# Fundamentals

An essential goal of modeling chromatographic processes is to describe the dynamics of concentration and temperature fronts traveling through chromatographic columns. Instructive models originate from differential mass balances for the fluid and solid phases. Model reduction based on evaluating just a limited number of moments of the profiles is known to be a powerful tool to simplify the description of band profiles.

We applied the well-established method of moments for different standard models. Then the method is extended to evaluate more complex and realistic column models. The cases of applying columns packed with core-shell particles, the quantitative description of radial concentration profiles and the propagation of temperature perturbations are analyzed.

#### References

[1] | Qamar, S., & Seidel-Morgenstern, A. (2016). Extending the potential of moment analysis in chromatography. Trends in Analytical Chemistry, 81, 87-101. doi:10.1016/j.trac.2016.01.007. |

[2] | Qamar, S., Sattar, F. A., Abbasi, J. N., & Seidel-Morgenstern, A. (2016). Numerical simulation of nonlinear chromatography with core–shell particles applying the general rate model. Chemical Engineering Science, 147, 54-64. doi:10.1016/j.ces.2016.03.027. |

[3] | Qamar, S., Akram, N., & Seidel-Morgenstern, A. (2016). Analysis of general rate model of linear chromatography considering finite rates of the adsorption and desorption steps. Chemical Engineering Research and Design, 111, 13-23. doi:10.1016/j.cherd.2016.04.006. |

[4] | Qamar, S., Perveen, S., & Seidel-Morgenstern, A. (2016). Numerical approximation of nonlinear and non-equilibrium two-dimensional model of chromatography. Computers and Chemical Engineering, 94, 411-427. doi:10.1016/j.compchemeng.2016.08.008. 235-247 |

[5] | Ahmed, M., Zainab, Q. U. A., & Qamar, S. (2017). Analysis of One-Dimensional Advection–Diffusion Model with Variable Coefficients Describing Solute Transport in a Porous medium. Transport in Porous Media, 118(3), 327-344. doi:10.1007/s11242-017-0833-0. |

[6] | Qamar, S., Sattar, F. A., Batool, I., & Seidel-Morgenstern, A. (2017). Theoretical analysis of the influence of forced and inherent temperature fluctuations in an adiabatic chromatographic column. Chemical Engineering Science, 161, 249-264. doi:10.1016/j.ces.2016.12.027. |

[7] | Qamar, S., Uche, D. U., Khan, F. U., & Seidel-Morgenstern, A. (2017). Analysis of linear two-dimensional general rate model for chromatographic columns of cylindrical geometry. Journal of Chromatography A, 1496, 92-104. doi:10.1016/j.chroma.2017.03.048. |

[8] | Qamar, S., Kiran, N., Anwar, T., Bibi, S., & Seidel-Morgenstern, A. (2018). Theoretical Investigation of Thermal Effects in an Adiabatic Chromatographic Column Using a Lumped Kinetic Model Incorporating Heat Transfer Resistances. Industrial and Engineering Chemistry Research, 57(6), 2287-2297. doi:10.1021/acs.iecr.7b04555. |

[9] | David, U. U., Qamar, S., & Seidel-Morgenstern, A. (2018). Analytical and numerical solutions of two-dimensional general rate models for liquid chromatographic columns packed with core–shell particles. Chemical Engineering Research and Design, 130, 295-320. doi:10.1016/j.cherd.2017.12.044. |

#### Ideal Adsorbed Solution Theory

Several theories have been developed in the last decades attempting to describe multicomponent adsorption. One of these theories, based upon classical chemical thermodynamic arguments, is Ideal Adsorbed Solution Theory (IAST) [1], [2].

Our research in this field has been a fruitful and successful enterprise with the Systems and Control Theory group of our institute (Prof. Dietrich Flockerzi).

This cooperation has led to the development of a robust generalized solution concept applicable to an *N* component system with nondecreasing single component adsorption isotherms [3].

The efficient solution method has also found successful application in more complex multicomponent adsorption models [4].

In order to test the application of the new solution method, a complex liquid chromatography system, where the single component adsorption isotherms display inflection points, has been chosen [3].

The calculation of this solution orbit is the core of the efficient approach proposed in [3].

#### Dynamic simulation of fixed-bed adsorbers

A very attractive feature of these achievements consists in overcoming time-consuming calculations by a sound mathematical understanding of the features of the IAST model. This results in robust and reliable implementations of the adsorption equilibrium computation for the dynamic simulation of liquid chromatography multi-column arrangements, e.g., SMB.

To this purpose, we implement suitable numerical schemes to solve the partial differential equations that describe fixed-bed adsorbers. We perform the necessary local adsorption equilibrium calculations with the efficient procedures described in [3] and [4].

This is one of several on-going topics within our research scope.

#### References

[1] | Myers, A.L., Prausnitz, J.M., (1965), Thermodynamics of Mixed-Gas Adsorption. AIChE Journal. 11, pp. 121-127 |

[2] | Radke, C.J., Prausnitz, J.M., (1972) Thermodynamics of Multi-Solute Adsorption from Dilute Liquid Solutions. AIChE Journal. 18(4), pp. 761-768 |

[3] | Rubiera Landa, H.O., Flockerzi, D., Seidel-Morgenstern, A., (2013) „A method for efficiently solving the IAST equations with an application to adsorber dynamics“ AIChE Journal, 59 (4), pp. 1263-1277 |

[4] | Heinonen, J., Rubiera Landa, H.O., Sainio, T., Seidel-Morgenstern, A., (2012) „Use of Adsorbed Solution theory to model competitive and co-operative sorption elastic ion exchange resins” Separation and Purification Technology, 95, pp. 235-247 |

[5] | Rubiera Landa, H. O., Flockerzi, D., Seidel-Morgenstern, A., (2013), A method for efficiently solving the IAST equations with an application to adsorber dynamics. AIChE-Journal, 59 (4), pp. 1263-1277 |

[6] | Mutavdzin, I., Seidel-Morgenstern, A., Petkovska, M., (2013), Estimation of competitive adsorption isotherms based on nonlinear frequency response experiments using equimolar mixtures–numerical analysis for racemic mixtures. Chemical Engineering Science, 89, pp. 21-30 |