72
2 Macroscopic Thermodynamics
in terms of the compression ratio r 12 defined as r 12 ≡ V 1 /V 2 . As can be seen from
Fig. 2.4a, the equality of the length of the constant-volume sides of the parallelogram
means that
ln P 3 − ln P 2 = ln P 4 − ln P 1 ,
or equivalently, that the pressures must satisfy the relation
P 3
P 2
=
P 4
P 1
.
(2.5.37a)
Moreover, as the slope of the initial adiabat is −γ (for an ideal classical gas), we
also have
ln P 2 − ln P 1
ln V 2 − ln V 1
= −γ ,
from which we obtain the pressure–volume relation
P 2
P 1
= r
γ
12 .
(2.5.37b)
Relations (2.5.37) enable us to re-express the efficiency of the Otto cycle as
η Otto = 1 −
1
r
γ −1
12
.
(2.5.38)
The lowest and highest temperatures, T min and T max , respectively, are assumed to
be known, with T min fixed by the ambient temperature and T max determined by the
combustion characteristics of the specific gasoline-air mixture being considered.
The intermediate temperatures, T high and T low , while not predetermined, may not
be varied freely, however, as the total entropy change, ((S) cycle , must be zero.
Moreover, as the two adiabatic steps require that S 2 = S 1 , or ((S) 1→2 = 0, and
S 3 = S 4 , or ((S) 3→4 = 0, the entropy constraint is thereby reduced to
((S) cycle = ((S) 2→3 + ((S) 4→1 = 0 ,
from which we may deduce directly that T high and T low are constrained to take values
such that
T high T low = T min T max .
(2.5.39)
We may employ this constraint on the temperatures to obtain a suggestive form for
the total work W (cycle) ≡ W cycle (fluid) for the fluid, namely,
2 Macroscopic Thermodynamics
in terms of the compression ratio r 12 defined as r 12 ≡ V 1 /V 2 . As can be seen from
Fig. 2.4a, the equality of the length of the constant-volume sides of the parallelogram
means that
ln P 3 − ln P 2 = ln P 4 − ln P 1 ,
or equivalently, that the pressures must satisfy the relation
P 3
P 2
=
P 4
P 1
.
(2.5.37a)
Moreover, as the slope of the initial adiabat is −γ (for an ideal classical gas), we
also have
ln P 2 − ln P 1
ln V 2 − ln V 1
= −γ ,
from which we obtain the pressure–volume relation
P 2
P 1
= r
γ
12 .
(2.5.37b)
Relations (2.5.37) enable us to re-express the efficiency of the Otto cycle as
η Otto = 1 −
1
r
γ −1
12
.
(2.5.38)
The lowest and highest temperatures, T min and T max , respectively, are assumed to
be known, with T min fixed by the ambient temperature and T max determined by the
combustion characteristics of the specific gasoline-air mixture being considered.
The intermediate temperatures, T high and T low , while not predetermined, may not
be varied freely, however, as the total entropy change, ((S) cycle , must be zero.
Moreover, as the two adiabatic steps require that S 2 = S 1 , or ((S) 1→2 = 0, and
S 3 = S 4 , or ((S) 3→4 = 0, the entropy constraint is thereby reduced to
((S) cycle = ((S) 2→3 + ((S) 4→1 = 0 ,
from which we may deduce directly that T high and T low are constrained to take values
such that
T high T low = T min T max .
(2.5.39)
We may employ this constraint on the temperatures to obtain a suggestive form for
the total work W (cycle) ≡ W cycle (fluid) for the fluid, namely,