Maximum Flow Rate in Open-Channel Flow for a Circular Pipe
This application determines the greatest attainable water flow rate in a partially filled circular pipe.
It uses the Manning formula to determine the flow rate in the open-channel flow of water:
Q = 1.49n⋅A⋅R23⋅S__012,
Q is the flow rate
n is an empirical coefficient
A is the cross-sectional area of flow
R is the hydraulic radius
S0 is the incline of the channel
An equation that represents the hydraulic radius of a partially filled circular pipe is derived and substituted into the Manning formula. The resulting equation is then optimized to find the maximum flow rate.
Manning Formula for a Circular Pipe
For a partially filled circular pipe, the flow area (the blue shaded area in the preceding diagram) is:
The wetted perimeter is given by the following formula.
Hence, the hydraulic radius, R, is:
The Manning formula then becomes:
Maximum Flow Rate for a Circular Pipe
Incline of the channel:
plotQ,θ=0.. 0.5 π,labels=θ,Flow rate,labeldirections=horizontal,vertical,labelfont=Calibri,title=Flow Rate in Circular Pipe,titlefont=Calibri,14,size=600,400,axesfont=Calibri,gridlines,color=ColorTools:-ColorRGB,0/255,79/255,121/255
res≔Optimization:-MaximizeQ,θ=0.. 0.5 π
The maximum flow rate is...
...when θ is ...
... and the flow depth is:
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