Linear driving force model

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 solution for (Eq. 9.9) requires two boundary conditions on c, one on v an initial condition on c and similarly one initial condition on q. Finally we must prescribe the sink/source term for the adsorption. This can be done in the most general case by writing another pde to describe adsorption, which is the transport of the adsorbing species into the crystal structure of the formed adsorbent. This model must be sufficiently broad to allow us to calculate the uptake at any location in the packed bed and at any time during the process. In many cases it wiU be found expedient and quite satisfactory to prescribe the uptake term as some kind of linear driving force model (LDF). 

A second and vitally important conclusion that came out of the literature in the 1970s was that linear driving force models also produced solutions that were indistinguishable from diffusion based models. 

That this was known in industry at least 15 years earlier is one of the unfortunate discrepancies between academic research and commercial industrial research and development. Not all that is known is necessarily published. This realization subsequently lead to the development of both solid phase and gas phase linear driving force models that each provide very good representations of measured data without the excess labor involved with the diffusion-based models. For trace systems there are quite a few analytical solutions that are available and quite tractable for both design work and the analysis of adsorption column performance. 

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. 

The linear driving force model has much more physical significance. It has been derived from a two-dimensional model of intra-particle diffusion, solution of which is a series development. The particle size appears explicitly. The effective diffusion coefficient is related to the particle porosity and to the size of the adsorbate molecule. Thus it makes sense to search for correlation of with these properties. However such relations are complex and it is rather difficult to predict for a given carbon and a given molecule. 

Recasens et al [41] Desorption EtAc P <13 MPa, T=300 to 338K (a) Regeneration of activated C Linear driving force model t 4... 

Goto et al. [50], also employed the linear-driving-force model for describing the high pressure delignification of wood with supercritical tert.-butyl alcohol, proving that it is a reasonable approximation. 

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... 

Thus the analogy between formal diffusion theory, the linear driving force model where the rate is proportional to (Ca — Ca), and chemical kinetics is very evident since the mathematical forms are the same. The difference arises in the interpretation of the gradient, namely, either in terms of a diffusion coefficient D (m s ), a mass... 

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... 

Fig. 6.4 Concentration profile for liquid film linear driving force models. 

In this book,. .transport dispersive model" always refers to the liquid film linear driving force model (Eqs. 6.71, 6.74 and 6.37). 

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 these models, the mass balance equation (Eq. 2.2) is combined vHth a kinetic equation (Eq. 2.5), relating the rate of variation of the concentration of each component in the stationary phase to its concentrations in both phases and to the equilibrium concentration in the stationary phase [80-93]. Although in principle kinetic models are more exact than the equilibrium-dispersive model, the difference between the individual band profiles calculated using the equilibrium-dispersive model or the linear driving force model, for example, is negligible when the rate constants are not very small i.e., when the column efficiency exceeds a few him-dred theoretical plates), as shown in Chapter 14 (Section 14.2). 

The other model is the liquid film linear driving force model ... 

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... 

The solution to this problem (Eqs. 14.2 to 14.4) has been derived by Glueckauf and Coates [7], and Michaels [8] derived the solution of a similar problem (Eqs. 14.2,14.4, and 14.5) assuming as kinetic model the liquid film linear driving force model ... 

According to Eqs. 14.6 and 14.7, the dimensionless plot of x versus — 1) depends on the single parameter R q, which indicates the deviation of the isotherm from linear behavior. Figure 14.1 illustrates this result by showing breakthrough profiles obtained for values of Req between 0 and 0.8, with the liquid film linear driving force model, under constant pattern conditions. 

The results obtained with the solid film linear driving force model, the pore diffusion model, and the micropore diffusion model were compared by Ruthven [14]. In contrast to linear chromatography, numerical solutions obtained with different models are different, especially in the initial time region. For moderate loadings i.e., for Req > 0-5), the differences remain small. As the loading increases, however, and Req becomes lower than 0.5, the differences between the numerical solutions derived from the various models studied increase. Accordingly, differences observed between experimental results and the profiles predicted by a kinetic model are most often due to the selection of a somewhat inappropriate 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]. 

Garg and Ruthven [16] also discussed the influence of axial dispersion on the shape of the constant pattern breakthrough curves. They used a numerical solution of the liquid film linear driving force model. They concluded that the linear addition rule is approximately valid for nonlinear isotherms. The deviation from the result of an addition of the two contributions becomes important only when... 

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... 

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