2.5 Thermodynamic Engines
77
ln P 1 − ln P 2
ln V 1 − ln V 2
= −γ , or
P 2
P 1
=
V 1
V 2
γ
.
(2.5.48b)
These two relations allow a further simplification of η Diesel to
η Diesel = 1 −
r
γ
32 − 1
γ (r 32 − 1)r
γ −1
12
(2.5.49)
in terms of the volume compression and expansion ratios r 12 and r 32 , namely,
r 12 ≡
V 1
V 2
,
r 32 ≡
V 3
V 2
,
(2.5.50)
respectively, that represent the volume ratios achieved in the initial compression step
and in the power expansion step of the Diesel cycle.
The total work for a Diesel cycle is given by
W cycle (system) = C P (T max − T high ) − C V (T low − T min ) ,
so that the net work, W (cycle), available for export from a Diesel engine is
W (cycle) = Q in − Q out
= C P (T max − T high ) − C V (T low − T min ) .
(2.5.51)
As for the Otto cycle, the net entropy change,
((S) cycle = C P ln
T max
T high
+ C V ln
T min
T low
≡ 0 ,
imposes the constraint
T min T
γ
max = T low T
γ
high
(2.5.52)
on the temperatures appearing in the Diesel cycle. We may then employ this
constraint to replace the temperature variable T low in Eq. (2.5.51) to obtain
W (cycle) = C P (T max − T high ) − C V T min
T
γ
max
T
γ
high
− 1
(2.5.53)
for the available work. Optimization of W (cycle) with respect to T high gives T high,opt
as
T high,opt = (T min T
γ
max )
f ,
f ≡ (γ + 1)
−1 ,
(2.5.54)