Enhanced framework for solving general energy equations based on metropolis-hasting Markov chain Monte Carlo

Abstract

Due to the widespread presence of heat and mass transfer phenomena in industrial applications, numerous studies have been devoted to the accurate solution of energy equations, providing a foundation for the analysis of heat and mass transfer processes in practical applications. In this study, a Green's Function Markov Superposition Monte Carlo (GMSMC) for solving general energy equations has been developed based on probability and statistical principles owing to its advantageous features of insensitivity towards dimension and geometric complexity as well as the capability to handle multiple integrals in complex domains. The energy equation is first decomposed, and corresponding probability models are established for each component, considering their interrelationships. Subsequently, a solution framework for solving the general energy equation is constructed by integrating these probability models based on a Markov chain structure. The mathematical principles and formulae of the proposed method are derived in detail. The performance of the proposed method is validated by several heat transfer systems with different combinations of boundary conditions and features, which mainly include the distribution of the internal heat source and whether the convection or transient term is included. Results of the validation show that the temperatures obtained by the proposed method are in good agreement with the FEM based on a fine grid, no matter whether the calculation is for a single point or a distribution.