Page 274 - Fluid Mechanics and Thermodynamics of Turbomachinery

Radial Flow Gas Turbines 255
in the tangential direction in the space between the vanes. Again, consider the
element of fluid and apply Newton’s second law of motion in the tangential direction
dp.dr D f t dm D 2w. rd dr/.

Hence,
1 ∂p
D 2rw (8.39)
∂
which establishes the magnitude of the tangential pressure gradient. Differentiating
eqn. (8.38) with respect to ,

1 ∂p ∂w
Dw . (8.40)
∂ ∂
Thus, combining eqns. (8.39) and (8.40) gives,

∂w
D2r (8.41)
∂
This result establishes the important fact that the radial velocity is not uniform
across the passage as is frequently assumed. As a consequence of this fact the
radial velocity on one side of a passage is lower than on the other side. Jamieson
(1955), who originated this method, conceived the idea of determining the minimum
number of blades based upon these velocity considerations.
Let the mean radial velocity be w and the angular space between two adjacent
blades be  D 2 /Z where Z is the number of blades. The maximum and minimum
radial velocities are, therefore,
1
w max D w C w D w C r .8.42a/
2
w min D w 1 w D w r .8.42b/
2
using eqn. (8.41).
Making the reasonable assumption that the radial velocity should not drop below
zero, (see Figure 8.6b), then the limiting case occurs at the rotor tip, r D r 2 with
w min D 0. From eqn. (8.42b) with U 2 D r 2 , the minimum number of rotor blades is

Z min D 2 U 2 /w 2 (8.43a)
At the design condition, U 2 D w 2 tan ˛ 2 , hence

(8.43b)
Z min D 2 tan ˛ 2
Jamieson’s result, eqn. (8.43b), is plotted in Figure 8.7 and shows that a large
number of rotor vanes are required, especially for high absolute flow angles at
rotor inlet. In practice a large number of vanes are not used for several reasons,
e.g. excessive flow blockage at rotor exit, a disproportionally large “wetted” surface