Open Drainage Channel Hydraulic Cross-Section Optimization (Environmental & Civil Engineering)

A water resources and civil environmental engineer at the UNLV Howard R. Hughes College of Engineering is designing an open, rectangular reinforced-concrete flood control drainage channel for the Clark County Regional Flood Control District in the Las Vegas valley.

Hydrologic peak runoff calculations dictate that the channel must maintain a fixed cross-sectional water flow area of:

where $x$ is the bottom channel width (in meters, $\text{m}$) and $y$ is the vertical water depth (in meters, $\text{m}$).

To minimize excavation costs, frictional head loss along the fluid boundary, and the expensive shotcrete concrete liner material across the bottom bed and both vertical sidewalls, the engineer must minimize the wetted perimeter:

1. Formulate the objective function $P(x, y)$ and the constraint equation $g(x, y) = k$.
2. Compute the gradient vectors $\nabla P(x, y)$ and $\nabla g(x, y)$.
3. Formulate the Lagrange multiplier vector equation $\nabla P = \lambda \nabla g$ and equate the individual scalar components.
4. Eliminate $\lambda$ to establish the optimal geometric aspect ratio between channel bottom width $x$ and fluid depth $y$.
5. Substitute into the flow area constraint to calculate the optimal channel dimensions $(x_{\text{opt}}, y_{\text{opt}})$ and the minimum wetted perimeter $P_{\min}$.
6. Verify why this optimal hydraulic ratio ($x = 2y$, where width equals twice the depth) represents the classic "most efficient rectangular open channel" in civil fluid mechanics.