1. Problem statement The maximum discharge in a circular culvert occurs, not when the culvert is full, but when the culvert is close to being full. The reason for this behavior is that a certain relative depth y/D, where y is the flow depth and D is the culvert diameter, the wetted perimeter begins to grow faster than the flow area and, consequently, the discharge starts to decrease.
In this article, the relative depth y/D which corresponds to maximum discharge in circular culvert flow is derived using differential calculus. 2. Derivation According to the Manning equation, the discharge in SI units is (Ponce, 2014):
in which Q = discharge, A = flow area, R = hydraulic radius, S = bottom slope, and n = Manning's n. Since R = A/P, it follows that:
In terms of r and θ, the flow area and wetted perimeter are, respectively (Fig. 1):
Fig. 1 Definition sketch. Therefore:
By geometry, the relative depth is (Fig. 1):
Following differential calculus, the maximum discharge occurs when dQ/dθ = 0. Using Eq. 2, for a constant slope and Manning friction, this condition implies that:
In Eq. 8, operating on the derivatives:
Simplifying:
Replacing Equations 36 into Equation 9 and simplifying:
Simplifying Eq. 11:
In Equation 12, solving for θ by trial and error:
θ = 302° 25' 51.96".
This angle is used in Eq. 7 to solve for relative depth, to give y/D = 0.938. 3. Summary The relative depth y/D required to achieve maximum discharge in a circular culvert has been derived using differential calculus. This value is shown to be y/D = 0.938. The reason for this hydraulic behavior is that when the flow depth increases beyond the value y = 0.938 D, the wetted perimeter begins to grow faster than the flow area, causing the flow rate to begin to decrease. References
Ponce, V. M. 2014. Fundamentals of openchannel hydraulics. Online text.

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