It is likely that most questions the reader might pose about turbulent flow have no satisfactory answer - questions like. What is its cause? How can equations as innocuous as the Navier-Stokes momentum equations produce such complex solutions? How can we describe it? How do we predict its properties? and so on. The phenomenon is common experience: turbulent eddying is seen in smoke billowing above a large fire, in dust clouds rising from an explosion, in the wake of a last-moving boat; it is heard in the roar of a jet engine, in the wind rushing over an automobile; it is fell when an airplane bobs up and down in it or when a stiff breeze blows in one's face. Turbulence is an essential element of many processes. A text on fluid mechanics is not complete without a chapter on turbulence. That said, we provide, in this chapter, an introduction to computation of turbulent flow. The reader interested in a more thorough treatment of the subject can consult books entirely devoted to turbulence, such as Pope (2000).

The word turbulence conjures up the notion of randomness. It has entered everyday vocabulary, divorced from the field of fluid mechanics. It evokes images of roiling, churning, and disorder. These are valid definitions, but in fluid flow it is often less severe than vernacular usage suggests. A 10% level of velocity fluctuation may be considered to be substantial. Turbulence is best defined as the irregular component of motion that occurs in fluids when the Reynolds number is sufficiently high. The irregularity may be mild or it may be severe.

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.

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).

In a cyclic heat engine, the mechanism plays a key and complicated role. Its main objective is to transport energy from the working substance to the output shaft. But it also functions to constrain and effect the movement of the piston in order that it carry out a certain thermodynamic cycle. This requires that the mechanism work in a bidirectional fashion. It must transport work from the piston to the flywheel and output shaft during some parts of the cycle, and from the flywheel to the piston in other parts. In practice, it is sometimes even more complex. For example, just after dead center in some engines, both the piston and the flywheel supply work to the mechanism, which is consumed by friction.

For analytic treatment, a comprehensive model of machines that reflects in detail all of the modes in which a mechanism is called upon to function in an engine is the natural first thought. However, such a model quickly becomes exceedingly complex, as the development in Appendix A shows. Rather, the main text of this monograph employs only very basic principles and examines best possible cases. As will be seen as the chapters unfold, a surprising number of interesting and practical insights about ultimate engine performance can be easily deduced through this simple approach.

Although the analysis presented in Chapter 7 is highly idealized, it is quite appropriate for providing some insight into the geometrical requirements of the ultra low temperature differential Stirling engine illustrated in Figures B.1–B.3. Nicknamed the P-19, this engine has proven itself capable of operating down to a temperature difference of just 0.5 °C (less than 1 °F) between its warm and cool sides. The P-19 was the first to run from heat absorbed while resting on the palm of a human hand. The P-19 was first publicly demonstrated at the 25th Intersociety Energy Conversion Engineering Conference held in Reno, Nevada, in August 1990.

BACKGROUND

A low temperature differential (LTD) Stirling engine may be characterized as one that operates more or less optimally with a temperature difference of less than 100 °C between its hot and cold end. Ivo Kolin was the first to design and build such an engine. At the Inter-University Center in Dubrovnik in 1983 he demonstrated the first of his engines operating with hot water as the heat source and cold water as the heat sink (Kolin, 1983). The engine continued to run until the temperature difference between the source and sink dropped to 15 °C.

Kolin's first engine inspired a number of research projects over the next decade to further develop LTD Stirling engines (Senft, 1996).

This book presents a general conceptual and basic quantitative analysis of the mechanical efficiency of heat engines. Typically, treatment of the mechanical efficiency of heat engines has been performed on a case-by-case basis. In ordinary practice, kinematic analysis and computer simulation of specific engine mechanisms coupled with calculated or measured pressure–volume cycles usually can indeed be effectively used for evaluating and locally optimizing engine designs. However, going beyond the specific and local requires broader insights that only a general theory can provide.

No general approach to mechanical efficiency of heat engines had been available until recently. This is in sharp contrast to the situation regarding the thermal efficiency of heat engines. Classical thermodynamics treats the subject of thermal efficiency in great generality. Its results, although obtained in a highly idealized setting, are of profound importance to engine theorists, designers, and practitioners. This book presents a theory of mechanical efficiency at a similar level of ideality and generality.

The first results in this area were published in 1985 and further developed in a series of papers up to the writing of this book. The work modeled the interaction between the mechanical section of an engine and its thermal section at a level compatible with that of classical thermodynamics.