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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 British 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 Metallurgical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1984 © CHRISTINE ELIZABETH HARRISON, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f METALLURGICAL ENGINEERING The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A p r i l 26 . 1984 DE-6 (3/81) i i ABSTRACT This investigation was undertaken to examine the heat flow characteristics of a liquid metal system in which f l u i d flow is present due to buoyancy forces. Previous investigations of heat flow in liquids has been confined to transparent materials, which have very different flow characteristics compared to liquid metals. Measurements were made on liquid t i n contained in a thin square cavity which had a temperature difference imposed across the c e l l to produce natural convection. The heat flow across the c e l l was calculated from the measured temperature difference across the cold end plate and the thermal conductivity of the plate. Using the calculated heat flow and the measured temperature difference across the melt the effective thermal conductivity of the melt was calculated. Two c e l l sizes were studied. The thermal conductivity of the cold end plate was found to have a significant effect on the heat transfer through the c e l l . Radioactive tracers were used to observe the flow pattern in the melt and to measure the flow velocity as a function of the temperature difference across the c e l l . The technique involved insertion of radioactive Sn^^3 into the melt, then quenching the sample after a given i i i length of time. The sample was then autoradiographed to determine the path of the tracer after insertion into the melt. The flow was found to be very fast for the smaller of the two c e l l sizes which exhibited three-dimensional flow characteristics. The larger c e l l produced laminar, two-dimensional flow. A correlation was observed between the time per cycle and the temperature difference across the large c e l l . The study also includes a finite-difference model which was developed to provide further insight into the thermal and f l u i d flow behaviour of the melt. The model examines the effect of nonuniform temperatures along the ends and bottom of the c e l l on the temperature and velocity fields and is used to compare the response of liquid t i n and liquid steel to identical temperature differences. Results from the model indicate that either a temperature drop along the hot and cold ends of the c e l l or the presence of a linear gradient along the bottom of the c e l l would decrease the maximum fluid velocity In the c e l l . The present investigation shows that the enhancement of the thermal conductivity due to the presence of natural convection in the liquid metal can be as high as ten times the stagnant thermal conductivity. However the degree of enhancement is influenced by the thermal resistance at the boundaries. i v TABLE OF CONTENTS 1.0 INTRODUCTION 1.1 Fluid 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 Cell I 24 2.1.2 Cell II 26 2.1.3 Cell III 27 2.2 Fluid Flow Measurement 29 2.3 Temperature Measurement 31 3.0 EXPERIMENTAL RESULTS 3.1 Results From Cell I 34 3.1.1 Calculation of effective thermal conductivity 34 3.1.2 Discussion of results from c e l l I 37 3.2 Results From Cell II 38 3.3 Results From Cell III 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 Cell 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 vorticity 65 4.2 I n i t i a l And Boundary Conditions 67 4.3 Finite Difference Equations 68 4.4 Results Of Computer Runs 75 5.0 SUMMARY REMARKS 97 REFERENCE LIST 100 APPENDIX I. List of Symbols 102 APPENDIX II. Chart for Argon Flow Rate Conversion 104 APPENDIX III. Computer Program 105 v i LIST OF TABLES PAGE TABLE I Comparison of Fluid Properties 5 TABLE II Properties of Liquid Tin 23 TABLE III Comparison of Thermal Resistances 39 TABLE IV Calculation of Effective Thermal Conductivity 58 ) v i i LIST OF FIGURES FIGURE 1.1 Fluid flow progression after start of casting. 1 FIGURE 1.2 Flow patterns induced by electromagnetic s t i r r i n g . 1 6 PAGE FIGURE 1.3 Effect of input stream on fl u i d flow in liquid pool (a) Flow patterns induced by different types of input streams 1 7 (b) Example of how input stream induces flow below the mould. 1 8 FIGURE 1.4 Fluid flow at solid-liquid i n t e r f a c e . 1 9 11 FIGURE 1.5 Convection currents in a solidifying ingot 20 11 FIGURE 1.6 Temperature isotherms for (a) purely conductive heat flow and (b) convective heat flow ( T 2 > Tj^). 8 13 FIGURE 1.7 V e r t i c a l (x) v e l o c i t y plotted versus horizontal position across mid-plane of cavity (x=0.5) according to (a) the analytical model of Batchelor and (b) the numerical model of Stewart which shows velocity profiles for Pr= 0.0127 and various Grashof numbers. 15 FIGURE 1.8 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 temperatures of 237°C, 260°C and 305°C . 8 melt 17 FIGURE 1.9 Radiographs from experiments of Stewart showing (a) two -dimensional laminar flow and (b) three-dimensional vortex flow. 8 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 in experimental cells a) c e l l I, b) c e l l II and c) c e l l III ( thermocouple location is marked by T). 32 v i i i FIGURE 3.1 Graph showing results from c e l l I with top of c e l l covered and uncovered. Slopes were calculated using linear regression of data points. The correlation coefficient is given in brackets. 35 FIGURE 3.2 Quenched samples from c e l l II (a) Tracer is 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 FIGURE 3.3 Examples of inadequate quenchs from c e l l III (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 FIGURE 3.4 Successful quench sample acheived using composite c e l l wall design. 44 FIGURE 3.5 Quenched samples from c e l l III (a) AT=7.6 °C, time prior to quench = 20 sec (b) £T=7.2 °C, time prior to quench = 21 sec. 45 FIGURE 3.5 (Continued) Quenched samples from c e l l III (c) AT = 1.4 °C, time prior to quench = 22 sec, (d) AT=4.4 °C , time prior to quench = 11 sec. 46 FIGURE 3.5 (Continued) Quenched samples from c e l l III (e) AT = 2.4 °C, time prior to quench = 18 sec. 47 FIGURE 3.6 Relationship between time required per cycle and the temperature difference across the melt (Temperatures in melt were measured using a thermocouple probe). 48 FIGURE 3.7 Graph of temperature differences across cold end versus temperature difference across melt for (a) experiment A which used a heater and argon cooling to produce the temperature difference across melt. Slopes indicated on graph were calculated using linear regression of data points. 50 ix FIGURE 3.7 (Continued) Graph of temperature differences across cold end versus temperature difference across melt for b) experiment B which only used argon cooling to produce the temperature difference across melt. Slopes indicated on graph were calculated using linear regression of data points. 51 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. 53 FIGURE 3.9 Graphs showing behaviour of temperature differences across (a) the cold end and (b) the melt versus argon flow rate for experiment B. 54 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. 55 FIGURE 4.1 System of reference for mathematical model. 61 FIGURE 4.2 Grid system used in mathematical model. 70 FIGURE 4.3 Isotherm distribution for Pr=0.0127 and (a) Gr=1.0X10\ (b) Gr=1.0X105 and (c) Gr=1.0X106. 77 FIGURE 4.4 Streamline plot for Pr=0.0127 and (a) Gr^L.OXlO1*, (b) Gr=1.0X105 and (c) Gr=1.0X106. 78 FIGURE 4.5 Nusselt number versus vertical (x) position along cold wall for Pr=0.0127 and Gr=1.0X10\ 1.0X105 and 1.0X106. 79 FIGURE 4.6 Normalized velocity f i e l d for Pr=0.0127 and (a) Gr=105 (maximum velocity =487.44) and (b) Gr=106 (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. 80 FIGURE 4.7 Isotherm plots with Pr=0.0127 and Gr=1.0X105. Solid line indicates profiles for isothermal ends and dashed line indicates those for (a) a 5% drop along ends and (b) a 10% drop along ends. 81 FIGURE 4.8 Isotherm plots with Pr=0.0127 and Gr=1.0X106. Solid line indicates profiles for isothermal ends and dashed line indicates those for (a) a 5% drop along ends and (b) a 10% drop along ends. 82 FIGURE 4.9 Streamline plot with Pr=0.0127 and Gr=1.0X106 with (a) isothermal ends and (b) 10% drop along both ends (with coldest temperature at top). 83 FIGURE 4.10 Isotherm plot with linear temperature gradient along bottom (Isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0Xl05 and (b) Gr=1.0X106. 85 FIGURE 4.11 Streamline plots with linear temperature gradient along bottom (isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0X105 and (b) Gr=1.0X106. 86 FIGURE 4.12 Isotherm plots with linear temperature gradient along bottom and 10% temperature drop along ends with Pr= 0.0127 and (a) Gr=1.0X105 and (b) Gr=1.0X106. 87 FIGURE 4.13 Isotherm distribution for a temperature difference of 0.01 °C across c e l l for (a)liquid t i n (Pr=0.0127 and Gr =3.6X10k) and (b)liquid steel (Pr=0.11 and Gr=6.0X103). 89 FIGURE 4.14 Isotherm distribution for a temperature difference of 0.5 °C across c e l l for (a) liquid t i n (Pr=0.0127 and Gr=1.8X 10 6) and (b) liquid steel (Pr=0.0127 and Gr=3.0X105). 90 FIGURE 4.15 Streamline distribution for a temperature difference of 0.01 °C across c e l l for (a)liquid tin (Pr=0.0127 and Gr =3.6X10**) and (b)liquid steel (Pr=0.11 and Gr=6.0X103). 91 FIGURE 4.16 Streamline distribution for a temperature difference of 0.5 °C across c e l l for (a)liquid t in (Pr=0.0127 and Gr =1.8X106) and (b)liquid steel (Pr=0.11 and Gr=3.0X105). 92 FIGURE 4.17 Vertical (x) velocity versus horizontal (y) position at x=0.5 for a temperature difference of 0.01 °C across c e l l in (a) liquid t in (Pr=0.0127 and Gr=3.6X101*) and (b) liquid steel (Pr=0.11 and Gr=6.0X103). 93 x i FIGURE 4.18 Vertical (x) velocity versus horizontal (y) position at x=0.5 for a temperature difference of 0.5 °C across c e l l in (a) liquid t i n (Pr=0.0127 and Gr=1.8X106) and (b) liquid steel (Pr=0.11 and Gr=3.0X105). 94 FIGURE 4.19 Plot of local Nusselt number versus position along cold wall for t i n and steel at temperature differences of 0.01 °C and 0.5 °C. 95 x i i ACKNOWLEDGEMENTS I would like to thank my supervisor, Dr. Fred Weinberg for his patience and guidance throughout this investigation. I would also like 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 and a grant from the National Science and Engineering Council. 1 1.INTRODUCTION An analysis of many m e t a l l u r g i c a l processes involves knowledge of the temperature d i s t r i b u t i o n i n the system. In recent years t h i s problem has been approached by using a heat transfer mathematical model analysis to estimate thermal p r o f i l e s . To do t h i s analysis requires data re l a t e d to boundary conditions and thermal properties of the materials involved. I f the system contains l i q u i d the task becomes more d i f f i c u l t since heat flow through a l i q u i d i s markedly influenced by f l u i d flow i n the l i q u i d . Such Is the case for s o l i d i f i c a t i o n processes i n large ingots, continuous casting and c r y s t a l growth as well as heat loss determinations i n ladles and tundishes. No quantitative information i s a v a i l a b l e for heat transfer i n a l i q u i d metal with f l u i d flow present. In p r a c t i s e , an estimated heat transfer c o e f f i c i e n t i s used which consists of the atomic thermal conductivity m u l t i p l i e d by an a r b i t r a r y number. The present i n v e s t i g a t i o n was undertaken to determine the heat transfer through a l i q u i d metal with known f l u i d flow i n the melt. 1.1 FLUID FLOW F l u i d flow i n a l i q u i d metal can r e s u l t from natural convection or be induced by mechanical or e l e c t r i c a l means to produce forced 2 convect ion i n the l i q u i d . 1.1.1 Natura l Convection Natura l convect ion i s caused by thermal or composit ional gradients which give r i s e to densi ty changes i n the f l u i d . The e f f ec t of the buoyancy force produces the f l u i d motion. Thermal convection can be present even at very low temperature d i f fe rences as shown by Cole and B o i l i n g 1 i n F igure 1.1. Composit ional d i f f e rences are almost always produced during s o l i d i f i c a t i o n as so lute segregates in to the melt . Convection from th i s cause i s most pronounced when the so lute and s o l v e n t a re s i g n i f i c a n t l y d i f f e r e n t i n d e n s i t y . Such d e n s i t y d i f f e rences cause complex flow patterns which change apprec iably with time and are therefore d i f f i c u l t to c l e a r l y de f ine . The geometry and phys i ca l proper t ies of the system great l y a f f e c t the f l u i d f low. The geometry re fe rs to the shape of the boundaries and the presence of any ba r r i e r i n the f l u i d . In many cases the boundaries can be very invo l ved , which i s e spec i a l l y true fo r s o l i d i f i c a t i o n where the s i z e and shape of the l i q u i d pool i s con t inua l l y changing with time. In add i t ion the boundary i t s e l f i s not c l e a r , cons i s t i ng of a p a r t i a l l y s o l i d and l i q u i d r eg ion . The system i s too complicated to inves t iga te d i r e c t l y and a simpler steady-state geometry i s r equ i red . 3 . 1 Fluid flow progression after start of casting. 1 4 Many of the studies on heat transfer and natural convection have been undertaken because of the interest in heat transfer in nuclear reactors, solar collectors and other industrial systems. Consequently the geometries studied have attempted to duplicate the pipes and ducts encountered in these applications. These geometries often cannot be directly related to those encountered in liquid pools in solidification or other applications where large volumes of liquid metal are contained in a vessel, which makes i t d i f f i c u l t to apply the heat transfer results. In addition, most studies have been made using transparent materials such as water, oils and gases. The physical properties of these fluids differ markedly from those of liquid metals, as shown in Table I. The disparity is particularly pronounced in the values of the density and thermal conductivity. To understand the significance of the differences in the physical properties i t is useful to look at the dimensionless parameters that are of importance to natural convection, namely the Grashof number and the Prandtl number. The Grashof number, Gr is the ratio of the buoyancy force times the inertia force over the shear force squared and is defined as follows: = B g L 3 AT v 2 ( 1 . 1 . 1 ) TABLE I COMPARISON OF FLUID PROPERTIES FLUID VISCOSITY (cp) SPECIFIC HEAT cal ( \ THERMAL CONDUCTIVITY cal ( 1 DENSITY gm 1 1 THERMAL COEFFICIENT OF EXPANSION 1 ( 1 PRANDTL NUMBER GRASHOF NUMBER 1 a ' cm-sec- C l ~ J cm \ ] Liquid t i n 8 1.88 0.054 8.0X10"2 6.953 1.02X10-^ 0.013 3.6X106 lead 8 2.39 0.038 3.9X10"2 10.62 1.15X10" *» 0.024 5.8X106 " s t e e l 8 6.5 0.12 7.0X10"2 6.95 2.0X10" *• 0.11 6.0X105 Al 4.5 0.259 2.0X10"1 2.37 - 0.058 -Water 1.38 1.0 1.4X10"3 1.00 I^XIO- 1* 10.0 1.3X106 Alr(50°C) 0.019 0.25 2.11X10"11 0.0011 - 0.225 -NHjCl 1 5 1.30 0.776 1.12X10"3 1.013 1.86X10-5 9.0 2.8X101' Oi l #1 2 1 2.5X101* 0.239 2.36X10"^ 1.54 1.98X10" *» 2.5X105 1.9X10"3 O i l #2 2 1 1.9X103 0.45 4.92X10"11 1.06 7.6X10-1* 1.7X101* 6.1X10" 1 6 where 8 = Coefficient of thermal expansion g = Acceleration due to gravity L = Characteristic length of the system AT = Temperature difference v = Kinematic viscosity The Prandtl number, Pr is the ratio of the momentum to the thermal diffusivity and is defined as: Pr=J±-£2. (1.1.2) k where u = viscosity Cp = Specific heat k = Thermal conductivity Table I shows the difference in the Prandtl number between the transparent and metallic materials. Also given in Table I is the Grashof number which has been calculated for a one degree temperature difference and an arbitrary characteristic length. The Grashof number and the Prandtl number are very important to the heat transfer characteristics of the system. The temperature distribution is related to the Rayleigh number which is the product of these numbers. In natural convection the Nusselt number is usually a function of the Grashof number and the Prandtl number. The Nusselt 7 number, Nu is a dimensionless parameter which is defined as: Nu = — (1.1.3) k where h is the heat transfer coefficient. 1.1.2 Forced Convection Forced convection refers to fl u i d motion which is caused by external forces rather than by buoyancy forces. Many examples of forced convection can be found in metallurgical systems. Figure 1.2 shows the range of flow patterns that can be produced by electromagnetic stirrers used in the continuous casting of steel. The exact extent and velocity of the fl u i d flow is not clear at present. In continuous casting the input stream of molten metal represents a considerable source of momentum to the liquid pool. The flow pattern produced and the extent of i t s effect is determined by the manner in which the stream enters the liquid pool, as shown in Figure 1.3. Below the mould as the momentum from the input stream is dissipated, i t Is unclear to what degree the flow is caused by the input stream or by natural convection. The flow region represents a situation where there is combined forced and free convection. FIGURE 1 . 2 Flow patterns induced by electromagnetic s t i r r i n g . FIGURE 1.3 Effect of input stream on f l u i d flow i n liquid pool (a) Flow patterns induced by different types of input streams 2 0 (b) Example of how input stream Induces flow below the mould. 2 0 10 1.2 CONVECTION IN INDUSTRIAL PROCESSES The interest in convective flow is best understood by looking at some of the applications where such flow is present. The behaviour of convecting liquid metal is of particular interest in the solidification of castings. The convecting f l u i d is a major determinant of the f i n a l cast structure due to i t s effect on heat and mass transfer. Figure 1.4 shows the pattern of fl u i d flow believed to be present at the sol i d -liquid interface. Natural convection produces counter current flow in the bottom end of large steel ingots which is thought to be responsible for the segregation pattern observed in these castings (see Figure 1.5). In continuous casting, the flow in the liquid pool is more complex than that in ingots. Work with radioactive tracers 2 has shown that mixing is most pronounced in and near the mould but in regions below the mould the behaviour of the f l u i d i s not clear. In mathematical models of the solidification profile in continuous casting, the liquid pool is assumed to be completely mixed everywhere. To account for the enhancement in the heat transfer due to convection in the liquid, the liquid is assumed to act as a conducting solid with an e f f e c t i v e thermal conductivity, k g f f The value of kg^^ has been taken i n the l i t e r a t u r e 3 ' 1 * ' 5 to be seven to ten times the stagnant thermal conductivity. FIGURE 1.5 Convection currents in a solidifying ingot. 12 Convection is also of concern in crystal growth where the uniformity of the crystal composition is directly related to product quality. Experiments 6 have shown that f l u i d flow due to convection is 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 in reference 7. Before discussing previous investigations some of the theory concerning natural convection w i l l be introduced. Theoretical analysis of natural convection is based on the solution of the energy equation and the Navier-Stokes equations. The energy equation for a two-dimensional system is as follows: o 2! 5% 9T oT ST -k( + ) + pCp (u — + v — ) = pCp — (1.3.1) ox 2 oy 2 ox dy ot where u and v are the x- and y-components of velocity, respectively. The f i r s t term in the equation represents the heat from conductive input. The second term is the convective transport contribution. Assuming no heat generation these two quantities are equal to the heat accumulation which is the term on the right-hand side of the equation. The result of the convective terms in this equation is that the isotherms for convective heat transfer are not straight as for heat transfer by conduction. Instead they are bent as shown in Figure 1.6. 13 FIGURE 1.6 Temperature isotherms for (a) purely conductive heat flow and (b) convective heat flow ( T 2 > T ^ . 8 14 Since the x- and y-components of velocity appear in the energy equation the velocity f i e l d must be known to solve for the temperature f i e l d . For forced convection the velocity f i e l d can be solved independant of the thermal f i e l d . However for natural convection, the temperature appears in the equation for momentum transport in the x-direction(note - the x-direction is taken to be vertical for this system of reference): du du du ldP d2u d2v — + u — +v -gB(T-T0) + v( + ) (1.3.2) dt dx dy pdx dx 2 dy 2 This coupling of the thermal and velocity fields makes theoretical analysis d i f f i c u l t . Analytical solutions for these equations have been developed by B a t c h e l o r 9 ' 1 0 for steady-state natural convection i n a rectangular cavity. The velocity profile predicted by the solution of Batchelor is shown in Figure 1.7(a). More recently, numerical solutions have been developed by W i l k e s 1 1 , Vahl D a v i s 1 2 and others 8' 1 3' 1 1 5 . Results reported by Stewart are shown in Figure 1.7(b) for a range of Grashof numbers with a fixed Prandtl number. According to the numerical results of Stewart, the maximum in the velocity profile shifts toward the boundary with increasing Rayleigh number whereas the profile remains constant according to the solution of Batchelor. Since the analytical model of Batchelor breaks down at large Rayleigh numbers numerical 15 ud Oi Ro O.OI5 0.010 -(a) 0.005-(b) - 3M0 - MOO - t400 - tooo - 1800 t-u o - too > 0.1 02 0.3 DISTANCE, Y 0.4 03 FIGURE 1.7 Vertical (x) velocity plotted versus horizontal position across mid-plane of cavity (x=0.5) according to (a) the analytical model of Batchelor and (b) the numerical model of Stewart which shows velocity profiles for Pr= 0.0127 and various Grashof numbers. 16 models are more useful for the modelling of metal systems because the Rayleigh number for many liquid metal systems is greater than 10**. In addition to the numerical model, Stewart conducted one of the few studies which used liquid metal for the experiments. This work established a correlation between the time per cycle and the temperature difference across a thin square cavity, as shown in Figure 1.8. The relationship was valid for the type of two-dimensional flow shown in Figure 1.9(a). 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 characteristics. Similar three-dimensional flow patterns were produced i f the c e l l thickness exceeded a certain value. For such flow i t is d i f f i c u l t to define the time per cycle and therefore d i f f i c u l t 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 flu i d velocity. Using such a c e l l , the temperature difference can be adjusted to study how the heat transfer across the c e l l is affected by the amount of flu 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, 260^ and 305^ . 8 18 FIGURE 1.9 Radiographs from experiments of Stewart showing (a) two-dimensional laminar flow and (b) three-dimensional vortex flow. 8 19 For this investigation, i t is necessary to characterize the heat flow across the c e l l . The idea of an effective thermal conductivity was adopted to describe the heat transfer across the c e l l . Assuming that no heat is lost from the top, bottom or side walls then 4 i j = 4 (1.4.1) n c o l d nmelt v ' where q -, is the rate of heat flow out of the cold end of the c e l l and cold q , „is the rate of heat flow across the melt. Assigning an effective nraelt & 6 thermal conductivity, k e ^ to the melt then q = k ,-VT (1.4.2) ^melt eff melt v ' Since q , . = k ,.71 , , (1.4.3) Hcold cold cold v ' then k T J V T u = k , C V T . (1.4.4) cold cold eff melt If we assume that the heat flow is primarily one-dimensional then, k n, '''cold = k ^melt (1.4.5) cold eff v ' cold melt 20 where d , , and d , ^  are the widths of the cold end and the melt, cold melt ' respectively and T - . and T are the temperature differences J cold melt across the cold end and the melt, respectively. Rearranging equation 1.4.5 yields T _ k e f f dcold (1.4.6a) cold k n , * d - " melt cold melt = k ... C . T _ (1.4.6b) eff melt v ' where C is a constant. Therefore the slope of the tangent to the curve of T ., versus T can be used to determine a value for k c,. cold melt eff Using this technique a value for kg^^ can be defined for any temperature difference across the melt. Once k CJ. is correlated to T , t h e n the eff melt relationship between the temperature difference across the c e l l and the flow velocity could be used to relate kg^^ to flow velocity. eff In the standard heat transfer notation the factor could be melt thought of as n a v g , the average heat transfer coefficient for the melt. The average Nusselt number would then be, „ ,, v ^melt ^eff ^melt ^eff / n . ,v Nu = (h ) . = , . , = (1.4.7) avg avg' k d k k Sn melt Sn Sn where k g is the stagnant thermal conductivity of t i n . To summarize, this study looked at heat transfer across liquid metal that was flowing due to natural convection. Using the temperature difference across a thin square cavity 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 effective thermal conductivity of the melt. This value when compared to the stagnant thermal conductivity indicated the magnitude of the enhancement in heat transfer due to the presence of convection. In addition to the experimental work a numerical model was developed to give insight into the thermal and flu i d flow behaviour of the c e l l . 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. The c e l l contained the liquid metal in a thin square enclosure which would produce laminar two-dimensional flow due to natural convection. The temperature of the ends of the c e l l were varied to change the convective flow velocity in the melt. The flow produced was measured by the flow techniques described in section 2.2. The temperature differences were measured across the melt and the cold end. These values could then be used to determine the heat flow across the melt. The liquid metal used in these experiments was 99.999% pure t i n . Table II gives the properties of liquid tin 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 . The thickness of the liquid c e l l was 0.32 cm which should have beeen small enough to ensure two-dimensional flow, according to Stewart. The heat transfer measurements required that heat flow occurred only through the end pieces therefore the bottom and side walls were made of insulating materials. TABLE II PROPERTIES OF LIQUID TIN TEMPERATURE VISCOSITY SPECIFIC HEAT THERMAL CONDUCTIVITY DENSITY COEFFICIENT OF THERMAL EXPANSION (°c) (c P) cal f ) cal f 1 gm ( } 1 r V gm- C 1 o ' cm-sec-UC cm3 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 Cell I To use the c o r r e l a t i o n between time per cycle and the temperature difference across the melt determined by Stewart, the c e l l geometry and dimensions which he used were adopted for this design, which is shown in Figure 2.1. The bottom and side walls were made from 0.32 cm glass sheet. The cold end piece was made of stainless steel. The low thermal conductivity of the stainless would produce a significant gradient across the end piece even for low temperature differences across the melt. This a b i l i t y was considered desireable since the heat transfer behaviour for low temperature differences across the melt was of particular interest because the most rapid change in flow velocity occurs in this range. The cold end was cooled by argon gas jets located in the assembly attached to the end (see Fig 2.1). The hot end of the c e l l consisted of a copper block with a T-shaped cross-section that had a hole to allow for the heating assembly. The heater consisted of chromel heating wire which was wrapped around a ceramic sheath. Power was supplied by a variac. Before assembling the c e l l the inside surfaces were coated with colloidal graphite to prevent liquid metal from attacking the c e l l walls and to seal any gaps between components. The walls of the c e l l were held together by bolts through the bottom piece and by the specially designed end pieces, as shown in Figure 2.1. The top of the c e l l was open to allow for 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 in the furnace by a hangar attached to the ends of the c e l l . The furnace temperature was controlled by a Honeywell temperature controller (model 5500101). 2.1.2 Cell II Since this c e l l was to be used for quench experiments the walls were made of 0.48 cm stainless steel instead of glass which would crack due to thermal shock i f quenched. Although the overall c e l l dimensions were the same as for c e l l I, new end pieces were designed due to problems with the end pieces in c e l l I. The cold end was made of copper instead of stainless steel. This copper piece had a U-shaped copper tube soldered to the back face. The tube was used to provide the Ar gas cooling instead of the jets used in c e l l I. Two thermocouples were soldered between the tube and the end piece. 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 through the bottom and clamps were used along the side edges. 27 2.1.3. CELL III Results from cells I and II lead to the design of the larger c e l l shown in Figure 2.2. A different furnace was necessary to accommodate the new c e l l . This c e l l design eliminated the need for bolts through the bottom of the c e l l . Instead, clamps were used along the bottom and sides of the c e l l , as shown in the figure. There was a special assembly at the cold end which incorporated the clamps and argon jets and provided a shield around the end to prevent argon from flowing into the furnace. To provide more reliable 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 in place by screws. However the inside thermocouples were s t i l l spotwelded in place since the use of screws would distort 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 c e l l wall consisted of an inner wall, made from 0.08 cm aluminum sheet and an outer wall, made from 0.016 cm stainless steel sheet, which were seperated by 0.016 cm teflon spacers at the edges. The resulting air gap between the walls provided an insulating layer in the wall. To Argon Inlet Tube Design of experimental c e l l I I I . 29 quench the melt, water was forced into this air gap from a thin-walled stainless steel tube (o.d.«1.2 cm) with the end flattened to allow insertion into the gap. Since only the aluminum sheet separated the cooling water from the melt, the molten metal could be quenched in less than two seconds using this technique. 2.2 FLUID FLOW MEASUREMENT Two methods were used to measure the flow velocities in the melt. The f i r s t method consisted of monitoring the activity of a small piece of radioactive copper added to the melt. The second procedure consisted of adding radioactive t i n (Sn 1 1 3) to the tin melt and then quenching the melt to establish the path of the radioactive tin in the melt. The general procedure for the copper particle experiment is as follows: i ) The temperature of the melt was monitored by a thermocouple probe. When the c e l l reached thermal equilibrium, a piece of radioactive copper (Cu 6 i +), less than 2 mm in diameter was inserted into the melt. i i ) Lead bricks were arranged so that the activity of the copper 30 particle 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 particle could be determined by estimating the absorption due to the t i n . The expected periodicity in the count rate would be directly related to the period of the fluid flow in the cavity. The general procedure which was followed for the experiments using the radioactive tin (Sn 1 1 3) as tracer was: i ) Before inserting the S n 1 1 3 into the melt, the temperature at end of the melt was measured using a thermocouple probe. Temperatures were taken at approximately the middle of the inside face of each end. i i ) The radioactive t i n (Sn 1 1 3) was added near the cold end of the c e l l and after a given time the c e l l was quenched. i i i ) After quenching, the solid block of tin was removed from the c e l l and placed on a sheet of X-ray film. Another sheet of film was placed on top and a glass sheet put on top to ensure good contact between the film and the sample. The film was exposed for two to three days then developed. The resulting dark areas on the film indicated the location of the S n 1 1 3 and thus showed the extent of flui d flow in the melt between the moment the radioactive t i n was added and the moment when the sample was quenched. 31 2.3 TEMPERATURE MEASUREMENT Chrome-alumel thermocouples were used for the temperature measurements in this study. For c e l l I, 0.1 mm uncoated thermocouple wire inside ceramic sheaths was used. For cells II and III, 0.025 cm wire was used since the finer 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 liquid metal from complete contact with the end piece. The locations of the thermocouples in each c e l l are shown in Figure 2.3. The use of spot-welded thermocouples instead of a thermocouple probe was preferred for the heat transfer measurements since the location of the probe could not be defined precisely and therefore i t would be d i f f i c u l t to reproduce i t s location for each measurement. However, for the quench tests, a thermocouple probe was used because i t would have been neccessary to replace spot-welded thermocouples after each sample was removed from the c e l l . The cold junctions of the thermocouples in cells I and II were maintained in an ice-water bath. The junctions in c e l l III were connected directly to the thermocouple switch. A thermometer attached to the thermocouple switch was used to determine the temperature of the (a) (b) 32 (c) FIGURE 2.3 Location of thermocouples in experimental cells a) c e l l I, b) c e l l II and c) c e l l III ( thermocouple location is marked by T). 33 cold junction. The output from the thermocouples was recorded on a Honeywell chart recorder (model no. 194). The range of the chart recorder was adjusted to be either one or two millivolts full-scale 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 partially covered. Sufficiently large temperature differences were produced without the use of the heater. Figure 3.1 shows the results plotted as AT g g, the temperature difference across the stainless steel end versus AT ^ , the temperature difference across the melt. The experiments with the top of the c e l l covered and uncovered both showed linear relationships between the temperature difference across the cold end and that across the melt. The constant slope in Figure 3.1 indicates that kgff> the effective thermal conductivity, was constant. 3.1.1 Calculation of the Effective Thermal Conductivity The value of k e f f can be calculated from m, the slope of the graph of AT versus AT . using equation 1.4.6(a). According to .1- d m »x* t • equation 1.4.6(a), ^eff ^cold / 0 , i v m = ^  . -j (3.1.1) cold melt Temperature Difference Across Cell (°C) FIGURE 3.1 Graph showing results from c e l l I with top of c e l l covered and uncovered. Slopes were calculated using linear regression of data points. The correlation coefficient is given in brackets. 36 Therefore, , , ^ cold * ^ melt x ,„ , „ v k e f f = m • ( — d — ; > ( 3- 1- 2) cold For c e l l I: k c o l d = T h e r m a l conductivity of stainless steel = 0.044 cal/cm-sec-°C ^melt = D * - S t a n c e across melt = 5.0 cm (3.1.3) ^cold = D ^ - S t a n c e across cold end = 1.25 cm From figure 3.1: m (uncovered) = 2.069 (3.1.4a) m (covered) = 1.705 (3.1.4b) Calculating the values for k g f f using equation 3.1.2 yields: k e££ (uncovered) = 0.3641 cal/cm-sec-°C = 4.5 X k c Sn k e f f ( c o v e r e d ) = 0.3003 cal/cm-sec-°C (3.1.5) s 3.8 X k 0 Sn k g n (stagnant) =0.08 cal/cm-sec-°C 37 3.1.2 Discussion of Results from Cell I There are several reasons why k^^ would be constant. It may have been constant because i t was independent of the degree of convection. This condition would be expected when the enhancement in heat transfer due to convective flow had reached i t s limit. Looking at the degree of convection produced, the largest temperature differences employed across the melt were big enough to ensure that the maximum flow was achieved, according to Figure 1.8. Therefore i t is possible that the heat transfer limit had been reached. Also according to Figure 1.8, the lowest temperature differences that were used would only produce flows that were marginally slower than the maximum flow. Therefore the reason k g^^ remained constant was probably because the change i n convective flow was too small to show an effect. The magnitude of the effective thermal conductivity is about four times the stagnant thermal conductivity of t i n . This factor is lower than the seven to ten times enhancement assumed i n the mathematical modelling of continuous casting. It is possible that this number represents the maximum enhancement acheivable with natural convection. However there may be another reason why kg^^ is low. The heat transfer through the c e l l is not completely determined by the characteristics of the molten bath. The rate of heat flow 38 through the c e l l i s affected by the thermal resistances of the boundaries as well as the thermal resistance of the melt. The thermal resistance of the hot end of the c e l l (Cu) was lower than that f o r l i q u i d t i n , as shown i n Table I I I . The thermal resistance of the cold end of the c e l l ( s t a i n l e s s s t e e l ) i s less than the thermal resistance of stagnent l i q u i d t i n but i s greater than that presented by the convecting melt. From t h i s analysis the heat transfer across the c e l l would appear to 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 piece rather than the behaviour of t h e m e l t . T h i s r e a s o n c o u l d a l s o e x p l a i n why k e f f remained constant. To avoid t h i s problem the next two c e l l designs employed copper for both ends of the c e l l . 3.2 RESULTS FROM CELL II The next series of experiments were conducted with the second c e l l which duplicated the dimensions of the f i r s t c e l l but used d i f f e r e n t materials for i t s construction. The f i r s t experiments with t h i s c e l l attempted to measure the degree of convective flow using the copper p a r t i c l e method. This method was preferred since the technique would provide information about the flow at the temperature measurement conditions rather than i n f e r r i n g the flow from a quench. As well, the change i n convective flow could be observed when the temperature differ e n c e was adjusted, providing a look at the transient and steady state behaviour. 39 TABLE III COMPARISON OF THERMAL RESISTANCES MATERIAL THERMAL CONDUCTIVITY LENGTH THERMAL RESISTANCE* L k L R = k A cal ( ) (cm) sec- °C ( 1 I ) n gm-sec-°C cal Copper 0.928 1.27 1.37 Stainless 0.044 1.27 28.97 steel Liquid t i n , 0.080 5.08 63.50 stagnant Liquid t i n , ** 0.30 5.08 16.93 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 failed to produce the expected periodicity the quench technique was used to look at the flow pattern in 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 particle experiments failed to show any periodicity. Other samples appeared to be very turbulent, as shown in Figure 3.2(c), but i t is 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 velocities than those expected from the results of Stewart. 3.3 RESULTS FROM CELL III 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 c e l l and to establish the relationship between the convective flow and the temperature difference across the c e l l by quenching the c e l l . (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 prior 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 f i r s t concern of the experiments on the new c e l l was the adequacy of the quench. The stainless steel 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 that some flow had occurred after the in i t i a t i o n of the quench. To verify this suspicion the f i r s t quench tests with c e l l III were performed using stainless steel walls which were made of two pieces of 0.16 cm stainless steel sheet glued together. The sampled was quenched by rapidly f i l l i n g up the Inside can of the furnace with cooling water. The quench was very slow and the cooler liquid from the sides would f a l l to the bottom and produce the flow pattern shown in Figure 3.3(a). When quenched in the manner used for 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 distorted during the quench producing the fork-like pattern shown in Figure 3.3(b). Figure 3.4 shows a successful quench obtained using the composite c e l l wall design described in section 2.1.3. The radiographs of the quenched samples used to determine the time per cycle are shown in Figure 3.5. The resulting graph of the time per cycle versus the temperature difference across the melt is given in Figure 3.6 and is very similar to that observed by Stewart. FIGURE 3.3 Examples of inadequate quenchs from c e l l III (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 III (c) AT-1.4 °C time prior to quench = 22 sec, (d) AT-4.4 °C , time prior to quench = 11 sec. 4 7 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 Relationship between time required per cycle and the temperature d i f f e r e n c e across the melt (Temperatures i n melt were measured using a thermocouple probe). 49 3.3.2 Temperature Measurements Having established that the flow was laminar and behaved in a predictable manner, detailed temperature measurements were performed. The measured temperatures were accurate to + 0.3 °C. The temperature difference across the melt was 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 in Figure 2.3(c). These temperatures were used to calculate the temperature difference across the cold end. Rather than using the average temperature of the outside and inside faces of the cold end to calculate a single temperature difference for the cold end, a temperature difference was calculated for each pair of thermocouples. Temperature 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 in 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 3.7 gives the plot of the temperature difference across the cold end versus the temperature difference across the melt for the two experiments. There are two distinct regimes in the results for experiment A, as shown i n Figure 3.7(a). I n i t i a l l y , the 50 FIGURE 3.7 Graph of temperature d i f f e rences across co ld end versus temperature d i f f e rence across melt for (a) experiment A which used a heater and argon coo l ing to produce the temperature d i f f e rence across mel t . Slopes ind i ca ted on graph were ca l cu l a ted using l i n ea r regress ion of data po ints• 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 for 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. 52 temperature d i f f erence across the melt does not appear to be re la t ed to the temperature d i f f erence across the cold end then there i s an abrupt t r a n s i s t i o n in to the second regime where there i s a l i n e a r r e l a t i o n s h i p between the two temperature d i f f e r e n c e s . The r e s u l t s for experiment B showed a gradual t r a n s i s t i o n in to the l i n e a r region as Indicated i n F igure 3 .7 (b ) . I t was found that the l i n e a r behaviour i n F igure 3.7 was associated with large argon flow r a t e s . To inves t iga te the dependence on flow r a t e , the temperature d i f ferences i n the c e l l were p lo t t ed as a func t ion of the argon flow r a t e . The argon flow rate i s expressed as a dimensionless number corresponding to the sca le reading on the flowmeter. Appendix II contains a chart which can be used to convert a given flow rate to m i l l i l i t r e s per minute. F igure 3.8 shows these p lo t s for experiment A. At low flow rates ne i ther the temperature d i f f erence across the co ld end nor that across the melt increased uniformly with i n c r e a s i n g argon flow r a t e . At one p o i n t , the temperature d i f f erence across the melt decreased with increas ing flow r a t e , as shown i n F igure 3 . 8 ( b ) . This behaviour appears to be associated with the use of the heater s ince i t d id not occur i n Figure 3.9 which shows the p lo t s for experiment B where only argon coo l ing was used to produce the temperature d i f f erence across the melt . F i g u r e 3 .10 shows the r e l a t i o n s h i p between the a c t u a l temperatures and the argon flow rate for the two experiments. 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 30 4 0 50 Argon F l o w R o l e T 1 1 1 1—< i A r g o n F l o w R a t e FIGURE 3.9 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 they are typical of a l l temperatures throughout the c e l l for a given experiment. In experiment A, the temperature showed an i n i t i a l decrease (region I) which was followed by a large increase (region II) afterwhich the temperature decreased steadily, as shown in Figure 3.10(a). In experiment B, the temperature curve was similar except for a small increase which preceded the i n i t i a l decrease in temperature, as shown in Figure 3.10(b). The temperature curve in Figure 3.10(a) could be explained i f some argon gas had leaked into the furnace. Although the cold end was shielded to prevent argon gas from flowing into the furnace, some gas may have escaped. The forced convection produced in the furnace atmosphere by this flow of gas would increase the heat transfer rate between the furnace walls and the c e l l . This phenomena would explain the increase observed in region II. Beyond a certain argon gas flow rate the enhancement in the heat transfer in the furnace would reach some limit and the temperature would then decrease with increasing flow rate as observed in region III. It is interesting to note that the linearity observed in Figure 3.7(a) occurred for the temperature differences corresponding to the region III temperature data. One additional observation should be made before proceeding to the next section. The temperatures at the bottom and middle of the inside face of the cold end were always within 0.2 °C of each other and were often the same. Therefore the disparity in the observed values of 57 the temperature difference across the bottom and middle sections of the cold end is due to the difference in the temperatures at the outside face. The argon gas flow should have been approximately the same for both sections although i f some nonuniformity existed i t is likely that the flow was higher to the middle section than to the bottom section. Consequently there does seem to be any obvious explanation for the difference in heat flow rates that is implied by the difference in the temperatures. 3.3.3 Calculation of the Effective Thermal Conductivity The values for k were calculated from the slopes in Figure 3.7 using equation 3.1.2. The results are given in Table IV. The slopes for Figure 3.7(a) were calculated using linear regression of the data points corresponding to region III in Figure 3.10(a). The slopes in Figure 3.7(b) were calculated by using linear regression of the last three data points although only two actually l i e in region IV of Figure 3.9(b). For both experimental cases, the value of k g j ^ calculated from the data for the bottom section of the cold end is much higher than that calculated for the middle section of the cold end. Looking at the ratio of k c , to k , the normal thermal conductivity of tin, the enhancement eff sn' i s seven to ten times k according to the data from the middle section sn 58 TABLE IV CALCULATION OF EFFECTIVE THERMAL CONDUCTIVITY EXPERIMENT LOCATION (COLD END) ^melt m k '« eff k e f f cold melt kSn cal f ] (cm) (cm) cal ( V 1 o ' cm-sec- C 1 • -o J cm-sec- C A-Using heater & Middle 0.927 10.0 1.25 0.0733 0.5435 6.8 Ar cooling Bottom 0.927 10.0 1.25 0.4625 3.430 ~ 43 B-Only argon Middle 0.927 10.0 1.25 0.1036 0.7685 9.6 cooling used Bottom 0.927 10.0 1.25 0.3472 2.575 ~ 32 59 and i s t h i r t y to forty times k g n according to data from the bottom section. Judging from these results, i t is not reasonable to use a plot of the temperature difference across the bottom section of the cold end versus the temperature difference across the mid-plane of the melt to calculate k e c . eff 60 4.0 MATHEMATICAL MODEL In previous mathematical models of fl u i d flow in a rectangular cavity, the hot and cold ends of the c e l l have been assumed to be isothermal. However in a real system i t is d i f f i c u l t to ensure uniform temperatures at the ends and there i s usually some temperature difference in the vertical direction. This model was developed to observe the effect of non-uniform boundary temperatures. In addition, the model was used to compare convective flow in liquid tin with that in liquid steel. 4.1 GOVERNING EQUATIONS The energy equation, the Navier-Stokes equations and the continuity equation are needed to solve for the temperature and velocity f i e l d s . The following are the form of these equations for the two-dimensional system shown in Figure 4.1: The energy equation: 5T* L *9T* . *6T* k , o2T* . o2T* . (4.1.1) _ _ + u — — + v w = ( — * j + ) St ox oy pCp ox oy FIGURE 4.1 System of reference for mathematical model. 62 The Navier-Stokes equations: -for momentum transfer in the x-direction, * * * * ? * 2 * cu" * du' * ou 1 SP . , 8 u • 5 u N h U + V = + V( + ) * * * o * *2 *2 St Sx Sy nm Sx Sx Sy -g8(T*- T*) (4.1.2) -for momentum transfer in the y-direction, * * * * 9 * 2 * Sv , * Sv • * Sv 1 SP , * S v , S v * . i . Q . r U r V = + V( + ) (4.1.3) * * * o * *2 *2 St Sx Sy K i by Sx Sy The continuity equation: - ^ + ^ L . = 0 (4.1.4) * * Sx Sy made: In applying these equations the following assumptions have been 1) Fluid properties have been assumed constant except for one term which accounts for the temperature dependence of the density 63 2) Viscous dissipation and compressibility effects have been neglected 3) There is no heat generation in the f l u i d 4) The applied temperature difference is small compared to 1/(3 (where 8 is the coefficient of thermal expansion) 4.1.1 Dimensionless Variables The asterisks used in equations 4.1.1 to 4.1.4 have been used to distinguish the marked variables from their dimensionless form. For computational purposes i t is easier to use the dimensionless variables in the governing equations therefore the following dimensionless variables were introduced: u = * u d * v d (4.1.5) T = T -T o * * » T -T h o * 2 P.d P = P v m 64 H.Cp . g&d ( T -T ) P r = c r = — — k » v 2 * where = Temperature of the hot end T q = Average temperature of the c e l l * * T , + T h c * T = Temperature of the cold end d = Width across c e l l The Grashof number as defined here is a modified version of the standard Grashof number since i t is calculated using one-half the temperature difference across the c e l l rather than the total 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. yields: oT dT oT 1 b2! — + u — + v — = — ( + ) at ax ay Pr ax 2 a y 2 (4.1.6) 65 ou du ou 5P d ^ 5^ — + u — + v — = - Gr .T - — + + (4.1.7) 5t ox dy ox dx 2 dy 2 av dv dv dP d ^ d ^ — + u — + v — = + + (4.1.8) dt dx dy dy dx 2 dy 2 du dv — + — =0 (4.1.9) dx dy Equations 4.1.7 and 4.1.8 can be combined by differentiating equation 4.1.7 with respect to y and equation 4.1.8. with respect to x, subtracting and using equation 4.1.9 to eliminate terms to give, 3 du dv d V d ^ d ^ d ^ ( ) + u + V u v d t d y d x dxdy dy dx"* dydx # dT d (V^) d ( v M - Gr . — + (4.1.10) dy dy dx 4.1.2 Stream Function and Vorticity To simplify equation 4.1.10, the concepts of the stream function and the vorticity were Introduced. The vorticity, E, is defined as: dv du dx dy (4.1.11) 66 The stream function, 4> is defined by the following equations: o<|> u = (4.1.12) oy 0(|» v = (4.1.13) o x Substituting the vorticity into equation 4.1.10 yields the vorticity equation: bi ac . ar — + u — +-v — = Gr — + V2C (4.1.14) 9t 8x oy by Equation 4.1.9 is automatically satisfied by the definition of the stream function. To solve for the stream function, equations 4.1.12 and 4.1.13 are substituted into equation 4.1.11 to give the stream function equation: a24» a2c|> 5 = - ( + ) (4.1.15) ax 2 a y 2 Equations 4.1.6, 4.1.12, 4.1.13, 4.1.14 and 4.1.15 form the set of equations used to solve the problem of natural convection in a square cavity. 67 4.2 INITIAL AND BOUNDARY CONDITIONS The temperature and velocity profiles calculated in this model are steady-state. However to reach a solution, the time-derivatives have been retained in the governing equations so that the computer w i l l produce a time-varying solution which converges to the steady-state solution. This technique requires the definition of i n i t i a l conditions for the temperature, stream function and vorticity. The i n i t i a l 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 of equations also requires the definition of boundary conditions. Since the velocity must be zero at the boundaries the gradient of the stream function was taken to be zero at the the boundaries. The top and bottom surfaces were assumed to be either insulating or perfectly conducting. The ends were taken to be either isothermal or having a linear temperature drop. Expressed numerically the boundary conditions were, o<l> t>0 x = 0.0 ,1.0 (|/ = =0 ox 68 i ) Insulated boundary oT — =0 ox i i ) Perfectly conducting T = -1.0 + 2.0*(1 - y) y=0.0 • «J, = — =0 (4.2.2) by 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 differential equations is achieved by using f i n i t e difference approximations to the governing equations. The 69 numerical technique used in this model was the implicit alternating technique, commonly referred to as the ADI method. The model used the ADI technique to solve for the temperature, vorticity and stream function equations. Other models 1 2' 1 5 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 for this model and had reported problems when using low Prandtl numbers. Since the model by Stewart was successful at the low Prandtl numbers and high Grashof numbers that would be needed for 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 implicit and a l l the y-derivatives are explicit. Implicit derivatives are evaluated at t=t 2 and explicit derivatives are evaluated at t=t^, where 1 2=t]+AT/2' The values for t=t ^  represent the values at the previous half of the time step and are known at t=t 2. The values at t=t 2 are 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 implicit and the x-derivatives are explicit. The f i n i t e difference approximations were generated from expansions based on the square grid system of points as shown in Figure 4.2. The subscripts I and j indicate the position of the node in the x and y directions respectively. The following equations use an asterisk FIGURE 4.2 Grid system used in mathematical model. 71 superscript (*) to denote variables that are evaluated at t=(n+l/2)AT, and the apostrophe superscript (') to denote variables that are evaluated at t=(n+l)Ax. Variables without superscripts are evaluated at t=nAx. 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 ^m rn fjt rn rn rj\ t>j i , j + u . t i + i ; j ' j - t , j + v x i , j + i - * i , j - i = Ax/2 i , J 2 Ax ± ' i 2 Ay 1 . T* .... ~ 2T* . + T* ., I T . , . . - 2T, , + T. ... 1-1,3 1,3 1+1,3 + l . j - l l , j '1,3+1 (4.3.!) Pr (Ax) 2 Pr (Ay) 2 For the second half of the time step, ' * * * » » rn rn ^n rn 1,3 1,3 H u 1+1,3 '1-1,3 , T 1,3+1 ' i , j - l _ Ax/2 i , J 2 Ax i , J 2 Ay * * * » » » 1 T . - 2T . + T - . 1 I... . -2T....+ T. ... _ i - l , j i,3 1+1,3 + _ i . J - l i,3 1,3+1 (4. 3. 2) Pr (Ax) 2 Pr (Ay) 2 The f i n i t e difference equation for the vorticity for the f i r s t half of the time step, * * * C l , 3 "  L±,3 + u C i + l , j " C i - l , j + v Ci,j+1 ~ C l , j - 1 Ax/2 x » j 2 Ax ^ 2Ay 72 « 75 75 j+1 " T - 1 , + g i - l , j " 2 g l , j + ^i+l,j + 2 Ay (Ax) 2 S j . j - l - ^ i . j +-£1,3+1 ( 4 > 3 > 3 ) (Ay) 2 For the second half of the time step, ' * * * » » " ^ i , J + u C i + l , j " C i - l , j ,h y " C i , j - I = AT/2 l , i 2 Ax 2Ay » • * * * T - T E - 2E + E C r«/i,j+l ' i . j - l , 4 H , j - 4+l.j + 2 Ay (Ax) 2 t » i g i , - j - l 2 C i , : j 1 - 5 t , j + l (4.3.4) ( A y ) 2 To solve the stream function equation, a time derivative is introduced to equation 4.1.15 to produce an unsteady state solution which converges to the steady state problem. 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 , A T /2 i , j (Ax) 2 73 (Ay) : For the second half of the time step, * * ic is <k ,--.<k., V i - - I ~ - 2 V - I - + * i>3 i>3 _ g + 1-1,3 i » 3 i+l»3 ( AT/2  ±>i ( A x ) 2 (Ay) 2 (4.3.5) ( 4 . 3 . 6 ) To solve the temperature, vorticity and stream function equations, the above equations are rearranged to collect a l l the unknown terms on one side of the equation and the known terms on the right hand side of the equation. A complete l i s t of the rearranged equations can be found in appendix II of reference 8. The resulting coefficient matrix for the unknown variables is a tridiagonal matrix which can be easily inverted for solution. Once the temperature, vorticity and stream function have been calculated for each node, the new velocities are calculated. Using equations 4.1.12 and 4.1.13, the following expansions were generated to solve for the velocities: , = A - *i.J-2 ~ 8 ( p i , j - l + 8 4 , i , j + l " +1,1+2 (4.3.7) 1 , 3 dy , J 12Ay 74 *1 j • " M l ^ _ ^-2,3- - ^ i - y ^ k . j - V l . j (4.3.8) ox j 12 Ax For points near the boundary a different expansion was used which was of the form: _ ~ 3»1,2 + H i , 3 " *1,4 (4.3.9) U i 2 X ' 6Ay v„ . =^ ^ (4.3.10) 6Ax Expansions of a similar form were used for the other boundaries. An additional calculation was performed to determine the boundary vo r t i c i t i e s since no boundary condition was defined for the vorticity. Equation 4.1.15 was used to solve for the boundary v o r t i c i t i e s . To satisfy the condition that the velocity must be zero normal to the boundary, b2<\> §2<\> at x=0,l = 0 and at y=0,l = 0 (4.3.11) by2 ox 2 Therefore at the boundary, equation 4.1.15 becomes, d2c|> a2<j> at x=0,l E = and at y=0,l E = (4.3.12) dx 2 dy 2 75 The expansions used for the boundaries were of the form, 2h 2 hi <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. This time step was larger than the time step used for the temperature and vorticity equations. In addition, convergence was much more rapid i f the latter time step was changed as the program progressed. For Grashof numbers > 1 0 5 even small changes in the temperature-vorticity time step would markedly affect the number of iterations that were required for convergence. No convergent solution was found for Grashof numbers >^  1 0 7 . From the results obtained, i t would appear that adjusting the value of the stream function time step might increase model sta 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 finer mesh size was used for Grashof number equal to 1 0 7 . The region of instability seems to occur for Ra > 1 0 5 . This limit would seem to concur with the stability limit observed in other models of conv e c t i o n i n a r e c t a n g u l a r c a v i t y 1 2 ' 2 1 . The Instability in the numerical model seems to associated with the onset of secondary flows in 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 tin (Pr = 0.0127) and various Grashof numbers. Figures 4.3, 4.4 and 4.5 show the computer plots that were produced. The results were in excellent agreement with the plots generated by the numerical model of Stewart for similar Prandtl and Grashof values. In addition, velocity plots were generated which show the velocity distribution in the c e l l . The results, given in Figure 4.6, show the development of small secondary flows in the upper l e f t and lower right corners of the velocity plot for Gr = 10 6. The temperatures of the .ends were adjusted so that there was a linear temperature drop from the top to the bottom of the ends. In the experiments, i t was found that the temperatures at the top of the c e l l were usually cooler than at the bottom therefore the gradient in 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=1.0X105. Figure 4.8 shows the effect of the same temperature drops for Pr=0.0127 and Gr=1.0X106. The effect on the isotherms is particularly pronounced near the ends. The presence of the temperature gradients in the ends also FIGURE 4.3 Isotherm distribution for Pr=0.0127 and (a) Gr^L.OXlO4 (b) Gr=1.0X105 and (c) Gr=1.0X106. 78 79 FIGURE A.5 Nusselt number versus vertical (x) position along cold wall for Pr=0.0127 and Gr-l.OXlO11, 1.0X105 and 1.0X106. 80 (a) ! 11' {{ \ \:'> < 11 -->>>//;; M \ \ \ w ^ - ' ^ ^ v ; ' ' \ \ \ \ / / / / • » \ \ w ^ - - — • * \ \ \ x ^ -1 * x N x. ~^ (b) * / / / / > ' ' ' \ J 1 if i i < ' \ \ » * \ \ W  v v x \ \ \ \ W v -\ \ \ \ \ ^ -v \ \ \ \ ' \ \ N "V i \ \, -v -«. -» -» • \ v s. — —' - •». v \ x V \ \ \ \ \ x X X \ \ \ W * \\ \ \\ >* J» * t 4 M t \ t t f f 1 f t FIGURE 4.6 Normalized velocity f i e l d for Pr=0.0127 and (a) Gr=105 (maximum velocity =487.44) and (b) Gr=106 (maximum velocity = 1185.09). Velocities given are dimensionless velocities. (a) (b) FIGURE 4.7 Isotherm plots with Pr=0.0127 and Gr=1.0X105. Solid line indicates profiles for isothermal ends and dashed line indicates those for (a) a 5% drop along ends and (b) a 10% drop along ends. FIGURE 4.8 Isotherm plots with Pr=0.0127 and Gr=1.0X106. Solid line indicates profiles for isothermal ends and dashed line indicates those for (a) a 5% drop along ends and (b) a 10% drop along ends. 83 FIGURE 4.9 Streamline plot with Pr=0.0127 and Gr=1.0X106 with (a) isothermal ends and (b) 10% drop along both ends (with coldest temperature at top). 84 affects the streamline distribution as shown in Figure 4.9. There is a slight effect on the shape of the streamlines. The maximum value of the stream function is lower when the gradients are present indicating slower velocities in the cavity. However the maximum velocity only decreased by about eight percent with a ten percent drop in the end temperatures. The velocities that were measured for c e l l II appeared to be much higher than the velocities measured by Stewart. One major difference in the c e l l design between c e l l II and the c e l l used by Stewart was the fact that Stewart used a U-shaped piece inside the c e l l which connected the cold end to the hot end. This design avoided the problem of liquid t i n leaking from the junction between the end piece and the bottom piece of the c e l l but i t also provided a thermal link between the end pieces through which heat could flow. For such a situation It is conceivable that a temperature gradient would exist along the bottom of the c e l l . To investigate the effect of a temperature gradient along the bottom of the c e l l computer runs were performed with a linear gradient imposed along the bottom of the c e l l . For the results shown in Figures 4.10 and 4.11 the ends were taken to be isothermal and the top surface was insulating. The effect on the isotherms seems to be limited to the bottom of the c e l l . The values of the stream function are lower than when the bottom was taken to be insulating, especially for the higher Grashof number, indicating that the velocities would be slower. Computer runs were also performed for the case when there was linear temperature drops in both ends combined FIGURE 4.10 Isotherm plot with linear temperature gradient along bottom (isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0X105 and (b) Gr=1.0X106. 86 FIGURE 4.11 Streamline plots with linear temperature gradient along bottom (isothermal ends and insulated top surface) with Pr=0.0127 and (a) Gr=1.0X105 and (b) Gr=1.0X106. 87 FIGURE 4.12 Isotherm plots with linear temperature gradient along bottom and 10% temperature drop along ends with Pr= 0.0127 and (a) Gr=1.0X105 and (b) Gr=1.0X106. 88 with the linear temperature gradient along the bottom. The results do not differ greatly from those produced when assuming isothermal end temperatures with the linear gradient along the bottom, as shown in Figure 4.12. The reduction in velocity due to the presence of the gradient along the bottom does not appear to be large enough to account for the difference in the results from c e l l II and the results of Stewart. Another factor which may have influenced the flow velocity is 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 steel. Therefore the walls for c e l l II were less conductive. A three-dimensional mathematical model would be necessary to estimate the effect on the velocities but such a model is beyond the scope of this study. Finally, computer runs were performed to compare the differences in the responses of liquid t i n and liquid steel to temperature differences of 0.5 °C and 0.01 °C. The results are shown in Figures 4.13 to 4.19. The isotherm plots do not differ greatly although there is more curvature in the isotherms for liquid steel. The difference in the streamline distribution i s more noticeable for the 0.5 °C temperature difference. The velocity gradients in steel are much steeper near the edges of the c e l l at this temperature difference, as shown in Figure 4.18. However looking at the vertical scale, the 89 0-75 (b) FIGURE 4.13 Isotherm distribution for a temperature difference of 0.01 °C across c e l l for (a) liquid t i n (Pr=0.0127 and Gr =3.6X10'*) and (b) liquid steel (Pr=0.11 and Gr=6.0X103). 90 FIGURE 4.14 Isotherm distribution for a temperature difference of 0.5 °C across c e l l for (a) liquid t i n (Pr=0.0127 and Gr=1.8X 10 6) and (b) liquid steel (Pr=0.0127 and Gr=3.0X105). 91 (b) FIGURE 4.15 Streamline distribution for a temperature difference of 0.01 °C across c e l l for (a) liquid t in (Pr=0.0127 and Gr =3.6X101*) and (b) liquid steel (Pr=0.11 and Gr=6.0X103). 92 FIGURE 4.16 Streamline distribution for a temperature difference of 0.5 °C across c e l l for (a) liquid t i n (Pr=0.0127 and Gr =1.8X106) and (b) liquid steel (Pr=0.11 and Gr=3.0X105). 93 700 (a) 0 4 06 Y Position 140 (b) 005 • 0 035 6 O020 " - 0 005 S 02 0 4 0 6 0 8 Y Position - -0 005 -0020 10 FIGURE 4.17 Vertical (x) velocity versus horizontal (y) position at x=0.5 for a temperature difference of 0.01 °C across c e l l in (a) liquid t i n (Pr=0.0127 and Gr=3.6X101*) and (b) liquid steel (Pr=0.11 and Gr=6.0X103). 94 2600 (a) -2400 04 0 6 Y Position 600 (b) -400, 02 04 06 08 Y Position FIGURE 4.18 Vertical (x) velocity versus horizontal (y) position at x=0.5 for a temperature difference of 0.5 °C across c e l l in (a) liquid t i n (Pr=0.0127 and Gr=1.8X106) and (b) liquid steel (Pr=0.11 and Gr=3.0X105). 95 FIGURE 4.19 Plot of local Nusselt number versus position along cold wall for tin and steel at temperature differences of 0.01 °C and 0.5 °C. 96 velocities in liquid t i n are much higher than those in liquid steel. The velocity scale for liquid tin is five times that for steel in Figure 4.17 and six times in 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 in a liquid metal. The interest was in determining at what convective flow rate the heat transfer across the c e l l became independent of the degree of convection. As long as the heat transfer was related to the flow velocity then the degree of flow could not be ignored in mathematical models which estimate a value for the effective thermal conductivity. However once the linear limit was reached then i t would only be important to know that fl u i d flow is present not the actual flow velocity. 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 linearity observed was probably due to the fact that the thermal resistance of the stainless steel end was higher than that for the convecting melt. In fact, the thermal conductivity of stainless is about one-half the thermal conductivity of stagnant liquid t i n . This situation is opposite to the situation encountered in liquid-solid heat transfer in s o l i d i f i c a t i o n . For most metals, the thermal conductivity of the solid state is higher that the thermal conductivity of the liquid state therefore the results from c e l l I were not representative of the behaviour during so l i d i f i c a t i o n . 98 The results from c e l l I raise some questions about a recent study which examined natural convective heat transfer across a parallelogrammic enclosure. In this study water and silicone 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 copper plate. Thermocouples were positioned between the glass plates to measure the temperature differences across the plates. Knowing the thermal conductivity of the glass, the plates served as a heat flux meter. Since the thermal conductivity of glass is much lower than that for copper, the thermal resistance of the glass heat flux meter would be much higher than thermal resistance of the copper. In fact the thermal conductivity of glass is comparable to the stagnant thermal conductivity of water so that 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 likely that the heat transfer across the parallelogram was influenced by the heat flux meter. To calculate a value for the effective thermal conductivity i t is assumed that a l l heat flow is through the cold end. The difference in the slopes for the covered and uncovered c e l l indicate that the heat l o s t from the top of the c e l l cannot be ignored. The value of k^^ is approximately twenty percent lower using data from the covered c e l l than when using data from the uncovered c e l l . However the effect of the heat loss appears to a constant fraction of the heat transfer across the 99 c e l l so that the shape of the AT versus AT . ^  curve i s not r cold melt affected. In the results from c e l l III, the behaviour at low flow rates was not clear. Using both argon cooling and the heater i t was d i f f i c u l t to achieve temperature differences across the c e l l that were less than 3.5 °C. Using only argon cooling, the data at the lower temperature differences i s hard to interpret because the temperature difference across the bottom section of the cold end was negative. 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 . Once the linear limit was reached, the magnitude of the effective thermal conductivity was calculated to be seven to ten times the thermal conductivity of stagnant liquid t i n . According to Figure 3.7, the temperature difference across the bottom section was higher than the temperature difference across the middle section implying higher heat transfer rates across the bottom of the c e l l . This situation is in direct conflict with the predictions of the computer model as reflected in the Nusselt number plots shown in 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 is necessary to resolve the disagreement in the heat flow distribution. 100 REFERENCE LIST 1. Cole, G.S. and G.F Boiling, The Solidification of Metals, 1968, London, The Iron and Steel Institute, 323-329. 2. Lait, 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, vol 239, 1747-1753. 4. Szekely, J. and V. Stanek, Met. Trans., 1970, vol 1, 119-126. 5. Lait, 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 British Columbia. 9. Batchelor, G.K., Q. Appl. Math., 1954, Vol 12, 209-233. 10. Batchelor, G.K., J. Fluid Mech., 1956, Vol 1, 177. 11. Wilkes, J.0. and S.W. Churchill, A.I.Che.J., 1966, Vol 12, 161-166. 12. De Vahl Davis, G., Int. 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. Sci. 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. Jassal, 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 SoldifIcation of Metals, 1968, London, The Iron and Steel Institute, 415-420. 19. Moore, J.J., and N.A. Shah, Int. Met. Review, 1983, Vol 28, 338-356. 20. Blank, J.R. and F.B. Pickering, The Solidification of Metals, 1967, London, The Iron and Steel Institute, 370-376. 21. Szekely, J. and M.R. Todd, Int. J. Heat Mass Transfer, 1971, Vol 14, 467-482. 22. Maekawa, T. and I. Tanasawa, Proc. Seventh Int. Heat Transf. Conf., Munchen, Fed Rep Germany, Vol 2, 227-232. APPENDIX I LIST OF SYMBOLS Specific heat Distance across c e l l Acceleration due to gravity Grashof number Modified Grashof number Heat transfer coefficient Thermal conductivity Effective thermal conductivity Thermal conductivity of t i n Characteristic length Nusselt number Pressure Dimensionless pressure Prandtl number Heat flow rate Rayleigh number Time Dimensionless time Temperature Dimensionless temperature Velocity in x-direction Dimensionless velocity in x-direction Velocity in y-direction Dimensionless velocity in y-direction Coordinate in vertical direction Dimensionless coordinate In vertical direction Coordinate in horizontal direction Dimensionless coordinate in horizontal direction Greek symbols Thermal coefficient of expansion Temperature difference Time step (computer program) Vorticity Viscosity Kinematic viscosity Density Mean density Stream function APPENDIX II 104 Chart for Argon Flow Rate Conversion NO. 2 STD. A/R ML./ I200-tooo-800-600-400-200-CAL/BRAT/ON CHART ^ FLOWMETER CATALOG NO. F 12.00 SERIAL NO. G277-G487-G4& Df = O. / £ 5 " = CM. pf=Z.53 GM./ML. *STD.= /ATM. AHO 70°F STD* WATER ML 25 : i. . .i :_.J.... .• -i • —1 -:r 1 1 1 1 1 M | 1II 1 • 1111 1 1 1 1 1 M | 1II 1 • 1111 :: i : •ii! . i . .: j - . ' 'i • ' I - i * i ....I..:. •-T- . . . -1 1 1 1 1 M | 1II 1 • 1111 : i ! : 1 1 1 1 1 M | 1II 1 • 1111 :•:: :::: :::: :: i: :t:: E Li!!::: _ _ L : ! 1 ! ' •:.:!••: •1 . j . ; . • 1 • -; — 1 1 1 1 1 M | 1II 1 • 1111 iiiiiip: i : ; : 1 1 1 1 1 M | 1II 1 • 1111 i;;:ji:;: :!!; ! •!»• •• 1 ' r • — — --j-- 1 1 1 1 1 M | 1II 1 • 1111 ::::|:::: i;:: 1 1 1 1 1 M | 1II 1 • 1111 :i:iji ; ;i i : :iji:ii - •-;•:'-.:....... • •! ' i 11 i 11 i 1 : i i i i ; i: • •• :"J "• i 11 i 11 i 1 iii! • 1 ! : . . . i 11 i 11 i 1 i 11 i 11 i 1 ..:] ::4:T . . •. !::•: :. i i • • :i: ;i- :• ; i :• j i : : j " : ....;.:. i - • •: -• • /o " : : I: .: .:.-t-J r -. : . . L : - ... L.. - . . . . . . . . . •:hl— :-:— . i : . _.: .. .:rni: 1 -- — j ii .:.:L... .... . . . . . . . . . -::-— ~i— .ii-!-' : • • i. . :^ n :::|-"--!:•--.'.-[•: ..!.. i ;- .j. -±" • : : ! : : : : • i - : -——: ~1~ to eo 30 <o so eo 70 eo so SCALE READ/MG AT CEA/TEA OP BAS-L. 30 25 20 15 too APPENDIX I I I - COMPUTER PROGRAM 1 C 2 C 3 C * * ****** * * * ** * THIS IS M0DEL5 *********************** 4 C 5 C THIS PROGRAM IS A MODIFICATION OF M0DEL4 6 C THIS VERSION ALLOWS THE TIME STEP TO BE CHANGED 7 C AT ANY ARBITRARY ITERATION NUMBER(S) 8 C 9 INTEGER FL.FLAG,IFLAG,IERR 10 C 1 1 REAL *8 TA,TB.GR,PR,DT,DX,DY,DIF,DIFF,CHECK,SFC 12 REAL*8 C1,C2,C3,C4.C5,C6,C7,C8,C9,C10.C11.C12.C13.C14 13 REAL PCTHOT.PCTCOL 14 C 15 C INPUT PRANDTL NUMBER (PR) ; GRASHOFF NUMBER (GR) ; 16 C DELTA T (DT) 17 c 18 READ(6,20)PR,GR,DTA,DTB.DTC.0T2 19 20 F0RMAT(6F10.3) 20 c 21 SFC=0.08D0 22 LIM=21 23 DX=0.05D0 24 DY=0.05D0 25 TA = - 1.ODO 26 TB= 1.0D0 27 N = 441 28 REAL*8 T1(21.21).T2(21.21).X1(21.21).X2(21.21).S1(21.21 29 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 ) 30 REAL +8 U(21.21 ) .V (2 1 .2 1 ) 31 c 32 K1=2/DT2 + 2/DX* k2 33 K2=2/DT2 - 2/DX**2 34 K3=2/DT2 + 2/DY**2 35 K4=2/DT2 - 2/DY**2 36 LI 1=LIM-1 37 LI2=LIM-2 38 FLAG=0 39 c 40 c INITIALIZE MATRICES 4 1 c 42 DO 50 0=1 ,LIM 43 DO 50 1=1.LIM 44 T1(I,d)=0.0 45 T2 ( I . J ) = 0 . 0 46 X 1 ( I , <J ) =o. 0 47 X 2 ( I . J ) =0.0 48 S1(I,J)=0.0 49 S2(I.J)=0.0 50 U ( I . J ) =0.0 51 50 V ( I , J ) =0.0 52 c 53 DO 54 1=1,19 54 A(I)=0.O 55 B(I)=0.0 56 C(I)=0.0 57 54 R(I)=0.0 58 DO 58 1=1.21 1 0 6 59 D(I ) =0.0 60 E ( I ) = 0 . 0 61 F ( I ) = 0 . 0 62 58 T ( I ) = 0 . 0 63 C 64 C 65 C LOADING FIXED TEMPERATURES 66 C 67 PCTHOT=0.0 68 PCTCOL=0.0 69 C 70 DO 60 1=1.LIM 7 1 T 1 (I , 1 ) = TA + PCTCOL*I/L IM 72 T2( I . 1 )=TA+PCTCOL*I/LIM 73 T1 ( I , L IM)=(1-PCTH0T)*TB+PCTH0T* I/L IM 74 60 T2 ( I . L IM)=(1-PCTH0T)*TB+PCTH0T* I/L IM 75 C 76 C SOLVING FOR THE TEMPERATURES 77 C 78 C FIRST HALF OF TIME STEP 79 C 80 90 FLAG = F LAG +1 8 1 C 82 OT=DTA 83 IF ( F L A G . G T . 4 ) DT =DTB 84 IF ( F LAG .GT .40 ) DT = DTC 85 C1=1/(2*DX) 86 C2=1/(2*DY) 87 C3=1/(PR*DX**2) 88 C4=1/(PR*DY**2) 89 C5=2/DT +2*C3 90 C6=2/DT -2*C3 91 C7=2/DT +2*C4 92 C8=2/DT -2*C4 93 C9= 1/DX**2 94 C10=1/DY**2 95 C1 1=2/DT + 2*C9 96 C12=2/DT - 2*C9 97 C13=2/DT + 2*C10 98 C14=2/DT - 2*C10 99 WRITE(6, 93 )FLAG 100 93 FORMAT( ' ' , ' START OF LOOP ' ,G6) 101 C 102 C 103 IF LAG =1 .0 104 C 105 C SOLVING FOR COLUMNS 2 TO LIM-1 106 C 107 DO 110 J = 2.L I 1 108 DO 100 1=2,LI1 109 96 D(I ) = -C1*U ( I . J ) -C3 1 10 E( I ) =C5 1 1 1 F ( I )= C1*U ( I , J ) -C3 1 12 100 T ( I ) = ( C 4 + V ( I , J ) * C 2 ) * T 1 ( I , J - 1 ) + (C4-V( I 1 13 1+C8*T1 ( I . d ) 1 14 D(1)=0 1 15 E(1)=C5 1 16 F( 1) = -2*C3 107 1 17 T( 1 )=C4*T1 ( 1 ,J- 1 )+C8*T 1 ( 1 ,d)+C4*T1( 1. d+1 ) 1 18 D(LIM)=-2*C3 119 E(LIM)=C5 120 F(L IM)=0 12 1 T (L IM )=C4*T1 (L IM .d-1 )+C8*T1 (L IM ,d )+C4 *T1(L IM, d+1 ) 122 CALL T R I S L V ( 2 1 . D , E , F . T , 0 , 9 0 0 ) 123 DO 105 1 = 1,LIM 124 105 T 2 ( I , d ) = T ( I ) 125 110 CONTINUE 126 C 127 C 128 IFLAG=IFLAG+1 129 C 130 C 131 C SECOND HALF OF TIME STEP (TEMPERATURES) 132 C 133 C SOLVING FOR ROWS 2 TO LIM-1 134 C 135 DO 200 1=2.LI 1 136 DO 175 d=3.L I2 137 K = d- 1 138 A (K )=-V ( I , d ) *C2-C4 139 B(K)=C7 140 C(K)= V ( I . d ) * C 2 - C 4 141 175 R (K )= (C3+U ( I , d ) *C1 ) *T2 ( I -1 ,d )+ (C3-U ( I , d ) * C 1 ) * T2(1+1 ,d) 142 1+C6*T2 ( I . d ) 143 A( 1 )=0 144 B(1)=C7 145 C (1 )= V ( I , 2 ) * C 2 - C 4 146 R ( 1 ) = ( C 3 + U ( I , 2 ) * C 1 ) * T 2 ( I - 1 , 2 ) + ( C 3 - U ( I ,2 )*C1 ) * T2(1+1 .2) 147 1+C6*T2 ( I .2 ) + ( V ( I , 2 )*C2+C4)*T1 (1 , 1 ) 148 A ( L I 2 ) = -V( I , L I 1)*C2-C4 149 B (L I2 )=C7 150 C ( L I2 )=0 151 L1=LI1 152 R ( L I2 ) = (C3+U ( I , L1 ) *C 1 )*T2( I - 1 ,L1 ) + (C3 -U( I , L1 ) *C1 )-*T2( 1+1 153 1+C6*T2 ( I , L I 1) + (C4-V ( I , L I 1 )*C2)*T1 ( I . L IM) 154 CALL T R I S L V ( 1 9 , A . B . C . R . 0 , 9 0 0 ) 155 DO 180 d = 2 . LI 1 156 K = d- 1 157 180 T 1 ( I , d ) = R ( K ) 158 200 CONTINUE 159 c 160 c TOP ROW 161 c 162 DO 220 d = 2.L I 1 163 K = d- 1 164 A(K)=-C4 165 B(K)=C7 166 C(K)=-C4 167 220 R ( K ) = C 6 * T 2 ( 1 . d ) + 2 * C 3 * T 2 ( 2 , d ) 168 A( 1 )=0 169 B(1)=C7 170 C(1)=-C4 171 R( 1)=C6*T2( 1 ,2 )+2*C3*T2 (2 ,2 ) +C4 + T1( 1 . D 172 A(L I2 )=-C4 173 B (L I2 )=C7 174 C ( L I2 )=0 108 175 R ( L I2 )=C6*T2 (1 , L I 1) + 2*C3*T2(2 . LI 1 ) +C4*T1(1 .L IM) 176 CALL TRISLV( 1 9 . A , B . C . R , 0 , 9 0 0 ) 177 DO 240 J=2,L I1 178 K=J-1 179 240 T1 (1 ,d )=R (K ) 180 C 181 C BOTTOM ROW 182 C 183 DO 250 J = 2.LI 1 184 K=J-1 185 A(K)=-C4 186 B(K)=C7 187 C(K)=-C4 188 250 R (K )=C6*T2(L IM,d )+ 2*C3*T2 (L I 1,d) 189 A(1)=0 190 B(1)=C7 191 C(1)=-C4 » 192 R( 1 )=C6*T2(L IM.2 ) + 2*C3*T2(L I 1,2 ) +C4*T1(L IM . 1) 193 A(L I2)=-C4 194 B (L I2 )=C7 195 C (L I2 )=0 196 R(L I2 )=C6*T2(L IM.L I 1 ) + 2*C3*T2(L I 1 ,LI 1 ) +C4*T1(L IM .L IM ) 197 CALL T R I S L V ( 1 9 , A , B . C , R , 0 . 9 0 0 ) 198 DO 260 <J = 2. LI 1 199 K=J-1 200 260 T1(L IM,d )=R(K ) 201 C 202 C 203 C 204 IFLAG=IFLAG+1 205 C 206 C INTERIOR VORTICITIES 207 C FIRST HALF OF TIME STEP 208 C 209 C SOLVING FOR COLUMNS 2 TO LIM-1 210 C 211 DO 310 J=2.L I1 212 DO 300 I=3,L I2 213 K=I-1 214 A (K )=-U ( I , J ) *C1-C9 215 B(K)=C11 2 16 C(K)= U ( I , d ) * C 1 - C 9 217 300 R (K )= (C10+V ( I , J )+C2 ) *X1 ( I . J -1 )+ ( -V ( I . d ) *C2+C10 ) *X1 ( I . J+1 ) 218 1+C14*X1 ( I , d ) + G R * C 1 * ( T 1 ( I , J + 1 ) - T 1 ( I , d - 1 ) ) 219 A(1)=0 220 B( 1 ) =C 1 1 221 C( 1 )= U ( 2 . d ) * C 1 - C 9 222 R(1) = (C10+V(2 , J ) *C2 )+X1 (2 , J -1 ) + ( -V (2 ,d ) *C2+C10 ) *X1 (2 ,d+1 ) 223 1+C14*X1(2 ,d ) 224 1+GR*C1*(T 1 (2 ,d+1 ) -T1 (2 ,d-1 ) ) + (C9+U(2 .d ) *C1 ) *X2 ( 1 , d l 225 A (L I2 ) = -U(LI 1 .d ) *C1-C9 226 B(L I2)=C11 227 C ( L I 2 )=0 228 R ( L I 2 ) = C 1 4 * X 1 ( L I 1 . d ) + ( C 9 - U ( L I 1 . d ) * C 1 ) * X 2 ( L I M . d ) 229 1 + ( C 1 0 + V ( L 1 . d ) * C 2 ) * X 1 ( L 1 , d - 1 ) + (- V(L1 ,d ) *C2+C10)*X1(L1 .d+1 ) 230 1+GR*C1* (T1 ( L I 1 . d+1 )-T1 ( L I 1 , d-1 ) ) 231 CALL T R I S L V ( 1 9 . A . B . C , R , 0 . 9 0 0 ) 232 DO 305 1=2,LI1 109 233 K=I-1 234 305 X 2 ( I , J ) = R ( K ) 235 310 CONTINUE 236 C 237 C 238 C 239 IFLAG =IFLAG+1 240 C 241 C SECOND HALF OF TIME STEP (VORT IC IT l ES ) 242 C 243 C SOLVING FOR ROWS 2 TO LIM-1 244 C 245 DO 350 1=2.LI1 246 DO 340 d=3.L I2 247 K = d- 1 248 > ( K ) = - V ( I , d ) * C 2 ~ C 1 0 249 B (K)=C13 250 C (K )= V ( I . J ) * C 2 - C 1 0 251 340 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) 252 1+GR*C1*(T1 ( I ,0+1 )-T1 ( I . d - 1 ) )+C12*X2 ( I . d ) 253 A(1 )=0 254 B (1 )=C13 255 C (1 )= V(I.2)*C2-C10 256 R ( 1 ) = ( C 9 + U ( I . 2 ) * C 1 ) * X 2 ( I - 1 , 2 ) + ( C 9 - U ( I , 2 ) * C D * X2(1 + 1 . .2)257 1+C12*X2<1.2) 258 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 > 259 A ( L I 2 ) = - V ( I , L I 1 ) * C 2 - C 1 0 260 B ( L I2 )=C13 261 C ( L I 2 ) = 0 262 L1=LI 1 263 R(L I 2) = (C9+U( I ,L1 )*C 1 )*X2(I - 1 ,L1 ) + (C9-U(I . L1 ) *C1 ) * X 2 ( 1 + 1 264 1 + C 1 2 * X 2 ( I , L I 1 ) + ( - V ( I , L I 1 ) * C 2 + C 1 0 ) * X 1 ( I , L I M ) 265 1+GR*C 1 * ( T 1 ( I , L I M ) - T 1 ( I . L I 2 ) ) 266 CALL T R I S L V ( 1 9 . A , B , C , R , 0 , 9 0 0 ) 267 DO 345 d=2.L I1 268 K = d- 1 269 345 X 1 ( I . d )=R(K) 270 C 27 1 350 CONTINUE 272 C 273 C 274 C 275 IFLAG=IFLAG+1 276 C 277 C STREAM FUNCTION 278 C F IRST HALF OF TIME STEP 279 C 280 FL=0 281 360 FL = FL+ 1 282 DO 400 d = 2 .L I 1 283 DO 375 1=3 ,L I2 284 K=I-1 285 A(K ) = -C9 286 B(K)=K1 287 C (K )=-C9 288 375 R(K)=X 1 ( I . d )+C10*S1 ( I , d-1 )+K4*S1 ( I , d)+C 10 * S1 ( I . d+1 ) 289 A(1 )=0 290 B(1)=K1 110 291 C( 1 ) = -C9 292 R( 1 )=X1(2 ,d )+C10*S1 (2 ,d-1 )+K4*S1(2, d ).+C 10*S 1 ( 2 , J+1) 293 A (L I2 )=-C9 294 B(LI2)=K1 295 C (L I2 )=0 296 R ( L I2 )=X1 (L I1 ,d )+C10*S1 ( L I1 , J -1 )+K4*S1 ( L I 1,d)+C10*S1(LI 1 .d+1 ) 297 CALL T R I S L V ( 1 9 , A , B , C . R , 0 , 9 0 0 ) 298 DO 380 1=2,LI1 299 K = I-1 300 380 S2 ( I , d )=R (K ) 301 C 302 400 CONTINUE 303 C 304 C 305 IFLAG=IFLAG+1 -(FL-1) 306 C 307 C SECOND HALF OF TIME STEP (STREAM FUNCTION) 308 DO 475 1=2,LI1 309 DO 450 d=3,L I2 310 K = d- 1 31 1 A(K)=-C10 312 B(K)=K3 313 C(K)=-C10 314 450 R (K )=X1 ( I ,d )+C9*S2 ( I - 1 ,d )+K2*S2 ( I . d)+C9*S2(1 + 1,d) 315 A( 1 )=0 316 B(1)=K3 317 C (1 )=-C10 318 R (1 )=X1 ( I . 2 )+C9*S2 (1-1 ,2 )+K2*S2( I ,2 )+C9* S2(I + 1,2) 319 A (L I2 )=-C10 320 B (L I2 )=K3 321 C ( L I 2 )=0 322 R ( L I2 )=X1 ( I , L I 1 )+C9*S2(I - 1 ,LI 1 )+K2*S2( I ,L I 1)+C9*S2( 1+1 , LI 1 ) 323 CALL T R I S L V ( 1 9 . A , B , C , R , 0 , 9 0 0 ) 324 DO 460 d = 2,L I 1 325 K = d- 1 326 460 S K I , d)=R(K) 327 C 328 475 CONTINUE 329 C 330 IFLAG=IFLAG+1 - (FL-1) 331 C 332 C CHECK FOR CONVERGENCE OF STREAM FUNCTION 333 C 334 IF ( F L . G T . 5 0 ) GO TO 1000 335 T E S T = C 9 * ( 4 * S 1 ( 2 , 2 ) - S 1 ( 2 , 3 ) - S 1 ( 3 . 2 ) ) 336 IF ( X 1 ( 2 , 2 ) .EQ.O) GO TO 520 337 CHECK= (TEST-X1 (2 .2 ) ) /X1 (2 .2 ) 338 IF ( F L A G . G T . 1 0 ) SFC=0.05 339 IF ( F L A G . G T . 2 2 ) SFC=0.02 340 IF (CHECK .GT .SFC ) GO TO 360 341 C 342 520 CONTINUE 343 C 344 C CALCULATION OF VELOCITIES 345 C 346 DO 550 1=2,LI1 347 DO 525 d=3,L I2 348 525 U ( I , d ) = ( S1 ( I ,d-2 )-8*S1 ( I , d-1 )+8*S1 ( I ,d+1 )-S1 ( I ,d+2) )/ ( 12 + DY ) I l l 349 U ( I , 2 ) = ( - 3 * S 1 ( I , 2 ) + 6 * S 1 ( I , 3 ) - S 1 ( I , 4 ) ) / ( 6 * D Y ) 350 550 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 ) 351 DO 600 J=2.L I1 352 DO 575 1=3,LI2 353 575 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 ' 354 V ( 2 . d ) = ( 3 * S 1 ( 2 . d ) - 6 * S 1 ( 3 . d ) + S 1 ( 4 . d ) ) / ( 6 * D X ) 355 6O0 V ( L I 1 . d ) = - ( 3 * S M L I 1 . d ) - 6 * S 1 ( L 11 - 1 . d ) + S 1 ( L I 1 - 2 . d ) )/(6*DX ) 356 C 357 C 358 C 359 C 360 C 361 C CALCULATION OF BOUNDARY VORTICITIES 362 DO 625 d = 2.L I 1 363 X 2 ( 1 , d ) = - 2 * S 1 ( 2 . d ) * C 9 364 X1( 1 ,d )=X2 (1 .d ) 365 X 2 ( L I M . d ) = - 2 * S 1 ( L I 1 . d ) * C 9 366 625 X1 ( L IM .d )=X2 ( L IM .d ) 367 DO 650 1=2.L11 368 X2( I , 1 ) = -2 + S K I , 2 ) * C 1 0 369 X 1 ( I , 1 )=X2( I . 1) 370 X 2 ( I , L I M ) = - 2 * S 1 ( I , L I 1 ) * C 1 0 371 650 X 1 ( I . L I M ) = X 2 ( I . L I M ) 372 C 373 C 374 C 375 C CONVERGENCE CHECK OF TEMPERATURES 376 C 377 DIF=0 378 IF ( F L A G . G T . 7 0 0 ) GO TO 720 379 DO 700 d=2.20 380 DO 700 1=2,20 381 IF ( T K I . d ) . N E . O ) D I F F = ( T 1 ( I . d ) - T 2 ( I . d ) ) / T 1 ( I . d ) 382 IF (0ABS(DIFF ) .GT .DIF) DIF=DABS(DIFF) 383 70O CONTINUE 384 IF ( D I F . G T . 0 . 0 0 5 ) GO TO 90 385 C 386 GO TO 725 387 720 WRITE(6,721 ) 388 721 FORMAT ( ' ' . ' * * * * * * * * * * * * * * INTERRUPT IN EFFECT ' ) 389 c 390 c 391 725 CONTINUE 392 c 393 c 394 c OUTPUT TEMPERATURES AND VELOCITIES 395 c 396 DO 880 1 = 1 .LIM 397 WRITE (6 .850)1 398 850 FORMAT( ' ' . ' D A T A FOR ROW NUMBER ' . G 5 ) 399 W R I T E ( 6 . 8 6 0 ) ( T 1 ( I . d ) , d = 1 . L I M ) 400 W R I T E ( 6 , 8 6 5 ) ( U ( I , d ) , d = 1 , L I M ) 401 W R I T E ( 6 . 8 7 0 ) ( V ( I , d ) , d = 1 , L I M ) 402 860 FORMAT( ' ' . ' T E M P E R A T U R E S : ' . 2 1 F 9 . 6 ) 403 865 FORMAT( ' ' . ' X-VELOCITY : ' .2 IF 10 .3 ) 404 870 FORMAT( ' ' , ' Y-VELOCITY : ' .2 IF 10 .3 ) 405 880 CONTINUE 406 c 112 407 GO TO 3000 408 C 409 C * * * * * * * * * * * * * * * * * * * * * * * * ^ * * * * * * * * + * * * * * * * * * ^ * * * * * * * * * * * * 410 C 41.1 C ERROR STATEMENTS ASSOCIATED WITH TRISLV 412 C 4^3 £ * + * * * * * * * * * * * # * * * * * + * * * + * * * * * * * * * * * * + * * * # * * * * * + + + * • * + * * * * 414 C 415 900 WRITE(6 .960) IFLAG 416 960 FORMAT(' ' . ' SOLVE COULD NOT FIND A SOLUTION, FLAG = ' . 1 6 ) 417 GO TO 3000 418 C 419 1000 CONTINUE 420 C INSERT WRITE STATEMENT WHICH SAYS STREAM FUNCTION NOT 421 C CONVERGING AFTER 10 ITERATIONS 422 C 423 WRITE (6 ,1200) 424 1200 FORMATC ' . ' S T R E A M FCN NOT CONVERGING') 425 C 426 DO 1600 J=1.L IM 427 W R I T E ( 6 , 1 5 0 0 ) S 1 ( 2 . J ) , S 2 ( 2 . J ) 428 1500 FORMAT(' ' . ' S 1 = ' . F 1 6 . 4 . ' S2= ' . F 1 6 . 4 ) 429 1600 CONTINUE 430 C 431 C 432 WRITE(6 ,1610)FLAG.CHECK 433 1610 FORMAT(' ' , ' NO. OF LOOPS RUN = ' . G 8 , ' CHECK = ' . F 1 6 . 5 ) 434 C 435 C 436 GO TO 3000 437 C 438 2000 CONTINUE 439 C INSERT WRITE STATEMENT WHICH SAYS TEMPERATURES NOT CONVERGING 440 C 441 WRITE ( 6 , 2 2 0 0 ) 442 2200 FORMAT(' ' . 'TEMPERATURES NOT CONVERGING') 443 C 444 C 445 DO 2300 1=1.21 446 W R I T E ( 6 . 2 2 5 0 ) ( T 2 ( I , d ) . J = 1 , 5 ) 447 2250 FORMATC ' . 5 F 1 6 . 5 ) 448 2300 CONTINUE 449 C 450 WRITE(6 .2325) 451 2325 FORMAT(' ' , ' S T R E A M FUNCTION' ) 452 C 453 DO 2400 1=2,20 454 W R I T E ( 6 , 2 3 5 0 ) ( S 1 ( I . J ) . J = 1 , 5 ) 455 2350 FORMAT( ' ' . 5 F 1 6 . 5 ) 456 2400 CONTINUE 457 C 458 GO TO 3000 459 C 460 3000 CONTINUE 461 C 462 STOP 463 END 

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