Page 78 - Entrophy Analysis in Thermal Engineering Systems

70 Entropy Analysis in Thermal Engineering Systems

removed, which will consequently lead to zero finite time power. On the
_
_
other hand, the power produced by the engine is _ W ¼ Q Q . Using
H L
Eqs. (6.2), (6.3), and (6.5), it can be shown that

ð K l T L T EH (6.8)
_ W ¼ K h T H T EH Þ K l
ð K h + K l ÞT EH K h T H T L
_
Applying ∂W =∂T EH ¼ 0 yields
p ffiffiffiffiffiffiffiffiffiffiffiffi
ð T EH Þ ¼ K l T L T H + K h T H (6.9)
opt K h + K l
Substituting Eq. (6.9) into Eq. (6.8) leads to an expression for the maximum
power production.
p p
ffiffiffiffiffiffiffi ffiffiffiffiffiffi 2
K h K l
_ W max ¼ K h + K l T H T L (6.10)
Curzon and Ahlborn [9] showed that the engine efficiency at maximum
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
power is η ð CA W max ¼ 1 T L =T H , where the subscript CA refers to the
Þ _
Curzon-Ahlborn engine (please refer to Appendix A for a discussion on
the efficiency at maximum power). To find out whether there is any rela-
tionship between the entropy production of the Curzon-Ahlborn cycle and
its thermal efficiency, we represent Eq. (6.7) in a dimensionless form by
dividing it by K l .

∗ ∗ 1 1
Φ ¼ r K r T Tð Þ ∗ (6.11)
EH
ð
1+ r K ÞT
EH r K r T r T
where
_
Φ
∗ K h T H ∗ T EH
Φ ¼ ; r K ¼ ; r T ¼ ; T ¼
EH T L
T L
K l
K l
Using Eq. (6.5), the thermal efficiency of the engine may be represented as
1
η ¼ 1 (6.12)
CA ∗
ð
1+ r K ÞT
EH r K r T
We may also rewrite Eq. (6.8) in a normalized form as
∗
∗ ∗ T EH (6.13)
ð
W ¼ 1+ r K r T T Þ ∗
EH
ð 1+ r K ÞT
EH r K r T
where W ∗ ¼ K l T L .
_ W