An Optimal Control Approach to Minimizing Entropy Generation in an Adiabatic Internal Combustion Engine

Abstract

Entropy generation due to combustion destroys as much as a third of the theoretical maximum work that could have been extracted from the fuel supplied to an engine. Yet, there is no fundamental study in the literature that addresses the question of how this quantity can be minimized so as to improve combustion engine efficiency. This paper fills the gap by establishing the minimum entropy generated in an adiabatic, homogeneous combustion piston engine. The minimization problem is cast as a dynamical system optimal control problem, with the piston velocity profile serving as the control input function. The closed-form switching condition for the optimal bang-bang control is determined based on Pontryagin’s maximum principle. The switched control is shown to be a function of the pressure difference between the instantaneous thermodynamic state of the system and its corresponding equilibrium thermodynamic state at the same internal energy and volume. At optimality, the entropy difference between these two thermodynamic states is shown to be a Lyapunov function. In thermodynamic terms, the optimal solution reduces to a strategy of equilibrium entropy minimization. This result is independent of the underlying combustion mechanism. It precludes the possibility of matching the piston motion in some sophisticated fashion to the nonlinear combustion kinetics in order to improve the engine efficiency. For illustration, a series of numerical examples are presented that compare the optimal bang-bang solution with the nonoptimal conventional solution based on slider-crank piston motion. Based on the solution for minimum entropy generation, a bound for the maximum expansion work that the piston engine is capable of producing is also deduced.