Change a centrifugal pump's speed and everything about it moves — but not by the same amount. Three relations, one lever, three exponents, and getting the exponent wrong is the entire failure mode of this lesson.

$\dfrac{Q_2}{Q_1} = \dfrac{N_2}{N_1}$, $\dfrac{H_2}{H_1} = \left(\dfrac{N_2}{N_1}\right)^{2}$ and $\dfrac{P_2}{P_1} = \left(\dfrac{N_2}{N_1}\right)^{3}$. Fix the names once: $Q$ is the flow, $H$ the developed head in metres, $P$ the shaft power in kilowatts and $N$ the shaft speed in rpm. The subscript convention runs through all three and never changes: 1 is the state you measured — the speed you are at and everything that goes with it — and 2 is the state you are moving to. Whichever of the six letters you are solving for, the ratio is always new over old.

Read aloud: Q-two over Q-one equals N-two over N-one; H-two over H-one equals N-two over N-one, squared; P-two over P-one equals N-two over N-one, cubed. Flow is linear. Head takes the square. Power takes the cube — and it must, because power is flow times head, so it inherits both exponents. That is not a coincidence to memorise; it is a check you can rebuild in three seconds if the numbers ever look wrong.

The cube is the whole economics of variable speed. Slow a pump to 80 % and it passes 80 % of the water, makes 64 % of the head and absorbs 51 % of the power. Slow it to half speed and it draws an eighth. Every variable-frequency drive ever fitted was paid for by that exponent, and every re-sheaved fan motor ever burnt out was killed by someone who thought the exponent was one.

One honest limit: the affinity laws move the pump along its own family of curves. They do not know what the SYSTEM will do. Where a circuit has real static lift, the operating point does not slide down the affinity curve exactly, and the savings are real but smaller than the pure cube promises.