## 3 Practical Regenerative Cycle

ideal cycle.

The Stirling cycle is a highly idealized thermodynamic cycle. It consists of four thermodynamic processes including two isothermal expansion and compression processes and two constant-volume processes.

This ideal cycle is discussed in detail in Chapter 2 along with a number of other alternative cycles all of which arc as idealized as the Stirling cycle. Lvngincs cannot be constructed to operate on these idealized cycles. Nevertheless they provide a model for comparison with the way in which practical engines may operate.

It was assumed in Chapter 2 that all the processes of the ideal cycles were thermodynamically reversible, i.e. that the lluid was everywhere at the same instantaneous equilibrium condition. It was assumed also that the processes of compression and expansion were isothermal oi iscn-tropic. The first assumption requires infinite rates of heat transfer between the cylinder walls and the working fluid. The latter assumption, of an isentropic process, requires zero heat transfer between the cylinder walls and the working fluid.

It was further assumed that the whole mass of working fluid in the cycle was, at any particular time, all in the compression space or the expansion space. I'he effects were neglected of any voids in the regenerative matrix, the cylinder clearance space, and any pockets in the cylinder or connecting ducts.

I'he two pistons were caused to move in some idealized discontinuous way to achieve the prescribed working lluid distribution. All aerodynamic and mechanical friction effects were ignored. Finally in most of the ideal cycles discussed in Chapter 2 regeneration was assumed to be perfect. This implies infinite rates of heat transfer between the working lluid and the regenerative matrix and. further, that the heat capacity of the regenerative matrix is infinite. Under such conditions the temperatures of the gas and the matrix arc, at any point, the same and remain constant regardless of the direction of fluid llow.

### Pit ACTICAl CYCLE

In any practical engine, all these factors and others combine to reduce the thermal efficiency to well below the Carnot value of the ideal cycle. The actual thermal efficiency may be quoted as a fraction of the theoretical Carnot efficiency; (his ratio is called the relative efficiency.

A value in excess of 0.4 for the relative efficiency is evidence of a well-designed machine. The maximum achievable value is about 0.7.

To illustrate the discussion of the ideal cycle, a mechanical arrangement was assumed of two opposed pistons, with an interposed regenerator. The two-piston machine is one of several different mechanical arrangements which are to be considered in detail later. One practical version of a two-piston machine is shown in big. 3.1 It consists of a Vee engine, with both pistons coupled to a common crankshaft. The spaces above the pistons constitute the compression and expansion volumes; they are coupled by a duel, containing the regenerator and additional heat exchangers. v

In the operation of this engine, a significant departure from ideality arises as a consequence of the continuous, rather than discontinuous, motion of the pistons. This results (as shown in Fig. 3.2) in a P- V

diagram which is a smooth continuous envelope. The four processes of the ideal cycle are not sharply defined.

The processes of compression and expansion do not take place wholly in one or other of the two spaces, so that three P-V diagrams may he drawn, one for the compression space, one for the expansion space, and one for the total enclosed volume, which includes the 'dead' space. The 'dead' space is defined as that part of the working space not swept by one of the pistons, and includes cylinder clearance spaces, void volumes of the regenerator and other heat exchangers, and the internal volume of associated ducts and ports. The P- V diagram for the expansion space represents the total positive work of the cycle, whereas the diagram for the compression space icpresenls the compression (or negative) work of the cycle. The difference in the areas of these diagrams is the net cycle output, lhe 'indicated' work available for overcoming mechanical friction losses and for providing useful power to the engine crankshaft.

In a cycle where the processes of compression and expansion are isothermal and there are no friction losses, the difference in the area of the expansion- and compression-space diagrams will be found to be exactly equal to the area of the P-V diagram for the. total working space. In a practical engine, of course, this equality does not obtain, because aerodynamic-flow losses in the regenerator and other heat-exchangers cause differences in the pressure of the working fluid in the compression and expansion spaces. Mow losses are important, because (as shown in Fig. 3.3), they cause a decrease in the area of the expansion-space P V diagram, resulting in (a) a decrease in the net cycle output (and, hence, in efficiency) of a prime mover and (b) a decrease in the cooling capacity and the COP of a refrigerating machine.

The sinusoidal piston motion results in the working fluid being distributed in a cyclically time-variant manner throughout various temperature ranges, and it is not possible to draw a meaningful T-S diagram for the tnrnl m:i.w of the work inn fluid. It is oossible to draw T-S diagrams for

Fio. 3.3 Effect of aerodynamic-How loss on engine work, (n) Prcssurr—time diagram for pressure variation in the expansion and compression spaces. The diffcrcnce in pressure is the How loss in the regenerator and other heat exchangers. (1») Pressure -volume diagrams lor the compression and expansion spaces. Ihe hntchcd area on the diagram for the expansion spacc represents Ihe work elfectively lost Ity flow losses in the regenerator and other heat exchangers.

Fio. 3.3 Effect of aerodynamic-How loss on engine work, (n) Prcssurr—time diagram for pressure variation in the expansion and compression spaces. The diffcrcnce in pressure is the How loss in the regenerator and other heat exchangers. (1») Pressure -volume diagrams lor the compression and expansion spaces. Ihe hntchcd area on the diagram for the expansion spacc represents Ihe work elfectively lost Ity flow losses in the regenerator and other heat exchangers.

lure range to another, but no eonvenierti way has been found to combine these multiple diagrams.

The processes of compression and expansion arc not isothermal, another major departure from ideality. In an engine, running at a reasonable speed (say, 1000 rev/min), it is likely that the processes are nearer adiabatic (no beat-transfer) than isothermal (infinite heat-transfer). In ordet to improve the situation special heat exchangers are often incorporated (as shown in Fig. 3.1). including (a) a heater, adjacent to the expansion space, imparting heal to the working fluid, and (b) a cooler, adjacent to Ihe compression space, abstracting heal from the working fluid. Despite the advantages of improved heal transfer, the provision of such heat exchangers imposes some penalties. Additional aerodynamic-How losses are likely, with the consequent deleterious elfect on performance, as discussed above. The dead volume will be increased by the void volume of the heater and cooler, and this has a critical effect on the heated, 1101 only when flowing from the regenerator to the expansion space, hut also when flowing from the expansion space to the regenerator. Similarly, the working fluid is cooled when flowing from, as well as to. the compression space. The provision of one-way How systems is possible, but adds much complication to the machine.

Considerations of increased llow loss and void volume (along with considerations of cost, size, and weight) combine to produce a compromise heat exchanger design. Consequently, substantial differences may exist between the temperatures at which heating (combustion products) and cooling (water or air) is available and the temperatures experienced by the working fluid. This is illustrated diagrammatical!)' in Tig. 3.4. and

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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