Page 109 - Entrophy Analysis in Thermal Engineering Systems

102 Entropy Analysis in Thermal Engineering Systems

7.7 Fixed heat input

Another special case is that the quantity of heat supplied to engine is
fixed—a design constraint. For an engine operating in a closed cycle, which
receives heat from a single high-temperature reservoir, and it rejects the
unused amount of heat to a low-temperature reservoir, it can be readily
shown that the minimization of entropy generation rate, maximization of
power output, and maximization of thermal efficiency are all equivalent.
The net power production is the difference between the rates of heat
_
_
supplied to/rejected from the engine, i.e., _ W net ¼ Q Q . For a given
H L
_
Q , the net power is maximized by minimizing the rate of heat rejected
H
_
from the cycle Q . From the definition of the thermal efficiency,
L
_
_
η ¼ W net =Q , maximization of efficiency would be equivalent to maximi-
H
_
zation of the power when Q is treated as a constant parameter. On the
H
other hand, the rate of total entropy generation given by Eq. (7.14) is min-
_
imized by minimization of the Q . Thus, the three optimization criteria
L
would be identical at the condition of fixed heat input.
One may argue with a similar reasoning that in an engine that exchanges
heat with two thermal reservoirs, maximization of thermal efficiency and
minimization of the total entropy generation rate would yield an identical
_ _
design at the condition of fixed power output W net . Eliminating Q
L
_
_
_
between W net ¼ Q Q and Eq. (7.14), we get
H L

1 1
_ _ _ W net (7.53)
Φ ¼ Q
H T L T H T L
For a fixed _ W net , to minimize the entropy generation rate would require one
_
to minimize Q , which would also maximize the thermal efficiency
H
_
η ¼ _ W net =Q .
H