---
title: "Specific speed of a turbine"
subject: "Fluid Machines"
chapter: 3
key: "semester_5/fluid_machines:specific-speed-turbine-derivation"
source: https://www.bctengineer.com/q/semester_5/fluid_machines/specific-speed-turbine-derivation
site: BCT Engineering
---

# Specific speed of a turbine

Fluid Machines · Chapter 3: Water Turbine

- Asked in: year not given, 6 marks
- Video: [Specific speed of turbine](https://www.youtube.com/watch?v=7F0BJhwc15g) — BEST MECHANICAL ENGINEERING
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/fluid_machines/specific-speed-turbine-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_5/fluid_machines/chapter-3

## Question

Define the specific speed of a turbine. Derive an expression for it and state the ranges of specific speed for Pelton, Francis and Kaplan turbines. What is its significance?

## Short answer

Specific speed Ns = N√P / H^(5/4) (rpm, kW, m) is the speed of a geometrically similar turbine producing 1 kW under 1 m head. It is derived from similarity laws: D ∝ √H/N and P ∝ N³D⁵, giving N ∝ √P / H^(5/4). Typical ranges: Pelton 10–35, Francis 60–400, Kaplan 300–1000. It identifies turbine type, aids selection, and predicts performance.

## Answer

**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)

| Turbine | Specific speed |
|---|---|
| Pelton (single jet) | 10 to 35; up to about 60 with multiple jets |
| Francis | 60 to 400 (slow 60 to 120, medium 120 to 180, fast 180 to 300) |
| Kaplan / propeller | 300 to 1000 |

#### 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Specific speed of a turbine is the speed of a similar turbine that:
   A. Develops 1 kW under 1 m head
   B. Discharges 1 m3/s
   C. Runs at 1 rpm
   D. Has D = 1 m
   Answer: A. Ns is defined at P = 1 kW and H = 1 m. (Key point: Definition)

2. The expression is:
   A. N sqrt(Q)/H
   B. N sqrt(H)/P^(5/4)
   C. N P/H^2
   D. N sqrt(P)/H^(5/4)
   Answer: D. From P proportional to H^(5/2)/N^2. (Key point: Formula)

3. The correct order of Ns ranges is:
   A. Pelton 300-1000, Francis 60-400, Kaplan 10-35
   B. Francis lowest
   C. Pelton 10-35, Francis 60-400, Kaplan 300-1000
   D. All equal
   Answer: C. Low Ns is impulse, high Ns is axial. (Key point: Ranges)

4. Higher specific speed means:
   A. Larger runner
   B. Higher speed, smaller machine, cheaper generator
   C. Higher head
   D. Lower speed
   Answer: B. Ns also acts as a type number for selecting turbines. (Key point: Significance)

## Related in Fluid Machines

- [Model testing and specific speed of turbine](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Aturbine-model-similarity-numerical) — Fluid Machines, chapter 3
- [Kaplan turbine construction and working](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Akaplan-turbine-working) — Fluid Machines, chapter 3
- [Kaplan turbine discharge and blade angles](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Akaplan-turbine-numerical) — Fluid Machines, chapter 3
- [Francis turbine: construction and working](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Afrancis-turbine-working-components) — Fluid Machines, chapter 3
- [Francis turbine velocity triangles and power](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Afrancis-turbine-numerical) — Fluid Machines, chapter 3
- [Pump specific speed and affinity laws](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Apump-specific-speed-affinity-numerical) — Fluid Machines, chapter 4

## Related in other subjects

- [Number of Francis/Kaplan units (300 m³/s, 30 m)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aturbine-units-number-francis-kaplan-300cumec) — Hydropower Engineering, chapter 7
- [Low specific speed reaction turbines](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Alow-specific-speed-reaction) — Power Plant Design, chapter 3
- [Pelton vs Francis performance; Francis setting](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Apelton-francis-performance-curves-setting) — Hydropower Engineering, chapter 7
- [Specific speed of a turbine](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Aspecific-speed-purpose) — Power Plant Design, chapter 3, asked 2×