Library / Hydraulics, open channels and fluid power / HYD-37
Hydraulics, open channels and fluid power
Pump operating flow from quadratic curves
Flow at the intersection of a pump curve Hp = H₀ − aQ² and a system curve Hsys = Hs + bQ².
- Ref
- HYD-37
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Q [m³/s]
With a pump curve Hp = H₀ − aQ² and a system curve Hsys = Hs + bQ², the pump runs where the heads are equal: H₀ − aQ² = Hs + bQ², so Q = √[(H₀ − Hs)/(a + b)]. It needs a positive flow, a + b above zero and a shut-off head above the static head. Use it only inside the range where the quadratic fits the real pump curve.
Flow at the intersection of a pump curve Hp = H₀ − aQ² and a system curve Hsys = Hs + bQ². It needs a + b above zero and a shut-off head above the static head.
Calculator
- Operating head, Hs + b Q²22.8 m
- Head available above the static head, H₀ − Hs18 m
- Hydraulic power for water, 9810 Q H13,420 W
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The pump settles at 0.06 m³/s (60 L/s), where both curves give 22.8 m of head.
- Raising the static head by 5 m would cut the flow to 0.05099 m³/s, showing how a fixed lift eats into the available head.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| H0 Pump shut-off head | 15 to 500 m | 0.02449 | 0.3124 | 0.2879 | +10% | +8.01 % |
| b System curve coefficient | 1 to 1,000,000 s²/m⁵ | 0.004238 | 0.09484 | 0.09061 | +10% | −2.87 % |
| a Pump curve coefficient | 1 to 1,000,000 s²/m⁵ | 0.004236 | 0.07745 | 0.07321 | +10% | −1.94 % |
| Hs Static head of the system | 0 to 15 m | 0.05477 | 0.07746 | 0.02269 | +10% | −3.39 % |
Sweep of System curve coefficient
| System curve coefficient (s²/m⁵) | Operating flow (m³/s) |
|---|---|
| 1 | 0.09484 |
| 5.623 | 0.09474 |
| 31.62 | 0.09413 |
| 177.8 | 0.09091 |
| 1,000 | 0.07746 |
| 5,623 | 0.04859 |
| 31,620 | 0.02314 |
| 177,800 | 0.01 |
| 1,000,000 | 0.004238 |
Sweep one variable
Variables and how each one contributes
Operating flow
What the equation gives you. Everything else feeds it.
Pump shut-off head
The pump shut-off head, in m. A stronger pump raises the flow, with a square-root dependence. Raising it by 10 % (to 33) moves the result by +8.01 %, from 0.06 to 0.06481 m³/s.
Static head of the system
The static head of the system, in m. A higher static head lowers the flow. Raising it by 10 % (to 13.2) moves the result by −3.39 %, from 0.06 to 0.05797 m³/s.
Pump curve coefficient
The pump-curve coefficient, in s²/m⁵. A steeper pump curve lowers the flow. Raising it by 10 % (to 2,200) moves the result by −1.94 %, from 0.06 to 0.05883 m³/s.
System curve coefficient
The system-curve coefficient, in s²/m⁵. A more resistive system lowers the flow. Raising it by 10 % (to 3,300) moves the result by −2.87 %, from 0.06 to 0.05828 m³/s.
Worked example with the values above
- Head available: H₀ − Hs = 30 − 12 = 18 m.
- Total curve coefficient: a + b = 2,000 + 3,000 = 5,000 s²/m⁵.
- Q = √(18 / 5,000) = 0.06 m³/s.
Assumptions and limits
- Steady incompressible flow with the pump curve and system curve both of quadratic form.
- Ideal blade-guided flow for the velocity-triangle results, so the real head is lower.
Common mistakes
- Using gauge pressure where absolute pressure is needed for NPSH.
- Forgetting that pump curves in different units, such as litres per second, need converting to m³/s.
- Adding the flows of parallel pumps at different heads.
Statistics and uncertainty
Pump curves are known only to within a few percent and wear them down in service, and available suction head changes with the level, temperature and flow, so use a safety margin on the requirement.
Where it is used
- Choosing pumps and finding the duty point on a pipe system.
- Checking cavitation margin and estimating head from impeller geometry.
Maths you need
- Pump curve: head against flow for a pump at a fixed speed
- System curve: head needed by the pipe system against flow
- Velocity triangle: the vector diagram of blade and fluid velocities