Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

decrease with the addition of further stages to the turbine. One of the earliest attempts
to assess the flow variation of a multistage turbine is credited to Stodola (1945), who
formulated the much used “ellipse law”. The curve labelled “multistage” in Figure 4.21
is in agreement with the “ellipse law” expression
[ ]
( ) = -( )
2
÷ 01 0 0 0
12
I II I ,
m T p k 1 p p
Axial-flow Turbines: Two-dimensional Theory 123
FIG. 4.21. Turbine flow characteristics (after Mallinson and Lewis 1948).
(4.32)
where k is a constant.
This expression has been used for many years in steam turbine practice, but an accurate
estimate of the variation in swallowing capacity with pressure ratio is of even
greater importance in gas turbine technology. Whereas the average condensing steam
turbine, even at part-load, operates at very high pressure ratios, some gas turbines may
work at rather low pressure ratios, making flow matching with a compressor a difficult
problem. The constant value of swallowing capacity, reached by the single-stage turbine
at a pressure ratio a little above 2, and the other turbines at progressively higher pressure
ratios, is associated with choking (sonic) conditions in the turbine stator blades.
Flow characteristics of a multistage turbine
Several derivations of the ellipse law are available in the literature. The derivation
given below is a slightly amplified version of the proof given by Horlock (1958). A
more general method has been given by Egli (1936) which takes into consideration the
effects when operating outside the normal low loss region of the blade rows.
Consider a turbine comprising a large number of normal stages, each of 50% reaction;
then, referring to the velocity diagram of Figure 4.22a, c 1 = c3 = w2 and c2 = w3.
If the blade speed is maintained constant and the mass flow is reduced, the fluid angles