Low specific speed reaction turbines

A very low specific speed reaction turbine is a slow Francis runner (about $N_s$ = 60–100, kW units) used near the top of the Francis head range. Its runner is large in diameter with long, narrow passages and nearly radial flow.

Disadvantages

1. Large runner diameter for the power produced, so the turbine itself is heavy and costly.
2. High friction and disc-friction losses in long narrow passages, so peak efficiency is lower than a medium-$N_s$ Francis.
3. Low running speed means a large, costly generator with many poles ($N = 120f/p$).
4. Leakage losses through seals rise at high head.
5. At heads where low-$N_s$ Francis is used, a Pelton may be cheaper and more efficient at part load.

Advantages

1. Low cavitation risk: Thoma's critical $\sigma$ is small at low $N_s$, so the runner can be set higher above tailwater with less excavation.
2. Can work under high heads (up to about 350–600 m), with small discharge.
3. Robust runner, stable operation and smaller draft tube.
4. Less sensitive to tailwater level changes.

Efficiency of Pelton wheel with speed

For a Pelton wheel with jet velocity $V_1$, bucket speed $u$, blade outlet angle $\phi$ and friction factor $k$, the hydraulic efficiency is

$$\eta_h = \frac{2u\,(V_1 - u)(1 + k\cos\phi)}{V_1^2}$$

Setting $\dfrac{d\eta_h}{du} = 0$:

$$\begin{aligned} V_1 - 2u &= 0 \\ u &= \frac{V_1}{2} \\ \eta_{h,max} &= \frac{1 + k\cos\phi}{2} \end{aligned}$$

• Efficiency is a parabola in speed ratio $u/V_1$: zero when the wheel is stationary ($u=0$), maximum at $u/V_1 = 0.5$ (in practice about 0.46 due to losses), and zero again at runaway ($u = V_1$).
• With $k = 1$ and $\phi = 15^\circ$, $\eta_{h,max} = (1+\cos 15^\circ)/2 \approx 0.983$ (theoretical).

 eff
  |      ___
  |    /     \
  |   /       \
  |  /         \
  | /           \
  +--------------+---> u/V1
  0     0.46    1.0

Since a Pelton runs at constant synchronous speed while the jet velocity is fixed by head, the speed ratio stays near optimum, and load changes are handled by the needle; this is why its efficiency is high over a wide load range.