A Fast Numerical Simulation for Modeling Simultaneous Metal Flow and Solidification in Thin Cavities Using the Lubrication Approximation

Abstract

A numerical algorithm for modelling steady flow of liquid metal accompanied by solidification in a thin cavity is presented. The problem is closely related to a die cast process and in particular to the metal flow phenomenon observed in thin ventilation channels. Using the fact that the rate of metal flow in the channel is much higher than the rate of solidification, a numerical algorithm is developed by treating the metal flow as steady in a given time-step while treating the heat transfer in the thickness direction as transient. The flow in the thin cavity is treated as two dimensional after integrating the momentum and continuity equations over the thickness of the channel, while the heat transfer is modelled as a one-dimensional phenomenon in the thickness direction. The presence of a moving solid-liquid interface introduces non-linearity in the resulting set of equations, and which are solved iteratively. The location and shape of the solid-liquid interface are found as a part of the solution. The staggered grid arrangement is used to discretize the flow governing equations and the resulting set of partial differential equations is solved using the SIMPLE algorithm. The thickness direction heat-transfer problem accompanied by phase change is solved using a control volume formulation. The results are compared with the predictions of the commercial software FLOW3D® which solves the full three-dimensional set of flow and heat transfer equations accompanied with solidification. The Reynolds's lubrication equations accompanied by the through-the-thickness heat loss and solidification model can be successfully implemented to analyze flow and solidification of liquid metals in thin channel during the die cast process. The results were obtained with significant savings in CPU time.

Notes

¹The typical velocity corresponds to the end of the die cast process, after the main cavity is filled and liquid metal is in the ventilation channel.

²FLOW3D® [23] is a general purpose commercial CFD software which solves three-dimensional fluid-flow and solidification problems using the finite different approximation. FLOW3D utilizes the volume-of-fluid technique and the FAVOR method to track free surfaces as well as solid-liquid interfaces. The two equation k-ϵ model is used to resolve the turbulent properties of the flow. The averaged Navier–Stokes equations coupled with the energy equation allow the software to achieve an accurate solution for turbulent metal flow undergoing solidification.

³Derivation of Eq. (16) is for two-dimensional variation of the cavity thickness h. The second term of the equations has a denominator of 6 instead of 12 in reference [12].

⁴The mesh densities are for solving the z-averaged velocity fields along x and y directions.