A process consists of two stirred tanks with input q and

Chapter 7, Problem 7.3

(choose chapter or problem)

A process consists of two stirred tanks with input \(q\) and outputs \(T_{1}\) and \(T_{2}\) (see Fig. E7.3). To test the hypothesis that the dynamics in each tank are basically first-order, a step change in \(q\) is made from 82 to 85 L/min, with output responses given in Table E7.3.

(a) Find the transfer functions \(T_{1}(s) / Q^{\prime}(s)\) and \(T_{2}(s) / T_{1}(s)\). Assume that they are of the form \(K_{i} /\left(\tau_{i} s+1\right)\)

\(\begin{array}{l}\text { Table E7.3 }\\ \begin{array}{ccc|ccc} \hline t(\min ) & T_{1}\left({ }^{\circ} \mathrm{C}\right) & T_{2}\left({ }^{\circ} \mathrm{C}\right) & t(\min ) & T_{1}\left({ }^{\circ} \mathrm{C}\right) & T_{2}\left({ }^{\circ} \mathrm{C}\right) \\ \hline 0 & 10.00 & 20.00 & 11 & 17.80 & 25.77 \\ 1 & 12.27 & 20.65 & 12 & 17.85 & 25.84 \\ 2 & 13.89 & 21.79 & 13 & 17.89 & 25.88 \\ 3 & 15.06 & 22.83 & 14 & 17.92 & 25.92 \\ 4 & 15.89 & 23.68 & 15 & 17.95 & 25.94 \\ 5 & 16.49 & 24.32 & 16 & 17.96 & 25.96 \\ 6 & 16.91 & 24.79 & 17 & 17.97 & 25.97 \\ 7 & 17.22 & 25.13 & 18 & 17.98 & 25.98 \\ 8 & 17.44 & 25.38 & 19 & 17.99 & 25.98 \\ 9 & 17.60 & 25.55 & 20 & 17.99 & 25.99 \\ 10 & 17.71 & 25.68 & 50 & 18.00 & 26.00 \\ \hline \end{array} \end{array}\)

(b) Calculate the model responses to the same step change in q and plot with the experimental data.