Melting Around a Horizontal Heated Cylinder: Part I—Perturbation and Numerical Solutions for Constant Heat Flux Boundary Condition

Abstract

The melting of a solid about a heated cylinder presents an irregularly shaped, moving boundary problem. A transformation is used to immobilize this boundary—replacing the problem of variable geometry by one of constant geometry. A constant heat flux boundary condition is used along the cylinder surface. Using perturbation and numerical methods, several solutions for this transient problem are generated for Stefan, Rayleigh, and Prandtl numbers of Stq = 0.374, Ra = 5000, and Pr = 54. Stq is the ratio of heat transfer rate to the thermal energy needed to melt the solid. Ra • B3 is the measure of the magnitude of the natural convection effect, where B is a dimensionless measure of the size of the melt region called the gap function. Ra itself can be thought of as a dimensionless heat flux, since it does not take the size of the melt region into account. The dimensionless groups Stq and Ra (based upon the surface heat flux) are used to determine two parameter expansions of the dependent variables for the regular perturbation method. The first three terms of the series solutions are determined. They provide accurate solutions for short times after the start of melting, for small values of Stefan and Rayleigh numbers. The accuracy of the perturbation method is verified using a numerical method, which is not limited to short initial time intervals or to small values of Stefan and Rayleigh numbers. Detailed predictions of the melt volume, shape, temperature field, global and local heat transfer rates are given for representative cases. Comparisons with earlier experimental results are made.