QUESTION $1$

A particular farm has two dams. When both are full the water levels are $25$ m verrically apart. The dars are connected with a $\backslash (\backslash$mathbf$\{2000$ m m$\}\backslash )$ internal diameter pipe of length $100$ m in series with two $150$ mm intermal diameter pipes in parallel $($see sketch$).$ Each of the $150$ mm pipes is $2$ km long. For most of the year the quantity of water that flows due to gravity is sufficient to keep the lower dam filled. However, it is sometimes necessary to connect a portable pump in the $\backslash (\backslash$mathbf$\{200\}\backslash$mathbf$\{$~ m m$\}\backslash )$ diameter pipe line to increase the flow rate. The efficiency of the pump is $\backslash (72\backslash \%\backslash ).$ All pipes are galvanized iron. Entry and exits are sharp. The $200$ mm diameter pipe is straight and contains two gate valves. During normal operation $($gravity flow and pumping$)$ both valves are fully open. There are two regular $\backslash (90^\{\backslash$circ$\}\backslash )$ flange bends in each of the $150$ mm pipes. Water temperature is $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).($Ignore the sudden reduction from the $2000$ mm to $150$ mm pipe$)$

$1\mathrm{.}1$ Assume for this question $(1\mathrm{.}1)$ that all flows are fully turbulent, Calculate the flow rate due to gravity only $($i$.$c$.$ without pump$).$ Convert your frictional coefficient into Darcy's coefficient to simplify your calculations.

$1\mathrm{.}2$ Is the assumption in $(1\mathrm{.}1)$ that the flow is fully turbulent in each pipe valid? Use calculations to support your answer.

$1\mathrm{.}3$ If the total flow rate of $\backslash (\backslash$mathbf$\{50\}\backslash )$ litres$/$s is found not to be enough and it must be incressed by $\backslash (110\backslash \%\backslash ).$ Calculate the power required to drive the pump with the new flow rate.

$1\mathrm{.}4$ Explain the term 'line correction'.