C$.$ Bisection method $-$ Exercise $1\mathrm{.}1$

For flow in the turbulent regime between completely smooth pipe surfaces and wholly rough pipe surfaces, Colebrook $(1939)$ developed the following empirical equation for the friction coefficient $f$ :

$\frac{1}{\frac{1}{f^{11}}}=-2lo{g}_{+-}(\frac{\frac{}{D}}{3\mathrm{.}7}+\frac{2\mathrm{.}51}{Re{I}^2})$

where $$ is the pipe surface roughness. Develop a procedure to determine $f$ for specified values of $\frac{}{D}$ and $Re.$ Use $f=0\mathrm{.}16R{e}^{-0\mathrm{.}16}$ and $f=0\mathrm{.}08R{e}^{-0\mathrm{.}16}$ for the first $2$ estimates of $f$ Solve for $f$ for a pipe having $\frac{e}{D}=0\mathrm{.}001$ when $Re={10}^n,$ where $n=4,4\mathrm{.}5,6,$ and $7.$ Stop your calculations after $3$ iterations

$8$