Problem $3(45$ points$)$

You are part of an engineering team tasked with creating a new channel to help transport post$-$rainfall runoff, known as stormwater. The existing stormwater channels have not been sufficient, leading to back flow and overflow, so you need to design a new stormwater channel that can bring the system back in to steady state during typical rain events.

Work through the steps below to determine A$.$ what typical flow rate the new channel must handle, B$.$ what the water depth of the new channel must be$,$ C$.$ what flow rate you would predict based on your design parameter choices, including error propagation, and D$.$ a short written interpretation of how your calculation in $C$ meets your system performance objective in $A($including any speculation of future performance or conditions of concern, drawing on any real$-$world context you would like$).$

Submit your responses to $A,B,C,$ and $D$ as well as the detailed steps of your methods and rationale for decision making $($including screen shots, excel files, python files with comments, etc which are preferably in a different file.$)$

System description

The layout of existing channels plus anticipated new channel are shown in figure $1.$ Measurements of typical uniform flow rate for existing channels are listed in table $1.$

Figure $1$ & Table $1.$ Layout of existing and anticipated new channel $($dashed outline$).$ Table lists flow rate during typical rain events.

Channel design:

Flow Rate, Q $(($:$\frac{m^3}{s}\},$ in an open channel can be predicted using the Manning equation:

$Q=\frac{1}{n}**A_c**R^{\frac{2}{3}}**S^{\frac{1}{2}}$

where $n=$ Manning roughness coefficient $($dimensionless$)$

$A_c=$ cross$-$sectional area of channel $m^2$

$S=$ channel slope $]$

$R=$ hydroulic radius $m$

For a trapezoidal channel, you can express Ac and R using the dimensions of the size slope, z$,$ the bottom width, b$,$ and water depth, $y$

Figure $2.$ Sketch of trapezoidal channel cross$-$section, with profile $($a$)$ showing base width b and water depth, $y,$ as well as side slope $z.$ In $($b$)$ channel side view, the direction of flow and the overall channel slope $S$ are shown, as well as the water depth $y.$

See chart below for equations to represent cross sectional area, Ac$,$ and hydraulic radius, R$,$ for different types of channel type based on geometry:

$\backslash$table$[[$Channel type,Area A$,$Wetted permiter $P,$Hydraulic radius R$,$Top width T$,$Hydraulic depth D$],[,$by$,\frac{by}{b+2y},\frac{by}{b+2y},$b$,y$