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\underline{Problem 1}: An autocatalytic reaction A → R takes place in a constant volume isothermal reactor. Component mass balances give the following reactor model:

\begin{aligned}\frac{dC_{A}}{dt} & = & \frac{F}{V} (C_{Ain} − C_{A}) − kC_{A}C_{R} \\ \frac{dC_{R}}{dt} & = & \frac{F}{V} (C_{Rin} − C_{R}) + kC_{A}C_{R} \end{aligned}  (1)

In the above model • C_{Ain} and C_{Rin} are the inputs

• C_{A} and C_{R} are the states

• At steady state C_{Ain_{s}} = 1 and C_{Rin_{s}} = 1

• \frac{F}{V} = 1

• k = 2

1. Find the steady state values of C_{A} and C_{R}

2. Linearize the model around the steady state and put it in the form

\frac{dX}{dt} = AX + BU  (2)

3. Suppose the system states are initially at their steady state values but both inputs suddenly increase to 1.5 from 1.0. How do C_{A} and C_{R} change with time ?