Page 164 - Fluid Mechanics and Thermodynamics of Turbomachinery

Axial-flow Compressors and Fans 145
The stage loading factor may also be expressed in terms of the lift and drag
coefficients for the rotor. From Figure 3.5, replacing ˛ m with ˇ m , the tangential
blade force on the moving blades per unit span is,

Y D L cos ˇ m C D sin ˇ m

C D
D L cos ˇ m 1 C tan ˇ m ,
C L
1
where tan ˇ m D .tan ˇ 1 C tan ˇ 2 /.
2
2
1
Now C L D L/. w l/ hence substituting for L above,
2 m
1
2
Y D c lC L sec ˇ m .1 C tan ˇ m C D /C L /. (5.15)
2 x
The work done by each moving blade per second is YU and is transferred to the
fluid through one blade passage during that period. Thus, YU D sc x .h 03 h 01 /.
Therefore, the stage loading factor may now be written
h 03 h 01 Y
D D . (5.16)
U 2 sc x U
Substituting eqn. (5.15) in eqn. (5.16) the final result is

D . /2/ sec ˇ m .l/s/.C L C C D tan ˇ m /. (5.17)
In Chapter 3, the approximate analysis indicated that maximum efficiency is obtained
when the mean flow angle is 45 deg. The corresponding optimum stage loading factor
at ˇ m D 45 deg is,
p
opt D . / 2/.l/s/.C L C C D /. (5.18)

Since C D − C L in the normal low loss operating range, it is permissible to drop
C D from eqn. (5.18).

Simplified off-design performance

Horlock (1958) has considered how the stage loading behaves with varying flow
coefficient, and how this off-design performance is influenced by the choice of
design conditions. Now cascade data suggests that fluid outlet angles ˇ 2 (for the
rotor) and ˛ 1 .D ˛ 3 / for the stator, do not change appreciably for a range of incidence
up to the stall point. The simplication may therefore be made that, for a given stage,

tan ˛ 1 C tan ˇ 2 D t D constant. (5.19)

Inserting this expression into eqn. (5.14b) gives
D 1 t. (5.20a)

An inspection of eqns. (5.20a) and (5.14a) indicates that the stagnation enthalpy
rise of the stage increases as the mass flow is reduced, when running at constant