2. Convective Mass Transfer: Theories of Mass Transfer Coefficient

2.1 Theories of Mass Transfer Coefficient

Film Theory

This is the simplest theory for interfacial mass transfer, proposed by Nernst in 1904. It assumes that a stagnant film exists near interface, and a steady mass transfer at the interface. \[k = \frac{D}{l}\] where $l$ is the film thickness. Here concentration profile is linear.

According to this model, $\text{Sh}=kl/D = 1$. This model says that the mass transfer coefficient $k$ is proportional to the diffusion coefficient $D$ and independent of the fluid velocity $v$.

Film theory is useful in the analysis of mass transfer with chemical reaction.

Penetration Theory

This theory was proposed by Higbie, in 1935. According to this model, fluid parcels are replaced at uniform interval $\theta$. \[k = 2\sqrt{\frac{Dv}{\pi l}}\] where $l/v=\theta$, is the contact time.

According to this model, each of liquid elements stays in contact with the gas for same period of time; and unsteady state mass transfer occurs.

Surface Renewal Theory

It was proposed by Dankwerts in 1951. \[k = \sqrt{D\,s}\] where $s$ = surface renewal rate (1/s).

According to this model, at any moment, each of the liquid elements at the surface has the same probability of being substituted by fresh element; and unsteady state mass transfer occurs.

Boundary Layer Theory

According to this model, \[\text{Sh}_x = \frac{x k_{c,x}}{D}= 0.332 \text{Re$_x$}^{0.5} \text{Sc}^{0.33}\] where, $x$ is the distance of a point from the leading edge of the plate; $k_{c,x}$ is the local mass transfer coefficient.

The average value of Sherwood number for the entire length $L$ of the plate is given by \[\text{Sh}_{\text{avg}} = \frac{L k_{c,{\text{avg}}}}{D} = 0.664 \text{Re}^{0.5} \text{Sc}^{0.33}\]

Mass transfer coefficient varies with the diffusion coefficient to the 0.5 to 0.7 power; and with the fluid velocity to the 0.7 power.

Two-film Theory

Experimentally the mass transfer film coefficients $k_y$ and $k_x$ are difficult to measure except for cases where the concentration difference across one phase is small and can be neglected. Under these circumstances, the overall mass transfer coefficients $K_y$ and $K_x$ are measured on the basis of the gas phase or the liquid phase.
$N_A = K_y(y_{A,G} - y_A^*) = K_x(x_A^*-x_{A,L})$
$\displaystyle \frac{1}{K_x} = \frac{1}{mk_y} + \frac{1}{k_x}$
$\displaystyle \frac{1}{K_y} = \frac{1}{k_y} + \frac{m}{k_x}$
$\displaystyle \frac{\text{Resistance in gas phase}}{\text{Total resistance}} = \frac{1/k_y}{1/K_y}$
$\displaystyle \frac{\text{Resistance in liquid phase}}{\text{Total resistance}} = \frac{1/k_x}{1/K_x}$
Liquid-side Controlling: Here, total resistance equals the liquid film resistance. It is due to low $k_x$, which is consequence of slow diffusion in the liquid state, or solute is relatively insoluble in liquid.

The absorption of a gas of low solubility, such as carbon dioxide or oxygen in water is of this type of system.
Gas-side Controlling: $K_y = k_y$. Solute is very soluble in the liquid.

The absorption of a very soluble gas, such as ammonia in water is an example of this kind.
For the equilibrium relation with $y^*=mx$, if the gas is highly soluble in liquid, then $m$ is small. For sparingly soluble gases, $m$ is very high. In general, substances that are gases at room temperature and pressure become less soluble with increased temperature, and the substances that are solids at room temperature and pressure tend to become more soluble when the temperature rises.
As a general rule, for highly soluble gases, the mass transfer process is likely to be gas-phase controlled, whereas for sparingly soluble gases, it might be liquid-phase controlled. These are general statements, with exceptions that can occur, depending on the relative magnitudes of the individual phase mass transfer coefficients and the equilibrium constant $m$. If the gas phase is pure component $A$, then there is no resistance to diffusion in the gas phase, and the entire resistance lies in the liquid phase.