Pump and Fan System Curve Calculator
Pump and Fan System Curve Calculator
Where a pump or fan actually operates: its own head-flow curve crossed with the system’s resistance curve, and the duty point where the two meet. Then the comparison that pays for the page — what it costs to trim the flow with a throttle valve against what it costs to trim it with a variable speed drive, computed from your own curves rather than asserted from the affinity laws.
Duty point: machine curve against system curve
a pump with a 40 m shut-off head that makes 32 m at 50 L/s, on a system with 10 m of static head and 22 m of friction at 50 L/s, 75% pump efficiency and 92% motor efficiency, asked to deliver 40 L/s
Two curves, one crossing
duty point: (c − k)·Q² + b·Q + (H₀ − Hstatic) = 0
affinity at speed ratio r: Hr(Q) = r²H₀ + b·r·Q + c·Q²
hydraulic power = Q · Δp, with Δp = ρ·g·H for a pump
- H₀
- shut-off head or pressure: what the machine makes at zero flow. The left-hand end of the manufacturer’s curve
- k
- the system’s friction coefficient. Friction loss goes as roughly the square of flow, so one measured point fixes it
- Hstatic
- the part of the system’s resistance that does not depend on flow — a lift, or a pressure being worked against. It is what breaks the affinity laws
- r
- speed ratio. Flow goes with r, head with r², power with r³ — but only ALONG a curve through the origin, which is not the system curve when there is static head
Worked example
a pump with a 40 m shut-off head that makes 32 m at 50 L/s, on a system with 10 m of static head and 22 m of friction at 50 L/s, 75% pump efficiency and 92% motor efficiency, asked to deliver 40 L/s
The machine curve through (50 L/s, 32 m) and the shut-off head is H = 40 − 3,200·Q², and the system curve is H = 10 + 8,800·Q², both with Q in cubic metres per second
Setting them equal gives Q = √((40 − 10) ÷ (8,800 + 3,200)) = 50 mm³/s, which is 50.0 L/s or 180.0 m³/h, at 32.00 m of head
That is 15.69 kW of hydraulic power, 20.92 kW at the shaft after 75% pump efficiency, and 22.74 kW from the supply
To get 40 L/s with a valve, the machine stays on its own curve and makes 34.88 m; the system only needs 24.08 m, so 10.80 m is burnt across the valve. Shaft power 18.24 kW
To get it by slowing down, solve r²H₀ + (c − k)Q² = Hstatic for r: 85.4% of full speed. The machine then makes exactly the 24.08 m the system needs and nothing is thrown away. Shaft power 12.59 kW
So speed control costs 69.0% of what throttling costs — a 31.0% saving, not the 50% the cube law is often quoted as promising. The cube of the speed ratio is 62.4%, and it applies to the power at the same point on a curve through the origin, which this system is not. With 10 m of static head, 31.3% of the duty head never falls with speed at all, and below 50.0% speed the flow stops completely
Why the cube law is not the comparison you want
| Question | What it compares | Answer for the page defaults |
|---|---|---|
| What does the cube law say? | shaft power at speed r against shaft power at full speed, ALONG a curve through the origin | r³ = 62.4% — but only if there is no static head |
| What does the drive actually cost against running wide open? | shaft power at the reduced speed against shaft power at the full-speed duty point | 60.2%, and the two differ because the static head pushes the reduced-speed operating point off the affinity parabola |
| What does the drive save against a throttle valve? | shaft power at the reduced speed against shaft power with a valve at the same flow | 69.0%, a 31.0% saving. The valve does not waste as much as people assume, because throttling also moves the machine back along its own curve to a lower flow |
| Where does the throttled power actually go? | the head the machine makes minus the head the system needs, times the flow | 10.8 m of head across the valve at 40 L/s |
The duty point, and what it costs to move it
A pump does not have a flow rate. It has a curve — head against flow, falling from its shut-off head as the flow rises — and the pipework it is connected to has a curve too, rising from whatever static lift there is as friction grows with flow. The installation runs where those two cross, and nowhere else. That point is the duty point, and it is the first thing to establish about any installed machine, because it is often nowhere near what the specification assumed. A fan is the same picture with pressure in pascals instead of metres of water.
The system curve has two parts and only one of them behaves. Friction loss rises as roughly the square of flow, so it is a parabola through the origin and it scales beautifully. Static head — a lift from a sump to a tank, a pressure the discharge is working into — does not depend on flow at all, and it is what spoils every neat rule about pumps. It puts a floor under the head the machine must make, so there is a minimum speed below which the machine delivers exactly nothing: the speed at which its shut-off head equals the static head. The page reports that speed, because it is a surprise the first time a variable speed drive is fitted to a lift application and the flow disappears at 50% speed instead of halving.
Throttling against speed control, computed rather than asserted. To reduce the flow with a valve you add friction: the system curve steepens, the duty point slides back up the machine’s own curve, and the difference between the head the machine makes and the head the system needs is burnt across the valve. To reduce it with a drive you lower the speed: the machine’s curve drops, and it lands on the unchanged system curve at exactly the head that system needs. Nothing is thrown away, and that is the whole argument. The saving is real and it is usually smaller than the sales figure, because throttling also moves the machine to a lower flow on its own curve and therefore uses less power than running wide open. On the page’s defaults the drive costs 69% of what the valve costs — a 31% saving, not half.
Where the cube law does and does not apply. The affinity laws say that flow goes with speed, head with speed squared and power with speed cubed. They are exact statements about a single machine at two speeds, and they trace a parabola through the origin in the head-flow plane. If your system curve is that parabola — no static head — then the operating point moves along it and power really does go as the cube of speed. If there is static head, the operating point does not stay on the affinity parabola and the cube law over-states the saving, sometimes enormously. The Hydraulic Institute and Europump guide calls this out explicitly. The page computes both numbers and labels which question each answers.
What this model leaves out. Efficiency is treated as constant, which it is not: a machine moved a long way from its best efficiency point loses several points, and at low speed the motor and drive lose more as well — so a large speed reduction saves a little less than this page says. The machine curve is fitted as a parabola or a quadratic through two or three points you read off the manufacturer’s chart, which is good near those points and poor far from them. Net positive suction head is not considered at all, and it is what actually limits the flow of many installations. For the shaft and electrical power on their own see the pump motor power calculator, for an enclosure fan sized by the heat it has to carry the fan airflow calculator, and for the motor’s own torque-speed behaviour on this square-law load the induction motor torque-speed calculator.
Frequently asked questions
What is the duty point?
The flow and head at which the machine’s own curve crosses the system’s resistance curve. It is the only point where the machine is producing exactly the head the pipework demands at exactly the flow the machine makes at that head, so it is where the installation settles regardless of what was specified.
Does a variable speed drive really cut the power by the cube of the speed?
Only if the system curve passes through the origin — no static head, all friction. Then the operating point slides down the affinity parabola and the cube law is exact. With static head the operating point leaves that parabola and the saving is smaller, sometimes much smaller. This page computes both and shows the gap.
How much does a drive save over a throttle valve?
The ratio of the head the system actually needs at the target flow to the head the machine makes at that flow on its own curve. For the page’s defaults it is 69%, a 31% saving. It is always less than the comparison with running wide open, because throttling itself reduces the power somewhat by moving back along the machine’s curve.
Why does my pump deliver nothing when I slow it down?
Because its shut-off head has fallen below the static head. Head goes as the square of speed, so at a speed ratio of √(static ÷ shut-off) the machine can just hold the column and deliver zero, and below that nothing at all. The page reports that speed. It is the single commonest surprise when a drive is fitted to a lift duty.
How do I get the machine curve if I only have the manufacturer’s chart?
Read the shut-off head at zero flow and one or two points further along it, and type them in. Two points force a pure parabola, which is a good fit for many centrifugal machines; three let the fit have a linear term, which most published curves need. Take the points near your expected duty point, because the fit is best there.
Does this work for a fan?
Yes — switch the machine selector to fan and work in pascals. The head-flow curve, the system resistance curve, the duty point and the affinity laws are the same; only the conversion from head to pressure differs, and in pascals there is no conversion. Almost every fan system has zero static pressure, which is why speed control works so well on fans.
Related calculators
References
- Hydraulic Institute, Europump and the U.S. Department of Energy Industrial Technologies Program. Variable Speed Pumping: A Guide to Successful Applications, May 2004. The affinity laws, the duty point, and the explicit warning that applying the affinity laws to estimate savings in a system with static head ‘can also lead to major errors’.
- Karassik I J, Messina J P, Cooper P, Heald C C (eds). Pump Handbook, 4th ed. McGraw-Hill, 2008. System-head curves, their intersection with the pump curve, and flow control by throttling, bypass and speed. Edition verified by search; the chapter number is not quoted because it was not.
- ANSI/AMCA Standard 210 / ASHRAE 51, Laboratory Methods of Testing Fans for Certified Aerodynamic Performance Rating. How a published fan curve is measured, and therefore what the points you read off it mean. Copyrighted; cited, not reproduced.