Exercise $3.$ There are two isothermal tanks connected to each other by a valve as shown in the following diagram.

The gas contained can be modeled as an ideal gas and the density is givien by$,$

$=\frac{P(MM)}{RT}$

Consider to options for the flow through the valves.

Critical Flow

If the flow through the valve is critical then,

$w_1(t)=k_{v,1}{p}_1(t),w_2(t)=k_{v,2}{p}_2(t)$

Find the transfer functions that relate the pressure on each tank with the inlet flow to the first tank.

Draw a diagram showing the pressures.

If you had $n$ identical tanks in series, all of them with the same volumen and connected with valves with the same coefficient

as that of the valve on the figure, what would be transfer function relating the pressure of the last tank with the flow in the

first tank, $P_n\frac{s}{W_i}(s)?$

Subcricitcal flow

If the flows through the valves are subcritical, then they are given by$,$

$w_1(t)=k_{v,1}\sqrt[\phantom{2}]{p_1(t)[p_1(t)-p_2(t)]},w_2(t)=k_{v,2}\sqrt[\phantom{2}]{p_2(t)[p_2(t)-p_3(t)]}$

Find the transfer function relating the pressure of each tank with the inlet flow of the first tank.

Draw the block diagram for all the tanks identifying the transfer function for each block. Write the global transfer function,

$P_2\frac{s}{W_i}(s),$ and the formulas for te effective time constants and the gain in steady state of the transfer functions is terms

of the process parameters.