Page 204 - Fluid Mechanics and Thermodynamics of Turbomachinery

Three-dimensional Flows in Axial Turbomachines 185

FIG. 6.7. Design and off-design velocity triangles for a free-vortex turbine stage.

The radial equilibrium equation at rotor outlet gives

dh 03 dc x3 c 3 d d
D c x3 C .rc 3 / D .rc 3 /, (6.38)
dr dr r dr dr
after combining with eqn. (6.33), nothing that dh 02 /dr D 0 and that .d/dr/.rc 2 / D 0
at all mass flows. From eqn. (6.37),

c 3 c x3 2 c x3
 C D  ,rc 3 D r 1 ,
r c Ł c Ł
x3 x3
which when substituted into eqn. (6.38) gives,
2
dc x3  2 c x3 r dc x3
D 2r 1 C .
dr c Ł c Ł c Ł dr
x3 x3 x3
After rearranging,
2 2
dc x3 d. r /
D . (6.39)
Ł
2 2
c .c Ł2 C  r /
c x3
x3 x3
Equation (6.39) is immediately integrated in the form
Ł2
2 2
c x3 c Ł x3 D c x3 C  r m (6.40)
2 2
c Ł c Ł2 C  r
c x3m
x3 x3
where c x3 D c x3m at r D r m . Equation (6.40) is more conveniently expressed in a
Ł Ł
non-dimensional form by introducing flow coefficients D c x3 /U m , D c /U m
x3