Specific speed (Ns) is a dimensionless or dimensional number that describes the shape and design characteristics of a turbine or pump. It is used to classify hydraulic machines and select the right type for a given head and flow rate.
Specific speed is defined as the speed at which a geometrically similar turbine would run if scaled to produce unit power under unit head, or a geometrically similar pump that would deliver unit discharge under unit head.
Specific Speed Formula for Turbines
Ns = N√P / H^(5/4)
Where:
Ns = Specific speed
N = Speed of the turbine (rpm)
P = Power output (kW or hp)
H = Net head (m)
In SI units (with P in kW and H in metres), typical Ns ranges:
Pelton turbine: 4–70
Francis turbine: 60–400
Kaplan turbine: 300–900
Propeller turbine: 500–1200
Specific Speed Formula for Pumps
Ns = N√Q / H^(3/4)
Where:
Q = Discharge / flow rate (m³/s or litres/s)
H = Total head developed (m)
Significance of Specific Speed in Turbine Selection
Turbine Type
Specific Speed (Ns)
Best for
Head Range (m)
Pelton
4–70
High head, low flow
200–2000+
Francis
60–400
Medium head, medium flow
40–600
Kaplan
300–900
Low head, high flow
2–40
Worked Example: Calculate Specific Speed
Problem: A Francis turbine runs at 300 rpm, produces 5000 kW under a net head of 80 m. Calculate its specific speed.
Specific speed (Ns) is a dimensionless (or dimensional) number that characterises the shape and type of a turbine or pump impeller. It allows engineers to select the most efficient machine type for a given application without building physical models.
Dimensional Specific Speed for Turbines
Ns = N × √P / H^(5/4)
where N = speed in rpm, P = shaft power in kW (or HP), H = net head in metres
In SI units (metric): Ns = N(rpm) × √P(kW) / H(m)^(5/4)
Dimensional Specific Speed for Pumps
Ns = N × √Q / H^(3/4)
where Q = flow rate in m³/s (or L/s), H = head in metres
Turbine Selection by Specific Speed
Turbine Type
Specific Speed Ns (SI, turbines)
Application
Pelton Wheel
4 – 70
High head (>300m), low flow
Francis Turbine
60 – 300
Medium head (40–600m)
Kaplan Turbine
300 – 900
Low head (<40m), high flow
Pump Selection by Specific Speed
Pump Type
Specific Speed Ns (SI, pumps)
Application
Centrifugal (radial flow)
10 – 70
High head, low flow
Mixed flow pump
70 – 170
Medium head and flow
Axial flow pump
170 – 500
Low head, very high flow
Solved Example, Kaplan Turbine
A Kaplan turbine develops 12,000 kW under a net head of 20 m at a speed of 150 rpm. Calculate the specific speed.
Ns = N × √P / H^(5/4) = 150 × √12000 / 20^(5/4)
= 150 × 109.5 / 37.97 = 432.5
This confirms a Kaplan turbine (Ns = 300–900) is correctly selected for this low-head, high-power application.
Unit Shape Number (Dimensionless Specific Speed)
The unit shape number Ω_s uses SI units throughout to give a truly dimensionless result:
Ω_s = ω√Q / (gH)^(3/4) for pumps, or Ω_s = ω√P / (ρ^(1/2)(gH)^(5/4)) for turbines
where ω = angular velocity in rad/s, g = 9.81 m/s², ρ = fluid density in kg/m³
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