Turbine stage design parameters
Three key nondimensional parameters are related to the shape of the turbine velocity triangles and are used in fixing the preliminary design of a turbine stage.
Design flow coefficient
This was introduced in Chapter 2. It is strictly defined as the ratio of the meridional flow velocity to the blade speed, φ 5 cm/U, but in a purely axial-flow machine, φ 5 cx/U. The value of φ for a stage determines the relative flow angles. A stage with a low value of φ implies highly staggered blades and relative flow angles close to tangential. High values imply low stagger and flow angles closer to axial. For a fixed geometry and fixed rotational speed, the mass flow through the turbine increases with increasing φ. This follows from the continuity equation for steady flow, which can be written for the turbine stage as
Stage loading coefficient
The stage loading is defined as the ratio of the stagnation enthalpy change through a stage to the square of the blade speed, ψ 5 Δh0/U2. In an adiabatic turbine, the stagnation enthalpy change is equal to the specific work, ΔW, and for a purely axial turbine with constant radius, we can use the Euler work equation (Eq. (1.19b)) to write Δh0 5 UΔcθ. The stage loading can, therefore, be written as
where Δcθ represents the change in the tangential component of absolute velocity through the rotor. Thus, high stage loading implies large flow turning and leads to highly “skewed” velocity triangles to achieve this turning. Since the stage loading is a nondimensional measure of the work extraction per stage, a high stage loading is desirable because it means fewer stages are needed to produce a required work output. However, as shown in later sections of this chapter, the stage loading is limited by the effects that high stage loadings have on efficiency.
Stage reaction
The stage reaction is defined as the ratio of the static enthalpy drop in the rotor to the static enthalpy drop across the stage. Thus,
Taking the flow through a turbine as nearly isentropic the equation of the second law of thermo- dynamics, Tds 5 dh 2 dp/ρ, can be approximated by dh 5 dp/ρ, and ignoring compressibility effects, the reaction can thus be approximated as
The reaction, therefore, indicates the drop in pressure across the rotor compared to that for the stage. However, as a design parameter, the reaction is more significant since it describes the asym- metry of the velocity triangles and is, therefore, a statement of the blade geometries. As will be shown later, a 50% reaction turbine implies velocity triangles that are symmetrical, which leads to similar stator and rotor blade shapes. In contrast, a zero reaction turbine stage implies little pressure change through the rotor. This requires rotor blades that are highly cambered, that do not accelerate the relative flow greatly, and low cambered stator blades that produce highly accelerating flow.