This chapter applies the Fundamental Efficiency Theorem to a central problem in basic Stirling engine design, that of identifying optimal engine geometry. This problem was treated in Chapter 7 for highly idealized engines having theoretical mechanisms, heat exchangers, etc. to produce cycles consisting of four distinct uniform thermodynamic processes. The results in Chapter 7 clearly showed the influence that the type of thermodynamic processes and the level of mechanism effectiveness have on optimum compression ratio and engine output potential.

In this chapter, a more realistic mechanical model of the Stirling engine is employed. It faithfully reflects practical and typical mechanical motions for the piston and displacer. In this setting, optimum values of two parameters are identified which yield maximum brake work output. In the interest of mathematical tractability, the thermal model used here is still highly idealized in that limitations in heat transfer are not considered. Accordingly, it yields best-case results, but allowing for this in a rational way when applying the optima in practical situations can provide an improved guide for first-order design of new engines.

THE GAMMA ENGINE

The analysis is limited here to a particular type of Stirling known as the gamma or split-cylinder. Illustrated in Figure 10.1, the split-cylinder is the simplest of the three main Stirling engine configurations.

Formula (4.2) for the indicated cyclic work of an ideal Stirling engine immediately suggests that output can be increased by charging the workspace with more working gas, keeping everything else the same. This is the motivation behind pressurizing or supercharging an engine. What matters in the end, of course, is whether shaft output improves, and this is a matter of mechanical efficiency.

An easy case to understand at this point is that of an ideal Stirling engine having a constant mechanism effectiveness and optimum buffer pressure. Its mean workspace pressure would be proportional to m, as Formula (3.9) explicitly shows. The Maximum Shaft Work Theorem (4.4) thus implies that if the engine has the charge of its working gas increased by a certain factor, and its buffer pressure adjusted to be optimal for the new charge (in fact, it will need to be increased by exactly the same factor, as Formula (3.4) shows), the shaft output will increase by the same factor. Hence, pressurizing an optimal ideal Stirling in this way will increase output in direct proportion to the charge factor. This kind of pressurization, called system charging, where the workspace and buffer pressure are charged together uniformly by the same factor, produces the same best possible results in many engine and buffer pressure combinations.

Crossley cycles are described by two isometric processes and two polytropic processes of the same kind. The ideal Stirling cycle and the twostroke Otto, or so-called adiabatic Stirling, are special cases. These two cases in fact bracket the spectrum of the four-step cycles that appear to be reasonable idealizations of the actual cycle of real Stirling engines.

Although the ideal Stirling cycle yields the best case analysis, it is a grand idealization of the actual state of affairs in real engines. The isothermal processes present the chief difficulty because of limited heat transfer rates in a real engine. A more realistic model is one in which the isothermal expansion and compression occur at temperatures somewhat displaced from the maximum and minimum engine hardware temperatures; this would model the temperature differential that is necessary to drive the heat transfer to and from the engine gas. This is treated in detail in Chapter 11. In many real engines the expansion and compression processes for the most part occur in engine spaces that have relatively little heat transfer area. Thus, it seems that the expansion and compression processes might be closer to adiabatic than to isothermal. Therefore, using the two-stroke Otto cycle has been advocated as a more faithful, but still idealized, cycle for representing real Stirling engines.

The tables in the booklet complement the property tables in the appendices to Thermodynamics: Concepts and Applications and Thermal-Fluid Science: An Integrated Approach by Stephen R. Turns. In addition to duplicating the SI tables in these books in both SI and US Customary units, the present booklet contains property data for the refrigerant R-134a and properties of the atmosphere at high altitudes.

This chapter continues the examination of the limits on Stirling engine performance by taking into consideration, with the mechanical losses already covered, thermal limitations and losses from which real Stirling engines suffer. First covered is limited heat transfer rate into and out of the working fluid of the engine. This is modeled here just as Curzon and Ahlborn did for Carnot engines (Curzon & Ahlborn, 1975). In addition, introduced later in the chapter is an internal heat leak through the engine from the hot to the cold section governed by the same heat transfer regime. This simulates in a general way the various internal thermal losses occurring in real Stirling engines.

HEAT EXCHANGE

Thermal energy must be transferred into and out of a Stirling engine via heat exchangers at the hot and cold ends. A temperature gradient is required to drive the transfer; in other words, there must be a temperature differential between the source reservoir and the working fluid when it receives thermal energy. Likewise, a temperature difference is required between the engine working substance and the sink reservoir in order for the engine to reject thermal energy. The larger these differences, the greater the rate of energy transfer. This aspect of heat transfer is modeled in a general way by Newton's Law of Cooling (Bejan, 1996b).