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Research Papers

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

[+] Author and Article Information
Kwee-Yan Teh

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305kteh@sandia.gov

Christopher F. Edwards

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305cfe@stanford.edu

J. Dyn. Sys., Meas., Control 130(4), 041008 (Jun 09, 2008) (10 pages) doi:10.1115/1.2936864 History: Received March 01, 2007; Revised February 19, 2008; Published June 09, 2008

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.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

The dynamical system model of a homogeneous, adiabatic piston engine, defined by the state variables U, V, N, and scalar control variable q (the rate of volume change)

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Figure 2

The slider-crank input qsc, states V, U, Nfuel, entropy difference LS∕LS,0=1−ξ, pressures P, Peq, and their difference, all as functions of time t

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Figure 3

The optimal solution for problem (9) without the minimum-volume constraint, showing control q*, states V*, U*, Nfuel*, entropy difference (also a Lyapunov function) LS*∕LS,0=1−ξ*, pressures P*, Peq*, and their difference, all versus time t

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Figure 4

The optimal solution for problem (9CR) with an active minimum-volume constraint, showing the optimal control q*, states V*, U*, Nfuel*, entropy difference LS*∕LS,0=1−ξ*, and pressures P* and Peq* as functions of time t

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Figure 5

(Top) The entropy generated by the optimal systems (solid lines) compared to that by the slider-crank system (dashed line), all plotted as functions of 1−ξ0, which, at optimality, equals the entropy difference LS*∕LS,0. The nonoptimal slider-crank Sgen,sc is not distinguishable from the optimal Sgen,CR* line at the scale used on the left plot. (Bottom) The quantity Sgen,sc−Sgen,CR*, which is the excess entropy generated in the nonoptimal slider-crank system.

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Figure 6

The diagram of U versus S showing the upper bound (the quasistatic limit Uqs) and the lower bound (the minimum-entropy limit USV−eq*) for the final internal energy Ufe* at optimality—and, thus, the corresponding limits for the maximum engine work output Wfe*. The limits are deduced based on the solution to the original entropy minimization problem, problem (9CR). Its final state is shown here as a circle at (UCR*(tf),SCR*[tf]). The convexity of the constant-volume equilibrium line is exaggerated to distinguish it from its tangent.

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