On the application of the double integral method with quadratic temperature profile for spherical solidification of lead and tin metals

Abstract

• High Stefan and Biot solidification problem considered and modelled. • The proposed method achieved a reduction of 92.73% compared to previous approaches. • Gaussian Quadrature is 3 orders of magnitude less expensive compared to Runge-Kutta. In this paper we developed the double integral method to solve the solidification problem of liquid tin and lead metals in spherical geometry with a high Stefan and Biot numbers. The solidification time solutions were obtained by deriving the heat conduction equation using a quadratic temperature profile. The set of equations were solved using a Gaussian Quadrature approach; a novelty for this problem. Hence, we intend to improve the numerical accuracy against the traditional simple integral methods in the literature. 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, 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.