Page 230 - Fluid Mechanics and Thermodynamics of Turbomachinery

Centrifugal Pumps, Fans and Compressors 211
Thus, using the relations developed earlier for T 01 /T 1 ,p 01 /p 1 and 01 / 1 , we obtain
3
3
P m 2 M cos ˇ 1 .tan ˇ 1 C tan ˛ 1 / 2
r1
f.M r1 / D 3 D 3 . (7.12)
k 01 a 01
1 1 C 2

1
2
2
2
1 C M cos ˇ 1 / cos ˛ 1
r1
2
Substituting
D 1.4 for air into eqn. (7.12) we get:
2
3
3
 Pm M cos ˇ 1 .tan ˇ 1 C tan ˛ 1 / 2
r1
f.M r1 / D D . (7.12a)
k 01 a 3 1 2 2 2 4
01 1 C M cos ˇ 1 / cos ˛ 1
5 r1
The rhs of eqn. (7.12a) is plotted in Figure 7.4 with ˛ 1s D 30 deg for M r1 D 0.8 and
2
0.9, showing that the peak values of Pm /k are significantly increased and occur
at much lower values of ˇ 1 .
EXAMPLE 7.2. The inlet of a centrifugal compressor is fitted with free-vortex guide
vanes to provide a positive prewhirl of 30 deg 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 1 kg/s, the stagnation pressure and temper-
ature are 101.3 kPa and 288 K. For air take R D 287 J/(kg K) and
D 1.4.
Assuming optimum conditions at the shroud, determine:
(1) the rotational speed of the impeller;
(2) the inlet static density downstream of the guide vanes at the shroud and the
axial velocity;
(3) the inducer tip diameter and velocity.
Solution. (1) From Figure 7.4, the peak value of f.M r1 / D 0.4307 at a
p
relative flow angle ˇ 1 D 49.4 deg. The constants needed are a 01 D .
RT 01 / D
3
2
340.2 m/s, 01 D p 01 /.RT 01 / D 1.2255 kg/m and k D 1 0.4 D 0.84. Thus, from
7
2
eqn. (7.12a),  D fk 01 a 3 D 5.4843 ð 10 . Hence,
01
 D 7405.6 rad/s and N D 70 718 rev/min.
1.2255
01 3
D D 0.98464 kg/m .
(2) 1 D 1 2.5 2.5
1 C .M r1 cos ˇ 1 / 2 1.06973
5
The axial velocity is determined from eqn. (7.11b):
2
 Pm 5.4843 ð 10 7
3
3
.w 1 cos ˇ 1 / D c D D ,
x 2
k 1 .tan ˇ 1 C tan ˛ 1 / ð 0.84 ð 0.98464 ð 3.0418
6
D 6.9388 ð 10 ,
∴ c x D 190.73 m/s.
P m
2
(3) A 1 D D kr ,
s1
1 c x
P m 1 3
2
∴ r D D D 2.0178 ð 10 ,
s1
1 c x k ð 0.98464 ð 190.73 ð 0.84
∴ r s1 D 0.04492 m and d s1 D 8.984 cm,
U D r s1 D 7405.6 ð 0.04492 D 332.7 m/s.