Free-vortex turbine stage
The off-design performance may be obtained by making the approximation that the rotor relative exit angle β3 and the nozzle exit angle α2 remain constant at a particular radius with a change in mass flow. This approximation is not unrealistic as cascade data (see Chapter 3) suggest that fluid angles at outlet from a blade row alter very little with change in incidence up to the stall point.
Although any type of flow through a stage may be successfully treated using this method, rather more elegant solutions in closed form can be obtained for a few special cases. One such case is out- lined next for a free-vortex turbine stage whilst other cases are already covered by Eqs (6.21)-(6.23).
Free-vortex turbine stage
Suppose, for simplicity, a free-vortex stage is considered where, at the design point, the flow at rotor exit is completely axial (i.e., without swirl). At stage entry, the flow is again supposed completely axial and of constant stagnation enthalpy h01. Free-vortex conditions prevail at entry to the rotor, rcθ2 5 rcx2 tan α2 5 constant. The problem is to find how the axial velocity distribution at rotor exit varies as the mass flow is altered away from the design value.
At off-design conditions, the relative rotor exit angle β3 is assumed to remain equal to the value β* at the design mass flow (* denotes design conditions). Thus, referring to the velocity triangles in Figure 6.7, at off-design conditions the swirl velocity cθ3 is evidently nonzero:
The foregoing analysis is only a special case of the more general analysis of free-vortex turbine and compressor flows (Horlock & Dixon, 1966), in which rotor exit swirl, rc* is constant (at design conditions), is included. However, from Horlock and Dixon, it is quite clear that even for fairly large values of α* , the value of φ is little different from the value found when α* 5 0, all other factors being equal. In Figure 6.8, values of φ are shown when α* 5 31:4o at φm 5 0.4 (φ* 5 0.8) for comparison with the results obtained when Δ0 5 ð1=2ÞðcxN1 2 cxN2Þ: It should be noted that the rotor efflux flow at off-design conditions is not a free vortex.