Circular Pipe Partial-Flow Depth and Velocity (Manning)
How deep a gravity pipe actually runs at a given flow, the number the full-bore capacity tile leaves out. Manning is applied to the circular segment (A = (D^2/8)(theta - sin theta), P = D theta/2) and solved for the depth by bisection, then reports d/D, the velocity there, the 2 ft/s self-cleansing check, and the boundary shear tau = 62.4 R S. An 8 in concrete pipe at 1% carrying 200 gpm runs 3.36 in deep (d/D 0.42) at 3.20 ft/s, so solids stay moving. The catch that trips people up: discharge is NOT monotonic with depth - it peaks about 7.6% ABOVE full-bore at d/D 0.938 and falls back at the crown, and velocity peaks at d/D 0.813, so most flows have two possible depths and this reports the smaller (physical) one. Constant n with depth. A design aid; the engineer of record and the local sewer code govern.
Formula and source
A = (D^2/8)(theta - sin theta); P = D theta/2; R = A/P; y = (D/2)(1 - cos(theta/2)); solve (1.486/n) A R^(2/3) sqrt(S) = Q for theta by bisection on the rising branch (0, 5.27811]; V = Q/A; tau = 62.4 R S. Max discharge at theta = 5.27811 (d/D = 0.9382), max velocity at tan theta = theta, theta = 4.49341 (d/D = 0.8128).
Manning's equation applied to circular-segment geometry for partial (part-full) gravity flow, the standard sewer-design relation as compiled in ASCE/WEF MOP FD-5 (Gravity Sanitary Sewer Design and Construction) and Chow's Open-Channel Hydraulics, by name; the 2 ft/s self-cleansing velocity is standard sewer practice.
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