166
4 Compressible Fluids
Examples ( η ∞ = 0.9): γ = 1.4; M 0 = 0.3: C = 0.998; M 0 = 0.5: C = 0.995,
γ = 1.3; M 0 = 0.3: C = 0.999; M 0 = 0.5: C = 0.996.
Factor C may be substituted by 1 with a very good approximation, so that the
Saint Venant formula, with a very good approximation becomes
(4.24)
This approximation means considering a polytropic process between states 00 and
1. The approximation is obviously not adequate if inlet kinetic energy is large with
respect to the enthalpy drop in the nozzle.
Similarly to (4.16), (4.17), (4.18), the flow evolution follows now from
From this we obtain
(4.25)
Further:
The last equation demonstrates that the Mach number change always has the opposite sign of the pressure change. With Venturi flow, the minima or maxima of
pressure and Mach number are still attained at the throat, just as in a nozzle without losses. The throat Mach number M th now plays the role of the critical Mach
number. We must distinguish between subcritical flow ( M < M th ) and supercritical
n 1
2
n
00
00
00
p
v
p
1
.
2
1
p
g
g
r
−
=
−
−
,
d
1 dp
n p
r
r
=
or
2
2
2
1
dp
dv
dp
1 dp
d v
dh
.
2
v
p
v
M
h
h
h
r
r
g
∞
∞
∞
−
= − =
−
=
−
=
2
dA
d
dv
1 dp ,
A
v
n p
M
h
r
r
g
∞
= −
−
=
−
with
.
2
2
th
th
2
M
dA 1
dp
n n 1
1
M
A n
p
1
M
h g g
∞
−
=
−
=
=
−
,
dc dp d
1 dp
2
1
c
p
n p
r
r
=
−
= −
.
2
dM dv dc
n 1 dp
M
v
c
2n
p
M
h
g
∞
−
=
−
= −
+
4 Compressible Fluids
Examples ( η ∞ = 0.9): γ = 1.4; M 0 = 0.3: C = 0.998; M 0 = 0.5: C = 0.995,
γ = 1.3; M 0 = 0.3: C = 0.999; M 0 = 0.5: C = 0.996.
Factor C may be substituted by 1 with a very good approximation, so that the
Saint Venant formula, with a very good approximation becomes
(4.24)
This approximation means considering a polytropic process between states 00 and
1. The approximation is obviously not adequate if inlet kinetic energy is large with
respect to the enthalpy drop in the nozzle.
Similarly to (4.16), (4.17), (4.18), the flow evolution follows now from
From this we obtain
(4.25)
Further:
The last equation demonstrates that the Mach number change always has the opposite sign of the pressure change. With Venturi flow, the minima or maxima of
pressure and Mach number are still attained at the throat, just as in a nozzle without losses. The throat Mach number M th now plays the role of the critical Mach
number. We must distinguish between subcritical flow ( M < M th ) and supercritical
n 1
2
n
00
00
00
p
v
p
1
.
2
1
p
g
g
r
−
=
−
−
,
d
1 dp
n p
r
r
=
or
2
2
2
1
dp
dv
dp
1 dp
d v
dh
.
2
v
p
v
M
h
h
h
r
r
g
∞
∞
∞
−
= − =
−
=
−
=
2
dA
d
dv
1 dp ,
A
v
n p
M
h
r
r
g
∞
= −
−
=
−
with
.
2
2
th
th
2
M
dA 1
dp
n n 1
1
M
A n
p
1
M
h g g
∞
−
=
−
=
=
−
,
dc dp d
1 dp
2
1
c
p
n p
r
r
=
−
= −
.
2
dM dv dc
n 1 dp
M
v
c
2n
p
M
h
g
∞
−
=
−
= −
+