The following Reaction will be Carried out in a poorly designed Batch Reactor:

$A-B$

$r_A=-k{C}_A$

Assume Constant Temperature

Assome Volume of Reactor is constant

Due to a poor agitation, there is a concentration gradient in the radial direction only $($No gradient in the $z$ direction$).$ the Concentration gradient at the beginning of the experiment is shown below:

you may assume the Concentration gradient is linear through the experiment and parallel to that at $t=.$

Derive the design equation.