Spherical solidification: An application of the integral methods

In this paper we developed and applied the simple and double integral methods to solve phase-change problems. We performed the double integral method considering a linear temperature profile applied to spherical geometry, thus obtaining an analytical solution for the solidification time, a novelty for this type of problem. The methods were evaluated considering different solidification conditions with finite and infinite Biot numbers (∞; 2, 5 and 10) and Stefan numbers (0.0061; 0,01). The results were compared against published data, and the performance between the linear analytical double integral method (DIM) and simple integral method (SIM) methods were compared against the computational quadratic SIM of Milanez and Ismail (1984). These results were further compared against our implemented computational quadratic DIM. Hitherto, similar comparisons have been neglected in the literature. Overall, the results indicated that both methods were capable of reproducing the physics of solidification. However, the linear double integral method achieved better agreement in terms of solidification time against the published data. The reduction in numerical error in computing the solidification time using the linear double integral method was up to 84.61% compared to the simple integral (for Bi = 2 and Ste = 0.01). For the infinite condition, the quadratic SIM in the literature did not present improvements against the analytical linear DIM. In general, for a Bi = 2 the analytical linear DIM and the quadratic DIM are equivalent. For higher Stefan and Biot numbers the computational quadratic methods are more accurate due to the inclusion of the Stefan number in the temperature coefficients. This study attested to the correctness of the mathematical development using the linear temperature profile in conjunction with the double integral method for solving the solidification process in spheres. This study can be extended in future work to the solidification of different material.