Problem $1.$ Concentration of CO Down the Channel of a Catalytic Converter $(30pts$

Use the equation derived in class to calculate the relative concentration of $CO$ at the exit of the

channel where $x=L,C_A\frac{L}{C_{Ao}}.$ Assume the following channel parameters; stagnant film

thickness $=100m,$ length of catalytic converter $L=20cm,$ and channel side dimension $a=4$

$mm.$ There is an instantaneous reaction at the channel wall. The diffusivity of $CO(A)$ through

exhaust gas $($C$)$ is $D_{AC}=0\mathrm{.}3{10}^{-4}\frac{m^2}{s}$ at $373K.$ In this problem, species $B$ is $C{O}_2.$ Further,

assume that the temperature inside the channel of the catalytic converter is $1,000K.$ Assume that

the exhaust pressure in the catalytic converter is $1$ atmosphere. Further, assume that $v_x$ is $400$

$c\frac{m}{sec}.$ Note: $5$ points of extra credit for anyone who can figure out how $v_x$ was calculated. I

assumed a $2-$liter engine running at $6,000rpm($revolutions per minute$).$ The engine is a $4-$stroke

engine. The total cross$-$sectional area through which the exhaust flows for all the channels in the

catalytic converter is $250c{m}^2.$ To meet emissions targets, the catalytic converter must reduce

the $CO$ concentration in the engine exhaust to a very low level, such that

a$)$ Prove that the solution given in class for the relative concentration $\frac{c_{AL}}{c_{AO}}=$exp$(-\frac{4D_{AC}L}{a{v}_x})$

is correct starting from $v_x\frac{del{c}_A}{\text{d}\text{e}\text{l}\text{x}}=-\frac{4D_{AC{c}_A}}{a},$ with boundary condition of $c_A=c_{Ao},$ at $x=0.$

b$)$ Is this criterion achieved for this catalytic converter design?

c$)$ Since the film thickness is not known with absolute certainty, repeat the analysis with $=$

$50m$ and then again for $200m.$ How sensitive is the value of $$ on the value of

$C_A\frac{L}{C_{Ao}},$ and is the criterion achieved for these new values of $?$