# Pump Head & Shaft Power Calculator (H, P=ρgQH/η)

Compute the total dynamic head H = (p_d - p_s)/(ρ·g) + Δz + (v_d² - v_s²)/(2g) of a centrifugal pump (m), then the fluid power P_fluid = ρ·g·Q·H and shaft power P_shaft = P_fluid/η. p_d and p_s are gauge pressures (suction may be negative for suction lift). Nozzle velocities v = 4Q/(π·D²) come from the flow rate and the discharge/suction nozzle diameters; if a nozzle diameter is left blank its velocity head is omitted. Pressure in Pa/kPa/bar/atm/psi, flow in m³/s/L/s/L/min/m³/h, density in kg/m³/g/cm³; power output in W/kW/hp.

> Canonical page: https://elysiatools.com/en/tools/pump-head-power

- **Category:** Math & Numbers

- **Keywords:** pump head, total dynamic head, shaft power, fluid power, pump efficiency, centrifugal pump, Bernoulli, velocity head, pressure head, pump power, ρgQH, hydraulics, pump performance, TDH

## Overview

The Pump Head & Shaft Power Calculator computes total dynamic head, fluid power, and shaft power for a centrifugal pump using gauge pressures, elevation difference, flow rate, fluid density, nozzle diameters, efficiency, and gravity.

## Inputs

- **Discharge Pressure p_d (gauge)** (number): Discharge gauge pressure p_d. Enter in the selected Pressure Unit.
- **Pressure Unit** (select)
- **Suction Pressure p_s (gauge)** (number): Suction gauge pressure p_s (may be negative for suction lift). Enter in the selected Pressure Unit.
- **Static Head Difference Δz** (number): Elevation difference Δz between the discharge and suction gauges (m). Positive if the discharge gauge is higher.
- **Flow Rate Q** (number): Volumetric flow rate Q. Enter in the selected Flow Unit.
- **Flow Unit** (select)
- **Density ρ** (number): Liquid density ρ. Enter in the selected Density Unit.
- **Density Unit** (select)
- **Discharge Nozzle Diameter D_d** (number): Discharge nozzle inner diameter D_d. Leave blank to omit the discharge velocity head. Enter in the selected Nozzle Diameter Unit.
- **Suction Nozzle Diameter D_s** (number): Suction nozzle inner diameter D_s. Leave blank to omit the suction velocity head. Enter in the selected Nozzle Diameter Unit.
- **Nozzle Diameter Unit** (select)
- **Efficiency η (%)** (number): Pump overall efficiency η as a percentage (0 < η ≤ 100). Used to compute shaft power P_shaft = P_fluid/η.
- **Gravity g (m/s²)** (number): Gravitational acceleration g in m/s².
- **Decimal Places** (number)

## When to use

- Estimate pump total dynamic head from discharge and suction gauge pressures.
- Calculate hydraulic fluid power and required shaft power at a known flow rate and efficiency.
- Include discharge and suction velocity heads when nozzle diameters are available.

## How it works

- Enter discharge pressure, flow rate, and their units; suction pressure and static head difference are optional.
- Set the liquid density, pump efficiency, gravity, and optional discharge and suction nozzle diameters.
- The calculator converts the inputs and applies H = (p_d − p_s)/(ρg) + Δz + (v_d² − v_s²)/(2g).
- It returns head in meters, fluid power, and shaft power in W, kW, and hp.

## Use cases

- Pump selection and preliminary sizing for water and other liquids.
- Checking operating conditions against a pump's expected head and power demand.
- Estimating motor or shaft power requirements from pressure, flow, density, and efficiency.

## Frequently asked questions

### What pressure type does the calculator use?

It uses gauge pressure for both discharge and suction. Suction pressure may be negative for suction lift.

### Which pressure units are supported?

You can enter pressure in Pa, kPa, bar, atm, or psi.

### Which flow-rate units are supported?

The calculator accepts m³/s, L/s, L/min, and m³/h.

### Are nozzle diameters required?

No. If a discharge or suction diameter is blank, the corresponding velocity-head term is omitted.

### How is shaft power calculated?

Shaft power equals fluid power divided by pump efficiency, with efficiency entered as a percentage.

## Related tools

- [AHU Coil Capacity Calculator (Cooling / Heating)](https://elysiatools.com/en/tools/ahu-capacity-calculator): Compute the total, sensible, and latent capacity of an air-handling-unit (AHU) coil from the entering/leaving air state and the dry-air mass flow ṁ_da. Total capacity Qt = ṁ_da·(h1 − h2); sensible capacity Qs = ṁ_da·cp_ma·(T1 − T2) with cp_ma ≈ 1.006 + 1.86·W \[kJ/(kg da·K)\]; latent capacity Ql = Qt − Qs; Sensible Heat Ratio SHR = Qs / Qt. Each state is described by dry-bulb T plus one humidity input (relative humidity φ, or humidity ratio W); W is derived from the Magnus saturation fit when RH is supplied, and enthalpy h = 1.006·T + W·(2501 + 1.86·T) \[kJ/kg da\]. Signed result — works for cooling or heating coils.
- [Bulk Density & Porosity Calculator](https://elysiatools.com/en/tools/bulk-density-calculator): Bulk density and porosity for granular or porous materials. Three-way solver: pick the unknown (ρ_bulk, m, or V) and supply the other two. ρ_bulk = m/V_bulk. Optionally enter the true particle density ρ_true to compute porosity ε = 1 − ρ_bulk/ρ_true, plus relative density against water. SI units (kg, m³, kg/m³).
- [Force Unit Converter (Extended: N / kN / dyn / lbf / kgf / poundal)](https://elysiatools.com/en/tools/force-unit-converter-extended): Convert force between newton (N, SI base), kilonewton (kN), dyne (dyn, CGS = 10⁻⁵ N), pound-force (lbf = 4.4482216152605 N), kilogram-force (kgf = 9.80665 N), and poundal (pdl = 0.138254954376 N). All factors derive from the exact defining constants 1 lb = 0.45359237 kg, 1 ft = 0.3048 m, and standard gravity g₀ = 9.80665 m/s². Converts via newton to the target unit and lists the equivalent value in all six units. Reference: 1 kg of mass on Earth ≈ 9.80665 N (1 kgf); an adult weighs ≈ 700 N.
- [Hardness Converter (HB / HV / HRC / HRB)](https://elysiatools.com/en/tools/hardness-converter): Convert between Brinell (HB), Vickers (HV), and Rockwell (HRC / HRB) hardness via ASTM E140 tabulated data with linear interpolation. Three material classes: carbon/alloy steel (HV 100–960), austenitic stainless (HV 100–600), and cartridge brass (HV 40–200). Any input scale produces all the others; a scale that does not apply in the current hardness band returns 'out of range'. Indicative only — not for acceptance testing per ASTM E140.
- [Pipe Pressure Drop Calculator (Darcy-Weisbach)](https://elysiatools.com/en/tools/pipe-pressure-drop-darcy): Compute the Darcy-Weisbach major (friction) pressure drop in a straight pipe: ΔP = f·(L/D)·(ρ·v²/2) Pa and head loss h_f = f·(L/D)·v²/(2g) m. The user supplies the Darcy friction factor f (not the Fanning factor). Length in m/km/ft, diameter in m/cm/mm/inch, density in kg/m³/g/cm³/lb/ft³ — all normalised to SI internally. Returns ΔP in Pa, kPa and bar, and head loss in m and ft. Gravity defaults to 9.81 m/s² and may be overridden.
- [Pump NPSH (Net Positive Suction Head) Calculator & Cavitation Check](https://elysiatools.com/en/tools/pump-npsh-calculator): Compute the available Net Positive Suction Head (NPSH_a) of a pump and compare it against the required NPSH (NPSH_r) from the pump curve. Head-balance form: NPSH_a = (p_surface_abs - p_vapor) / (ρ·g) + H_static - h_friction. Surface and vapor pressures are absolute; H_static is positive for flooded suction, negative for suction lift. Returns margin (NPSH_a - NPSH_r), ratio (NPSH_a/NPSH_r) and a classification: safe, marginal (margin < 0.5 m), or cavitation likely. If NPSH_r = 0, only NPSH_a is reported. Pressure in Pa/kPa/bar/atm/psi, length in m/ft, density in kg/m³/g/cm³/lb/ft³.
- [Sensible / Latent Heat Split (SHR = Sensible / Total)](https://elysiatools.com/en/tools/sensible-latent-heat-split): Split an air-conditioning load into sensible and latent components. Total cooling capacity Qt = ṁ_da·(h1 − h2); sensible capacity Qs = ṁ_da·cp_ma·(T1 − T2) with cp_ma ≈ 1.006 + 1.86·W \[kJ/(kg da·K)\]; latent capacity Ql = Qt − Qs; Sensible Heat Ratio SHR = Qs / Qt. Entering/leaving states are described by dry-bulb temperature T and humidity ratio W; enthalpy h = 1.006·T + W·(2501 + 1.86·T) \[kJ/kg da\]. Three modes: full air-state split (T1,W1)→(T2,W2), from Qt & Qs (solve SHR + Ql), or from Qt & SHR (solve Qs, Ql).
- [Torque Unit Converter (N·m / lbf·ft / kgf·m / lbf·in)](https://elysiatools.com/en/tools/torque-unit-converter): Convert torque (moment of force) between N·m (SI base), lbf·ft (pound-foot), kgf·m (kilogram-force-metre), and lbf·in (pound-inch). Factors based on standard gravity g₀ = 9.80665 m/s² and 1 lbf = 4.4482216152605 N. Converts via N·m to the target unit and also lists the equivalent value in all four units for reference. Used for mechanical specs, automotive, and fastener tightening torque.

## Samples

- [Web Image Processing Python Samples](https://elysiatools.com/en/samples/web-image-processing-python): Web Python image processing examples using PIL/Pillow including reading, saving, resizing, and format conversion
- [Android Image Processing Java Samples](https://elysiatools.com/en/samples/android-image-processing-java): Android Java image processing examples including reading/saving images, scaling, and format conversion
- [Android Image Processing Kotlin Samples](https://elysiatools.com/en/samples/android-image-processing-kotlin): Android Kotlin image processing examples including reading/saving images, scaling, and format conversion
- [macOS Image Processing Objective-C Samples](https://elysiatools.com/en/samples/macos-image-processing-objectivec): macOS Objective-C image processing examples including image reading/saving, image scaling, and format conversion