Page 95 - Fluid Mechanics and Thermodynamics of Turbomachinery

76 Fluid Mechanics, Thermodynamics of Turbomachinery
For a compressor cascade (i.e. diffusing flow),
Ł
2
m D 0.23.2a/l/ C ˛ /500, (3.40a)
2
where ˛ is the distance of maximum camber from the leading edge. For the inlet
guide vanes, which are essentially turbine nozzles (i.e. accelerating flow),
m D constant D 0.19 (3.40b)
EXAMPLE 3.1. A compressor cascade has a space chord ratio of unity and blade
inlet and outlet angles of 50 deg and 20 deg respectively. If the blade camber line is
a circular arc (i.e. a/l D 50%) and the cascade is designed to operate at Howell’s
nominal condition, determine the fluid deflection, incidence and ideal lift coefficient
at the design point.

0
Ł
Solution. The camber, D ˛ 0 ˛ D 30 deg. As a first approximation put ˛ D
1 2 2
Ł
20 deg in eqn. (3.40) to give m D 0.27 and, using eqn. (3.39), υ D 0.27 ð 30 D
Ł
8.1 deg. As a better approximation put ˛ D 28.1 deg in eqn. (3.40) giving m D
2
Ł
Ł
0.2862 and υ D 8.6 deg. Thus, ˛ D 28.6 deg is sufficiently accurate.
2
Ł
Ł
Ł
From Figure 3.16, with s/l D 1.0 and ˛ D 28.6 deg obtain ε D ˛ Ł 1 ˛ D
2
2
Ł
0
Ł
21 deg. Hence ˛ D 49.6 deg and the nominal incidence i D ˛ Ł ˛ D0.4 deg.
1 1 1
The ideal lift coefficient is found by setting C D D 0 in eqn. (3.18),
tan ˛ 2 /.
C L D 2.s/l/ cos ˛ m .tan ˛ 1
Ł
Ł
Ł
1
Ł
Ł
Putting ˛ 1 D ˛ , ˛ 2 D ˛ Ł and noting tan ˛ D .tan ˛ C tan ˛ / obtain ˛ D
1 2 m 2 1 2 m
Ł
40.75 deg and C D 2.1.172 0.545/0.758 + 0.95.
L
In conclusion it will be noted that the estimated deviation is one of the most
important quantities for design purposes, as small errors in it are reflected in large
changes in deflection and thus, in predicted performance.
Off-design performance
To obtain the performance of a given cascade at conditions removed from the
design point, generalised performance curves of Howell (1942) shown in Figure 3.19
Ł
Ł
may be used. If the nominal deflection ε and nominal incidence i are known the
off-design performance (deflection, total pressure loss coefficient) of the cascade at
any other incidence is readily calculated.
0
EXAMPLE 3.2. In the previous exercise, with a cascade of s/l D 1.0, a D 50 deg
1
Ł
Ł
0
and a D 20 deg the nominal conditions were ε D 21 deg and i D0.4 deg.
2
Determine the off-design performance of this cascade at an incidence i D 3.8 deg.
Ł
Ł
Solution. Referring to Figure 3.19 and with .i i //ε D 0.2 obtain C D + 0.017,
Ł
ε/ε D 1.15. Thus, the off-design deflection, ε D 24.1 deg.
From eqn. (3.17), the total pressure loss coefficient is,
3
1
2
D p 0 / c D C D /[.s/l/ cos ˛ m ].
2 x
0
Now ˛ 1 D ˛ C i D 53.8 deg, also ˛ 2 D ˛ 1 ε D 29.7 deg, therefore,
1
1
1 1
˛ m D tan f .tan ˛ 1 C tan ˛ 2 /gD tan f0.969gD 44.1 deg,
2