An incompressible Newtonian fluid of nearly pure species B flows down a vertical wall $(x=0)$ in a film of

thickness $.$ Species $A$ is transferred from the surrounding gas stream to the liquid where it undergoes an

irreversible first$-$order homogeneous reaction with a rate constant $k_1.$ Outside the immediate neighborhood

of the liquid film the gas stream has a uniform concentration. If the liquid were in equilibrium with this

gas stream, its concentration would be ${}_{A0}.$ Very far upstream, the liquid is pure species $B.$ Let us assume

that there is no mass transfer from the gas stream to the falling stream for

The symbols denote the following:

$x$ : horizontal direction

$z$ : vertical direction

$v_{\text{m}\text{a}\text{x}}$ : maximum velocity of the liquid in the vertically falling film

$D_{AB}$ : binary diffusivity of species $A$ in species $B$

In the limit of concentration distribution of A being limited to a thin boundary layer near the entrance $(x)$

for $Pe=\frac{v_{\text{m}\text{a}\text{x}}}{D_{AB}}1,$ show that the rate at which mass of species $A$ is absorbed per unit width of a film of

length $L$ is

$\frac{W}{{}_{A0}{v}_{\text{m}\text{a}\text{x}}}=\sqrt[\phantom{2}]{\frac{D_{AB}}{k_1}}[(0\mathrm{.}5+)erf\sqrt[\phantom{2}]{}+\sqrt[\phantom{2}]{\frac{}{}}{e}^{-})$

Also show that in the limit where there is no chemical reaction, this result reduces to

$W=2{}_{A0}\sqrt[\phantom{2}]{\frac{v_{\text{m}\text{a}\text{x}}{D}_{AB}L}{}}$