Applying Manning's Equation to Pipes

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 with a free water surface, using this equation.

Where:

Q = Flow, a.k.a. discharge (cfs)

n = The Manning's "roughness coefficient"

A = Flow area (ft2)

R = Hydraulic Radius, equal to A/P, where P is the wetted perimeter.

S = friction slope, but for uniform flow you can assume it's the channel slope.

Note: If there is no free water surface in the pipe, then Manning’s equation should normally not be used to calculate the flow, but there’s one exception. You can actually use Manning’s equation for a pipe flowing just full, but not technically pressurized. The assumption is that the pipe has just barely become full, and that any additional infinitesimal flow would make the pipe pressurized. The discharge of the pipe in this condition is usually called Qfull or Qfull capacity. Qfull is actually extremely important, because a pipe flowing under the Qfull condition has less discharge than a pipe flowing just below the full depth (a circular pipe conveys the most flow at about 94% of its full depth). The reason why Qfull is less than Q94% depth is because even though there is more flow area in the full-condition, there is even more friction (wetted perimeter) gained as a result of the pipe closing in on itself. This additional friction cancels out the additional flow area and slows down the water.

Take a look at this example to see how this concept applies to a nine foot pipe:

If you still can’t believe this is true (because I definitely didn’t at first!), check out this graph and look for where Q/Qfull is maximum:

Qfull is actually pretty usefull (pun intended). At work, I use a popular program called Flowmaster to calculate Qfull. During the beginning, planning stages of sizing a pipe, I’ll use Qfull to get an idea of my pipe’s maximum capacity rather than Q94% depth. Qfull is a safer number to use since there’s always a chance the pipe will seal up with water, especially if there’s a clog in the system or backwater effects.

But remember, gravity-drained systems, such as storm drains, should not be designed solely on the basis of Qfull. A more detailed hydraulic analysis, utilizing the energy equation and a whole lot of iterative calculations (standard-step method) is usually needed, especially if there are any transitions to different-sized pipes, tight curves, abrupt changes in the slope, and/or the pipe becomes pressurized.

As a shortcut for the PE exam, here’s the formula for calculating Qfull in a circular stormwater or sewage pipe. If the pipe is not full, use the circular pipe ratio graphs to calculate A and R for use in Manning’s equation:

*Side Note: Manning's equation actually can be used to calculate friction losses in a pressurized pipe, but you would need to know or assume what your discharge (Q) is in the first place.