Environmental Engineering Reference
In-Depth Information
In this section we consider simple models of heat engines that illustrate the principal features
of several practical devices. Of particular importance is the amount of work produced (
)in
proportion to the amount of heat that is added (
q
) to represent the combustion of fuel, whose ratio
is called the
thermodynamic efficiency
w
η
th
(
≡
w/
q
)
. But other features are of practical consequence
as well, and these are displayed in the analysis.
In all these analyses, we assume that the fluid that produces the mechanical work undergoes
reversible processes, so that the entropy change is related to the heat addition by equation (3.10).
3.10.1
The Carnot Cycle
The Carnot cycle is a prototype cycle that has little practical importance but is beautifully illustrative
of the second law limits on the simplest of heat engine cycles. It is sustained by two heat reservoirs,
a hot one of temperature
T
h
and a cold one of temperature
T
c
. (We may think of the hot reservoir as
one that is kept warm by heat transfer from a burning fuel source and the cold one as the atmosphere.)
Consider the heat engine to be a cylinder equipped with a movable piston and enclosing a fluid of unit
mass. The cycle consists of four parts, as illustrated in Figure 3.2: an isothermal expansion during
which an amount of heat
q
h
is added to the engine (1
2 in Figure 3.2); an adiabatic isentropic
additional expansion during which the fluid decreases in temperature from
T
h
to
T
c
(2
→
→
3); an
isothermal compression while the system adds a quantity of heat
q
c
to the cold reservoir (3
→
4);
and finally an isentropic compression to the initial state (4
→
1). For this cycle the net work
w
of
the piston per cycle is
pd
v
, and by the first law equation (3.9) we obtain
q
h
−
q
c
=
w
=
pd
v
(3.28)
In Figure 3.2 a temperature-entropy plot of the Carnot cycle shows that the entropy increase
(
s
2
−
s
1
) during heating by the hot reservoir is equal in magnitude to the decrease (
s
3
−
s
4
) during
cooling, and by the second law (3.10) it follows that
q
h
T
h
=
q
c
T
c
(3.29)
T
T
h
Hot reservoir
q
h
1
2
T
h
w
Heat engine
T
c
q
c
3
4
T
c
Cold reservoir
s
Figure 3.2
The Carnot cycle consists of isothermal and isentropic expansions
(
1
→
2
,
2
→
3
)
and
compressions
(
3
→
4
,
4
→
1
)
of a fluid in a cylinder while absorbing heat
q
h
from a hot reservoir
(
1
→
2
)
,
rejecting heat
q
c
to a cold reservoir
(
3
→
4
)
and producing work
w
.
