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Combined Coriolis and MHD Effects in a Melt-Crucible Model Of © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Numerical analysis of conjugate heat transfer in a melt-cruciblemodel of Czochralski systems: combined Coriolis and MHD effects on time-dependent3D melt convection E.M. Smirnov, A.G. Abramov, N.G. Ivanov, A.B. Korsakov, D.K. Zajtsev Department of Aerodynamics, ,%PetersburgState Technical Universiq, Russia Abstract 3D unsteady convection of low-Prandtl-number liquid (Pr = 0.015) in a model Czochralski crystal growth system is numerically simulated on the base of conjugate heat transfer formulation. Computations using the full Navier-Stokes equations with no turbulence model have been carried out for the Rayleigh number of 5x10’. Effects of crucible rotation and stationary axial magnetic field on the convection are investigated. 1Introduction The majority of monocrystalline silicon wafers used m electronics are produced by the Czochralski (CZ) method with the crystal pulled vertically from the melt. The Si-melt convection determines directly the heat transfer through the melt and the crystallisation rate at the melt-crystal interface (e.g. Muller [l]).Natural convection is combined with the effects induced by crystal rotation and by rotation of the crucible containing the melt. Axisymmetric models for the analyses of melt convection and heat transfer has been in wide use since the end of the 80’s. However, to predict flow instabilities and melt turbulence, only 3D time-dependent calculations of CZ melt flow may be applied. A number of attempts have been reported on this subject (see Basu et al. [2] and references therein). © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 As judged from the literature, previous computations of 3D unsteady phenomena during the CZ crystal growth were based on segregated formulations for melt convection. The segregated formulations require the use of proper approximations defining thermal boundary conditions at the melt-crucible interface. As a rule, an axisymmetric distribution of wall temperature is assumed. Solutions that are closer to the reality can be obtained considering a conjugate heat transfer problem. Setting such a problem may start from the suggestion that unsteady phenomena attributed to instability of the melt are concentrated in the hot zone including the melt, the silica crucible and the graphite support. To control the melt convection and heat transfer during the industrial CZ process the application of steady magnetic fields seems to be rather attractive. This is a motivation for performing both experimental and numerical studies of magnetohydrodynamic (MHD) effects on melt convection (e.g. Grabner et al. [3]). Sure, MHD effects add to the complexity of the problem. A great deal of effort has been put into numerical simulation of magnetoconvection for CZ configurations, but most of the contributions are restricted to the axisymmetric formulation only. However, the magnetic fields used in practice are usually not strong enough to suppress 3D instabilities in the melt completely (e.g. Vizman et al. [4]). The present work is aimed at getting an experience in numerical simulation of 3D unsteady magnetoconvection in a model CZ configuration on the base of conjugate heat transfer formulation. 2 Computational model The domain geometry and boundary conditions used for the present heat transfer calculations are typical for CZ crystal growth (Figure la). A low-Prandtl-number liquid (melt) is placed in a thin-wall crucible of radius R, made of material with a comparatively low thermal conductivity. We assume h,A, = 0.08, where h, and h, are the crucible and the melt thermal conductivities respectively. The crucible wall thickness, 6, is taken constant. A support of radius Rsupand height Hsupis also included into the computational domain. The ratio of the support thermal conductivity, hsup,to the melt conductivity is taken as 0.7. A concave melt- crystal interface of radius R, occupies the central part of the melt upper boundary. At Y > R, the melt free surface is flat, while for R, 5 r 5 R, the typical meniscus geometry is assumed. The vertical distance from the lowest bottom point to the flat free surface, H, is taken as the length scale. The crucible and crystal rotate about the vertical axis with the angular velocities Q and Cl,, respectively. To nondimensionalise heat fluxes, the reference heat flux, qreb is introduced, given by the following expression: qref= h,31(p2gP3H6),where p, is the melt density, g is the gravitational acceleration, p is the coefficient of thermal expansion. Consequently, the reference temperature difference, AT, may be taken as the ratio q,,,Hlh,,,. As one can see below, this choice of reference © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advmcmi CompututiodMethods in HatTrumf?v 109 values ensures that the normalised temperature, 0 = (T-TJIAT, is of the order of unity, here T, is the temperature of the melt-crystal interface. The buoyancy velocity, V, = (gj3ATH)”’, is considered as the velocity scale. The ratio HIV, serves as the time scale. Figure 1: (a) The solution domain (b) Multiblock computational grid. Considering as a prototype the EKZ-1300 CZ furnace used for 100 mm Si- crystal growth (e.g. Evstratov et al. [.5]), basic geometrical ratios can be taken as RJH = 1.7,RJH = 0.6, RJH = 0.5, WH = 0.08, R,,,/H = 2.0, Hsup/H=1S. Three-dimensional melt convection is computed on the base of the full Navier-Stokes equation with no turbulence model. The governing equations accounting for the effects of a stationary magnetic field on the melt motion are written under the assumption of small magnetic Reynolds number, Re, = pBoHVb<< 1, where pB is the magnetic permeability, o is the electric conductivity of the melt. With this conventional approximation the back influence of the melt motion on the magnetic field is neglected. In the present work the MHD effects on the melt motion are studied for the case of a pure vertical magnetic field, with constant axial component of the magnetic induction vector, B. With given geometry and thermal boundary conditions, the behaviour of the melt is determined by tie Pmndtl number, Pr= vla (v is the kinematic viscosity,a is the thermal diffusivity), tie Rayleigh number, Ra = gpATH’/(va), the Rossby number, Ro = Vb/RcH,the Stuart number, N = oB2H/(pmVb),and the angular velocity ratio, QJQ. Using the Boussinesq’s approach for incorporation of the gravity buoyancy effects, a vector formulation of the momentum equation can be written in a reference rotating together with the crucible as follows: ---vp*+*AV-@egdV -~(eaxV)+N[-V@+VxB]xB. - dt &G Ro © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Here p* is the reduced pressure, Q is the electric potential. Via the term depending on the normalised temperature, 0, eqn (1) is coupled with the energy equation (not written). Zero relative velocity conditions on the crucible wall are imposed, while on the rotating disk the velocity distribution is given by V,,= -(Qs + Q)Y.Zero shear stresses are imposed on the free surface. Heat flux boundary distributions are adopted to be uniform. The support is heated from the periphery with non- dimensionalised flux q,,= ~.OX~O-~.Heat flux leaving the free surface, g,, is twice g,,. This ratio of heat fluxes was chosen to get Q,,= 1.l& where Qhis the heat rate supplied to the support, and Q, is the heat rate leaving the free surface. At these conditions, the heat rate supplied to the melt-crystal interface is one order less than the heat rate passing through the melt, that is typical for the growth process. The Poisson equation governing the electrical potential distribution, A@ = B.(V x V), is solved on the assumption that all boundaries are electrically insulating. Computations in the framework of the above formulation have been carried out using a well-validated in-house code (named SINF). This advanced 3D Navier-Stokes solver is based on the second-order finite-volume spatial discretisation using the cell-centred variable arrangement and body-fitted block- structured grids. The discretisation of the time derivatives is done with a three- time-level, second-order implicit scheme. The artificial-compressibility method is applied at each time step. A QUICK-type upwind scheme is employed to compute convective fluxes. The code is in wide use for solving both fundamental and industrial problems, in particular, it has been used for simulation of 3D turbulent convection during industrial-scale CZ Si-crystal growth (Evstratov et al. [5]). Additional details for the solver can be found elsewhere [6]. The computations were performed with Ra= 5x105, Pr = 0.015. The domain occupied by the melt was covered by a three-block grid consisting of 79,872 cells, with nodes clustered to the wall and to the free surface. Six additional blocks of 29,952 cells covered the crucible and the support. The grid is shown in Figure lb. As a result of preliminary computations the non- dimensional time step of0.25 has been chosen. 3 Crucible rotation effects This section describes results obtained at computations of cases with no magnetic field action.
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