A two-dimensional differential transform method is applied to solve one-dimensional phase change problems in a slab of finite thickness, which is subjected to convective thermal loading at one surface and a constant prescribed temperature at the other. In the problems, the initial temperature of the slab does not necessarily have to be the same as the fusion temperature. A series solution is derived for the temperature profile in the melting or solidifying slab with temperature-dependent thermal conductivity and volumetric heat capacity. The latent heat effect of the phase change is incorporated into the temperature-dependent heat capacity. Numerical results demonstrate the effects of the temperature-dependent parameters on the transient temperature profile of the slab.
There is a strong demand for the analyses of heat conduction problems with phase change in a broad range of fields such as ice thermal storage, refrigeration and thawing of foods, freeze dehydration, freeze-drying, cryosurgery, and freeze preservation of living materials. In addition, this type of analysis is important in terms of the quality and productivity estimates of casting products. Unlike normal heat conduction problems, phase change problems are characterised by nonlinearity due to the motion of the change-of-phase front. Thus, exact solutions can be obtained only for a few cases. In particular, when heat flows in both the liquid and solid phases are considered (i.e., the two-phase problem) and/or the object to be analysed is confined to a finite region, the analysis difficulty increases. An excellent textbook and two review articles have been published on the mathematical modelling of phase change problems [
Thus far, only a limited number of researchers have studied the two-phase problem using analytical methods. It should be noted that an important study on a semi-infinite slab by Neumann [
Since the late 1990s, the differential transform method (DTM) has been attracting attention. It is a powerful tool based on Taylor series expansion and is used to explicitly solve not only linear differential equations but also nonlinear ones. This method yields a solution in a simple power-series form. The main advantage of this method is its direct applicability to nonlinear differential equations without requiring linearisation, discretisation, or perturbation. Although there is a criticism that the DTM is purely and solely the traditional Taylor series method [
A brief review of the relevant literature published before 2011 can be found in our previous paper [
In the present paper, the two-dimensional differential transform method is applied to solve one-dimensional transient heat conduction problems with phase change. In particular, the two-phase problem for a slab of finite thickness with temperature-dependent material properties is analysed using the apparent specific heat method [
The basic theory of the two-dimensional (2D) DTM is briefly described here. We consider
Table
Fundamental operations of two-dimensional differential transform.
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Let us consider a slab with finite thickness
Analytical model of one-dimensional phase change process for a finite-thickness slab.
In the present paper, the heat conduction in the slab is analysed on the basis of the apparent specific heat method [
If the volume change during the phase change is neglected (i.e., the density is assumed to be constant), then the heat conduction problem for the slab is formulated as follows:
Temperature dependencies of (a) apparent specific heat and (b) thermal conductivity.
Substituting (
Equation (
Additionally, applying the 2D differential transform to
To verify the presented series solution, the transient heat conduction in a finite slab with temperature-independent material properties is considered, for which an analytical solution exists [
The temperature distributions calculated from the presented series solution and the analytical solution [
Convergence behaviour of the series temperature solution for
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(0.01, 0.4) | 0.00278 | 0.00278 | 0.00043 | 0.00043 | 0.00043 | 0.00011 | 0.00011 |
(0.01, 0.6) | 0.02916 | 0.02916 | 0.00870 | 0.00870 | 0.00870 | 0.00304 | 0.00304 |
(0.01, 0.8) | 0.08296 | 0.08296 | 0.01346 | 0.01344 | 0.01344 | 0.01255 | 0.01255 |
(0.1, 0.0) | 0.00962 | 0.00962 | 0.00056 | 0.00031 | 0.00031 | 0.00003 | 0.00003 |
(0.1, 0.2) | 0.00733 | 0.00733 | 0.00119 | 0.00057 | 0.00057 | 0.00000 | 0.00000 |
(0.1, 0.4) | 0.00055 | 0.00056 | 0.00333 | 0.00078 | 0.00078 | 0.00004 | 0.00004 |
(0.1, 0.6) | 0.01020 | 0.01020 | 0.00974 | 0.00012 | 0.00012 | 0.00005 | 0.00005 |
(0.1, 0.8) | 0.00686 | 0.00686 | 0.00596 | 0.00009 | 0.00009 | 0.00002 | 0.00002 |
Comparison of transient temperature profiles for different Biot numbers: (a)
All the numerical results shown in this subsection are obtained using the following parameter set:
Transient temperature profiles in slab with temperature-dependent specific heat for
Figure
Temperature profiles at
Figure
Transient temperature profiles in slab with different latent heats for
The two-dimensional differential transform method has been employed to solve the nonlinear heat conduction problem with phase change in a finite-thickness slab. The slab was subjected to convective thermal loading at one boundary surface and a constant prescribed temperature at the other boundary surface. In addition, the slab had temperature-dependent thermophysical properties: the thermal conductivity and specific heat (or volumetric heat capacity). The treatment of the phase change was based on the apparent specific heat method. The presented analytical method gives an analytical solution in the form of a power series with easily computable components. Numerical results illustrated that the DTM is useful as a new analytical method for solving the phase change problem in a slab with temperature-dependent parameters.
Our future plans are (i) to apply the present analytical method to the cases of other geometries (e.g., inward and outward solidification in a cylindrical or spherical geometry) and (ii) to adopt a novel treatment for multiple complex nonlinear terms, as given by (
The relationship between the latent heat
Consider a slab of finite thickness
The author declares that there is no conflict of interests regarding the publication of this paper.