Page 144 - Advanced thermodynamics for engineers

130 CHAPTER 6 FINITE TIME (OR ENDOREVERSIBLE) THERMODYNAMICS

The power output of a combined cycle power plant is the sum of the power of the individual
engines, hence,

_
W CC 1 ðs 1 s 2 T C =T H Þ
_
_
¼ W H þ W C ¼ ½ð1 s 1 Þþð1 s 2 Þs 1 Š:

C H T H C H T H C H C H
s 1 s 2 1 þ þ
C 2 C C
This may be reduced to
_
W CC ðs 1 s 2 T C =T H Þ
¼ ½1 s 1 s 2 Š: (6.37)

C H T H C H C H
s 1 s 2 1 þ þ
C 2 C C
The efficiency of the combined cycle is defined by
_
W CC
h ¼ ¼ 1 s 1 s 2 (6.38)
th
Q _ H
Equation (6.38) shows that the expression for the efficiency of the combined cycle engine at
maximum power output is similar to that for the efficiency of a single heat engine, except that in this
case the temperature ratio of the single engine is replaced by the product of the two temperature ratios.
If the combined cycle device is considered to be two endoreversible heat engines connected by a
perfect conductor (i.e. the resistance between the engines is zero; C 2 ¼ N), then T 3 ¼ T 2 , and Eqn
(6.38) becomes
_
W CC T 1 T 3 T 1
h ¼ ¼ 1 s 1 s 2 ¼ 1 ¼ 1 (6.39)
th
Q _ H T 2 T 4 T 4
The efficiency given in Eqn (6.39) is the same efficiency as would be achieved by a single
endoreversible heat engine operating between the same two temperature limits, and what would be
expected if there was no resistance between the two engines in the combined cycle plant.
To determine the efficiency of the combined cycle plant, composed of endoreversible heat engines,
_
producing maximum power output requires the evaluation of the maxima of the surface, W CC , plotted
against the independent variables s 1 and s 2 . It is difficult to obtain a mathematical expression for this
and so the maximum work will be evaluated for some arbitrary conditions to demonstrate the
necessary conditions. This will be done based on the following assumptions:
Temperatures: T H ¼ 1600 K; T C ¼ 400 K
Conductivities: C H ¼ C 2 ¼ C C ¼ 1
While these assumptions are arbitrary, it can be shown that the results obtained are logical and
general. The variation of maximum power output with temperature ratios across the high- and low-
temperature engines is shown in Fig. 6.6. It can be seen that the maximum power occurs along a
ridge that goes across the base plane. Examination shows that, in this case, this obeys the equation
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s 1 s 2 ¼ 0:5 ¼ T C =T H . Hence, the efficiency of a combined cycle heat engine operating at maximum
power output is the same as the efficiency obtainable from a single heat engine operating between the
two same reservoirs. This solution is quite logical, otherwise it would be possible to arrange heat
engines as in Fig. 6.7, and produce net work output while transferring energy with a single reservoir.