[Paper Review] Explicit solution for Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face using Kummer functions

Julieta Bollati, Domingo A. Tarzia|arXiv (Cornell University)|Oct 28, 2016

Phase Change Materials Research4 citations

TL;DR

This paper presents an explicit analytical solution for the one-dimensional Stefan problem with position-dependent latent heat and a convective boundary condition at the fixed face, using confluent hypergeometric (Kummer) functions. The solution is derived by transforming the moving-boundary problem into a boundary value problem via a change of variables, and the key contribution is the closed-form expression of the moving interface and temperature distribution using special functions, enabling exact analysis of phase change dynamics under non-uniform latent heat and convective conditions.

ABSTRACT

An explicit solution of a similarity type is obtained for a one-phase Stefan problem in a semi-infinite material using Kummer functions. Motivated by [D.A. Tarzia, Relationship between Neumann solutions for two phase Lam\\'e-Clapeyron-Stefan problems with convective and temperature boundary conditions, Thermal Sci.(2016) DOI 10.2298/TSCI 140607003T, In press], and [Y. Zhou, L.J. Xia, Exact solution for Stefan problem with general power-type latent heat using Kummer function, Int. J. Heat Mass Transfer, 84 (2015) 114-118], we consider a phase-change problem with a latent heat defined as a power function of the position with a non-negative real exponent and a convective boundary condition at the fixed face $x=0$. Existence and uniqueness of the solution is proved. Relationship between this problem and the problems already solved by Zhou and Xia with temperature and flux boundary condition is analysed. Furthermore it is studied the limit behaviour of the solution when the coefficient which characterizes the heat transfer at the fixed boundary tends to infinity. Numerical computation of the solution is done over certain examples, with a view to comparing this results with those obtained by general algorithms that solve Stefan problems.

Motivation & Objective

• To address the lack of explicit solutions for Stefan problems with position-dependent latent heat and convective boundary conditions.
• To model phase change processes where latent heat varies spatially, reflecting real materials with temperature-dependent or spatially heterogeneous energy absorption.
• To extend classical Stefan problem solutions by incorporating convective heat transfer at the fixed boundary, improving physical realism.
• To derive a closed-form analytical solution using Kummer functions for both the moving interface and temperature distribution.
• To provide a mathematically rigorous framework for analyzing transient heat transfer in melting/freezing processes with complex thermal properties.

Proposed method

• Transform the moving-boundary Stefan problem into a fixed-domain problem using a change of variables that maps the moving interface to a fixed coordinate.
• Apply the method of lines or similarity transformations to reduce the partial differential equation to an ordinary differential equation in the transformed variable.
• Use confluent hypergeometric (Kummer) functions as fundamental solutions to the resulting ODE due to their analytical tractability and known asymptotic behavior.
• Incorporate the convective boundary condition at the fixed face by applying a Robin-type condition in the transformed domain.
• Derive the moving interface position as a function of time by enforcing the integral energy balance and Stefan condition in the transformed coordinates.
• Verify the solution by checking consistency with physical boundary conditions and limiting cases.

Experimental results

Research questions

• RQ1 Can an explicit analytical solution be derived for the Stefan problem when latent heat depends on position and a convective boundary condition is applied at the fixed face?
• RQ2 How do Kummer functions facilitate the solution of the transformed boundary value problem in this context?
• RQ3 What is the functional form of the moving interface position in time under position-dependent latent heat and convective boundary conditions?
• RQ4 How does the inclusion of convective heat transfer affect the dynamics of the phase change process compared to standard Dirichlet or Neumann conditions?
• RQ5 What are the conditions under which the solution remains physically valid and mathematically well-defined?

Key findings

• An explicit analytical solution is derived for the moving interface position as a function of time, expressed in terms of Kummer functions.
• The temperature distribution in the liquid phase is obtained in closed form using confluent hypergeometric functions, ensuring mathematical precision.
• The solution satisfies the Stefan condition and the convective boundary condition at the fixed face exactly, without numerical approximation.
• The position-dependent latent heat is incorporated into the energy balance via a variable coefficient in the heat equation, which is handled through the transformation method.
• The method allows for the determination of the interface velocity and temperature profiles without iterative or numerical schemes.
• The solution is valid under specific parameter constraints, and its convergence and physical consistency are verified through analytical checks.