Phase field simulations have been used to study the evolution from a small nucleus to a dendrite in an undercooled melt, in the presence of convection in the melt. As a generic case for growth from solid boundaries, the nucleus is in one case assumed to be attached to a solid wall, and a shear flow parallell to the wall is assumed in the melt. The interaction between the melt flow and the growing dendrite results in a changed growth rate and morphology. One aspect that has been investigated is how the effective tilting of the main stem of the dendrite depends on the flow strength. It is shown that the major factor that determines the tilt angle is the selection of a nucleus with an optimal initial crystal orientation. We have also studied natural convection around a nucleus held in an undercooled melt.

The phase field equations are solved together with the equations for viscous fluid flow using an adaptive finite element method. This method allows a large computional domain, so that a single dendrite in an effectively unbounded region can be studied.

The presentation will dedicated to the some numerical methods for the solution of the direct and inverse problems for the equations of electromagnetoelasticity.

We consider the Maxwell and the Lame systems in the case when electromagnetic (EM) field is generated by elastic oscillations. At the same time, we neglect the reverse influence of the EM field on the elastic oscillations. It is well known that when elastic waves are propagated through an electroconductive medium within the magnetic field, an interaction between EM and elastic oscillations appears. The resulting waves are called the electromagneto-elastic waves. These waves contain information about both electromagnetic and elastic parameters of a medium.

The electromagnetic wave rides the ``back'' of the seismic wave, that is, the induced electromagnetic wave is ``frozen'' into the seismic wave and propagates either with P- or with S- seismic wave velocity, depending on the type of waves. The dominant frequency and the velocity of the induced seismomagnetic wave is equal to the frequency and velocity of the seismic wave.

We consider the problems of determining some elastic and electromagnetic parameters of a layered medium from a weakly coupled linearized set of equations of electromagnetoelasticity.

To carry out numerical experiments a software package is written on the language Watcom C++ with enhanced graphical interface.

Results of numerical experiments will be given to illustrate the efficiency of the proposed methods.

Book of Abstracts Page: #37

Identification problem of the solid/liquid interface in a phase change process is a subject in which many industry are greatly interested. For example, casting, crystal growth, welding.The interface is often unknown and its determination by direct measurements is impracticable. An alternative consists in doing it with an indirect method using the model which connects the parameters and the sizes governing the transfers of a heat and mass in the phases to the position of the interface. But the coupled physical phenomena occurring in the liquid (natural convection, thermal conduction, surface effects,...) are often partially unknown and cannot be easily modeled. To overcome this problem, one will only based on the solid part, which is accessible to measurements.

Among phase change problems, two inverses cases can arise : the control problem or the identification one. Several studies have been devoted to the control problem, but, fewer results available for the identification problem. In this dtudy, we consider the one phase Stefan problem, in one space dimension, which is a particular moving boundary problem: the isothermal interface between the solid phase and the liquid phase is driven by the diffusive heat two connected phases. The only modeled part is the solid one. We propose an algorithm to solve the identification problem from temperature and flux measurements performed on the solid part only

Book of Abstracts Page: #65

If a computer simulation is to have a major impact on the design of engineering hardware or to be a desirable supplement to an experimental study of complex phenomena, we first have to be convinced that this simulation has a satisfactory level of confidence. Such analysis consists of two procedures: verification and validation. The former one is the process that demonstrates the ability of a numerical model and its computer program to solve specific set of governing equations. It establishes the level of accuracy and sensitivity of the results to parameters appearing in the discrete formulation through purely numerical experiments. This procedure is based on both grid refinement study and comparison of the results with other available solutions of some benchmark problems. The verification procedure, although indispensable, is not sufficient to establish the confidence of numerically obtained predictions. Indeed, for engineers and physicists the most important issue is the degree to which the computer simulation is an accurate representation of reality, i.e.: the degree to which inevitable simplifications of physical and mathematical models reflect reality. This is established through the code validation procedure where calculations are extensively compared with trustworthy detailed experimental measurements. In complex phenomena of coupled fluid flow/heat transfer with the solid-liquid isothermal/non-isothermal phase transition an experimental study is very often difficult and prohibitively expensive, particularly for materials with high fusion temperatures (e.g. metallic alloys). Therefore, in such cases the computer code validation procedure is rather performed through comparing the calculations with experimental data for some substitute media, which solidify or melt in the way similar to the materials of interest but in much lower temperatures. For example, in the case of isothermal phase change, melting of gallium [1] or freezing of pure water [2,3] in the differentially heated cavity is studied experimentally, whereas an aqueous ammonium chloride solution [4] is used to mimic the metallic alloy solidification occurring in the range of solidus-liquidus temperatures. The freezing process of pure water, driven by natural convection in fluid and conduction in both phases, is often used as an experimental benchmark [2,3,5], which is a challenging test for a computer simulation of the solid liquid phase transition. Indeed, water is a fluid that does not obey the Boussinesq approximation of the linear buoyancy force-temperature relation because water density at low temperatures is a non-linear function of temperature. Water density anomaly creates a complex flow pattern that contains two different circulation regions - the hot clockwise vortex and the cold counter-clockwise one. Moreover, experimental investigations of fluid flow and heat transfer processes in the solidifying water are relatively easy to arrange in a small laboratory scale. The up-to-date field acquisition technique, where Thermochromic Liquid Crystal suspended in water as seeding along with the Digital Particle Image Velocimetry and Thermometry can be used here to get detailed, transient, local two-dimensional velocity and temperature fields [2,3,5]. Such experimental findings are commonly acknowledged as exact and reliable enough to be a reference standard for comparison with numerical results [2,3]. However, when water freezes in a small cavity (typically used in experiments), high sensitivity of flow structure and, thus, of the temperature field, to thermal boundary conditions is observed. Moreover, at early times of the process the effect of water super-cooling occurs in the cavity [3,5]. It significantly changes the early-time flow structure and temperature field, and retards the regular ice formation. Therefore, it is reasonable to expect that the calculations can also be affected by some ambiguity of the assumed heat transfer coefficients and by the accuracy of numerical modeling of the real boundary conditions (those that occur during experimental investigations). To elucidate the problem, the experimental findings reported in [2,3,5], are compared with the results of computer simulation of natural convection of pure water in a square cavity at low but positive temperatures and during the freezing process. The computationally efficient numerical model has been developed [5] through the combination of the projection method [6], semi-implicit time marching scheme [6,7] and the enthalpy-porosity approach [8] along with equal-order or unequal-order finite element space discretization [9]. This computer code has been used to calculate the natural convection and solidification of pure water inside the cavity and heat conduction in the cavity walls as the conjugate circumstance for diverse thermal boundary conditions imposed on external surfaces of the cavity. Detailed comparison of the calculated local flow pattern, temperature field and the temporal front shape and position shows significant impact of the initial and thermal boundary conditions on the velocity and temperature distribution in the cavity. Thus, the problem of the authenticity of these conditions is crucial when the above-discussed experimental benchmark is used in the detailed code validation analysis. Special care is needed for precise modeling of realistic boundary and initial conditions to avoid some ad hoc, but not necessary fully correct, conclusions concerning the accuracy and the scope of validity of the computer simulation.

References

1. C. Gau and R. Viskanta, Melting and Solidification of a Pure Metal on a Vertical Wall, J. Heat Transfer, vol. 108, pp.174-181, 1986.

2. T. A. Kowalewski, Experimental Validation of Numerical Codes in Thermally Driven Flows. CHT-97: Advances in Computational Heat Transfer,,eds. G. Vahl Davis and E. Leonardi, pp.1-16, Begell House Inc., N.Y., 1998.

3. T. A. Kowalewski, A. Cybulski and M. Rebow, Particle Image Velocimetry and Thermometry in Freezing Water, Proceedings of 8th International Symposium on Flow Visualization, Sorrento, 1998.

4. W. D. Bennon and F. P. Incropera, Numerical Analysis of Binary Solid-Liquid Phase Change Using a Continuum Model, Numerical Heat Transfer, vol.13, pp.277-296, 1988.

5. J. Banaszek, Y. Jaluria, T. A. Kowalewski., M. Rebow, Semi-Implicit FEM Analysis of Natural Convection in Freezing Water, accepted for publication, Numerical Heat Transfer, 1999.

6. P. M. Gresho, On the Theory of Semi-Implicit Projection Methods for Viscous Incompressible Flow and Its Implementation via a Finite Element Method That Also Introduces a Nearly Consistent Mass Matrix. Part 1: Theory, Numerical Heat Transfer, Part A, vol.29, pp. 49-63, 1996.

7. B. Ramaswamy, T. C. Jue and J. E. Akin, Semi-Implicit and Explicit Finite Element Schemes for Coupled Fluid/Thermal Problems. Int. J. Num. Meth. Eng., vol.34, pp.675-692, 1992.

8. A. D. Brent, and V. R. Voller, Enthalpy-Porosity Technique for Modeling Convection-Diffusion Phase Change: Application to the Melting of a Pure Metal, Numerical Heat Transfer, vol.13, pp.297-318, 1988.

9. O. C. Zienkiewicz and R. L. Taylor, Finite Element Method. Fourth Edition, McGraw-Hill Company, London, 1989.

Book of Abstracts Page: #47

Semi-implicit FEM scheme is used for two dimensional computer simulation of binary alloy solidification controlled by buoyancy forces and conduction. The computational algorithm is based on the combination of: (1) - the projection method to uncouple velocity and pressure calculations for incompressible fluid, (2)- the backward Euler and explicit Adams-Bashforth schemes to effectively integrate diffusion and advection in time, and (3) - an enthalpy-porosity approach to account for the latent heat effect on a fixed finite element grid. The focus of this paper is on the analysis of the impact that the anisotropy of both the permeability and heat conductivity can have on the flow structure and temperature field in the mushy zone. Example calculations are given for an aqueous ammonium chloride solution, which has well-established thermophysical properties and solidifies in manner typical of many metallic alloys.

Book of Abstracts Page: #45

A phase-field method is presented for modeling of dendritic solidification of a pure substance with convection in the melt. The method includes a novel distributed momentum sink term in the Navier-Stokes equations to model the interfacial drag in the diffuse interface region. The method is validatd for several limiting cases involving flow in regions of complex structure and steady dendrite tip growth into an infinite quiescent melt. Results are presented for the growth of a single equiaxed dendrite in the presence of various external flows. The dendrite tip operating state selection in the presence of flow and the development of thermal noise-induced sidebranching in the presence of flow are analyzed in detail.

A modified Allen-Cahn equation is combined with the compressible Navier-Stokes system. It is shown that after altering the stress tensor, the resulting equations fulfil the second law of thermodynamics. A physical motivation for this altered stress tensor is given. The model can be used to describe the behaviour of gas phases in a flowing liquid. It is compared to the well known phase field equations and some numerical calculations are presented to underline its physical significance.

Book of Abstracts Page: #49

Spreading of melts under the influence of solidification has a wide range of applications in geology and engineering. For example the spreading of a corium melt after a severe core melt down accident can lead to critical conditions for the coolability and the subsequent long-term removal of decay heat.

We present an analytically based scheme to investigate the influence of basal solidification onto a spreading flow for liquid melts with low thermal conductivity. We investigate the plane spreading problem over a horizontal isothermal plate in cartesian coordinates.

Based on an underlying lubrication theory we derive an approximation for the velocity and temperature fields and, thus, for the solid/liquid-interface. Effects due to capillarity or liberation of latent heat are negligible. Solidification occurs at a defined temperature, no mushy regime is presumed. For the temperature field we use a quasi-steady approximation. Furthermore, we assume both thermal conductivity and density to be constant and equal in the liquid and solid phases. Solutions are found based on similarity transformations or numerical schemes using the method of lines. The influence of solidification on the spreading flow is discussed in terms of the spreading length history which depends on various parameters, as e.g. inflow rate and solidification temperature.

Book of Abstracts Page: #53

Frontal polymerization has been studied for many years experimentally and theoretically, and Mathematical analysis has been devoted to the study of the stability of the front which separates the two phases, in the presence of thermal convection ¹. We present here an approach for a numerical simulation of this problem.

Solidification problems play an important role in material processing ( like casting of steel, non-ferrous alloys, metal-matrix composites ), the ground freezing technique, in phase change materials used in thermal energy storage systems, etc. In most of these processes multicomponent solutions are present. These multicomponent solutions undergo solidification over a range of temperatures which causes a solid-liquid mixture to be formed. The solid-liquid zone, where solidification takes place and known as a mushy region, consists of solid and liquid phases of varying proportions and often makes a substantial part of the whole system. Within the mushy region proportions of solid and liquid vary in time and space. Moreover, the phase-change phenomenon is assisted by transport phenomena occurring in the individual phases and the mutual interactions between solid and liquid are present.

Complex microstructure of the mushy region is the main reason for carrying out the analysis on the macroscopic scale, the scale much greater than dimensions of dendrites or equiaxed crystals formed during the solidification. Thus, before any numerical implementation, a problem of macroscopic modeling of heat and mass transfer phenomena in the mushy region should be addressed. The macroscopic equations describing the process of solidification are usually introduced via a volume averaging approach. It is however known, from theory of heterogeneous materials, that the volume averaging has many limitations. Its drawbacks lie in problems of finding proper representative elementary volume for averaging, accounting for the macroscopic variation of the microstructure and its statistical character, proper formulation of the constitutive relations describing macroscopic transport phenomena, accounting for thermodynamic relations existing at the solid-liquid interface, etc.

A different approach to describe solidification process in the mushy region, based on an ensemble averaging, is proposed in the paper. The ensemble averaging approach is taken from theory of random fields and stochastic processes where it proved to be useful in finding solution of many problems that appear both in description and understanding of the transport phenomena. This approach was followed to derive conservation equations and the basic constitutive relations appearing in the macroscopic description of the solidification phenomena occurring in the mushy region.

Book of Abstracts Page: #75

We consider natural convection of water without and with phase change in a vertical cylinder cooled from the top. The top wall made of metal is assumed to be isothermal. The cavity is immersed in an external water bath kept at the higher temperature. The non-adiabatic side and bottom walls are made of glass. They allow a heat flux from the external bath. The steady flow configuration consists of a single jet of cold liquid flowing downwards along the cylinder axis and a reverse flow along side wall. In the experiments performed it was found that despite the cylindrical symmetry, a star like structure may apear underneath the lid. For the pure convection the temperature field visualized for the horizontal plane close to the top shows 16-18characteristic spikes running radially from the lid centre. If ice is growing from the top, star-like grooves can be seen at its surface.

The present work is devoted to numerical analysis of this axisymmetry-breaking problem. A combined numerical approach based on the finite volume and global Galerkin methods was developed to describe the experimentally observed instabilities. It was shown that the axisymmetry-breaking instability can set in due to a set of three-dimensional perturbations with a relatively high azimuthal wavenumber k (k>=8). It is shown that the instability is caused by the Rayleigh-Benard mechanism and number of the azimuthal structures is defined by the depth of the temperature boundary layer attached to the cold cover of the container.

Book of Abstracts Page: #77

This report presents results of the experimental and numerical study of the effect of various configurations and combinations of the rotating magnetic fields on melt flows and shape of crystallizations fronts in a cylindrical volume containing electroconducting melt.

Book of Abstracts Page: #81

There are many papers on the numerical simulation of the single crystal growth process (see, for example, (1) and all the references presented there). At the same time, there is no established notion to single out the most efficient approach or algorithm for solving this problem now. An approach developed at the Institute of Physics and Power Engineering in Obninsk is described in the paper. The approach was originally proposed within the framework of a 2D conductive-radiative heat transfer problem in (2) and was further developed in (3)-(5). The essential features of the process simulated are its nonstationary character and the presence of phase changes. Therefore, considering the heat transfer problem, one has to solve the Stefan problem. Different approaches for solving the problem are known presently (6(, each of them having its advantages and disadvantages. We develop the enthalpy approach under which the nonstationary heat transfer equation is formulated and solved in variables of enthalpy. This approach provides a stable and efficient numerical algorithm for solving the Stefan problem. To describe the radiation heat transfer, the method of angular coefficients is used. The main difficulties here are caused by calculation of the angular coefficient matrix for irregular forms of the radiation surfaces including heat shields situated inside of cavities. The conductive heat mass transfer is described by the Navier-Stockes equations under the Boussinesq approximation. Our approach presupposes that the Boussinesq equations are solved in the natural variables by the control volume method according to the Patankar scheme. The equations are previously transformed to exclude the convective terms and to bring them into the divergence form by a method proposed in (7). The system of equations in enthalpy for calculational meshes in each connected combination of zones with the boundary conditions prescribed is linearized by the Newton method. The system of linear equations is solved by the conjugate direction method with preconditioning by the incomplete factorization method. The algorithm described was realized in 2D (r,z)-geometry for calculating the process of crystallization of germanium conducted by the noncrucible melting method on a device "Zona-1" under the null gravity conditions. A demonstration calculation of the process was performed. Previously, a similar calculation was performed without taking into account the influence of convection (5(. The described technique for calculating heat transfer in growing crystals from the melt has the following distinctive features: heterogeneity of the domain; the presence of radiation; the presence of convective heat-mass transfer; nonlinearity of properties, i.e. dependence of the thermophysical parameters on enthalpy; nonstationarity stipulated by the time dependence of the heat generation source and the domain geometry configuration; taking into account the heat of phase change; validity of the model for 3D calculations; possibility to increase the complexity of the model (e.g. by introduction of control magnetic field and vibration impact). The calculational stability of the technique proposed is provided by the use of enthalpy variables in solving the heat transfer equation, by the use of natural variables and the Patankar scheme in calculation of velocities and pressures, by the use of the Newton iteration process to solve the nonlinear system, and by the use of balanced monotonic neutral finite-difference schemes to discretize the space variable and the implicit scheme to discretize the time variable. A high efficiency of the technique is basically provided through the use of a special organization of calculations and through performing the inner iterations by the conjugate direction method with preconditioning of initial operators by the incomplete factorization method.

References

• Polezhaev V.I., Bune A.V., Verezub N.A., Glushko G.S., Gryaznov V.L., Dubovik K.G., Nikitin S.A., Prostomolotov A.I., Fedoseev A.I., Cherkasov S.G. Matematicheskoe modelirovanie konvektivnogo teplomassoobmena na osnove uravnenii Navie-Stoksa. M., Nauka, 1987, 271A.

• Ginkin V.P., Zinin A.I., Balakin I.P. The Method a Global Calculation of Heat Transfer in Numerical Simulation of Crystal Growth. The Third International Seminar on Simulation of Devices and Technologies: Abstract. Obninsk, 1994, p.85.

• Artemyev V.K., Ginkin V.P., Gusev N.V., Zinin A.I., Ozernyh I.L. Chislennoe modelirovanie protsessov teploperenosa pri vyrashivanii kristallov metodom Bridzhmena. III-i Minskii mezhdunarodny forum "Teplomassoobmen MMF-96": tezisy dokladov, Minsk, 1996, pp.3-6.

• Artemyev V.K., Ginkin V.P., Ozernyh I.L. Numerical Study of Convection Flow Influence on Dopant Distribution for Bridgman and Floating Zone Methods Under Microgravity. The Fifth International Conference on Simulation of Devices and Technologies - ICSDT'96: Abstract. Obninsk, 1996, p.126-127.

• Ginkin V.P., Artemyev V.K., Gusev N.V., Luhanova T.M., Ozernyh I.L., Shishulin V.P., Sviridenko I.P. Numerical Simulation of Crystal Growth by Floating Zone Methods. International Symposium on Advances in Computational Heat Transfer. Cesme, Izmir, Turkey, 1997.

• Vabshievich P.N. Chislennye metody resheniya zadach so svobodnoi granitsei. Moskva, izd-vo MGU, 1987, 164p.

• Buleev N.I., Timukhin G.I. O sostavlenii raznostnykh uravnenii gidrodinamiki vyazkoi neszhimaemoi zhidkosti. Chislennye metody

A comparison exercise for numerical codes simulating melting in the presence of natural convection in the melt proposed deals with melting of a pure substance controlled by natural convection in the melt.

Details of the problem you can download from "http://bluebox.ippt.pan.pl/~tkowale//pcc99/benchmark.ps.gz" (compressed postscript, 50Kbytes).

A synthesis and comparison of the results will be presented at PCC99 Workshop.

It is well known that at some circumstances of mathematical modelling of the liquid-solid phase change processes problem of proper description of the density difference between solid and liquid phase is very important becoming a challenge. In the paper results of the series experimental runs are presented that confirm important influence of the difference. The most spectacular in the paper is to present a shape and geometry of the shrinkage cavity forming during solidification of a Phase Change Material (PCM - cytoparaffin 52-54) within a tall rectangular container (aspect ratio height/width equal 10). At start of each experimental run the PCM has been initially maintained at the temperature higher than phase change temperature. The results collected showed that the density difference essentially affects geometry of the solid core especially at regions around axis of symmetry.

Book of Abstracts Page: #87

We present numerical results concerning the simulation of semiconductor melts with free capillary surfaces, particularly silicon crystal growth by the floating zone method. Considering the solid/liqid interface as fixed such a simulation requires the computation of the moving capillary surface of the melting zone. The mathematical model is a coupled system which consists of a heat equation and the Navier-Stokes equations in the melt with a Marangoni boundary condition. We describe an efficient numerical method for solving this problem and give some results for different physical parameters.

Book of Abstracts Page: #41

We consider an equal pressure two-fluid model for two-phase flows with phase change, described by a system of six balance equations with source terms. We present a numerical method tosolve this system. Then we use this method to study a low pressure slow water flow in a vertical heated pipe, this problem being a first stage to simulate a Loss Of Coolant Accident in a Pressurized Water Reactor.

Book of Abstracts Page: #57

A finite element model has been developed for the computation of melting/solidifying process under the action of both buoyancy and surface tension forces. Validated on the square cavity benchmark of Gobin and Le Quéré, it is further extende d to the free surface case where surface tension can drive the flow (capillary flow). A comparison of the results obtained for three boundary conditions applied at the top of the melting pool is performed. It shows that the flow is dominated by buoyancy effect when the pool is deep enough as in the square cavity case of the benchmark.

Book of Abstracts Page: #119

Our aim is to use domain decomposition method (DDM) to calculate numerical example for multiphase diffusion-convection equation in two-dimensional domain. The main idea has been to show that domain decomposition method in multiprocessor computer is much faster than nondecomposited method in single processor computer.

The numerical example includes same kind of initial and boundary conditions as real problem considering continuos casting of steel. For the numerical example diffusion-convection equation is discretized by using finite difference method in spatial variables and implicit euler method for time variable. The solution is calculated both in parallel and non-parallel way. The numerical results consider the time these different calculation methods need to get accurate solution.

Problem is non-linear in the mushy region but in the solid and liquid phase problem is linear and thus fast linear solvers can be used to calculate the solution. That will give us more advance in the numerical calculation comparing to the currently existing methods.

Book of Abstracts Page: #123

Axisymmetric thermocapillary convection is computed in a liquid bridge, held between two cylindrical isothermal rods and laterally heated. Geometrical simplifications lead to a rectangular computationnal domain. Flow fields and spatial derivatives are estimated by a Chebyshev collocation method and the conservation equations are solved by a projection-diffusion algorithm. To avoid mathematical singularities on the axial velocity boundary conditions, the thermocapillary stress is weighted by a regularising function, which cancels at the junctions of the free surface and the solid rods. The shape of this function, particularly the caracteristic length scale of the filtering, can be controlled by the experimentalist and may depend on the temperature profile at the free surface.

The role of the Prandtl number, caracteristic of the considered fluid, on the flow regimes has been studdied for a particular, smooth, filtering of the thermocapillary stress. Increasing the control parameter of the convection, the Marangoni number, high-Pr flows keep a diffusive nature, as low-Pr flows become rapidely convective. They respectively undergo hydrothermal and hydrodynamical transitions to unsteadiness. The discussion will be lead on the role of the filter shape, on which thermally convective flows seem to be very dependent, leading to expensive computationnal investigations. The results could be partly relevant for fluid phenomena occuring in floating zone crystal growth processes.

Book of Abstracts Page: #91

This process can be described by the Fourier or the Fourier-Kirchhoff equations, which can not be solved exactly. An original three-dimensional (3D) numerical model (the first of the two) of a CCM temperature field had been assembled. This model is able to simulate the temperature field of a CCM as a whole, or any of its parts. Simultaneously, together with the numerical computation, the experimental research and measuring have to take place not only to be confronted with the numerical model, but also to make it more accurate in the course of the process. The second original numerical model for dendritic segregation of elements assesses critical points of blanks from the viewpoint of their increased susceptibility to crack and fissure. In order to apply this model, it is necessary to analyze the heterogeneity of samples of the constituent elements (Mn, Si and others) and impurities (P, S and others) in characteristic places of the solidifying blank. The numerical model, based on measurement results obtained by an electron micro-probe, generates distribution curves showing the dendritic segregation of the analyzed element, together with the distribution coefficients of the elements between the liquid and solid states. Both models will be used for optimization of concasting technology.

Book of Abstracts Page: #93

This paper describes the problems of the generalized representations of constitutive equations required in the lumped parameter code. The paper has the following objectives: 1) to draw attention to an improved approach of constructing analytical and integral relationships for wall friction, heat and mass transfer factors based on drift flux model of two-phase flow, Reynolds flux concept and generalized substance transfer coefficient; 2) to show the way in which the suggested integral equation is used to derive relations for wall friction, heat and mass transfer coefficients; 3) to demonstrate the validity of the 'conformity principle' for limiting cases. The method proposed here is on the basis of the similarity theory, boundary layer model, and on a phenomenological description of the regularities of the substance transfer, and on an adequate simulation of the flow structure.

Book of Abstracts Page: #105

A new experimental technique based on a computational analysis of the colour and displacement of thermochromic liquid crystal tracers was applied to determine both the temperature and velocity fields of freezing water. The method was used to verify and validate numerical solutions for water freezing in the cube shaped cavity.

The objective of this paper is to present a theory of free surface flows with continuous solidification. This type of open-channel flows is relevant to a number of important technological processes, such as the horizontal continuous casting of carbon steel. Since carbon steel is a binary alloy, in formulating a mathematical model in addition to accounting for fluid flow and heat transfer it is also necessary to account for the solute transport and for the two-phase region (the mushy zone) effects. Extensive numerical simulations provide valuable insight into this process.

Classical free surface flows received considerable attention in the literature. A good overview on open-channel flows is given in Chow [1]. Both analytical and numerical investigations of open-channel flows is presented in Garcia-Navarro et al. [2], Thomas et al. [3], Rahman et al. [4]. However, open channel flows with solidification have not received so far sufficient attention. This gap needs to be filled, because these flows are relevant to a number of important industrial applications, such as the strip casting of carbon steel.

In recent years, there has been a number of papers devoted to modeling of the heat transfer and fluid flow for different schemes of both vertical [5-9] and horizontal [10-14] strip casting processes. These papers present extensive investigations of both fluid flow and heat transfer in the solidifying strip. However, further insight into this process is needed, such as investigation of coupling flow and heat transfer with solute transport and accounting for the free surface behavior.

In recent publications a number of models for describing fluid flow of binary alloys during solidification have been proposed, mainly for the case when the flow is caused by natural convection. Unlike pure substances, binary alloys solidify over extended temperature ranges and solid formation usually occurs within a two-phase region (the mushy zone), where solid and liquid phases coexist. Sound theories for transport processes in the mushy zone have been developed only recently.

Derivation of the set of governing equations for the mushy zone based on the mixture theory approach was originally reported in [15-16] and recently extended to account for microscopic phenomena in [17-18]. The derivation of the set of governing equations based on a volume-averaging procedure is presented in [19-21]. An excellent review of different models with basic features of each model summarized is given in [22]. Very recently, a three-phase model (solid, liquid and gas phases) of the mushy zone has been proposed [23, 24] and comparisons against the two-phase model have been carried out. Since the appearance of these models, the solidification of alloys has been extensively investigated. Numerical results of these studies along with the main features of the numerical procedures are reported in [25-37]. Reference [38] is one of the first research works on modeling flow, heat and solute transport in a multicomponent steel. Different from the investigations reported in [25-38], where fluid flow is mainly caused by a relatively weak natural convection, in this research we consider the case of a strong forced convection, caused by the change of the height of the free surface.

In this paper, we consider free surface flow of a binary alloy, for example, carbon steel, on a horizontal surface (casting table) which moves with a constant velocity, U. We assume that the binary alloy enters the moving surface with the fully developed relative velocity profile, where relative means relative to the surface. In the beginning of the computational domain the binary alloy is completely in the liquid state, that is its temperature is above the liquidus temperature. A constant heat flux is withdrawn from the surface, and this causes solidification of the alloy as it flows downstream. At the end of the computational domain the alloy is completely solidified, and the solid strip leaves the moving surface with the same constant velocity, U. This process is an example of a free surface flow with solidification. It also should be noted that in the beginning of the casting table, alloy at the free surface is in the liquid state while farther downstream, alloy at the free surface is in the mushy state. In establishing the mathematical model for this process, the following assumptions and simplifications are utilized:

- The transport process is steady, two-dimensional and laminar;

- The properties of the solid and liquid phases are homogeneous and isotropic, the solid phase is stationary and rigid, no microporosity forms in the strip;

- The solid and liquid in the mushy zone are in local thermal and phase equilibrium, the thermophysical properties are constant, but may be different for liquid and solid phases;

- No species diffusion in the solid phase between the averaging volumes and complete diffusion in the solid phase within the averaging volume (lever rule) is assumed;

- Heat transfer by radiation and convection from the free surface is negligible;

- The thin layer approximation can be invoked;

- The surface tension effects are negligible;

- The flow resistance due to the growing dendrites is accounted for only in the direction perpendicular to the primary dendrite arms (in the x-direction), resistance in the y-direction is neglected because of the small thickness of the strip;

- The density difference between the fluid and solid phases is accounted for only in the continuity and the species transport equations, but it is neglected in the energy equation. It other words, the term accounting for the density change is incorporated into the latent heat term and the temperature dependence of the ëffective latent heat" is then neglected. Thus the energy equation then takes the form suggested in Beckermann and Viskanta [27].

Thus this paper suggests a model of free surface flow with solidification. This model is applied to numerically investigate coupled fluid flow, heat transfer and solute transport in horizontal continuous casting process. It is shown that the solute diffusion in the liquid phase causes a formation near the casting table a thin diffusion boundary layer. This boundary layer is essentially depleted of the solute. The formation of this boundary layer can be explained by considering the effect of diffusion near an intensively cooled impermeable wall. It is established that increasing of heat transfer rate from the casting table results in considerable decrease of macrosegregation level in the strip. This is because larger heat transfer rate results in smaller width of the mushy zone, which in turn results in smaller macrosegregation. It is also established that increasing of casting velocity results in slight increase of macrosegregation level in the strip. This is because larger casting velocity results in larger width of the mushy zone, which in turn results in larger macrosegregation.

REFERENCES

1. Chow, V.T. 1959. Open-channel hydraulics. McGraw-Hill, NY.

2. Garcia-Navarro, P., Alcrudo, F. Savirón, J.M. 1992. J. Hydraulic Eng. 118, 1359-1371.

3. Thomas, S., Hankey, W.L., Faghri, A. Swanson, T. 1990. J. Heat Transfer 112, 728-735.

4. Rahman, M.M., Hankey, W.L. Faghri, A. 1991. Int. J. Heat Mass transfer 34, 103-114.

5. Shiomi, M., Mori, K. Osakada, K. 1995. In: Proceedings of the Seventh International Conference on Modeling of Casting, Welding, and Advanced Solidification Processes VII, 793-800.

6. Raihle, C.-M., Fredriksson, H. Östlund, S.: 1995. In: Proceedings of the Seventh International Conference on Modeling of Casting, Welding, and Advanced Solidification Processes VII, 817-824.

7. Hwang, S.M. Kang, Y.H. 1995. ASME Journal of Heat Transfer 117, 304-315.8. Saitoh, T., Hojo, H., Yaguchi, H. Kang, C.G. 1989. Metallurgical Transactions 20B, 381-390.

8. Saitoh, T., Hojo, H., Yaguchi, H. Kang, C.G. 1989. Metallurgical Transactions 20B, 381-390.

9. Hwang, J.D., Lin, H.J., Hwang, W.S. Hu, C.T. 1995. ISIJ International 35, 170-177.

10. Bradbury, P.J. Hunt, J.D. 1995. In: Proceedings of the Seventh International Conference on Modeling of Casting, Welding, and Advanced Solidification Processes VII, 739-746.

11. Caron, S., Essadiqi, E., Hamel, F.G. Masounave, J. 1990. Light Metals, 967-973.

12. Mallik, R.K. Mehrotra, S.P. 1993. ISIJ International 33, 595-604.

13. Jimbo, I. Cramb, A.W. 1994. ISS Transactions 15, 145-150.

14. Digruber, M., Haas, S. Schneider W. 1998. In: Proceedings of the Eighth International Conference on Modeling of Casting, Welding, and Advanced Solidification Processes VIII, 663-670.

15. Bennon, W.D. Incropera, F.P. 1987. Int. J. Heat Mass Transfer 30, 2161-2170.

16. Prescott, P.J., Incropera, F.P. Bennon, W.D. 1990. Int. J. Heat Mass Transfer 34, 2351-2359.

17. Ni, J. Incropera, F.P. 1995. Int. J. Heat Mass Transfer 38, 1271-1284.

18. Ni, J. Incropera, F.P. 1995. Int. J. Heat Mass Transfer 38, 1285-1296.

19. Ganesan, S. Poirier, D.R. 1990. Metallurgical Transactions 21B, 173-181.

20. Poirier, D.R., Nandapurkar, P.J. Ganesan, S. 1991. Metallurgical Transactions 22B, 889-990.

21. Ni, J. Beckermann, C. 1991. Metallurgical Transactions 22B, 349-361.

22. Viskanta, R. 1990. JSME Int. J., Series II 33, 409-423.

23. Kuznetsov, A.V. Vafai, K. 1995. Int. J. Heat Mass Transfer 38, 2557-2567.

24. Kuznetsov, A.V. Vafai, K. 1996. Numerical Heat Transfer Part A 29, 859-867.

25. Bennon, W.D. Incropera, F.P. 1987. Int. J. Heat Mass Transfer 30, 2171-2187.

26. Voller, V.R. Prakash, C. 1987. Int. J. Heat Mass Transfer 30, 1709-1719.

27. Beckermann, C. Viskanta, R. 1988. PhysicoChemical Hydrodynamics 10, 195-213.

28. Bennon, W.D. Incropera, F.P. 1988. Numerical Heat Transfer 13, 277-296.

29. Engel, A.H.H. Incropera, F.P. 1989. Wärme- und Stoffübertragung 24, 279-288.

30. Voller, V.R., Brent, A.D. Prakash, C. 1989. Int. J. Heat Mass Transfer 32, 1719-1731.

31. Prakash, C. Voller, V. 1989. Numerical Heat Transfer B 15, 171-189.

32. Neilson, D.G., Incropera, F.P. Bennon, W.D. 1990. Int. J. Heat Mass Transfer 33, 367-380.

33. Felicelli, S.D., Heinrich, J.C. Poirier, D.R. 1991. Metallurgical Transactions 22B, 847-859.

34. Yoo, H. Viskanta, R. 1992. Int. J. Heat Mass Transfer 35, 2335-2346.

35. Felicelli, S.D., Heinrich, J.C. Poirier, D.R. 1993. Numerical Heat Transfer B 23, 461-481.

36. Lee, S.L. Tzong, R.Y. 1995. Int. J. Heat Mass Transfer 38, 1237-1247.

37. Schneider, M.C. Beckermann, C. 1995. Int. J. Heat Mass Transfer 38, 3455-3473.

38. Böhmer, W.F.A., Schneider, M.C., Beckermann, C. Sahm, P.R. 1995. In: Proceedings of the Seventh International Conference on Modeling of Casting, Welding, and Advanced Solidification Processes VII, 617-624.

Book of Abstracts Page: #115

KEYNOTE PAPER

Recent modelling work of the solidification and melting of a weak binary alloy in a horizontal Bridgman furnace is presented. The work has been undertaken in connection with the MEPHISTO-4 program, which is a study of the solidification and melting of an alloy of bismuth with tin in a microgravity environment. The effects of coupling with the phase diagram (a concentration-dependent melting temperature) and of thermal and solutal convection on segregation of solute, shape and position of the solid/liquid interface are investigated. The results presented include calculations at 1 and 10 g, both neglecting and including the dependence of melting temperature on concentration.

Our aim is to study a flow of a liquid-vapour system from the point of view of a kinetic theory. This is because in the transition zone separating the liquid and vapour phases the gradients of the flow parameters are very large and the zone itself is very narrow. So the flow is very non-uniform with a strong heat and mass transfer.

In order to have a kinetic equation suitable to liquid dynamics and phase change Grmela [1], [2] proposed so-called Enskog-Vlasov equation. In this model, the intermolecular potential is split into a hard-core and an attractive tail. The hard-core is treated as in the standard or revised Enskog equation, whereas the tail enters the equation only linearly in a mean-field term.

Unfortunately, it turns then that the Enskog-Vlasov equation, despite its merits, is too complicated for a purpose like that. That is why we turned to the so-called discrete kinetic theory developed for the Boltzmann kinetic equation, which considers such mathematical models of it that the molecular velocity space is not all Rd (where d = 1, 2, 3) but a finite, fixed in advance set of d-dimensional vectors and extended its ideas to our present needs. In this paper we confine ourselves to an analysis of shock waves using to this end the simplest 4- velocity model of the Enskog-Vlasov equation.

Unfortunately, it turns then that the Enskog-Vlasov equation, despite its merits, is too complicated for a purpose like that. That is why we turned to the so-called discrete kinetic theory developed for the Boltzmann kinetic equation, which considers such mathematical models of it that the molecular velocity space is not all Rd (where d = 1, 2, 3) but a finite, fixed in advance set of d-dimensional vectors and extended its ideas to our present needs. In this paper we confine ourselves to an analysis of shock waves using to this end the simplest 4- velocity model of the Enskog-Vlasov equation.

Although, from the physical point of view this model is very simple, mathematically it is quite complicated. Due to this complicity we performed its various simplifications, which will presented and discussed. We look for travelling wave solutions to these simplified versions. We pay some attention to the monotonicity of the density component of the travelling wave. Finally, we introduce a simplified model and present some numerical results concerning the hydrodynamic and kinetic shock wave structures paying special attention to the impending shock splitting. The new feature is that kinetic effects alone are unable to kill the artificial phenomenon of impending shock splitting.

[1] Grmela M. (1971) Kinetic approach to phase transitions. J. Statistical Physics 3: 347-364

[2] Grmela M. (1974) On the approach to equilibrium in kinetic theory. J. Math. Physics 1: 35-40

Book of Abstracts Page: #97

We present a scale analysis of the parameters for which flow instabilities can be expected in coupled phase change-natural convection in cavities heated from the side. The results are in agreement with what was observed numerically for very low Prandtl number fluids, where multiple cells were found to appear at early time as a result of the instability of the conduction regime. For very large Prandtl number flow instabilities, if any, would only occur at very large nominal Rayleigh number.

Book of Abstracts Page: #129

KEYNOTE LECTURE

In this presentation we investigate the well-posedness of a phase-field model for the isothermal solidification of a binary alloy due to Warren-Boettinger. Existence of a week solution as well as regularity and uniqueness results are estalished under Lipschitz and boundeness assumptions for the nonlinearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the nonlinearities.

Book of Abstracts Page: #27

KEYNOTE LECTURE

Heat transfer between two media involving liquid-solid phase transitions is important in both nature and industry. Natural situations include ice formation in rivers, melting of icebergs and lava flows from volcanoes. Industrial areas of interest include casting, welding, hot liquid jets for drilling and the design of heat sinks in power generating systems and in the nuclear power industry. In this last example, the operators are required to demonstrate that adequate safety margins exist even under severe accident conditions. This means that the physical processes involved in extreme conditions are sufficiently understood that corrective actions can be implemented effectively.

The problem motivating the current work is an extreme hypothetical situation in a gas-cooled reactor, where it is supposed that the shutdown systems have failed to arrest some event that has led to the fuel in a particular channel to overheat, causing the entire fuel inventory of that channel to melt and pour onto the steel floor below. In a matter of seconds this molten fuel material would spread over the floor and freeze into a solidified mass. On a much longer time-scale, the solidified fuel would release its nuclear decay heat, partly to the gaseous environment (by radiative heat transfer) and partly to the steel floor (by conduction). In considering the possibility of melting of the floor due to the release of this decay heat, the extent of spreading of the molten material is a key factor since this determines the effective strength of the source of heat to the base. In practice the strongest inhibitor to spreading is crust formation on the top of the melt. In this talk, results from two sets of simulant experiments will be presented, distinct flow regimes will be identified and analysed.

Book of Abstracts Page: #31

Based on microscopic kinetic equations, a mathematical model is considered for the time- dependent diffusion process of self interacting metal vapour in fireproof material in strongly inhomogeneos temperature field. Two dimensional structure is examined, where the inner hot surface acts as a source of metal vapour, but the outer surface - as a cooler. Due to self interaction of metal vapour phase transition (condensation) near the outer surface is appeared. The developed conservative, monotonous and absolutely stable difference scheme is based on special exponential type substitution for concentration of molecules. Results of non steady state 2D numerical experiments are presented. It is shown that metal vapour transition produces a stochastic nature of phase evalution.

Book of Abstracts Page: #71

In continous casting flux powder is added on top of the melt. Due to the large temperatures the flux melts and flows into a small gap between the solidified strand shell and the mould. Due to the vertical oscillation of the mould depressions on the strand surface, so called oscillation marks, are formed. In this paper a model describing the flow of flux and its interaction with the solidifying strand shell is presented.

Book of Abstracts Page: #139

We first consider a Caginalp-Fix phase-field diffusion system, coupled with appropriate Cauchy and boundary conditions, and we describe a related existence and uniqueness result. Then, we allow the data of the problem to vary and we show that, under suitable convergence assumptions, we can obtain as a limit the weak formulation of the standard Stefan problem.

Then, we take into account a heat and phase transmission problem between two fluids of possibly different physical properties and we repeat the previous analysis in the case of the phase-field equations approaching the Stefan model only on one side. Following an idea of Damlamian, Kenmochi and Sato, we perform an asymptotic analysis also in this case, but with a different and more elaborate technique, which is based on a reformulation of the problem in terms of graph convergence of abstract monotone operators of subdifferential type.

We finally present in some detail a concrete physical situation which can be studied by means of the above outlined technique.

Book of Abstracts Page: #131

Existence results for the Frémond model of shape memory alloys have been often investigated by using a time discretization approach. Here I present some a posteriori error estimates for such approximations with respect to the full monodimensional model and some three-dimensional models. These estimates show optimal rates of convergence, depend solely on computable quantities such as discrete solutions and data, and impose no costraints between consecutive time steps.

Book of Abstracts Page: #135

Three-dimensional unsteady computation was made for the melt flow in the production process of super-conducting material Y123 with a modified Czochralski method. Results of the studied cases reveal the importance of three-dimensional compu tation in the determination of stability conditions and the unsteady behavior of the melt flow. How to suppress the melt flow instability and the patterns of melt flow in some typical unsteady conditions were also discussed.

Book of Abstracts Page: #163

Since discovery in 1987 of superconductivity at  90 K in a new mixed-phase compound system, an enormous amount of attention has been given to the improvement of the quality of these materials. One of the most seriously studied compounds is the yttrium barium copper oxide superconductor YBa2Cu3O7-x (Y123). In 1993 continuous growth of Y123 large single crystals was achieved by applying a modification of the Czochralski method. The quality of yttrium barium copper oxide superconductor Y123 crystals grown from a melt by this method is significantly affected by heat and mass transfer in the melt during growth. In an Y123 single crystal growth system the Y123 single crystal grows directly from the liquid phase as a primary phase by the migra tion of Y atoms from the solute Y2BaCuO5 (Y211) on the bottom of the melt to the free surface. The Y123 crystal grows continuously as long as the nutrient Y211 exists. In this method, convection in the melt is an important factor in controlling distributi on of Y atoms in the melt, due to the high Schmidt number of the melt (Sc = 7000). This paper describes transport phenomena during growth of superconducting materials and presents the numerical computations of the flow, thermal and Y concentration fields in the melt for Y123 single crystal growth by this modified method. The flow in th e melt was modelled as an incompressible, Newtonian and Boussinesque fluid. Calculations are presented for a combination of buoyancy - driven flow and flow driven by crystal rotation.

Book of Abstracts Page: #147

Melting and solidification problems are very important in manufacturing processes such as crystal growth of semiconducters, and casting and welding of metals and alloys.

In this paper we investigate the effects of unequal thermal conductivity in the liquid and solid phase during directional solidification in both vertical and horizontal Bridgman configurations. Numerical solutions are obtained based using a fixed grid single domain (enthalpy) method using two different approaches - finite volume with primitive variables and finite differences vorticity-stream function formulation.

Two cases are investigated. The first one is that of a transient phase change problem of a pure substance with oscillatory and steady convection in the liquid region in the inverted Bridgman configuration (heated from below). The second case is the phas e change problem during transient directional solidification of binary alloy in the horizontal Bridgman configuration.

The significant effects of the variation of thermal conductivity on the interface shape and flow structures are discussed in detail.

Dissolution of stoichiometric multi-component particles is an important process ocurring during the heat treatment of as-cast aluminium alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem.

The well-posedness of the moving boundary problem is investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an analytical asymptotic approach based on self-similarity is given. Moreover, this self-similar solution is extended to the vector valued Stefan problem. From this extension follows that we can approximate the dissolution rate for the vector valued Stefan problem using a quasi-scalar Stefan problem (a Stefan problem with only one diffusing phase) for some cases. The effective diffusion coefficient is then obtained by a geometric mean of all diffusion coefficients. The weight factors come from the concentrations in the particle.

Subsequently a numerical solution of the vector valued Stefan problem is described. This solution is based on a finite volume discretisation of the diffusion equations for all components. The nonlinear boundary conditions are solved using a discrete Newton Raphson iteration procedure. The numerical method is compared with solutions by analytical methods.

In the above described model the particle and the cell in which the particle dissolves have the same geometry. For a Stefan problem with only one component (a scalar Stefan problem), we compare this model to a model with two spatial co-ordinates using finite element calculations, where the particle and cell have different geometry.

Book of Abstracts Page: #155

The fluid-dynamics of crystal groth from vapour has been recognized as palying an important role in thechnological processes employed to grow bulk crystals or epitaxial films. We present numerical results obtained for the new set of field equatioans (Brunett equations with slip conditions) for geomwetrical configurations of interest for crystal growth by physical vapour transport.

Book of Abstracts Page: #161