Based on the assumptions involved in the penetration theory of mass transfer across a phase boundary, the concentration $C_A$ of a solute $A$ at a depth $y$ below the interface at a time $t$ after the formation of the interface is given by:
$$
\frac{C_A}{C_{A i}}=\operatorname{erfc}\left[\frac{y}{2 \sqrt{(D t)}}\right]
$$
where $C_{A i}$ is the interface concentration, assumed constant and $D$ is the molecular diffusivity of the solute in the solvent. The solvent initially contains no dissolved solute. Obtain an expression for the molar rate of transfer of $A$ per unit area at time $t$ and depth $y$, and at the free surface $(y=0)$.
In a liquid-liquid extraction unit, spherical drops of solvent of uniform size are continuously fed to a continuous phase of lower density which is flowing vertically upwards, and hence countercurrently with respect to the droplets. The resistance to mass transfer may be regarded as lying wholly within the drops and the penetration theory may be applied. The upward velocity of the liquid, which may be taken as uniform over the cross-section of the vessel, is one-half of the terminal falling velocity of the droplets in the still liquid.
Occasionally, two droplets coalesce forming a single drop of twice the volume. What is the ratio of the mass transfer rate $(\mathrm{kmol} / \mathrm{s})$ at a coalesced drop to that at a single droplet when each has fallen the same distance, that is to the bottom of the column?
The fluid resistance force acting on the droplet should be taken as that given by Stokes' law, that is $3 \pi \mu d u$ where $\mu$ is the viscosity of the continuous phase, $d$ the drop diameter and $u$ its velocity relative to the continuous phase.
It may be noted that:
$$
\operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}} \int_x^{\infty} \mathrm{e}^{x^2} \mathrm{~d} x
$$