# Simulation of Adsorption Processes – Modeling Basics

Adsorption processes are well-established methods for gas separation, gaining increasing importance for direct air capture of CO2 and air drying in battery production. These processes can be dynamically simulated with TIL Adsorption. Here, we present the basics of the modeling.

Helena Vorspohl

|

February 22, 2024

egorovartem

Modelling and simulation is essential for the design and optimization of the operation of plants with adsorption processes, such as sorption wheels, TSA (Temperature Swing Adsorption) and plants for CO2 capture from the air (Direct Air Capture). Dynamic simulations enable the correct dimensioning of adsorption plants, the optimization of process times, the selection of the best possible components and the analysis of potential savings to improve energy efficiency.

The model library TIL Adsorption, offers a powerful and flexible basis for the simulation of various adsorption processes.in this blog article we would like to explain the modeling with TIL Adsorption in an understandable way. We will focus on the description of adsorption in the gas phase and consider the adsorption of a single gas component.

Adsorption is defined as the attachment of a molecule from the gas phase to a solid. Desorption, on the other hand, is the removal of an adsorbed molecule and is the opposite of adsorption. The adsorbent is the solid to which the molecule is bound. The molecules bound (adsorbed) to the solid are called adsorbate, the unbound molecules in the gas phase are called adsorptive.

For process simulation, however, the transition between different states is of particular interest. Only by considering the adsorption kinetics, which describe the heat and mass transfer between gas and adsorbent, technical adsorption processes can be modeled in a meaningful way.

## Modeling of the equilibrium state

The thermodynamic equilibrium state of adsorption is usually described by adsorption isotherms, as shown in Figure 2. The isotherms define the loading of the adsorbent x as a function of the partial pressure pi of the adsorptive at constant temperature T.

There are several models for the mathematical description of the adsorption equilibrium from which the isotherms can be calculated. Among the best known are the Langmuir, Sips, Toth or Dubinin equations. The type of isotherm or characteristic curve can usually be used to determine which model is appropriate for a given adsorptive/adsorbent pairing.

## Modeling of the adsorption kinetics

Adsorption kinetics describe the heat and mass transfer between the gas phase and the adsorbent. Accordingly, the adsorption kinetics are modeled in TIL Adsorption within the so-called adsorption cell. The adsorption cell essentially consists of a gas cell and an adsorbent cell, which are in heat and mass exchange with each other. Conservation equations for energy and mass are defined in the cells.

The mass transfer between the gas cell and the adsorbent cell determines whether adsorption or desorption takes place. The driving potential is the pressure difference between the partial pressure pi of the adsorbent in the gas cell and the equilibrium pressure peq of the adsorbate in the adsorbent cell. The equilibrium pressure peq can be calculated from the loading x and the temperature T of the adsorbent using the equilibrium models in the simulation.

Adsorption occurs when the partial pressure pi of the adsorbent in the gas cell is greater than the equilibrium pressure peq of the adsorbate. Desorption occurs when the partial pressure pi of the adsorbent in the gas cell is less than the equilibrium partial pressure peq of the adsorbate. If the partial pressure pi is equal to the equilibrium partial pressure peq, neither adsorption nor desorption takes place.

$$ṁ_{\rm i} \ = m_{\rm Ads} \ \cdot \ k_{\rm x,ef} \ (x_{\rm eq}-x)\ \ \ (Eq. 1)$$

The gas and adsorbent cells are also connected by an effective thermal resistance to model the heat exchange (see Figure 5). The effective resistance describes both the convective heat transfer between the gas and the surface and the heat conduction within the adsorbent. For the convective part, correlations (e.g. for flow debris) are usually used to describe the resistance, while the heat conduction is a material parameter. In practice it is difficult to separate the two components. The resistance is often dominated by the thermal conductivity.

For the modeling of adsorption systems, TIL Adsorption includes component models for adsorbers and adsorption wheels in various designs. These models are based on the adsorption cells and can be modified and integrated into larger system models.

The adsorber consists of adsorption cells and flow resistors connected in series (see Figure 6). The series connection models a 1D discretization in the direction of flow. The resolution of the discretization is freely selectable and thus determines the number of cells. The resistances are used to model the pressure drop along the length of the adsorber. Depending on the design of the adsorber, heat capacities of housings and support structures can also be taken into account. Convection between the casing and the adsorber as well as external heat input can also be modeled.

The adsorption wheel is also modeled with multiple adsorption cells. Figure 7 shows the structure of an adsorption wheel with three different chambers with different flow rates. Gas separation takes place in the "adsorption" chamber. The adsorbed gas molecules are typically desorbed in the "regeneration" chamber by applying heat. After regeneration, the "cooling" chamber is used to cool the adsorption wheel. The arrangement of the adsorption cells shown in Figure 7 allows both tangential and axial discretization in the model.

## Calibrating the model

Calibration of heat and mass transfer coefficients is recommended for accurate simulation of adsorption systems. This requires measurements of a dynamic operation of the adsorber, such as a breakthrough curve. The breakthrough curve describes the time course of the outlet concentration at the adsorber after a sudden change in otherwise constant inlet conditions. For the measurement of adsorption kinetics, the adsorber is usually not loaded at the beginning of the measurement.

To calibrate the transport coefficients, TIL Adsorption is used to create a digital twin of the experimental setup. Care must be taken in the model to ensure that the dimensions of the adsorber, the initial conditions, and the gas composition of the feed match the experiment.

In the example shown here, CO2 is adsorbed on a zeolite granulate. The experimental data are taken from the publication "Confinement effects facilitate low-concentration carbon dioxide capture with zeolites" by Fu et al (PNAS, Vol. 119, No. 39, 2022). Figure 9 shows the time courses of the adsorption concentrations in the adsorber, which can be determined from the difference between the inlet and outlet concentration of the measurement or simulation. The TLK Optimization Suite was used to fit the kinetic coefficient in order to achieve the best possible agreement between simulation and measurement data.

## Temperature Swing Adsorption (TSA) Process Modeling

TIL Adsorption can be used to perform comprehensive plant simulations. The following example considers temperature swing adsorption (TSA) for hydrogen drying. The TSA system shown in Figure 10 consists of two adsorbers filled with zeolite granules that alternately adsorb and desorb water. The system is operated cyclically and exploits the temperature dependence of adsorption. This means that water is adsorbed at low temperatures and desorbed at high temperatures.

By discretizing, in this example by 10 cells, the load curve in the adsorber can be represented. Figure 11 shows the loading curve at two different times. It is clear that with the selected adsorber size, the desorption time is not sufficient to completely desorb the adsorber. Simulation is therefore a valuable tool for adsorber sizing.

The dynamic model of the TSA can also be used to study the influence of various process parameters. Figure 12 shows the time course of the average load in Adsorber 1 for different desorption temperatures. It clearly shows, for example, that a desorption temperature of 70 °C is not sufficient to desorb the adsorber sufficiently at the selected process times.