**(English Units)**

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 (y _{n})** 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 (ft^{2}) (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 (S_{f}), 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 (S_{f} = S_{channel}). 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)

**y _{n}: **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.

P.S. A shoutout to Conrad at ReviewCivilPE.com. I don’t think he updates his site anymore, but his article on Manning’s equation inspired mine!