Modern Engineering Thermodynamics

460 CHAPTER 13: Vapor and Gas Power Cycles
Table 13.2 Definitions of Some Common Efficiencies
Mechanical efficiency (η m )
Isentropic efficiency (η s )
Relative efficiency (η r )
Thermal efficiency (η T )
ðη m Þ workproducing
= W actual
=
W reversible
device
ðη s Þ workproducing
= W actual
=
W isentropic
device
ðη r Þ workproducing
= W reversible
=
W isentropic
device
_W actual
_W reversible
_W actual
_W isentropic
_ W reversible
_W isentropic
ðη m Þ workabsorbing
actual _W actual
= W reversible W
= _ reversible
W
device
ðη s Þ workabsorbing
actual _W actual
= W isentropic W
= _ isentropic
W
device
ðη r Þ workabsorbing
reversible _W reversib1e
= W isentropic W
= _ isentropic
W
device
where
ðη T Þ isentropic
= ð W _ out Þ isentropic
_Q in
ðη T Þ actual
= ðη T Þ brake
=
_W out
actual
_Q in
=
ðη T
_ W out represents the magnitude of the net power output in each case:
Þ reversible
= ðη T Þ indicated
= ð W _ out Þ reversible
_W out
brake
_Q in
_Q in
isentropic efficiency η s is based on comparing the actual performance of the device with that which would occur
if the device were adiabatic as well as reversible (i.e., isentropic). Since most prime movers and pumps are
thermally insulated, we always assume that they are adiabatic when their heat loss is not given. The isentropic
efficiency of work-producing and work-absorbing devices is defined mathematically in Table 13.2.
13.5.3 Relative Efficiency
It is also possible to define a relative efficiency (or efficiency ratio) η r for these devices that relates the mechanical
and isentropic efficiencies. For a work-producing device, the relative efficiency is defined as the ratio of the
reversible work to the isentropic work:
ðη r Þ workproducing
device
and for a work-absorbing device, it is defined as
= W reversible
W isentropic
=
_W reversible
_W isentropic
ðη r Þ workabsorbing
device
= W isentropic
W reversible
= _ W isentropic
_W reversib1e
Then, with a little algebra, we can write η s = η m η r for either a work-producing or work-absorbing device. Notice
that, if a device is insulated and is therefore adiabatic, then η s = η m and η r = 1.0.
The terms shaft and brake arealsocommonlyusedtodescribetheactual work or power produced or absorbed
by a device. 7 These terms are interchangeable, but for clarity, the term actual is used most often throughout
this chapter. Further, the terms reversible and indicated are synonymous because the reversible work or power
produced inside a device can be determined from p V data provided by an indicator diagram; therefore, we can
write _W reversible = _W indicated . Consequently, for a work-producing device, we can always write
and for a work-absorbing device, we can always write
_W actual = _W shaft = _W brake =
_W actual = _W shaft = _W brake = ðη s Þ _W isentropic = ð ηm Þ _W reversible
_W isentropic / ð ηs Þ = _W reversible / ð ηm Þ
7 The term brake is a descriptive term that comes from an early method of measuring the power output of machines using a friction
band brake dynamometer called a Prony brake. It was named after Baron Gaspard Clair Francois Marie Riche de Prony (1755–1839)
and was developed in the 1830s to measure the power output of water wheels and steam engines.