Pump and System Curves: How to Find the Operating Point

A centrifugal pump has no flow of its own. Ask a pump engineer what a given machine "does" and the honest answer is a curve, not a number: for every flow there is one head it can add, and which point on that curve it actually runs at is decided entirely by the pipework, the valves and the height it is asked to lift against. Put the same pump into a different system and it delivers a different flow, at a different head, drawing different power.

Here is that fact worked through at the end of this article. One real pump — a nine-stage vertical multistage with a published 68 m shutoff head — and three installations that differ only in static lift and pipe length:

Installation A runs the pump at its nominal 5 m³/h duty. Installation B runs the identical machine close to the far right-hand end of its published curve, 64% over nominal — the classic result of choosing a pump because "68 metres is plenty" for a 30 m lift. Nothing about the pump changed. The system did.

The two curves

The pump curve

The manufacturer measures the head the pump adds against the flow through it, on a test rig, and publishes it as the H-Q curve. For a centrifugal pump it falls from a maximum at zero flow (the shutoff head) and keeps falling past the last point they print. That last published flow is runout, the end of the curve; the head there is still well above zero (30 m on the pump below). It is not a licence to run there: the manufacturer's allowable operating region, set by vibration, NPSH and motor load, can end well short of it, and beyond it there is no data at all. Somewhere in the middle is the best efficiency point (BEP), where the pump's efficiency peaks. The nominal flow in a series name is a size designation, not a promise about where that peak sits; read it off the efficiency curve.

For calculation the published points are fitted with a quadratic:

• $H_{pump}$ — head added by the pump (m)
• $Q$ — volumetric flow through the pump (m³/s)
• $C$ — the fitted shutoff head (m), which is close to but not necessarily the printed one
• $B$ — the linear coefficient (m per m³/s), often small and positive on a strongly curved droop, which makes the fitted curve rise slightly just above zero flow
• $A$ — the quadratic coefficient (m per (m³/s)²); its sign alone settles nothing, what must hold is that the slope $B + 2AQ$ is negative over the flows you use

The quadratic is a convenience, not physics. Its job is to pass through the published points closely enough over the range you care about; the traps that come from forgetting that are covered further down.

The system curve

The system curve is the head the pipework demands at each flow. It has two parts: a fixed part that does not depend on flow at all, and a friction part that rises roughly with the square of it.

• $H_{sys}$ — total head the system requires from the pump at flow $Q$ (m)
• $H_{static}$ — the flow-independent part: elevation gain plus any pressure difference between the two ends (m)
• $h_f(Q)$ — pipe friction plus fitting losses at that flow (m)

The static part is the one people under-count. It is not only the height difference:

• $z_1, z_2$ — liquid surface elevations at the source and the delivery end (m)
• $P_1, P_2$ — pressures on those two surfaces (Pa), both gauge or both absolute
• $\rho$ — liquid density (kg/m³)
• $g$ — gravitational acceleration (9.81 m/s²)

Delivering into a vessel held at 2 bar(g) adds about 20 m of water head to the static term before a single metre of pipe is counted. A closed recirculating loop, by contrast, has a static head of exactly zero: the pump only ever overcomes friction, and the system curve starts at the origin.

The friction part is Darcy-Weisbach plus the fittings, exactly as in the pressure-drop guide:

• $f$ — Darcy friction factor from Colebrook-White, which falls slowly as flow rises
• $L$ — pipe length (m)
• $D$ — inside diameter (m)
• $\sum K$ — sum of the fitting loss coefficients on the line (see minor losses)
• $V$ — mean velocity in the pipe (m/s)

Because $f$ is not quite constant, a real system curve is not quite a parabola. The common shortcut — assume $h_f = k\,Q^2$ with a single $k$ — is checked against the full Colebrook calculation in the worked example below. The answer is that it is a very good shortcut in turbulent flow, and a poor one in laminar.

How to find the pump operating point

The pump runs where the two curves cross: the one flow at which the head the pump adds equals the head the system demands. That intersection is the operating point — the duty point, in most pump catalogues — and finding it is the whole job.

• $Q$ — the operating flow (m³/s), the only unknown
• $C, B, A$ — the pump-curve fit coefficients from above
• $H_{static}$ — static head of the system (m)
• $h_f(Q)$ — friction and fitting loss at flow $Q$ (m)

With a true parabola for the friction term this is a quadratic you can solve by hand. With Colebrook in the loop it needs a few iterations, which is what the worked example does. Either way, the crossing is stable: if the flow were momentarily higher than the crossing, the pump would be adding less head than the pipe needed, and the flow would decelerate back; momentarily lower, and the surplus head would accelerate it forward. A falling pump curve and a rising system curve cross at most once, and they do cross whenever the shutoff head exceeds the static head.

That stability argument is also why a pump curve with a hump — one that rises from shutoff before it falls, as some low-specific-speed designs do — can cause trouble on a system that is mostly static head. A flat system curve can cut the hump twice, and operation near those crossings can be unstable; whether the pump actually hunts depends on the dynamics of the system. SimuPipe flags a curve that rises before it falls when the points are entered, and reports a solved operating point that sits left of the peak after the run.

The worked example

The pump. A Lowara e-SV 5SV09, a nine-stage vertical multistage rated at 5 m³/h nominal, taken from the manufacturer's 50 Hz two-pole performance table (e-SV Series technical brochure, Cod. 191002071 Rev. M, 01/2024, page 23). Eight of the nine tabulated points are used, flow in m³/h and head in metres: 0/68.0, 2.4/65.5, 3.0/64.0, 3.6/62.2, 4.4/59.3, 6.0/50.6, 7.2/41.9, 8.5/30.2 (the table also lists 2.7/64.8).

A least-squares quadratic through all eight, with flow in m³/h, gives

• 67.77 — the fitted shutoff head (m); the datasheet says 68.0
• 0.598 — the linear coefficient (m per m³/h); being positive, it makes the fitted curve rise 0.15 m over the first 0.5 m³/h before it falls, a fitting artefact far from any duty used here
• 0.586 — the quadratic coefficient (m per (m³/h)²), entering with a minus sign; the slope is negative everywhere above 0.51 m³/h

The largest miss against any published point is 0.34 m, on a 68 m curve — half a percent. That is a good fit, and it is a good fit because the points span the whole curve. Keep that in mind for the sparse-fit trap below.

The system. Water at 20 °C (density 998.2 kg/m³, viscosity 1.002 mPa·s) pumped from an open tank up 120 m of DN50 Schedule 40 steel (inside diameter 52.5 mm, roughness 0.045 mm) to a second open tank. Fittings from Crane TP-410 at $f_T = 0.019$ for 2-inch pipe: a sharp entrance (0.5), four 90° elbows at $30 f_T$, a gate valve at $8 f_T$, a swing check at $50 f_T$ and the exit (1.0), giving ΣK = 4.88. Friction factors from Colebrook-White at each flow.

For installation B (30 m static) the system curve tabulates as:

The curves cross at 8.20 m³/h and 33.28 m. The pump is delivering 30 m of lift and 3.28 m of friction, and it is doing so close to the end of its published curve: the last datasheet point is 8.5 m³/h. That is installation B in the opening table — a machine chosen on shutoff head, running 64% past its nominal duty and needing more NPSH than it does at 5 m³/h, because required NPSH climbs with flow.

Raise the delivery tank to 55 m (installation A) and the crossing moves to 4.96 m³/h at 56.29 m — the duty the manufacturer rates it at, with only 1.3 m of the head spent on friction. Keep the 30 m tank but run 600 m of the same pipe instead of 120 m (installation C) and it lands at 7.17 m³/h at 41.92 m, with friction now 11.9 m of the total.

How good is the parabola shortcut? Anchor $h_f = k\,Q^2$ at installation B's operating point and compare it with the full Colebrook system curve at other flows: the largest difference between 2 and 8 m³/h is 0.09 m. That is small in absolute terms because the friction itself is small on this line, 3.3 m at the anchor point. In relative terms the shortcut under-reads the friction by about 20% at 2 m³/h (0.20 m against 0.25 m), because the friction factor rises as the Reynolds number falls, and the 600 m line of installation C shows the same 20% at 2 m³/h when anchored at its own operating point. So the parabola is accurate near the point it was anchored at and degrades with turndown; whether that matters depends on how much of the total head is friction. It is wrong outright below a Reynolds number of about 2,000, where friction goes linear in flow and the shortcut over-states the curvature.

Throttling vs speed control vs static head

Installation B delivers 8.2 m³/h. Suppose the process actually wants 4. There are three levers on the operating point, and they are not equivalent.

Throttle. Close a valve on the discharge until the flow reads 4 m³/h. The system curve steepens and cuts the pump curve further left. At 4 m³/h the pump produces 60.78 m; the static lift is still 30 m and the pipe now only loses 0.87 m, so the valve must burn the remaining 29.91 m — a loss coefficient of about 2,200, and 49% of everything the pump produces thrown away as heat and noise across a valve seat. It works, and it is how most fixed-speed installations are run, but the pump is still working at 60 m of head to deliver 30 m of lift.

Slow the pump down. With a variable-speed drive the pump curve itself moves, by the affinity laws:

• $Q_1, H_1$ — a point on the published curve at rated speed $N_1$
• $Q_2, H_2$ — the corresponding point at the new speed $N_2$
• $N_1, N_2$ — shaft speeds (rev/min), the ratio is all that matters

Scaling the whole curve this way and re-solving (the figure below shows both routes), 4 m³/h is reached at 75.3% of rated speed — about 2,180 rev/min from a 2,900 rev/min two-pole motor — with the pump producing 30.87 m, which is the static lift plus the pipe friction and nothing else. Hydraulic power is ρgQH, so at the same 4 m³/h it scales with head and the slowed pump is doing 51% of the hydraulic work of the throttled one. The other 49% is exactly the valve loss from the previous paragraph. The electrical saving is a different number: it depends on the pump, motor and drive efficiencies at both operating points, can come out above or below the hydraulic figure, and is not estimated here. Note what the saving is: eliminating a throttling loss, not reducing a friction loss. This system is static-dominated (30 of the 30.87 m is lift), which also puts a floor under the speed — below about 66% of rated speed the slowed curve cannot reach 30 m at all and the flow stops. Variable speed pays most on friction-dominated systems, where the required head falls with the flow; on a static-dominated one the usable speed range is narrow.

Change the static head. Raise the delivery tank 10 m, to 40 m, and the operating point moves to 7.10 m³/h at 42.50 m with no valve and no drive. Static head is usually fixed by the plant, but it is worth seeing how strong a lever it is: 10 m of lift removed more flow than 480 m of extra pipe did in installation C. Getting all the way down to 4 m³/h this way would take about 60 m of static head.

Why parallel pumps rarely double the flow

Two identical pumps in parallel share the flow: at any head, the pair delivers twice what one pump would at that head. Graphically, the combined curve is the single curve stretched horizontally by two. In series they share the head: at any flow the pair adds twice the head, and the combined curve is stretched vertically by two.

What you gain from either is a system question, not a pump question. Running the same 5SV09 pair through all three installations:

† The series operating point lies beyond the last published flow (8.5 m³/h per pump), so these values are extrapolated from the fitted curve — indicative only.

Three things fall out of that table.

• Parallel pumps deliver less than twice the flow on any system with friction, and the steeper the system curve the less you get: ×1.81 on the short line drops to ×1.47 on the long one. Only a system of pure static head, a flat line on the chart, would give exactly double. The second pump pushes the operating point up the system curve, where every extra cubic metre costs more head, and both pumps back off to deliver it.
• Series pumps pay when the system needs most of a single pump's head, whether that head goes into lift or into a steep friction curve. On installation A, where 55 m of the head is committed to lift before any friction is paid, doubling the head nearly doubles the flow (×1.74). On installation B, where the 30 m lift is less than half the pump's shutoff head, the second one in series buys 20%, and the steeper line of installation C lifts that to 27%. The rule of thumb is the slope ratio: a system curve steep relative to the pump curve favours series, a flat one favours parallel.
• Each parallel pump runs to the left of where a single pump would. On installation B the pair sits at 7.41 m³/h apiece. Trip one and the survivor jumps to 8.20 m³/h — close to the end of its published curve. Parallel installations need to be checked with one pump running, not just with both.

Curve-fit traps

Some operating-point errors are not hydraulics errors at all. They are fitting errors, made before any system curve is drawn.

Sparse points. Fit the same pump through only three selected points (3.6, 6.0 and 8.5 m³/h) and the quadratic still lands installation B at 8.18 m³/h against 8.20 with the full fit, and installation A at 5.01 against 4.96. But its fitted shutoff head is 64.9 m against a real 68.0. Near the data a sparse fit is fine; away from it, it is a guess. Enter the points around the duty you expect, and enter the shutoff point too if the pump will ever run near it.

A fit that opens upward. Enter three points whose droop slows toward high flow, say 2/60, 6/50 and 8/48, which usually means one was misread, since a real curve steepens there. The parabola through them still falls across those points, but its curvature is positive and it turns upward just past them, at 9 m³/h. Within the data it is a usable fit; asked for any flow beyond it, a solver sees head rising with flow, which can cut a system curve twice, converge on the wrong intersection, or not converge at all. The check is the slope $B + 2AQ$ over the whole range the pump might see, and because that range is not known before the solve, a positive curvature is flagged at entry. The fix is manufacturer data covering the flows the pump will actually see, and an operating point that stays inside it.

Extrapolation. Every series case in the table above sits past 8.5 m³/h, where the manufacturer stopped publishing. The parabola carries on regardless. Beyond the last point the quadratic may over- or under-state the real curve by metres, and a pump that far out is usually also past its allowable operating region for vibration, NPSH and motor load. Treat any operating point outside the published range as a flag to re-select the pump, not as a result.

Where a network simulator fits

For one pump on one line, everything above fits on a page: fit the curve, tabulate the system, find the crossing. The reason to reach for a simulator is that real systems rarely give you one system curve.

A branched network with its valves set does reduce to one head-against-total-flow relation, but it is a different curve for every combination of valve positions, and constructing it by hand means solving the network anyway. Two pumps feeding a common header interact through it. A control valve holding a flow setpoint is redrawing it continuously. In those cases nobody draws the curve — the pump's H-Q relation is just one equation among all the others, and the operating point is whatever flow satisfies every branch at once.

That is how SimuPipe handles it: a pump in curve mode contributes its fitted $C + B\,Q + A\,Q^2$ to the network equations and the solved flow through it is the operating point. After a run, the pump's curve dialog marks that point on the H-Q chart and — when the static head is unambiguous from the boundary conditions, a single pump on a single supply and delivery — overlays a dashed system curve through it, using the same $H_{static} + k\,Q^2$ shortcut, which is why it is labelled indicative: honest near the operating point on a turbulent line, looser on the friction term far from it. The traps get flagged as you go: a fit that opens upward, rises at its high-flow end, or rises from shutoff before it falls is flagged when it is entered, a solved operating point left of such a peak is reported after the run, a solved operating point beyond the flows you supplied is reported as an extrapolation, and a pump driven past its curve to zero head is warned about explicitly. The pump used in this example is in the catalog, alongside the other published curves.

The upstream half of the story — whether the pump can actually be fed at that operating point — is NPSH, which the solver reports as NPSH available at every liquid pump from the solved suction conditions. Whether that is enough is a comparison against the NPSHr you enter from the datasheet at that flow; the catalog curves do not carry one. For sizing the line that sets the friction term, see the pipe sizing calculator and the friction loss calculator; the density and viscosity used here are from the water properties table, and there is a solved booster-pump network in the worked examples gallery.