Page 83 - Entrophy Analysis in Thermal Engineering Systems

Endoreversible heat engines 75

_ T EL , respec-
engine are given by η MN ¼1 T EL /T H and _ W ¼ Q 1
H T H
_
_
tively. Noting that Q ¼ _ W + Q and using Eq. (6.3), we find
H L

T L
ð
_ W ¼ K l T H T EL Þ 1 T EL (6.20)
p ffiffiffiffiffiffiffiffiffiffiffiffi
Maximization of the power output given in Eq. (6.20) with respect to T EL
ð
yields T EL Þ ¼ T L T H , which is the same as (T EH ) opt of the Novikov’s
opt
engine. The efficiency and the power output of the modified Novikov’s
engine at maximum power production are obtained by
r ffiffiffiffiffiffiffi
η ð MN W max ¼ 1 T L (6.21)
Þ _
p p T H
ffiffiffiffiffiffiffi ffiffiffiffiffiffi 2
(6.22)
T H T L
_ W max ¼ K l
Comparing Eqs. (6.22) and (6.10), it can be inferred that when the thermal
conductance at the hot-end side of the Curzon-Ahlborn model tends to
infinity K h !∞, the model of Curzon-Ahlborn reduces to the modified
model of Novikov’s engine. A further observation is that the efficiency of
all three engines that we have examined so far at maximum power output
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
is 1 T L =T H , whereas the maximum power of the Novikov’s engine is
the highest, and that of the modified Novikov’s engine is the lowest, and that
of the Curzon-Ahlborn engine in between.
To evaluate the production rate of entropy, we note that there is merely
one source of entropy generation due to the finite time heat exchange at the
cold-end side of the engine. The heat transfer at the hot-end side of the
_
_
engine takes place reversibly. So, Φ ¼ Q 1=T L 1=T EL Þ, which using
ð
L
Eq. (6.3), we get

_ T L
Φ ¼ K l T EL T L Þ 1 T EL (6.23)
ð
_
Solving ∂Φ=∂T EL ¼ 0 leads to (T EL ) opt ¼T L at which the power production
is zero; see Eq. (6.20). It can be concluded that the minimum entropy gen-
eration and the maximum power output are two different operational
regimes in the modified model of Novikov’s engine.
Like the Curzon-Ahlborn model, the relationship between the entropy
production and the thermal efficiency is graphically demonstrated in Fig. 6.6
for the Novikov’s engine and the modified Novikov’s engine. For the case
of the Novikov’s engine (Fig. 6.6A), the normalized power output W* and