Thermodynamics and Propulsion

12.5 Velocity Triangles for an Axial Compressor Stage

Velocity triangles are typically used to relate the flow properties and blade design parameters in the relative frame (rotating with the moving blades), to the properties in the stationary or absolute frame.

We begin by ``unwrapping'' the compressor. That is, we take a cutting plane at a particular radius (e.g. as shown in Figure 12.4) and unwrap it azimuthally to arrive at the diagrams shown in Figure 12.6. Here we have assumed that the area of the annulus through which the flow passes is nearly constant and the density changes are small so that the axial velocity is approximately constant.

Figure 12.6: Velocity triangles for an axial compressor stage. Primed quantities are in the relative frame, unprimed quantities are in the absolute frame.

In drawing these velocity diagrams it is important to note that the flow typically leaves the trailing edges of the blades at approximately the trailing edge angle in the coordinate frame attached to the blade (i.e. relative frame for the rotor, absolute frame for the stator).

We will now write the Euler Turbine Equation in terms of stage design parameters: $\omega$ , the rotational speed, and $\beta_b$ $\beta_c'$ , the leaving angles of the blades.

$$c_p (T_{Tc} - T_{Tb}) = \omega (r_c v_c - r_b v_b)$$

From geometry,

$$v_b = w_b \tan \beta_b$$

and

$$v_c = w_c \tan \beta_c = \omega r_c - w_c \tan\beta_c'$$

so

$$c_p(T_{Tc} - T_{Tb}) = \omega(\omega r_c^2 - w_c r_c \tan \beta_c' - r_b w_b \tan \beta_b)$$

or

$$\underbrace{\frac{T_{Tc}}{T_{Tb}}}_{\substack{\textrm{stagnation ... ...}_{\substack{\textrm{$\beta$'s set by} \textrm{blade design}}}\right)\right].$$

So we see that the total or stagnation temperature rise across the stage increases with the tip Mach number squared, and for fixed positive blade angles, decreases with increasing mass flow. This behavior is represented schematically in Figure 12.7.

Figure 12.7: Compressor behavior