Modelling and Simulation in Engineering

Volume 2019, Article ID 7468478, 9 pages

https://doi.org/10.1155/2019/7468478

## Optimal Stroke Path for Reciprocating Heat Engines

The Arab Academic Institute of Education, Beit-Berl College, Kfar Saba 44905, Israel

Correspondence should be addressed to Mahmoud Huleihil; moc.liamg@ana.dumham

Received 3 August 2018; Revised 30 October 2018; Accepted 21 November 2018; Published 2 January 2019

Academic Editor: Farouk Yalaoui

Copyright © 2019 Mahmoud Huleihil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By testing piston motion in reciprocating heat engines as a control variable, one could find piston trajectories, different from the conventional near sinusoidal motion that should increase power production. This results from minimizing frictional losses. The purpose of this study is to determine piston trajectories that are optimal for noncombustion strokes in reciprocating engines, in the sense of minimizing frictional dissipation and hence maximizing efficiency and power. The optimal piston traces for noncombustion strokes are determined by using a combination of optimal control theory and models for the thermodynamic irreversibilities. Hence, the results are germane to external combustion engines and to the noncombustion strokes of internal combustion engines. The optimal piston traces or trajectories obtained here can be viewed as some of the building blocks from which optimal overall cycles can be constructed.

#### 1. Introduction

Internal combustion engines [1–3] are studied using the methods of finite-time thermodynamics [4, 5]. Two modeling approaches are pursued to optimize heat engines for maximum power production: optimization for a given trajectory or thermodynamic cycle [6–16] and optimization seeking optimal trajectory by applying the methods of optimal control theory (Pontryagin’s maximum principle) [17, 18, 19].

One possible way to increase the power delivery of reciprocating heat engines is to vary the piston trajectory relative to its conventional near sinusoidal motion [17]. The idea of treating piston motion as a control variable in an attempt to minimize thermodynamic losses and hence maximize power production was raised in [17, 18], in which attempts were made to find the optimal piston trajectory for Otto and Diesel cycle engines. The exercise was performed by using optimal control theory in concerts with models for the key thermodynamic irreversibilities [17, 18]. An important finding in these works was that the potential improvement in the efficiency of the reciprocating engines is not negligible and could be as high as 10–15% of current engine efficiencies. The improvements stem from lowering frictional dissipation, and heat leaks in the internal combustion engine cycles considered.

These earlier analyses, however, adopted oversimplified models for the influence of combustion processes on engine dynamics [20, 21]. Two other limitations were that (1) the heat generated due to friction was modeled as being totally transferred to the engine cooling system, namely, none of that resulted in the heating of the engine working fluid and (2) the only type of friction considered was the rubbing friction of a piston well-lubricated cylinder surface, where power dissipation goes as the square of piston speed.

The aim of this study is to determine piston trajectories that are optimal for noncombustion strokes in reciprocating engines, in the sense of minimizing frictional dissipation and hence maximizing efficiency and power. In this study, we determine the optimal piston trajectories for the strokes of reciprocating heat engines, in the sense of maximizing power production. However, we restrict our analyses to noncombustion strokes due to the complex nature of modeling combustion and its influence on engine performance. The complex models dictate pure numerical solutions. Among the benefits of doing this is the capability of the analytical solution to enable more explicit and transparent results. Hence, our results are directly applicable to external combustion engines and to the noncombustion strokes of internal combustion engines.

Although we restrict the study to noncombustion strokes, the same methods of optimal control theory could be used to optimize combustion strokes as was done in [17, 18, 19].

The two key irreversibilities modeled are friction and heat leak, and a range of friction sources are considered: mechanical and/or fluid friction. In addition, our results account for frictional dissipation heating the engine working fluid.

The piston trajectories we determine could hence be viewed as some of the building blocks from which one can calculate the optimal piston motion for various engine cycles. In addition, one would have to calculate the fraction of the total cycle time allotted to each stroke to tailor the results to the particulars of any engine cycle under consideration. These are the cycle-specific calculations.

The achievable maximum power with these optimal piston trajectories will be compared to the power that can be attained with conventional near sinusoidal piston motion. Furthermore, sensitivity studies on important engine parameters such as compression ratio, maximum piston acceleration, type of friction (i.e., functional dependence on piston speed), and the degree to which frictional dissipation heats up the engine working fluid are performed. The potential improvements in engine power for the strokes analyzed here shown to be of the order of few percent.

#### 2. Formulation of the Problem and Modeling Assumptions

Figure 1 is a schematic of the system analyzed here: a piston moving inside a cylinder of fixed, given cross-sectional area *A*. The piston moves along the vertical axis, and the volume *V* of the working fluid can change from a maximum value of *V*_{max} to a minimum value of *V*_{min}, with the engine compression ratio *r* defined as . Piston motion as a function of time during the stroke is described by the time evolution of the gas volume , or, equivalently, piston position . Piston velocity is described by . For the convenience of analysis in what follows, piston velocity is defined as positive for a given stroke. The steady-state operation is only analyzed: namely, start-up and turnoff transients are not considered.