Question

Find the depth of flow (in m, rounded to two decimals) in a hydraulically efficient rectangular channel ($n = 1/80$) carrying 64 m$^3$/s at slope $S = 0.01$.

Hint:

A hydraulically efficient rectangular channel always has width = 2 × depth. Use Manning’s formula with $A = 2y^2$ and $R = y/2$.

Answer 1

Correct Answer: 1.95

For a hydraulically efficient rectangular channel: \[ \text{Hydraulic radius: } R = \frac{y}{2} \] \[ \text{Width: } B = 2y \] \[ A = By = 2y^2 \] Use Manning’s formula: \[ Q = \frac{1}{n} A R^{2/3} \sqrt{S} \] Substitute: \[ 64 = 80(2y^2)\left(\frac{y}{2}\right)^{2/3} \sqrt{0.01} \] \[ 64 = 80(2y^2)(0.1)\left(\frac{y^{2/3}}{2^{2/3}}\right) \] \[ 64 = 16 y^2 \cdot \frac{y^{2/3}}{2^{2/3}} \] \[ 64 \cdot 2^{2/3} = 16 y^{8/3} \] \[ y^{8/3} = \frac{64 \cdot 1.587}{16} = 6.35 \] \[ y = (6.35)^{3/8} = 2.00\ \text{m} \] Thus: \[ \boxed{2.00\ \text{m}} \]