Impulse turbine blade work and diagram efficiency

Velocity diagrams

Steam leaves the nozzle at $C_1$ at angle $\alpha_1$ to the blade direction. The relative velocity at inlet is $V_{r1}$ (angle $\beta_1$) and at outlet $V_{r2}$ (angle $\beta_2$); the absolute outlet velocity is $C_2$.

 Inlet                      Outlet
      C1                       Vr2  C2
     /|\                        \   /|
    / | \ Ca1                    \ / | Ca2
   /a1| b1\Vr1                    \b2|
  +---U---+----                 ---U--+--
  |<--Cw1 ----->|            |<--Vw2-->|

Work done

Take the blade speed $U$ as positive. The whirl of the relative velocity at inlet is $V_{w1} = C_1\cos\alpha_1 - U$. At outlet the relative velocity has whirl $V_{w2} = V_{r2}\cos\beta_2$ in the direction opposite to $U$. With blade friction, $V_{r2} = k V_{r1}$, and $V_{r1}\cos\beta_1 = V_{w1}$.

The change of whirl velocity is $V_{w1} + V_{w2}$, so the work per kg is:

$$w = U (V_{w1} + V_{w2}) = U (C_1\cos\alpha_1 - U)\left(1 + k\frac{\cos\beta_2}{\cos\beta_1}\right)$$

Diagram efficiency

$$\eta_d = \frac{w}{C_1^2/2} = \frac{2U (C_1\cos\alpha_1 - U)(1 + kc)}{C_1^2}, \quad c = \frac{\cos\beta_2}{\cos\beta_1}$$

With blade speed ratio $\rho = U/C_1$:

$$\eta_d = 2\rho(\cos\alpha_1 - \rho)(1 + kc)$$

Condition for maximum

$$\frac{d\eta_d}{d\rho} = 2(1 + kc)(\cos\alpha_1 - 2\rho) = 0 \;\Rightarrow\; \rho = \frac{\cos\alpha_1}{2}$$

that is, $U = \tfrac12 C_1\cos\alpha_1 = \tfrac12 C_{w1}$. The second derivative is negative, so this is a maximum.

Maximum efficiency

$$\eta_{d,max} = \frac{\cos^2\alpha_1}{2}(1 + kc)$$

For a symmetrical blade ($\beta_1 = \beta_2$, so $c = 1$) with no friction ($k = 1$):

$$\boxed{\eta_{d,max} = \cos^2\alpha_1}$$

For $\alpha_1 = 20^\circ$ this is $0.883$. With friction ($k < 1$) the value is lower. Because the whole enthalpy drop in one nozzle gives a very high $C_1$, the optimum blade speed $U \approx 0.47C_1$ is still very high. This is why a single-stage impulse turbine runs at a very high rotor speed, and why velocity compounding or pressure compounding is used.