Lumped pore diffusion model

Thus, two mass balance equations are written in the lumped pore diffusion model for the two different fractions of the mobile phase, the one that percolates through the network of macropores between the particles of the packing material and the one that is stagnant inside the pores of the particles ... 

Numerical Solution of the Lumped Pore Diffusion Model. 689... 

Several models use the mass balance in Eq. 2.2 (ideal and equUibrimn-disper-sive models. Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accoimts also for the effect of the mass transfer resistances. This is legitimate imder certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models. Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or FOR model, see later Sections 2.1.7 and 2.2.4). 

An alternative method to take surface diffusion into account consists in lumping pore diffusion and concentration-dependent surface diffusion together, thus creating an apparent effective diffusion coefficient, which is concentration dependent. This approach was used by Ma et al [53], by Pigtkowski et al. [28] and by Zhou et al. [10]. This method is also an approximation, but it is still an improvement over the simpler HSDM model. 

The lumped pore model (often referred to as the POR model) was derived from the general rate model by ignoring two details of this model [5]. The first assumption made is that the adsorption-desorption process is very fast. The second assumption is that diffusion in the stagnant mobile phase is also very fast. This latter assumption leads to the consequence that there is no radial concentration gradient within a particle. Instead of the actual radial concentration profile across the porous particle, the model considers simply its average value. 

Equation 6.138 defines a formal connection between the effective mass transport and the film transport, the pore diffusion and the adsorption rate coefficient. It illustrates that keff is a lumped parameter", composed of several transport effects connected in series. This also gives reasons for the use lumped rate models as it proves that the impact of the lumped parameters on the most important peak characteristics, retention time and peak width, is identical to the effect described by general rate model parameters in linearized chromatography. 

In the equilibrium dispersive model it is assumed - like in the ideal model - that the mobile and the stationary phases are permanently in the equilibrium state. In addition to adsorption and convection all band-broadening transport effects are considered. Thereby, all kinetic effects like axial dispersion, film and pore diffusion are lumped together in the apparent dispersion coefficient Djp. As the basis of the model the following partial differential equation can be written ... 

If the first two inequalities hold /8,9/, it can be expected that heat and mass transport outside the catalyst particle is not limiting and therefore there is no need for a heterogeneous model a pseudo-homogeneous model may be sufficient, in which the whole catalyst particle is regarded as a sink and/or source for heat and mass in the fluid phase. This results in a drastic reduction of the number of model equations, and the transfer terms in the fluid phase equations are lumped into effective reaction rate terms. If the second inequality holds /9/, pore diffusion is not limiting. The third inequality is valid if axial dispersion can be neclected /lO/ and the fourth if radial dispersion is of no importance /lO/. 

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

In the modeling of chromatography, the contributions of aU the phenomena that contribute to axial mixing are lumped into a single axial dispersion coefficient. Two main mechanisms contribute to axial dispersion molecular diffusion in the interparticle pores and eddy diffusion. In a first approximation, their contributions are additive, and the axial dispersion coefficient, Di, is given by... 

The linear driving force (LDF) model can be classified in the group of equilibrium transport dispersive models (Fig. 9.5). For this model it is no longer assumed that the mobile and the stationary phases are permanently in equilibrium state, so that an additional mass-balance equation for the stationary phase is required. Assuming a linear concentration gradient an effective mass-transfer coefficient keff is implemented, where all mass-transfer resistances and the diffusion into the pores of the particle are lumped together. In this model a constant local equilibrium between the solid and the liquid in the pores is assumed. 

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