fluid_mechanics

656 Chapter 12 ■ Turbomachines
Combining Eq. 12.12 with this, we get
a p out
r V 2
out
2 gz outb a p in
r V in 2
U
2 gz 2 V u2 U 1 V u1
inb loss
Dividing both sides of this equation by the acceleration of gravity, g, we obtain
where H is total head defined by
U 2 V u 2 U 1 V u1
g
H out H in h L
H p
rg V 2
2g z
and h L is head loss.
From this equation we see that 1U 2 V u2 U 1 V u1 2g is the shaft work head added to the fluid by the
pump. Head loss, h L , reduces the actual head rise, H out H in , achieved by the fluid. Thus, the ideal
head rise possible, h i , is
h i U 2V u2 U 1 V u1
g
(12.13)
The pump actual
head rise is less
than the pump ideal
head rise by an
amount equal to the
head loss in the
pump.
The actual head rise, H out H in h a , is always less than the ideal head rise, h i , by an amount
equal to the head loss, h L , in the pump. Some additional insight into the meaning of Eq. 12.13 can
be obtained by using the following alternate version 1see Eq. 12.82.
h i 1
2g 31V 2 2 V 1 2 2 1U 2 2 U 1 2 2 1W 1 2 W 2 2 24
(12.14)
A detailed examination of the physical interpretation of Eq. 12.14 would reveal the following. The
first term in brackets on the right-hand side represents the increase in the kinetic energy of the
fluid, and the other two terms represent the pressure head rise that develops across the impeller
due to the centrifugal effect, U 2 2 U 2 1 , and the diffusion of relative flow in the blade passages,
W 2 1 W 2 2 .
An appropriate relationship between the flowrate and the pump ideal head rise can be obtained
as follows. Often the fluid has no tangential component of velocity V u1 , or swirl, as it enters
the impeller; that is, the angle between the absolute velocity and the tangential direction is 90°
1a 1 90° in Fig. 12.82. In this case, Eq. 12.13 reduces to
From Fig. 12.8c
h i U 2V u2
g
cot b 2 U 2 V u2
V r2
(12.15)
h i
Q
so that Eq. 12.15 can be expressed as
The flowrate, Q, is related to the radial component of the absolute velocity through the equation
(12.16)
(12.17)
where is the impeller blade height at the radius r 2 . Thus, combining Eqs. 12.16 and 12.17 yields
b 2
h i U 2 2
g U 2V r2 cot b 2
g
Q 2pr 2 b 2 V r2
h i U 2 2
g U 2 cot b 2
2pr 2 b 2 g Q
(12.18)
This equation is graphed in the margin and shows that the ideal or maximum head rise for a centrifugal
pump varies linearly with Q for a given blade geometry and angular velocity. For actual