$($b$)$ In a pumped$-$storage hydropower project, water is delivered from the upper impounding reservoir

through a low$-$pressure tunnel and four high$-$pressure penstocks to the four pump$-$turbine units. The

elevation of the impounding reservoir $($headrace$)$ water level is $500m$ AOD $($above ordinary datum$),$ and

the elevation of the downstream reservoir $($tailwater$)$ water level is $200m$ AOD. The maximum reservoir

storage which can be utilized continuously for a period of $48h$ is $15{10}^6{m}^3.$

The low$-$pressure tunnel is constructed as follows: length $4km$; dia. $8m$; friction factor, $f=0\mathrm{.}028.$

The high$-$pressure penstocks $(4$ nos.$)$ are constructed as follows: length of each penstock $500m$;

diameter $2m,$ friction factor, $f=0\mathrm{.}016$;

turbine efficiency when generating power is $90\%$; generator efficiency $90\%,$

Pump efficiency when pumping back to the upper reservoir is $80\%.$

Draw a neat schematic diagram of the hydropower development scheme and make a detailed

analysis and determine:

$($i$)$ the discharge $Q(\frac{m^3}{s})$

$($ii$)$ maximum power output from the installation i$.$e$.$ by the turbines $.$

$($iii$)$ the specific speed and specify the type of turbine appropriate for the development by giving

reasons.

$($iv$)$ the total amount of power required to pump the water used to generate power by the turbine

back to the upper reservoir with the same discharge as used to generate power by the turbine

using the four pumps through the four penstocks and the low pressure pipe.