# (232v) A Theoretical Approach to Predict Adsorption Isotherms of Isomers Using Density-Functional-Theory and Lattice-Cluster-Theory

- Conference: AIChE Annual Meeting
- Year: 2016
- Proceeding: 2016 AIChE Annual Meeting
- Group: Engineering Sciences and Fundamentals
- Session:
- Time:
Monday, November 14, 2016 - 3:15pm-5:45pm

*Patrick Zimmermann ^{1}*

*, Thomas Goetsch*

^{2}, Tim Zeiner^{2}, Sabine Enders^{1}^{1}Karlsruhe Institute of Technology, Institute of Technical Thermodynamics and Refrigeration Engineering, Engler-Bunte-Ring 21, D-76131 Karlsruhe, Germany.

Email: patrick.zimmermann@kit.edu; sabine.enders@kit.edu

^{2}Technical University Dortmund, Laboratory of Fluid Separations, Emil-Figge-Straße 70, 44227 Dortmund, Germany.

Email: Thomas.Goetsch@bci.tu-dortmund.de, Tim.Zeiner@bci.tu-dortmund.de

Key words: Adsorption Isothermes of Binary Mixtures, Density Functional Theory, Separation of Close Boiling Mixtures

Abstract:

The separation of mixtures having components with very similar vapor pressures, like isomers, using distillation is always a very hard task and requires a large amount of invest and energy cost. For some purposes, e.g. exact measurements of thermophysical properties, a high grade of purity of the compound is required. One possible way to achieve almost pure substances is given by adsorption. For the development of industrial adsorption processes, adsorption isotherms, describing the relation of the excess fluid density in the pores over wide pressure ranges are applied. McCabe-Thiele plots showing the composition of the adsorbed phase against the feed composition can be helpful, too. Experiments in this area require a high effort of money and time so that it would be great to have the possibility to reduce the number of experiments and predict adsorption isotherms in a wide range of conditions using parameters that are adjusted just to a few experimental data points. Therefore, this contribution deals with the prediction of partial density profiles in narrow pores applying density functional theory (DFT) [1,2,3,4] in combination with a suitable equation of state, which should be able to describe different isomers. The analysis of these partial densities profiles results in the knowledge of the adsorption isotherms, phase transitions in pores, hysteresis effects and wetting as well as prewetting phenomena.

The DFT applied in the grand canonical ensemble starts with an expression for the grand potential as a functional of the density profile of the fluid, where we assume the fluid is inhomogeneous in only the * *direction perpendicular to the wall of the narrow slit pore. When considering a fluid in an external field at a fixed temperature and at fixed chemical potentials of the components in the fluid mixture, the density of the fluid is not spatially constant. At thermodynamic equilibrium the grand potential function is in its global minimum with respect to variations of the density profiles. The key step in DFT is to specify an approximate form for grand potential that is both tractable and accurate.

The properties of the fluid mixture can be described with the inhomogeneous Helmholtz energy, which can be calculated using any equation of state (EOS). Recently, the different equations of state (EOS) (e.g. the Peng-Robinson EOS [5], SAFT-VR-EOS [6], PC-SAFT-EOS [7]) were applied in the DFT framework for mixtures, whereupon the local density approach was used. However, all used equations are to join a common feature, they are not able to model isomers in terms of branching. In order to model mixtures of several isomers differing in the degree of branching the Lattice-Cluster-Theory (LCT) introduced by Freed et al. [8,9] and reformulated as equation of state (LCT-EOS) by Langenbach et al. [10] can be applied. The advantage of the LCT-EOS is given by the use of three pure component parameters to describe different isomers with the same number of united atom groups. These parameters can be fitted to the physical bulk properties of the e.g. n-alkane (vapor-pressure or/and liquid densities) and then the properties of the branched alkanes can be predicted by involving architecture coefficients given by the chemical formula. The binary interaction parameters of mixtures can be estimated by fitting to vapor-liquid equilibria. The external potential occurring in the grand potential describes the interaction of the fluid with solid wall and can be expressed by the Steele 10-4-3 potential [11]. To obtain the equilibrium density profile, the grand potential functional is minimized with respect to the local density.

The DFT using the Steele potential in combination with the LCT requires three parameters for one class of isomers to describe the bulk fluid and another triplet of parameters for the external field. Given a few points of an adsorption isotherm of a pure substance we are able to calculate the adsorption behavior of mixtures. By integration of the achieved density profiles and comparing this values with corresponding bulk densities, we can establish adsorption isotherms as well as McCabe-Thiele plots. Additionally, the impact of the porosity on the adsorption isotherm is implemented by the use of experimental pore size distributions.

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