Linear Driving Force Model Approach

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 

In the case of a step input, the numerical solution of the system of Eqs. 16.30 and 16.31 has been discussed in the literature for multicomponent mixtures [16]. The numerical solution of Eqs. 16.30 and 16.31 without an axial dispersion term i.e., with Di = 0) has been described by Wang and Tien [17] and by Moon and Lee [18], in the case of a step input. These authors used a finite difference method. A solution of Eq. 16.31 with D, = 0, combined with a liquid film linear driving force model, has also been described for a step input [19,20]. The numerical solution of the same kinetic model (Eqs. 16.30 and 16.31) has been discussed by Phillips et al. [21] in the case of displacement chromatography, using a finite difference method, and by Golshan-Sliirazi et al. [22,23] in the case of overloaded elution and displacement, also using finite difference methods. 

When the number of transfer units is large, the band profiles calculated in overloaded elution with the solid film driving force model are very similar or identical to those calculated with the equilibrium-dispersive model [22]. By contrast, when the number of mass transfer units is small i.e., at very low values of kf), very different profiles can be obtained. 

The adsorption isotherms of the pure enantiomers of 3-chloro-l-phenyl-l-propa-nol were measured by FA on a cellulose tribenzoate coated on silica, eluted with a 95/5 mixtture of n-hexane and ethyl acetate. These data were well accounted for by a simple Langmuir isotherm model. The adsorption data measured fitted well to the Langmuir isotherm model. The elution band profiles of large amoimts of 

Piqtkowski el al. measured the single-component and the competitive equilibrium isotherms of phenetole (ethoxy-benzene) and n-propyl benzoate on a 150 x 3.9 mm S3onmetry -Cig (endcapped) column (Waters), using a methanol/water (65 35, v/v) as the mobile phase [26]. The adsorption equilibrium data of the single-component systems were acquired by frontal analysis. For both compoimds. 

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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. 

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

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. 

To avoid the edmplexity of the diffusion solutions it is common practice to use the simple linear driving force model with the effective rate constant estimated from Eq. (8.41). Fortunately the error involved in this approach is smaller than might be expected. 

As mentioned in Section 6.2.2, the mass transfer term in Eq. 6.3 is defined by the linear driving force approach. Therefore, the transport dispersive model consists of the balance equations in the mobile phase (Eq. 6.71) written with the pore concentration... 

Several modifications of this model can be found in the literature. One that is frequently used considers the mass transfer resistance in the solid phase to be dominant. As proposed by Glueckauf and Coates (1947), an analogue linear driving force approach for the mass transfer in the solid can then be applied and Equations 6.73 and 6.35 are replaced by Equations 6.74 and 6.75. Mathematically, this linear driving force is modeled as the difference between the overall solid loading of Equation 6.17 and an additional hypothetical loading eq, which is in equilibrium with the liquid phase concentration ... 

As a first approach, the linear driving force (LDF) model is adopted with negligible temperature effect. Mass conservation of adsorbate is given in dimensionless form as... 

With a few exceptions, industrial applications of zeoUtes involve column operation in feeds containing more than two counterions. In general, therefore, the prediction of column performance involves the prediction of multicomponent equilibria and kinetics under dynamic flow conditions. Considering the complexity and diversity of these models (see Sects. 2.3 and 2.4), it is obvious that simplifications and approximations need to be made for practical coliunn modelling. For engineering pmposes, the most popular approach for coliunn modelling is the linear driving force - effective plate concept [97]. 

In our approach to membrane breakdown we have only taken preliminary steps. Among the phenomena still to be understood is the combined effect of electrical and mechanical stress. From the undulational point of view it is not clear how mechanical tension, which suppresses the undulations, can enhance the approach to membrane instability. Notice that pore formation models, where the release of mechanical and electrical energy is considered a driving force for the transition, provide a natural explanation for these effects [70]. The linear approach requires some modification to describe such phenomena. One suggestion is that membrane moduli should depend on both electrical and mechanical stress, which would cause an additional mode softening [111]. We hope that combining this effect with nonlocality will be illuminating. 

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