur i n F i g u r e  only  Figure the  3.7  in  was  dependence  c e l l were p l o t t e d as a  scale  F i g u r e 3.8  across  temperature d i f f e r e n c e  temperatures  in  the  expressed  reading  on  melt  as a the  temperature  associated  cooling  plots  difference  increased uniformly with  the  3.9  shows these  temperature  decreased w i t h i n c r e a s i n g flow r a t e ,  This since  that  Indicated  a c h a r t which can be used to c o n v e r t a  A t low flow r a t e s  argon flow  the melt  3.8(b). heater  cold  relationship  as  The argon flow r a t e i s the  an abrupt  experiment B  To i n v e s t i g a t e  to  to  for  region  i n the  is  a linear  behaviour  g i v e n flow r a t e to m i l l i l i t r e s per m i n u t e . for  is  there  The r e s u l t s  the  corresponding  Appendix I I  appear to be r e l a t e d  c o l d end then  temperature d i f f e r e n c e s  the argon flow r a t e .  dimensionless  does not  differences.  transistion  found  of  the melt  second regime where  was  on flow r a t e , function  the  gradual  It  across  as  with  difference  shown i n F i g u r e the  use  which shows the  was  used  to  of  the  plots  for  produce  the  melt.  relationship  and the argon flow r a t e  f o r the  between  the  two e x p e r i m e n t s .  actual Although  53  FIGURE 3.8  Graphs showing behaviour of temperature differences across (a) the cold end and (b) the melt versus argon flow rate for experiment A.  54  7  0  10  20 Argon  T  1  Argon  FIGURE 3.9  40  30 Flow  1  Flow  50  Role  1  1—< i  Rate  Graphs showing behaviour of temperature differences across (a) the cold end and (b) the melt versus argon flow rate for experiment B.  FIGURE 3.10  Graphs of temperature at centre of outside face of cold end versus argon flow rate for (a) experiment A and (b) experiment B.  56  the  curves are only given for one thermocouple  temperatures throughout the c e l l A,  the  temperature  followed  by  a  showed  large  they are t y p i c a l of a l l  for a given experiment.  an  initial  decrease  increase  (region  II) afterwhich the  decreased steadily, as shown i n Figure 3.10(a).  (region  In experiment I) which  was  temperature  In experiment B, the  temperature curve was similar except for a small increase which preceded the  i n i t i a l decrease i n temperature, as shown i n Figure 3.10(b).  The  temperature  curve i n Figure 3.10(a) could be explained i f  some argon gas had leaked into the furnace. shielded may  to prevent argon gas from flowing into the furnace, some gas  have  escaped.  atmosphere by  this  The  forced  convection produced  rate  in  the  furnace  flow of gas would increase the heat transfer  between the furnace walls and the c e l l . the  Although the cold end was  increase observed i n region I I . the enhancement i n the heat  rate  This phenomena would explain  Beyond a certain argon gas flow  transfer  i n the furnace would reach  some l i m i t and the temperature would then decrease with increasing flow rate as observed i n region I I I . linearity  observed  in  Figure  I t i s interesting  3.7(a)  occurred  to note that the  for  the  temperature  differences corresponding to the region III temperature data.  One the  next  additional observation should be made before proceeding to  section.  The  temperatures  at the bottom  inside face of the cold end were always within 0.2 were often the same.  and middle of the °C of each other and  Therefore the d i s p a r i t y i n the observed values of  57  the temperature cold  end  face.  difference across the bottom and middle sections of the  i s due  The  argon  to the difference i n the temperatures gas  at the outside  flow should have been approximately  the same for  both sections although i f some nonuniformity existed i t i s l i k e l y that the flow was  higher to the middle  Consequently  there  does  seem  section than to the bottom section.  to be  any  obvious  explanation for the  difference i n heat flow rates that i s implied by the difference i n the temperatures.  3.3.3  Calculation of the E f f e c t i v e Thermal Conductivity  The 3.7  values f o r k  using equation  3.1.2.  were c a l c u l a t e d from the slopes i n Figure The  results  are  given i n Table IV.  The  slopes for Figure 3.7(a) were calculated using linear regression of the data points corresponding to region III i n Figure 3.10(a).  The slopes  i n Figure 3.7(b) were calculated by using linear regression of the l a s t three data points although only two actually l i e i n region IV of Figure 3.9(b).  For both experimental cases, the value of k j ^ calculated from g  the data for the bottom section of the cold end i s much higher than that calculated for the middle section of the cold end.  Looking at the r a t i o  of k , to k , the normal thermal conductivity of t i n , the enhancement eff sn' c  is  seven  to ten times k  according to the data from the middle section sn  58  TABLE IV  EXPERIMENT  CALCULATION OF EFFECTIVE THERMAL CONDUCTIVITY  LOCATION  cold  melt  ^melt  m  k  '« eff  k  eff  k  Sn  (COLD END)  cal f ] cm-sec- oC '  (cm)  (cm)  Middle  0.927  10.0  1.25  0.0733  0.5435  6.8  Bottom  0.927  10.0  1.25  0.4625  3.430  ~ 43  Middle  0.927  10.0  1.25  0.1036  0.7685  9.6  Bottom  0.927  10.0  1.25  0.3472  2.575  ~ 32  1  A-Using heater &  cal (cm-sec• C-o V 1  J  Ar cooling  B-Only argon cooling used  59 and i s t h i r t y section.  to f o r t y times k  g n  a c c o r d i n g to data from the bottom  Judging from these r e s u l t s , i t i s not reasonable to use a plot  of the temperature difference  across the bottom section of the cold end  versus the temperature difference calculate k  . eff e c  across the mid-plane of the melt to  60  4.0 MATHEMATICAL MODEL  In previous mathematical models of f l u i d cavity,  the hot and cold  isothermal.  have been assumed  to be  However i n a r e a l system i t i s d i f f i c u l t to ensure uniform  temperatures difference  ends of the c e l l  flow i n a rectangular  at  the ends  and  i n the v e r t i c a l  there  direction.  observe the effect of non-uniform  i s usually  some  temperature  This model was developed to  boundary temperatures.  In addition,  the model was used to compare convective flow i n l i q u i d t i n with that i n liquid  steel.  4.1 GOVERNING EQUATIONS  The  energy  equation,  the Navier-Stokes  equations  and the  continuity equation are needed to solve f o r the temperature and v e l o c i t y fields.  The following  are the form of these equations f o r the two-  dimensional system shown i n Figure 4.1:  The energy equation:  5T* *9T* __ + — — St ox L  u  . *6T* + = oy v  w  k  , o T* . o T* . (— * j + ) pCp ox oy 2  2  (4.1.1)  FIGURE 4.1  System of reference for mathematical model.  62 The Navier-Stokes equations:  - f o r momentum transfer i n the x-direction,  *  cu" * St  *  h  *  * du' * ou U + V = * * Sx Sy  1  *  SP  ?  *  2  *  . ,8 u• 5u + V( + ) o * *2 *2 nm Sx Sx Sy -g8(T*- T*) (4.1.2) N  -for momentum transfer i n the y - d i r e c t i o n ,  *  Sv  *  *  , * Sv • * Sv r U r V  *  *  St  Sx  *  1 =  *  Sy  K  9  *  2  SP  *  , *S v , Sv * + V( + ) (4.1.3) o * *2 *2 i by Sx Sy . i  .  Q  .  The continuity equation:  -^+^L.= 0  * Sx  In applying  (4.1.4)  * Sy  these equations the following assumptions  have been  made:  1) F l u i d properties have been assumed constant except for one term  which  the density  accounts  f o r the temperature  dependence of  63  2) Viscous d i s s i p a t i o n and compressibility  e f f e c t s have been  neglected  3) There i s no heat generation i n the f l u i d  4) The  applied  temperature  difference  i s small compared to  1/(3 (where 8 i s the c o e f f i c i e n t of thermal  4.1.1  Dimensionless  Variables  The asterisks used i n equations 4.1.1 distinguish  the marked variables  computational purposes in  the  governing  expansion)  from  to 4.1.4  have been used to  t h e i r dimensionless form.  For  i t i s easier to use the dimensionless variables  equations  therefore  the  following  dimensionless  variables were introduced:  u =  * v d  * u d  T -T o T = * * » T -T h o  P =  * 2 P.d Pm v  (4.1.5)  64  H.Cp P  r  .  =  cr  k  »  g&d ( T -T ) =  — — v  2  * where  = Temperature of the hot end  T  = Average temperature of the c e l l  q  * T,  h  T  d  +  T  *  c  * = Temperature of the cold end  = Width across c e l l  The Grashof number as defined here i s a modified version of the standard Grashof  number  since  i t i s calculated  using  one-half the temperature  difference across the c e l l rather than the t o t a l temperature difference. The superscript  (•) has been used to distinguish the modified Grashof  number from the standard Grashof number.  Substituting these dimensionless parameters into equations 4.1.1 to 4.1.4. y i e l d s :  oT dT oT 1 b! — + u — + v — = — ( + ) at ax ay Pr ax ay 2  2  2  (4.1.6)  65  ou — + 5t  du u  —  ox  +  v  av dv — + u —+ dt dx  —  ou 5P = - Gr .T - — + dy ox dv  dP  v — = dy  dy  d^ +  dx  d^ + dx 2  5^ dy  (4.1.7) 2  d^ +  2  dy  (4.1.8) 2  du dv — + — =0 dx dy  Equations 4.1.7  4.1.7  with  (4.1.9)  and 4.1.8  respect  to  y  can be combined by d i f f e r e n t i a t i n g and  equation  4.1.8. with  equation  respect  to  x,  subtracting and using equation 4.1.9 to eliminate terms to give,  du dv d V d^ ( ) + u + V d t d y d x dxdy dy 3  dT d (V^) - Gr . — + dy dy #  u  d^ dx"*  v  d^ dydx  d (vM (4.1.10) dx  4.1.2 Stream Function and V o r t i c i t y  To simplify equation 4.1.10, the concepts of the stream function and the v o r t i c i t y were Introduced.  dv  The v o r t i c i t y , E, i s defined as:  du (4.1.11)  dx  dy  66 The stream function, 4> i s defined by the following equations:  o<|>  u =  (4.1.12) oy  0(|»  v =  (4.1.13) ox  Substituting  the v o r t i c i t y  into  equation  4.1.10 y i e l d s  the v o r t i c i t y  equation:  bi ac . ar — + u — +-v — = Gr — + 9t 8x oy by  Equation  4.1.9  i s automatically  stream function.  VC  (4.1.14)  2  satisfied  by  the d e f i n i t i o n  of the  To solve f o r the stream function, equations 4.1.12 and  4.1.13 are substituted into equation 4.1.11 to give the stream function equation:  a 4» 2  5 = - (  ax  Equations  2  +  ay  )  (4.1.15)  2  4.1.6, 4.1.12, 4.1.13, 4.1.14  equations used to solve cavity.  2  ac|>  and 4.1.15  the problem of natural  form  the set of  convection  i n a square  67  4.2 INITIAL AND BOUNDARY CONDITIONS  The are  temperature  steady-state.  and v e l o c i t y p r o f i l e s calculated i n this model  However to reach a solution,  the  time-derivatives  have been retained i n the governing equations so that the computer w i l l produce  a  time-varying solution  solution. for  the  which  converges  to  the steady-state  This technique requires the d e f i n i t i o n of i n i t i a l conditions temperature,  stream  function  and  vorticity.  The  initial  conditions used were,  t=0  0 = x = 1.0 }C=0,  <|, = 0, T = 0  (4.2.1)  0 = y = 1.0  The  solution  of  the  system  d e f i n i t i o n of boundary conditions.  of  equations  also  the the boundaries.  The  isothermal or  having  conducting. a  linear  The  o<l>  x = 0.0  ,1.0  ends were taken to be  temperature  numerically the boundary conditions were,  t>0  taken to be zero  top and bottom surfaces were assumed to be  either insulating or perfectly either  the  Since the velocity must be zero at  the boundaries the gradient of the stream function was at  requires  (|/ =  =0 ox  drop.  Expressed  68 i ) Insulated boundary oT — =0 ox i i ) Perfectly conducting T = -1.0 + 2.0*(1 - y)  y=0.0  • «J, = — = 0 by  (4.2.2)  i ) Isothermal end T = -1.0 i i ) With temperature drop T = -1.0 + (PCT/2)*x  y = 1.0  (\> = — =0 oy i ) Isothermal end T = +1.0 i i ) With temperature drop T = (1-PCT/2)*1.0 - (PCT/2)*x  where PCT = temperature drop along end expressed as percentage of ^ T ^ m e  t  4.3 THE FINITE DIFFERENCE EQUATIONS  The solution of the d i f f e r e n t i a l finite  difference  approximations  equations i s achieved by using  to the governing  equations.  The  69  numerical  technique  used  i n this  model was  the i m p l i c i t  technique, commonly referred to as the ADI method. ADI  technique  to  solve  f u n c t i o n equations.  f o r the  temperature,  Other m o d e l s ' 1 2  1 5  alternating  The model used the  vorticity  and  stream  have used a relaxation technique  to solve for the stream function equation (equation 4.1.15).  However  these models were developed for use with much higher Prandtl numbers and lower Grashof  numbers than those used f o r this model and had reported  problems when using low Prandtl numbers. successful  at the low Prandtl  numbers  Since the model by Stewart was and high Grashof  numbers  that  would be needed f o r this model the same approach was adopted.  The ADI technique divides the time step, A T into two parts.  For  the f i r s t half of the time step a l l the x-derivatives are i m p l i c i t and all  the y-derivatives are e x p l i c i t .  at  t=t  2  and  1 =t]+AT/ 2' 2  half  explicit  derivatives  Implicit derivatives are evaluated are  evaluated  at  t=t^, where  The values for t=t ^ represent the values at the previous  of the time step and are known at t = t . 2  The values at t = t are 2  unknown and must be solved before proceeding to the second half of the time step.  For the second half of the time step, the y-derivatives are  i m p l i c i t and the x-derivatives are e x p l i c i t .  The  finite  difference  approximations  were  generated  from  expansions based on the square grid system of points as shown i n Figure 4.2.  The subscripts I and j indicate the position of the node i n the x  and y directions respectively.  The following equations use an asterisk  FIGURE 4.2  Grid system used i n mathematical model.  71 superscript and  (*) to denote variables  the apostrophe  superscript  evaluated at t=(n+l)Ax. t=nAx.  that are evaluated at  (')  to  denote  t=(n+l/2)AT,  variables  that  are  Variables without superscripts are evaluated at  The f i n i t e difference approximation to the temperature equation  for the f i r s t half of the time step i s ,  rp  * ^  fjt  rn  m  t>j i,j Ax/2  +  1 .  u  *  * rn  t  i  ,  rj\  i,j+i '  -*i,j-i 2 Ay  x +  v  J  T* .... ~ 2T* . + T* ., 1-1,3 1,3 1+1,3 (Ax)  ±  i  =  I T . , . . - 2T, , + T. ... l . j - l l , j '1,3+1 (4.3.!) Pr (Ay)  +  Pr  rn  . i + i ; j ' j - t , j 2 Ax  2  2  For the second half of the time step,  '  *  *  *  rn  1,3  1,3  H  Ax/2  i  *  1 _  ,  '1-1,3 ,  *  i  *  (Ax)  ,  'i,j-l _ 2 Ay  J  »  »  »  I . . . . -2T....+ T. ... i.J-l i,3 1,3+1 ( 4 . . )  +  3  Pr  2  rn  1,3+1  T  2 Ax  J  »  ^n  T . - 2T . + T - . 1 i-l,j i,3 1+1,3 _  Pr  The  1+1,3  u  »  rn  (Ay)  f i n i t e difference equation for the v o r t i c i t y  f o r the f i r s t half of  the time step,  * C  l,3  * " ±,3 L  Ax/2  C +  u x  »  *  i+l,j " i - l , j C  j  2 Ax  C +  v  i,j+1 ~ l , j - 1 C  ^  2  2  2Ay  72  «  j+1 "  - 1 ,  T  75  i-l,j "  g +  2 g  2 Ay  (Ax)  For  ^i+l,j  +  2  +-£1,3+1  Sj.j-l - ^ i . j (Ay)  l,j  75 +  ( 4 > 3 > 3 )  2  the second half of the time step,  '  *  *  " ^i,J  C +  u  i+l,j " i - l , j ,  »  -  *  'i.j-l  ,  - 2E H,j 4  (Ax)  t  »  i,-j-l  2 C  i,:j  (Ay)  To introduced which  solve  the  *  + E - 4+l.j  +  2  i 1  - t,j+l  (4.3.4)  5  2  stream  function  to equation 4.1.15  converges  =  2Ay *  E  2 Ay  g  C  2 Ax  •  T  » " i,j-I  C  l , i  T «/i,j+l  » hy  AT/2  C r  *  equation, a time derivative i s  to produce  an unsteady  to the steady state problem.  state  solution  Using this approach the  f i n i t e difference equation for the stream function for the f i r s t half of the  time step, i s ,  AT/2  i , j  (Ax)  2  73  (4.3.5) (Ay)  :  For the second half of the time step,  *  *  ic  <k i>3,--.<k., i>3 _ g AT/2 >i  +  is  V i-1-1,3  I ~ -  ±  (  A  V - I -  i»3  2  x  )  +  i+l»3  * (  2  (4.3.6)  (Ay)  To  solve  2  the temperature,  vorticity  and  stream  function  equations, the above equations are rearranged to c o l l e c t a l l the unknown terms on one side of the equation and the known terms on the right hand side of the equation. be  found  i n appendix  A complete l i s t  of the rearranged equations can  II of reference 8.  The resulting c o e f f i c i e n t  matrix for the unknown variables  i s a tridiagonal matrix which can be  easily  Once the temperature,  inverted  f o r solution.  v o r t i c i t y and  stream function have been calculated f o r each node, the new v e l o c i t i e s are  calculated.  Using  equations  4.1.12  and 4.1.13, the following  expansions were generated to solve for the v e l o c i t i e s :  ,  = 1 , 3  A dy  - *i.J-2 ~ , J  8 ( p  i,j-l  +  8 4 ,  12Ay  i , j + l " +1,1+2  (4.3.7)  74  ^  _ ^-2,3- - ^ i - y ^ k . j  *1 j • " M l ox  - V l . j  (4.3.8)  12 Ax  j  For  points near the boundary a different expansion was used which was of  the  form:  _ ~ »1,2 3  U  i  +  H  i , 3 " *1,4  (4.3.9)  2 '  X  v„  6Ay  .  =^  ^  (4.3.10)  6Ax  Expansions of a similar form were used f o r the other boundaries.  An boundary  additional  vorticities  vorticity. vorticities.  calculation since  Equation  no boundary  4.1.15  To s a t i s f y  was  was  performed  to  determine  condition was defined  used  to  solve  f o r the  the  f o r the boundary  the condition that the v e l o c i t y must be zero  normal to the boundary,  b <\> = 0 by 2  at x=0,l  § <\> 2  and at y=0,l  ox  2  = 0  (4.3.11)  2  Therefore at the boundary, equation 4.1.15 becomes,  dc|> 2  at x=0,l  E =  dx  2  a<j> 2  and at y=0,l  E =  dy  2  (4.3.12)  75  The expansions used for the boundaries were of the form,  h  2  hi  2 <'' > 4 3 13  (Ay)  2  4.4 RESULTS OF COMPUTER RUNS  During the development  of the model i t was found that a separate  time step was needed for the solution of the stream function which was kept  constant for a l l runs.  step used  This time step was larger than the time  f o r the temperature  and v o r t i c i t y  equations.  convergence was much more rapid i f the l a t t e r  In addition,  time step was changed as  For Grashof numbers > 1 0 even small changes i n  the  program progressed.  the  temperature-vorticity time step would markedly affect the number of  5  iterations  that were required for convergence.  was  f o r Grashof numbers >^ 1 0 .  found  7  would appear might  No convergent  From the results  solution  obtained,  it  that adjusting the value of the stream function time step  increase model s t a b i l i t y  at the higher Grashof  numbers.  Close  examination of the data used by Stewart did not reveal the time step used but i t did show that a f i n e r mesh size was used for Grashof number equal to 1 0 . 7  This  limit  seems to occur for Ra > 1 0 .  would seem to concur with the s t a b i l i t y  o t h e r models Instability  The region of i n s t a b i l i t y  of c o n v e c t i o n i n a r e c t a n g u l a r  5  limit cavity  observed i n 1 2  '  2 1  .  The  i n the numerical model seems to associated with the onset of  secondary flows i n the c a v i t y .  2 1  To test  the computer program, isotherm, streamline and Nusselt  number plots were generated for a square cavity with isothermal ends and insulated top and bottom boundaries using the Prandtl number for t i n (Pr = 0.0127) and various Grashof numbers.  Figures 4.3,  the  The  computer plots  that were produced.  4.4  and 4.5 show  results were i n excellent  agreement with the plots generated by the numerical model of Stewart for similar Prandtl and Grashof values. generated results,  which  show  the  velocity  given i n Figure 4.6,  In addition, v e l o c i t y plots were distribution  show the development  i n the  cell.  The  of small secondary  flows i n the upper l e f t and lower right corners of the v e l o c i t y plot for Gr = 10 . 6  The temperatures of the .ends were adjusted so that there was a l i n e a r temperature drop from the top to the bottom of the ends. experiments, i t was were usually  In the  found that the temperatures at the top of the c e l l  cooler  than at the bottom  therefore the gradient i n the  model had the coolest temperatures at the top boundary.  The temperature  drop was expressed as a percentage of the temperature difference across the  melt.  For example, a 5% temperature drop at the end boundary would  mean a difference of 0.1  °C between top and bottom i f the temperature  difference across the c e l l was 2.0  °C.  Figure 4.7 shows the effect of a  5% and 10% temperature drop for Pr=0.0127 and Gr=1.0X10 .  Figure  4.8  for Pr=0.0127  and  5  shows  the  Gr=1.0X10 . 6  the  ends.  effect  of  the  same  temperature  drops  The effect on the isotherms i s p a r t i c u l a r l y pronounced near The presence of the temperature gradients i n the ends also  FIGURE 4.3  Isotherm d i s t r i b u t i o n f o r Pr=0.0127 and (a) Gr^L.OXlO (b) Gr=1.0X10 and (c) Gr=1.0X10 . 5  6  4  78  79  FIGURE A.5  Nusselt number versus v e r t i c a l (x) p o s i t i o n along cold wall f o r Pr=0.0127 and Gr-l.OXlO , 1.0X10 and 1.0X10 . 11  5  6  80  ! 11' {{ \ \:'> < 11  -->>>//;;  M  \ \ \ w  ' \ \ \ \  • » \ \  ^  -  '  ^  ^  v  ;  '  / / / /  w ^ - - —  • * \ \ \ x ^ 1  * x  x. ~^  N  (a)  -  •». v \ x  V \ \  * /// / > ' ' ' \ J 1 if i i < ' \ \  \ \ W  » v  \ \ \ \ W  v  x  x  v -  X *  J»  t  t  \  \ \ \ \\  f  W  \  * t 4 M  * >*  X  \ \  f  \\  t \ 1 f  t  \ \\ \ \ ^ \ ' \ i • v  (b)  FIGURE 4.6  \ \ \ \  \ \ N "V \, -v -«. -» -» v s. — —'  Normalized v e l o c i t y f i e l d f o r Pr=0.0127 and (a) Gr=10 (maximum v e l o c i t y =487.44) and (b) Gr=10 (maximum v e l o c i t y = 1185.09). V e l o c i t i e s given are dimensionless velocities.  5  6  (a)  (b)  FIGURE 4.7  Isotherm plots with Pr=0.0127 and Gr=1.0X10 . Solid l i n e indicates p r o f i l e s f o r isothermal ends and dashed l i n e indicates those f o r (a) a 5% drop along ends and (b) a 10% drop along ends. 5  FIGURE 4.8  Isotherm plots with Pr=0.0127 and Gr=1.0X10 . Solid l i n e indicates p r o f i l e s f o r isothermal ends and dashed l i n e indicates those f o r (a) a 5% drop along ends and (b) a 10% drop along ends. 6  83  FIGURE 4.9  Streamline plot with Pr=0.0127 and Gr=1.0X10 with (a) isothermal ends and (b) 10% drop along both ends (with coldest temperature at top). 6  84  affects the streamline d i s t r i b u t i o n as shown i n Figure 4.9. s l i g h t effect on the shape of the streamlines. stream  function  i s lower  slower v e l o c i t i e s decreased  by  when  the  i n the cavity.  about  eight  There i s a  The maximum value of the  gradients  are present  indicating  However the maximum v e l o c i t y  only  percent with a ten percent drop i n the end  temperatures.  The much  velocities  higher  difference  than  that were measured for c e l l  the  velocities  i n the c e l l  Stewart was  measured  design between c e l l  problem of l i q u i d the bottom  between  the  situation along  II and  piece of the c e l l  the c e l l  major  used  by  end  the  pieces  through which  bottom  of  gradient  the  heat  and  the  cell.  To  along the bottom  top  surface  could  a temperature  was  flow. gradient  investigate of the c e l l  the  insulating,  function  are lower  especially  would  exist  effect  of  the ends were taken to be  insulating.  The  effect  than when the bottom  was  on  v e l o c i t i e s would be slower.  the  case when there was  the  The values of taken to be  for the higher Grashof number, indicating  the  a  computer runs were  isotherms seems to be limited to the bottom of the c e l l . stream  For such a  along the bottom of the c e l l .  the results shown i n Figures 4.10 and 4.11  isothermal  This design avoided the  but i t also provided a thermal l i n k  performed with a linear gradient imposed  the  One  t i n leaking from the junction between the end piece  I t i s conceivable that  temperature  For  Stewart.  to be  the fact that Stewart used a U-shaped piece inside the c e l l  which connected the cold end to the hot end.  and  by  II appeared  that  Computer runs were also performed for  l i n e a r temperature drops i n both ends  combined  FIGURE 4.10  Isotherm plot with l i n e a r temperature gradient along bottom (isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0X10 and (b) Gr=1.0X10 . 5  6  86  FIGURE 4.11  Streamline plots with l i n e a r temperature gradient along bottom (isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0X10 and (b) Gr=1.0X10 . 5  6  87  FIGURE 4.12  Isotherm plots with l i n e a r temperature gradient along bottom and 10% temperature drop along ends with Pr= 0.0127 and (a) Gr=1.0X10 and (b) Gr=1.0X10 . 5  6  88 with the linear temperature gradient along the bottom.  The results do  not  isothermal  differ  greatly  temperatures Figure  from  those produced  with the linear  gradient  when assuming  along the bottom,  as  end  shown i n  4.12.  The reduction i n v e l o c i t y due  to the presence of the gradient  along the bottom does not appear to be large enough to account for the difference  i n the  results  Another factor which may  from  cell  II and  the results  of Stewart.  have influenced the flow v e l o c i t y i s the heat  flow out the side walls.  The walls of the c e l l used by Stewart were  made of 0.0625 inch aluminum sheet whereas the side wall for c e l l II was made of 0.1875 inch stainless s t e e l . were less conductive.  Therefore the walls for c e l l II  A three-dimensional mathematical model would  be  necessary to estimate the effect on the v e l o c i t i e s but such a model i s beyond the scope of this study.  F i n a l l y , computer runs were performed to compare the differences in  the  responses  differences 4.13  of  of 0.5  to 4.19.  liquid  °C and 0.01  t i n and °C.  The  liquid  steel  The isotherm plots do not d i f f e r greatly although there  streamline  temperature  distribution  difference.  The  is  more  velocity  steeper near the edges of the c e l l shown  temperature  results are shown i n Figures  i s more curvature i n the isotherms for l i q u i d s t e e l . the  to  i n Figure  4.18.  However  noticeable gradients  The difference i n for  the  i n steel  0.5  °C  are much  at this temperature difference, as  looking  at  the v e r t i c a l  scale,  the  89  0-75  (b)  FIGURE 4.13  Isotherm d i s t r i b u t i o n f o r a temperature difference of 0.01 °C across c e l l f o r (a) l i q u i d t i n (Pr=0.0127 and Gr =3.6X10'*) and (b) l i q u i d s t e e l (Pr=0.11 and Gr=6.0X10 ). 3  90  FIGURE 4.14  Isotherm d i s t r i b u t i o n f o r a temperature difference of 0.5 °C across c e l l f o r (a) l i q u i d t i n (Pr=0.0127 and Gr=1.8X 10 ) and (b) l i q u i d steel (Pr=0.0127 and Gr=3.0X10 ). 6  5  91  (b)  FIGURE 4.15  Streamline d i s t r i b u t i o n for a temperature difference of 0.01 °C across c e l l f o r (a) l i q u i d t i n (Pr=0.0127 and Gr =3.6X10 *) and (b) l i q u i d s t e e l (Pr=0.11 and Gr=6.0X10 ). 1  3  92  FIGURE 4.16  Streamline d i s t r i b u t i o n f o r a temperature difference of 0.5 °C across c e l l f o r (a) l i q u i d t i n (Pr=0.0127 and Gr =1.8X10 ) and (b) l i q u i d s t e e l (Pr=0.11 and Gr=3.0X10 ). 6  5  93  700  (a) 04 06 Y Position 140  005  • 0 035  O020  - 0 005  6 "  S  - -0 005  (b)  FIGURE 4.17  02  04 06 Y Position  -0020 10  08  V e r t i c a l (x) v e l o c i t y versus horizontal (y) position at x=0.5 f o r a temperature difference of 0.01 °C across c e l l i n (a) l i q u i d t i n (Pr=0.0127 and Gr=3.6X10 *) and (b) l i q u i d steel (Pr=0.11 and Gr=6.0X10 ). 1  3  94  2600  (a)  -2400  04 06 Y Position  600  (b)  FIGURE 4.18  -400,  02  04 06 Y Position  08  V e r t i c a l (x) v e l o c i t y versus horizontal (y) p o s i t i o n at x=0.5 for a temperature difference of 0.5 °C across c e l l i n (a) l i q u i d t i n (Pr=0.0127 and Gr=1.8X10 ) and (b) l i q u i d steel (Pr=0.11 and Gr=3.0X10 ). 6  5  95  FIGURE 4.19  Plot of l o c a l Nusselt number versus p o s i t i o n along cold wall f o r t i n and s t e e l at temperature differences of 0.01 °C and 0.5 °C.  96  velocities  in liquid  t i n are much higher than those i n l i q u i d  steel.  The v e l o c i t y scale for l i q u i d t i n i s f i v e times that for steel i n Figure 4.17  and six times i n Figure 4.18.  97  5.0 SUMMARY REMARKS  One  of  the  aims  of  this  study  was  to observe  how  the heat  transfer changed with increasing convective flow i n a l i q u i d metal. interest  was  i n determining  at  what  convective flow  rate  the  The heat  transfer across the c e l l became independent of the degree of convection. As long as the heat transfer was degree  of  flow  could  not  be  related to the flow v e l o c i t y then the ignored  i n mathematical  models  estimate a value for the e f f e c t i v e thermal conductivity. the linear  limit  was  reached  which  However once  then i t would only be important to know  that f l u i d flow i s present not the actual flow v e l o c i t y .  Looking at the data from c e l l I, the heat transfer rate appeared to be linear over the entire range of temperature differences employed. However the l i n e a r i t y  observed was  probably due  thermal resistance of the stainless s t e e l end was the convecting melt.  to the fact  that the  higher than that for  In fact, the thermal conductivity of stainless i s  about one-half the thermal conductivity of stagnant l i q u i d  tin.  This  s i t u a t i o n i s opposite to the s i t u a t i o n encountered i n l i q u i d - s o l i d  heat  transfer i n s o l i d i f i c a t i o n .  For most metals, the thermal conductivity  of the s o l i d state i s higher that the thermal conductivity of the l i q u i d state therefore the results from c e l l I were not representative of the behaviour during s o l i d i f i c a t i o n .  98  The study  results  which  from  examined  cell  I raise  natural  parallelogrammic enclosure.  some questions about  convective  heat  transfer  a recent across  a  In this study water and s i l i c o n e o i l were  used as the experimental media.  The cold end of the c e l l consisted of a  copper plate immediately adjacent to the convecting f l u i d followed by a series of four glass plates which were the same thickness as the copper plate  afterwhich there was another  positioned  between  the  glass  differences across the plates. glass,  the plates  served  copper  plates  plate.  to  Thermocouples were  measure  the  temperature  Knowing the thermal conductivity of the  as a heat  flux  meter.  Since the thermal  conductivity of glass i s much lower than that f o r copper, the thermal resistance  of the glass  heat  flux  thermal resistance of the copper. glass that  i s comparable  meter would  be much higher  than  In fact the thermal conductivity of  to the stagnant  thermal conductivity  of water so  the thermal resistance of the glass plates was probably the same  order of magnitude as the convecting f l u i d .  Based on the results from  c e l l I i t seems l i k e l y that the heat transfer across the parallelogram was influenced by the heat flux meter.  To calculate a value f o r the e f f e c t i v e thermal conductivity i t i s assumed that a l l heat flow i s through the cold end. The difference i n the slopes f o r the covered and uncovered c e l l indicate that the heat lost  from the top of the c e l l cannot be ignored.  The value of k ^ ^ i s  approximately twenty percent lower using data from the covered c e l l than when using data from the uncovered  cell.  However the effect  of the  heat loss appears to a constant f r a c t i o n of the heat transfer across the  99  cell  so  that  the  shape of  the AT  versus AT cold  r  . ^ curve i s not melt  affected.  In the results from c e l l was not c l e a r .  Using both argon cooling and the heater i t was  to achieve temperature 3.5  °C.  I I I , the behaviour at low flow rates  differences across the c e l l  Using only argon cooling,  differences  i s hard  to  interpret  that were less than  the data at the lower because  the  across the bottom section of the cold end was  difficult  temperature  temperature negative.  difference  The negative  sign means that heat should have been flowing into the cold end of the c e l l instead of out of the c e l l . magnitude of  the  Once the l i n e a r l i m i t was reached, the  e f f e c t i v e thermal conductivity  was  calculated  to be  seven to ten times the thermal conductivity of stagnant l i q u i d t i n .  According to Figure 3.7, bottom  section  was  the temperature  higher than the temperature  difference across the difference  across the  middle section implying higher heat transfer rates across the bottom of the c e l l .  This s i t u a t i o n i s i n d i r e c t c o n f l i c t with the predictions of  the computer model as r e f l e c t e d i n the Nusselt number plots shown i n Figures 4.5 and 4.19.  According to these plots, the heat transfer rate  at the bottom of the cold end should be less than that across the centre of the cold end.  The highest heat transfer rates should be encountered  at the top of the c e l l .  Unfortunately data was  not available for the  upper section of the cold end but further experimention i s necessary to resolve the disagreement  i n the heat flow d i s t r i b u t i o n .  100  REFERENCE LIST  1.  Cole, G.S. and G.F B o i l i n g , The S o l i d i f i c a t i o n of Metals, 1968, London, The Iron and Steel I n s t i t u t e , 323-329.  2.  L a i t , J.E., J.K. Brimacombe and F. Weinberg, Iron, and Steelmaking, 1974, No. 1, 35-42.  3.  Mizakar, E.A., Trans. Met. Soc. AIME, 1969, v o l 239, 1747-1753.  4.  Szekely, J . and V. Stanek, Met. Trans., 1970, v o l 1, 119-126.  5.  L a i t , J.E., J.K. Brimacombe and F. Weinberg, Iron, and Steelmaking, 1974, No. 2, 90-97.  6.  MacAuley, L.C. and F. Weinberg, Met. Trans., 1973, Vol 4, 2097-2107.  7.  Catton,I., Proc. Sixth Int. Heat Transfer Conf.,1978, 13-31.  8.  Stewart, Murray, PhD Thesis, 1970, University of B r i t i s h Columbia.  9.  Batchelor, G.K., Q. Appl. Math., 1954, Vol 12, 209-233.  10. Batchelor, G.K., J . F l u i d Mech., 1956, Vol 1, 177. 11. Wilkes, J.0. and S.W.  C h u r c h i l l , A.I.Che.J., 1966, Vol 12, 161-166.  12. De Vahl Davis, G., I n t . J . Heat Mass Tran., 1968, Vol 11, 1675-1693. 13. Gershuni, G.Z., E.M. Zhukhouiskii and E.L. Tarunin, 1966, Mech, Liquids Gases, Akad. S c i . USSR No. 5, 56-62. 14. Poots, G., Q. J . Mech. Appl. Math., 1958, Vol 11, No 3, 257-273.  101 15. Szekely, J . and A.S. J a s s a l , Met. Trans. B, 1978, Vol 9B, 389-398. 16. Marr, H.S., Iron & Steel Int., 1979, 29-41.  17. Heaslip, L. et a l . , Proc. 2nd Process Technol. Conf., Chicago, 1981, AIME, 54-63.  18. Kohn, A., The S o l d i f I c a t i o n of Metals, 1968, London, The Iron and Steel I n s t i t u t e , 415-420. 19. Moore, J . J . , and N.A. Shah, I n t . Met. Review, 1983, Vol 28, 338-356.  20. Blank, J.R. and F.B. Pickering, The S o l i d i f i c a t i o n of Metals, 1967, London, The Iron and Steel I n s t i t u t e , 370-376. 21. Szekely, J . and M.R. Todd, I n t . J . Heat Mass Transfer, 1971, Vol 14, 467-482. 22. Maekawa, T. and I. Tanasawa, Proc. Seventh Int. Heat Transf. Munchen, Fed Rep Germany, Vol 2, 227-232.  Conf.,  APPENDIX I  LIST OF SYMBOLS  S p e c i f i c heat Distance across c e l l Acceleration due to gravity Grashof number Modified Grashof number Heat transfer c o e f f i c i e n t Thermal  conductivity  E f f e c t i v e thermal conductivity Thermal conductivity of t i n Characteristic Nusselt  length  number  Pressure Dimensionless pressure Prandtl number Heat flow rate Rayleigh number Time Dimensionless time Temperature Dimensionless  temperature  Velocity i n x-direction Dimensionless v e l o c i t y i n x-direction Velocity i n y-direction Dimensionless v e l o c i t y i n y - d i r e c t i o n Coordinate i n v e r t i c a l d i r e c t i o n Dimensionless coordinate In v e r t i c a l d i r e c t i o n Coordinate i n horizontal d i r e c t i o n Dimensionless coordinate i n horizontal d i r e c t i o n  Greek symbols  Thermal c o e f f i c i e n t of expansion Temperature  difference  Time step (computer program) Vorticity Viscosity Kinematic v i s c o s i t y Density Mean density Stream function  104 APPENDIX I I  Chart for Argon Flow Rate Conversion  CAL/BRAT/ON  CATALOG  SERIAL  STD. A/R ML./  D  = O. / £ 5 "  f  *STD.=  /ATM.  =  •ii! :  :::: :•:: ::::  :: i: :t:: i:;:  iiiiiip:  70°F  25  •1  !  1 ! '•:.:!••:  i iji:ii  i:iji i  ::  ;  ;  ;  i  :• j  i  ::j":  -  r-  : • • i. .  to  --j--  :  '  "J "•  ...  —j  :::|:^n "--!:•-.'.-[•:  eo  ....;.:.  . _.: ..  ii  i  -  :  .... .........  ..!..  i ;  <o  so  READ/MG  •:hl— :-:—  .........  .:rni:  1  .:.:L...  30 SCALE  !::•: •• •/o •: "-: •.  ... L..-  . i: --  30  —  1  200-  i: i- :•  .:..L: J  • •!  . .  :4:T  :  :  ;  -  1 i  ..:]  ——  •• ! :  ...  i1  iii!  I: .: .:.-t-  •-T-  1 ' r •  • 1  -:r  •1 •  . j.;.  - •-;•:'-.:.......  : i i i i i: •  •  ....I..:.  •!»• ••  ;;  :.i •i  i  •  11  400-  i *  I-  !  800600-  '  —1  i  tooo-  •  ' 'i  Li!!::: _ _ L :  E  -i  .•  i;::  ::::|:::: :  . i . .: j - .  :  STD* WATER ML  GM./ML.  f  :!!;  i;;:ji:;:  I200-  i!  12.00  p =Z.53  CM.  : i. . .i :_.J.... :: i :  F  G277-G487-G4&  NO.  AHO  NO.  1111  FLOWMETER  1 •  ^  CHART  1 1 1 1 1 M | 1II  NO. 2  -±"  .j.  eo AT  70 CEA/TEA  ~i—  -::-— •::!:::: • i- -  .ii-!-'  ——:  :  eo  OP  BAS-L.  so  ~1~  too  25  20 15  APPENDIX I I I - COMPUTER PROGRAM 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  C C C C C C C C  * * * * * * * * * * * ** * THIS IS M0DEL5  ***********************  THIS PROGRAM IS A MODIFICATION OF M0DEL4 T H I S VERSION ALLOWS THE TIME STEP TO BE CHANGED AT ANY ARBITRARY I T E R A T I O N NUMBER(S) INTEGER F L . F L A G , I F L A G , I E R R  C REAL *8 TA,TB.GR,PR,DT,DX,DY,DIF,DIFF,CHECK,SFC REAL*8 C 1 , C 2 , C 3 , C 4 . C 5 , C 6 , C 7 , C 8 , C 9 , C 1 0 . C 1 1 . C 1 2 . C 1 3 . C 1 4 REAL PCTHOT.PCTCOL C C C  c  INPUT DELTA  20  PRANDTL T (DT)  NUMBER  ( P R ) ; GRASHOFF  NUMBER  (GR) ;  READ(6,20)PR,GR,DTA,DTB.DTC.0T2 F0RMAT(6F10.3)  c SFC=0.08D0 LIM=21 DX=0.05D0 DY=0.05D0 TA = - 1.ODO TB= 1.0D0 N = 441 REAL*8 T 1 ( 2 1 . 2 1 ) . T 2 ( 2 1 . 2 1 ) . X 1 ( 2 1 . 2 1 ) . X 2 ( 2 1 . 2 1 ) . S 1 ( 2 1 . 2 1 REAL*8 A ( 1 9 ) , B ( 1 9 ) , C ( 1 9 ) , R ( 1 9 ) , D ( 2 1 ) , E ( 2 1 ) . F ( 2 1 ) , T ( 2 1 ) REAL 8 +  c  U ( 2 1 . 2 1 ) . V ( 2 1 .2 1 )  K1=2/DT2 + K2=2/DT2 K3=2/DT2 + K4=2/DT2 L I 1=LIM-1 LI2=LIM-2 FLAG=0  2/DX* 2 2/DX**2 2/DY**2 2/DY**2  c  c  INITIALIZE  MATRICES  c  c  DO 50 0=1 ,LIM DO 50 1=1.LIM T1(I,d)=0.0 T2(I.J)=0.0 X 1 ( I , <J ) =o. 0 X 2 ( I . J ) =0.0 S1(I,J)=0.0 S2(I.J)=0.0 U ( I . J ) =0.0 50 V ( I , J ) =0.0 DO 54 1=1,19 A(I)=0.O B(I)=0.0 C(I)=0.0 54 R ( I ) = 0 . 0 DO 58 1=1.21  k  106  59 60 61 62 63 64 65 66 67 68 69 70 7 1 72 73 74 75 76 77 78 79 80 8 1 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 11 1 1 12 1 13 1 14 1 15 1 16  D(I ) =0.0 E(I)=0.0 F(I)=0.0 58 T ( I ) = 0 . 0  C C C LOADING FIXED TEMPERATURES C PCTHOT=0.0 PCTCOL=0.0 C DO 6 0 1 = 1 . L I M T 1 (I , 1 ) = TA + P C T C O L * I / L I M T2(I . 1 )=TA+PCTCOL*I/LIM T1(I,LIM)=(1-PCTH0T)*TB+PCTH0T*I/LIM 60 T2(I.LIM)=(1-PCTH0T)*TB+PCTH0T*I/LIM C C SOLVING FOR THE TEMPERATURES C F I R S T HALF OF TIME STEP C C 9 0 FLAG = F LAG +1 C OT=DTA IF ( F L A G . G T . 4 ) DT =DTB IF ( F L A G . G T . 4 0 ) DT = DTC C1=1/(2*DX) C2=1/(2*DY) C3=1/(PR*DX**2) C4=1/(PR*DY**2) C5=2/DT +2*C3 C6=2/DT - 2 * C 3 C7=2/DT +2*C4 C8=2/DT - 2 * C 4 C9= 1/DX**2 C10=1/DY**2 C1 1=2/DT + 2*C9 C12=2/DT - 2*C9 C13=2/DT + 2 * C 1 0 C14=2/DT - 2 * C 1 0 W R I T E ( 6 , 93 )FLAG 93 FORMAT( ' ' , ' START OF LOOP ' ,G6) C C IF LAG =1 . 0 C C S O L V I N G FOR COLUMNS 2 TO LIM-1 C DO 110 J = 2 . L I 1 DO 100 1 = 2 , L I 1 96 D ( I ) = - C 1 * U ( I . J ) - C 3 E ( I ) =C5 F(I)= C1*U(I,J)-C3 100 T ( I ) = ( C 4 + V ( I , J ) * C 2 ) * T 1 ( I , J - 1 ) + ( C 4 - V ( I 1+C8*T1(I.d) D(1)=0 E(1)=C5 F( 1) = - 2 * C 3  107  1 17 1 18 119 120 12 1 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  C C  T( 1 )=C4*T1 ( 1 ,J- 1 )+C8*T 1 ( 1 , d ) + C d+1 4*T1 ) ( 1. D(LIM)=-2*C3 E(LIM)=C5 F(LIM)=0 T ( L I M ) = C 4 * T 1 ( L I M . d - 1 ) + C 8 * T 1 ( L I M , d ) + C 4 * T 1 ( L I M , d+1 ) CALL T R I S L V ( 2 1 . D , E , F . T , 0 , 9 0 0 ) DO 105 1 = 1,LIM 105 T 2 ( I , d ) = T ( I ) 110 CONTINUE  IFLAG=IFLAG+1 C C C SECOND HALF OF TIME STEP (TEMPERATURES) C C SOLVING FOR ROWS 2 TO LIM-1 C DO 2 0 0 1 = 2 . L I 1 DO 175 d = 3 . L I 2 K = d- 1 A(K)=-V(I,d)*C2-C4 B(K)=C7 C(K)= V ( I . d ) * C 2 - C 4 175 R ( K ) = ( C 3 + U ( I , d ) * C 1 ) * T 2 ( I - 1 , d ) + ( C 3 - U ( I , d ) * C 1 ) * T2(1+1 , d ) 1+C6*T2(I.d) A( 1 )=0 B(1)=C7 C(1)= V(I,2)*C2-C4 R ( 1 ) = ( C 3 + U ( I , 2 ) * C 1 ) * T 2 ( I - 1 , 2 ) + ( C 3 - U ( I ,2 )*C1 ) * T2(1+1 . 2 ) 1 + C 6 * T 2 ( I . 2 ) + ( V ( I , 2 )*C2+C4)*T1 ( 1 , 1 ) A ( L I 2 ) = - V ( I , L I 1)*C2-C4 B(LI2)=C7 C(LI2)=0 L1=LI1 R ( L I 2 ) = ( C 3 + U ( I , L 1 ) * C 1 ) * T 2 ( I - 1 , L 1 ) +-(U C(3I , L 1 ) *C1 )-*T2( 1+1 1 + C 6 * T 2 ( I , L I 1) + ( C 4 - V ( I , L I 1 ) * C 2 ) * T 1 ( I . L I M ) CALL TRISLV(19,A.B.C.R.0,900) DO 180 d = 2 . LI 1 K = d- 1 180 T 1 ( I , d ) = R ( K ) 200 CONTINUE  c c c  TOP ROW DO 2 2 0 d = 2 . L I 1 K = d- 1 A(K)=-C4 B(K)=C7 C(K)=-C4 220 R ( K ) = C 6 * T 2 ( 1 . d ) + 2 * C 3 * T 2 ( 2 , d ) A( 1 )=0 B(1)=C7 C(1)=-C4 R( 1 ) = C 6 * T 2 ( 1 , 2 ) + 2 * C 3 * T 2 ( 2 , 2 ) +C4 + T1( 1. A(LI2)=-C4 B(LI2)=C7 C ( L I 2 )=0  D  108  175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 2 16 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232  C C C  C C C  R ( L I 2 ) = C 6 * T 2 ( 1 , L I 1) + 2 * C 3 * T 2 ( 2 . LI 1 ) CALL T R I S L V ( 1 9 . A , B . C . R , 0 , 9 0 0 ) DO 2 4 0 J = 2 , L I 1 K=J-1 240 T 1 ( 1 , d ) = R ( K )  +C4*T1(1.LIM)  BOTTOM ROW DO 2 5 0 J = 2 . L I 1 K=J-1 A(K)=-C4 B(K)=C7 C(K)=-C4 2 5 0 R ( K ) = C 6 * T 2 ( L I M , d ) + 2 * C 3 * T 2 ( L I 1,d) A(1)=0 B(1)=C7 C(1)=-C4 » R( 1 ) = C 6 * T 2 ( L I M . 2 ) + 2 * C 3 * T 2 ( L I 1,2 ) + C 4 * T 1 ( L I M . 1) A(LI2)=-C4 B(LI2)=C7 C(LI2)=0 R ( L I 2 ) = C 6 * T 2 ( L I M . L I 1 ) + 2 * C 3 * T 2 ( L I 1 ,LI 1 ) + C 4 * T 1 ( L I M . L I M ) CALL T R I S L V ( 1 9 , A , B . C , R , 0 . 9 0 0 ) DO 2 6 0 <J = 2. LI 1 K=J-1 260 T 1 ( L I M , d ) = R ( K )  IFLAG=IFLAG+1 C C INTERIOR V O R T I C I T I E S C F I R S T HALF OF TIME STEP C C SOLVING FOR COLUMNS 2 TO LIM-1 C DO 310 J = 2 . L I 1 DO 3 0 0 I = 3 , L I 2 K=I-1 A(K)=-U(I,J)*C1-C9 B(K)=C11 C(K)= U ( I , d ) * C 1 - C 9 300 R ( K ) = ( C 1 0 + V ( I , J ) + C 2 ) * X 1 ( I . J - 1 ) + ( - V ( I . d ) * C 2 + C 1 0 ) * X 1 ( I . J + 1 ) 1+C14*X1 ( I , d ) + G R * C 1 * ( T 1 ( I , J + 1 ) - T 1 ( I , d - 1 ) ) A(1)=0 B( 1 ) =C 1 1 C( 1 )= U ( 2 . d ) * C 1 - C 9 R(1) = (C10+V(2,J)*C2)+X1(2,J-1 ) + (-V(2,d)*C2+C10)*X1(2,d+1) 1+C14*X1(2,d) 1+GR*C1*(T 1 (2 ,d+1 ) - T 1 ( 2 , d - 1 ) ) + ( C 9 + U ( 2 . d ) * C 1 ) * X 2 ( 1 , d l A ( L I 2 ) = -U(LI 1 . d ) * C 1 - C 9 B(LI2)=C11 C(LI2)=0 R(LI2)=C14*X1(LI1.d)+(C9-U(LI1.d)*C1)*X2(LIM.d) 1 + ( C 1 0 + V ( L 1 . d ) * C 2 ) * X 1 ( L 1 , d - 1 ) + (- V ( L 1 , d ) * C 2 + C 1 0 ) * X 1 ( L 1 . d + 1 ) 1+GR*C1*(T1(LI1.d+1)-T1(LI1,d-1)) CALL T R I S L V ( 1 9 . A . B . C , R , 0 . 9 0 0 ) DO 305 1 = 2 , L I 1  109  233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 27 1 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  C C C C C C C C  C C C C  305 310  K=I-1 X2(I,J)=R(K) CONTINUE  I F L A G =IFLAG+1 SECOND HALF  OF TIME  STEP  S O L V I N G FOR ROWS 2 TO  (VORTICITlES)  LIM-1  DO 3 5 0 1 = 2 . L I 1 DO 3 4 0 d = 3 . L I 2 K = d- 1 >(K)=-V(I,d)*C2~C10 B(K)=C13 C(K)= V ( I . J ) * C 2 - C 1 0 340 R ( K ) = ( C 9 + U ( I , d ) * C D * X 2 ( I - 1 . d ) + ( C 9 - U ( I , J ) *C D * X2(1+ 1 ,d) 1 + G R * C 1 * ( T 1 ( I , 0 + 1 )-T1 ( I . d - 1 ) ) + C 1 2 * X 2 ( I . d ) A ( 1 )=0 B(1)=C13 C(1)= V(I.2)*C2-C10 . 2 ). R ( 1 ) = ( C 9 + U ( I . 2 ) * C 1 ) * X 2 ( I - 1 , 2 ) + ( C 9 - U ( I , 2 ) * C D * X2(1 + 1 1+C12*X2<1.2) 1 + G R * C 1 * ( T 1 ( I , 3 ) - T 1 ( I , 1 ) ) + ( V ( I , 2 ) * C 2 + C 1 0 ) * X 1 (•I 1 > A(LI2)=-V(I,LI1)*C2-C10 B(LI2)=C13 C(LI2)=0 L1=LI 1 R ( L I 2 ) = ( C 9 + U ( I , L 1 )*C 1 ) * X 2 ( I - 1 , L 1 ) + ( C 9 -.U L (1I ) *C1 ) * X 2 ( 1 + 1 1+C12*X2(I,LI1)+(-V(I,LI1)*C2+C10)*X1(I,LIM) 1+GR*C 1 * ( T 1 ( I , L I M ) - T 1 ( I . L I 2 ) ) CALL T R I S L V ( 1 9 . A , B , C , R , 0 , 9 0 0 ) DO 3 4 5 d = 2 . L I 1 K = d- 1 345 X 1 ( I . d )=R(K) 350  CONTINUE  IFLAG=IFLAG+1 C C STREAM FUNCTION C F I R S T H A L F OF TIME STEP C FL=0 3 6 0 F L = FL+ 1 DO 4 0 0 d = 2 . L I 1 DO 3 7 5 1 = 3 , L I 2 K=I-1 A ( K ) = -C9 B(K)=K1 C(K)=-C9 375 R ( K ) = X 1 ( I . d ) + C 1 0 * S 1 ( I , d - 1 ) + K 4 * S 1 ( I , d ) + C 10* S 1 ( I . d + 1 ) A ( 1 )=0 B(1)=K1  110  291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 31 1 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348  C C C  C( 1 ) = -C9 R( 1 ) = X 1 ( 2 , d ) + C 1 0 * S 1 ( 2 , d - 1 ) + K 4 * S 1 ( 2 , d ).+C 10*S 1 ( 2 , J + 1 ) A(LI2)=-C9 B(LI2)=K1 C(LI2)=0 R(LI2)=X1(LI1,d)+C10*S1(LI1,J-1)+K4*S1(LI 1 , d ) + C 1 0 * S 1 ( L I 1 .d+1 ) CALL T R I S L V ( 1 9 , A , B , C . R , 0 , 9 0 0 ) DO 3 8 0 1 = 2 , L I 1 K = I-1 380 S 2 ( I , d ) = R ( K ) 4 0 0 CONTINUE  IFLAG=IFLAG+1 -(FL-1) C C SECOND HALF OF TIME STEP (STREAM FUNCTION) DO 4 7 5 1 = 2 , L I 1 DO 4 5 0 d = 3 , L I 2 K = d- 1 A(K)=-C10 B(K)=K3 C(K)=-C10 4 5 0 R ( K ) = X 1 ( I , d ) + C 9 * S 2 ( I - 1 , d ) + K 2 * S 2 ( I . d ) + C 9 * S 2 ( 1 + 1,d) A( 1 )=0 B(1)=K3 C(1)=-C10 R ( 1 ) = X 1 ( I . 2 ) + C 9 * S 2 ( 1 - 1 , 2 ) + K 2 * S 2 ( I , 2 ) + C 9 * S 2 ( I + 1,2) A(LI2)=-C10 B(LI2)=K3 C(LI2)=0 R ( L I 2 ) = X 1 ( I , L I 1 ) + C 9 * S 2 ( I - 1 , L I 1 ) + K 2 * S 2 ( I , L I 1 ) + C 9 * S 2 ( 1+1 , LI 1 ) CALL T R I S L V ( 1 9 . A , B , C , R , 0 , 9 0 0 ) DO 4 6 0 d = 2 , L I 1 K = d- 1 460 S K I , d)=R(K) C 475 CONTINUE C IFLAG=IFLAG+1 -(FL-1) C C CHECK FOR CONVERGENCE OF STREAM FUNCTION C IF ( F L . G T . 5 0 ) GO TO 1000 TEST=C9*(4*S1(2,2)-S1(2,3)-S1(3.2)) IF ( X 1 ( 2 , 2 ) . E Q . O ) GO TO 5 2 0 CHECK=(TEST-X1(2.2))/X1(2.2) IF ( F L A G . G T . 1 0 ) S F C = 0 . 0 5 IF ( F L A G . G T . 2 2 ) S F C = 0 . 0 2 IF ( C H E C K . G T . S F C ) GO TO 3 6 0 C 520 CONTINUE C C C A L C U L A T I O N OF V E L O C I T I E S C DO 5 5 0 1 = 2 , L I 1 DO 5 2 5 d = 3 , L I 2 525 U ( I , d ) = ( S 1 ( I , d - 2 ) - 8 * S 1 ( I , d - 1 ) + 8 * S 1 ( I , d + 1 ) - S 1 ( I , d + 2 ) ) / ( 12 DY ) +  Ill  349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406  U(I,2)=(-3*S1(I,2)+6*S1(I,3)-S1(I,4))/(6*DY) 550 U ( I . L I 1 ) = - ( - 3 * S 1 ( I . L I 1 ) + 6 * S 1 ( I . L I 1 - 1 ) - S 1 ( I . L I 1 - 2 ) ) / ( 6 * D Y ) DO 6 0 0 J = 2 . L I 1 DO 5 7 5 1 = 3 , L I 2 575 V ( I . J ) = ( - S 1 ( I - 2 . d ) + 8 * S 1 ( I - 1 . J ) - 8 * S 1 ( 1 + 1 . d ) + S 1 ( I + 2 . J ) ) / ( 1 2 ' V(2.d)=(3*S1(2.d)-6*S1(3.d)+S1(4.d))/(6*DX) 6O0 V ( L I 1 . d ) = - ( 3 * S M L I 1 . d ) - 6 * S 1 ( L 1 1 - 1 . d ) + S 1 ( L I 1 - 2 . d ) ) / ( 6 * D X )  C C C C C C C A L C U L A T I O N OF BOUNDARY V O R T I C I T I E S DO 6 2 5 d = 2 . L I 1 X2(1,d)=-2*S1(2.d)*C9 X1( 1 , d ) = X 2 ( 1 . d ) X2(LIM.d)=-2*S1(LI1.d)*C9 625 X 1 ( L I M . d ) = X 2 ( L I M . d ) DO 6 5 0 1=2.L11 X2( I , 1 ) = -2 + S K I , 2 ) * C 1 0 X 1 ( I , 1 ) = X 2 ( I . 1) X2(I,LIM)=-2*S1(I,LI1)*C10 650 X 1 ( I . L I M ) = X 2 ( I . L I M ) C C C C CONVERGENCE CHECK OF TEMPERATURES C DIF=0 IF ( F L A G . G T . 7 0 0 ) GO TO 7 2 0 DO 7 0 0 d = 2 . 2 0 DO 7 0 0 1 = 2 , 2 0 IF ( T K I . d ) . N E . O ) DIFF=(T1(I.d)-T2(I.d))/T1(I.d) IF ( 0 A B S ( D I F F ) .GT . D I F ) DIF=DABS(DIFF) 70O CONTINUE IF ( D I F . G T . 0 . 0 0 5 ) GO TO 9 0 C GO TO 7 2 5 720 WRITE(6,721 ) 721 FORMAT ( ' ' . ' * * * * * * * * * * * * * * INTERRUPT IN E F F E C T  c c c c c c  725 CONTINUE OUTPUT  850  c  860 865 870 880  TEMPERATURES AND V E L O C I T I E S  DO 8 8 0 1 = 1 . L I M WRITE(6.850)1 FORMAT( ' ' . ' D A T A FOR ROW NUMBER ' . G 5 ) WRITE(6.860)(T1(I.d),d=1.LIM) WRITE(6,865)(U(I,d),d=1,LIM) WRITE(6.870)(V(I,d),d=1,LIM) FORMAT(' '.'TEMPERATURES:'.21F9.6) FORMAT( ' ' . ' X - V E L O C I T Y : ' .2 IF 1 0 . 3 ) FORMAT( ' ' , ' Y - V E L O C I T Y : ' .2 IF 1 0 . 3 ) CONTINUE  ')  112  407 408 409 410 41.1 412 4^3 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463  GO TO 3 0 0 0 C C ************************^********+*********^************ C C ERROR STATEMENTS ASSOCIATED WITH T R I S L V C £ *+***********#*****+***+************+***#*****+++*•*+**** C 900 WRITE(6.960) IFLAG 9 6 0 F O R M A T ( ' ' . ' SOLVE COULD NOT FIND A S O L U T I O N , FLAG = ' . 1 6 ) GO TO 3 0 0 0 C 1000 CONTINUE C INSERT WRITE STATEMENT WHICH SAYS STREAM FUNCTION NOT C CONVERGING AFTER 10 ITERATIONS C WRITE(6,1200) 1200 F O R M A T C ' . ' S T R E A M FCN NOT C O N V E R G I N G ' ) C DO 1600 J = 1 . L I M WRITE(6,1500)S1(2.J),S2(2 . J ) 1500 F O R M A T ( ' ' . ' S 1 = '.F16.4.' S2= ' . F 1 6 . 4 ) 1600 CONTINUE C C WRITE(6,1610)FLAG.CHECK 1610 F O R M A T ( ' ' , ' NO. OF LOOPS RUN = ' . G 8 , ' CHECK = ' . F 1 6 . 5 ) C C GO TO 3 0 0 0 C 2 0 0 0 CONTINUE C INSERT WRITE STATEMENT WHICH SAYS TEMPERATURES NOT CONVERGING C WRITE ( 6 , 2 2 0 0 ) 2 2 0 0 F O R M A T ( ' ' . 'TEMPERATURES NOT C O N V E R G I N G ' ) C C DO 2 3 0 0 1=1.21 WRITE(6.2250)(T2(I,d).J=1,5) 2250 FORMATC ' . 5 F 1 6 . 5 ) 2 3 0 0 CONTINUE C WRITE(6.2325) 2325 FORMAT(' ' , ' S T R E A M FUNCTION') C DO 2 4 0 0 1 = 2 , 2 0 WRITE(6,2350)(S1(I.J).J=1,5) 2350 FORMAT(' '.5F16.5) 2 4 0 0 CONTINUE C GO TO 3 0 0 0 C 3 0 0 0 CONTINUE C STOP END  

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