Mathematical model

The mathematical model here is a description of the mass transfer in the adsorption-desorption continuous system (fixed-bed or column) using mathematical concepts and language, predicting the breakthrough and elution curves. Compared to the experiment, mathematical modeling needs no experimental equipment and chemicals but gives accurate predictions of breakthrough and elution curves based on the isotherm and kinetics parameters. Currently, many mathematical models, reported for liquid-solid adsorption, have been simplified based on different rate-controlling steps like axial dispersion and sorption equilibrium.

Originally, the fixed-bed or column is assumed to be uni-dimensional and only have two independent variables, the time t and the column length z. To study the band profile of the components in the column, the differential mass balance is introduced. Guiochon et al.

(2006) provided the general differential mass balance in the bulk mobile phase and deduced the solution of the mass transfer (Equation 2.1) according to several fundamental assumptions: a) the column is radically homogeneous with constant axial dispersion; b) the mobile phase is hardly compressed and the sample components keep stable in both phases; c) the solvent or the weak solvent is not adsorbed in the isotherm; d) the thermal effects and the influence of the heat of adsorption on the band profile are neglected (Guiochon et al., 2006).

where F is the phase ratio, Vs/Vm, equal to (1-ε)/-ε. The first two terms on the left-hand side represent the accumulation in the mobile and stationary phases, respectively; the third term is the convection; the term on the right-hand side represents diffusion.

Because of the different properties of the chromatography, the mass transfer kinetics is different. When the mass transfer between two phases is fast, the concentration of the component in the stationary phase Cs,ican be represented by qi, and the stationary phase concentration in equilibrium with Cs,iis:

)

and when the mass transfer kinetics is slow, such as in the liquid-solid chromatography and ion-exchange chromatography, the Cs,ican be represented as:

)

Currently, the most precise model to describe the mass conservation and transport in the basic stage of the model is the general rate model, considering all the possible contributions to the mass transfer kinetics, which is widely used when the impact of particle size distribution cannot be neglected. This model includes two parts:

(a) the bulk model can be written as follows:

2

where DLis the axial dispersion coefficient, and the term q t is the rate of adsorption averaged over the particle. For a spherical particle, the equation is:

p

where kfis the external mass transfer coefficient and Cp is the concentration of the solute within the pores inside the particle (Guiochon et al., 2006).

(b) The particle model can be divided into micro-pore control (rc << 1 mm) and macro-pore control (rc = 1 - 5 mm) according to the size of adsorbents. For the micro-pore control model, the equation can be written as follows (Guiochon et al., 2006):

2 )

where Dc is the diffusion coefficient of the solute into the particle and q is the stationary phase concentration. For the macro-pore control model, the equation can be written as follows (Guiochon et al., 2006):

2 )

where εpis the internal porosity of the particle, Dpis the diffusion coefficient of the solute in the particle pores, and Cs is the concentration of the solute adsorbed by the stationary phase. When assuming the kinetics of adsorption-desorption is infinitely fast, Cs can be related to the linear adsorption isotherm equation:

)

where Cp*is concentration of solute in equilibrium, which is commonly obtained through the Langmuir isotherm or Freundlich isotherm.

It is proved that the predictions of general rate model were very consistent with the experimental data, such as the analysis of Solanesol adsorption on macro-porous Resins (Du et al., 2008), the adsorption of Pb²⁺ in a fixed bed of ETS-10 adsorbent (Lv et al., 2008). However, the problem of using this model is to make a detailed analysis of the significance of all the contributions, and then obtaining all the four parameters through experiments, which require numerous experiments and sufficient facilities.

2.4.2.2 Lumped kinetic model

Simpler models, such as the various lumped kinetic models, were developed by generally sacrificing accuracy in describing the suspended-particle batch adsorption process (Rowe et al., 1999). It is formed based on a kinetic equation, lumping all the slow kinetics in a single rate coefficient; so the mass balance equation is then written:

2

and when the kinetics of adsorption-desorption is slower than the other steps of the column process, the kinetic equation can be presented as:

s

while if the slowest step is the mass transfer kinetics, the kinetic equation is:

Though some steps are simplified in the lumped kinetic models, good agreement was found between the experimental and calculated data in the column with slow kinetics and the diffusion coefficient mostly equaled the axial dispersion (Bak et al., 2007;

Pérez-Martínez et al., 2015).

2.4.2.3 Equilibrium-Dispersive model

When the mass transfer kinetics is fast but not infinitely fast, it can be assumed that the mobile and the stationary phases are constantly in equilibrium. Thus, the equilibrium-dispersive model was created; the mass balance equation for component i is as follows:

where Da,i is the apparent dispersion coefficient which lumped the axial dispersion, non-equilibrium effects and finite kinetics of adsorption-desorption.

This model is usually applied into HPLC columns with high efficiency because of the assumptions of simplified process (Quiñones et al., 2000). Therefore in the lab-scale experiments this model is difficult to get good agreement between model and experiment.

2.4.2.4 Ideal model

When there is no axial dispersion and the two phases are constantly at equilibrium, which means the column efficiency is infinite; then the mass transfer equation can be simplified as the ideal model:

This simplest model focuses on the the effects of the nonlinear thermodynamics of phase-solid equilibrium, which are in good agreement with the experiment for large samples, highly efficient columns and small dispersive effects of the column efficiency (Guiochon et al., 2006). The application of this model is scarce in the literature because of its strict assumptions.