Among standard open-channel shapes, which cross-section is the most efficient hydraulically (i.e., provides the maximum discharge for a given area/wetted perimeter)?

Introduction / Context:
“Most efficient” channel sections minimize wetted perimeter for a given cross-sectional area, thereby maximizing the hydraulic radius R = A/P and reducing frictional head loss. This concept guides shape selection in sewers, aqueducts, and irrigation canals.

Given Data / Assumptions:

• Steady, uniform flow approximations with Manning or Chezy resistance laws.
• Comparison among geometric families for equal area or conveyance goals.
• No freeboard, stability, or constructability constraints considered—purely hydraulic efficiency.

Concept / Approach:

Among common shapes, the semicircular channel provides the least wetted perimeter for a given area because a circle encloses maximum area per unit perimeter. A half-circle retains this extremal property for an open channel with a free surface. Consequently, for equal area, semi-circular sections yield the largest hydraulic radius and hence the highest discharge at the same slope and roughness.

Step-by-Step Solution:

Verification / Alternative check:

Isoperimetric inequality shows circles maximize area for a given perimeter; this extends to the open half for channels.

Why Other Options Are Wrong:

Rectangular and triangular sections are simpler to build but are not hydraulically optimal. “Half hexagon” is a specific trapezoid and does not beat the semicircle in A/P performance. Parabolic channels can be efficient but still do not surpass the semicircle in the idealized comparison.

Common Pitfalls:

Confusing hydraulic efficiency with practical design constraints (erosion, lining, cost). Real projects often choose trapezoids for constructability despite lower hydraulic efficiency.

Final Answer:

Semi-circular