Page 272 - Fluid Mechanics and Thermodynamics of Turbomachinery

Radial Flow Gas Turbines 253
Solution. (1) From the gas tables, e.g. Rogers and Mayhew (1995), at T 01 D
1050 K, we can find values for C pD1.1502 kJ/kg K and
D1.333. Using eqn. (8.25),
S D W/.C p T 01 / D 230/.1.15 ð 1050/ D 0.2.

From Whitfield’s eqn. (8.31b),

2
cos ˛ 2 D 1/Z D 0.083333, ∴ ˛ 2 D 73.22 deg
and, from eqn. (8.31a), ˇ 2 D 2.90 ˛ 2 / D 33.56 deg.

(2) Rewriting eqn. (8.26),

/.
1/ 4
p 3 S 0.2 p 01
D 1 D 1 D 0.32165, ∴ D 3.109.
p 01 ts 0.81 p 3
(3) Using eqn. (8.32),

S 2 cos ˇ 2 0.2 2 ð 0.8333
2
M D D ð D 0.5460
02

1 1 C cos ˇ 2 0.333 1 C 0.8333
∴ M 02 D 0.7389.
Using eqn. (8.33),

M 2 02 0.546
2
M D D D 0.6006 ∴ M 2 D 0.775.
2 1 2
1 .
1/M 1 .0.333/2/ ð 0.546
2 02
To find the rotor tip speed, substitute eqn. (8.35) into eqn. (8.27) to obtain:
2
U S
2 cos ˇ 2 D
2
a 01
1
s
r
S 0.2
∴ U 2 D a 01 D 633.8 D 538.1 m/s,
.
1/ cos ˇ 2 0.333 ð 0.8333
p p
where a 01 D
RT 01 D 1.333 ð 287 1050 D 633.8 m/s, and T 02 D T 01 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, f r and f t respectively, of a particle moving in a radial plane
(Figure 8.6a) are:

2
f r DPw  r .8.36a/
P
f t D r C 2w .8.36b/