Open-channel hydraulics – section shape for maximum hydraulic efficiency For channels conveying open-channel flow on a given bed slope, the highest hydraulic efficiency (maximum discharge for a given cross-sectional area and roughness) is generally achieved with a channel of which section shape?

Introduction / Context:
Hydraulic efficiency of an open channel section is typically assessed using the hydraulic radius R = A / P, where A is the flow area and P is the wetted perimeter. For a given area and roughness, a shape that minimizes wetted perimeter at a given area maximizes R and therefore discharge capacity on a given slope.

Given Data / Assumptions:

• Uniform open-channel flow on a given slope with identical roughness characteristics.
• Comparison among standard geometric sections (circular run partly full, rectangular, trapezoidal, etc.).
• Friction formulae like Manning or Chezy apply; discharge increases with hydraulic radius.

Concept / Approach:
For a given area, the shape with the smallest wetted perimeter gives the largest hydraulic radius. Among common shapes, the circle has the smallest perimeter for a given area, hence the best perimeter-to-area ratio. In open channels, a semicircular trough is theoretically the most economical, and a circular conduit running at an appropriate depth closely approximates this optimum.

Step-by-Step Solution:
1) Objective: maximize R = A / P for given A.2) Geometric isoperimetric principle: of all plane curves with a given perimeter, the circle encloses maximum area (equivalently, for a given area the circle has minimum perimeter).3) Therefore P is minimized and R maximized by a circular profile (or semicircular channel in open flow).4) Higher R reduces boundary shear effects relative to area, increasing discharge capacity for the same slope and roughness.

Verification / Alternative check:
Design texts derive the most economical proportions for rectangular and trapezoidal sections (e.g., b = 2y for best rectangular). Even at these best proportions, the achievable R for a given A is still lower than that of a circular/semicircular form, confirming the circular section's superior efficiency.

Why Other Options Are Wrong:

• Square/rectangular: larger wetted perimeter for the same area than circular, hence smaller R.
• Trapezoidal: can be optimized but still does not beat the circular shape's isoperimetric advantage.
• Triangular: very large wetted perimeter for a given area; least efficient among those listed.

Common Pitfalls:
Confusing “most economical rectangular/trapezoidal” proportions (best within that family) with the globally best shape; the circle (or semicircle for open flow) remains superior in perimeter-to-area performance.

Final Answer:
circular