A batch reactor with constant volumetric holdup, $V,$ operating over a $1-$hour

period generates two products according to the parallel and irreversible reaction

mechanisms, as the following:

$AB$:$r_{A,1}=-k_1{c}_A[kmo\frac{l}{s*m^3}]$

$AC$:${}_{A,2}=-k_2{c}_A[kmo\frac{l}{s-m^3}]$

Notation:

$r_{A11}=$ reaction rate for species $A$ in reaction $1$;$k_1=$ rate constant for ${}_{A,1}$

$r_{A,2}=$ reaction rate for species $A$ in reaction $2$;$k_2=$ rate constant for $r_{A,2}$

The rate constants are provided to be dependent on temperature according to

the Arrhenius correlation for reaction $j,$ as the following:

$k_j=k_{j0}$exp$(\frac{E_j}{RT})$

$k_{j0}=$ pre$-$exponential factor of rate constant $k_j[k_{10}={10}^6{s}^{-1}$;$k_{20}={10}^{11}{s}^{-1}]$

$E_j=$ Activation energy of rate constant for reaction

$\{$:${10}^{3}ca\frac{l}{g}$mol$]$

The task assigned to you as a process engineer is to find the operating

temperature profile to maximize the yield of product B$.$ However, the temperature

cannot be operated beyond $300C.$ It is given that the component balance that

governs the concentration of species B in the batch reactor is as the following:

$\frac{d{c}_s}{dt}=k_1{c}_A$

Initial concentration of species A and temperature of the batch reactor is $c_{A,0}$ and

$T_{g,}$ respectively, while there is no $B$ and $C$ being produced at the beginning.

a$.$ Classify dependent variable $($s$),$ independent variable $($s$)$ and parameter $($s$)$ from the problem.

b$.$ Justify that it is an optimization problem by performing Degree of Freedom

$($DOF$)$ analysis.

c$.$ Identify equality and inequality constraints.