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11.2 Thermodynamic Modelling
11.2.3 Infinitesimal Efficiency; Polytropic Efficiency
The problem caused by the reheat effect can be avoided by defining efficiency on an
infinitesimal base. The relations for a compression are (with
2
d ½ v
0
= ):
For an infinitesimal part of the process, efficiency, in strict thermodynamic sense, is
This is termed small stage efficiency or infinitesimal efficiency.  The  subscript  ∞ 
indicates that the complete process is thought as being composed of an infinite
number of infinitesimal parts. With an ideal gas, we further have, with p = ρRT,
dh = c p dT:
With constancy of the specific heat ratio (perfect gas) and η ∞ over the entire compression follows
(11.10)
The process thus is polytropic with exponent n (  p ~ ρ
n
):
(11.11)
In the h-s diagram (Fig. 11.11, left) we see that n > γ > 1, so
For example: γ = 1.4, η ∞ = 0.95, n = 1.430; γ = 1.4, η ∞ = 0.9, n = 1.465.
Integration of (11.10) results in:
irr
1
dW dh, dW
dp dq .
r
=
=
+
1
s
dp
dh
.
dh
dh
r
h ∞ =
=
p
RT dp
1 dp / p .
c p dT
dT / T
g
h
g
∞
−
=
=
dp
dT .
p
1 T
g
h g
∞
=
−
or
,comp
n
n
(
) / (
).
n 1
1
n 1
1
g
g
h
h
g
g
∞
∞
=
=
−
−
−
−
n
,
1.
n 1
1
g
h
g
∞
<
<
−
−
2
1
2
1
1 ln( p / p )
ln( T / T ).
g
h
g
∞
−
=