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• Author: E. Węgrzyn-Skrzypczak
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Abstract

The paper presents an approach of numerical modelling of alloy solidification in permanent mold and transient heat transport between the casting and the mold in two-dimensional space. The gap of time-dependent width called "air gap", filled with heat conducting gaseous medium is included in the model. The coefficient of thermal conductivity of the gas filling the space between the casting and the mold is small enough to introduce significant thermal resistance into the heat transport process. The mathematical model of heat transport is based on the partial differential equation of heat conduction written independently for the solidifying region and the mold. Appropriate solidification model based on the latent heat of solidification is also included in the mathematical description. These equations are supplemented by appropriate initial and boundary conditions. The formation process of air gap depends on the thermal deformations of the mold and the casting. The numerical model is based on the finite element method (FEM) with independent spatial discretization of interacting regions. It results in multi-mesh problem because the considered regions are disconnected.

Open access

Abstract

The paper is focused on the modeling of the directional solidification process of pure metal. During the process the solidification front is sharp in the shape of the surface separating liquid from solid in three dimensional space or a curve in 2D. The position and shape of the solid-liquid interface change according to time. The local velocity of the interface depends on the values of heat fluxes on the solid and liquid sides. Sharp interface solidification belongs to the phase transition problems which occur due to temperature changes, pressure, etc. Transition from one state to another is discontinuous from the mathematical point of view. Such process can be identified during water freezing, evaporation, melting and solidification of metals and alloys, etc.

The influence of natural convection on the temperature distribution and the solid-liquid interface motion during solidification of pure copper is studied. The mathematical model of the process is based on the differential equations of heat transfer with convection, Navier-Stokes equation and the motion of the interface. This system of equations is supplemented by the appropriate initial and boundary conditions. In addition the continuity conditions at the solidification interface must be properly formulated. The solution involves the determination of the temporary temperature and velocity fields and the position of the interface. Typically, it is impossible to obtain the exact solution of such problem. The numerical model of solidification of pure copper in a closed cavity is presented, the influence of the natural convection on the phase change is investigated. Mathematical formulation of the problem is based on the Stefan problem with moving internal boundaries. The equations are spatially discretized with the use of fixed grid by means of the Finite Element Method (FEM). Front advancing technique uses the Level Set Method (LSM). Chorin’s projection method is used to solve Navier-Stokes equation. Such approach makes possible to uncouple velocities and pressure. The Petrov-Galerkin formulation is employed to stabilize numerical solutions of the equations. The results of numerical simulations in the 2D region are discussed and compared to the results obtained from the simulation where movement of the liquid phase was neglected.