Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

After differentiating the last term,
(6.30a)
the subscript 3 having now been dropped.
From eqn. (6.20) the radial equilibrium equation applied to the rotor exit flow is
After logarithmic differentiation of rccos a = constant,
(6.30b)
(6.31)
Eliminating successively dho between eqns. (6.30a) and (6.30b), dr/r between eqns.
(6.28) and (6.31) and finally da from the resulting equations gives
where Mx = Mcos a = ccos a/÷ (g RT) and the static temperature
(6.32)
(6.33)
The verification of eqn. (6.32) is left as an exercise for the diligent student.
Provided that the exit flow angle a3 at r = 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
Therefore
Three-dimensional Flows in Axial Turbomachines 191
Substituting cq3 = c x3tan b 3 -Wr into the above equation, then, after some simplification,
(6.34)
In a particular problem the quantities c x2, c q2, b 3 are known functions of radius and
W can be specified. Equation (6.34) is thus a first order differential equation in which