Flow Rate and Velocity in a Pipe

Finds the flow rate as the cross-sectional area × the velocity, taking the area from the inside diameter of the pipe. For a given flow rate, a narrower pipe means a faster stream.

Explanation

How much water a pipe delivers depends on how wide it is and how fast the water moves.

Q=AvA=π(d2)2Q = A v \qquad A = \pi \left( \dfrac{d}{2} \right)^2

Here QQ is the flow rate, AA the cross-sectional area, vv the velocity and dd the inside diameter.

Example

Water runs at 2 m/s through a pipe of 20 mm inside diameter.

A=π×0.012=3.14×10−4 m2=3.14 cm2A = \pi \times 0.01^2 = 3.14 \times 10^{-4}\ \text{m}^2 = 3.14\ \text{cm}^2
Q=3.14×10−4×2=6.28×10−4 m3/s=37.7 L/minQ = 3.14 \times 10^{-4} \times 2 = 6.28 \times 10^{-4}\ \text{m}^3\text{/s} = 37.7\ \text{L/min}

Almost 38 litres a minute, enough to fill a 200 litre bath in a little over five.

The continuity equation

With nothing leaking out, the same flow must pass every point, so

A1v1=A2v2A_1 v_1 = A_2 v_2

Narrow the pipe and the water must speed up. And because area goes as the square of the radius, halving the diameter quarters the area and quadruples the velocity.

This is what happens when you squeeze the end of a garden hose. Restricting the outlet cannot reduce the water arriving from behind, so the only way to balance the books is to go faster.

Rules of thumb

Domestic water pipes are typically 13 to 20 mm inside, carrying water at 1 to 2 m/s.

Higher velocities let you use thinner, cheaper pipes, but too fast brings noise, vibration and erosion of the pipe wall. Around 2 m/s is the usual ceiling. Forcing a large flow through a small pipe buys you plumbing that is both loud and short-lived.