(Solved): 1. An isothermal, constant-holdup, constant-throughput CSTR with a first-orde ...

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1. An isothermal, constant-holdup, constant-throughput CSTR with a first-order irreversible reaction is expressed as \[ \frac{d C_{A}}{d t}+\left(\frac{F}{V}+k\right) C_{A}=\frac{F}{V} C_{A 0} \] where \( C_{A} \) is the concentration of Component \( \mathrm{A} \) at the exit of the reactor. \( C_{A 0} \) is the concentration of Component \( \mathrm{A} \) at the entrance of the reactor. \( F \) is the volumetric flow rate of the inlet stream, as well as the outlet stream. \( V \) is the volume of the fluid inside the reactor. \( k \) is the reaction rate constant. The reactant concentration in the tank is initially zero \[ C_{A}(0)=0 \] At time equal zero, the inlet concentration \( C_{A 0} \) is now changed by the controller to hold \( C_{A} \) near its setpoint value \( C_{A}^{\text {set }} \) \[ C_{A 0}=C_{A M}+C_{A D} \] where \( C_{A D} \) is a disturbance composition. The controller has proportional and integral action \[ C_{A M}=\bar{C}_{A M}+K_{c}\left(E+\frac{1}{\tau_{I}} \int E d t\right) \] where \( K_{c} \) and \( \tau_{I} \) are constants, \( \bar{C}_{A M} \) is the steady-state value of \( C_{A M} \), and \( E=C_{A}^{\text {set }}-C_{A} \). Derive the second-order equation describing the closed-loop process in terms of perturbation variables. Show that the damping coefficient is \[ \zeta=\frac{1+k \tau+K_{c}}{2 \sqrt{K_{c} \frac{\tau}{\tau_{I}}}} \] What value of \( K_{c} \) will give critical damping? At what value of \( K_{c} \) will the system become unstable?