A Method for the Solution of Heat Transfer Problems With a Change of Phase

Abstract

A method of broad applicability is presented which can be used to obtain solutions to problems involving a phase change. The solution in one of the phases is specified as a known single-phase solution; an inverse analysis then determines the solution for the other phase. Two problems are studied: The first yields the similarity solution for the planar geometry and the second gives the exact solution to a more general problem. Convergence is shown and error bounds are given. The method can accommodate convection, heat generation, variable properties, nonplanar, and multidimensional systems.

An analytical self-consistent determination of a bubble with a deformed cap trapped in solid during solidification

“…[2]: Ϫ1Ϫ(␤ ln 2)/3, which are determined by introducing ϭ 0 at ϭ 0. Equation [15], with a positive sign, then becomes R* ϭ 1 (Ϫ1 ϩ 2 ln 1 2 ) ͬ ␤ ϭ 0, [17] dp g dt Ṽ ϩ p g dṼ dt ϭ dñ g dt R u T [23] namely, b ϭ 1 6 (1 ϩ 2 ln 2) where temperature is considered as the constant melting temperature. The total volume of the pore yields Equation [15], with positive and negative signs, should be Ṽ (t) ϭ Ṽ w (t) ϩ Ṽ c (t) [24] continuous at ϭ 1 ϩ ␤ (1 ϩ 2 ln 2)/6, where ϭ 1.…”

“…[11] Briefly speaking, they are (1) an axisymmetric system, (2) ideal gases and a lumped system in the bubble or pore, (3) ignorance of convection, [22,23,24] (4) physico-chemical equilibrium at the cap sur-…”

Heat Transfer Model for Thin Solidified Material in Continuous Casting

“…However, this method is limited in the assumption of Ste ( 1 to decompose the expansion function. Fredrick and Greif 19) proposed the inverse method for a general magnitude of Ste. However, in this method, the expansion function is based on the Taylor expansion on the S.F.…”

“…In the perturbation method, the expansion function is generally assumed to depend only on variable $ (¼ h s À x 2 ) under the assumption of Ste ( 1; [15][16][17][18][19] temperature distribu-tion T s ½$ was expressed as the n-th power of Ste. Because of the nature of a power function, n is limited in the second order at maximum from the view of stability of the function shape; 5) the shape of the final approximated function could be too simple to express the possible complex curve of T s distribution that is caused with, for example, the existence of impinging liquid flow on a specific area of the S.F.…”

“…2) the perturbation method, [15][16][17] the inverse method, [18][19][20] and the spectral method. 21) In these methods, the original governing equations are approximately expressed by a sum of a series of expansion functions of x 2 and t; each expansion function could be decided analytically under the given boundary conditions.…”