Page 182 - Fluid Mechanics and Thermodynamics of Turbomachinery

Axial-flow Compressors and Fans 163
after using eqn. (5.31). Substituting for L and rearranging,
2
c lZC L rdr cos.ˇ m
/
x
d D 2 Ð . (5.33)
2 cos ˇ m cos
Now the work done by the rotor in unit time equals the product of the stagnation
enthalpy rise and the mass flow rate; for the elementary ring of area 2 rdr,
d D .C p T 0 /d Pm, (5.34)

where  is the rotor angular velocity and the element of mass flow, d Pm D c x 2 rdr.
Substituting eqn. (5.33) into eqn. (5.34), then

l cos.ˇ m
/
U C x
C p T 0 D C p T D C L 2 . (5.35)
2 s cos ˇ m cos
where s D 2 r/Z. Now the static temperature rise equals the stagnation temperature
rise when the velocity is unchanged across the fan; this, in fact, is the case for both
types of fan shown in Figure 5.17.
The increase in static pressure of the whole of the fluid crossing the rotor row
may be found by equating the total axial force on all the blade elements at radius
r with the product of static pressure rise and elementary area 2 rdr,or
ZdX D .p 2 p 1 /2 rdr.
Using eqn. (5.32) and rearranging,
2
c l sin.ˇ m
/
x
p 2 p 1 D C L 2 (5.36)
2 s cos ˇ m cos
Note that, so far, all the above expressions are applicable to both types of fan shown
in Figure 5.17.

Blade element efficiency

Consider the fan type shown in Figure 5.17a fitted with guide vanes at inlet. The
pressure rise across this fan is equal to the rotor pressure rise (p 2 p 1 ) minus the
drop in pressure across the guide vanes (p e p 1 ). The ideal pressure rise across
the fan is given by the product of density and C p T 0 . Fan designers define a blade
element efficiency

p 1 / p 1 /g/. C p T 0 /. (5.37)
b Df.p 2 .p e
The drop in static pressure across the guide vanes, assuming frictionless flow for
simplicity, is

2
1
2
1
p 1 D .c 2 c / D c . (5.38)
p e 1 x y1
2 2
Now since the change in swirl velocity across the rotor is equal and opposite to the
swirl produced by the guide vanes, the work done per unit mass flow, C p T 0 is
equal to Uc y1 . Thus the second term in eqn. (5.37) is
p 1 //. C p T 0 / D c y1 /.2U/. (5.39)
.p e