What are the pump affinity laws?
The affinity laws are mathematical relationships that let you estimate how a pump performs at speeds other than those shown on its published curves. Many applications need exactly that, for example a pump on a variable frequency drive or a belt drive. The laws scale one known point on the curve to a new speed.
To use them you need one point on the existing curve with its speed (rpm), head, capacity and power draw. You then estimate a new speed and check the result against the condition point you want. Finding the right speed is often a short trial and error process. The method assumes pump efficiency does not change between the two speeds.
What are the affinity law formulas for a speed change?
For a speed change with a constant impeller diameter, capacity varies directly with the speed ratio, head varies with the square of the speed ratio, and absorbed power varies with the cube of the speed ratio. Any consistent units work, because each law is written as a ratio of new to original values.
Q / Q1 = n / n1
H / H1 = (n / n1)²
P / P1 = (n / n1)³
Where:
- n = new speed (rpm)
- n1 = speed of the published curve (rpm)
- Q = capacity at n; Q1 = capacity at n1 (gpm, m³/h or L/s)
- H = head at n and Q; H1 = head at n1 and Q1 (ft or m)
- P = absorbed power at n, Q and H; P1 = absorbed power at n1, Q1 and H1 (hp or kW)
How do you use the affinity laws in practice?
Pick a point on the published curve, apply the speed ratio to move it, and repeat for several points to sketch the new curve. Then see where the new curve crosses your system curve. If it misses the target condition point, adjust the speed estimate and try again until the two match.
- Choose several points (Q1, H1, P1) on the published curve at speed n1.
- Estimate a new speed n.
- Calculate Q, H and P for each point with the formulas above.
- Plot the new curve and compare it with the system head curve.
- Adjust n and repeat until the curves cross at the desired flow and head.
Because power rises with the cube of speed, a small speed increase raises power quickly. Always check the new power against the motor rating.
Can you use the affinity laws for impeller trimming?
Not in their simple form. Some handbooks and data sheets substitute the impeller diameter ratio for the speed ratio. That approach always produces an impeller too small to meet the new condition point, because trimming changes the pump geometry and affinity theory no longer holds. This approach should not be used.
Centrifugal pump impellers are usually trimmed to the diameter needed to meet the operating point while running at a synchronous motor speed. Slurry pumps are different:
- Elastomer lined and hard metal impellers are often impossible, or at least difficult, to trim.
- Their speed is usually adjusted instead, with belt and sheave drives or variable frequency drive (VFD) controls.
- Slurry pump impellers are rarely trimmed, and a speed change is preferred.
- Some installations require a fixed speed, and then the impeller must be trimmed.
How should a trimmed impeller curve be estimated?
Use a diameter correction method that accounts for the change in impeller geometry at constant speed, rather than the simple diameter ratio. A correct method predicts a trimmed curve that is steeper than the original curve, which matches what happens on the test stand. The manufacturer's trim curves are the most reliable source.
The terms used in a trim calculation are:
- D2 = original impeller outlet diameter
- Q1 and H1 = capacity and head at D2
- D2TR = trimmed impeller outlet diameter
- Q and H = capacity and head at D2TR
Correction factors for trimming vary by pump design, so confirm the trimmed diameter with the pump manufacturer before cutting. PCT works with you and the factory to set speed or trim for your condition point. Request a quote or see our process pumps.