Phase-change problems with variable surface temperatures.

Abstract

Simplified procedures are described for the approximate solution of heat-transfer problems with phase change for time-dependent surface temperatures. For constant surface temperature, the specified problem reduces to classical form and may be solved analytically by the method of Neumann. Comparisons between our approximate solutions and numerical solutions, for time-independent and for time-dependent surface temperatures, show that our techniques yield estimates for the rate of movement of the interface between the phases which become progressively better as the complexity of the simplified approximation is increased. I. Outline of Theoretical Considerations N ONSTEADY calculations dealing with high-temperature ablation may be formulated either as problems with specified heat flux or as problems in which the surface temperature is a known (measured) function of the time. Both types of problems are generally not solvable analytically. Furthermore, when the nonsteady heat-transfer problems are coupled to the (steady) external flow, a complex iteration to establish matching of the interface may be required for the time-dependent heat transfer. In problems of this type, it may be justified to utilize simplified approximations in order to reduce the complexity of the required (numerical) analysis. In this paper, we consider the development of simplified, approximate procedures for a classical phase-change problem, namely, that of heat transfer to a system with phase change in which the surface temperature is an arbitrary function of the time. The one-dimensional problem involving surface heating and a phase change is described schematically by the diagram sketched in Fig. 1. In the notation of Carslaw and Jaeger,1 we let vi(x, t) represent the temperature profile of phase 1 and v%(x, t) that of phase 2. Here x denotes the linear distance, and t is the time; the constant thermal diffusivities are identified as KI and /c2, respectively, whereas the constant thermal conductivities are KI and K2. The heat absorbed or liberated, per unit mass of material, at the phase boundary is =FL, respectively; the density is assumed to be the same for both phases and is denoted by p. Hence, the enthalpy increment per unit volume for the phase change becomes =Fl/p. As is fully discussed in Ref. 1, the requisite boundary-value problem is described by the following differential equations, boundary conditions, and interface conditions: