(Solved): 2. A stabilization tank, shown in Figure 2 below, is installed in a water treatment facility to av ...

2. A stabilization tank, shown in Figure $2$ below, is installed in a water treatment facility to avoid discharging component A directly to the environment. Despite a series of reactions in the tank at constant volume and temperature, the tank outflow contains considerable amount of toxic compound which needs to be controlled and managed by manipulating the feed flow rate, $f_{\text{in }}(t)$. The concentration of component $\mathrm{A}$ at the inlet, $C_{\mathrm{A},\mathrm{in}}$, varies, and the conversion of $\mathrm{A}$ in the tank follows the reaction rate of $r_A=k_1{C}_A+k_2{C}_A^2$. Figure 2: A typical stabilization tank. (a) Under a perfectly mixed condition, develop an unsteady mass balance for component A over the tank. (5 marks) (b) Linearize the non-linear terms in the dynamic equation in part (a). (6 marks) (c) Alter the mathematical equation obtained above and express the equation in the form of deviation variables. (5 marks) (d) Determine the transfer function for the concentration of A. (6 marks) (e) Under the test condition, component A flows at constant flow rate into the stabilization tank. A change in the upstream causes a step change given by $0.05\mathrm{u}(\mathrm{t})$ in the concentration of $\mathrm{A}$. Derive an equation describing the output concentration as a function of time. (8 marks)