Pump Curve & Affinity Laws Calculator
Enter your pump's test points to fit a curve, then use the speed slider to see how affinity laws predict performance at different speeds. Find the operating point where your pump curve meets the system curve.
Pump Test Data
Enter at least 3 flow/head pairs from the pump's performance curve.
Variable Speed Drive
System Curve
R² = 0.998
a0 = 120.0
a1 = -0.05
a2 = -0.0008
The Affinity Laws
When you change pump speed (using a VFD), the performance changes predictably:
Why Variable Speed Matters
The cubic power law is the key insight. If you reduce pump speed by 20%, power consumption drops by ~49%. This is why variable frequency drives (VFDs) can dramatically reduce pumping energy costs.
In irrigation: Rather than throttling valves (which wastes energy as pressure drop), a VFD slows the pump to match the exact demand. When fewer zones are running, the pump slows down automatically.
The optimizer's role: Droughtless's hydraulic model calculates the exact pump speed needed for each combination of open valves, ensuring the system always runs at peak efficiency.
Pump Curve Fitting
This calculator fits your test data to a quadratic model H = a₀ + a₁Q + a₂Q², which matches the physics of centrifugal pumps. The quadratic form arises from the Euler turbomachine equation.
The system curve H = H_static + k×Q² represents the pipe network's resistance. The operating point is where these curves intersect. Changing pump speed shifts the pump curve while the system curve stays fixed.
Related: Water Savings Calculator · How Differentiable Physics Powers Irrigation
How to Read a Pump Curve
A pump performance curve (also called a pump curve or H-Q curve) shows the relationship between flow rate and head (pressure) that a pump can deliver. Understanding how to read it is essential for proper pump selection and system design.
The Head-Flow Relationship
At zero flow (shutoff), the pump produces maximum head. As flow increases, head decreases. This inverse relationship follows a quadratic curve: H = a0 + a1Q + a2Q². The curve shape comes from the Euler turbomachine equation - faster impeller tip speeds produce more head, but higher flow rates reduce the effective velocity triangle.
The operating point is where the pump curve intersects the system curve. The system curve represents the total head required by your pipe network: static lift plus friction losses (which increase with the square of flow).
Practical Example: Sizing a Pump for Irrigation
For a 10-acre irrigation system with 6 zones:
- Design flow: Each zone needs ~40 GPM, running 2-3 zones simultaneously = 80-120 GPM required
- Static head: 30 ft elevation change from pump to highest sprinkler
- Friction losses: Calculate using the Hazen-Williams equation for your pipe sizes and lengths
- Sprinkler pressure: Most rotors need 45-65 PSI (104-150 ft head) at the nozzle
- Total dynamic head: Static head + friction losses + sprinkler pressure = typically 140-200 ft for irrigation
Select a pump whose curve passes through (or near) the design point of 100 GPM at your calculated total dynamic head. The pump should operate near its best efficiency point (BEP), typically at 60-80% of maximum flow.
Why Variable Speed Drives Save Energy
The affinity laws show that power consumption scales with the cube of speed. This means:
| Speed | Flow | Head | Power | Energy Savings |
|---|---|---|---|---|
| 100% | 100% | 100% | 100% | - |
| 90% | 90% | 81% | 73% | 27% |
| 80% | 80% | 64% | 51% | 49% |
| 60% | 60% | 36% | 22% | 78% |
In irrigation, pump demand varies constantly as different zone combinations run. Without a VFD, the pump always runs at full speed and excess pressure is wasted through throttling valves. With a VFD, the pump slows to match demand, capturing the cubic power savings.
Frequently Asked Questions
What is a pump operating point?
The operating point is where the pump curve intersects the system curve on a head vs. flow chart. At this point, the head produced by the pump exactly matches the head required by the system. The operating point determines the actual flow rate and pressure the pump delivers in your specific installation. If you add more zones (lower system resistance), the operating point shifts right (more flow, less head). If you close zones, it shifts left.
How do the pump affinity laws work?
The affinity laws relate pump performance at different speeds: Flow scales linearly (Q∝N), head scales with speed squared (H∝N²), and power scales with speed cubed (P∝N³). These laws assume geometric similarity and are accurate for centrifugal pumps operating within their normal range. The cubic power relationship is why variable frequency drives (VFDs) are so effective - a 20% speed reduction saves nearly 50% on energy.
How do I get pump test data for this calculator?
Pump test data comes from three sources: (1) The manufacturer's published pump curve (available in product catalogs and data sheets). Read 4-5 points along the H-Q curve. (2) Field testing with a pressure gauge at the pump discharge and a flow meter. Throttle the discharge valve to different positions and record pressure and flow at each setting. (3) Certified pump test reports from the manufacturer, which provide the most accurate data. Enter at least 3 points for a reasonable fit; 5 or more points give the best results.