391
11.2 Thermodynamic Modelling
Results for η ∞ = 0.9 and γ = 1.4 are shown in Table 11.1. For a given infinitesimal efficiency, the difference with the other efficiencies increases with the pressure ratio.
Isentropic efficiency decreases with increasing pressure ratio. The efficiency based
on re-expansion work increases with increasing pressure ratio.
Infinitesimal efficiency may be introduced in the same way for an expansion. We
note (with
2
d ½ v
0
= ):
Infinitesimal efficiency is defined by
So:
(11.13)
In the h-s diagram (Fig. 11.11, right) we see that n < γ, so
For example: γ = 1.4, η ∞ = 0.95, n = 1.373; γ = 1.4, η ∞ = 0.9, n = 1.346.
Further:
(11.14)
Assuming that the expansion can be represented as polytropic, we calculate for an
ideal and perfect gas (Fig. 11.11, right) the following quantities.
Isentropic work (  isentropic head ):
irr
1
dh
dW ,
dp
dW dq .
r
− = −
−
= −
+
p
1
s
c dT
dh
dh
( dT / T ) .
dh
dp
RT( dp / p )
1 ( dp / p )
r
g
h
g
∞
−
−
−
−
=
=
=
=
−
−
−
− −
,exp
n
(
) / (
).
1
n 1
g
h
g
∞
=
−
−
n
,
1.
n 1
1
g
h
g
∞
>
<
−
−
01
02
p
01
02s
ln( h ) ln( h ) .
ln( h ) ln( h )
h
−
=
−
1
2s
2s
2
1
1
1
s
1
1
1
1
1
1
1
1
1
1
2
1
1
p
p
1
p
p
p
h
dp
d(
)
(
) d(
)
p
p
p
p
p
[ 1 (
)
].
1
p
g
g
g
r
D
r
r
r
r
g
g
r
−
−
= −
=
−
=
−
=
−
−
∫
∫
∫
r = p 2 /p 1 =2 r = 10
r = 40
η s = Δh s /Δh
0.8898
0.8641
0.8398
η ∞ = Δh r /Δh
0.9000
0.9000
0.9000
η sre = Δh sre /Δh 0.9096
0.9296
0.9442
Table 11.1 Comparison of
efficiency definitions of a
compression with η ∞ = 0.9 and
γ= 1.4