Specific speed of a turbine

Definition: The specific speed $N_s$ of a turbine is the speed of a geometrically similar turbine that develops 1 kW of power under a head of 1 m.

Derivation

For geometrically similar turbines, using the dimensionless groups with $D$ = diameter, $N$ = speed, $H$ = head:

• Head coefficient: $\dfrac{gH}{N^2D^2}=\text{constant}\ \Rightarrow\ D\propto\dfrac{\sqrt H}{N}$
• Power coefficient: $\dfrac{P}{\rho N^3D^5}=\text{constant}\ \Rightarrow\ P\propto N^3D^5$

Substituting $D$:

$$P\propto N^3\left(\frac{\sqrt H}{N}\right)^5=N^{-2}H^{5/2}$$

so

$$P\propto\frac{H^{5/2}}{N^2}\ \Rightarrow\ N\propto\frac{\sqrt P}{H^{5/4}}$$

Writing the constant of proportionality as $N_s$ (the value when $P=1$ kW and $H=1$ m):

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

with $N$ in rpm, $P$ in kW (shaft power) and $H$ in m (net head). (If $P$ is in horse power, as in some old books, the numbers differ.)

For a Pelton wheel with more than one jet, $P$ is the power per jet when comparing a single-jet equivalent.

Typical ranges (metric units: rpm, kW, m)

Significance

• It is a type number for the turbine: it identifies the shape of the runner (low Ns: impulse or narrow radial; high Ns: axial).
• Helps to select the type of turbine for given head, power and speed.
• Allows comparison and performance prediction of different turbines, since two similar turbines have the same Ns.
• Gives the speed at which the turbine will run: higher Ns means a higher speed, smaller machine and cheaper generator for the same power.