Page 227 - Fluid Mechanics and Thermodynamics of Turbomachinery

208 Fluid Mechanics, Thermodynamics of Turbomachinery
From eqn. (7.8a), Df0.303/.2 ð 1.303/g 0.5 D 0.3410,
3
4
∴ r D 0.025/. ð 0.91 ð 151.84 ð 0.341/ D 1.689 ð 10 ,
s1
∴ r s1 D 0.05528 m.
The required diameter of the eye is 110.6 mm.
(iii) c x1 D r s1 D 0.341 ð 151.84 ð 0.05528 D 2.862 m/s.
(iv) From eqn. (7.8b),
0.75 b c 2 x1 0.75 ð 0.303 ð 2.862 2
H s D D D 1.632 m.
g 2 9.81 ð 0.341 2

Optimum design of a centrifugal compressor inlet

To obtain high efficiencies from high pressure ratio compressors it is necessary
to limit the relative Mach number at the eye.
The flow area at the eye can be written as
2
2
A 1 D r k, where k D 1 .r h1 /r s1 / .
s1
2
Hence A 1 D kU / 2 .7.9/
s1
with U s1 D r s1 .
With uniform axial velocity the continuity equation is Pm D 1 A 1 c x1 .
Noting from the inlet velocity diagram (Figure 7.1) that c x1 D w s1 cos ˇ s1 and
U s1 D w s1 sin ˇ s1 , then, using eqn. (7.9),

P m 2 3 2
D w sin ˇ s1 cos ˇ s1 . (7.10)
s1
1 k
For a perfect gas it is most convenient to express the static density 1 in terms of
the stagnation temperature T 01 and stagnation pressure p 01 because these parameters
are usually constant at entry to the compressor. Now,
p T 0
D .
0 p 0 T
1 2
With C p T 0 D C p T C c and C p D
R/.
1/
2
T 0
1 2 a 2 0
then D 1 C M D
T 2 a 2
where the Mach number, M D c/.
RT/ 1/2 D c/a, a 0 and a being the stagnation and
local (static) speeds of sound. For isentropic flow,
y/.y 1/
p T
D .
p 0 T 0
Thus,
1 y/.y 1/ 1/.y 1/
1 T 1
1 2
D D 1 C M 1
0 T 0 2