Two-Film Theory & Overall Mass Transfer Coefficients

The Interface Reality

In our previous article on Film Theory, we established that a stagnant boundary layer limits mass transfer. However, in any real gas-liquid contactor (like a scrubber or stripper), there is not one film—there are two.

1. The Stagnant Gas Film (with coefficient $k_y$)
2. The Stagnant Liquid Film (with coefficient $k_x$)

For a molecule of $CO_2$ to be absorbed from natural gas into a liquid amine solution, it must first diffuse through the gas film, cross the physical interface, and then diffuse through the liquid film.

The Fundamental Assumption of Two-Film Theory: The two phases are assumed to be in perfect thermodynamic equilibrium exactly at the microscopic interface. (i.e., $y_i = m \cdot x_i$, where $m$ is Henry's Law constant).

The Concept of the Overall Coefficient

To calculate the total flux, an engineer needs to know the interface concentrations ($y_i$ and $x_i$). Unfortunately, measuring the concentration of a chemical at a microscopic boundary layer inside a pressurized steel tower is physically impossible.

We only know the bulk gas concentration ($y$) and the bulk liquid concentration ($x$).

To solve this, engineers created the Overall Mass Transfer Coefficient ($K_y$ and $K_x$). This allows us to calculate flux based entirely on bulk concentrations!

$$ N_A = K_y \cdot (y - y^*) = K_x \cdot (x^* - x) $$

Where $y^*$ is the theoretical gas concentration that would be in equilibrium with the bulk liquid $x$.

Adding Resistances in Series

Just like electrical resistors in series ($R_{total} = R_1 + R_2$), mass transfer resistances are additive. The inverse of the mass transfer coefficient is the resistance ($R = 1/k$).

The Overall Resistance ($1/K_y$) is the sum of the gas film resistance and the liquid film resistance (adjusted by the equilibrium slope $m$):

$$ \frac{1}{K_y} = \frac{1}{k_y} + \frac{m}{k_x} $$

This is arguably the most important equation in gas absorption design!

Gas-Phase Controlled vs. Liquid-Phase Controlled

By analyzing the equation above, we can determine which film dictates the design of the tower:

• Highly Soluble Gases (Gas-Phase Controlled): E.g., Ammonia absorbing into water. The equilibrium slope ($m$) is very small. Therefore, the term $m/k_x \approx 0$. The equation reduces to $1/K_y \approx 1/k_y$. The entire resistance is in the gas film. To improve the tower, you must increase gas velocity and turbulence. Modifying the liquid flow will achieve almost nothing!
• Slightly Soluble Gases (Liquid-Phase Controlled): E.g., Oxygen absorbing into water (aeration). The slope ($m$) is massive. The term $m/k_x$ dominates the equation. The entire resistance is in the liquid film. To improve the tower, you must increase liquid agitation and turbulence.