164
4 Compressible Fluids
expansion. Entropy increases due to losses, causing a decrease of the total pressure
( 01
00
p
p
<
).
Figure 4.5 (right) shows similarly the h-s diagram of a compression with losses
between station 1 and station 2. Total pressure decreases here as well, due to losses
( 02
01
p
p
<
). We use the terms expansion or compression to indicate processes with
pressure decrease or increase (here without work exchange). This terminology is
typical for compressible fluids and indicates that density decreases with expansion
and increases with compression.
We consider an infinitesimal part of an expansion, as sketched in the h-s diagram
in Fig. 4.5 (left). We define the efficiency of this infinitesimal process (subscript ∞), 
using the concept of isentropic efficiency (see thermodynamics) by
For an ideal and perfect gas:
Assuming that η ∞ = cst during the whole process, it is polytropic.
From
~
follows
~
or
.
n
n
n 1
dp
n dT
p
p T
p n 1 T
r
−
= −
.
s
dh
dh
h ∞
−
= −
With
, it follows that
and
.
s
1
1
dh
Tds dh
dp
dh
dp
dp
r
h
r
r
∞
= −
=
=
/ .
/
p
p
p
c dT
c dT c
p
dT T
dp
RT dp
R dp p
r
h ∞ =
=
=
Thus :
.
1
dp
dT
p
T
g
h
g
∞
= −
Fig. 4.5 Expansion and compression with losses: decreasing total pressure