Page 88 - Entrophy Analysis in Thermal Engineering Systems

80 Entropy Analysis in Thermal Engineering Systems

Note that the efficiency of the power-producing compartment at maximum
_
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
power is 1 T l,in =T h,in . Also, as the heat input Q is constant, the max-
in
imum efficiency occurs at the same optimum T EH that the power output
is maximized, i.e., Eq. (6.29).
The entropy production rate due to the heat exchange between the
power-producing compartment and the hot and cold streams is
_
_
_
_
_
Φ ¼ ΔS h + ΔS l , where ΔS h and ΔS l denote the net change in the entropies
of the hot and cold streams, respectively, due to the heat exchange with the
working fluid of the cycle.
To evaluate the total entropy generation rate associated with the oper-
ation of the engine model of Fig. 6.7 and based on the arguments of Bejan
[8], we need to account for additional sources of entropy generation due to
(i) the transfer of heat from the exhaust of the hot stream to the surrounding,
(ii) the rejection of heat from the exhaust of the cold stream to the surround-
ing, and (iii) heating the hot stream to increase its temperature from T l,in to
T h,in ; see Fig. 6.8.
As the hot stream is provided from the ambient, its temperature first rises
from T l,in to T h,in . It then reduces to T h,out within the hot-end side heat
exchanger where it loses part of its energy to evaporate the working sub-
stance of the engine. Finally, it cools down to T l,in after it returns back to
the ambient. Thus, the net change in the entropy of the hot stream fluid
is zero. Likewise, the net change in the entropy of the cold stream is zero
as it is supplied from the ambient and eventually discharged to the ambient.
So, a more accurate way to determine the total entropy generation rate of
the system designated with the dashed rectangle in Fig. 6.8 is to evaluate the
total heat transfer to the hot stream as well as the total heat rejected from the
cycle to the ambient. Thus, by accounting for all possible sources related to
the operation of the Carnot vapor cycle, the total entropy production rate is
_ _ 1 1
_ Q Q _ _ W
Φ tot ¼ out in (6.33)
¼ Q
in T l,in T H T l,in
T H
T l,in
_ was eliminated
where T H is the heat source temperature. In Eq. (6.33), Q
out
_
_
_
using W ¼ Q Q .
in out
As the rate of heat input is constant; see Eq. (6.31), the minimum total
entropy generation rate occurs at exactly the same optimum T EH that the
power output is maximized, i.e., Eq. (6.29). Thus, the maximum power,
maximum thermal efficiency, and minimum entropy production rate
become coincident for the Carnot vapor cycle when the heat input is