On the Development and Validation of the Double Integral Method for Spherical Solidification of Metals

In this paper we developed the double integral method to solve the solidification of liquid metals in spherical geometry with a high Stefan number, a novelty for this type of problem. We provided extensive numerical validation against published numerical and experimental data of Milanez (1982). Using a quadratic temperature profile, we obtained solidification time solutions by deriving the heat conduction equation. Hence, we intend to improve the numerical accuracy against the traditional simple integral method in the literature. Initially, the derived equations were compared against the analytical solidification solutions of Tao (1967) for problems with different Biot (∞; 2; 10) and Stefan (0,5; 1,0; 2,0) numbers. Overall, the double integral method with the quadratic profile performed better than the single integral method with a quadratic profile; for Bi = 10 and Ste = 2, the reduction in solidification time relative error using the double integral method was 92,73% compared to the simple integral method. Compared to the experimental data of lead and tin, both cooled by air and water, the double integral method performed better than the single integral method for all Biot and Stefan conditions; for tin cooled by water, the double integral method was 20,06% more accurate in predicting the solidification time compared to the single integral method. Furthermore, we evaluated that using a Gaussian Quadrature integrator led to a reduction in computational cost up to 3 orders of magnitude compared to traditional Runge-Kutta methods. Therefore, our studies attested to the correctness and suitability of the developed double integral method for applications in the solidification of metals.