Computation of spatio-temporal temperatures in simple bodies with thermal radiation cooling by means of the numerical method of lines (NMOL)

The objective of the present study is to examine an approximate numerical solution of unsteady heat conduction in simple bodies (wall, cylinder and sphere) with thermal radiation cooling to a nearby sink environment. The finite difference method to be employed is the Numerical Method Of Lines (NMOL), instead of the traditional finite difference methods. One component of NMOL discretizes the spatial derivatives of first and second order in the one–dimensional heat conduction equation using two-point and three-point central finite-difference approximations; both involve truncation errors of second order. This step generates an adjoint system of n ordinary differential equations of first order, which consists in n−1 ordinary differential equations of first order at the interior lines and one nonlinear ordinary differential equation of first order at the surface line. Another component of NMOL deals with the numerical solution of the adjoint system of n ordinary differential equations of first order along with the initial condition with the potent fourth order Runge-Kutta algorithm. The two controlling parameters are the radiation-conduction parameter N_RC and the dimensionless sink temperature ϕ_s. The numerically-computed temperature distributions are channeled through the center, surface and mean temperature distributions for suitable combinations of N_RC and ϕ_s.