We wish to model for the mass transfer

coefficient in a chemical vapor

deposition process. The flow of a

reactant$-$laden gas through a duct of

rectangular cross section can be

assumed laminar with velocity profile;

$u(y)=\frac{3}{2}U(1-(\frac{y}{H}{)}^2)$

Deposition of molecule $A,$ present in small concentrations $C_{A0}$ in the gas phase with majority component $B,$

can be assumed to proceed through a very fast reaction at the deposition surface $(y=H).$ Please note that

the $y=-H$ is an inert impermeable surface. The concentration gradients of $A$ in the gas phase are expected

to exist only in a thin layer near the deposition surface $(y=H).$ Show that the concentration profile of $A,$

$C_A(x,y)$ can be described as

$1-\frac{C_A}{C_{A0}}=\frac{{\displaystyle \underset{}{\overset{}{}}}\text{e}\text{x}\text{p}{}^{3}d}{{\displaystyle \underset{0}{\overset{}{}}}\text{e}\text{x}\text{p}{}^{3}d}$;$=(1-\frac{y}{H})(\frac{U{H}^2}{x{D}_{AB}}{)}^{\frac{1}{3}}$

Here, $D_{AB}$ is the binary diffusivity of species $A$ in species $B,$ and $$ is a dummy integration variable.