Q is the flow rate

n is an empirical coefficient

A is the cross-sectional area of flow

R is the hydraulic radius

S₀ is the incline of the channel

$\mathrm{restart}\:$

$Q≔\frac{1.49}{n}\cdot A\cdot R^{\frac{2}{3}}\cdot {\mathrm{S\_\_0}}^{\frac{1}{2}}\:$

$A≔\mathrm{\π}\cdot r^2-r^2\cdot \frac{\left(\mathrm{\θ}-\sin \left(\mathrm{\θ}\right)\right)}{2}\:$

$P≔2\cdot \mathrm{\π}\cdot r-r\cdot \mathrm{\θ}\:$

$R≔\frac{A}{P}$

$R≔\frac{\mathrm{\pi }{r}^2-r^2\left(\frac{\mathrm{\theta }}{2}-\frac{\sin \left(\mathrm{\theta }\right)}{2}\right)}{2\mathrm{\pi }r-r\mathrm{\theta }}$

$Q$

$\frac{1.49\left(\mathrm{\pi }{r}^2-r^2\left(\frac{\mathrm{\theta }}{2}-\frac{\sin \left(\mathrm{\theta }\right)}{2}\right)\right){\left(\frac{\mathrm{\pi }{r}^2-r^2\left(\frac{\mathrm{\theta }}{2}-\frac{\sin \left(\mathrm{\theta }\right)}{2}\right)}{2\mathrm{\pi }r-r\mathrm{\theta }}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{S\_\_0}}}{n}$

$r≔3\:$

$\mathrm{S\_\_0}≔0.0001\:$

$n≔0.013\:$

$\mathrm{plot}\left(\mathrm{Q}\,\mathrm{\θ}\=0.. 0.5 \mathrm{\π}\,\mathrm{labels}\=\left[''\theta ''\,''Flow\; rate''\right]\,\mathrm{labeldirections}\=\left[\mathrm{horizontal}\,\mathrm{vertical}\right]\,\mathrm{labelfont}\=\left[\mathrm{Calibri}\right]\,\mathrm{title}\=''Flow\; Rate\; in\; Circular\; Pipe''\,\mathrm{titlefont}\=\left[\mathrm{Calibri}\,14\right]\,\mathrm{size}\=\left[600\,400\right]\,\mathrm{axesfont}\=\left[\mathrm{Calibri}\right]\,\mathrm{gridlines}\,\mathrm{color}\=\mathrm{ColorTools}:-\mathrm{Color}\left(''RGB''\,\left[0\/255\,79\/255\,121\/255\right]\right)\right)$