Lump Kinetic Model

We can obtain the lump composition of the feedstock directly via GC/MS, H NMR, NMR, HPLC and ASTM methods. However, this is infeasible on a regular basis for refineries, given the changing nature of the feedstock. Aspen HYSYS Petroleum Refining includes a method that uses existing feed analysis to infer feed composition using routinely collected data. However, we have developed an alternative scheme to infer feed composition. We detail this method in Section 4.8. 

343 -510 °C (Heavy VGO) Heavy paraffin (PH) Heavy naphthene (NH) Heavy aromatics with side chains (AHs) One-ring heavy aromatics (AHrl) Two-ring heavy aromatics (AHr2) Three-ring heavy aromatics (AHr3) 

510-e °C (Residue) Residue paraffin (PR) Residue naphthene (NR) Residue aromatics with side chains (ARs) One- ring Residue aromatics (ARrl) Two-ring Residue aromatics (ARr2) Three-ring Residue aromatics (ARr3) 

Coke Kinetic coke (produced by reaction scheme) Metal coke (produced by metal activity on catalyst) 

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Radial density gradients in FCC and other large-diameter pneumatic transfer risers reflect gas—soHd maldistributions and reduce product yields. Cold-flow units are used to measure the transverse catalyst profiles as functions of gas velocity, catalyst flux, and inlet design. Impacts of measured flow distributions have been evaluated using a simple four lump kinetic model and assuming dispersed catalyst clusters where all the reactions are assumed to occur coupled with a continuous gas phase. A 3 wt % conversion advantage is determined for injection feed around the riser circumference as compared with an axial injection design (28). 

Lee, L.S., Chen, T.W., Haunh, T.N., and Pan, W.Y. (1989) Four lump kinetic model for fluid catalytic cracking process. Canadian Journal of Chemical Engineering, 67, 615. 

The lumped kinetic model can be obtained with further simplifications from the lumped pore model. We now ignore the presence of the intraparticle pores in which the mobile phase is stagnant. Thus, p = 0 and the external porosity becomes identical to the total bed porosity e. The mobile phase velocity in this model is the linear mobile phase velocity rather than the interstitial velocity u = L/Iq. There is now a single mass balance equation that is written in the same form as Equation 10.8. 

In the lumped kinetic model, various kinetic equations may describe the relationship between the mobile phase and stationary phase concentrations. The transport-dispersive model, for instance, is a linear film driving force model in which a first-order kinetics is assumed in the following form ... 

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. 

Cao, G., Viola, A., Baratti, R., Morbidelli, M., Sanseverino, L., and Cruccu, M., Lumped kinetic model for propene-butene mixtures oligomerization on a supported phosphoric acid catalyst. Adv. Catal. 41,301 (1988). 

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. 

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). 

In linear chromatography, these last two linear kinetic models are particular cases of the model used by Lapidus and Amundson [85] (Eq. 2.22). By contrast, the different lumped kinetic models give different solutions in nonlinear chromatography. Investigations of the properties of these models and especially of the relationship between the band profiles and the value of the kinetic constant have been carried out for many single-component problems. Numerous studies of the influence of the mass transfer kinetics on the separation of binary mixture have been published in the last ten years. These results are discussed in Chapters 14 and 16, respectively. 

The GRM is the most comprehensive model of chromatography. In principle, it is the most realistic model since it takes into account all the phenomena that may have any influence on the band profiles. However, it is the most complicated model and its use is warranted only when the mass transfer kinetics is slow. Its application requires the independent determination of many parameters that are often not accessible by independent methods. Deriving them by parameter identification may be acceptable in practical cases but is not easy since it requires the acquisition of accurate band profile data in a wide range of experimental conditions. This explains why the GRM is not as popular as the equilibrium-dispersive or the lumped kinetic models. 

From the Lumped Kinetic Model back to the Equilibrium-Dispersive Model. . 300... 

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. 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

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 ... 

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. 

The study of the lumped kinetic models shows that, as long as the equilibration kinetics is not very slow and the column efficiency exceeds 25 theoretical plates (a condition that is satisfied in all the cases of practical importance), the band profile is a Gaussian distribution. We can thus identify all independent sources of band broadening, calculate their individual contributions to the variance of the Gaussian distribution, and relate the column HTTP to the sum of these variances. The method is simple and efficient. It has been used successfully for over 40 years [29]. We may want a more rigorous approach. 

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. 

The above equation can be compared with Eq. 6.56, which was obtained with the lumped kinetic model. The two expressions are identical only if n = Nm, i-e. the average number of adsorption-desorption events in the microscopic model and the number of mass transfer units in the macroscopic kinetic model are analogous terms. Felinger et al. showed that not only the first and the second moments but also the whole band profiles obtained as the solutions of the microscopic model and the macroscopic lumped kinetic model are completely identical [9]. 

Because it is possible to calculate the shock layer thickness with a lumped kinetic model and with the equilibrium-dispersive model, a comparison of these two expressions provides an attractive method of investigation of the range of validity of the latter model. In the equilibrium-dispersive model, the apparent dispersion coefficient is assumed to be given by the equation... 

We assume a linear equilibrium isotherm q = aC) and a lumped kinetic model with linear driving force mass transfer, we have at steady state ... 

It would be very attractive to derive analytical expressions for the optimum experimental conditions from the solution of a realistic model of chromatography, i.e., the equiUbriiun-dispersive model, or one of the lumped kinetic models. Approaches using analytical solutions have the major advantage of providing general conclusions. Accordingly, the use of such solutions requires a minimum number of experimental investigations, first to validate them, then to acquire the data needed for their application to the solution of practical problems. Unfortunately, as we have shown in the previous chapters, these models have no analytical solutions. The systematic use of these numerical solutions in the optimization of preparative separations will be discussed in the next section. 

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