Sewer hydraulics – head loss per unit length when flowing full versus half-full For the same circular sewer and the same mean velocity V, what is the ratio of head loss per unit length (full flow : half-full flow)? Assume identical roughness and steady uniform conditions.

Introduction:
Design of sanitary and storm sewers often compares performance at different depths of flow. A classic result is that for a circular conduit, certain hydraulic properties are unchanged at half-full depth. This question checks your understanding using resistance equations commonly employed in sewer design (e.g., Manning).

Given Data / Assumptions:

• Same pipe, same roughness n (Manning) or friction factor.
• Same mean velocity V in both cases.
• Steady uniform open-channel flow for the half-full case; full pipe treated analogously.

Concept / Approach:

With Manning’s formula, slope S (which equals head loss per unit length) satisfies V = (1/n) * R^(2/3) * S^(1/2). For fixed V and n, S ∝ 1 / R^(4/3). For a circular pipe flowing full, hydraulic radius R_full = A/P = (π D² / 4) / (π D) = D/4. For half-full flow, A = π D² / 8 and wetted perimeter P = π D / 2, giving R_half = (π D² / 8) / (π D / 2) = D/4. Hence R_half = R_full.

Step-by-Step Solution:

Verification / Alternative check:

The same conclusion follows with Darcy–Weisbach if V and f are the same and equivalent hydraulic radius is equal, reaffirming equal head gradients.

Why Other Options Are Wrong:

Values different from 1.00 assume a change in hydraulic radius or friction not supported at half-full depth for a circular section.

Common Pitfalls:

Confusing discharge Q with velocity V (Q does halve at the same slope, but here V is fixed); mixing up area and wetted perimeter calculations for half-full geometry.

Final Answer:

1.00