Pump Calculations

Total head from pressure gauges

The head a pump adds is the difference in total energy between its discharge and suction nozzles, expressed as a height of the liquid pumped:

H = (p_d − p_s) / (ρ g) + (v_d² − v_s²) / (2 g) + (z_d − z_s)

where p is gauge pressure, v the velocity in each nozzle and z the height of each gauge. The velocity term matters when the suction and discharge nozzles differ in size; the height term matters when the gauges are not at the same level. Getting the density right matters as much as the gauges: the same pressure rise is twice the head for a liquid half as dense.

Hydraulic power, shaft power and efficiency

P_hyd = ρ g Q H η_pump = P_hyd / P_shaft

Example. Water at 1000 kg/m³, 100 m³/h and 50 m head: P_hyd = 1000 × 9.81 × (100/3600) × 50 = 13.6 kW. At 75 % pump efficiency the shaft needs 18.2 kW, and the motor draws more again (divide by the motor efficiency). Comparing the efficiency from field readings with the maker's curve at the same flow shows how far the pump has worn.

NPSH available

Net positive suction head available is the margin of pressure above the vapour pressure at the pump suction, as a height of liquid:

NPSH_a = (p_{abs, surface} − p_v) / (ρ g) + h_static − h_friction

Example. Water at 30 °C from an open tank at sea level (101.3 kPa), liquid level 2 m above the pump centreline, 0.5 m suction-line friction. The vapour pressure of water at 30 °C is 4.25 kPa (steam tables) and its density 995.6 kg/m³, so NPSH_a = (101.3 − 4.25) × 1000 / (995.6 × 9.81) + 2 − 0.5 = 11.4 m. NPSH_a must exceed the pump's NPSH_r by a margin; ANSI/HI 9.6.1 recommends margin ratios that depend on the service and the pump's suction energy. Hot liquids, high suction lifts and long suction lines are where the margin disappears.

System curve and operating point

The system curve is the head the piping needs at each flow: a static part (level and pressure difference) plus friction that rises roughly with the square of flow, H_sys = H_static + k Q². The pump runs where its curve crosses the system curve. Throttling a valve steepens the system curve; changing speed moves the pump curve.

Affinity laws

Q₂/Q₁ = N₂/N₁ H₂/H₁ = (N₂/N₁)² P₂/P₁ = (N₂/N₁)³

At 80 % speed a centrifugal pump gives 80 % of the flow, 64 % of the head and needs about 51 % of the power. In a system with a large static head the operating point does not follow the affinity curve, so the real saving from a variable-speed drive must be read from the system curve — the affinity laws calculator does this.

Reciprocating and metering pumps

A positive-displacement pump delivers its swept volume per stroke times the stroke rate, less slip (volumetric efficiency). Its suction must also supply the acceleration head needed to start the liquid column moving on every stroke — often the deciding term for metering pumps on long suction lines.

Calculators for this

Free calculators

Pump efficiency — Total head, hydraulic and shaft power, efficiency from field readings — and what a worn pump costs a year.
NPSH available — NPSH available and margin, with water vapour pressure from steam tables and suction-line friction.
Affinity laws and VFD savings — Flow, head and power at a new speed or impeller size — and the yearly saving from slowing a pump or fan.
Water hammer surge pressure — Joukowsky surge, wave speed for steel, iron, PVC and HDPE pipe, critical closing time and column separation.

Online tools and Excel tools

Pump Head, Power and Efficiency — Head, efficiency and power curves for any impeller diameter and speed from the vendor's curves, the motor size with API 610 margins, low-flow alerts and a field test against the curve. Use online · Excel tool, USD 39
Pump NPSH and Cavitation — NPSH available for four cases side by side — tank or suction-gauge route, altitude, hot liquid, low level — against the vendor NPSH3 curve with margins, suction specific speed and vortex submergence. Use online · Excel tool, USD 29
Pump System Curve and Operating Point — System curve from one known point or from the pipe, the operating point for 1–4 pumps in parallel or series, and throttle vs VFD vs impeller trim to a target flow, with the yearly energy cost. Use online · Excel tool, USD 29
Reciprocating and Metering Pumps — Capacity, slip and brake power for simplex to nonuplex pumps, flow pulsation from the crank motion, acceleration head and NPSH with or without a suction dampener, and metering stroke settings. Use online · Excel tool, USD 29
Pump Troubleshooting Guide — Pick the pump kind and the symptom, answer simple yes/no questions and add any readings — get the likely causes ranked, with what to check, what to do and the source. Use online · Excel tool, USD 19

Field guide

Pumps Made Simple — A 118-page plain-language field guide for operators and young engineers: head and power, curves, systems, NPSH, minimum flow, motors, vertical, reciprocating, rotary and metering pumps, seals, testing, troubleshooting and energy.

Questions

How do I calculate pump efficiency from field readings?

Measure flow, suction and discharge pressure and the motor input power. Work out total head, then hydraulic power ρgQH, and divide by the shaft power (motor input × motor efficiency).

What NPSH margin do I need?

NPSH available must exceed the NPSH required on the pump curve. ANSI/HI 9.6.1 recommends margin ratios that depend on the service and the pump's suction energy; the PlantCalcKit NPSH tool applies them.

Why does a VFD save less than the cube law suggests?

The cube law applies along a system curve with no static head. With static head the operating point moves along the real system curve, and the power falls more slowly.

More calculation guides

These guides and calculators give engineering estimates for screening, troubleshooting and training. Check important decisions against site data, your equipment maker and your plant's approved procedures.