Page 226 - Fluid Mechanics and Thermodynamics of Turbomachinery

Centrifugal Pumps, Fans and Compressors 207
suction surfaces there is normally a rapid velocity increase which produces a further
decrease in pressure. At cavitation inception the dynamic action of the blades causes
2
b .1/2 w /. The parameter
the local pressure to reduce such that p D p v D p 1
1
b which is the blade cavitation coefficient corresponding to the cavitation inception
point, depends upon the blade shape and the flow incidence angle. For conventional
pumps (see Pearsall 1972) operating normally this coefficient lies in the range 0.2 6
b 6 0.4. Thus, at cavitation inception.
1 2 1 2
p 1 D p 01 c x1 D p v C b . w /
1
2 2
1 2
1
1
1 2
2
p v // D c C b . w / D c .1 C b / C b U 2
∴ gH s D .p 01 1 s1
2 x1 2 2 x1 2
where H s is the net positive suction head introduced earlier and it is implied that
this is measured at the shroud radius r D r s1 .
To obtain the optimum inlet design conditions consider the suction specific
speed defined as  ss D Q 1/2 /.gH s / 3/4 , where  D U s1 /r s1 and Q D c x1 A 1 D
2
kr c x1 . Thus,
s1
2
 2 ss U c x1
s1
D D (7.7)
1
1
1 2
1
2
2 3/2
k f c .1 C b / C b U g f .1 C b / C b g 3/2
s1
2 x1
2
2
2
where D c x1 /U s1 . To obtain the condition of maximum  ss , eqn. (7.7) is differ-
entiated with respect to and the result set equal to zero. From this procedure the
optimum conditions are found:

1/2
b
D , .7.8a/
2.1 C b /
1
2
3
gH s D b . U /, .7.8b/
2 2 s1
2 k.2/3/ 1.5 3.420k
2
 D D . .7.8c/
ss 0.5 0.5
b .1 C b / b .1 C b /
EXAMPLE 7.1. The inlet of a centrifugal pump of the type shown in Figure 7.1 is
3
to be designed for optimum conditions when the flow rate of water is 25 dm /s and
the impeller rotational speed is 1450 rev/min. The maximum suction specific speed
 ss D 3.0 (rad) and the inlet eye radius ratio is to be 0.3. Determine
(i) the blade cavitation coefficient,
(ii) the shroud diameter at the eye,
(iii) the eye axial velocity, and
(iv) the NPSH.
Solution. (i) From eqn. (7.8c),
2
2
4
.1 C b / D .3.42 k/ / D 0.1196
b ss
2
2
with k D 1 .r h1 /r s1 / D 1 0.3 D 0.91. Solving iteratively (e.g. using the
Newton Raphson approximation), b D 0.3030.
2
(ii) As Q D kr c x1 and c x1 D r s1
s1
3
then r D Q/. k / and  D 1450 /30 D 151.84 rad/s.
s1