Page 86 - Entrophy Analysis in Thermal Engineering Systems

78 Entropy Analysis in Thermal Engineering Systems

at temperature T EL . The evaporation process takes place by receiving heat
from the hot stream through a heat exchanger. The inlet and outlet temper-
atures of the hot stream are T h,in and T h,out , respectively. At the cold-end side
of the engine, the condensation heat is rejected through another heat
exchanger to the cold stream, which enters the heat exchanger at temper-
ature T l,in and exits at temperature T l,out .
Assume that both heat exchangers at the hot-end and the cold-end sides
are operating ideally. In other words, the temperature of the hot stream leav-
ing the engine is equal to the evaporation temperature, i.e., T h,out ¼T EH , and
the exit temperature of the cold stream is the same as the condensation tem-
perature, i.e., T l,out ¼T EL . In this case, the heat rate supplied from the hot
stream for evaporation of the working fluid is

_ _
ð
Q ¼ C h T H,in T EH Þ (6.24)
H
_
where C ¼ _mc p . The rate of heat rejected by the engine at the cold-end side
heat exchanger is absorbed by the cold stream. Hence,
_ _
ð
Q ¼ C l T EL T l,in Þ (6.25)
L
The power-producing compartment (the rectangle in Fig. 6.7) is internally
_
_
reversible, so Q =Q ¼ T EL =T EH . Using Eqs. (6.24) and (6.25), a relation
L H
can be established between T EL and T EH .
_
(6.26)
C l T l,in T EH
_ _ _
T EL ¼
C h + C l T EH C h T h,in
The power output of the engine is obtained as follows.
_ _ _ _
ð
ð
_ W ¼ Q Q ¼ C h T H,in T EH Þ C l T EL T l,in Þ (6.27)
H L
Substituting Eqs. (6.26) into Eq. (6.27) yields
" #
_
_ ð _ C l T l,in T EH (6.28)
_ _ _ T l,in

_ W ¼ C h T H,in T EH Þ C l
C h + C l T EH C h T h,in
For fixed values of the inlet temperatures and heat capacitances of the hot
and cold streams, the power produced by the engine has only one degree
of freedom, T EH . Maximization of the power with respect to T EH yields
_ _ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð T EH Þ ¼ C h T h,in + C l T l,in T h,in (6.29)
_ _
opt C h + C l