Technical Brief

Solution of the Phase Change Stefan Problem With Time-Dependent Heat Flux Using Perturbation Method

[+] Author and Article Information
Mohammad Parhizi

Mechanical and Aerospace Engineering Department,
University of Texas at Arlington,
500 W First Street, Room 211,
Arlington, TX 76019

Ankur Jain

Mechanical and Aerospace Engineering Department,
University of Texas at Arlington,
500 W First Street, Room 211,
Arlington, TX 76019
e-mail: jaina@uta.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 20, 2018; final manuscript received October 22, 2018; published online December 13, 2018. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 141(2), 024503 (Dec 13, 2018) (5 pages) Paper No: HT-18-1243; doi: 10.1115/1.4041956 History: Received April 20, 2018; Revised October 22, 2018

Theoretical understanding of phase change heat transfer problems is of much interest for multiple engineering applications. Exact solutions for phase change heat transfer problems are often not available, and approximate analytical methods are needed to be used. This paper presents a solution for a one-dimensional (1D) phase change problem with time-dependent heat flux boundary condition using the perturbation method. Two different expressions for propagation of the phase change front are derived. For the special case of constant heat flux, the present solution is shown to offer key advantages over past papers. Specifically, the present solution results in greater accuracy and does not diverge at large times unlike past results. The theoretical result is used for understanding the nature of phase change propagation for linear and periodic heat flux boundary conditions. In addition to improving the theoretical understanding of phase change heat transfer problems, these results may contribute toward design of phase change based thermal management for a variety of engineering applications, such as cooling of Li-ion batteries.

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Grahic Jump Location
Fig. 1

Schematic of the one-dimensional phase change problem with time-dependent heat flux. The schematic shows solid-to-liquid phase change, but the opposite process can also be analyzed in the same framework.

Grahic Jump Location
Fig. 2

Comparison of the present analytical result with past results for the special case of constant heat flux: (a) plot of nondimensional phase change front location, y(t) as a function of nondimensional time, t for the present work and three past results [9,10,19]. Results from FEM simulation are also shown for comparison. (b) Comparison of present model with Tao [9] for a larger time period for two different values of α.

Grahic Jump Location
Fig. 3

Validation of the present work with finite element simulation for linear, time-varying heat flux: (a) phase change front y(t) as a function of t for linear gt=A+Bt. The value of A is taken to be 5000 and values of B are shown in the legend. (b) Temperature distribution as a function of x for the specific case of B = 4.56 × 107. Both plots show very good agreement between the analytical model and finite element simulation.

Grahic Jump Location
Fig. 4

(a) Plot of phase change front y(t) as a function of time for periodic gt=A(1+cos(εt)) for multiple values of the nondimensional frequency, ε. (b) Plot of the variation in phase change front propagation for different values of thermal diffusivity for constant heat flux case.



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