Page 273 - Fluid Mechanics and Thermodynamics of Turbomachinery

254 Fluid Mechanics, Thermodynamics of Turbomachinery

FIG. 8.6. Flow models used in analysis of minimum number of blades.

where w is the radial velocity, Pw D .dw//.dt/ D w.∂w//.∂r/ (for steady flow),  is
P
the angular velocity and  D d/dt is set equal to zero.
Applying Newton’s second law of motion to a fluid element (as shown in
Figure 6.2) of unit depth, ignoring viscous forces, but putting c r D w, the radial
equation of motion is,

.p C dp/.r C dr/d prd pdrd Df r dm

where the elementary mass dm D rd dr. After simplifying and substituting for f r
from eqn. (8.25a), the following result is obtained,
1 ∂p ∂w 2
C w D  r. (8.37)
∂r ∂r
Integrating eqn. (8.37) with respect to r obtains

2
1
p/ C w 2 1 U D constant (8.38)
2 2
which is merely the inviscid form of eqn. (8.2).
The torque transmitted to the rotor by the fluid manifests itself as a pressure
difference across each radial vane. Consequently, there must be a pressure gradient