Iterative Pipe sizing:

A circular pipe must be sized to convey water $500m.$ Gravity flow will carry the water through the pipe and the available change in head along the length of the pipe is $4m.$ The required design flowrate to be conveyed through the pipe is $0\mathrm{.}345\frac{m^3}{s}.$ The equivalent sand roughness of the pipe is $0\mathrm{.}0006m,$ and the water is at $15C($kinematic viscosity, $v=1\mathrm{.}1386{10}^{-6}\frac{m^2}{s}).$ There is a valve installed at the midpoint of the pipe for maintenance purposes. Assume a pipe inlet loss coefficient $C_{in}=0\mathrm{.}50,$ a valve loss coefficient $C_{\text{v}\text{a}\text{l}\text{v}\text{e}}=2\mathrm{.}5,$ and a pipe outlet loss $C_{\text{o}\text{u}\text{t}}=0\mathrm{.}8.$ Perform the following calculations:

a$)$ Iteratively determine the required pipe diameter to convey the specified flow. Provide final answer in meters to two decimal places.

b$)$ What pipe diameter $($in meters$)$ would you select from the table of commercially available pipe sizes for your final design $($Table $5\mathrm{.}1M)?$

c$)$ What is the characteristic pipe loss $(P_L)$ for your final selected pipe diameter?