Page 271 - Fluid Mechanics and Thermodynamics of Turbomachinery

252 Fluid Mechanics, Thermodynamics of Turbomachinery
TABLE 8.1. Variation of ˛ 2 for several
values of ˇ 2 .
ˇ 2 (deg) 10 20 30 40
˛ 2 (deg) 85 80 75 70

From eqns. (8.27) and (8.30), after some rearranging, a minimum stagnation Mach
number at rotor inlet can be found:

S
2 2 2 2 cos ˇ 2
M 02 D c /a 01 D (8.32)
2

1 1 C cos ˇ 2
and the inlet Mach number can be determined using the equation
2 M 2
2
M D c 2 D 02 (8.33)
2 1 2
a 2 1 .
1/M
2 02
assuming that T 02 D T 01 , the flow through the stator is adiabatic.
Now, from eqn. (8.28)
c 2 1
D .
U 2 1 C tan ˇ 2 / tan ˛ 2
After rearranging eqn. (8.31) to give
1 (8.34)
tan ˇ 2 / tan ˛ 2 D sec ˇ 2
and combining these equations,

c 2 /U 2 D cos ˇ 2 D 1 2/Z. (8.35)
Equation (8.35) is a direct relationship between the number of rotor blades and the
relative flow angle at inlet to the rotor. Also, from eqn. (8.31a),

cos 2˛ 2 D cos.180 ˇ 2 / D cos ˇ 2
2
so that, from the identity cos 2˛ 2 D 2 cos ˛ 2 1, we get the result:
2
cos ˛ 2 D .1 cos ˇ 2 //2 D 1/Z, (8.31b)
using also eqn. (8.35).
EXAMPLE 8.3. An IFR turbine with 12 vanes is required to develop 230 kW from
a supply of dry air available at a stagnation temperature of 1050 K and a flow rate of
1 kg/s. Using the optimum efficiency design method and assuming a total-to-static
efficiency of 0.81, determine:
(1) the absolute and relative flow angles at rotor inlet;
(2) the overall pressure ratio, p 01 /p 3 ,
(3) the rotor tip speed and the inlet absolute Mach number.