Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

(ii) Rewriting eqn. (8.26),
(iii) Using eqn. (8.32),
Using eqn. (8.33),
Radial Flow Gas Turbines 263
To find the rotor tip speed, substitute eqn. (8.35) into eqn. (8.27) to obtain
where a01 = g RT01
= 1. 333 ¥ 2, 871, 050 = 633. 8 ms,
and T02 = T01 is assumed.
Criterion for minimum number of blades
The following simple analysis of the relative flow in a radially bladed rotor is of considerable
interest as it illustrates an important fundamental point concerning blade
spacing. From elementary mechanics, the radial and transverse components of acceleration,
fr and f t respectively, of a particle moving in a radial plane (Figure 8.6a) are
(8.36a)
(8.36b)
where w is the radial velocity, w . = (dw)/(dt) = w(∂w)/(∂r) (for steady flow), W is the
angular velocity and W . = dW/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 cr = w, the radial equation of motion
is
where the elementary mass dm = rrdqdr. After simplifying and substituting for fr from
eqn. (8.25a), the following result is obtained,
Integrating eqn. (8.37) with respect to r obtains
(8.37)