The Adsorption command models adsorption by simulating either tracer particles or a concentration field moving through the structure and interacting with porous voxels. Within the porous active zone, the governing adsorption equations are solved based on the transported concentration and the equilibrium concentration.

Isotherm Models

An adsorption isotherm model describes how a solute distributes between the liquid or gas phase and the surface of a solid (adsorbent) at constant temperature. The adsorbed solute is called adsorbate. The model mathematically relates the amount of adsorbate to the equilibrium concentration of the solute in the surrounding phase.

Isotherm models are used to characterize adsorption capacity, surface properties, and interaction mechanisms. They help to predict how much solute can be removed under different conditions and are essential for designing and optimizing adsorption processes. Common isotherm models include Langmuir, Freundlich, and Toth, each based on different assumptions about surface behavior and adsorption patterns. Already adsorbed species can detach from the surface and return to the solute phase, making the process reversible. Additionally, the solute is assumed to behave like an ideal gas, and interactions between adsorbates are neglected. The adsorption reaction can be formulated as such:

where is a solute species, is an adsorbent site, and is the adsorbate species.

Below, a schematic representation of the adsorption process is shown. Here, (A) depicts the adsorbent surface, (B) indicates the adsorbate at the interface, and (C) shows the solvent containing the solute.

Currently three isotherm models are implemented in the Adsorption command: Freundlich, Langmuir, and Toth. If you require different adsorption isotherm models to accurately simulate your reactions, please contact us to discuss the possibility of implementation.

Langmuir

The Langmuir isotherm model describes monolayer adsorption on a surface with a finite number of identical sites, where each site can hold only one solute molecule. Adsorption is reversible, with the adsorption rate proportional to the solute concentration and available sites, and the desorption rate proportional to the amount already adsorbed. This model assumes no interaction between adsorbed molecules and predicts a maximum adsorption capacity when all sites are occupied. For the Langmuir model the equilibrium loading of the adsorption sites is calculated by:

where c is the concentration (mol/m3), K is the equilibrium constant (m3/mol), and sm is the maximum loading (mol/kg).

Freundlich

The Freundlich isotherm model describes adsorption on heterogeneous surfaces with sites of varying adsorption energies and does not assume monolayer coverage or a finite number of identical adsorption sites. When adsorption does not exhibit a strictly bounded capacity, it can be modeled using a power law, where the adsorption rate depends non-linearly on the solute concentration while desorption remains linear. As a result, adsorption increases continuously with solute concentration. For the Freundlich model the equilibrium loading of the adsorption sites is calculated by:

where c is the concentration (mol/m3), K is the equilibrium constant (m3/mol), and m is the empirical exponent.

Tóth

The Tóth isotherm model is a modification of the Langmuir isotherm that introduces a heterogeneity parameter t to better fit experimental data, especially at low and intermediate pressures. The model reduces to the Langmuir equation when the heterogeneity parameter equals one, making it useful for systems that deviate from ideal monolayer adsorption behavior. Here, the equilibrium loading of the adsorption sites is computed from:

where c is the concentration (mol/m3), K is the equilibrium constant (m3/mol), sm is the maximum loading (mol/kg), and t is the Tóth heterogeneity parameter:

The calculation of the Tóth heterogeneity parameter requires Tóth coefficients (t0, c0) that are valid for given adsorbent/adsorbate pairs and can be derived from experiments or found in literature. Refer to Coker and Knox (2014) for exemplary Tóth coefficients.

Tóth Legacy

Adsorption Rate Models

In the Adsorption command, the adsorbent is always represented by a porous material with unresolved porosity. Adsorption then occurs on internal interfaces in this unresolved adsorption zone.

The amount of available adsorption sites is given as a maximum loading under the defined equilibrium conditions. Inside the porous zone the solute can loose or gain concentration , while the porous voxel of the adsorbent gains or looses the corresponding adsorbate loading . Here, the adsorption speed is defined by the solute concentration and the number of free adsorption sites on the adsorbent surface. The desorption speed is determined by already occupied number of adsorption sites.

Linear Driving Force

The Linear Driving Force model represents the net reaction rate as directly proportional to the deviation from equilibrium. It reflects the concept that sorption moves toward equilibrium, with the rate depending on how far the current adsorbed amount is from the equilibrium value. The rate is positive when net adsorption occurs, negative during net desorption, and zero when the system reaches equilibrium.

Using the rate constant kldf, the selected isotherm equation φ(c), and the current loading inside the voxel the local reaction rate R (mol m-3 s-1) can be calculated according to: