Numerically solving a transient heat conduction problem with convection and radiation

Abstract

The transient surface temperature distribution is determined for the flat plate and sphere subjected to cooling by combined convection and radiation. In the study, the initial boundary value problem is reduced to a singular nonlinear Volterra integral equation of the second kind using the integral transform method. Several numerical techniques are introduced in an attempt to find an approximate solution of the problem: The method of successive approximations, the Runge-Kutta method, and the finite difference method. The integral equation is solved numerically by the Runge-Kutta method of orders 1, 3, and 5. In addition, the finite difference method is implemented to solve the initial boundary value problem, and the solutions are compared with those generated by the Runge-Kutta method. All the numerical results are presented graphically. Limitations and difficulties involved in these schemes are discussed. At the end, a numerical algorithm for solving the problem is proposed.