170
4 Compressible Fluids
The slope of an isobar in a h-s diagram is
p
dh
( )
T
ds
= . So, the isobars diverge in
the sense that the isentropic enthalpy difference between two isobars increases with
increasing entropy. So it follows: a b
as
1s
h h h
h
− >
−
. This inequality expresses that
a part of the loss incurred on the s
a
a
→ path, causing heating of the gas, is recovered during the following expansion, as the isentropic enthalpy drop increases. This
effect is termed reheat effect, meaning heating due to internal loss.
Further breakdown of the 00 1
→ path in isentropic and isobaric parts results in
a polytropic representation as a limit case. This demonstrates that, with the polytropic representation, the energy used to generate the kinetic energy (4.28) is larger
than with the isentropic-isobaric process. We also understand that losses intervene
in two ways with the real process: the useful result 1 2 1
2
v is smaller than with the
isentropic process 1 2 1
2
v s and the energy converted to achieve this result
r
h
D
−
is
larger than with the isentropic process 1 2 1
2
v s .
The infinitesimal representation describes the real process better than the isentropic-isobaric representation. Both representations constitute models however, as
a correct assessment of efficiency is impossible without full details of the process
path. Infinitesimal efficiency has the advantage of an unchanged value when two
processes are set in series, as demonstrated in Fig. 4.6. Connecting two processes in
series changes the isentropic efficiency. Figure 4.6 demonstrates that the isentropic
efficiency of a series of two expansions with the same isentropic efficiency generates a result with better isentropic efficiency.
The different behaviour of isentropic efficiency and polytropic efficiency also
applies to expansions and compressions with work (see Chap. 11). Because of the
preservation of efficiency with connection in series, use of polytropic efficiency is
advisable with cycle studies. Both definitions may be used when assessing a single
machine component. Isentropic efficiency is the most visual, as it implies a clear
representation of losses in the h-s diagram. We therefore apply isentropic efficiency
with the fundamental study of turbomachine components, as with the study of steam
turbines in Chap. 6. The Saint Venant formula (4.24) is used to calculate expansion
processes within steam turbine components and infinitesimal efficiency indirectly
intervenes when defining the polytropic exponent.
For completeness, we mention that with a constant density fluid, there is no difference between isentropic and polytropic efficiencies. The definition of infinitesimal efficiency is still
with
,
irr
s
dh
1
dq
Tds dh
dp
dh
h
r
∞
−
=
=
= −
−
1
1
1
so that
and
.
irr
s
dp dq
d
dh
d
p
p
r
r
h
r
∞
−
−
−
=
=
Further:
and
,
s
irr
1
1
h
p
q
h
p
r
r
∆ = ∆
= ∆ − ∆
4 Compressible Fluids
The slope of an isobar in a h-s diagram is
p
dh
( )
T
ds
= . So, the isobars diverge in
the sense that the isentropic enthalpy difference between two isobars increases with
increasing entropy. So it follows: a b
as
1s
h h h
h
− >
−
. This inequality expresses that
a part of the loss incurred on the s
a
a
→ path, causing heating of the gas, is recovered during the following expansion, as the isentropic enthalpy drop increases. This
effect is termed reheat effect, meaning heating due to internal loss.
Further breakdown of the 00 1
→ path in isentropic and isobaric parts results in
a polytropic representation as a limit case. This demonstrates that, with the polytropic representation, the energy used to generate the kinetic energy (4.28) is larger
than with the isentropic-isobaric process. We also understand that losses intervene
in two ways with the real process: the useful result 1 2 1
2
v is smaller than with the
isentropic process 1 2 1
2
v s and the energy converted to achieve this result
r
h
D
−
is
larger than with the isentropic process 1 2 1
2
v s .
The infinitesimal representation describes the real process better than the isentropic-isobaric representation. Both representations constitute models however, as
a correct assessment of efficiency is impossible without full details of the process
path. Infinitesimal efficiency has the advantage of an unchanged value when two
processes are set in series, as demonstrated in Fig. 4.6. Connecting two processes in
series changes the isentropic efficiency. Figure 4.6 demonstrates that the isentropic
efficiency of a series of two expansions with the same isentropic efficiency generates a result with better isentropic efficiency.
The different behaviour of isentropic efficiency and polytropic efficiency also
applies to expansions and compressions with work (see Chap. 11). Because of the
preservation of efficiency with connection in series, use of polytropic efficiency is
advisable with cycle studies. Both definitions may be used when assessing a single
machine component. Isentropic efficiency is the most visual, as it implies a clear
representation of losses in the h-s diagram. We therefore apply isentropic efficiency
with the fundamental study of turbomachine components, as with the study of steam
turbines in Chap. 6. The Saint Venant formula (4.24) is used to calculate expansion
processes within steam turbine components and infinitesimal efficiency indirectly
intervenes when defining the polytropic exponent.
For completeness, we mention that with a constant density fluid, there is no difference between isentropic and polytropic efficiencies. The definition of infinitesimal efficiency is still
with
,
irr
s
dh
1
dq
Tds dh
dp
dh
h
r
∞
−
=
=
= −
−
1
1
1
so that
and
.
irr
s
dp dq
d
dh
d
p
p
r
r
h
r
∞
−
−
−
=
=
Further:
and
,
s
irr
1
1
h
p
q
h
p
r
r
∆ = ∆
= ∆ − ∆