$(25\%)$ Flows through pipe networks. The Colebrook equation, as shown below, is used to estimate the

friction factor $(f_D,$ dimensionless$)$ between a fluid and the internal walls of a pipe where such fluid flows. The

friction factor in a fully developed turbulent pipe flow depends on the Reynolds number $($Re$,$ dimensionless$)$

and the relative roughness $(\frac{}{D}),$ where $,$ in mm $,$ is the roughness of the pipe and $D,$ in mm $,$ is the internal

diameter of the pipe.

$\frac{1}{\sqrt[\phantom{2}]{f_D}}=-2lo{g}_{10}(\frac{\frac{}{D}}{3\mathrm{.}7}+\frac{2\mathrm{.}51}{Re\sqrt[\phantom{2}]{f_D}})$

Note that $lo{g}_{10}$ is the logarithm base $10($i$.$e$.,$ not the natural logarithm, ln $).$ Consider the flow of liquid water

though a pipe made of stainless steel. The roughness of the pipe is $=0\mathrm{.}002mm$ and the internal diameter

of the pipe is $D=10mm.$ The flow is considered fully developed and turbulent with an associated Reynolds

number of $Re=8{10}^5.$ Estimate the friction factor in this configuration using the Secant method and

considering the initial guesses of $f_{D_{i-1}}=0\mathrm{.}008$ and $f_{D_i}=0\mathrm{.}021.$ Show $4$ iterations of the Secant method.

Compute the absolute relative approximate percentage error $(|{}_a|)$ at the end of each iteration. Use at least $4$

significant digits and rounding in all your calculations. Provide a plot of the equation, showing labels and

highlighting the solution of the equation. You are allowed to use MATLAB to create the plot.