We can determine an effectiveness of the heater or cooler heat exchanger in a similar way to that of the regenerator in terms of the following equation:
where e 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 results 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 results
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.
From the basic equation for convective heat transfer we obtain:
where 
(watts) is the heat transfer power,
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 = hk Awgk (Twk - Tk) / freq
Qh = 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 freq / (hk Awgk)
Th = Twh - Qh 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. 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
On to Pumping
Loss Simple AnalysisOn to the Computer
Program function set 'simple'
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