Calculations with context

Pump operating-point calculator

Calculate the intersection of pump and system curves; inspect flow, head and model assumptions together.

Build the model

Find where the pump and system curves meet. Flow, head and conditional power calculations share the same explicit assumptions.

Hpump = H₀ − aQ² Hsystem = Hstatic + kQ²

Inputs and units

Q: m³/h · H: m · a and k: m/(m³/h)². Use a decimal point or comma; do not use thousands separators.

Shut-off head H₀ m10⁻⁹–10,000 m.

Pump coefficient a m/(m³/h)²0 or 10⁻¹² to 10,000. Positive values define a falling curve.

Static head Hstatic m0 or 10⁻⁹–10,000 m. Flow-independent elevation and pressure head difference.

System coefficient k m/(m³/h)²0 or 10⁻¹² to 10,000. Must use the same Q unit.

Curve lower flow limit m³/h0–100,000 m³/h; must be below the upper limit.

Curve upper flow limit m³/hGreater than 0, up to 100,000 m³/h.

Density ρ · optional kg/m³1–30,000 kg/m³. 1,000 is the illustrative water assumption only.

Pump efficiency η · optional %0.1–100%. Total pump efficiency at this operating point.

The flow envelope defines where these assumed curves may be used in this calculation. It is not the manufacturer’s permissible operating range.

Power requires density. Shaft power also requires total pump efficiency at this point. Motor and drive efficiency are not used.

The ideal parallel example uses the equivalent curve of two identical pumps: each carries half the total flow, so a = 0.02/4 = 0.005. Branch losses and unequal pumps are not modeled; power is total hydraulic power.

Calculated result

The calculator is loading. The worked example below remains readable without JavaScript.

This calculator processes inputs and creates the CSV locally in your browser. It does not send or store calculator data.

How the calculation works

The operating point is where the head delivered by a fixed-speed pump equals the head required by the system at the same flow. This tool uses one pump characteristic (or the stated ideal equivalent curve) and one system curve, with steady, single-phase, incompressible flow.

Here the pump curve is represented by a falling parabola. Real pump curves do not all have this shape. The kQ² system loss is an approximation for roughly constant loss coefficients; laminar flow, changing viscosity and complex networks can exceed its scope.

Rearranging the equality gives (a + k)Q² = H₀ − Hstatic. For positive a + k, Q = √[(H₀ − Hstatic)/(a + k)]. Substitute into both curves to check H. The negative root is outside this forward-flow model.

For power, convert flow to m³/s: QSI = Q/3600. Phydraulic = ρ × 9.81 × QSI × H / 1000 gives kW. At a valid positive point, and only if efficiency is supplied, pump shaft power is Phydraulic/(η/100). This is not electrical consumption or a motor selection.

Worked example: increasing resistance

Original invented curves for this site, not measurements from any manufacturer. Assume H₀ = 30 m, a = 0.02, Hstatic = 6 m, a curve envelope of 0–30 m³/h and ρ = 1,000 kg/m³.

Baseline, resistance and ideal equivalent curve
Case	Q · m³/h	H · m
Initial: k = 0.04	20	22
Higher resistance: k = 0.08	15.4919	25.2
Ideal parallel: a = 0.005	23.0940	27.3333

Initially, Q² = (30 − 6)/(0.02 + 0.04) = 400; Q = 20 m³/h. H = 30 − 0.02 × 400 = 22 m.
With higher resistance, Q² = 24/0.10 = 240; Q ≈ 15.4919 m³/h. H = 25.2 m. Flow falls by about 22.5% while head rises.
Initial hydraulic power is 1.199 kW. With higher resistance it is about 1.06383 kW. Adding an assumed pump efficiency of 70% for the second example gives about 1.51976 kW shaft power. Efficiency is prescribed for this example, not calculated from a curve.

Before using the result

An intersection does not prove the pump can operate safely there. Manufacturer curves, permissible minimum/maximum flow, best efficiency point and motor limits need separate assessment.
Suction conditions, NPSH and cavitation margin are not calculated. Transients, water hammer, entrained air/gas, multiphase flow, speed control, detailed parallel/series network behavior and branch losses are outside scope.
Negative static head is not supported. Gravity-assisted, reverse-flow or turbine-mode systems require another model. Zero-flow and zero-head boundaries are not operating recommendations.
Range checks and numerical tests verify software behavior; they do not validate this physical model for an installation. This educational tool cannot replace manufacturer curves, a qualified engineering review or design approval.

Method references

Sources support the method principles. Coefficients and examples are this site’s educational assumptions. References checked 8 October 2026.

Full guide: Pump curves and system resistance

Read the method and its limits

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