Open channel flow: derive function for energy slope on friction?

[MrS]

TL;DR Summary

Manning's formula is not usable when the energy slope is unknown in a horizontal system. Can I determine this slope if I only know a discharge and some system characteristics (pipe diameter & roughness)?

I want to determine the normal flow depth in a perfectly horizontal circular conduit. The system characteristics are known (Internal pipe diameter, Mannings roughness, Discharge). However, I am not sure how to calculate the normal flow depth. When using Manning's equation one can find the normal flow depth, however this formula consists of the energy slope which is unknown (and dependent on h). How should I solve this? I think there should be, with assuming normal flow conditions, a way to write the energy slope as function of gravity and friction. Is there anyone who has ideas on how to do that or has another solution?

Thanks in advance!

 

[jrmichler]

In order to have gravity flow, there must be a gradient. If the conduit is level, then the fluid surface will have a gradient. Use the fluid surface slope in the Manning equation. If the gradient is only in the fluid depth, then the flow cross sectional area will decrease in the downstream direction. If the change in cross sectional area is greater than about 10% or so, the solution would be done in parts.

You might get some good ideas from HDS 5: Hydraulic Design of Highway Culverts, 3rd Edition. It's available online: https://www.fhwa.dot.gov/engineering/hydraulics/pubs/12026/hif12026.pdf. Culvert flows can be under inlet control, outlet control, or in a transition regime. Your case has an energy slope with zero bed slope. Be aware that at least one of the nomograms in that document is wrong by more than an order of magnitude. The text and equations are good, though.