$2.$ A hydropower plant is planned to be built using an old dam already in place and constructed for irrigation purposes in a certain district in Uganda. A $2$m diameter concrete pipeline could be used as a penstock. The length of the penstock is $1100$m$.$ The friction losses are $5\%$ of the gross head. The gross head is $900$m$.$ As a Renewable energy MSc$.$ Student, you are involved in a feasibility study as a consultant to establish an operation plan for the plant. For the project to be feasible the payback time should be less than $20$ years. It is your task to evaluate if this will be the case. Assumptions: The electrical generator efficiency is $95\%$ The plant operates $8760$ hours$/$year Parasitic losses are $1\%$ Transmission losses are $1\%$ Downtime losses are $4\%$ A gas turbine using natural gas has a CO$2$ emission factor of $56$ kg$/$GJ of fuel and an electrical efficiency of $40\%$ The price of electricity is $0\mathrm{.}3210$ USD$/$kWh The capital cost of the turbine is COST $=31060\mathrm{.}71(0\mathrm{.}3048)0\mathrm{.}423106$P $0\mathrm{.}71(0\mathrm{.}3048$ H $)0\mathrm{.}42,$ where P is the net power output $($MW$)$ and H is the net head $($in meters$)$ The capital cost of the plant is twice the capital cost of the turbine The operational and maintenance cost of the plant is Cop $=2\mathrm{.}41040\mathrm{.}82\mathrm{.}4104$P $0\mathrm{.}8,$ where P is the net power output $($MW$)$ Calculate the following: a$)$ What is the design flow of the hydropower scheme $($m$^3/$s$)?$ b$)$ What is the net head of the hydropower scheme $($m$)?$ c$)$ What is the efficiency of the hydropower turbine $(\%)?$ d$)$ What is the hydropower plant power output at design condition $($MW$)?$ e$)$ What is the annual energy production $($MWh$/$year$)?$ f$)$ What would be the annual revenue of the power plant in USD? g$)$ What is the payback time $($years$)?$ h$)$ What is the amount $($annual$)$ of CO$2$ emissions avoided by using the hydropower instead of fossil fuel plant? i$)$ What is your conclusion, is the project feasible?