Problem $1-$ Carlsbad Caverns ventilation $(35$ pts$)$

A stream of air is being moved out of Carlsbad Caverns $($in New Mexico$)$ by a blower in a duct to enhance the amount of air exchange and make sure that the amount of people touring the caves does not run low on oxygen. The general conditions in the cave are

You are to design a galvanized steel $(\backslash (\backslash$mathrm$\{$e$\}=0\mathrm{.}00049\backslash$mathrm$\{$ft$\}\backslash ))$ duct to transmit this stream of air. The air flow should be $\backslash (4000\backslash$mathrm$\{$ft$\}^\{3\}/\backslash$mathrm$\{$min$\}\backslash )$ in terms of volume at the temperature and pressure in the cave. The length of the section will be $1000$ ft $,$ and the overall head loss on this section should be at a rate of $\backslash (0\mathrm{.}2\backslash$mathrm$\{$ft$\}/\backslash$mathrm$\{$tt$\}^\{1\}\backslash ).$

Your design in this problem is to determine the diameter $($D$)$ of the a circular duct.

$($a$)$ Determine air density and air viscosity at this condition using either reference information from your book or calculation. If you make any assumptions about pressure or temperature beyond what is given, you should state them. $(\backslash (\backslash$rho $\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$blt$\}\_\{\backslash$mathrm$\{$m$\}\},\backslash$mu $\backslash$mathrm$\{$lb$\}\backslash$mathrm$\{$b$\}/\backslash$mathrm$\{$ft$\}-\backslash$mathrm$\{$s$\},10\backslash$mathrm$\{$pts$\}\backslash ))$

$($b$)$ Use a Type $3$ solution technique to determine the diameter for the duct. The constraint you should concern yourself with is that of ensuring that the head loss rate $\backslash ((0\mathrm{.}2\backslash$mathrm$\{$ft$\}/\backslash$mathrm$\{$t$\})\backslash )$ is achieved. To do this calculation, be sure to $(15$ pts $)$:

a$.$ Perform at least two iterations using a table similar to the one you see below.

b$.$ Use any initial guess for friction factor that you wish.

c$.$ Remember that you must preserve the $\backslash (\backslash$mathrm$\{$L$\}$: $\backslash$mathrm$\{$W$\}=2$: $1\backslash )$ aspect ratio for every iteration.

d$.$ You can use solvers or other equations to help you, but you still must show the iteration process in the end.

Iteration table suggested structure