168
4 Compressible Fluids
The
2
1 2
d v term is the useful part of the energy conversion. This term may be integrated to
2
1 1
2 v , if the total state (00) is considered as the initial state. The pressure
energy used to generate the kinetic energy is
(4.28)
So, this term can only be determined if the path of the expansion process is known.
A fundamental problem arises here since the internal details of a thermodynamic
process are mostly unknown. Moreover, we wish to make a statement about the efficiency of a thermodynamic process, exclusively based on the initial and the final
states. From the above, we learn that this actually is impossible for a process with
a compressible fluid.
Both isentropic efficiency and infinitesimal efficiency, also called polytropic
efficiency, are efficiency assessments, for which a process path has been agreed
on for the calculation of (4.28). The isentropic efficiency corresponds to the path
s
00 1
1
→ → (Fig. 4.5, left). To the
s
00 1
→ part applies
The energy used is
2
1 1s
2 v . To the s
1
1
→ part applies
As this path is isobaric, there is no contribution to (4.28) and integration is possible
with result
(4.29)
The isentropic efficiency definition (4.27) corresponds to this interpretation. Dissipation during the process is considered as the enthalpy difference between states
1 and 1s:
With infinitesimal efficiency, the path is determined by defining the efficiency of
an infinitesimal part, which is indisputably possible, and by assuming a constant
infinitesimal efficiency on the total path. With the approximations introduced in the
preceding section, the infinitesimal efficiency follows from expressions (4.24) and
(4.21).
1
r
00
1
h
dp.
D
r
−
= − ∫
2
1
1
dW 0 d v
dp.
2
r
= =
+
2
irr
1
1
dW 0 d v
dp dq .
2
r
= =
+
+
2
2
1
1
irr
1s
1
2
2
q
v
v .
=
−
irr
1
1s
q
h h .
= −