Library / Fluid Mechanics / FLU-06

Fluid Mechanics

Pump shaft power

Power a pump shaft needs to deliver a flow at a given head.

Ref: FLU-06
Rev: v1
Field: Fluid Mechanics
Result: P_shaft [kW]

P_{hyd} = ρ g Q h_{p}, P_{shaft,in} = \frac{P_{hyd}}{η_{p}}

Use actual volumetric flow Q in m³/s, pump head h_p in metres and efficiency as a fraction. Motor electrical input also depends on motor efficiency. Q is entered in L/s.

Lifting a flow of liquid to a head needs power set by the flow, the head and the fluid weight, and the pump wastes some of it as heat.

Calculator

Solve for

P_shaftShaft powerComputed

ρDensity

kg/m³

6001,500

QFlow rate

L/s

0.11,000

h_pPump head

1300

η_pPump efficiency

0.20.95

Hydraulic power5.884 kW
Heat dumped into the fluid2.522 kW

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Shaft power (kW) current value 8.406 kW

QFlow rateKey driver

0.042420

h_pPump head

0.2884.1

η_pPump efficiency

6.1929.4

ρDensity

5.0412.6

Power rises with flow × head, and a pump at 70% efficiency wastes 30% as heat.
Motor selection should add a margin on the shaft power and account for motor efficiency.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (kW)	Local step	Change in result
Q Flow rate	0.1 to 1,000 L/s	0.04203	420.3	420.2	+10%	+10 %
h_p Pump head	1 to 300 m	0.2802	84.06	83.78	+10%	+10 %
η_p Pump efficiency	0.2 to 0.95	6.194	29.42	23.23	+undefined%	—
ρ Density	600 to 1,500 kg/m³	5.043	12.61	7.565	+10%	+10 %

Sweep of Flow rate

Flow rate (L/s)	Shaft power (kW)
0.1	0.04203
0.3162	0.1329
1	0.4203
3.162	1.329
10	4.203
31.62	13.29
100	42.03
316.2	132.9
1,000	420.3

Sweep one variable

Vary

Shaft power (kW)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

P_shaft

Shaft power

Result · kW

What the equation gives you. Everything else feeds it.

Density

kg/m³ · typical 600 to 1,500

Fluid density. Directly proportional.

Flow rate

L/s · typical 0.1 to 1,000

Volumetric flow. Directly proportional.

h_p

Pump head

m · typical 1 to 300

Pump head. Directly proportional.

η_p

Pump efficiency

· typical 0.2 to 0.95

Pump efficiency as a fraction. Inversely proportional.

Worked example with the values above

Flow: 20 L/s = 0.02 m³/s.
Hydraulic power: ρgQh_p = 1,000 × 9.807 × 0.02 × 30 = 5,884 W.
P_shaft = 5.884 / 0.7 = 8.406 kW.

Assumptions and limits

Steady operation at the duty point.
Incompressible fluid.

Common mistakes

Using L/s in an equation that needs m³/s.
Using hydraulic power as the motor rating.

Statistics and uncertainty

Efficiency is read from the pump curve with about ±3–5% accuracy, which adds directly to the power estimate.

Where it is used

Sizing pump motors.
Energy-use estimates for cooling-water circuits.

Maths you need

Pump head: m
Power: W

Related formulas

Practise this formula