74
2 Macroscopic Thermodynamics
If, in Eq. (2.5.36a) for η Otto , we employ the adiabaticity constraint (2.5.39) on
the non-externally-fixed temperatures T high and T low to eliminate T low , we may also
express η Otto in a form analogous to that for the Carnot efficiency, that is,
η Otto = 1 −
T min
T high
.
(2.5.42a)
Note, however, that as the highest temperature achieved in an Otto cycle is T max
and the lowest temperature is the initial temperature T min , both T high and T low
(with T high > T low ) lie between these two values. The Otto efficiency given by
Eq. (2.5.42a) will thus be smaller than the maximal efficiency η max , given by
η max = 1 −
T min
T max
,
(2.5.42b)
that would be obtained for a Carnot cycle operating between thermal reservoirs
characterized by temperatures T max and T min . We may also express the total work,
W (cycle), given by Eq. (2.5.40a) as a function of the efficiency η Otto as
W (cycle) = C V T max η Otto
η max − η Otto
1 − η Otto
,
(2.5.43)
from which we also find that η Otto is given by Eq. (2.5.41), as might be anticipated.
Moreover, as we require W (cycle) ≥ 0, η Otto is restricted to values such that 0 ≤
η Otto < 1.
If we normalize W (cycle) by dividing Eq. (2.5.43) by W max (cycle), then we
obtain
W (cycle)
W max (cycle)
=
η Otto (η max − η Otto )
η 2
opt (1 − η Otto )
(2.5.44)
for the normalized total work for the Otto cycle. From this result, it is clear that
W (cycle) goes to zero both for an Otto cycle having zero efficiency (η Otto = 0)
and for one having the Carnot efficiency (η Otto = η max ). Also, as the normalized
work per cycle has only a single maximum, this means that for each value of
W (cycle)/W max (cycle) other than 0 or 1, an Otto engine can perform work at two
different efficiencies, one lying above, the other lying below η opt .
The traditional compression volume ratio r 12 for the initial step of the Otto cycle
may be obtained from Eq. (2.5.36a) as a function of the efficiency as
r 12 ≡
V 1
V 2
= (1 − η Otto )
−
1
γ +1 .
(2.5.45)
As r 12 is an increasing monotonic function of η Otto on the physically relevant
interval 0 ≤ η Otto ≤ η max < 1, its minimal value r min
12 = 1 is attained for
η Otto = 0, and its maximal value
2 Macroscopic Thermodynamics
If, in Eq. (2.5.36a) for η Otto , we employ the adiabaticity constraint (2.5.39) on
the non-externally-fixed temperatures T high and T low to eliminate T low , we may also
express η Otto in a form analogous to that for the Carnot efficiency, that is,
η Otto = 1 −
T min
T high
.
(2.5.42a)
Note, however, that as the highest temperature achieved in an Otto cycle is T max
and the lowest temperature is the initial temperature T min , both T high and T low
(with T high > T low ) lie between these two values. The Otto efficiency given by
Eq. (2.5.42a) will thus be smaller than the maximal efficiency η max , given by
η max = 1 −
T min
T max
,
(2.5.42b)
that would be obtained for a Carnot cycle operating between thermal reservoirs
characterized by temperatures T max and T min . We may also express the total work,
W (cycle), given by Eq. (2.5.40a) as a function of the efficiency η Otto as
W (cycle) = C V T max η Otto
η max − η Otto
1 − η Otto
,
(2.5.43)
from which we also find that η Otto is given by Eq. (2.5.41), as might be anticipated.
Moreover, as we require W (cycle) ≥ 0, η Otto is restricted to values such that 0 ≤
η Otto < 1.
If we normalize W (cycle) by dividing Eq. (2.5.43) by W max (cycle), then we
obtain
W (cycle)
W max (cycle)
=
η Otto (η max − η Otto )
η 2
opt (1 − η Otto )
(2.5.44)
for the normalized total work for the Otto cycle. From this result, it is clear that
W (cycle) goes to zero both for an Otto cycle having zero efficiency (η Otto = 0)
and for one having the Carnot efficiency (η Otto = η max ). Also, as the normalized
work per cycle has only a single maximum, this means that for each value of
W (cycle)/W max (cycle) other than 0 or 1, an Otto engine can perform work at two
different efficiencies, one lying above, the other lying below η opt .
The traditional compression volume ratio r 12 for the initial step of the Otto cycle
may be obtained from Eq. (2.5.36a) as a function of the efficiency as
r 12 ≡
V 1
V 2
= (1 − η Otto )
−
1
γ +1 .
(2.5.45)
As r 12 is an increasing monotonic function of η Otto on the physically relevant
interval 0 ≤ η Otto ≤ η max < 1, its minimal value r min
12 = 1 is attained for
η Otto = 0, and its maximal value