Session 9

Chairman: B. Milton

The abstract is not available.

Energy storage applications are based on either sensible or latent heat principles. Sensible heat storage can be described as heating and cooling of water for liquid-based systems or rock for air-based systems. The storage density of systems based on the latent heat method is larger than that based on the sensible heat method. Interest in the latent heat method of solar or electrical energy storage has been widely studied, since available heat may be stored at the temperature required in subsequent applications.

Fundamental studies have been done on heat transfer during melting and solidification of a Phase Change Material (PCM) in cylindrical geometries. The objectives in most of the experimental work were to measure the temperature distribution in both phases and to locate the interface position during melting and solidification. Many attempts were also carried out in previous experimental works to study the effect of natural convection on the process. However, no attempts were made involving phase change heat transfer during cyclic solidification and melting in the presence of two interfaces.

Simulating phase change heat transfer problems have been mostly dealt with by solving analytically or numerically the unsteady heat conduction equation. The problem was treated as pure conduction in both phases or conduction in both phases but with the liquid thermal diffusivity being replaced be an effective thermal diffusivity to account for the natural convection that takes place during melting. However, this type of treatment was found to suffer from major limitations mainly when applied to multidimensional systems and/or when more than one interface was present. A method called the "Effective Heat Capacity" (EHC) developed by Farid overcomes these drawbacks. In this method, the unsteady heat conduction equation is solved for the solid, melt and the two phase region by treating them as one phase composed of three regions. Each region has different thermal diffusivity. In the liquid phase, an effective thermal conductivity is used. For the two-phase region, effective heat capacity replaces the value of the heat capacity to account for the latent heat. This method has been successful in simulating melting and solidification of PCM's in different geometries, and in multidimensional problems with more than one interface. This method is used in this study to simulate the cyclic solidification and melting of paraffin wax in a vertical cylinder.

In our work, the cyclic variation of the lower boundary temperature of a PCM during solidification and remelting was investigated. The PCM was paraffin wax with a melting range (58-60 ^oC). The experimental apparatus consisted of a Teflon cylinder containing the PCM material which was fixed between two copper boxes, through which water from two tanks, each at fixed temperature, was circulated. Each experimental run was started by having the PCM completely liquid at a uniform initial temperature then was exposed to a sudden drop in temperature, below the melting range, at the lower plate. The upper plate was kept constant at its initial temperature. As a result, a solid phase appeared and a moving solid-liquid interface was formed. The cooling span was terminated by returning the lower plate temperature to the initial value which was greater than the melting range of the PCM, leading to remelting of the solidified layer and forming a second moving solid-liquid interface.

The presence of an intermediate solid layer in between the two liquid phases in the heating span, led to apperance of flat temperature variation at positions that had not undergone solidification in the cooling span. Also, this led to the slow movement of the upper interface, since the solid layer acted as a resistance to the heat flow from below. The temperature variation with time during solidification and remelting was recorded via a multi-channel recorder.

The uni-dimensional transient conduction equation was solved numerically side by side with the EHC method. The effect of natural convection during remelting was considered by lumping it in an effective thermal diffusivity. The predictions of the temperature distribution and the solid-liquid interface positions were in good agreement with the experimental measurements.

The movement of the interface during the cooling span was found to be dependent on the lower plate temperature; as the difference between the melting point and the lower plate temperature increases, the velocity of the interface movement and the rate of heat discharge from the PCM increase. In the heating span, the velocity of the lower interface movement and the time for complete melting of the solid layer, as well as the heat charge to the PCM from the lower plate were dependent on both the lower plate temperature and the initial thickness of the solid layer formed in the cooling span.

Numerical investigation of complicated transport phenomena is important for many technological processes. A characteristic example of such a process is casting.

Over the recent years, there has been increased interest in producing thin strips by a strip casting technology. In recent publications both vertical and horizontal continuous strip casting processes have been investigated. Though both the fluid flow and heat transfer in a solidifying strip have already been extensively studied, the solute transport during this process, to the best of the author's knowledge, has not yet received any attention.

In this paper we concentrate on the modification of the strip casting process when the molten metal leaves the casting wheel slightly overheated and all solidification occurs on the casting table. This process is characterized by a strong forced convection. Despite of the small thickness of the strip, the strong forced convection can result in a noticeable redistribution of the solute over the strip thickness.

In this paper a coupled fluid flow and heat and species transport problem is solved numerically. In addition to the importance of this investigation for a better understanding of the strip casting process, the results presented in this paper may also be valuable for understanding of the fundamentals of the species transport caused by the strong forced convection.

In recent years, considerable attention has been concentrated on developing sophisticated models capable of predicting transport phenomena during solidification of alloys. A number of models for describing the coupled fluid flow and heat and species transport have been proposed. Despite of the extensive investigations performed in modeling the coupled fluid flow and heat and species transport during the solidification of alloys, little attention was given to species transport occurring during strong forced convection. In the previous studies basically the case of a relatively weak natural convection was considered.

In the present investigation the case of the strong forced convection is considered. We proceed from the governing equations obtained by the volume averaging technique presented in refs. The set of basic equations is modified to account for the specifics of the strip casting process. A dendritic columnar structure of the mushy zone is assumed.

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 is assumed;
• Heat transfer by radiation and convection from the free surface is negligible;
• The thin layer approximation can be invoked. This approximation makes it possible to parabolize the governing equations for this problem. On the other hand, proceeding from this approximation, the natural convection effects are neglected compared to the forced convection effects;
• 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 x-direction); the flow resistance in the direction parallel to the primary dendrite arms are neglected since the thickness of the strip is small;
• The density difference between the fluid and solid phase is accounted for only in the continuity and the species transport equations, but it is neglected in the energy equation. In other words, the term accounting for the density is incorporated into the latent heat term and the temperature dependence of the "effective latent heat" is then neglected. Thus the energy equation then takes the form suggested in Beckermann and Viskanta.

The main results obtained in this research are given in what follows. In is shown that the mushy zone effects lead to the velocity profiles which are essentially non-parabolic. Forced convection as well as shrinkage-induced fluid flow influences the solute concentration distribution in the strip. Four concentration zones are found in the strip. In an inverse segregation zone very near the bottom of the strip the solute concentration is essentially higher than at the inlet. The inverse segregation zone is followed by a zone where concentration is lower than at the inlet. This zone is formed by the transport of the solute from the upper to the lower part of the mushy zone with the shrinkage-induced flow. Near the outlet at the free surface there is a zone where the carbon concentration is higher than at the inlet. This zone is formed by transport of the solute with the forced flow in the direction of the free surface. Finally, in the triangular region at the inlet boundary occupied by liquid metal concentration of carbon equals the inlet concentration.

1. Shiomi, M., Mori, K., and Osakada, K., Finite element and physical simulations of non-steady state metal flow and temperature distribution in twin roll strip casting, Proceedings of the Conference Modeling of Casting, Welding and Advanced Solidification Processes VII, pp. 793-800, 1995.
2. Raihle, C.-M., Fredriksson, H., and Östlund, S., Modeling of heat flow and solidification in a strip caster, Proceedings of the Conference Modeling of Casting, Welding and Advanced Solidification Processes VII, pp. 817-824, 1995.
3. Bradbury, P.J. and Hunt, J.D., A coupled fluid flow, deformation and heat transfer model for a twin roll caster, Proceedings of the Conference Modeling of Casting, Welding and Advanced Solidification Processes VII, pp. 739-746, 1995.
4. Ganesan, S. and Poirier, D.R., Conservation of mass and momentum for the flow of interdendritic liquid during solidification, Metallurgical Transactions, vol. 21B, pp. 173-181, 1990.
5. Poirier, D.R., Nandapurkar, P.J., and Ganesan, S., The energy and solute conservation equations for dendritic solidification, Metallurgical Transactions, vol. 22B, pp. 889-990, 1991.
6. Beckermann, C. and Viskanta, R., Double-diffusive convection during dendritic solidification of a binary mixture, PhysicoChemical Hydrodynamics, vol. 10, pp. 195-213, 1988.

Plasma facing components for tokamak fusion reactors will be exposed to fast neutron and plasma fluxes, and it will be under severe heat load condition. Especially during the plasma disruption, the armor material will be subjected to high thermal load in the range from several to several tens of MJ/m². For a duration of several milliseconds, consequently the surface of armor materials will melt and evaporate. In the design of plasma facing components for a tokamak reactor, the erosion loss induced by plasma disruption is regarded as dominant factor which limit the lifetime of armor materials. In order to predict the lifetime of armor materials, it is important to estimate its erosion depth. However, the heat transfer process between the plasma particles and the solid wall, and the behaviors of evaporating surface that is exposed to high energy particle fluxes are not well understood. We adapt molecular dynamics method for the simulation of the process.

The molecular dynamics simulations have been performed for argon and helium molecular system. We simulate the process of melting and evaporation of a solid argon wall, with injecting helium molecules with high kinetic energy to the surface of this wall. The molecular behavior near the interface between a gas phase and a solid/liquid mixture phase is analyzed. The interaction between molecules is decided with Lennard-Jones potential function. In order to integrate the Newton's equation of motion, a main time step and a divided time step is used. The main time step is applied to the interaction of Ar-Ar , and the divided time step is applied to the interaction of Ar-He and He-He.

Schematic of calculating system is shown in figure 1. Cartesian coordinate x, y, z is introduced with the x- and y-axis parallel to, the z-axis normal to the surface of the argon wall. Boundary (1) is a free-boundary, and boundary (2) is an insulated boundary. For the other four boundaries, the periodic boundary conditions are applied. We simulated two conditions of incident He particle as below.

1. Inject helium molecules periodically which have 1.29 eV of kinetic energy. The argon wall is formulated by 2160 argon molecules, and the main time step is 5.0 fs.
2. Inject helium molecular that has 100 eV of kinetic energy. The argon wall is formulated by 1170 argon molecules, and the time step is 0.2 fs.

The molecular arrangement of the calculation cell of simulation (ii) is shown in Figure 3. In this simulation, it was observed that a few argon molecules jump out from the surface as soon as a helium molecule enter the wall, then local melting and evaporation occurred.

Figure 3 and 4 shows the velocity distribution function of each component for the simulation (i) and (ii).The velocity distribution function for particles evaporating from the surface is similar to Maxwell - Boltzmann distribution for all components in case of simulation (i). For the z-axis is normal to the evaporating surface, only positive components appear in the velocity distribution function. In case (I), because incident energy are relatively low, kinetic energy introduced by helium molecules is transferred to the surface almost uniformly. Therefor the velocity distribution function is getting near to the equilibrium distribution of Maxwell-Boltzmann. For the simulation (ii), x- and y-component of velocity distribution are similar to the Maxwell - Boltzmann distribution, like in the case of simulation (i). On the other hand, z-component of velocity distribution largely slide from Maxwell - Boltzmann distribution. In this case, local evaporation occurs before energy introduced by helium molecule transfer the surface molecules uniformly. As a result of it, it is considered that the vibration of the surface molecules is strongly exited only for x-component.

In the case of simulation (i), all incident helium molecules to the argon wall are reflected at the first atomic layer of the surface. But in the case of simulation (ii), some helium molecule penetrate into the second or third atomic layer. Figure 5 shows the kinetic energy transitions of typical two helium molecules in simulation (ii). The incident helium molecular that scattered at the first atomic layer is lost about 30 % of initial kinetic energy after one or two collision with surface argon molecules of the wall. On the other hand, the molecules that penetrate into the wall lost almost all kinetic energy after many collisions with the lattice atoms of argon wall. The ratio of incident helium energy to the energy transferred to the wall is rather different from each case of simulation (i) and (ii), and the distribution ratio of the energy absorbed into the wall to vaporization heat, melting heat and temperature rising is also different from each case.

The process of melting and evaporation of the solid argon wall was simulated by using the molecular dynamics method. The behavior of evaporating molecules in the difference of incident particle energy is considered. It is shown that the profile of velocity distribution may largely depend on the kinetic energy of incident particle. Therefore the general assumption for the velocity distribution at evaporating surface will be necessary to reconstruct.

THERMAL MODELLING OF A TRANSFERRED PLASMA ARC FURNACE

J-M Gatt L. Buffe C. Jegou G. Cognet
CEA
DRN/DER/SERA
CE Cadarache
13108 St Paul lez Durance (France)

INTRODUCTION

Within the framework of PWR severe accident studies, scenarios leading to partial or whole core melting are envisaged. In this case, a molten mixture, called --corium-- and essentially composed of highly refractory materials (U0₂, ZrO₂) and metals (Fe Zr) can flow down towards the lower head (as during the TMI2 accident) and if there is no intervention, melt through the vessel and spread into the reactor pit.

The main goal of the VULCANO program is to study corium spreading and cooling, and its interactions with different materials in order to qualify core-catcher concepts. Three tests have been performed:

• interaction with core-catcher materials (thermal ablation, physico-chemical interaction),
• spreading test,
• cooling test with sustained heating.

The VULCANO facility is mainly composed of a furnace and a test section of which the geometry depends on the specific objectives of each experiment. The furnace has been designed to heat up to 3000 K roughly one hundred kilograms of corium composed of representative materials: U0₂, Zr0₂, Zr and Fe.

The chosen technique to melt the corium constitutive materials is to heat them, in a rotary plasma arc furnace. Thanks to the rotation of the furnace, powders are centrifugated allowing the creation on the furnace axis of a 200 kW plasma arc at very high temperature (10000 to 20000 K depending on gases) which heats and melts the load surface layer by thermal radiation; then, deeper layers are melted by conduction. The furnace is cooled by external water circulation in a double jacket.

The aim of this paper is to present, after a short description of the furnace, the study of its thermal behaviour.

The transient calculations are performed with the CASTEM 2000 finite elements code and compared with experimental tests. This study has three goals: first, the understanding of the power distribution in the furnace, second, parametric calculations to prepare the test and third, performing real time calculation. We try to optimize the melted volume and the reached temperature as a function of time without endangering furnace integrity. These three goals lead us to propose three different modellings:

• a complete modelling accounting for plasma arc radiation,
• a simplified modelling based on conduction in the whole fizrnace,
• a very simplified one-dimensional modelling describing the powder load.

In each of these modellings, we take into account physical properties (conductivity, specific mass, thermal capacity) variation with temperature.

COMPLETE MODELLING

The whole furnace with the plasma arc (see Figure 1 ) is modelled. The axisymmetric calculation is performed accounting for radiation in the closed cavity. Then the calculation is compared with test results.

Furthermore, we study the evolution of the maximum and mean temperatures reached in the load versus the energy injected in the furnace. This study shows the influence of the heating strategy (level, plateaus, kinetics). We describe the furnace behaviour using three parameters: maximum temperature, mean temperature and energy, knowing that maximum temperature and energy are experimental data.

PARTIAL MODELLING

The first modelling requires very long calculation times (24 hours on a Sun Sparc 20 workstation to calculate 1 hour of test), so in this part we study the furnace, only considering the structure and the powder (the electrodes and the plasma arc are not considered). The boundary conditions (power distribution versus time) are taken from the first calculation. In this study we propose parametric calculations and comparisons with the tests (see Figure 2).

SIMPLIFIED MODELLING

This modelling allows to simulate furnace behaviour in real time. During the test, we can adapt the parameters according to the thermocouple measurements. The goal of this calculation is to estimate the mean temperature in the melting zone to know if the spreading procedure can be started.

This modelling is very simplified as we consider only the powder in one dimension (axisymmetric). The boundary conditions are taken from the two previous modellings.

Figure 1
Furnace meshing

Figure 2
Comparison between test and calculation