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 $\frac{?}{?}$ mole.

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 $?$