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Fluid flow and heat transfer in continuous casting processes Matys, Paul 1988

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FLUID FLOW A N D H E A T T R A N S F E R IN CONTINUOUS C A S T I N G PROCESSES Pau l Ma tys Diplom-lngenieur , Universi ty of Essen, West Germany, 1985 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F M E C H A N I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A July 1988 © Paul Matys> f 1 9 8 8 In presenting this thesis in part ial fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia , I agree that the Libra ry shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my wri t ten permission. Department of Mechanical Engineering The Universi ty of Br i t i sh Co lumbia 1956 M a i n M a l l Vancouver, Canada Date: A b s t r a c t A three-dimensional finite difference code was developed to simulate fluid flow and heat transfer phenomena in continuous casting processes. The mathematical model describes steady state transport phenomena in a three dimensional solution domain that involves: turbulent fluid flow, natural and forced convection, conduction, release of latent heat at the solidus surface, and tracing of unknown location of l iquid /so l id interface. The governing differential equations are discretized using a finite volume method and a hybrid central, 'upwind differencing scheme A fully three-dimensional A D I -like iterative procedure is used to solve the discretized algebraic equations for each dependent variable. The whole system of interlinked equations is solved by the S I M P L E algori thm. The developed computer code was used for parametric studies of continuous casting of a luminum. The results were compared against available experimental data. This numerical simulation enhances understanding of the fluid flow and heat transfer phenomena in continuous casting processes and can be used as a tool to optimize technologies for continuous casting of metals. ii T a b l e of C o n t e n t s Abstract ii List of Tables v i List of Figures v:M-Acknowledgement xiv Nomenclature xv 1 I N T R O D U C T I O N 1 1.1 Literature Review 4 1.1.1 Exper imental Studies 4 1.1.2 Theoretical Studies 5 1.1.3 Mot iva t ion and Scope of Present Work 9 2 M A T H E M A T I C A L F O R M U L A T I O N 13 2.1 Basic Assumptions 13 2.2 Governing Differential Equations 14 2.2.1 Instantaneous Conservation Equations 14 2.2.2 T ime Averaged Equations 15 2.2.3 Turbulence Model l ing 16 2.2.4 General Transport Equations 18 i i i 2.3 Boundary Condit ions 18 2.3.1 Inlet and Outlet 18 2.3.2 Free Surface and Symmetry Planes 19 2.3.3 Walls 20 2.3.4 Solidus Surface 23 2.3.5 Buoyancy 23 3 S O L U T I O N P R O C E D U R E 26 3.1 G r i d System 26 3.2 Discretization of Differential Equations 27 3.3 Discretized Boundary Condit ions 31 3.3.1 Inlet 31 3.3.2 Free Surface and Symmetry Planes 32 3.3.3 Walls 32 3.3.4 Solidus Surface 34 3.4 S I M P L E Algo r i t hm 35 3.5 Solution of the Linear Algebraic Equations 36 3.6 Numerical Considerations 38 4 R E S U L T S A N D D I S C U S S I O N 55 4.1 T w o Dimensional Heat Flow 55 4.1.1 Description of the Problem 55 4.1.2 Numerical Simulation 55 4.1.3 Discussion of Results 57 4.2 Two Dimensional F lu id Flow 59 4.2.1 Description of the Problem 59 4.2.2 Numerical Simulation 60 iv 4.2.3 Results and Discussion 60 4.3 Three Dimensional F l u i d Flow and Heat Transfer 62 4.3.1 Description of the Problem 62 4.3.2 Numerical Simulation 62 4.3.3 Results and Discussion 64 4.3.4 Parametric Studies 66 5 S U M M A R Y A N D R E C O M M E N D A T I O N S 124 5.1 Summary 124 5.2 Recommendation 125 B i b l i o g r a p h y 128 v List of Tables 2.1 Values of turbulence model constants 17 4.2 Parameters and physical properties used for simulation of two-dimensional axis-symmetrical continuous casting of a luminum alloy (6063) billet. 56 4.3 Parameters and physical properties used for simulation of two-dimensional axis-symmetrical continuous casting of a luminum alloy (7049) billet. 61 4.4 Parameters and physical properties used for simulation of continu-ous casting of a luminum slab in three-dimensional Cartesian coor-dinate system 63 v i List of F igures 1.1 Flow diagram for production of coiled strip by conventional and continuous casting route (numbers indicate metal thickness in mm) . 11 1.2 Schematic diagram of vertical continuous casting 12 2.3 Nozzle configuration in symmetrical quarter of cast 25 3.4 Discretization domain for Cartesian, three-dimensional coordinate system 42 3.5 F in i te volume grid in x-y plane with associated notation 43 3.6 Control volume used to obtain the finite volume equation 44 3.7 Control volume and notation for u-momentum analysis 45 3.8 Control volume and notation for v-momentum analysis 46 3.9 Control volume and notation for transport of scalar quantity in z-plane 47 3.10 G r i d notation for a plane x-y defined by constant K in a domain wi th irregular boundaries 48 3.11 G r i d notation for a plane x-z defined by constant J in a domain wi th irregular boundaries 49 3.12 Control volume for all scalar variables defined at the grid node (1,J\K) : (a) constant K-plane: (b) constant J-plane 50 3.13 Control volume for velocity u defined at the grid node ( I . J .K) ; (a) constant K-plane; (b) constant J-plane 51 v i i 3.14 Control volume for velocity v defined at the gr id node ( I , J ,K) ; (a) constant K-plane; (b) constant I-plane 52 3.15 Control volume for velocity w defined at the grid node ( I , J ,K) : (a) constant J-plane; (b) constant I-plane 53 3.16 Representation of the line-by-line method 54 4.17 Temperature field in a luminum billet obtained by finite element method ; inlet temperature 690 C , casting speed 160 m m / m i n (Grandfield and Baker, courtesy of C O M A L C O Research and Tech-nology) 69 4.18 Surface temperature of a luminum billet (Grandfield and Baker, courtesy of C O M A L C O Research and Technology) 70 1.19 Comparison between temperature field obtained by Grandfield and Baker (left) and temperature field obtained by finite volume method assuming uniform flow field (right); inlet temperature 690 C , casting speed 160 m m / m i n 71 4.20 Temperature field in a luminum billet obtained by finite volume method assuming inflow through nozzle and laminar flow; inlet tem-perature 690 C , casting speed 160 m m / m i n 72 4.21 Temperature field in a luminum billet obtained by finite volume method assuming inflow through nozzle, laminar flow with buoy-ancy; inlet temperature 690 C , casting speed 160 m m / m i n 73 4.22 Temperature field in a luminum billet obtained by finite volume method assuming inflow through nozzle, turbulent flow wi th buoy-ancy; inlet temperature 690 C , casting speed 160 m m / m i n 74 v i i i 4.23 Velocity field in sump of aluminum billet wi th a 320 m m dia. mea-sured by magnet probe (Ricou and Vives) 75 4.24 Velocity field in sump of aluminum billet wi th a 320 m m dia. ob-tained by finite volume method assuming turbulent flow wi th buoy-ancy 76 4.25 Solidus surface in symmetrical quarter of cast, casting speed 1 m m / s , inlet temperature 680 C . solidus temperature 660 C . . . . 77 4.26 Isothermal surface for 500 C in symmetrical quarter of cast, casting speed 1 m m / s , inlet temperature 680 C , solidus temperature 660 C . 78 4.27 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C: z - 0.02m 79 4.28 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C : z 0.09m 80 4.29 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C : z=0.17m 81 4.30 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C : z=0.21m 82 4.31 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C: x=0.03m 83 4.32 Velocity and temperature fields . casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C : x=0.19m 84 4.33 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C : x=0.30m 85 4.34 Velocity and temperature fields , casting speed 1 mm/s , inlet tem-perature 680 C , solidus temperature 660 C : x = 0.52m 86 ix 4.35 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C : y=1.17m 87 4.36 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C: y=0.94m 88 4.37 Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C : y=0.73m 89 4.38 Solidus surface in symmetrical quarter of cast, casting speed 0.75 m m / s , inlet temperature 680 C , solidus temperature 660 C. . . . 90 4.39 Isothermal surface for 500 C in symmetrical quarter of cast, casting speed 0.75 m m / s , inlet temperature 680 C , solidus temperature 660 C 91 4.40 Velocity and temperature fields , casting speed 0.75 m m / s , inlet temperature 680 C, solidus temperature 660 C : z -0.02m 92 4.41 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 680 C, solidus temperature 660 C: z=0.09m 93 4.42 Velocity and temperature fields , casting speed 0.75 m m / s , inlet temperature 680 C , solidus temperature 660 C : z—0.17m 94 4.43 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 680 C, solidus temperature 660 C: z=0.21m 95 4.44 Velocity and temperature fields , casting speed 0.75 m m / s , inlet temperature 680 C , solidus temperature 660 C: x=0.03m 96 4.45 Velocity and temperature fields , casting speed 0.75 m m / s , inlet temperature 680 C , solidus temperature 660 C: x—0.19m 97 4.46 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 680 C , solidus temperature 660 C : x -0.30m 98 x 4.47 Veloci ty and temperature fields , casting speed 0.75 m m / s , inlet temperature 680 C , solidus temperature 660 C : x=0.52m 99 4.48 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 680 C , solidus temperature 660 C : y=1.17m 100 4.49 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 680 C , solidus temperature 660 C : y=0.94m 101 4.50 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 680 C , solidus temperature 660 C : y=0.73m 102 4.51 Solidus surface in symmetrical quarter of cast, casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C ; solution domain defined by I x J x K 15x20x10-3000 grid points 103 4.52 Isothermal surface for 500 C in symmetrical quarter of cast, casting speed 0.75 mm/s , inlet temperature 700 C , solidus temperature 660 C ; solution domain defined by I x J x K 15x20x10=3000 grid points. 104 4.53 Velocity and temperature fields , casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C : z—0.02m 105 4.54 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 700 C , solidus temperature 660 C : z - 0 . 0 9 m 106 4.55 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 700 C , solidus temperature 660 C : z=0.17m 107 4.56 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 700 C , solidus temperature 660 C : z—0.21m 108 4.57 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 700 C , solidus temperature 660 C : x - 0 . 0 3 m 109 4.58 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 700 C , solidus temperature 660 C : x=0.19m 110 xi 4.59 Velocity and temperature fields , casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C : x=0.30m I l l 4.60 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 700 C, solidus temperature 660 C: x=0.52m 112 4.61 Velocity and temperature fields , casting speed 0.75 mm/s , inlet temperature 700 C, solidus temperature 660 C: y = 1.17m 113 4.62 Velocity and temperature fields , casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C : y=0.94m 114 4.63 Velocity and temperature fields . casting speed 0.75 mm/s , inlet temperature 700 C, solidus temperature 660 C: y=0.73m 115 4.64 Turbulent kinetic energy field, casting speed 0.75 mm/s , inlet tem-perature 700 C, solidus temperature 660 C : z=0.02m 116 4.65 Turbulent kinetic energy field, casting speed 0.75 mm/s , inlet tem-perature 700 C , solidus temperature 660 C: z=0.09m 117 4.66 Turbulent kinetic energy field, casting speed 0.75 mm/s , inlet tem-perature 700 C, solidus temperature 660 C : z=0.17m 118 4.67 Effective viscosity field, casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C: z=0.02m 119 4.68 Effective viscosity field, casting speed 0.75 m m / s , inlet temperature 700 C, solidus temperature 660 C : z^0 .09m 120 4.69 Effective viscosity field, casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C: z=0.17m 121 4.70 Solidus surface in symmetrica] quarter of cast, casting speed 0.75 m m / s . inlet temperature 700 C , solidus temperature 660 C; solution domain defined by I x J x K 22x32x18=12,672 grid points 122 xn 4.71 Isothermal surface for 500 C in symmetrical quarter of cast, casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C ; solution domain defined by I x J x K 22x32x18=12,672 grid points. 123 x i i i A c k n o w l e d g e m e n t The author would like to express his deep gratitude to Professor M . E . Salcudean for her guidance, invaluable suggestions and constant encouragement throughout this research. The author is also indebted to Dr . Ned Di j l a l i , Prof. F . Weinberg and Zia Abdu l l ah for their assistance and fruitful discussions that contributed to this re-search. The cooperation of C O M A L C O Research Centre in providing unpublished data is gratefully acknowledged. Final ly , the author wishes to thank his family and friends who provided moral support and encouragement. Financial support for this research was provided by the Nat ional Science and Engineering Research Counci l of Canada. xiv N o m e n c l a t u r e a area of cell b o u n d a r y A total convect ive and diffusive flux coefficient; area c convect ive flux coefficient Ci,C2 e m p i r i c a l constants in d i s s ipat ion equat ion 0^ e m p i r i c a l constant in turbulent viscosity equat ion C D empi r i ca l constant in turbulent energy equat ion c p specific heat content D diffusive flux coefficient E e m p i r i c a l constant in the "law of the wal l" G generat ion t e r m in the turbu lent kinetic energy equat ion g grav i ta t iona l accelerat ion h specific enthalpy h' heat transfer coefficient k turbulence kinet ic energy; t h e r m a l c o n d u c t i v i t y p pressure Pe Peclet n u m b e r p' pressure correc t ion Pr P r a n d t l n u m b e r Q to ta l t ranspor t across cell boundar ie s Re R e y n o l d s n u m b e r xv S v o l u m e t r i c source t e r m u x -d irec t ion ve loc i ty c o m p o n e n t uT fr ic t ion veloci ty v y -d irec t ion veloci ty c o m p o n e n t V v o l u m e w z -d irect ion velocity c o m p o n e n t y + d imensionless d is tance f r o m the wall Greek Symbols (3 vo lumetr i c t h e r m a l expans ion coefficient t d i ss ipat ion rate of turbulent kinet ic energy 4> general var iab le r diffusion coefficient AC von K a r m a n n constant ti d y n a m i c viscosity v k inemat i c viscosity p density o P r a n d t l n u m b e r Ok turbulent P r a n d t l n u m b e r in k -equat ion o( turbulent P r a n d t l n u m b e r in e-equation Superscripts ' local correc t ion; fluctuating c o m p o n e n t * guessed value 4- d imensionless xv i eff N,S,E,W,F,B,P n, s. e, w, f, b, p t S Subscripts effective value coordinate directions grid points cell faces turbulent value at solidus xv i i Chapter 1 INTRODUCTION Metal casting is a process which is widely used in various manufacturing industries to obtain semifinished shapes and components. An alternative to the traditional ingot casting method is continuous casting. This method is very attractive because it has many advantages over the ingot route; it involves fewer processing steps (see fig. 1.1), reduces investment and labour costs and allows for increased productivity. In addition, continuous casting reduces energy consumption and improves quality. The continuous casting process is shown schematically in fig. 1.2. Molten metal is continuously poured from the tundish through the nozzle-float into the liquid pool inside the mould. Due to the cooling through the mould walls, a solid shell is created and grows towards the center of the cast. As the cast emerges from the mould, it is cooled further with the water spray. The solidified metal slab is withdrawn continuously downwards. The major controlling factor in the continuous casting process is heat transfer. The conversion of liquid metal into a solid semi-finished shape involves the removal of: the superheat from the liquid entering the mould from the tundish, the latent heat of fusion at the solidification front, and the sensible heat from the solid metal. In the liquid pool, heat transfer takes place through convection and conduction, whereas in the solid region heat transfer is through conduction only. The liquid 1 Chapter 1. INTRODUCTION 2 metal is subjected to primary cooling by conduction of heat through a water-cooled mould and to secondary cooling by direct application of water to the solid shell as it emerges from the mould and by radiation. The heat transfer process at the external surface of the solid shell is very complex. At some point above the base of the mould, the shell shrinks away from the mould wall leaving an air gap which reduces drastically the extraction of heat which occurs through conduction across the air gap. This can lead to partial remelting of the shell and break-outs. The extent of this depends on ferrostatic pressure and on the composition of the lubricant. The rate of heat extraction by the water spray is primarily determined by the boiling mechanism of the water on the cast surface. Depending on metal temper-ature, surface roughness, water flow, water temperature, water composition and spray geometry, the heat is removed through nucleate, transition or film boiling. The nozzle geometry, shape of the solidification front and casting speed con-tribute to a very complex, turbulent, buoyancy driven flow field in the sump. The flow pattern in the molten metal has a significant effect on the quality of the cast because it controls the removal of non-metallic inclusion as well as the convective heat transfer and solidification processes. The nature of mass transfer in the neigh-borhood of the solidification front has a primary effect on the motion of crystals and crystal growth, on the possible development of dendrites, on the homogeneity and the volume of the grain, and, consequently, on the mechanical properties of the metal. Recognizing the growing importance of the continuous casting process for in-dustry, research in this field has expanded rapidly in recent years in order to improve the understanding of this process. Some of the areas researchers have concentrated on are: Chapter 1. INTRODUCTION 3 1. Electromagnetic stirring (EMS). In the primary cooling zone, E M S reduces non-metallic inclusion content in the surface and sub-surface through their better distribution and improves surface quality. In the secondary cooling zone, E M S results in a more uni-form and refined solidification structure, reduces center line porosity and segregation, thus yielding more uniform mechanical properties. 2. Strand bulging. This phenomenon causes cracking and center line segregation. It occurs be-tween back-up rolls of a casting machine, and is due to the static pressure in the liquid core of the strand. For a better understanding of this phenomenon it is necessary to study heat transfer in the primary and secondary cooling zones. 3. Continuous casting mould powders. A liquid slag can be formed by introducing powder that infiltrates between the metal shell and the mould walls. This liquid slag, which acts as a lubri-cant, results in better surface quality of the cast and reduced friction forces. Heat removal from the metal to the mould can be better controlled. 4. Metal powder injection. This consists of injecting powdered or granular metal into molten metal as it enters the mould from the tundish. It helps in extracting part of the superheat of the metal, and creates new centers of heterogeneous nucleation. In all above mentioned research areas the scientific understanding of fluid flow and heat transfer phenomena in the cast is a prerequisite to any progress in con-tinuous casting technology. Chapter 1. INTRODUCTION 4 1.1 Literature Review In this section, previous experimental and theoretical investigations of phase change phenomena, part icularly those emphasizing continuous casting processes, are re-viewed. Depending on the approach used, the studies are divided into few groups and their contr ibution is briefly discussed. In conclusion the motivation and scope of the present work are outl ined. 1.1.1 Experimental Studies In molten metals metallurgy, experimental studies of transport phenomena are difficult. Measurement techniques of classical fluid mechanics are ineffective due to the hostile environment, the opacity and high temperatures of the molten metals (e.g. l iquid steel 1550 ° C , l iquid a luminum 700 °C). Water modelling can provide useful but only qualitative information about the general nature of the flow field wi th in the mould . There are l imitations due to simili tude criteria and such models can reflect properly only selected parameters of the original system. For mercury flows, hot-wire and hot-film sensors [23] can measure average and fluctuating velocities. This technique has, however, been unsuccessful for temperatures higher than 100 °C. Addi t ional ly , it is difficult to keep constant temperature during measurements because of the high conductivity of mercury. Lehner and Hsiao [20] calculated velocities in argon stirred liquid steel from the drag force exerted on the graphite rods immersed in the pool . Weinberg and M o r t o n [25], obtained autoradiographs of the billet by adding radioactive gold into the l iquid pool during the continuous casting of steel. They established the poo] profile and the extent of l iquid mixing in the pool . Tungsten pellets containing radioactive cobalt were dropped into the pool to measure the pool Chapter 1. INTRODUCTION 5 depth independently of l iquid mixing . One of the most promising measurement techniques is the magnet-probe method. This technique permits the determination of both the magnitude and direction of the velocity inside a molten metal . The working principle is based on the electro-magnetic and hydrodynamic phenomena caused by the flow of l iquid metal around a cyl indr ical magnet. Using this magnetic-method, Vives and Ricou [42] obtained the l iquid pool flow field pattern during the continuous casting of a luminum billets in conventional and electromagnetic moulds. 1.1.2 Theoretical Studies For a complete understanding of the continuous casting process, there is, no substi-tute for experimental investigations. The mathematical modelling and numerical simulation may, however, be helpful in shedding some light on the fluid flow and heat transfer phenomena and in guiding experimental research. M i z i k a r [24] developed a numerical model for the continuous casting of steel slabs. His model consisted of a one-dimensional transient conduction equation and associated boundary conditions. The solidification profiles and temperature distr ibut ion were computed for different secondary cooling systems. The author concluded that the most effective method of heat extraction were multibank spray cooling and radiant cooling. Fahidy [8] found existing numerical models too complicated and burdensome, and proposed, as an alternative, a semi-rigorous analytical technique. This tech-nique allows a quick approximate estimation of the time dependent variation of the average mould zone temperature in a continuous cast steel billet . Hi l l s [13] developed a general integral-profile method for the analysis of heat Chapter 1. INTRODUCTION 6 transfer during solidification. The solid metal layer was characterized in terms of two non-linear differential equations for heat fluxes crossing the boundaries of the solid shell. The method was used to predict solidification rates. Gaut ier [9] neglected axial conduction in continuous casting of steel, and de-veloped a one-dimensional mathematical model . The finite difference formulation was used to calculate the profile of the solidification front and the results were compared wi th measurements. Szekely and Stanek [38] used Miz ika r ' s [24] approach to heat transfer and solidification processes wi th the major modification of assigning a definite flow pattern to the molten core. It was shown that the solidification processes and the profile of the solidus line are relatively insensitive to the flow pattern in the molten phase, but the rate at which superheat is removed is markedly affected. Szekely and Yadoya [39] developed a mathematical model of the turbulent flow field in the mould region in the continuous casting systems. The two-dimensional turbulent flow equations were solved numerically. The computed results were found to be in qualitative agreement wi th the experimental results from water model studies. Asa i and Szekely [l] proposed a mathematical representation for a continuous casting system exhibi t ing axial symmetry. The formulation involved solidification phenomena and turbulent transport of momentum and thermal energy. The gov-erning differential equations were discretized and solved numerically. Predictions were made for solidification profiles, velocity patterns and trajectories of inclusion particles. Shamsundar and Sparrow [35] analyzed multidimensional conduction phase change v ia the enthalpy model . It was shown that the mathematical representation of the enthalpy model is equivalent to the conventional conservation equation in Chapter 1. INTRODUCTION 7 the solid and l iquid regions at the solid-liquid interface. A fully implic i t finite-difference scheme was used to solve the solidification problem in a convectively cooled square mould . G r i l l et al . [ l l ] investigated heat flow and gap formation in the mould of a continuous slab caster using a mathematical model . The ultimate purpose of their investigation was to predict the casting conditions which can lead to break-outs. The mathematical formulation simulated transient, one-dimensional heat flow in a transverse slice of a steel slab and between the shell surface and the mould wal l . It was shown, that due to the shape of the gap, a hot spot forms on the surface of the slab. Under these conditions, the gap may collapse and this could lead to break-outs or bleedings of the steel. Shibani and Ozisik [36] used a matched asymptotic technique to investigate the freezing of low Prand t l number liquids in turbulent flow between parallel plates. The location of the solid-liquid interface and the heat transfer rate were established as a function of the axial position along the channel. Sparrow, Patankar and Ramadhyani [37] analyzed mult idimensional melt ing around a vertical tube in the presence of natural convection. Solutions were ob-tained by an implici t finite difference procedure modified to account for the move-ment of the solid-liquid interface. The findings indicated that natural convection effects are very important in many phase change problems. Voller , Markatos and Cross [44] presented an enthalpy method for convec-tion/diffusion phase change which is consistent wi th existing control volume tech-niques for heat and mass flow problems. They examined different methods for the treatment of the moving interfaces. The introduction of the Darcy source was recommended as a most promising technique for the simulation of the gradual slow down of the fluid flow in the mushy region. Chapter 1. INTRODUCTION 8 L a i , Salcudean, Tanaka and Guthr ie [14] presented numerical solutions of the three-dimensional turbulent Navier-Stokes equations, in conjunction wi th the k - e turbulence model. They considered the problem of turbulent flow wi th in a tundish of high aspect ratio. Calculated flow fields were shown to be similar to corresponding experimental flow fields obtained via Laser-Doppler anemometry and flow visualization techniques. Salcudean and A b d u l l a h [33] carried out a numerical study of the effects of natural convection during the solidification of cast iron in cyl indrical moulds. Free convection was modelled in the l iquid region by solving the momentum, continuity and energy equations using the Boussinesq density model . The flow in the mushy region was simulated based on the principles of flow through porous media. The study concluded that natural convection causes alterations of temperature field and enhances heat transfer. Grandfield and Baker [10],[2] carried out an experimental and numerical study of the heat flow in gas pressurized and conventional a luminum continuous casting moulds. The numerical solutions were obtained using a finite element method. Heat flow through the gas layer was found to be controlled by the physical prop-erties of the gas and external factors such as applied metallostatic head pressure. F i l m boi l ing affecting heat flow, solidification and the cast microstructure was pr i -mari ly determined by spray geometry and, to some extent, by changes in water temperature and composit ion. Bommaraju , in his P h D thesis [5], studied the influence of electromagnetic stir-ring ( E M S ) on mould behavior, mould related quality and columnar-to-equiaxial t ransi t ion. Trials were performed at steel companies, billet samples were examined and mathematical modell ing was carried out. The effect of E M S on mould heat extraction was found to be negligible. A two dimensional heat flow model was Chapter 1. INTRODUCTION 9 used to establish mould heat flux profiles which were further employed in a one-dimensional transient heat flux model to predict removal of the superheat from the l iquid pool . 1.1.3 Motivation and Scope of Present Work One can divide the theoretical investigations of phase change phenomena and continuous casting processes reviewed in the previous section into the following groups: 1. Ana ly t i ca l studies of conduction (e.g. [8], [13], [36]). 2. Numerica l studies of conduction involving: one dimensional transient or two dimensional steady state model (e.g. [24], [9], [11], [10]). 3. Numerica l studies of heat transfer and fluid flow in two dimensional domain (e.g. [38], [39], [1], [44], [33]). T w o dimensional models of transport phenomena can only be applied to a few real continuous casting processes that exhibit axial symmetry of cast (e.g. billet) and nozzle configuration (e.g. simple axial nozzle). In all other situations, a realistic s imulat ion of the heat transfer and fluid flow requires a fully three-dimensional model . To the best knowledge of the author, no such simulation is available in the literature. The main objective of this work was to develop a fully three-dimensional com-puter code which takes into account turbulent flow, buoyancy forces, and heat transfer through conduction, forced and natural convection. The heat transfer due to convection may be small as compared wi th conduction in the case of l iquid Chapter 1. INTRODUCTION 10 metals wi th low Prand t l numbers. Convection is important in removing impur i -ties and reducing centerline porosity and segregation. The test case chosen in the present study is that of the continuous casting of an a luminum slab, geometries and different metals. C h a p t e r 1. INTRODUCTION 11 20 500 DC U caster Homogenization Scalp Scalp Homogenlzation Preheat -BD hot H rolling Edge (. shear Shear Tandem hot / warm rolling 6 I h Holder Thin strip caster i Shear Coil as reron Thick strip caster Mill 1 Mill 2 Shear 4 " 22 6 Figure 1.1: Flow diagram for production of coiled strip by conventional and con-tinuous casting route (numbers indicate metal thickness in mm). Chapter 1. INTRODUCTION DOWN SPOUT NOZZLE FLOAT WATER COOLED s MOULD WATER SPRAY WITHDRAWAL SPEED Figure 1.2: Schematic diagram of vertical continuous casting. Chapter 2 MATHEMATICAL FORMULATION In this chapter, we present the mathematical statement of the physical laws which govern the flow and heat transfer in the continuous casting process. The governing equations and the underlying assumptions are presented, followed by a discussion of the boundary conditions relevant to the problem considered here. 2.1 Basic Assumptions The complexity of the continuous casting process was highlighted in the first chap-ter. A realistic simulation of this process requires the solution to be carried out in a three dimensional domain. The model must take into account turbulent flow motion, buoyancy effect and a varying location of the solid/liquid interface. In this study the liquid metal is assumed to be incompressible and non-reacting, and solidification is assumed to take place isothermally. The change of dimensions of the slab as it is cooled, the curvature of the meniscus, and the thickness of the nozzle walls are neglected. The simplified geometry of the solution domain is shown in fig. 2.3. Finally, it is assumed that the process is a steady state one. 13 Chapter 2. MATHEMATICAL FORMULATION 14 2.2 Governing Differential Equations Transport phenomena in continuous casting processes can be described mathemat-ically by the differential equations expressing conservation of mass, momentum and energy. In the case of turbulent flows, the scale of the smallest turbulent eddies is too small to be solved economically by a numerical scheme, and turbulence modell ing is necessary. A l l instantaneous quantities are decomposed into statis-t ical averages and random fluctuations. B y applying the averaging procedure to the instantaneous equations, the corresponding time-averaged equations can be obtained. The effect of the turbulence on the mean flow is obtained by using the two-equation k — e model [19]. 2.2.1 Instantaneous Conservation Equations The governing equations describing the instantaneous properties of incompressible, non-reacting transport phenomena are given as follows [4]: Mass Conservation Equation | + A ( ( , „ , ) = 0 (2.1) Momentum Conservation Equation d , d , . dP d , , dui., . -(,„,) +-(,„,„,) = _ _ + [ „ ( ) ) (2.2) Chapter 2. MATHEMATICAL FORMULATION 15 Energy Conservation Equat ion llMJ + A ^ ^ ^ g ) , (2.3) 2.2.2 Time Averaged Equations A l l instantaneous values of any variable <j> can be split into an average component 4> and fluctuating component <f> i.e.: <p = 4> + 4>' (2.4) where 4> = / t r o ^ o o T f 4>{xi,t)dt (2.5) I Jo This time-average procedure can be applied to instantaneous equations. The time-mean equations describing conservation of mass, momentum and energy can be formulated in Cartesian tensor notation as follows: Mass Conservation Equat ion 8 dx(puA = 0 (2.6) M o m e n t u m Conservation Equat ion (pUjiii) = - — + — \ i t t t J { — )\ (2.7) dxj dxi dxj dxj where the overbars are omitted for convenience, and Boussinesq's eddy viscosity concept is used to substitute for turbulent stresses. The effective viscosity Htfj 15 given by: Chapter 2. MATHEMATICAL FORMULATION i 16 Energy Conservation Equat ion a ink) = 4r\£ + r L ) ( ^ ) l (") If the temperature is desired in the final solution, one can rewrite the above equation in the following way: ± f r l T ) = ± \ £ + ( 2 . 9 ) OXj OXj Oh &h,t Ox) where: Oh P r and t l number for the energy equation, Oh,t turbulent Prand t l number for the energy equation, 2.2.3 Turbulence Modelling In this work the two-equation k — e turbulence model is employed to characterize turbulent flow. The relation between turbulent viscosity fit and turbulent kinetic energy k and dissipation rate of kinetic turbulent energy e is given as follows: Jfc2 lit = CMp— (2.10) where: k, the turbulent kinetic energy, is defined as: £, the dissipation rate of kinetic turbulent energy, is defined as: 1 = 7<5r) (2-12) Chapter 2. MATHEMATICAL FORMULATION 17 Constant C u C l C2 K E °k °t C3 value 0.09 1.44 1.92 0.4187 9.793 1.0 1.0 Table 2.1: Values of turbulence model constants. The local values of the mean turbulence properties k and c are obtained through the solution of the modelled transport equations (see [18], [19]). Turbulent Kinet ic Energy Equat ion {pUjk) = ^ e ! f / 0 k { jj + c-pe (2.13) dxj dxj dxj where: G, is a generation te rm defined as: „ , dui.,, dui. , dui.. . Dissipation Rate of Kinet ic Turbulent Energy dx~ipUjt) = dx~yeff/°i{dtj)] + ( C z G " Cspt)k ( 2 - 1 5 ) The values of the turbulence model constants are listed in table 2.1. Chapter 2. MATHEMATICAL FORMULATION 18 2.2.4 General Transport Equations A l l differentia] equations introduced in the previous section have a similar form as they are expressing the same conservational principle. The generalized differential equation for the variable <j> can be formulated as follows: 3 (P«;*) = £ [ r , ( # ) ] + S, (2.16) dxj dxj dxj where: <l> - dependent variable - exchange coefficient for the variable <j> 2.3 Boundary Conditions To solve the differential equations, boundary conditions for the momentum and energy equations are required. 2.3.1 Inlet and Outlet Inlet and outlet velocities are known and can be imposed directly. One of the most important parameters of continuous casting is the withdrawal rate or casting velocity Utast or Uout. It is assumed that the metal leaving the solution domain is completely solidified. This allows to set the velocities in the solid region to the known casting speed. The inlet velocity of the liquid metal flowing into the domain from the nozzle Uin can be calculated from the continuity equation: Uin = Uout(pArea)out/{pArea)tn Chapter 2. MATHEMATICAL FORMULATION 19 The turbulent kinetic energy k is obtained from the turbulence intensity / at the inlet defined as: (2.17) The dissipation rate of turbulent kinetic energy e can be specified through the characteristic turbulent length scale / at the inlet of the solution domain. For isotropic turbulence the values of k and e can be calculated as: kin = 1 . 5 / 2 t / £ (2.18) and ein = C3J4k^/l (2.19) The value of J has to be obtained from experiments. It is otherwise assumed to be in the range 0.02-0.05. The turbulence length scale / can be evaluated as a small fraction of the dimension of the inlet (e.g. / = 0.1<f tn). Another important parameter of continuous casting is the temperature Tj„ of the l iquid metal flowing into the l iquid pool from the nozzle. The temperature at the outlet is unknown, but one can assume that at a distance far enough from the solidification front the axial heat transfer is negligible and the outlet plane can be considered as quasi-adiabatic: dT 2.3.2 Free Surface and Symmetry Planes If the free surface is assumed to be undisturbed, and the friction between the l iquid metal and the slag-layer is negligible, the boundary conditions can be defined as: Chapter 2. MATHEMATICAL FORMULATION 20 For the energy equation, the free surface is assumed to be adiabatic: — - 0 dy For planes of symmetry, the same conditions as outlined above are val id i.e. x-y symmetry plane du dv w = 0, — = 0, — = 0 oz oz dT z-y symmetry plane dv div v = 0, — = 0, — = 0 dx ox dT n d~x=° 2 .3.3 Walls M o m e n t u m Equations The treatment of the wall region requires a special approach (see [18]). F low conditions are specified as functions of the dimensionless distance from the wall y + defined as: where: u T friction velocity defined as: y + = ^ (2.20) ur = \jrw',p (2.21) The wall flow region can be divided into three layers: • viscous sublayer, 0 < Y 4 < 5; • buffer layer, 5 < l ' + < 30 Chapter 2. MATHEMATICAL FORMULATION 21 • inertial sublayer 30 < Y+ < 400 For numerical convenience, the buffer layer can be neglected and the dimensionless distance from the wall Y + = 11.63 (see [19],[18]) can be regarded as a new dividing point between the viscous and inertia] sublayers. In the viscous sublayer the velocity profile is linear. In the fully turbulent inertial sublayer, one can use the logarithmic law of the wall to relate the velocity to the distance from the wal l . This can be summarized as follows: U/ur = Y + for Y + < 11.63 U/uT = l n ( £ Y + ) / / c for Y+ > 11.63 where: E integration constant, E = 9.793; /c von K a r m a n constant, K = 0.4187. Dissipation Rate of Turbulent Kinetic Energy In the inertial sublayer, convection and diffusion of the turbulent kinetic energy k are negligible and production rate of k equals dissipation rate of k i.e.: this implies V ( f £ ) 2 = e2 (2.23) Us ing the logarithmic law of the wal l one gets: c 2 = C ^ 2 ( u T / / c y ) 2 (2.24) E l imina t ion from above equation of u T gives the final expression for dissipation rate of turbulent kinetic energy in the inertial sublayer: ' ' " o ^ - ( 2 ' 2 5 ) Chapter 2. MATHEMATICAL FORMULATION 22 Turbulent Kinet ic Energy For the derivation of boundary conditions for turbulent kinetic energy k, the generation rate G of k is evaluated using the calculated shear stress i.e.: 8U du: du{ , Thermal Energy Equations Thermal boundary conditions at the outer surface of the metal can be imposed in three different ways depending on available data. 1. Dir ichlet Boundary This is the simplest case when the temperature at the wal l TB is known, then: T = TB 2. Neumann Derivative Boundary If the heat flux at the boundary qB is known then one can use the source term to impose boundary conditions: Sh = QB 3. Convective Boundary Condit ions In this case the heat flux qB is specified v ia a heat transfer coefficient h~ and the temperature of the surrounding fluid 7y, then: qB = h'(TF - TB) Chapter 2. MATHEMATICAL FORMULATION 23 In the present study the temperature at the outer surface of the cast is assumed to be known. However, if instead of the temperature the heat flux or the heat transfer coefficient are known, the boundary conditions can be imposed v i a the source term as outlined in points 2 and 3. 2.3.4 Solidus Surface In the coordinate system which is fixed wi th respect to the mould-wall the l iquid-solid interface is stationary. However, the metal which solidifies at this interface is continuously moving downwards wi th the casting velocity. A t the solidus surface, velocities are set as follows: v = vcatt u = w = 0 For isothermal solidification, the full amount of latent heat hi is released at the l iquid-solid interface. Th i s physical phenomenon can be modelled mathematically through the introduction of an additional source term 5/, in the thermal energy equation. This term is obtained from: dSh = vcattphLdA where A is the area perpendicular to vca!t. 2.3.5 Buoyancy The buoyances forces FB due to temperature gradient are expressed as: FB = gPp{T - Trtf) (2.27) where /?, the coefficient of cubical expansion, is given by: Chapter 2. MATHEMATICAL FORMULATION 24 (2-28) The axial pressure gradient and body force that appear in the y-momentum equation: + (2.29) can be rewritten as: - |£ + g[3p{T - Tref) (2.30) This expression represents forces due to the axial pressure gradient and buoyancy forces due to natural convection. Chapter 2. MATHEMATICAL FORMULATION 25 Figure 2.3: Nozzle configuration in symmetrical quarter of cast. C hapter 3 SOLUTION PROCEDURE In this chapter the governing differential equations are discretized using the finite volume method. The resulting algebraic equations are cast into a form suitable for a numerical solution. The TEMA code of Lai and Salcudean [15], specially modified to incorporate the present models, is used to solve the governing equa-tions. The solution method is based on the SIMPLE algorithm coupled with an iterative ADI-like procedure. 3.1 Grid System The partial differentia] equations and boundary conditions introduced in the pre-vious chapter are very complex and can not be solved analytically. The numerical solution procedure used in this work is based on the finite volume method. The solution domain is divided into a finite number of six-sided control volumes or cells (see fig. 3.4). At the geometric center of each volume a grid point is placed. This arrangement has the following advantages: 1. The value of the general variable <p directly available at the center of the cell is a good representation of the average value over the control volume and can be used without interpolation to calculate the source terms and physical 26 Chapter 3. SOLUTION PROCEDURE 27 properties. 2. Discontinuities at the boundaries can be conveniently handled by locating boundary cells where the discontinuities occur. Fol lowing Patankar [27], a staggered grid arrangement (fig. 3.5) is used. Scalar quantities (i.e.p, T, c, k) are calculated at the geometric center of the control volume (see fig. 3.9), whereas the velocity components u , v, w are calculated at the scalar cell faces (see fig. 3.7, 3.8). In this way velocities are directly available for evaluation of convection through the boundaries of the control volume. 3.2 Discretization of Differential Equations The finite volume equations are derived by approximate integration of the general transport equation (see equation 2.16). The integral form of the differential equa-t ion over a typical control volume (see fig. 3.6) for a general dependent variable (f> can be wri t ten as: where: A, V are the area and volume of the control volume, n is the outward normal unit vector, r^, is the diffusion coefficient for the variable <j>. The left hand side of equation 3.33 expresses convective and diffusive in-flow/outflow through the area A of the scalar cell and can be rewrit ten, for a Cartesian coordinate system, as follows: (3.31) Chapter 3. SOLUTION PROCEDURE 28 fVn r*j 86. - / / \pu4> - I * — \Zwdzdy Jy. Jzh OX [*• f'f, 86. + / / \pvtj> - T^ — \Vndzdx fXr fZf, 96 - / / \pv<f>-T^ — ]v.dzdx Jz„ Jzt, oy [*• fVn 86. + / / [pw6 - T^ — \2fdydz Jzw Jy. OZ rx' rvn 86 - / / \pw6-T^)Zldydz (3.32) Jzy, Jy. OZ The first term on the left hand side describes total transport in the x-direction and can be discretized as follows: fVn t't, 86, , 86, , 36, / / \Pu6-T^}xdzdy=\pu6-T^}e-\pu6-r^}w (3.33) Jy. Jzb OX OX OX where: Sx is the grid spacing in the x-direction, the subscript e denotes the value of the variable at the east face of the scalar cell. With the exception of the velocity components, which are directly available at the scalar cells faces, all other variables have to be determined by linear interpo-lation between neighboring cells e.g.: Pe = ftPE + (1 - f«)pp where ft is a spatial weighting factor, defined as: Axp OXp Total transport Q( through the east cell boundary consists of convective and diffusive fluxes; Ce = \pu6)e and De — |r^||]e. Due to the staggered grid ar-rangement, fluxes are calculated at the scalar cell faces. However, the value of the Chapter 3. SOLUTION PROCEDURE 29 variable <f> is directly available only at the nodal points. There are a few possible approximation schemes to obtain fluxes at the scalar cell boundaries. The central difference scheme assumes a piecewise-linear profile of the variable <j> between nodal points. The convective flux can be expressed as: C. = ( P u U ^ ± ^ ) (3.34) This formulation is second order accurate, and gives satisfactory results for flows characterized by low Peclet number, Pee = \{pu)/(f^)]e, i.e. for flows dominated by diffusion. However, in cases when convection is much larger than diffusion the central difference method leads to numerical instabilities and yields unrealistic results. These difficulties can be overcome by using the upwind scheme approximation for convection dominated flows. The value of <f> at the scalar cell face is assumed to be the same as the value of <j> at the upwind grid point. The convective flux at the east scalar cell boundary is evaluated as: Ce = {pu)e<j>p if ue > 0 Ce = {pu)e<f>E if u e < 0 The hybrid differencing scheme [27] combines the advantages of central dif-ferencing for small Peclet numbers (|Pe| < 2) and upwind differencing for large Peclet numbers (|Pe| > 2). As a result, the numerical stability and accuracy of the solution are enhanced. The discretized term representing convective and diffusive transport across the east boundary Qe can be approximated as: S x d(j) Qt = me(4> - — — ) e = m e |(l - 7e)<k + lt<f>p (3.35) Chapter 3. SOLUTION PROCEDURE 30 The coefficient 7e is defined for hybrid central/upwind differencing scheme as: 7e = {(1 - / ) + m o z [ - ( l - f)Pe,fPe,l]/Pe}e (3.36) Assuming a linear dependence between the source term and the dependent variable ci>, the right hand side of equation 3.31 can be rewritten in the following way: f SfdV = r f" fZf S+dxdydz = Si- SUP (3-37) where: Sy is the constant part of linearized source term, Sp is the coefficient of (f>p. Now the final form of discretized transport equation can be formulated: me\{l - ~jt)4>E + lt4>p) + mw\{\ - 7t)<f>w + 1W4>P] + m n [ ( l - 7„)t>iv + ln4>p] + m,[(l - 7e)<f>s + i,4>p] + - 7j)<t>F + lf4>p] + mb\(l - lb)<t>B + IbM = 4 ~ SUP (3-38) For numerical convenience a compact form of the above equation can be intro-duced: Ap<pP = Ae4>E + Aw<f>w + AK<f>N •+ AS(f>s + Ap<t>F + AB4>B + Sfj (3.39) or in short form: Ap<f>P = ZcAc<f>c + 5tt (3.40) where: Chapter 3. SOLUTION PROCEDURE 31 the subscript c denotes neighboring grid points, i.e. E, W, N, S, B, F; -the coefficients A are defined as: AE = max\\Ce/2\,De]-Ce/2 Aw = max\\Cw/2\,Dw] + Cw/2 AN = max\\Cn/2\,Dn} - C„/2 As = max[\Ct/2\,D.] + Cl/2 AF = max\\Cf/2\,Dj] - C//2 AB = max[\Cb/2lDb}-rCb/2 Ap = T,CAC -f Sp 3.3 Discretized Boundary Conditions In this section, the boundary conditions that have been formulated analytically in the previous chapter are given in their discretized form. 3.3.1 Inlet The values of the dependent variables at the inlet are known or can be assumed (see section 3.2.1). These values can be directly assigned at the boundary points. In the case of an internal grid point, the given value of any dependent variable 6 can be imposed via the source terms Sp and Sp in the following way: Si- = tfW, 1030 (3.41) SP = -10 3 n (3.42) Introducing these terms into the discretized governing equation (see equation 3.34) makes all other terms negligible, and the equation simplifies to: <f>P SP w - S v (3.43) Chapter 3. SOLUTION PROCEDURE 32 this forces the following value to be returned by the numerical procedure: 4>P = - <t>given (3-44) bp 3.3.2 Free Surface and Symmetry Planes The heat transfer from the free surface of the liquid pool is unknown, however, it is expected to be small compared with the heat loss through the mould. In the present study an adiabatic condition (zero gradient) is used. A free slip (zero gra-dient) condition is used for the momentum equations at the free surface boundary. The zero gradient condition for the free surface and symmetry planes, can be imposed, for any dependent variable </>, by cancelling the respective flux term; e.g. for the north boundary: A* = 0 (3.45) 3.3.3 Walls Momentum Equations In the near wall region, with the assumption of an one-dimensional Couette flow, the turbulence is in local equilibrium. For the fully turbulent layer (Y+ > 11.63) the wail shear stress can be expressed as: TW = T = fit^ (3.46) where: fit turbulent viscosity, |^ tangential velocity gradient. The turbulent viscosity is much larger than the laminar viscosity and the latter can be assumed negligible. Chapter 3. SOLUTION PROCEDURE 33 The balance between the local production and dissipation of the turbulent kinetic energy k can be expressed as: dxi M^r)2 = Pe (3-47) oy This allows (using equation 2.10) for a new expression for the wall shear stress: TW = C'J2pk (3.48) Fina l ly , as U/uT = ln[EY*/K, (logarithmic law of the wall) one gets: •* = i ^ r (3-49) where the dimensionless distance from the wal l Y + is evaluated as: y + = CLJ4(pk^Y)P/n (3.50) The resultant shear stresses have to be decomposed into the two directions. For a boundary normal to the x-direction: TVz = TwV/U (3.51) T,Z = TWW/U (3.52) where: U is the velocity tangential to the wal l . The boundary conditions are introduced into the numerical formulation by changing the flux terms AE, AW etc., and by modifying the source terms. A t the wal l , because of friction, there is a momentum sink. The resultant source terms due to the wall stress can be wri t ten, for the north boundary, as: SN — O-WTxy (3.53) Chapter 3. SOLUTION PROCEDURE 34 for u-momentum, and: zz (3.54) for w-momentum where aw is the area of the cell boundary in contact wi th the solid wal l . Similar expressions are derived for other solid boundaries. Turbulent Kinet ic Energy The differential expression representing the generation rate G in the wall region (see equation 3.29) can be approximated by an algebraic equation. For a solid boundary in the y-z plane, the expression takes the form: Dissipation Rate of Turbulent Kinet ic Energy The calculat ion of the dissipation rate c for the cell adjacent to the wall was outlined in Section 2.3.3 . The computed value of £ is imposed v i a the source terms as described in Section 3.3.1 . In a coordinate system fixed w i th respect to the mould-wall the l iquid-solid inter-face is stationary. However, metal that solidifies at this interface is continuously moving downwards wi th the casting velocity. A t the solidus-surface, velocities are set as follows: Uw — U p . 2 (3.55) 3.3.4 Solidus Surface Chapter 3. SOLUTION PROCEDURE 35 For isothermal solidification the full amount of latent heat hL is released at the l iquid-solid interface. This physical phenomenon can be modelled mathematically by introducing a source term 5/, in the thermal energy equation at the solidus surface. A s we move down the solution domain along a slab of cells defined by constant I and K , the temperature in the cells is checked. If the temperature drops below solidus temperature Ts, the latent heat is released v ia the source term. The source term is calculated as: Sv = vcaatphLA/cp for the energy equation wi th temperature as a dependent variable, and: Su — vcai,tphiA for the energy equation wi th enthalpy as a dependent variable. The value of this source term is relatively high compared wi th other terms in the discretized equation. In order to achieve convergence, a carefully set underre-laxation factor is necessary (see section 3.6). 3.4 SIMPLE Algorithm The SIMPLE (Semi Impl ic i t M e t h o d for Pressure L i n k e d Equations) algori thm of Patankar and Spalding [27] is used to solve the system of nonlinear coupled equations. This algori thm uses an iterative solution procedure which involves the following steps: 1. The fields of dependent variables u , v, w, p, T , k, i are guessed for the first iteration loop. Chapter 3. SOLUTION PROCEDURE 36 2. The momentum equations in the x-, y- , z-directions are solved, and the first estimates of the velocity field u* ,v* ,w* are obtained from the guessed pressure field. 3. A pressure correction equation is solved. 4. Corrections are made to the velocity components and pressure: u = u" + u (3.56) v = v' + v (3.57) w — rv" + w (3.58) p = p' +p (3.59) 5. The equations for the scalar quantities T, k, e are solved. 6. The results are checked for convergence. If convergence is not attained then the new values of the dependent variables are used as the starting values, and the procedure is repeated from step 2. 3.5 Solution of the Linear Algebraic Equations Using i,j,k indices instead of the compass convention, the general finite volume equation (3.39) can be rewritten as: + Ai,i+i,k<i>i,j+l,k + •4t1j-l,*0ij-l,* + A-1;,fc+l&1;.*+l + Aj.fc-l&j./t-l + (3.60) Chapter 3. SOLUTION PROCEDURE 37 It is, in principle, possible to solve the above system of equations by matr ix inver-sion. This is, however, a computationally costly operation. A n alternative solution procedure is to solve the system using an iterative line by line method. For the line of constant i and constant k equation 3.59 can be expressed as: <t>i = a A + i + M j-i + ci (3-61) where: a, = Aj+ i , i M.j , k (3.62) bj = Aij-i^/Aj.k (3-63) C3 — (•^i+lj.fc^t + lj.fc + A-l,j,t^t-l,;,fc + A j ' + l , + 1,* + A,;'-l,t<^i,>-l,t + A'j.t+i^t j.t+i + Aijyk-\<f>i,j,k-i + St)lAiJth (3.64) The nonzero coefficients of the set of equations defined by equation 3.60 form a tri-diagonal matr ix . Such a system of equations can be solved directly by a Gaussian-elimination method known as tri-diagonal matr ix algori thm ( T D M A ) or Thomas-algori thm. When the values of 4> on the line are found, the same procedure is carried out for all lines in the y-direction and can be repeated for the x- and z-directions as well (see fig. 3.16). The convergence of the above method is fast because the boundary conditions information is transmitted at once to the nodal points lying inside the solution domain. Chapter 3. SOLUTION PROCEDURE 3 8 3.6 Numerical Considerations Underrelaxation In order to handle nonlinearities in the iterative solution of the algebraic equa-tions, it is necessary to underrelax or slow down the iteration process. The de-pendent variable, e.g. temperature T, can be underrelaxed by the introduction into the general discretized equation ApTp — \ZCACTC + Sy of an underrelaxation factor a . Thus: —TP = T.CACTC + Si + (1 - a ) — TP ( 3 . 6 5 ) a a where TP is the value of Tp from the previous iteration. A t convergence, Tp — Tp, and the equation wi th the underrelaxation factor is identical to the discretized equation. In the present work, because of the nonlinear-ity introduced by the source term simulating the release of latent heat, a relatively small underrelaxation factor is used for the energy equation, i.e. Q j = 0 . 1 . The fol-lowing underrelaxation factors are used for other equations: au = 0 . 2 5 , av = 0 . 2 5 , aw ' 0 . 2 5 , aP = 1 .0 , ak = 0 . 5 , a< - 0 . 5 . The source term Su can be underrelaxed as follows: Sv = aSP + (1 - a)Sv ( 3 . 6 6 ) where: S[r denotes the source term from the previous iteration. The value of a for the thermal energy equation was set to 0 . 0 1 . This very low value was necessary to stabilize the numerical solution of the energy equation. Another way of controll ing convergence could be the use of a transient ap-proach. The practice of solving a steady state problem v i a an unsteady formulation Chapter 3. SOLUTION PROCEDURE 39 can be regarded as a special k ind of underrelaxation. C o n v e r g e n c e C r i t e r i a A n iterative procedure is normally converged when further iterations do not produce any significant changes in the values of the dependent variables. In addi-t ion , it is necessary to have a convergence criterion which measures the degree to which a computed solution satisfies the original set of transport equations. In the present work, this is done by using the 'residual source' method. The residual over one cell is defined as: J2, = £ Ac4»c + St, - Ap<pP (3.67) c and the sum of the residuals for the whole field is: *J = E 1**1 (3-68) field A t convergence Rf becomes 0. For practical purposes, it is sufficient for the sum of the residuals to drop below a specified small number i.e.: < ^ E / A (3.69) where: R™* is the reference value for variable <f>, A is the level of convergence for all dependent variables, ( e.g. A=0.000l). The value of Rf can be monitored during the iteration process and any un-expected behavior can indicate the source of error in the program. For example, a divergent behavior could mean errors in the implementation of some boundary conditions. False Source Chapter 3. SOLUTION PROCEDURE 40 In order to ensure convergence and stabilize the numerical behavior of the solution, one may introduce into the finite volume equations so called 'false source' term Sf defined as: Sj = CP{<t>i-'i-4>i) (3.70) where: Cp is the mass imbalance for a control volume defined as: Cp = | mc\ — \me — mw -+- m n — mt -+- rrtj — mj,| (3-71) c The false source term can be incorporated into the discretized equation as follows: [Ap + CP)6P = £ AC6C + S* + Cpdp-1 (3.72) c A t convergence, Cp and Sp become 0 and have no effect at the final converged solution. The addit ion of false source term allows to avoid the singularity problem which may arise dur ing the iteration process when Ap take the value 0 (corre-sponding to zero <j> transport into the cell) . False Diffusion For high Peclet numbers (i.e. \Pe\ > 2) the hybrid scheme reduces to the upwind scheme. The upwind scheme, though numerically stable, has the disad-vantage of being only first order accurate and introduces a discretization error referred to as false diffusion. The effect of false diffusion is an artificial increase in the (physical) diffusion coefficients which results in smearing of the gradients in the flow field. The errors introduced in the solution can be quite important , par-t icularly when the grid lines are inclined wi th respect to the local velocity vectors [29]. False diffusion can be limited by the following: fine mesh, grid lines aligned to the flow direction, discretization schemes accounting for the multidimensional Chapter 3. SOLUTION PROCEDURE 41 nature of the flow and involving more neighboring cells, curvilinear coordinate system. Numerica l Errors and Instabilities The most likely cause of numerical errors are: unrealistic in i t ia l values and too coarse grids. Numerica l instabilities are often due to: inappropriate under-relaxation factors (usually too large); too few sweeps in the iterative line-by-line method. The correct evaluation of numerical parameters can be based on previous experience or can be achieved through tr ia l and error. The errors in the results are usually caused by mistakes in the implementa-t ion of boundary conditions or by wrong input data (e.g. physical properties, geometry). Chapter 3. SOLUTION PROCEDURE 42 Figure 3.4: Discretization domain for Cartesian, three-dimensional coordinate sys-tem. Chapter 3. SOLUTION PROCEDURE LOCATION VARIABLES STORED t u Y///A CONTROL VOLUME FOR J ^  R S X S M CONTROL VOLUME FOR « | | CONTROL VOLUME FOR u Figure 3.5: F in i te volume grid in x-y plane wi th associated notation. Chapter 3. SOLUTION PROCEDURE 44 Chapter 3. SOLUTION PROCEDURE Figure 3.7: Control volume and notation for u-momentum analysis. Chapter 3. SOLUTION PROCEDURE UNW 1 1 ^ •NE ! * l vW I, • - J w u w HP ^ P E • T • s • Figure 3.8: Control volume and notation for v-momentum analysis. Chapter 3. SOLUTION PROCEDURE 47 Figure 3.9: Control volume and notation for transport of scalar quantity in z-plane. Chapter 3. SOLUTION PROCEDURE 48 JN(l,K)-« J -IWU.K) -2 6 T T VIJ) 1 - 1 2 a JS(I. K) - • S 4 2 XU(I) X(l) -•J • • • • • • • • • • • 1 • 4UJ.K) • • • • • • < • • • • • < lE(J.K) • 4 6 2 2 7 Nl 2 Figure 3.10: G r i d notation for a plane x-y defined by constant K in a domain wi th irregular boundaries. Chapter 3. SOLUTION PROCEDURE 49 KF(M) - s s B e • • — — xutn »1 K - X(l) -H IEU.K) • NK rnw.K) -s • • • • • • 7 3 6 • • • • • • > 7 a 4 • • • • • e T ZfllK) 2 T 3 < • • • — • • 6 1 41U.K) tt(U.K) Z(K) 2 I 2 < • • • • • • 6 1 1 - 1 2 3 4 6 6 7 N I KBII.J) 2 2 2 2 2 6 Figure 3.11: G r i d notation for a plane x-z defined by constant J in a domain wi th irregular boundaries. Chapter 3. SOLUTION PROCEDURE 50 RV{J) Q) VV(J) DXEP0-1) (- SEW(I) ^ Figure 3.12: Control volume for all scalar variables defined at the grid node (I,J,K) : (a) constant K-plane; (b) constant J-plane. Chapter 3. SOLUTION PROCEDURE 51 © U) —xu( i ) © FtV(J) YV(J) DXEP(I) - H r*SEW(l)«+*— SEW(t + 1) - H ® (b) T ZW(K) © - XU{I) Z(K) X(H1) DXEP(I) («SEW(I)«4*- SEW(I-H) — H Figure 3.13: Control volume for velocity u defined at the grid node ( I , J , K ) ; (a) constant K-plane; (b) constant J-plane. Chapter 3. SOLUTION PROCEDURE 52 Figure 3.14: Contro l volume for velocity v defined at the grid node (1,J ,K); (a) constant K-plane; (b) constant I-plane. Chapter 3. SOLUTION PROCEDURE 53 (•) © K DZFP(K) «H SFB(K+1) »1"SFB(KH (b) RV(J) ^ YVU) © ZW(K) 2(K+1) -N 1 ////) IP B I S **• DZFP(K) ** »SFB(K+1) »I«SFB(KH T OYNP(J) < 1 SNS(J) J^ OYNP(J-I) 1 Figure 3.15: Cont ro l volume for velocity w defined at the grid node (1,J ,K): (a) constant J-plane; (b) constant 1-plane. Chapter 3. SOLUTION PROCEDURE y 4 \ CHOSEN LINE Figure 3.16: Representation of the line-by-line method. Chapter 4 RESULTS AND DISCUSSION In this chapter the developed computer code is applied to a number of test cases in two-dimensional and three-dimensional solution domains. The numerical results are compared with available experimental measurements. 4.1 Two Dimensional Heat Flow 4.1.1 Description of the Problem The problem considered in this section is that of two-dimensional heat flow in aluminum billets. This problem has been investigated by Grandfield and Baker ( see [10], [2]). The surface temperature was measured and a two-dimensional finite element method was used to compute the temperature field and the solidification front inside the cast (see fig. 4.17,4.18). The fluid flow inside the liquid pool was neglected by Grandfield and Baker. 4.1.2 Numerical Simulation The program was set up using the following set of assumptions and conditions: 1. Two-dimensional, axi-symmetric solution domain . 55 Chapter 4. RESULTS AND DISCUSSION 56 CONTINUOUS CASTING OF ALUMINUM BILLET PROBLEM SPECIFICATION DATA (6063) Mass flow rate <kg/s) E 0.1281 Casting v e l o c i t y (m/s) = 0.0026 In l e t v e l o c i t y (m/s) • 0.0091 In l e t temperature (C) = 690.0 Liquidus temperature (C) = 655.0 Solidus temperature (C) = 615.0 Length of s o l u t i o n domain (m) • 0.150 Radius of s o l u t i o n domain (m) « 0.0775 In l e t area of nozzle (m*m) « 0.0055 Outlet area <m*m) « 0.0189 Latent heat (kJ/kg) « 395.0 Heat capacity (J/kg/K) • 1008.7 Conduction c o e f f i c i e n t (W/m/m/K) « 219.0 Density (kg/m**3) «= 2548.0 Dynamic v i s c o s i t y (kg/m/s) * 1.1E-03 Expansion c o e f f i c i e n t 9.0E-06 Prandtl number • 5.6E-03 Mesh s i z e IxJ - 22x22 Number of nodal points i n domain » 484 Table 4.2: Parameters and physical properties used for simulation of two-dimensional axis-symmetrical continuous casting of aluminum alloy (6063) billet. 2. Uniform grid 22 x 22 nodal points covering symmetrical half of the billet; dimensions: radius 77.5 mm, length 150 mm. 3. Physical properties for aluminum alloy 6063 as listed in table 4.2. 4. Casting parameters as listed in table 4.2. The calculations were performed for following cases: Chapter 4. RESULTS AND DISCUSSION 57 1. A uniform velocity field is assumed in the solution domain; the a luminum billet is moving downwards with casting speed. Only the energy equation is solved (fig. 4.19). 2. Laminar fluid flow without buoyancy is assumed in the l iquid pool of the cast. The momentum and energy equations are solved (fig 4.20). 3. Laminar fluid flow wi th buoyancy is assumed in the l iquid pool of the cast. The momentum and energy equations are coupled by buoyancy terms and solved iteratively (fig 4.21). 4. Turbulent fluid flow wi th buoyancy is assumed in the l iquid pool of the cast. The momentum, thermal energy, turbulence kinetic energy and dissipation rate of turbulence kinetic energy are solved iteratively (fig 4.22). 4.1.3 Discussion of Results Due to the large inflow area and low inflow velocity, the flow pattern inside the relatively shallow l iquid pool is not very complex. There are no significant differ-ences between cases #2, #3 and #4. Natura l convection changes the flow pattern only slightly because the temperature gradient is small (see fig. 4.20, 4.21). Near the mould wal l , the downwards acting buoyancy force slows the counter-clockwise recirculating cel l . Near the axis, the upward acting buoyancy force decreases the velocity of the downwards flowing l iquid. The turbulent calculation indicates that for turbulent flow the inlet jet has larger entrainment (compared to laminar flow). The differences are, however, small (see fig. 4.21, 4.22). The Reynolds number based on the orifice diameter is: Re = (0.0091m/* * 0.074m)/(0.432 * 1 0 _ 6 m 2/5) = 1559, Chapter 4. RESULTS AND DISCUSSION 58 The Reynolds number based on the cast diameter and casting speed is: Re = (0.0026m/s * 0.155m)/(0.432 * 1 0 ~ 6 m 2 / s ) = 946 Thermal energy is transported in the l iquid pool mostly by means of conduction due to the very high conductivity of a luminum and the relatively low velocities. The order of magnitude of conduction D and convection C in the radial direction can be estimated (for laminar flow) as: £> = £ = (O.218fc0/ms)/(O.OO39m) = 56kg/s C = pu = (2547fcg/m s) * (0.0026m/s) = 6.6kg/s and Peclet number Pe: ^ = ! ( / > " ) / ( £ ) ] = 6.6/56 = 0.12 A low value of Peclet number indicates that the hybr id differencing scheme is reduced in this case to the central differencing scheme. The temperature field in the l iquid pool is pr imari ly affected by conduction and only very slightly by convection. Thermal energy in the solid region close to the solidification front is transported mostly in the axial direction. Further downwards, the preferred direction changes from axial to radia l . Th i s is mostly due to the relatively low heat removal rate in the upper region of the billet where an air gap drastically reduces heat transfer from the cast to the mould . The sensible heat hs which has to be removed from the liquid metal is much smaller than the latent heat hs being released at the solidification front. They can be estimated as: hs = cP{Tin - TL) = {89ZJ/kgK) * (690°C - 655°C) = 31 * 103J/kg hL = 393 * 103J/kg This large difference in specific heat flux is i l lustrated on the graphs where the isotherms in the solid region are much denser than those in the l iquid pool . The temperature fields obtained in the present work are very close to those pub-lished in the literature [10]. The addit ional calculation of fluid flow and convective Chapter 4. RESULTS AND DISCUSSION 59 heat transfer in the liquid pool does not seem to result in any significant changes. This is due to the physical properties of a luminum (i.e. high conduct ivi ty) , nozzle configuration (low inlet velocity, low radial velocities), cooling conditions (shallow liquid pool) . Neglecting the fluid flow inside the pool seems, therefore, to be a reasonable assumption in this case. 4.2 Two Dimensional Fluid Flow 4.2 .1 Description of the Problem Vives and Ricou [42] measured the local velocities inside the sump of a continuously cast billet of a luminum alloy 7049 (see figure 4.23). The temperature inside the sump ranged from 630°C to 670°C . The l iquid metal jet was flowing radially and horizontally from the nozzle-float into the l iquid pool . The magnetic probes used for the measurements were capable of sampling at a rate of 200 readings per minute and measured average values and standard deviations. The instantaneous measurements indicated the existence of two zones: 1. A n unstable zone in the upper part of the sump where standard deviations of velocity were 30% bigger than average values. 2. A stable zone in the lower part of the sump where standard deviations were relatively small and average values of velocity were less than 3 cm/ s . The plotted velocity field revealed the presence of two main recirculation cells and a secondary vortex inside the upper part of the liquid pool . Chapter 4. RESULTS AND DISCUSSION 60 4 . 2 . 2 Numerical Simulation The program was set up under the following assumptions and conditions: 1. Two-dimensional , axisymmetric-symmetric solution domain . 2. Uniform grid 18 x 18 nodal points covering symmetrical half of the billet; dimensions: radius 160 m m , length 240 m m . 3. Physical properties for a luminum alloy 7049 as listed in table 4.3. 4. Cast ing parameters as listed in table 4.3. Turbulent fluid flow wi th buoyancy is assumed in the l iquid pool of the cast. The momentum, thermal energy, turbulence kinetic energy and dissipation rate of turbulence kinetic energy are solved iteratively. 4 . 2 . 3 Results and Discussion The flow field obtained from the numerical simulation is presented in fig. 4.24. The molten a luminum flows into the l iquid pool wi th a relatively high inlet velocity (one order of magnitude larger than the withdrawal rate). Due to buoyancy, shortly after leaving the nozzle-float the l iquid metal flows upwards towards the free surface of the sump and towards the wal l . Close to the mould , the l iquid a luminum is cooled and flows downwards along the solidification front towards the symmetry axis of the billet where it turns and flows upwards. One can recognize two recirculation cells: 1. In the upper region of the l iquid pool , between the nozzle-float and the mould-wall . Chapter 4. RESULTS AND DISCUSSION 61 CONTINUOUS CASTING OF ALUMINUM BILLET PROBLEM SPECIFICATION DATA (7049) Mass flow rate (kg/s) * 0.2184 Casting v e l o c i t y (m/s) = 0.001 Inlet v e l o c i t y (m/s) = 0.031 Inlet temperature (C) = 670.0 l i q u i d u s temperature (C) = 630.0 Length of soluti o n domain (m) * 0.240 Radius of sol u t i o n domain (m) •= 0.160 Inlet area of nozzle (m*m) = 0.0026 Outlet area (m*m) «= 0.0804 Latent heat (kJ/kg) * 395.0 Heat capacity (J/kg/K) « 1008.7 Conduction c o e f f i c i e n t (W/m/m/K) « 219.0 Density (kg/m**3) * 2548.0 Dynamic v i s c o s i t y (kg/m/s) • 1.1E-03 Expansion c o e f f i c i e n t • 9.0E-06 Prandtl number • 5.6E-03 Mesh s i z e IxJ * 18x18 Number of nodal points i n domain * 324 Table 4.3: Parameters and physical properties used for simulation of two-dimensional axis-symmetrical continuous casting of aluminum alloy (7049) billet. Chapter 4. RESULTS AND DISCUSSION 62 2. In the lower part of the l iquid pool , between the solidification front and the nozzle-plate. Al though the experimental results show the existence of other recirculation cells, these are not apparent in the numerical simulation. This is probably because the relatively coarse grid used for the numerical solution does not allow for sufficient resolution. It should be pointed out that there is some uncertainty in the exper-imental results, part icularly in measuring the velocities in the upper part of the sump. It is also important to note that the present simulation assumes a free slip flat surface. Experiments , however show that the level and shape of the free surface change wi th the t ime. These conditions are difficult to simulate numerically. 4.3 Three Dimensional Fluid Flow and Heat Trans-fer 4.3.1 Description of the Problem The continuous casting of a luminum slabs was chosen for the simulation of three-dimensional transport phenomena. The cross-section of the a luminum block has the following dimensions: length 1.422m (56in), width 0.609 (24in). L iqu id alu-minum flows horizontally from the nozzle-float into the l iquid pool (see fig. 2.3). 4.3.2 Numerical Simulation The three-dimensional computer code was set using the following set of assump-tions and conditions: 1. Three-dimensional solution domain covers a symmetrical quarter of the slab. Chapter 4. RESULTS AND DISCUSSION 63 CONTINUOUS CASTING OF ALUMINUM SLAB PROBLEM SPECIFICATION DATA FOR SYMMETRICAL QUARTER OF CAST Mass flow rate (kg/s) = 0.5164 Casting v e l o c i t y (m/s) = 0.0010 In l e t v e l o c i t y (m/s) « 0.2240 In l e t temperature (C) = 680.0 Solidus temperature (C) * 660.0 Length of domain (m) = 0.7112 Depth of domain (m) * 1.2500 Width of domain (m) = 0.3048 Length of nozzle (m) = 0.2188 Depth of nozzle (m) «= 0.0127 Width of nozzle (m) « 0.0762 In l e t area of nozzle (m*m) = 0.0010 Outlet area (m*m) « 0.2168 Latent heat (JtJ/kg) « 390.0 Heat capacity (J/Jtg/K) = 1003.0 Conduction c o e f f i c i e n t (W/m/m/K) * 220.0 Density (Jtg/m**3) •= 2382.0 Dynamic v i s c o s i t y (kg/m/s) • 1.239E-03 Expansion c o e f f i c i e n t « 9.000E-06 Prandtl number * 5.670E-03 Mesh siz e IxJxK « 15x20x10 Number of nodal points i n domain « 3000 Table 4.4: Parameters and physical properties used for s imulat ion of continuous casting of a luminum slab in three-dimensional Cartesian coordinate system. 2. G r i d arrangement 15x20x10; nonuniform for nozzle, uniform otherwise. 3. Isothermal solidification. 4. Physica l properties for pure a luminum as listed in table 4.4. 5. Cas t ing parameters as listed in table 4.4. Turbulent flow wi th buoyancy is assumed in the liquid pool of the cast. The equations for momentum, thermal energy, turbulent kinetic energy and dissipa-t ion rate of turbulent kinetic energy are solved iteratively. The location of the Chapter 4. RESULTS AND DISCUSSION 64 l iqu id / so l id interface is tracked. Release of latent heat at the solidification front is simulated through the source term in the thermal energy equation. 4.3.3 Results and Discussion Visual iza t ion of the velocity field in a three-dimensional domain is a very challeng-ing task. Three-dimensional graphics for velocity are often difficult to interpret as vectors in different points of the field overlay each other on a two-dimensional piece of paper. In this work, the results are presented as follows: • solidus surfaces are shown in three-dimensional views of a symmetrical quar-ter of the cast (e.g. see fig. 4.25); • temperature fields are shown using isothermal surfaces in a symmetrical quarter of the cast or using isotherms on planes cutt ing through the cast (see fig. 4.26 and 4.27); • velocity fields are shown using two-component vectors on planes cutt ing through the cast, the thi rd component of velocity is shown in a three-dimensional view of the solution domain (see fig. 4.27). Figure 4.25 shows the solidus surface of the cast. The shell thickness and shape of the l iquid pool are visualized. This allows to estimate the danger of break-outs which can be caused by a very thin shell. Isothermal surfaces plotted in a symmetrical quarter of the cast (see fig. 4.26) can be helpful in the analysis of heat flux in a three dimensional domain. The velocity vectors and isotherms in the x-y plane cut t ing through the cast at z=0.02m are shown in fig. 4.27. The l iquid/sol id interface is plotted as a thick solid line representing the isotherm corresponding to the solidus temperature ( Ts = 660°C). The temperature drop from the inlet Chapter 4. RESULTS AND DISCUSSION 65 to the solidification front is relatively small ( 20°C). After leaving the nozzle, the l iquid metal impinges on the mould wal l . The l iquid metal is deflected downwards and flows along the solidus surface. Two zones wi th distinct flow patterns can be distinguished in the l iquid pool : (i) an upper zone where the flow of the molten metal maintains a high velocity and flows upwards, towards the nozzle-float and the free surface of the sump, forming a recirculating pattern; (ii) a lower zone, at the bo t tom of the pool , where the l iquid metal flows downwards wi th essentially the same velocity as the solidified metal which is continuously wi thdrawn. The z-components of velocity are relatively small because this plane is very close to the x-y symmetry plane. Similar flow patterns are present in the planes corresponding to z=0.09, 0.17 and 0.21m shown in fig. 4.28-4.30. The depth of the l iquid becomes smaller as we move from the symmetry plane towards the mould wal l , and the recirculating flow becomes weaker. Large z-component velocities are observed, especially near the free surface, indicating a lateral spreading of the l iquid metal as it is ejected from the nozzle-float. F i g . 4.31 shows the velocity pattern and temperature field in the y-z plane close to the symmetry plane and cut t ing through the nozzle. The inlet velocity can be recognized as the very large r-velocity near the top of the domain in the three-dimensional view of the cast. Features s imilar to those observed in the x-y plane are present, but w i th weaker and smaller recirculation cells. A s we move along the x-axis (fig. 4.32-4.34), the flow pattern becomes increasingly more complex and difficult to interpret due to complicated three dimensional interactions. It appears that the y-z plane flow is dominated by entrainment caused by the spreading nozzle jet at small x-values. A t larger x-values, the flow behaves like an impinging jet constrained by the solidified metal , and is deflected downwards. Chapter 4. RESULTS AND DISCUSSION 66 Figures 4.35-4.37 show the x-z plane results. Close to the free surface, velocities are relatively large and one counterclockwise recirculation cell can be identified. A s we move downwards the recirculation cell disappears and the area of the l iquid pool becomes smaller. The overall flow pattern in a three dimensional domain can only be construed after analyzing all graphs. 4.3.4 Parametric Studies The withdrawal rate was decreased by 25% from l .Omm/s to 0.75mm/s in order to determine the importance of this casting parameter on the continuous casting process. The most important change is best visualized on fig. 4.38; the l iquid pool defined by the solidus surface is smaller than in the previous case; the depth is decreased from 1.04m in the previous case to 0.86m in the present case, represent-ing a 17% decrease. This is caused by the smaller amount of latent heat released at the solidification front, which is directly proportional to the casting speed (see section 2.3.1). The flow pattern is very similar to the previous case, however, the velocities are generally smaller due to smaller inlet velocity and smaller withdrawal rate. In a second test, the inlet temperature was increased from 680 C to 700 C , and the withdrawal rate was maintained at 0 .75mm/s. The amount of heat to be removed from the l iquid pool before metal solidifies is larger, and as a result, the depth of the l iquid pool is slightly larger as well (see fig. 4.51). Al though the depth of the pool is affected by the withdrawal rate, it is relatively insensitive to the convection wi th in the pool . Similar observations were made by Szekely and Stanek [38] in their two-dimensional simulation. They noted that, though the Chapter 4. RESULTS AND DISCUSSION 67 solidification profile is not much affected by the flow pattern, superheat removal is. This is because of the relative magnitude of latent heat to superheat. Latent heat, which controls solidification, is much larger than superheat for a luminum. The flow is more sensitive to buoyancy forces. This is clear if we compare the velocity vectors along the solidus surface and close to the symmetry planes in fig. 4.40-4.47 (previous case) wi th fig. 4.53-4.60. This indicates that inlet temperature controls, to a certain extent, the flow pattern in the pool . The predicted levels of the turbulent kinetic energy ( T . K . E . ) for the x-y planes are shown in figures 4.64-4.66. The plotted values of the T . K . E . are normalized over the incoming kinetic energy, i.e. k" = k/u*n. In the upper region of the l iquid pool , the turbulent energy is considerably larger than in the rest of the flow field. The spatial distr ibution can be explained wi th reference to the graphs showing the flow field. High values of T . K . E . are generated by steep gradients of velocities which are specially noticeable in the neighborhood of the l iquid metal jet and the nozzle-float. In the lower part of the l iquid pool and away from the nozzle-float the values of T . K . E . decrease rapidly. The influence of turbulence on the flow field is illustrated by the increase in the effective viscosity. In figures 4.67-4.69 the effective viscosity normalized over the laminar viscosity (i.e. p." = p.ejf/n) is presented for the x-y planes. The effective viscosity and turbulent kinetic energy are related by the following formula: fiejj = fit + p. =• C^pk^^/t + p.. This relation is easily recognized by comparing the spatial distr ibution of these quantities; they are both concentrated in the upper part of the liquid pool where the effective viscosity is roughly up to two orders of magnitude larger than the laminar viscosity. In order to analyze the dependence of the numerical results on the mesh size, the following computations were carried out: Chapter 4. RESULTS AND DISCUSSION 68 1. Input data: casting speed 0.75mm/s, inlet temperature 700 C ; mesh size ( I x J x K ) : 15x20x10, i.e. 3000 nodal points. 2. Input data: casting speed 0.75mm/s, inlet temperature 700 C ; mesh size ( I x J x K ) : 22x32x18, i.e. 12,672 nodal points. The solidus surface in the solution domain defined by the fine mesh is pre-sented on figure 4.70. Compar ing it w i th the coarse mech results shown in figure 4.51, we observe that, though the predictions are not grid independent, there are no significant differences in the predictions. The increase in the number of nodal points in the solution domain changes the convergence rate of the code, and more iterations are required to reach the convergence criteria. This is because more dif-ference equations and therefore larger arrays have to be solved, but also because of the slower rate of information transfer wi th in the solution domain. The fine-mesh computations are l imited by the C P U - t i m e requirements which are part icularly extensive for a three-dimensional domain. For example, this code ( for 3000 grid points) required about 24 hours of C P U - t i m e on an Apol lo workstation and about 300 seconds on a C R A Y supercomputer. Chapter 4. RESULTS AND DISCUSSION 69 Figure 4.17: Temperature field in a luminum billet obtained by finite element method ; inlet temperature 690 C, casting speed 160 m m / m i n (Grandfield and Baker , courtesy of C O M A L C O Research and Technology). Chapter 4. RESULTS AND DISCUSSION 70 TEMPERATURE (C) X 100 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 POSITION (mm)x 100 Figure 4.18: Surface temperature of aluminum billet (Grandfield and Baker , cour-tesy of C O M A L C O Research and Technology). Chapter 4. RESULTS AND DISCUSSION 71 Figure 4.19: Comparison between temperature field obtained by Grandfield and Baker (left) and temperature field obtained by finite volume method assuming uniform flow field (right); inlet temperature 690 C , casting speed 160 m m / m i n . Chapter 4. RESULTS AND DISCUSSION 72 8 o o D o D o d o i n * 4 * * ' r n T n V * / / ; i , co / / « • , • i i i • O.DD 0.01 0.02 0.03 0.M 0.05 RADIUS IN M 0.05 0.07 Figure 4.20: Temperature field in a luminum billet obtained by finite volume method assuming inflow through nozzle and laminar flow; inlet temperature 690 C , casting speed 160 m m / m i n . Chapter 4. RESULTS AND DISCUSSION 73 D.DD DIDI 0.02 0.03 0.04 0.05 0.06 0.07 0 00 0.01 02 03 04 05  RADIUS IN M FiKure 4 21: Temperature field in aluminum billet obtained by finite volume method assuming inflow through nozzle, laminar flow with buoyancy; inlet tem-perature 690 C , casting speed 160 m m / m i n . Chapter 4. RESULTS AND DISCUSSION 74 O.DD 0.01 0.02 0.03 0.04 0.D5 0.05 0.07 Ramus IN M Figure 4.22: Temperature field in aluminum billet obtained by finite volume method assuming inflow through nozzle, turbulent flow with buoyancy; inlet tem-perature 690 C , casting speed 160 m m / m i n . Chapter 4. RESULTS AND DISCUSSION 75 Figure 4.23: Velocity field in sump of aluminum billet with a 320 mm dia. mea-sured by magnet probe (Ricou and Vives). Chapter 4. RESULTS AND DISCUSSION 76 Figure 4.24: Velocity field in sump of a luminum billet wi th a 320 m m dia. obtained by finite volume method assuming turbulent flow with buoyancy. 0.0000 SYMMETRICAL QUARTER OF CAST •a fl> to Co H Co g co O Co CO O 55 X-axis, length In m 0.7112 <£> Figure 4.27: Velocity and temperature fields , casting speed I mm/s, inlet tem-perature 680 C , solidus temperature Gf>() C: z~().02rn . ! f / / I \ \ \ \ SYMMETRICAL QUARTER OF CAST 0.0000 0.7112 X-axis, length in m Figure 4.28: Velocity and temperature fields , casting speed 1 m m / s , inlet tem-perature 680 C , solidus temperature 660 C: z=0.09rn . Figure 4.29: Velocity and temperature fields , casting speed 1 mm/s, inlet perature 680 C , solidus temperature 660 C: z=0.17m . Figure 4.30: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C, solidus temperature 660 C: z=0.21m . Figure 4.31: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C , solidus temperature 660 C: x=0.03m . Z-axIs, length In m Figure 4.32: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C , solidus temperature 660 C: x=0.19m . Figure 4.33: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C , solidus temperature 660 C: x=0.30m . Z-axis, length in m °° Figure 4.34: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C , solidus temperature 660 C: x=0.52m . 5 •8 re - 1 SYMMETRICAL QUARTER OF CAST gj Figure 4.35: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C , solidus temperature 660 C: y=1.17m . Figure 4.36: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C , solidus temperature 660 C: y=0.94m . SYMMETRICAL QUARTER OF CAST X-axis, length In m Figure 4.37: Velocity and temperature fields , casting speed 1 mm/s, inlet tem-perature 680 C , solidus temperature 660 C: y=0.73m . CA, *Ptef. OA> SYMMETRICAL QUARTER OF CAST 0.0000 0.7112 X-axis, length in m Figure 4.40: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: z=0.02m . 0.0000 0.7112 X-axis, length in m Figure 4.41: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: z=0.09m . r / — t f r F T f t ! J ! J f ! f f t ! ! f 1 T ? T f f f T T I T ? f f f T 1 T T t - l f f » f f t 0.0000 X-axis, length in m SYMMETRICAL QUARTER OF CAST 5 •s ft cn C t-H to to *—. CO O C to cn O 0.7112 Figure 4.42: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C, solidus temperature 660 C: z=0.17m . 9 SYMMETRICAL QUARTER OF CAST 0.0000 0.7112 X-axis, length in m Figure 4.43: Velocity and temperature fields , casting speed 0.75 mm/s, i temperature 680 C , solidus temperature 660 C: z=0.21m . Z-axis, length in m to Figure 4.44: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: x=0.03m . Figure 4.45: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: x=0.19m . Figure 4.46: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: x=0.30m . Figure 4.47: Velocity and temperature fields , casting speed 0.75 mm/s, in temperature 680 C , solidus temperature 660 C : x=0.52m . Figure 4.48: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: y=1.17m . Figure 4.49: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: y=0.94m . Figure 4.50: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 680 C , solidus temperature 660 C: y=0.73m . E c a a> -o V) *X O I >-o q d. 0.0000 0.7112 X-axis, length in m SYMMETRICAL QUARTER OF CAST 9 •8 tt rt to H CO >: to to >—. CO O C Co CO O o Figure 4.53: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C, solidus temperature 660 C: z=0.02m . 0.0000 0.7112 X-axis, length In m ^ o 05 Figure 4.54: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C , solidus temperature 660 C: z=0.09m . 9 •S SYMMETRICAL QUARTER OF CAST 0.0000 0.7112 X-axis, length In m £ Figure 4.55: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C , solidus temperature 660 C: z=0.17m . V V ^ V \ f -f"f' ! ^ O f !,-'! \ \,y T / T T ,f-r''f l ^ T 1,-1' I !?1>QP f ?,-'? f f ! f ,-f'f ! / ! , { SYMMETRICAL QUARTER OF CAST 0.0000 X-axis, length In m 0.7112 Figure 4.56: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C, solidus temperature 660 C: z=0.21m . Z-axis, length In m Figure 4.57: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C , solidus temperature 660 C: x=0.03m . Figure 4.58: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C, solidus temperature 660 C: x=0.19m . 9 "a Figure 4.59: Velocity and temperature fields , casting speed 0.75 mm/s, l temperature 700 C , solidus temperature 660 C: x=0.30m . 9 Z-axis, length in m Figure 4.60: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C , solidus temperature 660 C: x-0.52m . SYMMETRICAL QUARTER OF CAST r •A-J.. 1 a-i i i 7* 0.0000 0.7112 X-axis, length in m Figure 4.61: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C, solidus temperature 660 C: y=1.17m . SYMMETRICAL QUARTER OF CAST X-axis, length In m Figure 4.62: Velocity a n d temperature fields , casting speed 0.75 mm/s, in temperature 700 C , solidus temperature 660 C : y=0.94m . 5 0.0000 0.7112 X-axis* length in m Figure 4.63: Velocity and temperature fields , casting speed 0.75 mm/s, inlet temperature 700 C, solidus temperature 660 C: y=0.73m . 0.0000 0.7112 X-axis, length in m Figure d.Ci: Turbulent kinetic energy field, casting speed ().7.r» m m / s , inlet tem-perature 700 C , solidus temperature 000 C: / • 0.0!>m . T 1 1 0 . 0 0 0 0 0.7112 X-axis, length in m KiRiirr 4.00: Turbulent, kinetic cnerRy field, casting speed 0.75 tnm/s, inlet tem-perature 700 C , solidus temperature (>(»() C: /, 0.17m . 00 0.0000 0.7112 X-axis, length in m Figure 4.07: Effective viscosity field, rasling sprrd 0.7.r> miii/s, inlrl Irmprraf lire 700 C , solidus temperature C>(>0 C: z 0.02m . 0.0000 0.7112 X-axis, length in m 1—4 O Figure \.OS: Effective viscosity field, casting speed ().7.r» inm/s. inlcl tcinperal lire 700 C, solidus temperature (i(>0 C: /. ().()<)ui . Chapter 4. RESULTS AND DISCUSSION es-. Chapter 4. RESULTS AND DISCUSSION 123 Figure 4.71: Isothermal surface for 500 C in symmetrical quarter of cast, casting speed 0.75 m m / s , inlet temperature 700 C , solidus temperature 660 C ; solution domain defined by I x J x K 22x32x18=12,672 grid points. Chapter 5 SUMMARY AND RECOMMENDATIONS 5.1 Summary A three-dimensional mathematical model was developed to simulate numerically continuous casting processes. The model was incorporated in a code which uses a finite volume method to solve the Reynolds averaged Navier-Stokes equations in conjunction wi th the k — e turbulence model , and the thermal energy equation. A treatment of the latent heat release during isothermal solidification was derived and incorporated in the code. Th i s physical phenomenon is highly non-linear and required a special treatment, through the use of a source term in the energy equation, in order to achieve convergence. The computer code was fully tested and used to analyze the following processes: • continuous casting of axi-symmetric a luminum billets, • continuous casting of a three-dimensional a luminum slab. Comparisons between the present numerical results and available experimental data show satisfactory agreement. 124 Chapter 5. SUMMARY AND RECOMMENDATIONS 125 The following conclusions can be made: 1. In the case of the axisymmetric billet , it is found that the fluid flow inside the pool plays a negligible role when the liquid metal is injected in the axial direction wi th small velocities. 2. For the three-dimensional slab, the flow inside the pool is composed of two distinct zones: a high velocity recirculatory flow zone in the upper part of the pool , and a low velocity zone where the l iquid metal follows the solidified metal wi th a velocity equal to the rate of withdrawal , at the bot tom of the pool . The shell of the cast is found to grow faster along the longer side of the mould . 3. A parametric study shows that varying the inlet temperature of the molten metal has a significant impact on the flow pattern inside the pool . 4. The present results suggest that convection is not very important in deter-mining the solidification profile. Th i s , however, is specific for a luminum; it is expected that the solidification front would be more sensitive to convection in the case of steel and other alloys, where higher Peclet numbers would be encountered inside the l iquid pool . It should be stressed that, from a technological point of view, convection can play a very important role in the solidification process. 5.2 Recommendation The work reported here provided some insight into the complex phenomena present in the continuous casting process. Possible extensions of this work Chapter 5. SUMMARY AND RECOMMENDATIONS 126 and further developments which could extend the usefulness and range of applicabil i ty of the code are: • The assumptions of constant properties is only valid when the range of temperature present in the l iquid pool is small . The range of ap-plicabil i ty of the code could be extended by taking into account the variation of physical properties wi th temperature. This is a relatively simple task, but it may result in substantially slower convergence rates because of the introduction of addit ional nonlinearities in the governing equations. • Some of the physical phenomena observed in the continuous casting pro-cess are of an unsteady nature. A transient formulation would enable the simulation of unsteady state behaviors such as start up conditions and oscillations of free surface level. • The conditions prevailing at the free surface of the l iquid pool can play an important role in controll ing the quality of the cast. Alternat ive methods of modell ing and implementing the boundary conditions for the free surface should be investigated. • Promis ing methods for improving surface quality are electromagnetic s t i r r ing and mouldless electromagnetic casting. The simulation of these processes would be most useful for design and opt imizat ion purposes. • The model can be extended to simulate the continuous casting of steel and other alloys in a relatively straightforward manner. In this case, the latent heat treatment should be extended to cover a mushy region where latent heat would be released over a range of temperatures. Such Chapter 5. SUMMARY AND RECOMMENDATIONS 127 a treatment would result in smaller local source terms and would re-duce the nonlinearities, therefore improving the convergence rate of the numerical procedure. F ina l ly the present numerical results have reproduced satisfactorily experi-mentally observed features of continuous casting, but addit ional model evalu-ation should be carried out wi th new experimental data sets as these become available. B ib l iography [l] Asai , S. & J. Szekely. 1975. Turbulent flow and its effects in continuous casting . Ironmaking and Steelmaking ,1975 No.3: 205-213. [2] Baker P.W. and J.F. Grandfield; Heat Transfer in Aluminum DC Cast-ing. Part I: Mould Heat Flow. Solidification Processing 1987, Sheffield UK, Sept. 21-24, 1987. (3] Benodekar, R. W., Gosman, A. D. & Issa, R. I. (1983). The TEACH-II code for the detailed analysis of two-dimensional turbulent recirculating flow. Dept. Mech Engng, Imperial College, Rept. FS/83/3. [4] Bird, R.B., W.E. Stewart & E.N.Lightfoot. (1960). Transport Phenom-ena , John Wiley, New York. [5] Bommaraju R.P.V.; Mold Behavior, Heat Transfer and Quality of Billets Cast with In-Mold Electromagnetic Stirring. PhD Thesis, University of British Columbia, Feb. 1988. [6] Brimacombe J.K., I.V. Samarsekera, J.E. Lait; Continuous Casting, Volume Two: Heat Flow, Solidification and Crack Formation. The Iron and Steel Society of AIME, 1984. [7] Emley, E.F.. 1976. Continuous casting of aluminium. Int. Metals Re-view, June 1976, 75-115. 128 Bibliography 129 [8] Fahidy T . Z . ; Unsteady-State Heat Transfer in Continuous Cast ing Moulds . Journal of The Iron and Steel Institute, 1969, 1373-1376. [9] Gauthier J . J . , Mor i l lon Y . , J . Dumont -F i l lon ; Mathemat ica l Study of The Continuous Cast ing of Steel. Journal of The Iron and Steel Institute , 1970, 1053-1059. [10] Grandfield J . F . and P . Baker; Var ia t ion of Heat Transfer Rates in the Direct C h i l l Water Spray of A l u m i n u m Continuous Cast ing. Solidifica-t ion Processing 1987, Sheffield U K , Sept. 21-24, 1987. [ l l ] G r i l l A . , K . Sorimachi, J . K . Brimacombe; Heat F low, Gap Formation and Break-Outs in the Continuous Cast ing of Steel Slabs. Metal lurgical Transactions B , V o l . 7 B , 1976, 177-189. [12] Heasilp L . J . , A . M c L e a n , I .D . Sommerville ; Continuous Cast ing, V o l -ume One: Chemica l and Physical Interactions Dur ing Transfer Opera-tions. The Iron and Steel Society of A I M E , 1984. [13] Hi l ls A . W . D . ; A Generalized Integral-Profile M e t h o d for the Analysis of Unidirect ional Heat F low Dur ing Solidification. T M S of A I M E , V o l . 245, Ju ly 1969, 1471-1479. [14] L a i K . Y . M . , M . Salcudean, S. Tanaka and R . I . L . Guthr ie ; Mathemat ica l Mode l ing of Flows in Large Tundish Systems in Steelmaking. Meta l lur -gical Transaction B , Sep. 1986, 449-459. [15] L a i , K . Y . M . & M . Salcudean. (1985). A computer program for two-phase , multi-dimensional, turbulent recirculating flows Dept. of Mech . Eng . , U n i v . of Ot tawa, Canada. Bibliography 130 [16] La i t J . E . , J . K . Br imacombe, F . Weinberg. Mathemat ica l Mode l l ing of Heat F low in the Continuous Cast ing of Steel. Ironmaking and Steel-making, 1974 No. 2, pp. 90. [17] Lancker, M . (1967). Metallurgy of Aluminium Alloys. W i l l i a m Cloves L t d . , London . [18] Launder , B . E . & D . B . Spalding. 1974. Matematical Models of Turbu-lence, Academic Press, London. [19] Launder , B . E . & Spalding, D . B . (1974). The numerical computat ion of turbulent flow. Comp. Meths. Appl. Mech. Engng 3, 269-289. [20] Lehner T . , T . C . Hsiao, Investigation of Fluid flow and Transport Phe-nomena in Agitated Flows in Heat and Mass Transfer in Metallurgical Systems, E d . Hemisphere Publ ishing Corporat ion, 1981, 341-347, Wash-ington, London . [21] Leonard, B . P . . A Survey of Fini te Differences of Opinion on Numerical M u d d l i n g of the Incomprehensible Deflective Confusion Equat ion . Int. J . Heat Mass Transfer 15:1787-1806 [22] Leschnizer, M . A . . Pract ica l evaluation of three finite difference schemes for the computat ion of steady-state recirculating flows. Computer M e t h -ods in Appl i ed Mechanics and Engineering 23(1980) 293-312 [23] Lykoudis L . A . , P . F . D u n n , International Journal Heat Mass Transfer, 1973,16,1439-1452. [24] M i z i k a r E . A . Mathemat ica l Heat Transfer M o d e l for Solidification of Continuous Cast ing of Steel Slabs. Trans T M S - A I M E , 1 9 6 7 , V o l . 239, pp. 1747-1753. Bibliography 131 [25] Mor ton S .K . , F . Weinberg; Continuous Cast ing of Steel. Par t 1: Poo l profile, l iquid mix ing , and cast structure in the continuous casting of mi ld steel. Journal of The Iron and Steel Institute, Jan . 1973, 13-23. [26] Patankar, S .V . & D . B . Spalding. 1974. A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows. Int. J . Heat Mass Transfer 15:1787-1806 [27] Patankar S. V . ; Numerical Heat Transfer and Fluid Flow. M c G r a w - H i l l (Hemisphere), New York 1980. [28] Proceedings of Fourth International Iron and Steel Congress on C o n -tinuous Cast ing. The Meta ls Society, London M a y 1982. [29] Rai thby, G . D . . A Cr i t i ca l Evaluat ion of Upstream Differencing A p -plied to Problem Involving F l u i d F low. Computer Methods in Appl i ed Mechanics and Engineering 9(1976) 75-103. [30] Reynolds, A . J . (1974). Turbulent Flows in Engineering. John Wiley & Sons, London. [31] R icou R . , C h . Vives , Int. J . Heat Mass Transfer, 1982, 25 , 10 , 1579-1588. [32] R o d i , W . (1984). Turbulence models and their application in hydraul ics—a state of the art review. International Associat ion for H y -draulics Research, Delft, the Netherlands. [33] Salcudean M . , Z . A b d u l l a h . Mathemat ica l Mode l l ing of Cas t ing Pro-cesses. University of Ot tawa, Report for D S S , 1987. [34] Schlichting, H . (1968). Boundary Layer Theory, 6th edit ion. M c Graw-H i l l , New York . Bibliography 132 [35] Shamsundar N . , E . M . Sparrow; Analysis of Mul t id imensional Conduc-tion Phase Change V i a the Enthalpy M o d e l . Journal of Heat Transfer, August 1975, 333-340. [36] Shibani A . A . and Ozisik M . N . ; A Solution of Liquids of Low Prand t l Number in Turbulent Flow Between Parallel Plates. Journal of Heat Transfer, Feb. 1977, 21-23. [37] S p a r r o w , E . M . , S . V . Patankar, S. Ramadhyani . Analysis of Mel t ing in the Presence of Na tu ra l Convection in the M e l t Region. Transaction of the A S M E V 99, Nov. 1977. [38] Szekely J . , V . Stanek; O n Heat Transfer and Liqu id M i x i n g in the Con-tinuous Cast ing of Steel. Metal lurgical Transactions, 1970, V o l . 1, 119-126. [39] Szekely, J . & R . T . Yadoya. 1973. The Physical and Mathemat ica l M o d -elling of the Flow Fie ld in the M o l d Region in Continuous Cast ing Sys-tems. Meta l lurg ica l Transactions,V.4, 1379-1388. [40] Touloukian Y . S . , Thermophysical Properties of Matter, Thermophysical Properties Research Centre. Purdue University, 1970. [41] Townsend A . A . , The Structure of Turbulent Shear Flow, Cambridge Universi ty Press, 1956. [42] Vives , C . & R . Ricou . 1984. Velocity and electromagnetic parameters measurements in the sump of a luminium alloy billets. Int. J . Heat Mass Transfer 15:1787-1806 [43] Vol ler , V . R . , N . C . Markatos , M . Cross. A n Enthalpy Me thod for Con-vective Diffusive Phase Change. Int. J . of Numerical Methods in E n g i -neering, 1986. Bibliography 133 [44] Voller V . R . , N . Markatos and M . Cross; Techniques for Account ing for the M o v i n g Interface in Convection/Diffusion Phase Change. Pro-ceedings of the Fourth Internationa] Conference: Numerica l Methods in Thermal Problems. Pineridge Press, Swansea, July 1985. [45] Weinberg F . . The Cast ing of Steel. Metal lurgical Transaction A , 1975 V o l . 6 A , pp.1971. 

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