PICENG$331$: Case Study $1$

Submission: $8^{th}$ of November, $2023$

The level$-$flow system in Figure Q$1$ is to be analyzed. The flow into the system, $F_0,$ is independent of the system pressures. The feed is entirely liquid, and the first vessel is closed and has a non$-$soluble gas in the space above the nonvolatile liquid. The flows $F_1$ and $F_2$ depend only on the pressure drops, because the restrictions in the pipes are fixed. The system is isothermal.

$1-$ Derive the differential equations that represent the dynamic response of $h_1$ and $h_2.$

$2-$ Obtain the two transfer functions for effect of $F_0$ on $L_1$ and $L_2.$

$3-$ Sketch the dynamic responses for effect of $F_0$ on $L_1$ and $L_2$ using Matlab and Simulink due to a step change in $F_0$ from $10$ to $20\frac{m^3}{m}in.$

$4-$ At $2$min, what will be height of the first and second tank.

$5-$ What are the cases where the second tank will overflow? State at least one possible case.

Density of liquid $=1k\frac{g}{m^3},$ density of gas $=1\mathrm{.}204k\frac{g}{m^3},$

The height and diameter of the first tank is $7m$ and $2m$ respectively,

The height and diameter of the second tank is $5m$ and $3m,$

The Molecular weight is $28\mathrm{.}97k\frac{g}{k}$mol

The steady state values are $h_{1,ss}=2m,h_{2,ss}$

$=3m,T=25C,F_O=10\frac{m^3}{m}in$

Assume $($realistic assumptions$)$ for any missing data.