554
15 Axial and Radial Turbines for Gases
Characteristics are thus similar as rendered in Fig. 15.7.
Conditions for choking within the stator are given by (15.8) and (15.9), those
within the rotor by (15.10) and (15.11). The difference with an axial turbine is that
u 2 is smaller than u 1 . The consequence is that the pressure ratio p 01 /p 02 for choking
within the rotor is higher. Which of the two conditions is the most stringent strongly
depends on the value of h 01 . In case of a high h 01 value, the rotor determines the
choking and the corresponding pressure ratio p
p
2
v
v
tg
01
02
2m
1m
1
/
:
≈
=
=
With
ψ φ α
, and this value slightly
depends on the rotational speed. For a low h 01 value, the stator may determine the
choking and p
p
4
v
v
tg
u
u
tg
u
u
01
02
2m
1m
1
2
1
2
2
2
1
2
/
:
.
≈
=
=
−
−
With
ψ φ α
β
, independent of the rotational speed. The rotor determines
the choking in absence of stator vanes. The pressure ratio for choking then depends
on the rotational speed and may be rather high in case of a low h 01 . No universal
value can be attributed to this pressure ratio. E.g. with T 01 = 500 K, u 1 = 400 m/s,
u 2 = 200 m/s, it corresponds to (15.10):
15.2.4 Radial Turbine Applications
The work coefficient with radial turbines is lower than with axial turbines. The
efficiency is lower as well. As a consequence hereof a radial turbine is only applied if an axial type is impossible, this means with small dimensions. Efficiency
then becomes disadvantageous with axial turbines, due to relatively high clearances
and relatively high blade thickness at the trailing edge compared to the spacing. A
typical application is the turbine in turbochargers, except for the big ones. Radial
turbines are also used with very small gas turbines. Due to their ability to cope with
high pressure ratios before choking, they also are applied in expansion processes in
chemical plants. Efficiency is not very crucial with these applications.
15.3 Dimensional Analysis with Compressible Fluids
15.3.1 Independent and Dependent Π-groups
A flow within a turbomachine with a compressible fluid features eight independent
parameters: R, C p , μ, p 00 , T 00 ,
m , Ω, D. The gas is determined by the gas constant R,
specific heat C p and viscosity µ. Gas properties C p and µ are temperature-dependent
and thus should be given at a reference temperature (e.g. at T 00 ). Dynamic viscosity
µ may be substituted by kinematic viscosity ν. The initial state is determined by p 00
With
:
.
2
2
2
2m
1m
1
2
2
1
1
u
u
v
v
tg
tg
u
u
y f a
b
=
=
−
−
n
n
2
2 n 1
n 1
02
01
p
2
0.8 400 0.5 200
1
0.20 1 5 .
p
1
1150 500
g
−
−
×
− ×
≈
−
≈
=
+
×
15 Axial and Radial Turbines for Gases
Characteristics are thus similar as rendered in Fig. 15.7.
Conditions for choking within the stator are given by (15.8) and (15.9), those
within the rotor by (15.10) and (15.11). The difference with an axial turbine is that
u 2 is smaller than u 1 . The consequence is that the pressure ratio p 01 /p 02 for choking
within the rotor is higher. Which of the two conditions is the most stringent strongly
depends on the value of h 01 . In case of a high h 01 value, the rotor determines the
choking and the corresponding pressure ratio p
p
2
v
v
tg
01
02
2m
1m
1
/
:
≈
=
=
With
ψ φ α
, and this value slightly
depends on the rotational speed. For a low h 01 value, the stator may determine the
choking and p
p
4
v
v
tg
u
u
tg
u
u
01
02
2m
1m
1
2
1
2
2
2
1
2
/
:
.
≈
=
=
−
−
With
ψ φ α
β
, independent of the rotational speed. The rotor determines
the choking in absence of stator vanes. The pressure ratio for choking then depends
on the rotational speed and may be rather high in case of a low h 01 . No universal
value can be attributed to this pressure ratio. E.g. with T 01 = 500 K, u 1 = 400 m/s,
u 2 = 200 m/s, it corresponds to (15.10):
15.2.4 Radial Turbine Applications
The work coefficient with radial turbines is lower than with axial turbines. The
efficiency is lower as well. As a consequence hereof a radial turbine is only applied if an axial type is impossible, this means with small dimensions. Efficiency
then becomes disadvantageous with axial turbines, due to relatively high clearances
and relatively high blade thickness at the trailing edge compared to the spacing. A
typical application is the turbine in turbochargers, except for the big ones. Radial
turbines are also used with very small gas turbines. Due to their ability to cope with
high pressure ratios before choking, they also are applied in expansion processes in
chemical plants. Efficiency is not very crucial with these applications.
15.3 Dimensional Analysis with Compressible Fluids
15.3.1 Independent and Dependent Π-groups
A flow within a turbomachine with a compressible fluid features eight independent
parameters: R, C p , μ, p 00 , T 00 ,
m , Ω, D. The gas is determined by the gas constant R,
specific heat C p and viscosity µ. Gas properties C p and µ are temperature-dependent
and thus should be given at a reference temperature (e.g. at T 00 ). Dynamic viscosity
µ may be substituted by kinematic viscosity ν. The initial state is determined by p 00
With
:
.
2
2
2
2m
1m
1
2
2
1
1
u
u
v
v
tg
tg
u
u
y f a
b
=
=
−
−
n
n
2
2 n 1
n 1
02
01
p
2
0.8 400 0.5 200
1
0.20 1 5 .
p
1
1150 500
g
−
−
×
− ×
≈
−
≈
=
+
×