Problem $2.(100$ points$)$ An appealing partial solution to the greenhouse gas problem is to convert biomass into liquid fuels. Most of biomass is composed of linked C$6$ sugars $(C_6{H}_{12}{O}_6)$ and $C5$ sugars $(C_5{H}_{10}{O}_5),$ which can be broken down by an enzyme secreted by fungi. Ideally, the sugars could then be converted into ethanol or other liquids in subsequent reactions.

This mechanism is proposed for the enzymatic breakdown:

$C_{11}{H}_{22}{O}_{11}+$ Enzyme $$ Complex $,($reaction $1)$

Complex $$ Enzyme $+C_6{H}_{12}{O}_6+C_5{H}_{10}{O}_5,($reaction $2)$

Reaction $1$ is expected to be reversible under some conditions, but reaction $2$ is

expected to be irreversible.

Experimental rate data on this enzyme from low$-$conversion batch$-$reactor experiments can be fit to this expression:

$d\frac{C_5{H}_{10}{O}_5}{d}t=r=a[C_{11}{H}_{22}{O}_{11}{]}_0\frac{[\text{E}\text{n}\text{z}\text{y}\text{m}\text{e}{]}_0}{1+b^{**}[C_{11}{H}_{22}{O}_{11}{]}_0}$

$($Eqn $3)$

$a=2{10}^4$ liter $\frac{?}{?}$ mole$-$second $,b={10}^8$ liter $\frac{?}{\text{m}\text{o}\text{l}\text{e}}.$

where $[C_{11}{H}_{22}{O}_{11}{]}_0$ and $[$ Enzyme ${]}_0$ are the initial concentrations added to the mixture, i$.$e$.$

$($moles added $/$ volume of solution$),$ not necessarily the actual concentrations of these species in the beaker when the reaction is running, since some of the enzyme will exist in the form of the complex.

$($a$)(20$ points$)$ Is the observed rate law $($Eqn $3)$ consistent with the mechanism shown above? If so$,$ give an expression for $b$ in terms of $k_1,k_{-1},$ and $k_2.$ If not consistent, explain.

Suppose we could tether ${10}^{-9}$mole of enzyme within a $4mm$ diameter porous particle without affecting the rate law $($i$.$e$.r$ is given by Eqn. $3).$ The diffusivity inside the porous particles is ${10}^{-10}\frac{m^2}{s}.$ In the bulk fluid $D=7{10}^{-10}\frac{m^2}{s}.$ Suppose we then filled a packed bed reactor $($internal diameter $2cm,$ length $30cm)$ with many particles like this, and flowed an aqueous solution of $C_{11}{H}_{22}{O}_{11}$ through the reactor at rate of $1$ liter$/$minute$.$ The void fraction of the packed bed $=0\mathrm{.}4.$ For concentrations of $C_{11}{H}_{22}{O}_{11}$ below $0\mathrm{.}5M,$ the viscosity and density of the solution is essentially the same as that of water.

$($b$)(30$ points$)$ Write an equation for the Thiele modulus for this system, as a function of $C_{11}{H}_{22}{O}_{11}.$ Over what range of $C_{11}{H}_{22}{O}_{11}$ is it reasonable to neglect diffusive transport limitations$?$

$($c$)(30$ points$)$ If $C_{11}{H}_{22}{O}_{11}$ is always in the range where transport limitations are

negligible, what differential equation$($s$)$ should be solved to compute the conversion?

What Matlab program would you use to solve the equation$($s$)$ numerically? Write the differential equation$($s$)$ in the form $d\frac{Y}{d}t=F(Y)$ required by the Matlab solvers.

$($d$)(20$ points$)$ Write $($but do not attempt to solve$)$ the differential equation$($s$)$ with

boundary conditions that would have to be solved to compute the effectiveness factor $$ if $[C_{11}{H}_{22}{O}_{11}{]}_{\text{b}\text{u}\text{l}\text{k}}$ had a value outside the range specified in part $($b$).$ Are you missing any data needed to calculate $?$