Mathematical Problems in Engineering

Volume 2015, Article ID 637852, 8 pages

http://dx.doi.org/10.1155/2015/637852

## Determination of One Unknown Thermal Coefficient through a Mushy Zone Model with a Convective Overspecified Boundary Condition

^{1}CONICET, Departamento de Matemática, Facultad de Ciencias Empresariales, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina^{2}Departamento de Matemática, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Pellegrini 250, S2000BTP Rosario, Argentina

Received 8 May 2015; Accepted 9 June 2015

Academic Editor: Mohsen Torabi

Copyright © 2015 Andrea N. Ceretani and Domingo A. Tarzia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A semi-infinite material under a solidification process with the Solomon-Wilson-Alexiades mushy zone model with a heat flux condition at the fixed boundary is considered. The associated free boundary problem is overspecified through a convective boundary condition with the aim of the simultaneous determination of the temperature, the two free boundaries of the mushy zone and one thermal coefficient among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat, and the two coefficients that characterize the mushy zone, when the unknown thermal coefficient is supposed to be constant. Bulk temperature and coefficients which characterize the heat flux and the heat transfer at the boundary are assumed to be determined experimentally. Explicit formulae for the unknowns are given for the resulting six phase-change problems, besides necessary and sufficient conditions on data in order to obtain them. In addition, relationship between the phase-change process solved in this paper and an analogous process overspecified by a temperature boundary condition is presented, and this second problem is solved by considering a large heat transfer coefficient at the boundary in the problem with the convective boundary condition. Formulae for the unknown thermal coefficients corresponding to both problems are summarized in two tables.

#### 1. Introduction

Heat transfer problems with a phase-change such as melting and freezing have been studied in the last century due to their wide scientific and technological applications. Some books in the subject are [1–9].

In this paper we consider a phase-change process for a semi-infinite material, which is characterized by , that is initially assumed to be liquid at its melting temperature (which without loss of generality we assume to be equal to C). We consider this material under a solidification process with the presence of a zone where solid and liquid coexist, known as “mushy zone,” with a heat flux boundary condition imposed at the fixed face . We follow [10, 11] in considering three different regions in this type of solidification process:(1)Liquid region at temperature : .(2)Solid region at temperature : .(3)Mushy region at temperature : . and are the functions that characterize the free boundaries of the mushy zone. We also follow [10] in making the following assumptions on the structure of the mushy zone, which is considered as isothermal:(1)The material contains a fixed portion of the total latent heat per unit mass (see condition (3)).(2)Its width is inversely proportional to the gradient of temperature (see condition (4)).Thermal coefficients involved in the solidification process are assumed to be constant. They are : latent heat by unit mass, : thermal conductivity, : mass density, : specific heat, : one of the two coefficients which characterize the mushy zone, : one of the two coefficients which characterize the mushy zone, : coefficient that characterizes the heat flux at , : coefficient that characterizes the heat transfer at , : bulk temperature at .

We suppose that five of the six thermal coefficients , , , , , and of the solid phase are known and that, by means of a change of phase experiment (solidification of the material at its melting temperature), we are able to measure the quantities , , and .

Encouraged by the recent works [12, 13] and with the aim of the simultaneous determination of temperature , the two free boundaries and , and one unknown thermal coefficient among , , , , , and , we impose an overspecified boundary condition [2] which consists of the specification of a convective condition at the fixed face (see condition (7)) of the material undergoing the phase-change process. This leads us to the following free boundary problem:

This problem was first studied in [11] with a temperature boundary condition at instead of the convective condition (7) considered in this paper. Moreover, the determination of one unknown thermal coefficient for the one-phase Lamé-Clapeyron-Stefan problem with an overspecified heat flux condition at the fixed face without a mushy zone was done in [14]. Other papers related to determination of thermal coefficients are [15–40].

The goal of this paper is to obtain the explicit solution to the phase-change process (1)–(7) with one unknown thermal coefficient independent of position and time and the necessary and sufficient conditions on data in order to obtain an explicit formula for the unknown thermal coefficient. In addition, we are interested in analysing the relationship between problem (1)–(7) and the phase-change process given by (1)–(6) besides the Dirichlet boundary condition overspecified at given by (31) (see below). In particular, we are interested in solving the problem with Dirichlet boundary condition through problem with convective boundary condition when large values of the coefficient that characterizes the heat transfer at are considered.

The organization of the paper is as follows. In Section 2 we prove a preliminary result where necessary and sufficient conditions on data for the phase-change process (1)–(7) are given in order to obtain the temperature and the two free boundaries and . Based on this preliminary result, in Section 3, we present and solve six different cases for the phase-change process (1)–(7) according to the choice of the unknown thermal coefficient among , , , , , and . In Section 4 we discuss the relationship between the phase-change process (1)–(6) with the Dirichlet boundary condition (31) and the same process with the convective boundary condition (7). We show that temperature , free boundaries and , and the explicit formula for the unknown thermal coefficient , , , , , or for the phase-change process (1)–(6) with the Dirichlet condition (31) can be obtained through the phase-change process with convective condition given by (1)–(7) when tends to . Explicit formulae for the unknown thermal coefficient for problems (1)–(7) and (1)–(6) and (31), besides restrictions on data that guarantees their validity, are summarized in Tables 1 and 2, respectively.