An Error Analysis of Numerical Solutions of the Transient Heat Conduction Equation

摘要

In this investigation the transient temperatures in a semi-infinite slab with a convective boundary layer are determined numerically using the Crank-Nicolson and Crandall Methods. These temperatures are compared with the exact solution to determine if the Crandall technique, as theoretically predicted, has smaller truncation error. A finite difference boundary equation is derived for the convective boundary condition which is consistent with the internal node formula developed by Crandall. The latter formula and the derived boundary equation are used to obtain six node temperatures at six time intervals for values of M (inverse Fourier modulus) equal to 1, 2, 3, 4, and the square root of 20; and N (Nusselt modulus) equal to 2, 1/2, 1/4, and 1/8. The results show that these temperatures are always more accurate than those obtained using the Crank-Nicolson Method if the nodes are closely spaced (N equal to 1/8). In addition, the solutions for those values of M and N which permit the boundary equation truncation error to be minimized indicate that the Crandall Method is more accurate if N is less than or equal to 1/2. The accuracy improvement factors for the two numerical methods are also determined; however, the results do not agree with those predicted theoretically and show no consistent trends. Also, the equation required to apply the Crandall technique to bodies with variable thermal properties is derived. (Author)