78
2 Macroscopic Thermodynamics
while the entropy constraint (2.5.52) gives T low,opt = T high,opt . The second
derivative of W (cycle),
d 2 W
dT 2
high
= −γ (γ + 1)C V
T min T
γ
max
T
γ +2
high
< 0 ,
so that the optimized value for W (cycle) is, as expected, a maximum. The final
expression for the optimized available work for a Diesel cycle is somewhat more
complicated than that for the Otto cycle, and takes the form
W max (cycle) = C V T max [γ (1 − τ
f ) − τ
f (1 − τ
1−f )] ,
(2.5.55)
in which τ is the temperature ratio τ ≡ T min /T max between the highest and lowest
temperatures in the Diesel cycle. The efficiency of the optimized Diesel cycle is
given by Eq. (2.5.47), with T low = T high = T high,opt , as
η
opt
Diesel = 1 −
T high,opt − T min
γ (T max − T high,opt )
or, expressed in terms of the temperature ratio τ , as [14]
η
opt
Diesel = 1 − τ
f 1 − τ 1−f
γ (1 − τ f )
.
(2.5.56)
The optimized expressions (2.5.55) and (2.5.56) for W max (cycle) and η
opt
Diesel are
more complex than those obtained for the Otto cycle. In particular, the Diesel cycle
efficiency depends upon the heat capacity ratio γ for the working fluid, while those
for the Carnot, Otto, and Curzon–Ahlborn cycles depend solely upon the ratio of the
highest and lowest temperatures for the cycle.
2.5.5 Counterclockwise Otto Cycles
In this subsection, we shall follow fairly closely a recent discussion of counterclockwise (CCW) pressure–volume working cycles other than Carnot cycles due to
Dickerson and Mottmann [16]. A pair of CCW Otto cycles are shown in Fig. 2.5.
Panel (a) shows a CCW Otto cycle that functions as a refrigeration cycle that takes
the working substance (for convenience, an ideal gas) through a counterclockwise
sequence of steps via the closed cycle 1 → 2 → 3 → 4 → 1. The first step of the
cycle begins with the working substance at point 1, (V 1 , P 1 ) lying on the isotherm
corresponding to the temperature T high of the high-temperature heat reservoir.
Following its isolation from the two heat reservoirs, the system (ideal gas) is
expanded adiabatically from point 1 to point 2, (V 2 , P 2 ), at which the system
2 Macroscopic Thermodynamics
while the entropy constraint (2.5.52) gives T low,opt = T high,opt . The second
derivative of W (cycle),
d 2 W
dT 2
high
= −γ (γ + 1)C V
T min T
γ
max
T
γ +2
high
< 0 ,
so that the optimized value for W (cycle) is, as expected, a maximum. The final
expression for the optimized available work for a Diesel cycle is somewhat more
complicated than that for the Otto cycle, and takes the form
W max (cycle) = C V T max [γ (1 − τ
f ) − τ
f (1 − τ
1−f )] ,
(2.5.55)
in which τ is the temperature ratio τ ≡ T min /T max between the highest and lowest
temperatures in the Diesel cycle. The efficiency of the optimized Diesel cycle is
given by Eq. (2.5.47), with T low = T high = T high,opt , as
η
opt
Diesel = 1 −
T high,opt − T min
γ (T max − T high,opt )
or, expressed in terms of the temperature ratio τ , as [14]
η
opt
Diesel = 1 − τ
f 1 − τ 1−f
γ (1 − τ f )
.
(2.5.56)
The optimized expressions (2.5.55) and (2.5.56) for W max (cycle) and η
opt
Diesel are
more complex than those obtained for the Otto cycle. In particular, the Diesel cycle
efficiency depends upon the heat capacity ratio γ for the working fluid, while those
for the Carnot, Otto, and Curzon–Ahlborn cycles depend solely upon the ratio of the
highest and lowest temperatures for the cycle.
2.5.5 Counterclockwise Otto Cycles
In this subsection, we shall follow fairly closely a recent discussion of counterclockwise (CCW) pressure–volume working cycles other than Carnot cycles due to
Dickerson and Mottmann [16]. A pair of CCW Otto cycles are shown in Fig. 2.5.
Panel (a) shows a CCW Otto cycle that functions as a refrigeration cycle that takes
the working substance (for convenience, an ideal gas) through a counterclockwise
sequence of steps via the closed cycle 1 → 2 → 3 → 4 → 1. The first step of the
cycle begins with the working substance at point 1, (V 1 , P 1 ) lying on the isotherm
corresponding to the temperature T high of the high-temperature heat reservoir.
Following its isolation from the two heat reservoirs, the system (ideal gas) is
expanded adiabatically from point 1 to point 2, (V 2 , P 2 ), at which the system