pump lecture notes02 2 up new
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in order to minimise energy loss the fluid should impinge on the bladetangentially, in terms of the vector triangle this means that the angle
between u1andR1should equal the blade angle, (defined as the angle
between the tangent to the pitch circle and the leading edge of the
impeller blade).
Figure 4(a) shows the fluid entering the runner tangentially, this is called shockless
entry, and Figure 4(b) shows the fluid entering the impeller non-tangentially i.e.under
non design conditions. When this happens the follow occur
impact losses occur boundary layer separation takes place eddies arise which give rise to some back flow into the inlet pipe, this
causes the incoming flow to have some whirl velocity
The result of this non tangential entry is a dramatic drop in the efficiency of the pump.
4.2 Analysis of centrifugal pump behaviour
Consider the inlet triangle in Figure 3, under design conditions we can immediately
write down
11
11
1
1
1
1Vand
60,tan
BD
QV
NDu
u
Vf
where
D1 diameter of the impeller at inletN speed of rotation in RPM
Q flow through the pump
B1 width of the impeller at inlet
Therefore we can write
NBD
Q
ND
BDQ
121
2
1
1
11
60
tan
60tan
This defines the entry angle of the vane if 01wV .
For the exit triangle three cases have to be considered as follows
(i) Forward facing blades < /2
Forward facing blades are ones which face in the same direction as the
rotation as shown in Figure 5(a), the symbols have the same meaning as
for the inlet triangle with the subscripts changed to 2
8
V2
Vt2
Vw2
U2
R2
Figure 5(a)
(ii) Radial blades = /2
This case is shown in Figure 5(b), the vector triangle is right angled
hence22 22
and fw VRuV
V2 Vt2
Vw2U2
R2
=
=
= 90
Figure 5(b)
(iii) Backward facing blades > /2
This is shown in Figure 5(c), for the time being all that needs be noted
from this triangle is that V2is smaller than in the other two cases.
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V2
Vt2
Vw2
U2
R2
For analytical purposes it is easiest to consider the triangle resulting from the forward
facing blades however the results obtained will apply to all the cases with no changes
in the signs.
Using equation 2.1 with1w
V set to zero and recalling that since we are now
considering pumps the sign will change then we can say that for an inviscid fluid the
head difference across the pump would be
g
uVw 22
from the triangle in Figure 5(a) we can write
cot22 2 fw
VuV
cot22 2 fw
VuV (4.1)
The head imposed on the fluid
5
is the energy given to it g
uVw 22less any losses, hi, in
travelling through the impeller. As the fluid leaves the impeller and enters the volute
a relatively small amount of the total energy is potential (i.e.pressure) energy much of
it is kinetic; this has to be converted to potential energy by the volute and diverging
delivery pipe. However efficiently the volute converts the kinetic energy to potential
there is still a head loss, hv.
We can now write the energy conservation equation in the form
5Remember that head is energy per unit weight of fluid.
10
g
vhhH
g
uV pvi
w
2
222
where vpis the velocity of flow in the outlet pipe. In order to advance this analysis wemust be able to evaluate the losses hiand hv, it is not possible to do this analytically
therefore the same method will be used as for minor losses in pipes. We assume that
the loss in the impeller is proportional to 22R since this is the velocity of flow relative
to the impeller, similarly the loss in the volute is assumed to be proportional to 22V
hence we can now write the energy equation as
g
v
g
Vk
g
Rk
g
uVH
pvi
w
222
222
2222
vpis very small compared to the other terms and can safely be ignored.We can write
2222 22 fw
VVV
Using equation 4.1 we can now write
g
VVuukVkVuuH
VV
R
VVuu
VVuuV
VVuV
ffvfif
ff
ff
ff
ff
2
coseccot2coseccot22
cosecsin
also
coseccot2
cot1cot2
cot
222
22
222
22
22
2
222
22222
222
22
22
222
22
2222
2
2
22
22
22
Tidying this expression up yields
22
22
222
22
2
22
and60
but
2
cosec1cot22
BD
QV
DNu
g
kkvkVuku
H
f
ivfvfv
aswritten-rebelyconvenientcanequationthehence
22 QCQNBNAH
whereA,Band Care constants defined by the properties of the pump.
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Matching Pump to Pipeline
0.00
10.00
20.00
30.00
40.00
50.00
60.00
0 0.02 0.04 0.06 0.08 0.1 0.12
Flow
Head
0
10
20
30
40
50
60
70
80
90
Pump
Pipeline
Operating point
Figure 7 An inefficient system
Matching a pump to a pipeline
0.00
10.00
20.00
30.00
40.00
50.00
60.00
0 0.02 0.04 0.06 0.08 0.1 0.12
Flow
Head
0
10
20
30
40
50
60
70
Pump
Pipeline
Operating point
hstatic
Figure 8 An efficient system
From these diagrams it is clear that the part of theH Qthat lies under the point of
maximum efficiency is quite short thus the remainder of the curve is of little interest.
Pump manufacturers use this property to put the efficient part of theH- Qfor all the
pumps in an homologous series onto one graph. An example7of this is shown in
Figure 9. The short length of curve is that part of the curve under the highest point of
the efficiency curve, this enables the user, once theH Qfor the pipeline is known to
read off the model number of the pump most suitable for the job.
7 This diagram is reproduced by kind permission of Weir Pumps Ltd., Glasgow.
16
Figure 9
If one pump cannot produce sufficient head then two or more pumps may be used in
series; for the great majority of pipelines this would not be considered a good
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