3. (b) Interacting Systems in Series

Tank: 1

Material balance: \[q-q_1 = A_1\frac{dh_1}{dt} \label{eqnInteract1}\] Flow from tank-1 depends on the level difference \(h_1-h_2\). It can be written as \[q_1 = \frac{h_1-h_2}{R_1} \label{eqnInteract2}\]

Tank: 2

Material balance: \[q_1-q_2 = A_2\frac{dh_2}{dt} \label{eqnInteract3}\]

Flow from tank-2 depends on the level \(h_2\). It can be written as \[q_2 = \frac{h_2}{R_2} \label{eqnInteract4}\]

From Eqns.[eqnInteract1] and [eqnInteract2], we get \[A_1R_1\frac{dh_1}{dt} + (h_1-h_2) = R_1 q \label{eqnInteract5}\] Similarly, from Eqns.[eqnInteract3], [eqnInteract4] and [eqnInteract2]we get \[A_2R_2\frac{dh_2}{dt} + \left(1+\frac{R_2}{R_1}\right)h_2-\frac{R_2}{R_1}h_1 = 0 \label{eqnInteract6}\] After rearranging the terms and taking Laplace transforms for the above equations, we get \[\begin{aligned} (A_1R_1s+1)H_1(s) - H_2(s) &= R_1 Q(s) \label{eqnInteract7} \\ -\frac{R_2}{R_1}H_1(s) + \left[A_2R_2s+\left(1+\frac{R_2}{R_1} \right) \right]H_2(s) &= 0 \label{eqnInteract8} \end{aligned}\]

Eliminating \(H_2(s)\) from Eqns.([eqnInteract7]) and ([eqnInteract8]), and using \(A_1R_1=\tau_1\), \(A_2R_2=\tau_2\) we get, \[\frac{H_1(s)}{Q(s)} = \frac{\tau_2 R_1s + (R_1+R_2)}{\tau_1\tau_2s^2+(\tau_1+\tau_2+A_1R_2)s+1} \label{eqnInteract9}\] Similarly, eliminating \(H_1(s)\), from Eqns.([eqnInteract7]) and ([eqnInteract8]), we get \[\frac{H_2(s)}{Q(s)} = \frac{R_2}{\tau_1\tau_2s^2+(\tau_1+\tau_2+A_1R_2)s+1} \label{eqnInteract10}\] From Eqn.[eqnInteract9], we could see that tank-1 is also showing second-order behavior. This is because of the effect of interaction.

The response of tank-2 is second order, which is as expected. This response is slower in comparison with non-interacting system {Compare Eqns.<span id="eqnInteract10" data-label="eqnInteract10"></span> and ... }, because of the term \(A_1R_2\) to the coefficient of \(s\).