Open Channel Flow – Economical Trapezoidal Section Key Property: For a most economical trapezoidal channel section (minimum perimeter for a given discharge and area), the hydraulic mean depth equals one-half of the flow depth.

Introduction:
This question checks the characteristic proportion of a most economical (efficient) trapezoidal channel section. The most economical section minimizes wetted perimeter for a given discharge, thereby reducing frictional losses.

Given Data / Assumptions:

• Prismatic trapezoidal channel with side slope z:1 (generic).
• Uniform, steady open-channel flow conditions.
• Conventional definition: hydraulic mean depth m = A / T, where A is area of flow and T is top width.

Concept / Approach:

For the most economical section, geometric conditions arise from minimizing wetted perimeter P for given A. These lead to proportional relationships between depth and widths. A key result typically cited is that the hydraulic mean depth equals half the depth of flow for the economical trapezoidal section.

Step-by-Step Solution:

Verification / Alternative check:

Textbook results also show companion conditions such as the half of the top width equalling the length of one sloping side in the optimal case. Using these and substituting into m = A / T reproduces m = y / 2.

Why Other Options Are Wrong:

1/2 breadth: Breadth does not directly govern m; m depends on A and T. 1/2 sloping side: Side length influences wetted perimeter but m is not half of a side length. 1/4(depth + breadth): No standard optimality condition gives this relation.

Common Pitfalls:

Confusing hydraulic radius R = A / P with hydraulic mean depth m = A / T; mixing properties of rectangular, triangular, and trapezoidal optimal sections.

Final Answer:

1/2 depth