Problem $2$: $(40)$

Given: The head loss due to friction in a pipe is given by$,h_{L_2}=K{Q}^n$ where $K$ includes the effects

of the pipe diameter, length, and roughness, as well as the fluid viscosity. For the Darcy$-$

Weisbach formula $($Delleur$,2002),K$ is given as: $K=8f\frac{L}{{}^{2}g{D}^5}$ and $n=2.L$ is the pipe

length, $Q$ is the discharge, and $D$ is the pipe diameter. The pipe length is $30ft.,$ the discharge is

normally distributed with mean $1\mathrm{.}5ft\frac{.{?}^3}{s}$ and standard deviation $0\mathrm{.}7ft\frac{.{?}^3}{s},$ the pipe diameter is

$1\mathrm{.}2ft.,$ and the $f$ factor ranges from $0\mathrm{.}1$ to $1\mathrm{.}2$ and could take any value within this interval.

Find:

Run a simulation of head loss in Excel using $100$ trials and the above parameters. Show

results in a table including formula used to run the simulation. $(10)$

Plot a frequency distribution $($approximate pdf$)$ of the resulting head loss through the pipe

using data analysis histogram function in Excel. $(10)$

Determine an approximate cdf of head loss using the trend line functions in Excel. Show

Equation and R$-$square value on the chart. $(10)$

Using the approximate cdf$,$ determine the probability that this pipe will experience greater

than $1\mathrm{.}0ft.$ of head loss. $(10)$