Page 77 - Entrophy Analysis in Thermal Engineering Systems

Endoreversible heat engines 69

(the rectangle region in Fig. 6.1) and the thermal reservoirs. Hence, the rate
of entropy generation is
_ _ _ _

_ Q Q Q Q
Φ ¼ T EH T H + T L L T EL (6.1)
H
L
H
_
where Q is the heat rate received by the engine from the high-temperature
H
_
thermal reservoir, and Q denotes the rate of heat rejected by the engine to
L
the low-temperature thermal reservoir.
_
Q ¼ K h T H T EH Þ (6.2)
ð
H
_
ð
Q ¼ K l T EL T L Þ (6.3)
L
where K h and K l denote thermal conductance (assumed to be constant) at
the hot-end and the cold-end sides of the engine, respectively. Also, T EH
and T EL denote the highest and the lowest temperatures of the engine.
As the engine is endoreversible, we have
_
ð
Q K l T EL T L Þ
T EL
¼ L ¼ (6.4)
_ K h T H T EH Þ
ð
Q
T EH
H
Solving Eq. (6.4) for T EL gives
T EL ¼ K l T L T EH (6.5)
ð K h + K l ÞT EH K h T H
Using Eq. (6.4), Eq. (6.1) reduces to
_ _
_ Q L Q H
Φ ¼ T L T H (6.6)
A combination of Eqs. (6.2), (6.3), (6.5), and (6.6) allows expressing
Eq. (6.6) as

2 3
1
_ 1 7
6
ð
6
Φ ¼ K h T H T EH Þ 7 (6.7)
4 5
1+ K h T EH K h T H T H
K l
K l
_
Solving ∂Φ=∂T EH ¼ 0 leads to (T EH ) opt ¼T H . Substituting this result into
Eq. (6.5), we also find (T EL ) opt ¼T L . Minimization of the entropy genera-
tion rate associated with the Curzon-Ahlborn model suggests that any irre-
versibility between the thermal reservoirs and the engine should be