Page 112 - Fluid Mechanics and Thermodynamics of Turbomachinery

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 pitch-
line 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 “blade-to-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 c 1 at angle ˛ 1 and accelerates to an absolute
velocity c 2 at angle ˛ 2 (Figure 4.1). All angles are measured from the axial (x) direc-
tion. 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 w 2 , at an angle ˇ 2 , is found by subtracting, vectorially,
the blade speed U from the absolute velocity c 2 . The relative flow within the rotor
accelerates to velocity w 3 at an angle ˇ 3 at rotor outlet; the corresponding absolute
flow (c 3 ,˛ 3 ) is obtained by adding, vectorially, the blade speed U to the relative
velocity w 3 .
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