Chapter 4d - Ideal Adiabatic Solution - The D-90 Ross Yoke-drive Engine

The 90 cc D-90 engine is fully described in Andy Ross' fascinating book Making Stirling Engines (pages 41 - 45). At Ohio University we have a laboratory model of the D-90 Yoke drive engine which is heated electrically in order to accurately determine the heat power input. In this section we examine the results of performing an Ideal Adiabatic simulation of the D-90 engine under specific typical operating conditions as follows

• Mean operating pressure - pmean = 2 bar.
  (The crankcase is sealed, and the output shaft power is obtained by a magnetic coupling. Andy typically pressurizes the crankcase with a bicycle pump to about 2 bar.)

• Cooler temperature 27 degrees Celsius (300 K), and heater temperature 650 degrees Celsius (923 K)

• Operating frequency 50 Hz. (Note that the Ideal Adiabatic model is independent of operating speed - all results are presented per cycle)

In order to simulate the engine by means of the Ideal Adiabatic model equation set given previously, we require the equations for the Yoke-drive volume variations and derivatives Vc, Ve, d Vc and d Ve (all functions of crank angle θ), as well as the void volumes of the heat exchangers Vk, Vr, and Vh.

The cyclic convergence behavior of the Ideal Adiabatic model is extremely good, and using 360 increments over the cycle, the system effectively converges within 5 cycles. The convergence criterion chosen is that after a complete cycle both variable temperatures Te and Tc must be within one degree Kelvin of their initial values. We now consider the solution of the temperature variables Tc and Te, the heat energy variables Qk, Qr, Qh, and the work energy variables Wc, We, and the net work done W. These results are presented as plots showing the variation of these parameters with the crank angle θ.

In the temperature-theta diagram we observe a large cyclic temperature variation of the gas in the expansion space (> 100 K), its mean value being less than that of the heater temperature of 923 K. Similarly the mean gas temperature in the compression space is higher than the coller temperature. This suggests that the adiabatic working spaces effectively reduce the temperature limits of operation, thus reducing the thermal efficiency to less than that of the Carnot efficiency.

The energy-theta diagram shows the accumulated heat transferred and work done over the cycle. Notice that the work done W starts with the (positive slope) expansion process then the compression process, and again returning to the expansion process, Thus the total work excursion is almost 15 joules, however the net work done at the end of the cycle is only 3 joules. The most significant aspect of the energy-theta diagram is the considerable amount of heat transferred in the regenerator over the cycle, almost ten times that of the net work done per cycle. This tends to indicate that the engine performance depends critically on the regenerator effectiveness and its ability to accomodate high heat fluxes. This aspect will be revisited in the section on the Simple analysis , when we examine the effect of imperfect heat exchangers on Stirling engine performance. Significantly the energy rejected by the gas to the regenerator matrix in the first half of the cycle is equal to the energy absorbed by the gas from the matrix in the second half of the cycle, thus the net heat transfer to the regenerator over a cycle is zero. It is for this reason that the importance of the regenerator was not understood for about 100 years after Stirling's original patent describing the function and importance of the regenerator. The Lehmann machine on which Schmidt did his analysis was apparently not fitted with a regenerator, and it is conceivable that Schmidt did not appreciate its importance, He refers to the textbook by Zeuner as containing a "complete, simple and clear theory" of air engines, but in the same textbook Zeuner decries the use of regenerators for air engines (Finkelstein, T., 1959, Air Engines in The Engineer part 1, 27 March.)

It is of interest to examine the two components, Wc and We, which added together gives the net work done W. These are shown as dashed lines in the following diagram.

Notice in particular that the expansion space work done (We) undergoes a vastly different process from that of heat transferred to the heater (Qh), however at the end of the cycle they have equal values (Qh = We). Similarly for the compression space work done (Wc) and the heat transferred to the cooler (Qk). In retrospect this must be so in order to retain an energy balance, however it did catch us unawares and surprised us when we first noticed this. The ideal regenerator thus behaves as the perfect isolator, isolating the energy balance of the heater and expansion space from that of the cooler and compression space. Thus for the Ideal Adiabatic model over a complete cycle

Qh = We; (Qe = 0)
Qk = Wc; (Qc = 0)
W = Wc + We

Recall that for the Ideal Isothermal model

Qe = We; (Qh = 0)
Qc = Wc; (Qk = 0)
W = Wc + We

Furthermore the Ideal Adiabatic model in itself does not give results which are significantly different from those of the Ideal Isothermal model. The pressure-volume diagram is of similar form, and the power output and efficiency are quantitatively similar (albeit the efficiency of the Ideal Adiabatic model is about 10% lower for reasons described above). However the the behaviour of the Ideal Adiabatic model is more realistic, in that the various results are consistent with the expected limiting behavior of real machines. Thus the heat exchangers become necessary components without which the engine will not function. The required differential equation approach to solution reveals the considerable amount of heat transferred in the regenerator, indicating its importance in the cycle, and provides a natural basis for extending the analysis to include non-ideal heat exchangers ( Simple analysis ). Thus the solution of the Ideal Adiabatic model equations is equivalent to a simulation of the engine behavior in all respects, from setting up the initial conditions until convergence to cyclic steady state is attained. Throughout this process all the variables of the system are available as by-products of the simulation and can be used for extending the analysis. Thus for example the mass flow rates through all the heat exchangers can be used in order to evaluate the heat transfer and flow friction effects over the cycle.

Stirling Cycle Machine Analysis by Israel Urieli is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License