Manning’s equation is perhaps the most popular formula for open channel flow. You can calculate the flow and velocity (Q/A) of a channel or non-pressurized conduit, such as a circular pipe, using this equation.
This formula can also be rearranged to solve for the normal depth (yn) of an open-channel, such as a rectangular channel:
(rectangular channel; solve by trial & error)
Here’s a summary of each term below:
Q: Flow, a.k.a. discharge (cfs)
n: The Manning’s “roughness” coefficient of the channel. This value shows how much resistance is acted upon the water by the channel. A lower n-value means less roughness, and usually implies a higher velocity and smaller depth in the channel (and vice-versa), with all else being equal. Concrete, which is valued for its hydraulic “smoothness”, has an n-value between 0.013-0.015. For comparison, a natural stream with little to heavy vegetation can have an n-value ranging anywhere between 0.025 to 0.150. Now just imagine riding your road bike on concrete vs. a grassy field, and which surface is much easier to ride on. That’s how the water feels.
A: Flow area (ft2) (Note: not necessarily the full area of your channel section!). For example, if a rectangular channel is flowing half-full, the area would be the base x ½ height, not base x height.
w: width (base) of a rectangular channel (ft).
R: Hydraulic radius (ft), or R = A / P. P is the wetted perimeter, or the length of water that is in contact with the physical channel (i.e. receiving friction). For example, in a rectangular channel flowing half-full, wetted perimeter is the base plus the length of both vertical sides touching the water (see equation above).
S: Technically it’s the friction slope (Sf), but for most applications (and on the P.E. exam) it is the channel’s slope, in decimal form (e.g 0.003 or 0.2). Channel slope is assumed because the prime assumption of Manning’s equation is that the channel is flowing under uniform flow. In uniform flow, the gravitational forces (i.e the weight of the water) cancel out the frictional (resisting) forces, which causes the friction slope to equal the channel slope (Sf = Schannel). Do an energy balance calculation between two points on a uniform-flow channel and prove it to yourself (I will cover this in a more nerdy, in-depth discussion of uniform flow in the near future)
yn: Normal depth (ft), or the depth of flow the water would normally take in the channel assuming there are no changes in the channel’s shape, friction, or backwater effects in either the upstream or downstream direction from the channel for a good distance. In other words, this is uniform flow, as will be discussed in a future post.
Manning’s equation can also be used for non-pressurized pipes, or those flowing with an exposed water surface, as I’ve discussed in a separate post.