(Solved): The differential balance equations (dynamic models) for the two-stage stirred-tank reactor system ( ...

The differential balance equations (dynamic models) for the two-stage stirred-tank reactor system (Figure Q1) have the forms given in equations (1.1) and (1.2). $\begin{array}{l}4\frac{d{C}_1}{dt}+2C_1=2C_i\\ 3\frac{d{C}_2}{dt}+2C_2=2C_1\end{array}$ Figure Q1: Two-stage stirred-tank reactor system. (i) Derive the following transfer functions for the Two-Stage Stirred-Tank Reactor system and define the gains and time constants for each. [10 marks] $G_1(s)=\frac{C_1^{?}(s)}{C_i^{?}(s)}\text{ and }{G}_2(s)=\frac{C_2^{?}(s)}{C_1^{?}(s)}$ (ii) Deduct the overall transfer function: $G(s)=\frac{C_2^{?}(s)}{C_i^{?}(s)}$ Write the transfer function in standard form and deternine if this system behaves in an underdamped or over-damped or critically-damped manner for the given values of the parameters. [10 marks] (iii) Solve for $C_2^{?}{}_2(t)$ in response to a step change in $C_i^{?}(t)$ of magnitude $M=2\mathrm{moles}/{\mathrm{m}}^3$. Sketch the shape of the dynamic behavior of $C^{?}{}_2(t)$. [10 marks]