You are a chemical engineer employed at Distell $($Heineken Beverages Home$),$ and you are responsible for overseeing the distillation. You are tasked by your line manager to control the flow of wine on the trays of the distillation column. The figure below shows the schematic diagram of the tray. The flow of the tray is given by the Francis weir formula

$F(t)=0\mathrm{.}4w{h}^{1\mathrm{.}5}\sqrt[\phantom{2}]{(2g)}$

Where $,h(t),=$ liquid level on tray above the top of the weir, $ft$

$w=$ width of the weir over which the liquid overflows, $ft$

$g=$ acceleration due to gravity, $32\mathrm{.}2f\frac{t}{s^2}$

The steady$-$state inlet flow and process parameters are as follows:

Tray cross$-$sectional area $=11\mathrm{.}2f{t}^2$

$w=30ft$

$F_0(0)=30f\frac{t^3}{sec}$

You perform a unit step change to the inflow to monitor the change in the height. The following changes of height were observed during your investigation.

$\backslash$table$[[$Time $($s$),$Change in Height $($m$)],[0,0\mathrm{.}0081],[0\mathrm{.}05,0\mathrm{.}00837],[0\mathrm{.}1,0\mathrm{.}0085],[0\mathrm{.}15,0\mathrm{.}00852],[0\mathrm{.}2,0\mathrm{.}0086],[0\mathrm{.}25,0\mathrm{.}0088],[0\mathrm{.}3,0\mathrm{.}0089],[0\mathrm{.}35,0\mathrm{.}00897],[0\mathrm{.}4,0\mathrm{.}00894],[0\mathrm{.}45,0\mathrm{.}00897],[0\mathrm{.}5,0\mathrm{.}00898],[0\mathrm{.}55,0\mathrm{.}008975],[0\mathrm{.}6,0\mathrm{.}008998],[0\mathrm{.}65,0\mathrm{.}00899],[0\mathrm{.}7,0\mathrm{.}008999],[0\mathrm{.}75,0\mathrm{.}00897],[0\mathrm{.}8,0\mathrm{.}00895],[0\mathrm{.}85,0\mathrm{.}00898],[0\mathrm{.}9,0\mathrm{.}00899],[0\mathrm{.}95,0\mathrm{.}00899],[1,0\mathrm{.}009]]$

Develop an empirical model for a first order system from the process data, that relates the change height in the tank to the inlet flow.Using process modelling, show how the height changes with the change in inlet flow