Example #$4$ : Pipe Diameter Required

A main pipeline that was taping, and a little pipe duct that is taking water to a Factory. Water is being taken for along a distance of $40$ meters through a pipe, what diameter of cast pipe $(0\mathrm{.}26$ mm $)$ is required to achieve $300\frac{L}{s}?$

$45$ m of headloss is acceptable: this is the amount of energy that we can afford to lose if the water is flowing through this pipe.

Iteration Process steps:

Guess an $f$ value

Calculate an initial $D=(\frac{8L{Q}^2}{h_{f}g{}^2}f{)}^{\frac{1}{5}}$

Calculate flow velocity, Re

Calculate $\frac{K_S}{D}$

Calculate and Update the new $f$ value using either:

$\frac{1}{\sqrt[\phantom{2}]{f}}=-2log[\frac{k_s}{3\mathrm{.}7D}+\frac{2\mathrm{.}51}{Re\sqrt[\phantom{2}]{f}}],or,f=\frac{1\mathrm{.}325}{[ln(\frac{k_s}{3\mathrm{.}7D}+\frac{5\mathrm{.}74}{R{e}^{0\mathrm{.}9}})]}$

Repeat starting with the new $f,$ and repeat over and over until the difference between input and output $f$ is very small.

Can you solve this with clear steps