Euler’s pump equation

A hammer thrower spins round and round, then lets go. How far the hammer flies depends on two things: how fast the thrower’s hands move, and how much of that spinning motion the hammer carries away.

An impeller is the same. The energy each kilogram of water receives depends on how fast the vane tip moves and how much swirl the water leaves with.

In 1754 the mathematician Leonhard Euler wrote it down. If the water enters without any swirl, the ideal head is:

H = u₂ × cᵤ₂ ÷ g — blade tip speed × the swirl speed of the water at the outlet ÷ 9.81.

This is Euler’s pump equation. Every impeller, fan, compressor and turbine obeys it.

The dashed green line picks out cᵤ: the part of c that points along the rotation. The readout multiplies it by u and divides by g.

No density in it. The head doesn’t depend on the liquid — exactly what you saw in the head lesson. Water, petrol and brine all reach the same height.

Radial vanes: cᵤ = u, so H = u²/g. A 200 mm impeller at 2900 RPM has u ≈ 30 m/s, so at most about 94 m. Faster or bigger impellers give more head, with the square of the tip speed.

With backward-curved vanes, more flow tilts c backward and shrinks cᵤ — so the head falls as the flow rises. That is the downward pump curve you know.

With forward-curved vanes, more flow makes cᵤ bigger and the head rises with flow. That makes a pump unstable and hungry for power — which is why pumps use backward vanes.

Euler’s head is the best the vanes could ever do. The real pump always gives less: the water doesn’t follow the vanes perfectly, and friction eats some energy. That is the next-but-one lesson.