Technical Note
Exact solution for a Stefan problem with latent heat a power function of position

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Abstract

A one-phase Stefan problem with latent heat a power function of position is investigated. The second kind of boundary condition is involved, and the surface heat flux is considered as a corresponding power function of time. The problem can be viewed as a special case of the shoreline movement problem under the conditions of nonlinear variation of ocean depth and a surface flux that varies as a power of time. An exact solution is constructed using the similarity transformation technique. Theoretical proof for the existence and the uniqueness of the exact solution is conducted. Solutions for some special cases presented in the literature are recovered. In the end, computational examples of the exact solution are presented, and the results can be used to verify the accuracy of general numerical phase change algorithms.

Introduction

There are many problems that involve a moving boundary in industrial processes, and this type of problem is usually named as the Stefan problem (or the moving boundary problem). Many analytical and numerical solutions for these problems can be found in the monographs [1], [2], [3], [4], [5].

Recently, a special type of Stefan problem with space-dependent latent heat attracts much attention. This type of problem generally arises from the study of the shoreline movement. Numerical solutions can be found in [6], [7]. Voller et al. [8] presented an exact solution for a one-phase Stefan problem with linearly distributed latent heat. Lorenzo-Trueba et al. [9] generalized Voller’s problem by considering the nonlinearity of the diffusivity and the appearance of two moving boundaries; they presented analytical solutions for simple cases and numerical solutions for the general condition. Salva et al. [10] extended Voller’s solution to a two-phase Stefan problem, and they also considered the linearly distributed latent heat.

Motivated by these works, we consider a one-phase Stefan problem with latent heat a power function of position. Mathematical equations for this problem are given byTt=ν2Tx2,0<x<s(t),t>0,Tx=s(t)=0,t>0kTxx=0=-ct(n-1)/2,t>0,kTxx=s(t)=-γs(t)ns(t),t>0where T is the temperature, x is the position coordinate, t is the time coordinate, s(t) is the moving interface, v is the thermal diffusion coefficient, k is the thermal conductivity, ct(n-1)/2 is the time-varying surface heat flux (c > 0 for melting, c < 0 for freezing), γxn is the variable latent heat per unit volume (γ > 0 for melting, γ < 0 for freezing), n is an arbitrary non-negative integer, and the phase-transition temperature is zero.

The Stefan problem described by Eqs. 1,2,3,4can be viewed as a special case of the shoreline movement problem under the conditions of nonlinear variation of ocean depth and a surface flux that varies as a power of time. In this problem, time-dependent surface flux of the form (3) is considered so that a similarity solution can be obtained; this is similar to Lombardi & Tarzia [11], in which the authors also uses the time-dependent surface flux in arriving a closed-form solution.

The main objective of this paper is to obtain an exact solution for the one-phase Stefan problem presented by Eqs. 1,2,3,4. Section 2 constructs an exact solution using the similarity transformation technique, proves the existence and the uniqueness of the solution, and recovers solutions of some special cases. Section 3 presents computational examples of the exact solution, followed by Section 4, with some conclusions.

Section snippets

Solution procedure

Using the similarity transformationf(η)=T(x,t)tn/2,withη=x2νtThe partial differential equation for T(x, t) becomes the following ordinary differential equation for f(η)f+2ηf-2nf=0The solution for this ordinary differential equation can be written asf(η)=Ainerfc(η)+Binerfc(-η)where A and B are arbitrary real constants, inerfc(  ) are the repeated integrals of the complementary error function [12] defined byerfc(ε)=2πεexp(-x2)dx,erf(ε)=1-erfc(ε),i0erfc(ε)=erfc(ε)i-1erfc(ε)=2πexp(-ε2),inerfc(ε)=

Computational examples

The exact solution for Eqs. 1,2,3,4 is constructed in Section 2. From the solution procedure, we know that λ is important. Once λ is determined, the exact solution can be obtained from Eqs. (10),(11), (14), (15).

As indicated in [8], the exact solution provides a worthwhile benchmark for verifying general numerical algorithms. In order for other researchers to use the solution more conveniently, we present predictions of λ in this section.

The nonlinear equation of λ is solved by Newton’s method,

Conclusions

In this short communication, a special type of one-phase Stefan problem is investigated. The novel feature in the problem is a latent heat that distributes as a power of position. In order to obtain a closed-form solution of a similarity type, time-dependent surface flux of the form ct(n-1)/2 is considered. The problem can be viewed as a special case of the shoreline movement problem under the conditions of nonlinear variation of ocean depth and a surface flux that varies as a power of time. An

Acknowledgments

This research was supported by National Natural Science Foundation of China (Grant Nos. 51204164, 51204170), National Science Foundation for Post-doctoral Scientists of China (Grant Nos. 2011M500969, 2011M500974), Fundamental Research Funds for the Central Universities (Grant Nos. 2013QNA18, 2011QNA16). Special thanks to the reviewers, for their suggestions which have greatly improved the paper.

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