A Numerical Algorithm Using Multizone Adaptive Grid Generation for Multiphase Transport Processes with Moving and Free Boundaries

The primary objective of this study is to develop a numerical scheme for accurate and eecient simulation of phase change and transport processes of industrial importance. These processes may include a variety of heat transfer and ow mechanisms in irregularly shaped domains with moving and/or free boundaries. Based on the multizone adaptive grid generation (MAGG) technique 1], a curvilinear nite volume scheme has been developed to discretize the governing equations. The combination of these two techniques provides a powerful tool for numerical modeling of complex transport processes. Several problems are considered to demonstrate the applicability and accuracy of the proposed method. They are (a) natural convection in a diierentially heated eccentric annuli, (b) solidiication of a pure material in a rectangular enclosure, (c) solidiication in an open cavity with shrinkage due to volume change, and (d) Czochralski crystal growth of silicon. The predictions show a good agreement with experimental data, much better than the previously reported numerical solutions. 1 Nomenclature Ar aspect ratio, H=L b partial source term Bo Bond number, gL 2 == C p speciic heat (J kg ?1 K ?1) D diameter ~ e unit vector F integral function Fr Froude number, U 2 i =gL g square of the Jacobian, acceleration due to gravity (m s ?2) g ij covariant metric tensor G weight function for grid orthogonality Gr Grashof number, gL 3 (T h ? T f)== 2 H mean curvature h length of the curve I unit tensor J uxes Ja Jacobian k interface index, thermal conductivity (W m ?1 K ?1) L distance in x-direction (m) M grid inertial coeecient Ma Marangoni number, (@=@T)(T w ? T f)L=== Max maximum Min minimum n normal unit vector p pressure (Pa) P pressure diierence (Pa) Pr Prandtl number, == 2 r radial distance, r =L s solidiication interface or melt meniscus S source R radius of the cylindrical enclosure (m) Re Reynolds number, U i L== Ste l liquid Stefan number, C pl (T w ? T f)=h f Ste s solid Stefan number, C ps (T f ? T si)=h f T temperature (K) t time, t U i =L t tangential unit vector u velocity in axial direction, u =U i u velocity vector v velocity in radial direction, v =U i W weight function x; y Cartesian coordinate, x =L and y =L Greek Symbols geometric coeecient, thermal diiusivity (m 2 s …