Unsteady contact melting of a rectangular cross-section material on a flat plate

Abstract

The work in this paper concerns a mathematical model of the contact melting process of a rectangular material in contact with a hot plate. The problem is described by a coupled system of heat equations in the solid and melt layer, fluid flow in the melt, a Stefan condition at the melt interface, and a force balance between the weight of the solid and the fluid pressure. Since the melt layer remains thin throughout the process, we use the lubrication approximation to the fluid equations and assume that the heat flow in the fluid is dominated by conduction across the thin film. In the solid we employ a heat balance integral method. Results show that the film height has initial and final rapid increases, whereas for intermediate times the height slowly increases. The quasisteady state of previous models is never attained: This is shown to be an effect of neglecting the change in mass and conduction in the solid. The previously observed initial infinite velocity of the melt is shown to be a result of the perfect thermal contact assumption. For a water-ice system the melting rate is shown to be approximately linear, this allows us to reduce the problem to solving a single first order differential equation for the liquid layer thickness. The main analysis is carried out in two dimensions, but we briefly highlight the extension to three dimensions. The method is verified by comparison with previously published experimental results on the melting of n-octadecane.

Optimal exponent heat balance and refined integral methods applied to Stefan problems

“…The Schrödinger equation is related to the heat equation but has a complex diffusivity. The HBIM has recently been applied to modelling the temperature in a thermistor [10], the process of contact melting [19], the Korteweg-de-Vries equation [18] and the ignition time of wood [25]. A much more comprehensive list is given by Hristov [9].…”

Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems

“…However, in the limit β → 0, the exact solution (23) has α ∼ ln(1/β), see [1], but the HBIM/RIM profiles do not predict this dependence. Again using (30,31), it can be shown that as β → 0, a/β ∼ 6. This explains why the accuracy in Tables 2 and 3 significantly increases as β decreases, which can also be seen in Figure 2.…”

“…Note the difference between this form and that obtained from the standard analysis (30). Again we could write down another six formulations but the solution given by (55) is the most accurate for realistic β.…”

“…The analysis subject to cooling conditions follows in the same manner. A similar finite domain problem is the analysis of the contact melting process of a phase change material (PCM) in contact with a hot plate [30]. This leads to melting of the PCM, so a fluid layer forms between the two surfaces.…”

“…Given that for ice we are only likely to consider ∆u < 100 K, using data from Table 1, this requires β > 0.73. In which case the method described above, leading to equations (30), (31), is generally the most accurate.…”