$4-12.$ Absorption with a Nearly Irreversible Reaction

Assume that gaseous species $A$ is absorbed in a stagnant liquid, resulting in the formation of species $B$ in the liquid via the reversible reaction $A,2B.$ The reaction rate is given by $R_{VA}=-k_V(C_A-K{C}_B^2).($Note that the equilibrium constant $K$ has units of inverse concentration.$)$ The reaction equilibrium strongly favors $B,$ but not quite to the extent that the reaction can be considered irreversible. The system is at steady state and the liquid layer is indefinitely thick. The liquid concentrations at the gas$-$liquid interface $)=(0$ are $C_A(0)=C_0$ and $C_B(0)=0.$ For simplicity, assume equal diffusivities, $D_A=D_B=D.$

$($a$)$ State the differential equations and boundary conditions governing the concentrations $()=\frac{C_A}{C_0}$ and $()=\frac{C_B}{C_0}.$ What is the length scale needed for converting $x$ to the dimensionless coordinate $?$

$($b$)$ Assuming that $=K{C}_{0}1,$ use a perturbation analysis to calculate $()$ and $()$ and find the flux of A into the film, $N_A(0).$ Determine the first two terms in the expansion for the flux and discuss the effect of reaction reversibility. $($Hint: Expand both concentration variables as regular perturbation series.$)$

Note: use regular perturbation. For theta$($eta$),$ determine the first two terms of expansion. Use theta$($eta$)$ to determine the flux of A into the film. For psi$($eta$),$ you only need to determine the first term of the expansion.

For and ODE that looks like f$"-$ f$=C_1+C_2{e}^{-x}+C_3{e}^{-2x},$ where C$1,$C$2,$ C$3$ are known constants, you should guess the solution of the form f$($x$)=f_h(x)+f_p(x),$ where $f_h$ is the homogeneous solution and $f_p(x)$

is the particular solution. Guess the particular solution of the form $f_p(x)$

$=$A $+$Bx$e^{-x}+$D$e^{-2x},$ where A$,$B$,$ and D are constants to be determined. Substitute this form into your ODE and compare terms with non$-$homogeneous right$-$hand side $($RHS$)$ to get the constsnts A$,$B$,$ and D$.$ This strategy is called "Methods of Undetermined Coefficients". You can use "Variation of Parameters," if you want to instead.