Mass transfer linear driving force model

For adsorption rate, LeVan considered four models axial dispersion (this is not really a rate model but rather a flow model), external mass transfer, linear driving force approximation (LDF) and reaction kinetics. The purpose of this development was to restore these very compact equations with the variables of Wheeler equation for comparison. 

An iterative procedure using the solid film linear driving force model has been used with a steric mass action isotherm to model displacement chromatography on ion exchange materials and the procedure applied to the separation of horse and bovine cytochrome c using neomycin sulfate as the displacer.4 The solid film linear driving force model is a set of two differential equations imposing mass transfer limitations. 

The distinction here is that the kK calculated from Eq. (9.19) would be used in a linear driving force model for the actual uptake rate expression and an axial dispersion coefficient would be substituted into the pde. If however one simply desires to match the adsorption response or breakthrough curves then the kK calculated according to Eq. (9.20) would provide very satisfactory results for estimation of the length of the mass transfer zone. 

Analytical equations for adsorbate uptake and radial adsorbent temperature profiles during a differential kinetic test are derived. The model assumes that the mass transfer into the adsorbent can be described by a linear driving force model or the surface barrier model. Heat transfer by Fourier conduction inside the adsorbent mass in conjunction with external film resistance is considered. 

To complete the required set of equations, it is necessary to incorporate the adsorption rate of the solute or contaminant, which can be described by the linear driving force model in terms of the overall liquid-phase mass-transfer coefficient [8,103,104]... 

While Eq. (103) was obtained from first principles, the approach of quantifying mass transfer with a mass transfer rate coefficient obtained from the linear driving force model, and expressing the mass transfer rate coefficient in terms of other dimensionless numbers, is the basis for many empirical models used to describe mass transfer. This is further discussed in the next section. 

The linear driving force model (a form of Fick s first law) is often used to determine interphase mass transfer rate coefficients. This equation is expressed as... 

According to the assumptions in Section 6.2.1, the liquid phase concentration changes only in axial direction and is constant in a cross section. Therefore, mass transfer between liquid and solid phase is not defined by a local concentration gradient around the particles. Instead, a general mass transfer resistance is postulated. A common method describes the (external) mass transfer mmt i as a linear function of the concentration difference between the concentration in the bulk phase and on the adsorbent surface, which are separated by a film of stagnant liquid (boundary layer). This so-called linear driving force model (LDF model) has proven to be sufficient in... 

Mass transfer kinetics are given by simple linear driving force models. The fraction of catalyst within the whole fixed bed is described by the factor Xcat- Assuming... 

In this second lumped kinetic model and in contrast to the first one, we assume that the kinetics of adsorption-desorption is infinitely fast but that the mass transfer kinetics is not. More specifically, the mass transfer kinetics of the solute to the surface of the adsorbent is given by either the liquid film linear driving force model or the solid film linear driving force model. In the former case, instead of Eq. 6.41, we have for the kinetic equation ... 

If we instead use the solid film linear driving force model of mass transfer kinetics, we have... 

Mass transfer kinetics are accurately accounted for either by the solid film or by the liquid film linear driving force model. 

The asymptotic solutions given in Eqs. 14.6 and 14.7 were derived assuming that axial dispersion is negligible. Acrivos [15] has discussed the influence on the shape of the constant pattern breakthrough curve of the combination of axial dispersion and mass transfer resistance. An exact analytical solution can be derived only in the case of an irreversible adsorption isotherm (Req = 1/(1 - - bCo) = 0, or b infinite), and assuming a liquid film linear driving force model [15]. 

The most comprehensive study of the combined effects of axial dispersion and mass transfer resistance under constant pattern conditions has been done by Rhee and Amimdson [17,18] using the shock layer theory. These authors assumed a solid film linear driving force model (Eq. 14.3) and wrote the mass balance equation as... 

The overloaded band profiles of the two enantiomers were recorded and compared to those calculated using the ED model. This simple model did not produced profiles in good agreement with the experimental ones. A much better agreement was observed between the experimental chromatograms and those calculated with the TD model, using the solid film linear driving force model to account for a slow mass transfer kinetics on both types of adsorption sites. In this case, the kinetic model is summarized as... 

This result means that if the mass transfer kinetics follows the liquid film linear driving force model, the breakthrough curve can be fitted to the Thomas model [23], provided that the apparent parameter given by Eq. 14.85 be used. Again, the apparent rate parameter is concentration dependent. 

A more comprehensive analysis of constant pattern behavior for a binary system has been given by Rhee and Amundson [3]. In this work, these authors have extended to binary systems the analysis of the combined effects of mass transfer resistance and axial dispersion that they had previously made in the case of single-component bands [4]. Rhee and Amundson [3] assumed the solid film linear driving force model, finite axial dispersion, and no particular isotherm model. The system of equations becomes... 

In these kinetic models of chromatography, all the sources of mass transfer resistance are lumped into a single equation. In the case of the solid film linear driving force model, we have for each component i... 

The profiles of individual zones in displacement chromatography have also been calculated using the solid film linear driving force model [23]. Again, when the number of mass transfer units of the column is high, the results are very similar to those obtained with the equilibrium-dispersive model (Chapter 12). As an example. Figure 16.10 shows the displacement chromatogram calculated with kpi = kfg = = 50 s . The bands in the isotachic train are clearly formed... 

This solution is valid for plug flow assuming a linear driving force model for mass transfer. 

The solution for a linear plug flow system in which the mass transfer rate is controlled by intraparticle diffusion rather than by the linear driving force model has been derived by Amimdson and Kasten [29]. 

Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. Eor the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boimdary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. 

Liquid film linear driving force model Simple model of mass transfer in chromatography, where it is assumed that the rate of variation of the stationary phase concentration is proportional to the difference between the local concentration of the component in the mobile phase and the mobile phase concentration which would be in equilibrium with the local stationary phase concentration (Eq. 2.43). 

Fast sorption rates were assumed, thereby closely approaching the instantaneous equilibrium case. Mass transfer is described by the linear driving force model. 

Because of its simple mathematical form and its physical consistence, the Linear Driving Force Model (LDFM) is commonly used to describe intraparticular mass transfer kinetics. Glueckauf and Coates first Introduced LDFM [18], which stated that the uptake rate of a species in the particle is proportional to the difference between the concentration of that species at the outer surface of the particle and its average concentration in the interior of the particle ... 

The most important mass transfer resistance is pore diffusion in the adsorberrt pellets. This depends on the diflusivity of the component / in the pores of the stationary phase (s) and the particle diameter = 2R. The LDF (linear driving force) model resirlts in the Gliickairf eqiration (see previous section) ... 

In Section 15.4. the engineering approach to mass transfer, the linear driving-force model introduced in Eq. fl-4T is explored in more detail, particularly for mass transfer between two phases. This third approach is applicable to any situation because correlations for the mass-transfer coefficients can be developed on the basis of dimensional analysis, and the constants in the correlations can be fit to experimental mass transfer data. In Section 15.5. a few correlations for the mass-transfer coefficient based on Fickian diffusivity are presented. Additional correlations are presented when needed in Chapters 16 to IS. If you have had a chemical engineering mass-transfer course. Sections 15.1 to 15.5 will contain familiar material. If you have not had a mass-transfer course. Sections 15.1 to 15.5 are the minimum material required to proceed to Chapter 16. 

Linear Driving-Force Model of Mass Transfer for Binary Systems... 

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