Open-Channel Design – “Best” (most economical) trapezoidal section For a given area and given side slopes (m:1, horizontal:vertical), which statement is characteristic of the most economical trapezoidal channel section?

Introduction / Context:
The “best” (most economical) section of an open channel is the shape that conveys a given area of flow with the minimum wetted perimeter, thereby minimizing frictional losses for a given discharge. For a trapezoidal section with fixed side slope m (horizontal:vertical), specific geometric relationships characterize this optimal condition.

Given Data / Assumptions:

• Steady, uniform flow; side slope m is fixed.
• Goal: minimize wetted perimeter for a given area → maximize hydraulic radius R = A/P.
• No freeboard or lining constraints considered.

Concept / Approach:

For the most economical trapezoid, two well-known results hold: (1) the hydraulic radius equals half the flow depth (R = y/2), and (2) the half top width equals the sloping side length, meaning the water surface bisects the sloping side. These conditions lead to a balanced distribution of perimeter among the base and sides, minimizing P for the given A.

Step-by-Step Solution:

Verification / Alternative check:

Setting R = y/2 and substituting A and P confirms the equivalence of the geometric and hydraulic conditions. Standard hydraulics texts present the same pair of criteria for the most economical trapezoidal channel.

Why Other Options Are Wrong:

(b) is dimensionally inconsistent and not a known criterion; (c) corresponds to the best rectangular section (b = 2y), not trapezoidal; (d) compares a length (top width/2) to a depth-derived quantity (R) incorrectly for trapezoids.

Common Pitfalls:

Confusing rectangular and trapezoidal optimality conditions, or misinterpreting “side slope” (a ratio) as a length.

Final Answer:

Half the top width equals the length of one sloping side (i.e., the water surface bisects the sloping side)