Modelling counter-current chromatography: a chemical engineering perspective

doi:10.1016/S0021-9673(02)01088-9

Journal of Chromatography A

Volume 973, Issues 1–2, 11 October 2002, Pages 39-46

https://doi.org/10.1016/S0021-9673(02)01088-9 Get rights and content

Abstract

In conventional chromatography, a solute is usually viewed to be longitudinally transported only in the mobile phase, remaining longitudinally motionless in the stationary phase. In counter-current chromatography, both phases undergo intense mixing in the variable force field of a coil planet centrifuge and longitudinal dispersion of matter in the stationary phase is not to be excluded. To take into account longitudinal mixing in both phases, a cell model of chromatographic process is proposed in which the number of perfectly mixed cells n is determined by the rates of mixing in stationary (D_S) and mobile (D_m) phases by the equation $n= LF/(2A_{c} D_{m}) (1+S_{f} (λ−1))$ with λ=K_DD_S/D_m (F, L, A_c and K_D are the mobile phase flow-rate, column length, column cross-section and distribution ratio, respectively). This equation has been derived by comparing the discontinuous cell model with continuous diffusion assuming equilibrium conditions. Parameter determination and their relationships are discussed.

Introduction

Counter-current chromatography (CCC) is a new technology for analytical and preparative scale separations of chemical and pharmaceutical substances; it combines the features of liquid–liquid extraction and partition chromatography [1], [2]. For scaling up, optimisation of device design and operation parameters, it is necessary to describe the chromatographic column hydrodynamics and non-steady state mass transfer between the stationary and mobile phases.

This paper is an attempt to apply approaches used in chemical engineering for modelling of mass transfer processes [3], [4], in particular solvent extraction columns [5], to simulate and scale-up the chromatographic process.

The chromatographic column is considered to be a very high (long) extraction column with an extremely high length L to diameter d ratio $(L d ≫100)$ , operating under special conditions: one of the contacting phases is held stationary and mass transfer takes place under non-steady state conditions. In extraction columns, light and heavy phases move countercurrently through a vertical apparatus and they operate under steady-state conditions.

In conventional chromatography, it is assumed that a solute is transported along the column only while it is in the mobile phase and remains longitudinally motionless in the stationary phase. In CCC, because of the lack of a solid support, both liquids undergo intense mixing in the variable force field of a coil planet centrifuge (mixing and settling zones in the coils [6] and wave mixing [1], [7] have been observed) and the axial transport of a solute in the stationary phase cannot be ignored. Thus, in CCC the chromatographic behaviour is influenced by longitudinal mixing in stationary and mobile phases and mass transfer between them. In the modelling and scale up of CCC there is a need for treating as separate phenomena the contributions of dispersion of matter in stationary and mobile phases.

To predict residence time (or the elution profile) of a solute in a chromatographic column, it is essential for there to be a quantitative analysis (or mathematical model) of longitudinal mixing and mass transfer. Furthermore, dispersion phenomenon must be represented by means of a set of equations. A large number of empirical functions have been proposed and used for the description and interpretation of chromatographic peaks. Recently, about 90 of these functions have been reviewed [8]. Since the parameters of these mathematical models are not directly related to the characteristic features of a real chromatographic process, their practical application in the process simulation and scale up is problematic. It is well known that for the reliable simulation and scale up of a mass transfer process the mathematical model applied is to be able to reflect the actual physical (or physical-chemical) picture of the process. If the model replicates, even in the simplified form, the mechanism of the phenomenon, it can be used to simulate the process and analyse the effects of different process variables. In our case, as mentioned above, the spreading of the injected solute in the chromatographic column is caused by the axial mixing in the phases and the mass transfer between them; in addition, extremely high ratio of column length to diameter allows one-dimensional models to be used.

Section snippets

Description of the models

Longitudinal dispersion takes place by a complicated interaction of different mechanisms: non-uniform velocity profile of mobile phase flow, turbulence and molecular diffusion in both phases. Two simplified model schemas: 1—discrete (staged)-cell model (a cascade of well mixed equal-size vessels) and, 2—continuous-diffusion model, are shown in Figs. 1 and 2. According to the first model, the axial mixing in the chromatographic column is characterised by one parameter—number of perfectly

Cell model

Mass balance equation for the current i–cell is: $V_{m} n d x_{i} d τ + V_{S} n d y_{i} d τ =F(x_{i−1} −x_{i})$ with i=1, 2,…, n, and where F is the volumetric flow-rate of mobile phase and τ is the time.

The solution of the set of n equations (1) with boundary and initial conditions (2): $x_{0} =0. τ=0: x_{1} = nQ V_{c} (1−S_{f} +S_{f} K_{D}); x_{2} =x_{3} =⋯=x_{n} =0$ (the inlet concentration of the mobile phase flow is zero; at τ=0, the amount Q of the solute in the sample is impulsively injected into the first ideally mixed cell) for any i-cell is obtained as: $x_{i} x ̄ = n^{i}$

Parameter determination from chromatograms

The rates of mixing in the phases can be calculated using Eq. (35) from known retained volume fraction of stationary phase S_f and from measurements of n₁ and n₂, K_D₁ and K_D₂ taken directly from chromatograms: $D_{S} D_{m} = 1−S_{f} S_{f} n_{1} −n_{2} n_{2} K_{D}_{2} −n_{1} K_{D}_{1} D_{m} = LF/A_{c} 2 n_{1} (1−S_{f} +λ_{1} S_{f})$ For experimental estimation of the distribution ratio, it is appropriate to determine K_D from the whole chromatographic curve using Eq. (18) and the relationship: $m_{k} = ∫ 0 ∞ τ^{k} x_{n} d τ≈ Δ τ ∑ 1 m x_{i} τ_{i}^{k}$ The main advantage of this way of measuring

Conclusion

The chromatographic process is appropriate to describe on the modified cell model basis using n determined by Eq. (35). Thus, the process model involves eight dimensional parameters: distribution ratio K_D, longitudinal mixing rates in stationary D_S and mobile D_m phases, column length L, mobile phase flow-rate F, column volume V_c, stationary phase volume V_S and the amount of the solute introduced Q. These eight parameters are reduced to four: two dimensional τ_c=V_c/F, $x ̄ =Q/V_{c}$ , and two

Nomenclature

A_c

column cross-section, cm²

column diameter, cm

D_m, D_S

axial dispersion coefficient, cm²/s

flow-rate of mobile phase, ml/s

K_D

distribution ratio (partition coefficient), dimensionless

current cell number, dimensionless

column length, cm

m_k

kth moment of distribution function, s^k+1 g/ml

M_k

kth moment of normalised distribution function, dimensionless

number of perfectly (ideally) mixed cells, dimensionless

n_c

number of perfectly mixed cells in the case, when the column is filled with mobile phase only,

Acknowledgements

This work is supported by a grant received for the realisation of Project INTAS 00–00782.

References (8)

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W.D. Conway
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J.M. Menet, D. Thiebaut (Eds.), Countercurrent Chromatography, Chromatographic Science Series 82, Marcel Dekker, Inc.,...
O. Levenspiel
Chemical Reaction Engineering
(1965)

There are more references available in the full text version of this article.

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