Modern Engineering Thermodynamics

214 CHAPTER 7: Second Law of Thermodynamics and Entropy Transport and Production Mechanisms
WHAT IS A CARNOT ENGINE?
A Carnot engine is a reversible heat engine that operates between a high-temperature heat source at T H
heat sink at T L . The thermal efficiency of a Carnot engine is simply ðη T Þ Carnot
= 1 − T L =T H .
and a low-temperature
It turns out that no one has ever actually made a running Carnot engine (although Rudolph Diesel thought he did, but he
invented the Diesel engine instead). The reversible constant temperature heat transfers T H and T L require the engine to
have infinite heat transfer surface areas, and this is not practical.
The “concept” of a Carnot engine is important because it turns out that no other heat engine can have a thermal efficiency
higher than that of a Carnot engine operating between the same two temperature limits. So the value of the Carnot engine
is only as a benchmark with which to compare the thermal efficiencies of other actual operating heat engines. The Carnot
cycle has become the universal standard by which the performance of other heat engine cycles can be measured.
and so forth. However, any simple power function of the form fðT 3 /T 1 Þ = ðT 3 /T 1 Þ n does satisfy Eq. (7.13), since
n
T 3
= T n n
2 T 3
T 1 T 1 T 2
The simplest such power function is a linear one (n = 1), and this is what Thomson chose to establish his absolute
temperature scale. Therefore, if we take
T
f 3
= T
3
= T
2 T3 T
= f 2 T
f 3
(7.14)
T 1 T 1 T 1 T 2 T 1 T 2
then Eq. (7.10) becomes
jQ out j
Q in rev
= jQ
Lj
= T L
(7.15)
Q H rev
T H
It should be noted that Eq. (7.14) is not the only function that accurately defines an absolute temperature scale
(but it is the simplest). Many other functions also work. However, they produce nonlinear temperature scales in
which the size of the temperature unit is not constant but depends on the temperature level. This might be a
useful technique to expand or condense a temperature scale in certain temperature regions, but the additional
complexity associated with a nonlinear temperature scale makes it generally unsuitable for common usage. 6
Now, clearly, the maximum possible thermal energy conversion efficiency of any real irreversible closed system
cyclic heat engine is equal to the thermal energy conversion efficiency that the same heat engine would have if
it were somehow made to run reversibly like a Carnot engine. Then, from Eq. (7.9),
ðη T
Þ max
= ðη T
Þ rev
= ðη T Þ Carnot
= 1 − T L
(7.16)
T H
EXAMPLE 7.1
If a heat engine burns fuel for its thermal energy source and the combustion flame temperature is 4000.°F, determine the
maximum possible thermal efficiency of this engine if it exhausts to the environment at 70.0°F.
Solution
First, draw a sketch of the system (Figure 7.7).
The unknown is the maximum possible thermal energy conversion efficiency of any heat engine. The “maximum” efficiency
occurs when an engine operates reversibly (i.e., with no internal losses due to friction, etc.). Since all reversible engines must
have the same thermal energy conversion efficiency when operated between the same high- and low-temperature reservoirs,
we can apply the results of the reversible Carnot engine analysis to this problem. Equation (7.16) gives the maximum possible
thermal efficiency as
ðη T Þ max
= ðη T Þ Carnot
= 1 − T L ð70:0 + 459:67Þ R
= 1 − = 0:881 = 88:1%
T H ð4000: + 459:67Þ R
6 It has been suggested that, since many thermal phenomena are inherently nonlinear, the use of a nonlinear (e.g., logarithmic)
temperature scale might have some engineering merit.