$(36$ points$)$ In the water$-$distribution network given below, all pipes have a friction factor of $f=0\mathrm{.}03$ and have diameters of $200mm.(g=9\mathrm{.}81\frac{m}{s^2})$

$($a$)(12$ points$)$ Determine the initial $($estimated$)$ flow distribution in the pipes. State your assumptions clearly. Indicate the flow directions in the pipes.

$($b$)(12$ points$)$ Perform one iteration for Loop $4$ and apply Hardy Cross method to calculate the corrected flows in this loop.

$($c$)(12$ points$)$ By using your corrected flowrates obtained in part $($b$),$ calculate the water pressure at each node around Loop $4,$ if the pipe pressure at Node P is $450$kPa and each node is located at the same elevation.