Page 202 - Fluid Mechanics and Thermodynamics of Turbomachinery

Three-dimensional Flows in Axial Turbomachines 183
and hence the gradient in stagnation enthalpy after the rotor is
dh o3 /dr Dd[U.c 2 C c 3 /]/dr Dd.Uc 2 //dr d.Uc 3 sin ˛ 3 //dr.

After differentiating the last term,
dh o D d.Uc 2 / C U.c sin ˛dr/r C sin ˛dc C c cos ˛d˛/ (6.30a)

the subscript 3 having now been dropped.
From eqn. (6.20) the radial equilibrium equation applied to the rotor exit flow is
2
2
dh o D c sin ˛dr/r C cdc. (6.30b)
After logarithmic differentiation of c cos ˛ D constant,
d / C dc/c D tan ˛ d˛. (6.31)
Eliminating successively dh o between eqns. (6.30a) and (6.30b), d / between
eqns. (6.28) and (6.31) and finally d˛ from the resulting equations gives

dc c 2 d.rc / c 2 dr
1 C D sin ˛ C 1 C C M x (6.32)
c U rc U r
p
where M x D M cos ˛ D c cos ˛/ .
RT/ and the static temperature
2
T D T 3 D T o3 c /.2C p /
3
1 2
[U.c 2 C c 3 / C c ]/C p . .6.33/
D T o2
2 3
The verification of eqn. (6.32) is left as an exercise for the diligent student.
Provided that the exit flow angle ˛ 3 at r D r m and the mean rotor blade speeds
are specified, the velocity distribution, etc., at rotor exit can be readily computed
from these equations.

Off-design performance of a stage

A turbine stage is considered here although, with some minor modifications, the
analysis can be made applicable to a compressor stage.
Assuming the flow is at constant entropy, apply the radial equilibrium equation,
eqn. (6.6), to the flow on both sides of the rotor, then
d c 3 d
dh 03 dh 02 dc x3
D  .rc 2 C rc 3 / D c x3 C .rc 3 /.
dr dr dr dr r dr
Therefore

dc x2 c 2 d dc x3 c 3 d
c x2 C  .rc 2 / D c x3 C C  .rc 3 /.
dr r dr dr r dr
r into the above equation, then, after some simpli-
Substituting c 3 D c x3 tan ˇ 3
fication,

dc x2 c 2 d dc x3 c x3 d
c x2 C  .rc 2 / D c x3 C tan ˇ 3 .rc x3 tan ˇ 3 /
dr r dr dr r dr
2c x3 tan ˇ 3 . .6.34/