A series of four well$-$mixed reactors operate isothermally in the steady state. All the tanks do not

have the same volume, but the sum of $V_i=20m^3.$ The component whose concentration is

designated by $C$ reacts according to the following mechanism: $r=-k{C}^n$ in each tank.

Figure $1$: A series of four well$-$mixed reactors

Determine the values of the tank volumes $($really residence times of the component$)$ in each of the

four tanks for steady$-$state operation with a fixed fluid flow rate of $q$ so as to maximize the yield of

product $C_4.$ Note $(\frac{V_i}{q_i})={}_i,$ the residence time. Use the following data for the coefficients in

the problems;

$n=2\mathrm{.}5$

$k=0\mathrm{.}00625[\frac{m^3}{(kg)(mol)}{]}^{-1\mathrm{.}5}(s{)}^{-1}$

$C_0=20$kgmo$\frac{l}{m^3}$

$q=71\frac{m^3}{h}$

The units for $k$ are fixed by the constant $0\mathrm{.}00625.$

$($a$)$ Formulate an optimization model to minimize the total cost of the system.

i$.$ The objective function

ii$.$ The independent and dependent variables.

iii. The equality constraint

iv$.$ The inequality constraint.

$($b$)$ For the above problem, which model type is the most suitable to complete the calculation?