What a Pump Curve and a System Curve Do Together

Why this matters

The operating point does not belong to the pump and it does not belong to the piping. It is the one flow at which the head the pump can make equals the head the piping demands, and neither party gets a vote on its own. That matters commercially because it means one gauge reading plus one published curve is enough to predict what a change will do before you make it. You can tell a customer what closing that valve will cost them in flow, in numbers, standing in the mechanical room, instead of doing it and finding out.

Before you gauge a running pump

• Keep the coupling guard on. Reading a differential means standing beside a rotating shaft with a hose in your hand. Route the hose so it cannot be drawn in, no loose sleeves, no lanyards.
• Isolate and confirm zero on a gauge before opening a port, and shut the heat source and let the fluid drop below 120 F, or discharge to a drain while standing clear with a face shield and heat-rated gloves. Water above 212 F held under pressure flashes to steam at the opening.
• Never throttle the suction side to move the operating point. Restricting suction lowers the head available at the impeller eye, which is the direct route to cavitation and to a destroyed impeller. Throttle on the discharge if you must throttle at all.
• If any part of this work goes inside an energized enclosure, that is 29 CFR 1910.333(b)(2), with the instrument proved on a known live source before and after per NFPA 70E-2021, 120.5.

Two curves, two different kinds of object

• The pump curve is a property of a machine. It comes from the manufacturer, it is measured on a test stand, and it describes how much head that impeller at that speed makes at each flow. It falls as flow rises. You do not calculate it, you look it up, and you look up the curve for the actual impeller diameter and speed installed.
• The system curve is a property of the piping. For a closed circuit it is head = k x flow squared, passing through the origin, where k is everything the fluid has to fight. You cannot look it up. You construct it, and one measured point is enough to do so. An open circuit that lifts fluid adds a static intercept, which changes the shape and is covered by the closed-versus-open card.

The pump curve is fixed until the machine changes. The system curve moves every time a valve moves or a strainer loads.

Building the artifact from one measured point

Everything below comes out of a single reading. A closed heating loop with a circulator whose published curve is on the wall.

Measured: 14.6 psi across the pump. Convert to feet, since curves are plotted in feet: 14.6 x 2.31 = 33.7 ft.

Read flow off the pump curve at 33.7 ft. Between the curve's 40 GPM point (37 ft) and its 50 GPM point (30 ft), the curve falls 0.7 ft per GPM. So 33.7 ft sits at 40 + (37 - 33.7) / 0.7 = 40 + 4.7 = 44.7 GPM. That is the pump being used as a flow meter, and it works here because this part of the curve is steep enough to resolve.

Solve for k, since the operating point is on the system curve too:

k = head / flow squared = 33.7 / (44.7 x 44.7) = 0.01687

Both curves are now known, from one gauge reading.

The table

Pump head is read off the published curve. System head is computed as 0.01687 times flow squared. The last column is the one that does the work.

Reading the crossing out of the sign change

The last column crosses zero between 40 and 50 GPM. That sign change is the operating point, and it is worth being explicit about what each side of it means:

• Left of the crossing, the pump makes more head than the piping demands. That surplus does not sit there. It accelerates the fluid, flow rises, and the system's demand rises with the square of it. The system self-corrects to the right.
• Right of the crossing, the piping demands more head than the pump can make, flow decays, and the system falls back to the left.

There is exactly one flow where the two are equal, and any real system sits on it within seconds. That is why a pump cannot be "too big for the flow" in the way people mean it. A larger pump does not push more flow through than the curve crossing allows; it moves the crossing, and the crossing may land somewhere you did not want.

Interpolating the sign change between +10.0 at 40 GPM and -12.2 at 50 GPM puts the zero at about 44.5 GPM, which agrees with the 44.7 GPM the curve read directly. Those two ways of getting the same answer disagreeing by more than a gallon per minute or so means the arithmetic slipped somewhere, and it is worth the ten seconds to check.

Using the same table to predict a change

The customer wants to know what closing the branch balancing valve two more turns will cost. Say that change raises the loop's resistance coefficient by 40 percent:

new k = 0.01687 x 1.4 = 0.02361

Recompute the system column near the crossing and find the new sign change. Pump head between 30 and 40 GPM rises 0.6 ft per GPM going left.

New operating point: 39.7 GPM at 37.2 ft.

Compare each against its own pre-change value, both taken on the same loop before the valve moved:

• Flow: 44.7 down to 39.7 GPM. A loss of 5.0 GPM, which is 11 percent of the 44.7.
• Head: 33.7 up to 37.2 ft. A rise of 3.5 ft, which is 10 percent of the 33.7.

A 40 percent increase in resistance cost 11 percent of the flow. That is the sentence worth carrying out of this article, because almost nobody guesses it correctly. People expect flow to fall roughly in step with resistance, and it does not come close.

Running the same table backwards

Opposite question, same two curves: the wing needs 50 GPM and today it gets 44.7. What has to change, and can adjustment get there at all?

Read the pump curve at 50 GPM: 30 ft. For the operating point to land at 50 GPM, the system curve has to pass through that point, so:

required k = 30 / (50 x 50) = 0.0120

Today's k is 0.01687. The loop's resistance has to fall to 0.0120 divided by 0.01687, which is 0.71 of what it is now: a 29 percent reduction in system resistance.

That single number answers "can we just open the balancing valve." If the valve's own share of the loop's resistance is less than 29 percent, opening it wide cannot get there, and the honest answer to give before you touch anything is that this pump will not deliver 50 GPM into this loop by adjustment alone. If its share is more than 29 percent, adjustment can do it, and the segment walk tells you whose share is whose.

Notice what the reverse solve did not need: any knowledge of what the resistance is made of. You get a target before you know where it will come from, which is the right order, because the target is what decides whether the afternoon of hunting is worth starting.

Why the pump curve's slope decides how much flow you lose

Run the same 40 percent resistance increase against a hypothetical pump whose curve is flat at 33.7 ft, so that head does not change no matter what flow does. Then flow is set entirely by the system curve:

new flow = square root of (33.7 / 0.02361) = 37.8 GPM

That is a loss of 6.9 GPM, 15 percent, against the 11 percent the real drooping curve produced. The real pump kept 1.9 GPM that the flat one would have given up, purely because its head rose as the point moved left.

The general result, and it applies to fans identically:

• A steep pump or fan curve gives you stable flow and sensitive head. Resistance changes show up on the gauge and barely in the delivery.
• A flat curve gives you stable head and sensitive flow. Resistance changes barely move the gauge and take the delivery with them.

That is the practical reason a flat-curve machine is a poor flow meter, and it is also the reason a flat-curve machine on a system with modulating two-way valves needs a differential-pressure control strategy rather than a fixed speed.

Where the constructed system curve stops being valid

The construction above is one measured point and an assumption. Four conditions break it, and all four occur routinely:

• A two-way control valve is a variable k, not a constant one. On a system with modulating valves, the system curve you constructed is valid for the valve position that existed when you took the reading, and for nothing else. Note the position on the reading.
• An open circuit has a static intercept. head = static lift + k x flow squared. Fitting a curve through the origin to an open system puts your predicted crossing in the wrong place and, at low flow, predicts flow where there would be none.
• The published curve has to be this pump's curve. Wrong impeller trim, a worn impeller, a different speed, or a drive running below nameplate all move the machine off the sheet you are holding. If your constructed crossing and your measured point disagree badly, suspect the pump curve before you suspect the arithmetic.
• k is only constant while the flow regime is. A valve closed to a slit, or a system running far below design flow, can leave the fully turbulent range where the square law holds cleanly. Treat predictions far away from the measured point as directional rather than quantitative, and re-measure once you get there.

References

• Manufacturer pump curve for the installed impeller diameter and operating speed
• Hydraulic Institute standards for centrifugal pump application and system head curve construction
• OSHA 29 CFR 1910.333(b)(2) with NFPA 70E-2021, 120.5, where instrument work enters an energized enclosure
• See related: Reading Pump Curves Reference; How an Operating Point Moves; What a Closed Loop Does That an Open One Does Not