Upstream of a rectangular concrete channel, n $=0\mathrm{.}014,$ width b $=2\mathrm{.}5[$m$]$ and slope i $=0\mathrm{.}0008$ where a flow of $8\mathrm{.}5[$m$3/$s$]$ flows, there is a thin$-$walled side weir of length $12[$m$]$ that evacuates $2\mathrm{.}5[$m$3/$s$].$ At $5000[$m$]$ upstream of the side weir, a free$-$flowing thick$-$walled weir is to be designed.

$($a$)$ Determine the height w of the side weir. For this, consider that the flow discharged by a thin$-$walled side spillway with river regime over its entire length corresponds to:

$Q_v=C_{dD}L(h_b-w)\sqrt[\phantom{2}]{2g(h_b-w)}$

where, L is the length of the weir, w the height of the weir, hb the runoff height at the end of the weir $($downstream$),$ and CdD the discharge coefficient, which is calculated as$.$

$C_{dD}=0\mathrm{.}345+\frac{0\mathrm{.}000685}{(h_b-w{)}^2{L}^{\frac{1}{2}}}$

$($b$)$ For the thick$-$walled weir consider sharp edge and friction losses.

$($c$)$ At what distance upstream of the thick$-$walled weir is the normal runoff head reached?

$($d$)$ If at a distance greater than that calculated in $($c$)$ you are asked to place a bottom side gate $2\mathrm{.}5[$m$]$ wide and $0\mathrm{.}5[$m$]$ opening. Calculate the flow rate and head of water immediately upstream of the side gate operating with a rejected shoulder.

$($e$)$ Draw the hydraulic axis to the inlet of the side gate.