Push water down a pipe and the wall takes a toll. Darcy–Weisbach is the rigorous way to price it: $h_f = f\,\dfrac{L}{D}\,\dfrac{v^{2}}{2g}$ — read aloud h-f equals f, L over D, v squared over two g.

Name every letter before it works. $h_f$ is the friction head loss, in metres of the fluid itself — not pascals, not bar: metres, the height of a column of this water that would push just as hard. $f$ is the Darcy friction factor, a bare number with no units, typically 0.015 to 0.04 for water in steel. $L$ is the developed length of the run in metres, and $D$ is the inside diameter, also in metres once it reaches the formula. $v$ is the average velocity in metres per second, and $g$ is 9.81 m/s². You are solving for $h_f$ going down, or for $f$ coming back when a commissioning test has measured the loss for you.

Read it as two levers and a fee. $\dfrac{L}{D}$ is the geometry lever — a 150 m run of 100 mm pipe is 1500 diameters long, and every one of them takes its cut. $\dfrac{v^{2}}{2g}$ is the velocity head, the kinetic energy of the stream expressed as a height, and it is the currency the whole subject is paid in. $f$ is the fee rate. Note that L and D arrive as a RATIO, so as long as both wear the same unit the unit cancels — which is why you can leave both in millimetres if you like, and why leaving only ONE of them in millimetres is the most expensive keystroke in this chapter.

The square on $v$ is the fact worth carrying out of the room. Double the flow through the same pipe and the head loss does not double — it quadruples. Halve the velocity by going one size up and you keep a quarter of the loss. Every argument for generous pipe sizing, in one exponent.