Hydraulic mean depth for pipes not running full (open-channel condition) For a circular conduit flowing partially full (i.e., as an open channel), the hydraulic mean depth m is defined as which of the following?

Introduction / Context:When a circular pipe does not run full, it behaves as an open channel. In open-channel hydraulics, two related geometric parameters are used: hydraulic radius R = A / P (area over wetted perimeter) and hydraulic mean depth m = A / T (area over top width). Understanding the correct definition is essential for using Chezy or Manning equations in partially filled conduits.

Given Data / Assumptions:

• Partially full circular conduit (free surface present).
• A = cross-sectional flow area.
• P = wetted perimeter (in contact with water).
• T = top width of the free surface.

Concept / Approach:By definition for open-channel flow: hydraulic mean depth m is the ratio of area to top width, m = A / T. Hydraulic radius R is a different but related parameter R = A / P. Students often confuse these two, but they are used in different empirical relations.

Step-by-Step Solution:

Verification / Alternative check:For a very wide channel, T ≫ depth, so m approaches the actual flow depth, consistent with intuition that “mean depth” ≈ depth for wide channels.

Why Other Options Are Wrong:

• m = r(θ − sin θ) is an expression (up to constants) related to sector geometry, not the general definition of m.
• m = A / P defines hydraulic radius R, not mean depth.
• m = P / A or any inverted form has wrong dimensions.

Common Pitfalls:Using A / P when asked specifically for hydraulic mean depth; mixing up R and m in Manning/Chezy applications.

Final Answer:m = A / T