Lessons · Engineering · continuity: flow is area times speed
Continuity: what flows in must flow out
Flow rate is area times speed, Q = A × v, and along a pipe with no leaks A_1 v_1 = A_2 v_2. A narrower pipe means faster fluid.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
A pump is rated in litres per second and the pipe is sized in millimetres. The speed in the pipe decides the noise, the wear and the pressure loss, and it comes straight from the area. Halve the diameter and the water goes four times as fast.
How to think about it
Area from the diameter: π d² / 4, in square metres. Multiply by speed for flow, or divide flow by area for speed. Where the pipe changes size, the flow is the same on both sides, so the speed scales by the inverse of the area.
Worked example
Q = A × v, and A_1 v_1 = A_2 v_2 along the pipeThe definition, and what it means when the pipe changes size.
A 100 mm pipe at 1.2 m/s: A = π × 0.1² / 4 = 0.007854 m²Area first, from the diameter in metres.
Q = 0.007854 × 1.2 = 0.009425 m³/s = 9.42 L/sArea times speed. A cubic metre is a thousand litres.
Into a 50 mm pipe: a quarter of the area, so v_2 = 4 × 1.2 = 4.8 m/sSame flow, quarter of the area, four times the speed.
Your turn
A 200 mm pipe at 0.5 m/s. Write the flow.
Q = (π × 0.2² / 4) × = 0.0157 m³/s
Solve one, graded on the server
The trap
Halving instead of squaring. Half the diameter is a quarter of the area and four times the speed, not twice. The pipe that sounds like a jet engine was sized by someone who halved.