A$/$ Pumped Hydro Energy Systems $($PHES$)$

Good news! The company Pump$-$DENSYS has hired you to design the pump to be used in a PHES. You're in luck, because prof. Sdanghi already has a pump in stock which he can lend you and which you can use for your design. The pump's characteristic curve is as follows $($figure on the left$)$:

Searching through his storeroom, he also found some pipes to build the penstock. The total length of the pipe is $\backslash (\backslash$mathrm$\{$L$\}=50\backslash$mathrm$\{$~m$\}\backslash ),$ and the diameter is $\backslash (\backslash$mathrm$\{$d$\}=20\backslash$mathrm$\{$~cm$\}\backslash ).$ For your design, you should:

$1.$ Draw the orientation of the blades in the middle figure. Justify your choice.

$2.$ Assuming laminar flow in the pipe, give the expression of the total head losses, $\backslash ($ H$\_\{$c$\}\backslash )($minor losses can be neglected$).$

$3.$ According to point $2),$ assuming no water movement in the upper tank and a useful head $\backslash ($ H$\_\{$u$\}=\backslash )40$ m $,$ determine the flow rate $\backslash ($ Q$\_\{$v$\}\backslash )$ supplied by the pump. What is the total hydraulic load $\backslash ($ H $\backslash )$ that the pump can cover?

$4.$ How can you calculate the diameter of the runner inlet? After explaining the formula that you used, give a range of values for the optimized $\backslash ($ D$\_\{1\}\backslash ).$

$5.$ What is the distance $\backslash ($ h $\backslash )$ between the upper and lower tanks? Justify your answer.

$6.$ What is an optimized value for $\backslash ($ h$\_\{$t$\}\backslash )?($Look at the figure on the right$).$ Justify your answer.