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Description
We propose a mathematical model of a high-performance liquid chromatography column across three length scales. We assume a column packed with porous particles, which adsorb the solute on their internal surfaces. We consider three scales: inside the porous particles, the packed particles and interstitial fluid scale, and the column scale (see figure). Chemical interactions are taken into account through adsorption isotherms on the internal surfaces of the porous particles.
Using asymptotic expansions we derive effective equations across the three scales. The effective equations on the column scale agree with standard models in the field, but now cell problems at the smaller scales provide values for parameters at the column scale. In particular, the apparent diffusion coefficient at the column scale depends not only on dispersion effects related to fluid velocity, but also on the concentration of the solute, through the adsorption isotherm. These effects are to the best of our knowledge poorly understood and often neglected.
Our asymptotic expansions give an explicit non-linear dependence of the apparent diffusion constant on the fluid velocity and solute concentration as well as pore geometry and particle packing. The resulting equations are exemplified and validated using lattice Boltzmann simulations in real [1] and simulated 3D geometries, and the effects on macroscopic parameters are investigated. The model can be generalized to multiple solutes, considering multi-component isotherms, and inhomogeneous particles.
| References | [1] A. J. Fijneman, M. Goudzwaard, A. D. Keizer, P. H. Bomans, T. Gebäck, M. Palmlöf, M. Persson, J. Högblom, G. de With, and H. Friedrich. Local quantification of mesoporous silica microspheres using multiscale electron tomography and lattice Boltzmann simulations. Microporous and Mesoporous Materials, 302:110243, 2020. |
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| Country | Sweden |
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