Hybrid computer solution of thermal energy equations in porous fuels

AbstractA serial (discrete-time continuous-space) method is employed to solve the unsteady-state energy equations in porous systems on a hybrid computer. The nonlinear, coupled partial-differential equations are solved by replacing the time derivatives with backward finite-difference approximations. To increase the order of accuracy in the time increment of the solution, the Crank-Nicholson scheme is used. The resulting difference-differential equations are solved in the direction opposite to that of the fluid flow to eliminate computational instability. The average temperatures over the consecutive time steps are solved on the analog portion of the hybrid computer. Solutions of the present time step are obtained by combining the analog solutions with those of the previous time step stored in the digital computer. The commonly encountered, mixed boundary conditions are satisfied by using a steepest descent iteration scheme based on least-squares-error minimization. A so-called binary-search technique provides reasonable initial trial values from which the iteration process converges. The trial values are improved by making use of the parameter influence coefficients that are obtained by taking finite differences through a number of test runs at the beginning of the solution and are taken to be constant during the entire solution time. In most cases, the iteration process converges in two to three iterations per boundary value searched. Comparisons of the hybrid computer solutions agree with those obtained by other numerical methods on a digital computer within 1%.