Worked example: pump and system curve

A pump does not have a flow rate of its own. Put the same pump on different pipework and it delivers different flows, because the flow is set by the system it pushes against. The flow settles where the head the pump can produce equals the head the system demands, the static lift plus the friction at that flow. That intersection is the operating point, and finding it is the central task in pump selection. This worked example shows it directly: one pump, one pipe, and the point where the two curves cross.

The setup

A centrifugal pump lifts water from a low reservoir into a higher one through a single pipe:

• Low reservoir at 5 m head, high reservoir at 30 m head, so the static lift is 25 m.
• Discharge pipe: 800 m of 200 mm pipe at 0.5 mm roughness.

The pump carries a head-flow curve defined by three points: 50 m of head at no flow, 40 m at 0.03 cubic metres per second, and 20 m at 0.06. It also carries an efficiency curve: 62 per cent at 0.02 cubic metres per second, rising to 78 per cent at 0.04, then easing to 70 per cent at 0.06. Those two curves are what turn a flow result into power and an efficiency check.

The physics and the method

The system curve is the head the pipework needs at any given flow: a constant 25 m static lift plus a friction term that grows with the square of the flow, with the friction factor from the Churchill (1977) correlation. The pump curve is the head the pump can supply, falling as flow rises. The operating point is where they meet. Fluid Network Studio fits a least-squares head-flow curve through the three pump points, assembles the network with the Global Gradient Algorithm (Todini and Pilati, 1988), and solves for the flow and heads that satisfy the pump relation, the pipe head loss and continuity at the same time. The conservation residual is reported on every solve, and this incompressible mode is cross-checked against EPANET, part of the published verification cases.

Because the example also supplies an efficiency curve, the solver does more than locate the flow. It reads the efficiency at the operating flow, reports the hydraulic power delivered to the water and the shaft power the pump draws, marks the best-efficiency point, and checks whether the duty point sits in a sensible operating region.

The solved result

The two curves cross at 40.93 L/s and 33.87 m of pump head:

The system-curve split is the first thing to read: of the 33.87 m the pump develops, 25.00 m is static lift that no amount of pipework will remove and only 8.869 m is friction. That is a lift-dominated duty, so changing the discharge diameter moves the operating point far less here than it would on a long flat line. The second thing is the efficiency check. The duty sits at 94.5 per cent of the best-efficiency flow, inside the 70 to 120 per cent preferred operating region, and the efficiency penalty for being off the peak is one tenth of a percentage point (78.2 against 78.3). This pump and this pipe are well matched.

Nothing here is hand-worked. Open the example, press Solve, and the same operating point should come back.

What you learn

Solving the example gives you the operating flow, the head the pump produces, the velocity in the discharge pipe, and the pump's efficiency, hydraulic power and shaft power at duty. You can see how close the operating point sits to the best-efficiency point, the difference between a pump that runs economically and one that wastes energy or runs rough. The revealing experiment is to change the pipe diameter or length and re-solve: a longer or narrower discharge line steepens the system curve, pushes the operating point to a lower flow and higher head, and shifts the pump away from its best-efficiency point.

This example runs on the free plan. For the wider context of pump selection, NPSH and duty diagnostics, see pump system design, and the glossary explains the head and efficiency terms used here.

Open this example in FNS and change the discharge pipe length to watch the operating point move along the pump curve.