Flowrate equals the area of the flow times the velocity of the flow.

Q = AV

Q = Flowrate, A=Area, V = Velocity

So if the velocity in a rectangular channel, 2 metres wide and 1 metre deep was 3 metres per second, the flowrate would be 6 cubic metres per second

A = 1 m x 2 m = 2 m2

Q = A x V = 2 m2 x 3 m / second = 6 m3 / second

Manning’s Equation

Robert Manning was an Irishman who did zillions of experiments with water running through a flume to see if he could determine the relationship between the various factors that effect water as it runs down hill. He did find a relationship and his reward for all the hard work was to get an equation named after him. He became immortal.

Manning found that Velocity in an Open Channel is proportional to the slope of the Channel, the roughness of the channel bottom, the shape of the channel and the depth of flow.

Q = A s1/2 R2/3 / n

s = slope of the channel

R = Hydraulic Radius = Area of the water flowing in the channel divided by the wetted perimeter. Hydraulic radius is kind of a tricky concept. The wetted perimeter of the water flowing in the channel is the part of the channel that touches (and therefore slows down the water by friction) the water. It is the part of the channel that is wetted. Hydraulic radius is even more confusing in a circular pipe. I have included a table of the hydraulic radius, flow are and wetted perimeter at the bottom of the page.

n = Manning’s “n” which is a coefficient of the roughness of the channel bottom and sides. Smooth concrete = 0.013

The equation is a good (although not perfect) estimator for the probable flow in a channel at a given depth and slope. It can be used for estimating the flow in stormwater channels for erosion and sedimentation control plans, for sizing gutters and spacing inlets in a stormwater collection system or for sizing sewer or stormwater pipes.

Rectangular Channel

 depth of flow bottom width Mannings “n” slope Velocity Flowrate 0.1 m 600 mm 0.013 1% 1.4 m/sec 0.082 m3/sec (82 L/sec) 0.2 m 600 mm 0.013 1% 1.9 m/sec 0.22 m3/sec (220 L/sec) 0.1 m 600 mm 0.013 2% 1.9 m/sec 0.12 m3/sec (120 L/sec) 0.1 m 600 mm 0.026 1% 0.7 m/sec 0.041 m3/sec (36 L/sec)

Flow in a Pipe

 depth of flow pipe diameter Mannings “n” slope Velocity Flowrate 0.1 m 600 mm 0.013 1% 1.1 m/sec 0.023 m3/sec (23 L/sec) 0.2 m 600 mm 0.013 1% 1.5 m/sec 0.076 m3/sec (76 L/sec) 0.1 m 600 mm 0.013 2% 1.6 m/sec 0.033 m3/sec (33 L/sec) 0.1 m 600 mm 0.026 1% 0.6 m/sec 0.012 m3/sec (12 L/sec)

Wetted Perimeter, Flow Area and Hydraulic Radius for a 1 metre diameter pipe flowing partially full

 Flow depth Wetted perimeter Flow Area Hydraulic Radius 0.9 2.50 0.745 0.298 0.8 2.22 0.674 0.304 0.7 1.98 0.587 0.296 0.6 1.77 0.492 0.278 0.5 1.57 0.393 0.250 0.4 1.37 0.293 0.214 0.3 1.16 0.198 0.171 0.2 0.93 0.112 0.121 0.1 0.65 0.041 0.063

It is interesting to note that water will flow faster out of a pipe that is 80% full than if it is 90% full. This is reflected in the hydraulic radius. The greater the hydraulic radius, the greater the velocity, assuming slope and “n” are unchanged.

Manning’s equation for rectangular channels

Q = A R2/3 s1/2 / n

A = b x d, in a rectangular channel

b = bottom width

d = depth of flow

R = flow area / wetted perimeter

wetted perimeter = b + 2d, in a rectangular channel

Manning’s equation resolves to:

Q = bd (bd / b + 2d )2/3 s1/2 / n

The complexity of even this simplest of channel types, means that the solution is usually best found using trial and error with a spreadsheet.