Page 162 - Fluid Mechanics and Thermodynamics of Turbomachinery

Axial-flow Compressors and Fans 143
Reaction ratio

For the case of incompressible and reversible flow it is permissible to define the
reaction R, as the ratio of static pressure rise in the rotor to the static pressure rise
in the stage

R D .p 2 p 1 //.p 3 p 1 /. (5.10a)

If the flow is both compressible and irreversible a more general definition of R is
the ratio of the rotor static enthalpy rise to the stage static enthalpy rise,

R D .h 2 h 1 //.h 3 h 1 /. (5.10b)

1
2
From eqn. (5.2), h 2 h 1 D .w 2 1 w /. For normal stages (c 1 D c 3 ), h 3 h 1 D
2
2
h 03 h 01 D U.c y2 c y1 /. Substituting into eqn. (5.10b)
w 1 2 w 2 2
R D .5.10c/
2U.c y2 c y1 /
w y2 /
.w y1 C w y2 /.w y1
D ,
2U.c y2 c y1 /
where it is assumed that c x is constant across the stage. From Figure 5.2, c y2 D
U w y2 and c y1 D U w y1 so that c y2 c y1 D w y1 w y2 . Thus,

R D .w y1 C w y2 //.2U/ D .c x /U/ tan ˇ m , (5.11)

where

1
tan ˇ m D .tan ˇ 1 C tan ˇ 2 /. (5.12)
2
An alternative useful expression for reaction can be found in terms of the fluid
outlet angles from each blade row in a stage. With w y1 D U c y1 , eqn. (5.11) gives,

R D 1 C .tan ˇ 2 tan ˛ 1 /c x /.2U/. (5.13)
2
Both expressions for reaction given above may be derived on a basis of incompress-
ible, reversible flow, together with the definition of reaction in eqn. (5.10a).

Choice of reaction
The reaction ratio is a design parameter which has an important influence on stage
efficiency. Stages having 50% reaction are widely used as the adverse (retarding)
pressure gradient through the rotor and stator rows is equally shared. This choice
of reaction minimises the tendency of the blade boundary layers to separate from
the solid surfaces, thus avoiding large stagnation pressure losses.
If R D 0.5, then ˛ 1 D ˇ 2 from eqn. (5.13), and the velocity diagram is symmet-
rical. The stage enthalpy rise is equally distributed between the rotor and stator rows.