Modern Engineering Thermodynamics

13.5 Operating Efficiencies 461
13.5.4 Thermal Efficiency
We also have to deal with three types of thermal efficiencies for work-producing or work-absorbing systems:
isentropic, reversible (or indicated), and actual (or brake). These thermal efficiencies are defined mathematically in
Table 13.2.
Using these definitions, the isentropic efficiency of the adiabatic piston-cylinder work-producing prime mover of
the Rankine cycle heat engine shown in Figure 13.9 is
ðη s Þ prime
mover
ðh 1 − h 2 Þ
= ðη s Þ pm =
actual
= h 1 − h 2
ðh 1 − h 2 Þ isentropic h 1 − h 2s
(13.4a)
or
h 1 − h 2 = ðh 1 − h 2
Þ actual = ðh 1 − h 2s Þðη s Þ pm (13.4b)
where h 2s is determined from p 2 (but not T 2 ) and the condition s 2s = s 1 , as shown in Figure 13.9. Similarly,
the isentropic efficiency of the adiabatic work-absorbing condensate pump in the heat engine shown in
Figure 13.9 is
ð
ðη s Þ pump
= ðη s Þ p
= h 4 − h 3 Þ isentropic
= h 4s − h 3
(13.5)
ðh 4 − h 3 Þ actual
h 4 − h 3
where h 4s is determined from p 4 (but not T 4 ) and the condition s 4s = s 3 , as shown in Figure 13.9. If the fluid
being pumped is an incompressible liquid (v = constant) with a constant specific heat c, then Eq. (7.33) of
Chapter 7 clearly shows that any isentropic process that it undergoes must also be isothermal. That is, T 4s = T 3
and consequently u 4s = u 3 . Then, for v 4s = v 4 = v 3 and (note that, for an isentropic pump, points 3 and 4s coincide
on a T–s diagram but not on a p–v diagram, see Figure 13.9) p 4s = p 4 and
h 4s − h 3 = u 4s − u 3 + p 4s v 4s − p 3 v 3
= cT ð 4s − T 3 Þ+ v 3 ðp 4s − p 3 Þ
= v 3 ðp 4 − p 3 Þ
(13.6)
Equation (13.5) can now be written as
ðη s Þ incompressible
liquid
pump
= v 3ðp 4 − p 3 Þ v 3 ðp 4 − p 3 Þ
=
h 4 − h 3 cT ð 4 − T 3 Þ+ v 3 ðp 4 − p 3 Þ
(13.7)
or
h 4 − h 3 = ðh 4 − h 3 Þ actual = v 3 ðp 4 − p 3 Þ/ ðη s Þ p (13.8)
Substituting Eqs. (13.4b) and (13.8) into the Rankine cycle thermal efficiency, Eq. (13.3) gives
ð
ðη T Þ Rankine = h 1 − h 2s Þðη s Þ pm − v 3 ðp 4 − p 3 Þ/ ðη s Þ p
(13.9a)
h 1 − h 3 − v 3 ðp 4 − p 3 Þ/ ðη s Þ p
where (η s ) pm is the isentropic efficiency of the prime mover. The maximum possible Rankine cycle thermal
efficiency occurs when both the prime mover and the condensate pump are isentropic: (η s ) pm = (η s ) p = 1.0, or
ðη T Þ maximum
Rankine
= ðη T Þ isentropic
Rankine
= h 1 − h 2s − v 3 ðp 4 − p 3 Þ
h 1 − h 3 − v 3 ðp 4 − p 3 Þ
(13.9b)
The phrase isentropic Rankine cycle thermal efficiency is used here to denote that the prime mover and the condensate
pump are both isentropic (i.e., reversible and adiabatic). Clearly, the entire cycle is not isentropic, since
thermal irreversibilities are associated with reheating the cold condensate returned to the hot boiler. This
notation is necessary to distinguish between reversible Rankine cycles, in which the prime movers and pumps are
modeled as reversible but are not adiabatic, and those Rankine cycles in which these items are modeled as both
reversible and adiabatic (i.e., isentropic). This same notation is used in referring to other power and refrigeration
cycles that contain isentropic components.