Pump Affinity Laws Calculator
Predict how a centrifugal pump's flow, head, and power change when its speed or impeller size is altered, using the pump affinity laws. Ideal for engineers optimising pump operation or evaluating variable-speed drives.
About this calculator
The pump affinity laws (also called fan laws) describe how the three key performance parameters of a centrifugal pump scale with rotational speed. When speed changes from N₁ to N₂: Flow scales linearly — Q₂ = Q₁ × (N₂/N₁). Head scales with the square — H₂ = H₁ × (N₂/N₁)². Power scales with the cube — P₂ = P₁ × (N₂/N₁)³. The same relationships apply when impeller diameter changes instead of speed. The cubic relationship for power is particularly important: reducing speed by just 20% cuts power consumption by (0.8)³ = 0.512, saving nearly 49% of energy. These laws assume geometric and dynamic similarity, and are most accurate for centrifugal pumps operating near their best-efficiency point. They are the theoretical basis for energy savings achieved with variable-frequency drives (VFDs).
How to use
A pump runs at 1,450 rpm, delivering Q = 120 m³/h, against a head H = 40 m, consuming P = 18 kW. The speed is increased to 1,600 rpm. Speed ratio = 1600/1450 ≈ 1.103. New flow: Q₂ = 120 × 1.103 ≈ 132.4 m³/h. New head: H₂ = 40 × 1.103² = 40 × 1.217 ≈ 48.7 m. New power: P₂ = 18 × 1.103³ = 18 × 1.342 ≈ 24.2 kW. So a 10.3% speed increase raises flow by 10.3%, head by 21.7%, and power demand by 34.2%.
Frequently asked questions
Why does pump power increase so dramatically with speed according to the affinity laws?
Power consumed by a centrifugal pump is proportional to the cube of rotational speed (P ∝ N³). This arises because power equals the product of flow rate and head — flow scales with N and head scales with N², so their product scales with N³. In practical terms, a 10% speed increase raises power demand by about 33%, while a 20% increase more than doubles power consumption. This strong cubic relationship also works in reverse: slowing a pump with a variable-frequency drive yields enormous energy savings, which is why VFDs are among the most cost-effective energy efficiency investments in pumping systems.
How accurate are the pump affinity laws in real-world applications?
The affinity laws assume perfect geometric and dynamic similarity, which holds well for moderate speed changes (within ±30–40% of design speed) on centrifugal pumps operating near their best-efficiency point (BEP). Accuracy decreases at very low or very high speeds, near the ends of the pump curve, or when pump efficiency changes significantly with speed. Viscous fluids further reduce accuracy because the laws assume turbulent, kinematically similar flow. For precise performance prediction, manufacturers' curves at multiple speeds or validated simulation models should be used. The affinity laws provide excellent first-order estimates for selection and energy analysis purposes.
Do the pump affinity laws apply to impeller diameter changes as well as speed changes?
Yes — the affinity laws apply equally to impeller diameter changes, with speed N replaced by diameter D in each formula: Q₂ = Q₁ × (D₂/D₁), H₂ = H₁ × (D₂/D₁)², and P₂ = P₁ × (D₂/D₁)³. Impeller trimming (reducing diameter) is a common way to permanently reduce a pump's duty point without changing speed. However, the diameter laws are less accurate than the speed laws because trimming changes the impeller's hydraulic geometry, causing efficiency to drop more than the speed analogy predicts. As a rule of thumb, trimming accuracy degrades noticeably beyond about 10–15% reduction from the original diameter.