Fluid Mechanics

Pump Hydraulic Power Calculator

Calculate hydraulic power, shaft power, flow rate, total head, or pump differential pressure rise using either head-based or pressure-based power relationships.

Formula P_h = rho g Q H; P_h = Delta p Q (Delta p across pump)Reviewed Sep 8, 2026

Pump hydraulic power is the useful power transferred from the pump to the liquid. In head form, that useful power is P_h = rho g Q H. In pressure-rise form, it is P_h = Delta p Q, where Delta p is the differential pressure increase across the pump. If you also account for pump efficiency, the required shaft input power becomes P_s = P_h / eta. This calculator covers both formulations so you can work from either total dynamic head or pump pressure rise, depending on how your pump duty is specified.

Calculation Bench

Calculation Basis

Solve for

01Flow Rate

Q · Volumetric flow delivered by the pump at the duty point.

02Fluid Density

rho · Liquid density used in the head-based power relation.

03Gravity

g · Local gravitational acceleration magnitude.

04Total Head

H · Total pump head added to the liquid.

Hydraulic power is the useful power delivered to the liquid. Shaft power is larger because pump efficiency losses are included. For reverse solves, you can enter hydraulic power directly or enter shaft power with efficiency. Always enter a positive delivered flow rate and enter efficiency as a decimal such as 0.75, not 75.

Solution

Enter the required values to calculate hydraulic power.

P_h = rho g Q H

Formula Sheet

P_h = \rho g Q H

P_s = \dfrac{\rho g Q H}{\eta}

Q = \dfrac{P_h}{\rho g H}

H = \dfrac{P_h}{\rho g Q}

P_h = \Delta p Q

P_s = \dfrac{\Delta p Q}{\eta}

Q = \dfrac{P_h}{\Delta p}

\Delta p = \dfrac{P_h}{Q}

P_hHydraulic Power
P_sShaft Power
QFlow Rate
rhoFluid Density
gGravity
HTotal Head
Delta pPump Pressure Rise
etaPump Efficiency

Variables & Units

Symbol	Variable	Description	Common Units
P_h	Hydraulic Power	Useful power actually transferred by the pump to the liquid.	W, kW, hp
P_s	Shaft Power	Mechanical power required at the pump shaft before pump losses are removed by efficiency.	W, kW, hp
Q	Flow Rate	Volumetric flow rate delivered by the pump.	L/min, m³/h, US gpm
rho	Fluid Density	Mass density of the pumped liquid.	kg/m³, g/cm³, lb/ft³
g	Gravity	Local gravitational acceleration magnitude.	m/s², ft/s², g
H	Total Head	Total pump head added to the liquid, typically including static and dynamic components.	m, ft
Delta p	Pump Pressure Rise	Differential pressure increase across the pump used in the pressure-based power form.	kPa, bar, psi
eta	Pump Efficiency	Pump efficiency entered as a decimal ratio between 0 and 1.

How to Use This Calculator

01Choose the calculation basis first. Use Head-Based when the pump duty is given as total head. Use Pressure-Based when you know the differential pressure rise across the pump instead.
02Select which quantity to solve for. Hydraulic Power and Shaft Power modes calculate power directly. Flow Rate and Total Head or Pressure Rise modes back-calculate those variables from a known hydraulic power, or from shaft power when pump efficiency is also known.
03Enter pump efficiency as a decimal between 0 and 1 when solving for shaft power. For example, enter 0.72 for 72%. If you only know a rough duty, 0.60-0.80 is a common first-pass range for many centrifugal pump estimates, but the manufacturer curve at the duty point should control final work.
04Use the fluid density and gravity values that match the pumped liquid and site conditions. In pressure mode, gravity and density are not needed because the pressure rise already captures the fluid-energy increase directly.
05Enter a positive delivered flow rate for pump-duty calculations. A zero-flow shutoff condition is outside the normal operating scope of this tool.

How the Formula Works

A pump adds energy to a liquid. When that energy increase is expressed as head, the hydraulic power delivered to the liquid is P_h = rho g Q H, where rho is fluid density, g is gravity, Q is volumetric flow rate, and H is total dynamic head. The same hydraulic power can also be written in pressure form as P_h = Delta p Q, where Delta p is the pressure increase across the pump, because differential pressure rise times volumetric flow rate also gives the rate of fluid-energy transfer.

Hydraulic power is the useful power imparted to the fluid. The pump shaft must supply more than that because real pumps are not perfectly efficient. If eta is the pump efficiency, shaft input power is P_s = P_h / eta. This means lower efficiency always increases the required shaft power for the same flow and duty point.

For equipment selection, shaft power is the number that usually gets compared against a pump driver or motor nameplate. Select the next standard motor size above the calculated shaft power after applying any required service factor, margin, and motor-efficiency checks.

Worked Example 01

Hydraulic power from flow and total head

Known

Flow Rate (Q): 50 m³/h
Fluid Density (rho): 1000 kg/m³
Gravity (g): 9.81 m/s²
Total Head (H): 30 m

Formula

P_h = rho g Q H

Substitution

P_h = 1000 x 9.81 x (50/3600) x 30

Result

P_h ≈ 4.09 kW

At 50 cubic meters per hour and 30 meters of head, the pump transfers about 4.09 kW of useful hydraulic power to the water.

Worked Example 02

Shaft power from pressure rise and efficiency

Known

Flow Rate (Q): 40 m³/h
Pump Pressure Rise (Delta p): 200 kPa
Pump Efficiency (eta): 0.75

Formula

P_s = Delta p Q / eta

Substitution

P_s = (200,000 x 40/3600) / 0.75

Result

P_s ≈ 2.96 kW

The liquid receives about 2.22 kW of hydraulic power, so the shaft must deliver about 2.96 kW when pump efficiency is 75%.

Worked Example 03

Flow rate from known hydraulic power and head

Known

Hydraulic Power (P_h): 4.0875 kW
Fluid Density (rho): 1000 kg/m³
Gravity (g): 9.81 m/s²
Total Head (H): 30 m

Formula

Q = P_h / (rho g H)

Substitution

Q = 4087.5 / (1000 x 9.81 x 30)

Result

Q ≈ 0.01389 m³/s (50 m³/h)

Rearranging the head-based hydraulic-power equation gives the pump flow rate at the stated duty point.

Applications

01Estimating useful hydraulic power delivered to a liquid at a known pump duty point
02Checking the shaft power required after accounting for pump efficiency
03Back-calculating flow, head, or pump differential pressure rise from known hydraulic power, or from shaft power with efficiency, in pump sizing and screening work
04Estimating the minimum pump shaft power before selecting the next larger standard motor size

Pump Power Input Hints

Input	Typical starting value	Why it matters
Fresh water density	about 1000 kg/m³	Use when specific gravity is close to 1.0
Sea water density	about 1025-1030 kg/m³	Higher density increases power for the same flow and head
Centrifugal pump efficiency	0.60-0.80	Use manufacturer duty-point efficiency when available
Motor selection margin	next standard size above shaft power	Shaft power, not hydraulic power, is the minimum driver load

Assumptions

01The calculation represents steady pump operation at a single duty point.
02Efficiency is treated as a known scalar value when converting between hydraulic power and shaft power.
03The pumped liquid density is treated as constant over the operating condition entered.

Where This Model Stops

01Does not calculate motor electrical input power, NPSH, cavitation margin, or pump curve position; it only evaluates power relationships at a stated duty point.
02Flow-rate and head or pressure-rise solve modes require either hydraulic power directly or shaft power together with pump efficiency.
03Total head and pressure rise are not interchangeable without the proper density relationship; use the mode that matches your available data.

References

[1]
Hydraulic Performance Acceptance Tests - Rotodynamic Pumps
ISO 9906 preview
Defines pump hydraulic power output with the head-based relation P_h = rho Q g H and pump efficiency as hydraulic power divided by pump input power.
[2]
System Curves: Pressure and Head Relationship
Hydraulic Institute Data Tool
Explains the engineering relationship between pressure and head in pump systems and why head is commonly used on pump curves.
[3]
Uniform Test Method for the Measurement of Energy Consumption of Pumps
U.S. e-CFR
Provides an official U.S. field-units form for pump power output based on flow, head, and specific gravity.

Frequently Asked Questions

What is the difference between hydraulic power and shaft power?

Hydraulic power is the useful power imparted to the liquid. Shaft power is the mechanical input power required at the pump shaft before internal pump losses are removed by efficiency. Shaft power is therefore larger than hydraulic power unless efficiency were perfect.

When should I use head-based mode instead of pressure-based mode?

Use head-based mode when pump duty is specified as total head, which is common in pump curves and hydraulic design. Use pressure-based mode when you know the differential pressure rise across the pump directly, such as from process pressure measurements or pressure specifications.

Can I use specific gravity instead of density?

Yes, if you first convert it to density. Multiply specific gravity by the reference density of water, typically about 1000 kg/m³, before entering it in this calculator.

Is shaft power the same as motor electrical power?

No. Shaft power is the mechanical power the motor must deliver to the pump shaft. Electrical input power is higher again because the motor also has losses. To estimate motor input, divide shaft power by motor efficiency, then compare the required shaft power with the motor nameplate and service factor.

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