Consider the gas storage tank in the figure below, where $p_i$ and $p_o$ are the upstream

and downstream pressures respectively; and $F_i$ and $F_o$ are the inlet and outlet molar

flow rates. The temperature $T$ is constant, and the ideal gas law, $pV=nRT,$ may

be assumed to hold.

$($a$)$ If the gas flow rates are determined by the pressure differences across the valves

according to

$F_i={}_i\sqrt[\phantom{2}]{p_i-p},F_o={}_o\sqrt[\phantom{2}]{p-p_o},$

perform an unsteady$-$state balance to derive a dynamic model of the following

form for this process:

$\frac{dp}{dt}={}_i\sqrt[\phantom{2}]{p_i-p}-{}_o\sqrt[\phantom{2}]{p-p_o}$

where ${}_i$ and ${}_o$ are constants related to the process parameters $($such as ${}_i,$

$V,$ etc.$).$ Expressions for ${}_i$ and ${}_o$ should also be provided.

$($b$)$ Derive a transfer function model for the gas storage tank that relates the tank

pressure $p$ to the upstream and downstream pressures, $p_i$ and $p_o,$ linearizing if

and where necessary.

$($c$)$ For parameters ${}_i={}_o=1\mathrm{.}41kN\frac{0\mathrm{.}5}{m}*s,$ and nominal steady$-$state conditions

$p_{is}=2000k\frac{N}{m^2},p_s=1800k\frac{N}{m^2}{p}_{os}=1600k\frac{N}{m^2},$ determine the

response of the tank pressure $p^{\text{'}}(t)$ to the unit impulse input $p_i^{\text{'}}(t)=(t).$