Problem $1$:

A tank with a bottom cross$-$sectional area $\backslash ($ B $\backslash$times Y $\backslash )$ is to be emptied by means of a pipe as shown in the figure below. It is obvious that as the tank drains, the height $\backslash ($ h $\backslash )$ decreases. Also, at the same time the velocity of flow in the pipe $($hence Re$)$ also decreases leading to a change in the friction factor. A design engineer wants you to use an average friction factor $\backslash ($ f$\_\{\backslash$text $\{$av $\}\}\backslash ).$

$($i$)$ Derive a formula for the time to empty the tank from an initial level $\backslash (\backslash$mathrm$\{$h$\}\_\{0\}\backslash ).$

$($ii$)$ Carry out a dimensional analysis for the problem and establish the $\backslash (\backslash$Pi $\backslash )$ parameters.