Scaling and perturbation analyses for a bimolecular reaction. Consider an irreversible, bimolecular reaction in a liquid film at steady state, as shown in Fig. P$2-12$ of Deen.

Extend the analysis of Problem $2-12$ in Deen, which was limited to "infinitely fast" reaction kinetics.

$($a$)$ Sketch qualitatively $C_A(x),C_B(x),$ and $C_C(x)$ for a very small value of the reaction rate coefficient $k,$ a moderate value of $k,$ and a large $($but finite$)$ value of $k.$

$($b$)$ For the special case of equal reactant diffusivities $(D_A=D_B)$ and equal boundary concentrations of A and $B(C_{A0}=C_{BL}),$ show that the concentration of $A$ is governed by

$\frac{d^2}{d{}^2}=(+2-1)Da$

$(0)=1,(1)=0,$

where $=\frac{C_A}{C_{A0}},=\frac{x}{L},$ and $Da=k{C}_{A0}\frac{L^2}{D_A}$ is the Damk$$hler number. Hint:

First combine the conservation equations for A and $B$ and solve for $C_B(x)$ in terms of

$C_A(x).$

$($c$)$ For slow reaction kinetics $(Da1),$ a regular perturbation scheme can be used. Determine $()$ for slow kinetics, including terms of $O(Da).$

$($d$)$ For fast $($but not infinitely fast$)$ reaction kinetics the reaction will be confined largely to a zone of thickness $$ at the center of the liquid film. In this region, $C_A{C}_B{C}^{*}.$

Use order$-$of$-$magnitude considerations to determine $$ and $C^{*}.$ Specifically, determine the dependence of $\frac{}{L}$ and $\frac{C^{*}}{C_{A0}}$ on $Da,$ for $Da1.$ These results show when the approximation used for infinitely fast kinetics $(=0)$ is reasonable.