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To calculate the diameter of a pipe given a certain run-off flow, you’ll first need to convert the flow rate to cubic meters per second since the rest of the units are in the metric system.

1 \text{ liter}/second = 0.001 \text{ cubic meters} /second

The Manning formula for full pipe flow can be rearranged to solve for the diameter:

Q = \frac{1}{n}A R^{2/3} S^{1/2}

For a full pipe, the hydraulic radius ( R ) is equal to


where ( D ) is the diameter of the pipe, and the cross-sectional area ( A ) is

\frac{\pi D^2}{4}

Substituting these into the equation gives:

Q = \frac{1}{n} \left( \frac{\pi D^2}{4} \right) \left( \frac{D}{4} \right)^{2/3} S^{1/2}

To explicitly solve for the diameter ( D ) in the Manning equation for a full pipe, we start expand this out:

Q = \frac{1}{n} \left( \frac{\pi}{4} \right) D^{2} \left( \frac{1}{4} \right)^{2/3} D^{2/3} S^{1/2}

This simplifies to:

Q = \frac{1}{n} \left( \frac{\pi}{4} \right) \left( \frac{1}{4} \right)^{2/3} D^{8/3} S^{1/2}

Finally, to isolate ( D ), we rearrange the equation:

D^{8/3} = \frac{Q \cdot n}{\left( \frac{\pi}{4} \right) \left( \frac{1}{4} \right)^{2/3} S^{1/2}}

And finally:

D = \left( \frac{Q \cdot n}{\left( \frac{\pi}{4} \right) \left( \frac{1}{4} \right)^{2/3} S^{1/2}} \right)^{3/8}

This is the equation to calculate the diameter ( D ) in meters, of a full pipe for a given flow rate ( Q ), Manning roughness coefficient ( n ), and slope ( S ).

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