Manning's equation for uniform open-channel flow, Q = (k/n)·A·R^(2/3)·√S, is empirical — fitted to measurements rather than derived — and n is its one fudge factor, deliberately absorbing everything messy about a real channel boundary. Finished concrete typically runs around n = 0.013; corrugated metal, gravel beds, and vegetated banks all carry larger values. Because n sits in the denominator, discharge falls in exact proportion as roughness rises: doubling n halves the flow a given channel conveys.
Choosing n is where the engineering judgment lives, and the working method is a published table rather than a formula — Ven Te Chow's Open-Channel Hydraulics is the classic source, listing values by lining material and condition. One subtlety keeps those tables universal: the dimensional constant k in the equation is 1.0 for metres and m³/s but 1.49 for feet and cfs, chosen precisely so the very same n values serve both unit systems unchanged.
In a worked rectangular concrete channel — 3 m wide, 1 m deep, bed slope 0.001, n = 0.013 — the equation delivers 5.19 m³/s at a mean velocity of 1.73 m/s, numbers that would drop immediately if the lining were allowed to roughen with age or vegetation.