Half-hexagon channel dimensions and Manning n

Properties of the half-hexagon section

The most economical trapezoid with the best side slope is half of a regular hexagon: side slope $60^\circ$ ($z=1/\sqrt3$), bed width $b$ = each sloping side $=\dfrac{2y}{\sqrt3}$, $R=\dfrac y2$.

$$A=(b+zy)y=\left(\frac{2y}{\sqrt3}+\frac{y}{\sqrt3}\right)y=\sqrt3\,y^2,\qquad P=3b=\frac{6y}{\sqrt3}=2\sqrt3\,y$$

   b
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 /60  \     b = side = 2y/sqrt(3)
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Given data

$Q=20.2$ m³/s, $S_0=1/2500$, $C=60$ m$^{1/2}$/s.

Depth from Chezy's equation

$$Q=CA\sqrt{RS_0}=60\,(\sqrt3\,y^2)\sqrt{\frac y2\times\frac{1}{2500}}$$ $$y^{5/2}=\frac{20.2}{60\times\sqrt3\times\sqrt{0.5/2500}}=13.7444\;\Rightarrow\;y=2.853\ \text{m}$$ $$b=\frac{2y}{\sqrt3}=3.294\ \text{m},\qquad \text{top width}=b+2zy=2b=6.588\ \text{m}$$ $$A=\sqrt3y^2=14.095\ \text{m}^2,\qquad R=\frac y2=1.426\ \text{m}$$

Check: $Q=60\times14.095\times\sqrt{1.426/2500}=20.20$ m³/s ✓.

Manning's roughness coefficient

$$C=\frac{R^{1/6}}{n}\;\Rightarrow\;n=\frac{R^{1/6}}{C}=\frac{1.426^{1/6}}{60}=\frac{1.0610}{60}=0.0177$$

Answer: depth $y=2.85$ m, bed width = each sloping side $=3.29$ m, top width $=6.59$ m; Manning's $n=0.0177$.