7 Dynamic Similitude
256
resulting in
(7.7)
Each family of similar flows within a turbomachine is characterised by a certain
value of Ω s and D s . The specific speed and the specific diameter of the machine
are the values for flows with optimum efficiency. The numbers Ω s and D s are more
convenient numbers than Φ* and Ψ*, as a machine is designed for target values of Q
and ∆E m . So the specific speed determines the corresponding rotational speed. The
specific diameter determines the corresponding diameter and as a consequence, the
machine dimensions. According to the similitude theory, only one dimensionless
group is independent. Generally, the specific speed is preferred. The three values
applied in this group, flow rate, energy change and rotational speed can always
be correctly established externally. A rotor diameter can only be approximately
determined externally, which implies that an exact determination of the numbers
Φ*, Ψ* and D s is unfeasible without information about the real value of the rotor
diameter.
An interpretation of Ω s and D s follows from considering a reference machine,
yielding the required flow rate and the required energy change with Φ = 1 and Ψ = 1.
The rotational speed and the diameter of the reference machine meet
3
r r
D Q
W
= and
2 2
r r
m
D
E
W
D
=
, resulting in
So:
The adjective ‘specific’ follows from this interpretation. More general terms are
speed number and diameter number. These terms are sometimes used. Numerical
factors are sometimes included into the definitions of Ω s and D s , namely when this
also applies to Φ and Ψ. The former definitions of Φ ′ and Ψ ′ imply for the rotational speed and the diameter of the reference machine:
Speed number ( )
s and diameter number ( δ) are then defined by
2
2
2 2
2
2 6
4
m
m
Q
D
Q
.
,
D
E
D E
F
W
Y W
D
D
=
=
D
D E
Q
s
m
=
(
) .
/
∆
1 4
and
3/ 4
m
r
r
1/ 4
m
Q
( E )
D
.
( E )
Q
D
W
D
=
=
s
s
r
r
D
,
D
.
D
W
W W
=
=
and
3 4
m
r
r
1 4
m
2 Q
( 2 E )
'
D'
.
Q
( 2 E )
D
W
p
p D
=
=
s
3 4
r
m
Q
0.335 ,
'
( 2 E )
W
W
s
W
W
p D
=
=
≈
256
resulting in
(7.7)
Each family of similar flows within a turbomachine is characterised by a certain
value of Ω s and D s . The specific speed and the specific diameter of the machine
are the values for flows with optimum efficiency. The numbers Ω s and D s are more
convenient numbers than Φ* and Ψ*, as a machine is designed for target values of Q
and ∆E m . So the specific speed determines the corresponding rotational speed. The
specific diameter determines the corresponding diameter and as a consequence, the
machine dimensions. According to the similitude theory, only one dimensionless
group is independent. Generally, the specific speed is preferred. The three values
applied in this group, flow rate, energy change and rotational speed can always
be correctly established externally. A rotor diameter can only be approximately
determined externally, which implies that an exact determination of the numbers
Φ*, Ψ* and D s is unfeasible without information about the real value of the rotor
diameter.
An interpretation of Ω s and D s follows from considering a reference machine,
yielding the required flow rate and the required energy change with Φ = 1 and Ψ = 1.
The rotational speed and the diameter of the reference machine meet
3
r r
D Q
W
= and
2 2
r r
m
D
E
W
D
=
, resulting in
So:
The adjective ‘specific’ follows from this interpretation. More general terms are
speed number and diameter number. These terms are sometimes used. Numerical
factors are sometimes included into the definitions of Ω s and D s , namely when this
also applies to Φ and Ψ. The former definitions of Φ ′ and Ψ ′ imply for the rotational speed and the diameter of the reference machine:
Speed number ( )
s and diameter number ( δ) are then defined by
2
2
2 2
2
2 6
4
m
m
Q
D
Q
.
,
D
E
D E
F
W
Y W
D
D
=
=
D
D E
Q
s
m
=
(
) .
/
∆
1 4
and
3/ 4
m
r
r
1/ 4
m
Q
( E )
D
.
( E )
Q
D
W
D
=
=
s
s
r
r
D
,
D
.
D
W
W W
=
=
and
3 4
m
r
r
1 4
m
2 Q
( 2 E )
'
D'
.
Q
( 2 E )
D
W
p
p D
=
=
s
3 4
r
m
Q
0.335 ,
'
( 2 E )
W
W
s
W
W
p D
=
=
≈