solve this question for group $5.$Find optimum diameter with trial error in SI un$$ts for minimize total cost. A pipe$-$line system is to be designed in a petroleum refinery to pump crude oil from the storage area to the atmospheric distillation unit. The entrance pressure can be taken to be atmospheric and at the exit the gauge pressure will correspond to a head of $20m$ of crude oil. On the piping system there will be $3$ gate valves fully open and $2$ gate valves half open, $2$ globe valves fully open, $10$ elbows, $5$ U$-$turns and $3$ tees with flow through run and $2$ tees with flow through branch.

$($a$)$ Make the necessary evaluations to find the optimum pipe diameter and the type of pump if the piping system operates $350$ days per year and has a service life of $10$ years.

$($b$)$ Calculate the yearly total cost of this system in U$.$S$.$ dollars.

Necessary information

Table $1.$ Distribution of Parameters for each group

$\backslash$table$[[$Group$,\backslash$table$[[$Volumetric$],[$Flow Rate$],[(m^{\frac{3}{h}})$