Model lumped rate

The basic material balance of the mobile phase for all lumped rate models is based on Eqs. 6.3, 6.4 and 6.13-6.17 and can be derived in the same manner as the equilibrium dispersive model (Eq. 6.58) ... 

The lumped rate models are distinguished on the basis of different equations for the particle phase, considering either adsorption kinetics or mass transfer. In the former case the concentration inside the particle pores cv is identical to the mobile phase concentration c, while they are different for the latter. 

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. 

The next level of detail in the model hierarchy of Figure 6.2 is the so-called lumped rate models. They are characterized by a second parameter describing rate limitations apart from axial dispersion. This second parameter subdivides the models into those where either mass transport or kinetic terms are rate limiting. No concentration distribution inside the particles is considered and, formally, the diffusion coefficients inside the adsorbent are assumed to be infinite ... 

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. 

Note that this kinetic equation is rather similar to Equation 10.15. The major difference between Equations 10.15 and 10.19 is that the general rate and the lumped pore models assume that adsorption takes place from the stagnant mobile phase within the pores, while the lumped kinetic model assumes that the mobile phase concentration is the same in the pores and between the particles. 

The connection of the overall mass transfer coefficient of the lumped kinetic and the parameters of the general rate model is... 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

It is clear that modeling of the coke deposition is essential for a satisfactory description of FCC unit operation. It is not useful to describe this phenomena based on a simple power rate law. A more sophisticated model has to be derived in which the initial effects are accounted for. It is shown that coke deposition and catalyst activity have to be divided in an initial process (typically within 0.15 s) and a process at larger residence times. A simplified lumped kinetic model can be adequately used for this purpose, but a realistic coke formation model has to be developed. 

The other subgroup of the lumped rate approach consists of the reaction dispersive model where the adsorption kinetic is the rate-limiting step. It is an extension of the reaction model (Section 6.2.4.3). Like the mass transfer coefficient in the transport dispersive model, the adsorption and desorption rate constants are considered as effective lumped parameters, kads,eff and kdes.eff- Since no film transfer resistance exists (Cpi = q), the model can be described by Eq. 6.79 ... 

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). 

The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

Note that the rate coefficient kf used in Eq. 6.82 was defined in Eq. 5.70 and has dimensions LT. By contrast, in the lumped kinetic models, the rate coefficient km in Eq. 6.43 or fcy in Eq. 14.3) has dimensions T . The third, fourth, and fifth moments are given by more complicated expressions and can be formd in the literature [30,31], In practice, only the first and second moments of a band are determined, the first to characterize its retention and calculate the equilibrium constant, the second to characterize and study the band spreading, hence the mass transfer kinetics. 

Lee also extended the non-equilibrium theory developed originally by Gid-dings [10] to obtain H in/ the plate height contribution due to the mass transfer resistances and to axial dispersion, the non-equilibrium contribution. He started from the kinetic equation of the lumped rate constant kinetic model ... 

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