A Series Solution for Heat Conduction Problem with Phase Change in a Finite Slab

Ryoichi Chiba

doi:10.1155/2014/684293

Abstract

A two-dimensional differential transform method is applied to solve one-dimensional phase change problems in a slab of finite thickness, which is subjected to convective thermal loading at one surface and a constant prescribed temperature at the other. In the problems, the initial temperature of the slab does not necessarily have to be the same as the fusion temperature. A series solution is derived for the temperature profile in the melting or solidifying slab with temperature-dependent thermal conductivity and volumetric heat capacity. The latent heat effect of the phase change is incorporated into the temperature-dependent heat capacity. Numerical results demonstrate the effects of the temperature-dependent parameters on the transient temperature profile of the slab.

Highlights

There is a strong demand for the analyses of heat conduction problems with phase change in a broad range of fields such as ice thermal storage, refrigeration and thawing of foods, freeze dehydration, freeze-drying, cryosurgery, and freeze preservation of living materials
The heat conduction in the slab is analysed on the basis of the apparent specific heat method [25]. This formulation allows for a continuous treatment of a system undergoing phase transition because the latent heat effect of the phase change is included in the apparent specific heat of the substance
The temperature distributions calculated from the presented series solution and the analytical solution [31] are shown in Figure 3 for different Biot numbers

Summary

Introduction

There is a strong demand for the analyses of heat conduction problems with phase change in a broad range of fields such as ice thermal storage, refrigeration and thawing of foods, freeze dehydration, freeze-drying, cryosurgery, and freeze preservation of living materials. It is a powerful tool based on Taylor series expansion and is used to explicitly solve linear differential equations and nonlinear ones This method yields a solution in a simple power-series form. Ndlovu and Moitsheki applied the one-dimensional DTM and two-dimensional DTM to steady [21] and transient [22] heat conduction analyses for fins of variable thickness with temperature-dependent parameters, respectively. The two-dimensional differential transform method is applied to solve one-dimensional transient heat conduction problems with phase change. The two-phase problem for a slab of finite thickness with temperature-dependent material properties is analysed using the apparent specific heat method [25].

Two-Dimensional Differential Transform Method

Heat Conduction Problem with

Numerical Results and Discussion

Concluding Remarks

Analytical Solution for Linear Case