7.3 Interphase mass transfer

Interphase Mass Transfer Principles

Interphase mass transfer describes how a chemical species moves from one phase to another across their shared boundary. This process is central to many chemical engineering operations: oxygen dissolving into a fermentation broth, a solvent pulling antibiotics out of a liquid mixture, or activated carbon pulling pollutants from a gas stream. Designing equipment for any of these requires you to predict how fast mass transfer happens and what controls that rate.

The rate depends on three main factors: the concentration driving force, the interfacial area available for transfer, and the mass transfer coefficients on each side of the interface. Engineers use theoretical models (two-film theory, penetration theory, and others) to estimate these quantities and size real equipment.

Fundamentals of Interphase Mass Transfer

Interphase mass transfer is driven by a concentration gradient (or, more precisely, a chemical potential gradient) between two phases. A component always moves from the phase where its concentration is higher toward the phase where its concentration is lower.

Transfer occurs across the interface between two immiscible phases. The three most common configurations are:

• Gas-liquid: oxygen absorbing into a fermentation broth, $CO_2$ scrubbing in amine solutions
• Liquid-liquid: solvent extraction of antibiotics from a fermentation broth
• Solid-fluid: adsorption of pollutants onto activated carbon from a gas or liquid stream

Net mass transfer continues until equilibrium is reached. At equilibrium, the chemical potentials of the transferring species are equal in both phases, so there's no longer a driving force for transport.

Factors Influencing Interphase Mass Transfer Rates

Four factors largely control how fast interphase mass transfer occurs:

• Concentration difference (driving force): The bigger the gap between the actual concentration and the equilibrium concentration, the faster the transfer.
• Interfacial area: More contact area between phases means more "room" for transfer. That's why packed columns, tray columns, and dispersed systems (bubbles, droplets) are used to maximize area.
• Mass transfer coefficients: These capture how easily a species moves through each phase. They depend on fluid properties (density, viscosity, diffusivity) and flow conditions (turbulence, mixing intensity). Higher coefficients mean faster transfer.
• Equilibrium relationship: The equilibrium distribution (e.g., Henry's law constant for gas-liquid systems) determines where the driving force "aims." A very soluble gas, for instance, has a large capacity to absorb before equilibrium limits the process.

Mass Transfer Rates Across Interfaces

Gas-Liquid Mass Transfer

Gas-liquid mass transfer happens when a soluble gas dissolves into a liquid or when a dissolved volatile compound strips out of a liquid into a gas. The rate depends on the gas solubility, the interfacial area, and the mass transfer coefficients on both the gas side and the liquid side.

Common examples:

• Oxygen transfer into fermentation broths (the liquid-side resistance usually dominates because $O_2$ has low solubility in water)
• $CO_2$ absorption into amine solutions for gas treating
• Air stripping of volatile organic compounds (VOCs) from contaminated wastewater

Liquid-Liquid Mass Transfer

Liquid-liquid mass transfer involves a solute moving from one liquid phase into a second, immiscible liquid phase. The rate depends on the distribution coefficient (also called the partition coefficient), which describes how the solute splits between the two liquids at equilibrium, along with the interfacial area and mass transfer coefficients in both liquid phases.

Examples include extraction of antibiotics from fermentation broths using an organic solvent and removal of metal-ion contaminants from wastewater via solvent extraction.

Solid-Fluid Mass Transfer

Solid-fluid mass transfer covers cases where a solute moves between a solid surface and a surrounding gas or liquid. The rate is governed by the solute's solubility (or adsorption capacity), the solid's surface area, and the mass transfer coefficient in the fluid phase.

Examples include adsorption of pollutants from flue gas onto activated carbon and leaching of a valuable mineral from crusite ore into an acid solution.

Modeling Interphase Mass Transfer

Several theoretical models exist to predict mass transfer coefficients. Each makes different assumptions about how fluid moves near the interface.

Two-Film Theory

The two-film theory (proposed by Whitman in 1923) is the simplest and most widely used model. Its key assumptions:

1. All resistance to mass transfer is concentrated in two thin, stagnant films, one on each side of the interface.
2. The bulk of each phase is well-mixed, so concentration gradients exist only within the films.
3. At the interface itself, the two phases are in equilibrium with each other.
4. Transport through each film occurs by steady-state molecular diffusion.

Because the resistances act in series, the overall mass transfer coefficient $K$ is found by adding the individual resistances:

$\frac{1}{K} = \frac{1}{k_1} + \frac{1}{k_2}$

where $k_1$ and $k_2$ are the individual-phase mass transfer coefficients (adjusted for equilibrium via the appropriate distribution coefficient). Whichever film has the larger resistance controls the overall rate.

Penetration and Surface Renewal Theories

Penetration theory (Higbie, 1935) takes a different approach. Instead of a permanent stagnant film, it assumes that fresh fluid elements from the bulk are periodically swept to the interface, stay in contact for a fixed time $t_c$, and are then replaced.

During that contact time, the solute diffuses into the element by unsteady-state diffusion. The resulting mass transfer coefficient is:

$k = 2\sqrt{\frac{D}{\pi t_c}}$

where $D$ is the diffusivity. A key prediction: $k \propto \sqrt{D}$, compared to $k \propto D$ in two-film theory. This square-root dependence often matches experimental data better for turbulent systems.

Surface renewal theory (Danckwerts, 1951) extends penetration theory by recognizing that not every fluid element stays at the interface for the same time. Instead, elements have a random distribution of contact times characterized by a surface renewal rate $s$. The mass transfer coefficient becomes:

$k = \sqrt{D \cdot s}$

This still gives the $k \propto \sqrt{D}$ relationship but removes the need to know a single fixed contact time.

Film-Penetration Theory

The film-penetration theory combines elements of both the two-film and penetration models. It accounts for steady-state diffusion through a thin film and unsteady-state diffusion in the bulk fluid beyond that film. This hybrid approach provides a more realistic picture when neither pure film resistance nor pure penetration alone captures the system behavior. In practice, it's most useful for systems where both near-interface and bulk-phase resistances matter.

Factors Influencing Interphase Transfer

Interfacial Area and Specific Interfacial Area

The mass transfer rate is directly proportional to the area of contact between phases. Equipment design focuses heavily on maximizing this area:

• Packed columns use structured or random packing to create a large wetted surface.
• Tray columns use perforated plates to generate bubbles, increasing gas-liquid contact.
• Dispersed systems break one phase into small bubbles, droplets, or particles to boost area.

The specific interfacial area $a$ (interfacial area per unit volume, typically in $m^2/m^3$) is a key design parameter. You'll often see mass transfer rates reported as $k \cdot a$ (the volumetric mass transfer coefficient) because $k$ and $a$ are difficult to measure independently in real equipment.

Phase Equilibrium and Driving Force

The equilibrium relationship tells you the maximum extent of transfer. Common equilibrium descriptions include:

• Henry's law for dilute gas-liquid systems: $p = H \cdot x$, where $p$ is partial pressure, $x$ is mole fraction in the liquid, and $H$ is Henry's law constant
• Partition (distribution) coefficients for liquid-liquid systems: $K_D = C_{phase\,1} / C_{phase\,2}$

The actual driving force for mass transfer is the departure from equilibrium. If the bulk concentrations are far from equilibrium values, the driving force is large and transfer is fast. As the system approaches equilibrium, the driving force shrinks and transfer slows.

Mass Transfer Coefficients and Correlations

Mass transfer coefficients depend on both fluid properties and flow conditions. In practice, you estimate them using dimensionless correlations that relate the Sherwood number to other dimensionless groups:

$Sh = \frac{k \cdot L}{D}$

where $k$ is the mass transfer coefficient, $L$ is a characteristic length, and $D$ is diffusivity. Typical correlations take the form:

$Sh = C \cdot Re^a \cdot Sc^b$

where $Re$ is the Reynolds number (flow regime), $Sc = \nu / D$ is the Schmidt number (ratio of momentum to mass diffusivity), and $C$, $a$, $b$ are empirical constants specific to the geometry.

Higher turbulence (larger $Re$), higher diffusivity (smaller $Sc$), and shorter diffusion distances all increase the mass transfer coefficient.

Temperature Effects

Temperature influences mass transfer through several competing mechanisms:

• Diffusivity increases with temperature, which raises $k$ and speeds up transfer.
• Viscosity decreases with temperature, which also tends to increase $k$ (thinner boundary layers, better mixing).
• Solubility and equilibrium shift with temperature. For many gases, solubility in liquids decreases at higher temperatures, which can reduce the driving force for absorption.

Choosing the right operating temperature requires balancing these kinetic benefits against the thermodynamic effects on equilibrium. For gas absorption, running cooler often helps because higher solubility provides a larger driving force, even though the mass transfer coefficient itself is somewhat lower.