Page 278 - Fluid Mechanics and Thermodynamics of Turbomachinery

Radial Flow Gas Turbines 259
From eqn. (8.45):

0.25
c m3 r 2
cot ˇ 3av D D D 0.5102 ∴ ˇ 3av D 62.97 deg
U 2 r 3av 0.49
0.25
c m3 r 2
cot ˇ 3s D D D 0.3571 ∴ ˇ 3s D 70.35 deg
U 2 r 3s 0.7
0.4544
sec ˇ 3s
∴ D D ð 2 D ð 2 D 2.702.
w 3s w 3av
w 3s
w 2 w 3av w 2 sec ˇ 3av 0.3363
The relative velocity ratio will increase progressively from the hub to the shroud.
EXAMPLE 8.5. Using the data and results given in the examples 8.3 and 8.4
together with the additional information that
(a) the static pressure at rotor exit is 100 kPa, and
(b) the nozzle enthalpy loss coefficient, N D 0.06, determine:
(1) the diameter of the rotor and its speed of rotation;
(2) the vane width to diameter ratio, b 2 /D 2 at rotor inlet.
Solution. (1) The rate of mass flow is given by

2
p 3 c m3 r 3s 2 2
P m D 3 c m3 A 3 D U 2 .1 /r .
2
RT 3 U 2 r 2
From eqn. (8.25), T 03 D T 01 .1 S/ D 1050 ð 0.8 D 840 K.
2 2
2 c m3 U 2
T 3 D T 03 c /.2C p / D T 03
m3
U 2 2C p
2 2
D 840 0.25 ð 538.1 /.2 ð 1150.2/.
Hence, T 3 D 832.1K.
Substituting values into the mass flow equation above,

5 2 2 2
1 D [10 /.287 ð 832.1/] ð 0.25 ð 538.1 ð 0.7 ð ð .1 0.4 /r 2
2
∴ r D 0.01373 and r 2 D 0.1172 m, ∴ D 2 D 0.2343 m
2
∴  D U 2 /r 2 D 4591.3 rad/s .N D 43 843 rev/min/.
(2) The rate of mass flow equation is now written as

2
P m D 2 c m2 A 2 , where A 2 D 2 r 2 b 2 D 4 r .b 2 /D 2 /
2
P m
b 2
∴ D .
4 2 c m2 r 2
D 2
2
Solving for the absolute velocity at rotor inlet and its components,
c 2 D SC p T 01 /U 2 D 0.2 ð 1150.2 ð 1050/538.1 D 448.9 m/s,