$($a$)$ Derive the steady$-$state oxygen mole balance for an incremental volume of $Ax($ A being the column cross$-$sectional area$)$ and show that the liquid phase balance is

$(1-)D_e\frac{d^{2}C}{d{x}^2}-u_{0L}\frac{dC}{dx}+k_{c}a(C^{**}-C)=0$

$($b$)$ Lump parameters by defining a new dimensionless length as

$z=\frac{k_{c}ax}{u_{0L}}$

and define the excess concentration as $y=(C-C^{**}),$ and so obtain the elementary, second order, ordinary differential equation

$\frac{d^{2}y}{d{z}^2}-\frac{dy}{dz}-y=0$

where

$=\frac{(1-)D_e{k}_{c}a}{u_{0L}^2},(dimensionless)$

Note, the usual conditions in practice are such that $1.$

$($c$)$ Perform a material balance at the entrance to the column as follows. Far upstream, the transport of solute is mainly by convection: $A{u}_{0L}{C}_0.$ At the column entrance $)=(0,$ two modes of transport are present, hence, show that one of the boundary conditions should be:

$u_{0L}{C}_0=-(1-)D_e\frac{dC}{dx}{|}_x|=0+u_{0L}C(0)$

The second necessary boundary condition is usually taken to be

$\frac{dC}{dx}=0,at,x=L$