Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

220 Fluid Mechanics, Thermodynamics of Turbomachinery
and
Thus, using the relations developed earlier for T01/T 1, p 01/p 1 and r 01/r 1, we obtain
Substituting g = 1.4 for air into eqn. (7.12) we get
(7.11b)
(7.12)
(7.12a)
The right-hand side of eqn. (7.12a) is plotted in Figure 7.4 with a1s = 30deg for M r1 =
0.8 and 0.9, showing that the peak values of m . W 2 /k are significantly increased and occur
at much lower values of b1.
EXAMPLE 7.2. The inlet of a centrifugal compressor is fitted with free-vortex guide
vanes to provide a positive prewhirl of 30deg at the shroud. The inlet hub–shroud radius
ratio is 0.4 and a requirement of the design is that the relative Mach number does not
exceed 0.9. The air mass flow is 1kg/s, the stagnation pressure and temperature are
101.3kPa and 288K. For air take R = 287J/(kgK) and g = 1.4.
Assuming optimum conditions at the shroud, determine
(i) the rotational speed of the impeller;
(ii) the inlet static density downstream of the guide vanes at the shroud and the axial
velocity;
(iii) the inducer tip diameter and velocity.
Solution. (i) From Figure 7.4, the peak value of f(Mr1) = 0.4307 at a relative flow
angle b1 = 49.4deg. The constants needed are , r01 = p01/(RT01) = 1.2255 kg/m3 and k = 1 - 0.42 = 0.84. Thus, from eqn. (7.12a),
W2 = pfkr01a3 01 = 5.4843 ¥ 107 a01 =÷( g RT01)=
340. 2 ms
. Hence,
(ii)
The axial velocity is determined from eqn. (7.11b):