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.

Internal combustion engines [

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 [

These earlier analyses, however, adopted oversimplified models for the influence of combustion processes on engine dynamics [

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 [

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.

Figure _{max} to a minimum value of _{min}, with the engine compression ratio

Schematic of the reciprocating engine cylinder.

The general picture could also include a regenerator inside the cylinder, as well as two-piston cylinders, as illustrated in Figure

Schematic of alternative design (Stirling cycle), with a regenerator inside the two-piston cylinder.

The ideal-gas approximation is adequate for most engine operating conditions and permits closed analytical solutions to be derived. The formalism described below can also be used for real gas behavior, but results would then have to be generated numerically [

The types of noncombustion engine strokes included in our analysis are compression, expansion, and constant volume. The two former types of strokes are found in all engine cycles, and the latter is relevant for Stirling cycles, among others.

The heat leak is assumed to occur with constant thermal conductance

The rate at which work is dissipated as friction is taken to be comprised of a static term and a dynamic term and is given by

In its most general form, the optimization problem can be stated as follows. We aim to maximize the useful work _{1} of three variables: piston position

The expression for

A constant volume stroke need not trivially mean a stationary piston. For example, in the Stirling and related cycles, which are double-piston engines for each cylinder, both pistons can move together at the same speed such that no net change occurs in gas volume, yet there are frictional dissipation and heat leak on these strokes (Figure

The dynamic constraint for the time evolution of _{2} of

We will also need to incorporate a realistic upper bound,

The problem can be solved with optimal control theory [

Following the procedures of optimal control theory [

These equations can be arranged in the two coupled equations and are given by

Although the initial state of our system is known, the energy at the end of the stroke is not known. Therefore, an additional boundary condition is required [

In summary, we know the initial conditions

Now, we have a set of ordinary differential equations and boundary conditions that can be solved to yield the optimal motion

For most practical reciprocating engines, on noncombustion strokes, heat leaks are negligible compared to the friction losses and/or

Static friction does not affect the optimal solution; only the dynamic friction term can influence the optimal path. To see this in the governing equations, integrate equations (

As a basis for later comparison, consider the conventional piston motion, which is approximated here as having a sinusoidal velocity profile (see Figure

Piston velocity plotted against time, for one engine (expansion) stroke for (1) conventional sinusoidal motion; (2) optimal motion, externally dissipative friction, unbounded acceleration; (3) optimal motion, externally dissipative friction,

Consider the case of externally dissipative friction; namely,

Strictly constant piston velocity, however, would require infinite acceleration at the start of the stroke and infinite deceleration at the end of the stroke. As shown in [

The optimal piston motion for externally dissipative friction is described by the velocity profile [

The work dissipated due to friction can now be evaluated for conventional piston motion and optimal piston motion. It is convenient to define nondimensional acceleration,

The optimal switching time

The optimal switching time for this case depends on maximum permissible acceleration only (Figure

Switching time as a function of

As noted above in Section

Figure

Ratio of work dissipated as friction with conventional piston motion relative to optimal piston motion, plotted against maximum piston acceleration, for two values of frictional exponent

Ratio of work dissipated as friction with conventional piston motion relative to optimal piston motion, plotted against frictional exponent

Figure

This case corresponds to solving the governing equations with

On constant volume strokes, of the type often encountered in Stirling cycles, with

In the more general case of expansion or compression strokes, the solution to equation (

Illustrations of optimal motion and conventional piston motion, when friction is dissipated internally, are presented in Figures

Piston velocity plotted against time, for one engine (expansion) stroke for (1) conventional sinusoidal motion; (2) optimal motion, internally dissipative friction, unbounded acceleration; (3) optimal motion, internally dissipative friction,

Example of sensitivity of optimal piston motion to compression ratio

Recall that to determine the potential improvement due to optimal piston motion, one needs to evaluate only the reduction in dynamic frictional dissipation. Dynamic frictional dissipation for the conventional and the optimal trajectories can be evaluated via numerical integration or analytically with the evaluation of hypergeometric functions.

Graphical illustrations of the magnitude of this improvement and how it depends on key system parameters are presented in Figures

Sensitivity to compression ratio of reduction in frictional losses. Calculations are for unbounded acceleration. Ordinate is dynamic frictional losses for conventional piston motion relative to that for optimal piston motion, for internally dissipative friction. Abscissa is the compression ratio. The two curves are for different frictional exponents

Sensitivity of reduction in frictional losses to maximum piston acceleration. Ordinate is dynamic frictional losses for conventional piston motion relative to that for optimal piston motion, for internally dissipative friction. Abscissa is maximum piston acceleration relative to its maximum value for the conventional trajectory. The two curves are for different frictional exponents

The potential improvement in engine efficiency is slightly larger than that for the externally dissipative case. The influence of compression ratio is modest, whereas the functional form of frictional losses, and piston maximum acceleration, can have a marked effect on the relative improvement in engine power.

One prospective method to increase the efficiency and power of reciprocating engines is to reduce frictional losses by modifying piston motion. The purpose of this paper (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 potential improvements in overall engine efficiency have also been evaluated and should lie in the range of around 2–10%, depending on how severe frictional losses were prior to the modification of piston motion. The optimized engine strokes that we have calculated could serve as building blocks in the construction of optimal cycles in external combustion of optimal cycles in the external combustion engines and for noncombustion strokes in internal combustion engines.

Several qualifications are in order. First, no experimental verification has yet been attempted. Second, specific functional forms for frictional losses have been assumed. However, these forms appear to cover the behavior of real engines [

Optimal piston motion turns out to be sensitive to where the heat generated by frictionally dissipated work goes: either to an external cooling system (externally dissipative), or to heating the engine working fluid (internally dissipative), or, in reality, somewhere between these two extremes. We have derived closed-form analytic solutions for optimal piston motion for both cases and compared them to conventional sinusoidal piston paths.

The improvement in engine efficiency for the optimal paths, relative to conventional piston motion, turns out to be mildly sensitive to compression ratio, with a significant effect arising from the functional form of frictional losses and from maximum piston acceleration. Only when piston accelerations in excess of those achievable with conventional sinusoidal motion can be reached will the improvements in engine efficiency be substantial. Fortunately, these improvements in engine efficiency increase rapidly with maximum attainable acceleration for values just above those of conventional motion and then asymptote (Figures

The mechanisms to achieve the optimal piston motion and the experimental verification of the optimal piston motion are two good research ideas for future research.

The data used to support the findings of this study are included within the article.

The author declares that he has no conflicts of interest.