(Solved): A circular water tank, \( 3 \mathrm{~m} \) in diameter, tapers into a cone as ...

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A circular water tank, \( 3 \mathrm{~m} \) in diameter, tapers into a cone as shown in the figure below. It is open to the atmosphere at the top. The tank is used to store water and distribute it to a local town. It is supplied with water through a horizontal pipe \( 200 \mathrm{~m} \) long and \( 220 \mathrm{~mm} \) diameter. A pump feeds the pipe and maintains a constant gauge pressure of \( 150 \mathrm{kPa} \) at the entry to the pipe. The pipe may be assumed to have a constant friction factor \( \lambda=0.02 \) and losses other than those due to wall friction may be neglected. The Tank discharges water to the town threw an opening at the base of the cone, that has a diameter of \( 300 \mathrm{~mm} \) and a \( c_{d}=0.68 \). Assuming the outflow is closed: (a) What is the piezometric head maintained by the pump at the inlet to the pipe? (b) Write an expression for the frictional head loss along the pipe due to a flow rate \( Q \), and hence find an expression for \( Q \) when the water level of the tank is a height \( h \) above the pipe If the tank is filled to a height of \( 7 \mathrm{~m} \) prior to the outflow being opened: (c) Find the time taken for the water level to reduce to \( 2 \mathrm{~m} \), assuming the inflow is maintained throughout.