Numerical Calculation of the Critical Parameters of Frank-Kamenetskii Equation in Spontaneous Combustion Theory

Abstract

The problem of the critical parameters of the Frank-Kamenetskii equation is a typical bifurcation-point problem. Previously, much research has been done with the direct method for the bifurcation point. However, as this method is too inefficient, it is almost impossible to use it for calculating the critical parameters of three-dimensional spontaneous combustion problems. Here, we develop another, highly efficient, indirect method to solve the problem. In this method, the equation at the central point is replaced by another one, which successfully turns the bifurcation-point problem into a normal one. In the indirect method, what we do in the next step is just correct the critical parameters. At the same time, the matrix equation, Richardson extrapolation, and the alternating-direction implicit method can be used for solving the nonlinear equations obtained by the indirect method. Most important, with this highly efficient indirect method, we are the first to obtain some numerical solutions of critical parameters in three-dimensional spontaneous combustion.