Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

CHAPTER 4
Axial-flow Turbines:
Two-dimensional Theory
Power is more certainly retained by wary measures than by daring counsels.
(TACITUS, Annals.)
Introduction
The simplest approach to the study of axial-flow turbines (and also axial-flow compressors)
is to assume that the flow conditions prevailing at the mean radius fully represent
the flow at all other radii. This two-dimensional analysis at the pitchline can
provide a reasonable approximation to the actual flow, if the ratio of blade height to
mean radius is small. When this ratio is large, however, as in the final stages of a steam
turbine or in the first stages of an axial compressor, a three-dimensional analysis is
required. Some important aspects of three-dimensional flows in axial turbomachines
are discussed in Chapter 6. Two further assumptions are that radial velocities are zero
and that the flow is invariant along the circumferential direction (i.e. there are no “bladeto-blade”
flow variations).
In this chapter the presentation of the analysis has been devised with compressible
flow effects in mind. This approach is then applicable to both steam and gas turbines
provided that, in the former case, the steam condition remains wholly within the vapour
phase (i.e. superheat region). Much early work concerning flows in steam turbine
nozzles and blade rows are reported in Stodola (1945), Kearton (1958) and Horlock
(1960).
Velocity diagrams of the axial turbine stage
The axial turbine stage comprises a row of fixed guide vanes or nozzles (often called
a stator row) and a row of moving blades or buckets (a rotor row). Fluid enters the
stator with absolute velocity c1 at angle a1 and accelerates to an absolute velocity c2 at
angle a2 (Figure 4.1). All angles are measured from the axial (x) direction. The sign
convention is such that angles and velocities as drawn in Figure 4.1 will be taken as
positive throughout this chapter. From the velocity diagram, the rotor inlet relative
velocity w2, at an angle b2, is found by subtracting, vectorially, the blade speed U from
the absolute velocity c2. The relative flow within the rotor accelerates to velocity w3 at
an angle b3 at rotor outlet; the corresponding absolute flow (c3, a3) is obtained by
adding, vectorially, the blade speed U to the relative velocity w3.
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