$(1)[40$ pts$]$ Costa Rica generates more than $95\%$ of its electricity from renewable sources.

Approximately $75\%$ comes from hydropower. To further lower its dependency on fossil$-$fuels down

to $0\%,$ the Costa Rican Institute of Electricity is studying a site along the Ayil River to build a new

hydropower plant.

The Ayil River has a mean annual discharge of $71\frac{m^3}{s}$ and a slope of $0\mathrm{.}00035$ within the reach of

interest. The cross$-$section of the river is $8m$ wide and relatively rectangular in shape. The bed of

the river is covered in gravel with a median grain size D $504mm$ and density ${}_s=2,650k\frac{g}{m^3}.$

As part of the feasibility studies and preliminary design tasks, you have been hired to determine the

distance away from the proposed $15m$ high dam where sediment entering the reservoir will deposit

and form a delta. This location will help them design road access for future dredging operations.

You also need to estimate the annual volume of sediment that will accumulate there so they can plan

the future dredging efforts.

$(10$ pts$)$ Where will the delta form? $($Hint: Compute the average bed shear stress along the

river. Use the energy slope $($not the channel slope$)$ to estimate the shear stresses. Where

does the local shear stress drop below the critical shear stress for the material found in the

river?$).$

As part of the hydraulic structures in the project, a group of hydraulic engineers has designed a

spillway to discharge excess water from the reservoir during high flow periods. The design has been

made for a specific discharge of $7\frac{m^2}{s}.$ The spillway produces a water depth of $50cm$ immediately

downstream; the channel receiving the water from the spillway has a slope of $0\mathrm{.}001$ and a Manning's

roughness of $0\mathrm{.}024$;$800m$ downstream of the spillway, the slope changes and becomes steep.

Assume that the channel is wide and rectangular for this part of the analysis $($in other words, you

do not need to know the width of the spillway and the channel receiving the flow$).$

$(10$ pts$)$ What will be the water surface elevation throughout this downstream channel? At

what distance from the spillway should the channel walls become taller? How tall should

they be$?$