ON THE APPLICATION OF ADI METHODS TO PREDICT CONJUGATE PHASE CHANGE AND DIFFUSION HEAT TRANSFER

Abstract

In this article, the process of phase-change heat transfer is investigated. To simplify the representation of the interface between phases, an enthalpy method is implemented to describe the energy conservation process. This equation describes an equivalence of the time rate of change of the continuous-phase quality x to the diffusive energy transport via a Laplacian of the continuous temperature field T. Because of the mixture of terms (x , T) , relaxation techniques are usually employed to solve the discrete energy equation. In the present study these equations are extended for implementation of an alternating direction implicit (ADI) method with the main goal of decreasing computational wall clock time. The phase quality formulation allows a discretized model to possess both phase-change nodes and solid diffusion nodes, the former defaulting to basic diffusion above and below the melt point of the material in question. In the present study, one set of equations is used to represent all thermal nodes. By relating the temperature to the phase quality as a linear function of x, the discretized equations are converted to the matrix form Ax = f, where x is the phase quality vector and is of a form suitable for ADI implementation. This produces the sought-after conversion from a temperature solution to a quality-based solution. The sweep equations are developed for both the 3-D splitting and 3-D Brian ADI methods. Because the overall heat transfer process involves transient solidification or melting, the time-space variations in the quality field cause the diagonals of A to change in time, forcing an additional LDU decomposition at each time step. These incremental expenses of an ADI implementation can be significant; however, for the case study presented, the overall ADI technique is nearly 30 times faster than conventional relaxation methods. The current method is validated against analytical solutions of 2-D flat and cylindrical solidification front propagation. A pseudo-code is presented to assist the reader in algorithmic implementation.