Exercise $4.-$ Let's consider the isothermal stirred$-$tank blending system shown in the following figure. The

control objective is to blend the two inlet streams to produce an outlet stream that has the desired composition.

Each input stream is a mixture of two chemical species, A and B$,$ has a mass flow rate $w_i(t)$ and a mass fraction

of $A,x_i(t),$ which can vary over time. The volume of liquid in the tank $V$ can vary with time, because the exit

flow rate is not necessarily equal to the sum of the inlet flow rates.

Assuming a constant density $$ and a perfect mixing, the dynamic model of this process obeys the following

equations:

$\frac{dV}{dt}=\frac{1}{}(w_1+w_2-w)$

$\frac{dx}{dt}=\frac{w_1}{V}(x_1-x)+\frac{w_2}{V}(x_2-x)$

where $x_1(t),x_2(t),w_1(t),w_2(t),x(t),w(t)$ and $V(t)$ are functions of time.

$($a$)$ Determine the steady$-$state equilibrium relationships among the different variables.

$($b$)$ Calculate the corresponding linearized equations around these steady$-$state values.