Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

The torque exerted by one blade element at radius r is rdY. If there are Z blades the
elementary torque is
after using eqn. (5.31). Substituting for L and rearranging,
(5.33)
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 2prdr,
where W is the rotor angular velocity and the element of mass flow, dm . = rc x2prdr.
Substituting eqn. (5.33) into eqn. (5.34), then
(5.34)
(5.35)
where s = 2pr/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 2prdr, or
Using eqn. (5.32) and rearranging,
(5.36)
Note that, so far, all the above expressions are applicable to both types of fan shown
in Figure 5.17.
Blade element efficiency
Axial-flow Compressors and Fans 171
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 (p2 - p 1) minus the drop
in pressure across the guide vanes (pe - p 1). The ideal pressure rise across the fan is
given by the product of density and CpDT0. Fan designers define a blade element
efficiency
(5.37)
The drop in static pressure across the guide vanes, assuming frictionless flow for simplicity,
is