7 Dynamic Similitude
270
(7.11)
It is further found that the flow rate and the energy rise are related as with a rotational speed change, according to
(7.12)
Corrections applied to flow rate and energy rise imply a correction of the kinematic
similitude condition. This means that, with different fluids, operation points with
optimum efficiency do not exactly correspond according to kinematic similitude.
From the quoted study emerges that the calculation method is very accurate on
condition that the loss change (1 − η i ) is limited to about 20 %.
Possible values for compressors are u 2 = 300 m/s; d 2 = 400 mm; b 2 = 20 mm;
R a = 2 µm (milled surfaces); ν = 15 · 10
−6
m
2
/s (air). This results in Re = ( u 2 b 2 )/ν = 4 · 10
5
,
ε = 5R a /b 2 = 5 · 10
− 4
. A point in the transition area (both effect of Re and ε) corresponds
in the Moody diagram. Possible values with pumps are u 2 = 30 m/s; d 2 = 200 mm;
b 2 = 10 mm; R a = 20 µm (smooth sand cast surfaces); ν = 10
−6
m
2
/s (water), resulting in Re = ( u 2 b 2 )/ν = 3 10
5
, ε = 5R a /b 2 = 10
−2
. The point in the Moody chart is in the
rough surfaces area (no effect of Re). A point within the transition area is obtained
with a fluid that is ten times as viscous (light oil).
7.4.4 Rotor Diameter Reduction: Impeller Trimming
The rotor diameter may be reduced by turning on a lathe in order to lower the head
of a pump (impeller trimming). When designing a pump installation, it would be a
mere coincidence if an exactly appropriate pump were found with a manufacturer.
In general, a pump with a slightly too high head shall be chosen. The head may then
be diminished by reduction of the rotor diameter. This way, a manufacturer can
cover a certain head and flow rate range with the same pump. Figure 7.15 renders an
application field for a given pump type. The meaning of the coding is the diameter
of the suction pipe combined with the rotor diameter. When trimming, normally
only the blade tips are cut. The hub disc and the shroud, if any, often must be kept,
as they contribute to the rotor sealing and the guiding of the fluid to the post-connected stator. Assuming that trimming does not change the rotor blade outlet angle,
which is normally met with a good approximation, flow rate and energy rise change
according to
2
2
2
2
m
2
2
Q ~ u ~ d , E ~ u ~ d ,
D
on the basis of kinematic similitude on
the outlet velocity triangle. Similitude is not perfect however. A decrease of the
blade length diminishes the work done on the fluid. In other words, the slip between
flow and geometry increases. This causes the energy rise to be lower than predicted
by the simple reasoning. Furthermore, flow rate reduction at a constant rotational
speed generates incidence at the rotor inlet and the post-connected stator. So, the
it
t
i
W 0.50 0.50 .
W
h
D
D
h
=
+
.
m
t
mt
E
Q
Q
E
∆
= ∆
270
(7.11)
It is further found that the flow rate and the energy rise are related as with a rotational speed change, according to
(7.12)
Corrections applied to flow rate and energy rise imply a correction of the kinematic
similitude condition. This means that, with different fluids, operation points with
optimum efficiency do not exactly correspond according to kinematic similitude.
From the quoted study emerges that the calculation method is very accurate on
condition that the loss change (1 − η i ) is limited to about 20 %.
Possible values for compressors are u 2 = 300 m/s; d 2 = 400 mm; b 2 = 20 mm;
R a = 2 µm (milled surfaces); ν = 15 · 10
−6
m
2
/s (air). This results in Re = ( u 2 b 2 )/ν = 4 · 10
5
,
ε = 5R a /b 2 = 5 · 10
− 4
. A point in the transition area (both effect of Re and ε) corresponds
in the Moody diagram. Possible values with pumps are u 2 = 30 m/s; d 2 = 200 mm;
b 2 = 10 mm; R a = 20 µm (smooth sand cast surfaces); ν = 10
−6
m
2
/s (water), resulting in Re = ( u 2 b 2 )/ν = 3 10
5
, ε = 5R a /b 2 = 10
−2
. The point in the Moody chart is in the
rough surfaces area (no effect of Re). A point within the transition area is obtained
with a fluid that is ten times as viscous (light oil).
7.4.4 Rotor Diameter Reduction: Impeller Trimming
The rotor diameter may be reduced by turning on a lathe in order to lower the head
of a pump (impeller trimming). When designing a pump installation, it would be a
mere coincidence if an exactly appropriate pump were found with a manufacturer.
In general, a pump with a slightly too high head shall be chosen. The head may then
be diminished by reduction of the rotor diameter. This way, a manufacturer can
cover a certain head and flow rate range with the same pump. Figure 7.15 renders an
application field for a given pump type. The meaning of the coding is the diameter
of the suction pipe combined with the rotor diameter. When trimming, normally
only the blade tips are cut. The hub disc and the shroud, if any, often must be kept,
as they contribute to the rotor sealing and the guiding of the fluid to the post-connected stator. Assuming that trimming does not change the rotor blade outlet angle,
which is normally met with a good approximation, flow rate and energy rise change
according to
2
2
2
2
m
2
2
Q ~ u ~ d , E ~ u ~ d ,
D
on the basis of kinematic similitude on
the outlet velocity triangle. Similitude is not perfect however. A decrease of the
blade length diminishes the work done on the fluid. In other words, the slip between
flow and geometry increases. This causes the energy rise to be lower than predicted
by the simple reasoning. Furthermore, flow rate reduction at a constant rotational
speed generates incidence at the rotor inlet and the post-connected stator. So, the
it
t
i
W 0.50 0.50 .
W
h
D
D
h
=
+
.
m
t
mt
E
Q
Q
E
∆
= ∆