Pipe network project

Consider a pipe network shown below. It has two reservoirs with known head $H_A$ and $H_G.$ The network supplies water to three facilities located at junctions $($nodes$)C,E$ and $F$ with a known flow rate of $Q_C,Q_E$ and $Q_F$ as shown. Using the pipe characteristics tablulated below, calculate the unknown flow rate and velocity in each pipe segment using the iterative multi$-$equation Newton$-$Raphson method.

$\backslash$table$[[$Pipe$,$Length $($m$),$Diameter $($m$),\frac{e}{D},f,$Junction,Elev. $($m$)],[AB,300,0\mathrm{.}30,0\mathrm{.}00087,0\mathrm{.}019,A,48],[BC,350,0\mathrm{.}20,0\mathrm{.}00130,0\mathrm{.}021,B,46],[BF,350,0\mathrm{.}20,0\mathrm{.}00130,0\mathrm{.}021,C,43],[CF,125,0\mathrm{.}20,0\mathrm{.}00130,0\mathrm{.}021,D,48],[DC,300,0\mathrm{.}20,0\mathrm{.}00130,0\mathrm{.}021,E,44],[EF,300,0\mathrm{.}20,0\mathrm{.}00130,0\mathrm{.}021,F,48],[DE,125,0\mathrm{.}20,0\mathrm{.}00130,0\mathrm{.}021,G,60],[GD,250,0\mathrm{.}25,0\mathrm{.}00104,0\mathrm{.}020,,]]$

Known flow rates: $])=[0\mathrm{.}1]=[0\mathrm{.}25]=([0\mathrm{.}10.$

Reservoir heads: $])=[85]=([102.$

Using conservation principles develop the simultaneous equations required to solve this problem $($see the note below$).$

Solve the system using the multi$-$equation Newton$-$Raphson method.

Develop a MATLAB program to solve the problem iteratively. Use the following trial solution $])=[0\mathrm{.}20]=[0\mathrm{.}50]=[0\mathrm{.}10]=[0\mathrm{.}05]=[0\mathrm{.}50]=[0\mathrm{.}10]=[0\mathrm{.}30]=([0\mathrm{.}25$ as a starting point and iterate till the solution converges, i$.$e$.,$ the relative approximate error $|lo{n}_a|$ drops below a certain desired value that you choose.

After solving the unknown flow rates $(Q),$ use it to calculate the unknown velocities $(V)$ in each pipe.

Conduct an analysis by starting with a different set of trial solutions and see if the solution still converges. If it does, how many iterations does it take?

Report your findings in a formal report. At a minimum, the report should contain the following sections

Description of the engineering problem addressed,

Description of the numerical method used,

Advantages and limitations of this method compared to other way to solve pipe network problems $($e$.$g$.$ Hardy$-$Cross method$)$ that you may have learned in the water resources class,

Solution of the problem $($Q and V$),$

Matlab code developed, and

A screenshot from Matlab to prove that the code works without throwing any error.