Lipase, immobilized on beads of density, ${}_b,2000kg{m}^{-3}$ at an activity of $1000KLUk{g}^{-1},$ is used

in a packed bed bioreactor to produce biodiesel. Ten $m^3$ per day $n-$hexane solution containing $15$

$mol{m}^{-3}$ triglycerides and $3$:$1$ molar ratio methanol are fed into the reactor, which is loaded with

beads with a void fraction, $,$ of $0\mathrm{.}2.$

Neglecting mass transfer limitations$,$ prepare a graph showing the conversion vs the volume of

the reactor and then determine the reactor volume needed for a desired conversion of $70\%.$

The triglycerides transesterification reaction rate, per gram of catalyst used, can be described by

the ping pong bi bi kinetic model with both substrates inhibition, given by:

$v=\frac{k_{\text{c}\text{a}\text{t}}[E]}{1+(\frac{K_{IA}}{[A]})(1+\frac{[S]}{K_S})+(\frac{K_{IS}}{[S]})(1+\frac{[A]}{K_A})}$

Where, $E$ is the enzyme activity $),S$ is the substrate concentration given as the ester

bonds on the triglycerides $(mol{m}^{-3}),[A]$ is the alcohol concentration $(mol{m}^{-3}),$ and the kinetics

parameters are given below:

$k_{\text{c}\text{a}\text{t}}$:$,0\mathrm{.}041molmi{n}^{-1}L{U}^{-1}$

$Ks$:$430mol{m}^{-3}$

$K_A$:$350mol{m}^{-3}$

$K_{IS}$:$4\mathrm{.}45{10}^{4}mol{m}^{-3}$

$K_{IA}$:$3\mathrm{.}30{10}^{4}mol{m}^{-3}$

Use Polymath $($or any other computer package$)$ to solve the differential equation$($s$).$

Clearly state any assumptions made.

Hint: Relate volume of the bed to the weight of the catalyst