Open channel and weir flow

Manning's equationFrancis formulaV-notch weirParshall flumechannel flowflume

Manning's equation, Francis rectangular weir, V-notch weir and Parshall flume — measuring flow in a channel with no meter in it.

Manning's Equation for Velocity

v = \frac{1}{n} R^{2/3} S^{1/2}

Open-channel velocity by Manning's equation in SI form, from the roughness coefficient, hydraulic radius and channel slope.

Manning's Equation for Flow

Q = \frac{1}{n} A R^{2/3} S^{1/2}

Open-channel discharge by Manning's equation in SI form, from flow area, roughness, hydraulic radius and slope.

Francis Formula: Rectangular Weir

Q = 3.33 \, L \, H^{3/2}

Flow over a suppressed rectangular weir by the Francis formula, with the trade constant 3.33 for crest length and head in feet.

V-Notch (Triangular) Weir Flow

Q = \frac{8}{15} C_d \sqrt{2g} \, \tan\!\frac{\theta}{2} \, H^{5/2}

Discharge over a sharp-crested triangular weir from the notch angle, head and discharge coefficient, in the standard theoretical form.

Parshall Flume Free Flow

Q = 4 \, W \, H^{1.522 \, W^{0.026}}

Free-flow discharge through a Parshall flume of 1 to 8 ft throat width, with W and the head H in feet and Q in cubic feet per second.

How they fit together

With no closed pipe there is no pressure to measure, so open channel flow is inferred from geometry and a depth. Manning does it for a channel running uniformly on a slope, using the hydraulic radius and a roughness coefficient n. Weirs and flumes do it by forcing a controlled constriction, so that a single head reading upstream maps to a flow: the rectangular weir goes as H^1.5, the V-notch as H^2.5, and the Parshall flume as H^1.522 for a 1-ft throat.

Pick by flow range and by how much head you can afford to lose. A 90° V-notch is the choice for low and variable flows — that 2.5 power means a small flow still produces a readable head — while a rectangular weir suits larger, steadier flows and a Parshall flume is what you use when solids would settle behind a weir plate, since it is self-cleaning and costs little head. Manning is for sizing and for sewers, not for measurement. Two cautions. Manning's n is the whole answer in disguise: 0.013 for concrete pipe is the standard number, but a sewer with grease and slime is nearer 0.015, and since Q scales as 1/n that is a 15% error before you start — and the equation assumes steady uniform flow it will not have in a surcharged or backed-up line. On weirs, measure the head upstream at least four times the maximum head back from the plate, in the pool and not in the drawdown, and make sure the nappe is fully aerated; a clinging nappe reads high every time.