Lattice Boltzmann method for heat conduction problems with chemical reactions

Mathematical modeling of heat conduction in a polymer material is carried out taking into account exothermic and endothermic reactions in a three-dimensional formulation. The boundary value problem of conductive heat transfer was formulated in terms of the mesoscopic lattice Boltzmann method (LBM). The discretization of the Boltzmann equation was carried out using the D3Q7 scheme. In this case, the single relaxation time scheme proposed by Bhatnagar-Gross-Krook was used to approximate the collision integral. The numerical simulation results were verified by comparison with the reference data obtained by the traditional finite difference method (FDM). The heat conduction equation was approximated by explicit schemes of the second order in space. For the convenience of analyzing exothermic and endothermic reactions, the power of internal volumetric sources of heat release and absorption was set in the range of -105 ≤ qv ≤ 105. It was revealed that the temperature in the core of the material increases under the conditions of an exothermic reaction and decreases when the endothermic one is taken into account over time. It was found that at a relatively short time of 100 s, the heat effect of the endothermic reaction is insignificant. As a result, the value of the temperature in the core is practically equal to the initial one and the energy transfer occurs only at the boundaries of the material. It is shown that the atypical lattice Boltzmann method reproduces the temperature fields similar to the typical finite difference method. In all considered cases, LBM gives correct temperature profiles, and deviations of local values are within 5%. Such an error may be due to an implicit conversion of macroscopic boundary conditions of the first kind by means of a mesoscopic distribution function. It was also found that the lattice Boltzmann method significantly loses in the execution speed of the computational program to the traditional method of finite differences when the number of nodes is more than 613