Maximum Flow Rate in Open-Channel Flow for a Circular Pipe - Maple Help

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Maximum Flow Rate in Open-Channel Flow for a Circular Pipe

Introduction

This application determines the greatest attainable water flow rate in a partially filled circular pipe.

It uses the Manning formula to determine the flow rate in the open-channel flow of water:

where:

 • 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

An equation that represents the hydraulic radius of a partially filled circular pipe is derived and substituted into the Manning formula. The resulting equation is then optimized to find the maximum flow rate.

Manning Formula for a Circular Pipe

 > $\mathrm{restart}:$

Manning formula:

 > $Q≔\frac{1.49}{n}\cdot A\cdot {R}^{\frac{2}{3}}\cdot {\mathrm{S__0}}^{\frac{1}{2}}:$

For a partially filled circular pipe, the flow area (the blue shaded area in the preceding diagram) is:

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

The wetted perimeter is given by the following formula.

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

Hence, the hydraulic radius, R, is:

 > $R≔\frac{A}{P}$
 ${R}{≔}\frac{{\mathrm{\pi }}{}{{r}}^{{2}}{-}{{r}}^{{2}}{}\left(\frac{{\mathrm{\theta }}}{{2}}{-}\frac{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{{2}}\right)}{{2}{}{\mathrm{\pi }}{}{r}{-}{r}{}{\mathrm{\theta }}}$ (2.1)

The Manning formula then becomes:

 > $Q$
 $\frac{{1.49}{}\left({\mathrm{\pi }}{}{{r}}^{{2}}{-}{{r}}^{{2}}{}\left(\frac{{\mathrm{\theta }}}{{2}}{-}\frac{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{{2}}\right)\right){}{\left(\frac{{\mathrm{\pi }}{}{{r}}^{{2}}{-}{{r}}^{{2}}{}\left(\frac{{\mathrm{\theta }}}{{2}}{-}\frac{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}{{2}}\right)}{{2}{}{\mathrm{\pi }}{}{r}{-}{r}{}{\mathrm{\theta }}}\right)}^{{2}}{{3}}}{}\sqrt{\mathrm{S__0}}}{{n}}$ (2.2)

Maximum Flow Rate for a Circular Pipe

 > $r≔3:$

Incline of the channel:

 > $\mathrm{S__0}≔0.0001:$

Roughness coefficient:

 > $n≔0.013:$
 >

 >
 ${\mathrm{res}}{≔}\left[{45.6796864427174}{,}\left[{\mathrm{\theta }}{=}{1.00507814259573}\right]\right]$ (1)

The maximum flow rate is...

 > $\mathrm{Q__maxflow}≔\mathrm{res}\left[1\right]$
 $\mathrm{Q__maxflow}{≔}{45.6796864427174}$ (2)

...when $\mathrm{θ}$ is ...

 > $\mathrm{θ__maxflow}≔\mathrm{rhs}\left(\mathrm{res}\left[2,1\right]\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$
 $\mathrm{θ__maxflow}{≔}{1.00507814259573}$ (3)

... and the flow depth is:

 >
 ${5.62908731621774}$ (4)
 >