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, Eq. (6.6a), to the flow on both sides of the rotor:
In a particular problem, the quantities cx2, cθ2, β3 are known functions of radius and Ω can be specified. Equation (6.34) is thus a first-order differential equation in which cx3 is unknown and may best be solved, in the general case, by numerical iteration. This procedure requires a guessed value of cx3 at the hub and, by applying Eq. (6.34) to a small interval of radius Δr, a new value of cx3 at radius rh 1 Δr is found. By repeating this calculation for successive increments of radius, a complete velocity profile cx3 can be determined. Using the continuity relation
this initial velocity distribution can be integrated and a new, more accurate, estimate of cx3 at the hub is then found. Using this value of cx3, the step-by-step procedure is repeated as described and again checked by continuity. This iterative process is normally rapidly convergent and, in most cases, three cycles of the calculation enable a sufficiently accurate exit velocity profile to be found.