Part VII – July 1968 - Papers - A Modified Heat of Fusion for Use in the Mathematical Formulation of Solidification Processes

The accuracy of the method of steady-state approximation applied to the problems of heat transfer involving phase change (London and Seban's solution) is improved by defining a "modified heat of fusion" which accounts for the effect of specific heat. An analytical expression for this modified heat of fusion is derived in terms of the actual heat of fusion, specific heat, and surface temperature. Numerical solution for the freezing of iron indicates that the ratio of the modified heat of fusion to the actual heat of fusion increases from 1.04 to 1.4 as the surface temperature decreases from 2700° to 1600°F. The heat transfer process accompanying solidification is complicated owing to the moving solid-liquid interface where the latent heat of fusion is released. Exact analytical solution is available only for a few special cases of freezing problems such as Stefan's1 problem. In a commonly used steady-state approximation method, London and seban's2 solution, the effect of specific heat is neglected. This approximation, however, may introduce some degree of error in mathematical analysis when the temperature of the solid surface is much lower than the freezing temperature, as in the case of freezing metals. In a recent work by Robertson and Schenck,3 the problem of freezing a semi-infinite region with finite surface conductance was programed in a digital computer through the finite difference method. A family of curves for the correction of the London-Seban solution was given in a dimensionless plot for different ratios of heat of fusion and specific heat effect; it was shown that the London-Seban solution when applied to the freezing of a mild steel slab under some particular conditions gave results in error by about 10 pct in the solidification rate. Although the Robertson-Schenck computer result has improved the London-Seban solution to a great extent, lack of generality and the tedious computing procedure have restricted its application. In this paper a "modified heat of fusion" which accounts for the effect of specific heat is defined, and is used instead of the actual heat of fusion in the London-Seban solution. An analytical expression of this modified heat of fusion in terms of the specific heat, surface temperature, and actual heat of fusion is derived by comparing the solutions of Stefan and London-Seban for the case of freezing of a semi-infinite region with constant surface temperature. It can also be extended to the case with finite surface conductance, and the results show good agreement with the Robertson-Schenck computer solution. The expression of the modified heat of fusion, the accuracy of the London-Seban solution, and the temperature distribution within the solid are presented graphically in dimensionless form. Numerical solution for the freezing of iron indicates that the ratio of the modified heat of fusion to the actual heat of fusion increases from 1.04 to 1.40 as the surface temperature decreases from 2700° to 1600°F.