Specific speed of a turbine

Specific speed ($N_s$) of a turbine is the speed at which a geometrically similar model turbine would run if it produced 1 kW of power under a head of 1 m. It is a type number that describes the shape of the runner.

Derivation (outline)

For similar turbines, the jet/flow velocity varies as $V \propto \sqrt{H}$ and the peripheral speed $u = \pi D N/60 \propto \sqrt{H}$, so $D \propto \sqrt{H}/N$. Discharge $Q \propto D^2\sqrt{H}$ and power $P \propto QH \propto D^2 H^{3/2}$. Substituting $D$:

$$\begin{aligned} P &\propto \frac{H}{N^2}\,H^{3/2} = \frac{H^{5/2}}{N^2} \\ \Rightarrow N &= K\,\frac{H^{5/4}}{\sqrt{P}} \end{aligned}$$

When $P = 1$ kW and $H = 1$ m, $N = K = N_s$. Hence

$$N_s = \frac{N\sqrt{P}}{H^{5/4}}$$

with $N$ in rpm, $P$ in kW and $H$ in m.

Purpose

1. Selection of turbine type: each type works best in a band of $N_s$:

2. Comparison of turbines: turbines of different size are compared on one scale.
3. Model testing: a model and prototype with the same $N_s$ are geometrically and dynamically similar, so model results can be scaled up.
4. Fixing speed and unit size: for a given head and power, choosing $N_s$ fixes the rotational speed, and hence the generator poles and cost.
5. Cavitation check: the Thoma critical $\sigma$ rises with $N_s$, so $N_s$ fixes how low the runner must be set.

Example: $N = 600$ rpm, $P = 5000$ kW, $H = 200$ m gives $N_s = 600\sqrt{5000}/200^{1.25} \approx 56.4$, which lies at the boundary of multi-jet Pelton and slow Francis, so both would be compared.