Hamilton-Jacobi-Bellman equations and dynamic programming for power-optimization of a multistage heat engine system with generalized convective heat transfer law

A multistage endoreversible Carnot heat engine system operating between a finite thermal capacity high-temperature fluid reservoir and an infinite thermal capacity low-temperature environment with generalized convective heat transfer law q ^∝(ΔT)^m] is investigated in this paper. Optimal control theory is applied to derive the continuous Hamilton-Jacobi-Bellman (HJB) equations, which determine the optimal fluid temperature configurations for maximum power output under the conditions of fixed initial time and fixed initial temperature of the driving fluid. Based on the universal optimization results, the analytical solution for the Newtonian heat transfer law (m=1) is also obtained. Since there are no analytical solutions for the other heat transfer laws (m≠1), the continuous HJB equations are discretized and dynamic programming algorithm is performed to obtain the complete numerical solutions of the optimization problem. The relationships among the maximum power output of the system, the process period and the fluid temperature are discussed in detail. The results obtained provide some theoretical guidelines for the optimal design and operation of practical energy conversion systems.