formula.expert

The pump and fan affinity laws

pump lawsfan lawssimilarity laws

How flow, head and power scale with impeller speed and diameter — the first, second and third power relationships behind every VFD retrofit.

Pump Affinity Law — Flow vs Speed

Q2Q1=N2N1\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

First affinity law: a centrifugal pump's capacity changes in direct proportion to shaft speed when the impeller diameter is unchanged.

Pump Affinity Law — Head vs Speed

H2H1=(N2N1)2\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^{2}

Second affinity law: pump head varies with the square of shaft speed, so a 20% speed cut costs 36% of the developed head.

Pump Affinity Law — Power vs Speed

P2P1=(N2N1)3\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

Third affinity law: absorbed power varies with the cube of shaft speed — the single relation that pays for every variable-frequency drive.

Pump Affinity Law — Flow vs Impeller Diameter

Q2Q1=D2D1\frac{Q_2}{Q_1} = \frac{D_2}{D_1}

Capacity scales directly with trimmed impeller diameter at constant speed, the classic way to de-rate an oversized centrifugal pump permanently.

Fan Affinity Law — Airflow vs Speed

Q2Q1=N2N1\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

Fan airflow in CFM changes in direct proportion to wheel speed, the first law used when re-sheaving a belt-driven air handler.

Fan Affinity Law — Static Pressure vs Speed

SP2SP1=(N2N1)2\frac{SP_2}{SP_1} = \left(\frac{N_2}{N_1}\right)^{2}

Fan static pressure rises with the square of wheel speed, the reason a modest re-sheave can overpressurise ductwork and blow out flex connections.

Fan Affinity Law — Power vs Speed

P2P1=(N2N1)3\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

Fan brake power varies with the cube of wheel speed — the law behind variable-air-volume energy savings and behind burnt-out re-sheaved motors.

How they fit together

Flow follows speed, head follows speed squared, and power follows speed cubed. That cube is the entire economic case for variable-frequency drives: slow a fan to 80% and it moves 80% of the air against 64% of the pressure while drawing 51% of the power. Halve the speed and you are at one-eighth the power.

The catch is that the laws describe the machine, not the system. They hold along a curve of geometric similarity, which is a fair approximation when the resistance is mostly friction — but a loop with real static head does not follow the cube, because part of the work is lifting water a fixed height no matter how slowly you turn the impeller. Trim the diameter rather than the speed and efficiency falls away too, since the impeller no longer matches its casing.