The pump affinity laws are three simple proportionalities that tell you exactly how a centrifugal pump responds when you change its speed — and they matter to a solids-control engineer because every desander, desilter and centrifuge is fed by one of these pumps. Get the feed-pump speed wrong and the cyclones starve; understand the affinity laws and you can size, set and troubleshoot the feed system on the back of an envelope. Here are the formulas, a worked example, and why the cube law is the one to remember.
The three laws
For a centrifugal pump at a fixed impeller diameter, changing the speed changes three things in a fixed way. Flow scales directly with speed: Q2/Q1 = N2/N1. Head scales with the square of speed: H2/H1 = (N2/N1)². And power scales with the cube of speed: P2/P1 = (N2/N1)³. Those three ratios are the whole toolkit — they let you predict a new operating point without any testing, just from the speed change.
The same pattern applies to trimming the impeller diameter at constant speed (flow with D, head with D², power with D³), which is how a fixed-speed pump is matched to a duty. The laws assume geometric similarity and roughly constant efficiency, so they are most accurate for moderate changes — within about ±30% of the design point — and they drift where efficiency shifts or viscosity is high. Knowing which law governs which variable is the starting point for how Rig IQ reasons about a feed-pump change.
Why the cube law is the one to remember
The power-cubed relationship is where the money and the mistakes are. Because power goes with the cube of speed, small speed changes move horsepower dramatically: run a pump at 80% speed and you get 80% of the flow, but only 0.8² = 64% of the head and 0.8³ ≈ 51% of the power — nearly half the energy for four-fifths of the flow. That cubic curve is exactly why variable-frequency drives save so much: trimming speed to match the real duty, rather than throttling a full-speed pump, collapses the power draw.
It cuts the other way too, and that is the trap. Push a feed pump faster to force more head and the power climbs with the cube — a 20% speed increase needs about 73% more power (1.2³ ≈ 1.73), loading the motor hard and risking the mechanical limits. So the cube law is both the opportunity (dial speed down, save power) and the warning (dial it up, pay steeply). Working that trade for the feed system — enough head for the cyclones without overspending on power or motor life — is the kind of calculation Rig IQ is built to run.
What it means for solids-control feed pumps
This is where the laws stop being abstract. A hydrocyclone only separates if it is fed at the right head — a desilter bank typically wants around 75 feet of feed head to develop its cut. The head law tells you immediately how bowl or pump speed sets that: because head goes with speed squared, a pump running too slow doesn’t just lose a little pressure, it loses it quadratically, and the whole cyclone bank under-performs. Set the feed-pump speed for the head the cones need, and the affinity laws tell you the flow and power that come with it.
The laws also frame the classic feed-pump failures. A centrifugal feed pump handling gas-cut mud loses prime and can’t develop head at all — which is why the degasser sits upstream. And matching a feed pump to a manifold is a head-versus-flow balance the affinity laws make predictable: too much speed floods the cones and wastes power on the cube curve; too little starves them. The same speed-to-force logic links to the centrifuge bowl, where RPM sets the G. Reading the feed system through the affinity laws — head where the cones need it, flow to match, power under control — is exactly what Rig IQ is built to do.
The affinity laws, in one line
Flow ∝ speed: Q2/Q1 = N2/N1 · Head ∝ speed²: H2/H1 = (N2/N1)² · Power ∝ speed³: P2/P1 = (N2/N1)³
Same pattern for impeller diameter (Q∝D, H∝D², P∝D³). Accurate within ~±30% of design.
At 80% speed: 80% flow, 64% head, 51% power — the cube law is why VFDs save energy.
For solids control: head ∝ speed² sets whether the cyclones get the ~75 ft feed head they need.
The pump affinity laws: flow scales with speed (Q∝N), head with speed squared (H∝N²), and power with speed cubed (P∝N³) — with the same pattern for impeller diameter. The cube law dominates: at 80% speed you get 80% flow, 64% head and only 51% power, which is why VFDs save so much energy. For solids control it matters because a hydrocyclone needs the right feed head (~75 ft) to cut, and head falls with the square of speed — so a slow feed pump under-performs the whole cyclone bank.
Common questions
What are the pump affinity laws?
They are three proportionalities for a centrifugal pump at constant impeller diameter: flow scales directly with speed (Q2/Q1 = N2/N1), head scales with the square of speed (H2/H1 = (N2/N1)^2), and power scales with the cube of speed (P2/P1 = (N2/N1)^3). The same pattern applies to impeller diameter changes. They let you predict a new operating point from a speed or diameter change without testing.
Why does pump power change with the cube of speed?
Because power is the product of flow and head: flow rises linearly with speed and head rises with the square of speed, so their product rises with the cube. That is why running a pump at 80% speed draws only about 51% of the power (0.8 cubed), and why variable-frequency drives that trim speed to the real duty save so much energy — and equally why over-speeding a pump loads the motor very quickly.
Why do the affinity laws matter for solids control?
Because desanders, desilters and centrifuges are fed by centrifugal pumps, and a hydrocyclone only cuts if it gets the right feed head — a desilter bank typically needs around 75 feet. Since head scales with the square of speed, a feed pump running too slow loses head quadratically and the whole cyclone bank under-performs. The affinity laws let you set feed-pump speed for the head the cones need and know the flow and power that follow.
