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#### Estimation of higher order frequency response functions from the nonlinear frequency response of a chromatographic column for a binary mixture

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##### Citation

Ilic, M., Petkovska, M., & Seidel-Morgenstern, A. (2007). *Estimation
of higher order frequency response functions from the nonlinear frequency response of a chromatographic column for a binary
mixture*. Poster presented at ProcessNet-Jahrestagung 2007, Aachen, Germany.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-971C-1

##### Abstract

In order to investigate the adsorption of components from a binary mixture, the nonlinear frequency response (FR) of a chromatographic column, packed with the adsorbent of interest, is analyzed using the concept of higher order frequency response functions (FRFs). At first the same approach is used to study the single solute adsorption. By applying the nonlinear FR method for estimation of single solute adsorption isotherms the first, second and third local derivatives of the isotherm for chosen steady-states can be estimated from the low frequency asymptotic behaviour of the corresponding FRFs. This method appears to be somewhat more complex compared to other classical ones, but very useful for estimation of isotherms possessing complex shapes (e.g. inflection points) ([1]).
Analysis of the theoretically derived FRFs for the binary adsorption case has shown that the local derivatives of the competitive isotherms can be also estimated from the low frequency asymptotes of the corresponding FRFs. In this case, the periodical concentration change of one and both components at the input should be considered ([2]). The aim of this investigation is to find the best way to estimate these FRFs from the inlet and outlet concentration changes. The first step is to perform the harmonic analysis over the selected input and output data corresponding to a certain frequency and amplitude of the inlet concentration change. Afterwards the FRFs are calculated from the input and output harmonics, whereby one or more input amplitudes can be used. It has been shown at the single solute adsorption case, that the nonlinearity of the investigated system is more pronounced at higher input amplitudes ([1]). If it is necessary to use higher input amplitude for very accurate estimation of the third order FRF, then contributions of higher harmonics can not be neglected and they should be properly accounted. One way to do that is to estimate the first order FRF from lower and the third order FRF from higher input amplitudes, whereby the second order FRF can be estimated from both input amplitudes. Alternatively it is possible to estimate the FRFs from more input amplitudes considering contributions of higher harmonics up to the seventh. Since it is possible to change periodically the concentration of only one component at the input, keeping the concentration of the other component constant, one should be careful regarding the input function which is used for calculation of the FRFs. The applicability of this method will be evaluated.
[1] Ilić M., Petkovska M., Seidel-Morgenstern A. (2007), Nonlinear frequency response method for estimation of single solute adsorption isotherms, submitted
[2] Ilić M., Petkovska M., Seidel-Morgenstern A. (2007), Investigation of binary adsorption in a chromatographic column using the nonlinear frequency response technique, submitted