One normally determines the effectiveness of a heater or cooler heat exchanger in a similar way to that of the regenerator in terms of the following equation:
where ε is the effectiveness of the heat exchanger and NTU is the "Number of Transfer Units" (Refer to "Compact Heat Exchangers", Kays & London). Both concepts are described in the section on 'regenerator Simple analysis'. Unfortunately, we are unable to determine a simple relation between the heater and cooler effectiveness and the engine efficiency, as we were able to do with the regenerator. Referring to the temperature profile diagram below we observe that the non-ideal heater and regenerator result in the mean effective temperature of the gas in the heater space (Th) being lower than that of the heater wall (Twh). Similarly the non-ideal cooler and regenerator result in the mean effective temperature of the gas in the cooler space (Tk) being higher than that of the cooler wall (Twk). This has a significant effect on the engine performance, since it is effectively operating between lower temperature limits than those of the heater and cooler walls. Thus the Simple analysis of the heater and cooler iteratively determines these temperature differences using the convective heat transfer equations, the values of Qh and Qk being evaluated by the Ideal Adiabatic analysis and the value of the regenerator enthalpy loss Qrloss being evaluated in terms of the regenerator effectiveness.
From the basic equation for convective heat transfer we obtain:
where
(watts)
is the total heat transfer power (including the regenerator net
enthalpy loss), h is the convective heat transfer coefficient, Awg
refers to the wall/gas, or "wetted" area of the heat
exchanger surface, Tw is the wall temperature, and T the gas
temperature. In order to reduce the units of this equation to the net
heat transferred over a single cycle Q (joules/cycle) we divide both
sides by the frequency of operation (freq), thus:
Qk - Qrloss = hk Awgk (Twk - Tk) / freq
Qh + Qrloss = hh Awgh (Twh - Th) / freq
where, as shown in the diagram above, the suffix h refers to the heater, and the suffix k refers to the cooler. We now rewrite these equations to evaluate the respective gas temperatures Tk and Th:
Tk = Twk - (Qk - Qrloss) freq / (hk Awgk)
Th = Twh - (Qh + Qrloss) freq / (hh Awgh)
The Simple solution algorithm requires iterative invoking of the Ideal Adiabatic simulation, each time with new values of Tk and Th, until convergence is attained. After each simulation run values of Qk and Qh are available and Qrloss is determined in terms of the regenerator effectiveness. The mass flow rates through the heater and cooler are used to determine the average Reynolds numbers and thus the heat transfer coefficients in accordance with the methods in the section on Scaling Parameters. Substituting these values in the above equations yields Tk and Th, and convergence is attained when their successive values are essentially equal.
The Simple simulation of our D90 Ross Yoke-drive engine case study results in the temperature distribution as shown below. Notice that the mean temperature of the gas in the heater space is 59 degrees below that of the heater wall, and similarly the mean temperature of the gas in the cooler space is 15 degrees above that of the cooler wall. This lower temperature range of operation reduced the output power from 178 W to 147 W
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Stirling Cycle Machine Analysis by Israel
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