Dual-site Langmuir Model

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

Mathias, P.M., et al.. Correlation of multicomponent gas adsorption by the dual-site Langmuir model Application to nitrogen/oxygen adsorption on 5A-zeolite, Ind. Eng. Chem. Res., 35(7), 2477-2483 (1996). 

The simulated isotherms for 2-methylalkanes at 300K temperature are shown in figure 4.19. The continuous lines in this figure are fits of the CBMC simulations using the dual-site Langmuir model which will be discussed in section 4.6. 

Figure 4.22 Parameters k and S of the dual-site Langmuir model for linear and branched alkanes as a function of the number of carbon atoms.

Fitting of simulated isotherms with dual-site Langmuir model... 

There is an important advantage in being able to describe the inflection behavior accurately with the help of the dual-site Langmuir model (DSL) this is because it would then be possible to predict the mixture isotherm from only pure component data. For single components, we have... 

We see from figures 5.2-5.4 that the simulated isotherms conform very well to the mixture rule II based on the dual-site Langmuir model. For alkanes with carbon atoms in the 5-7 range, we need to set up the mixture rule considering the total Silicalite matrix (including sites A and B) as one entity. This is because the branched alkanes do not easily occupy site B (channel interiors) and for some pressure range the channel interiors are completely devoid of the branched isomers. The simulated isotherm for the 50%-50% mixture of butane-isobutane behaves differently, however. Neither mixture rule, I or II, is completely successful. An average of the two mixture rules, on the other hand, is very successful. 

The mixture isotherms can be predicted quite accurately by applying the appropriate mixture rule to the dual-site Langmuir model. This model allows the mixture isotherm to be predicted purely on the basis of the parameters describing the isotherms of the pure components, linear and branched alkane. 

The model considers the noble-metal catalyzed oxidation reactions of CO, two hydrocarbons of differing reactivities and H2, and the reaction kinetics was described by the global rate expressions of the dual-site Langmuir-Hinshelwood type [2]. 

The error model used in the minimization is based on the hypothesis that the residuals have zero mean and are normally distributed. The first is easily checked, the latter is only possible when sufficient data points are available and a distribution histogram can be constructed. An adequate model also follows the experimental data well, so if the residuals are plotted as a function of the dependent or independent variable(s) a random distribution around zero should be observed. Nonrandom trends in the residuals mean that systematic deviations exist and indicate that the model is not completely able to follow the course of the experimental data, as a good model should do. This residual trending can also be evaluated numerically be correlation calculations, but visual inspection is much more powerful. An example is given in Fig. 12 for the initial rate data of the metathesis of propene into ethene and 2-butenc [60], One expression was based on a dual-site Langmuir-Hinshelwood model, whereas the other... 

Equilibrium model lAST (Langmuir) Extended Dual-Site Langmuir LRC... 

Carbonization seems to be an effective method to adjust the pore size of PAF-1 to increase the gas selectivity. PAF-1-450 (PAF-1 carbonized at 450 °C), with a narrow micropore distribution of 0.8 nm, shows obvious increased CO2 sorption. Besides, on the basis of single component isotherm data, the dual-site Langmuir-Freundlich adsorption model-based lAST prediction indicates that the CO2/N2 adsorption selectivity may be as high as 209 at a 15 85 CO2 N2 ratio. Also, the CO2/CH4 adsorption selectivity should be in the range of 7.8-9.8 at a 15 85 C02 CH4 ratio at 0adsorption selectivity could be about 392 at 273 K and 1 bar for the 20 80 CO2 H2 mixture (Figure 10.2). ... 

A second possibility to arrive to a similar first-order kinetics is starting from a dual site Langmuir-Hinshelwood model with oxygen adsorbed strongly in dissociative form while methane is adsorbed weakly ... 

In this chapter we present the results of computer simulations of lineeir cind branched alkcines in the zeolite Silicalite. We focus on the development of the model and a detailed comparison with experimental data for the linear and brcinched alkcines. In addition we demonstrate that these isotherms can be described quantitatively with a dual-site Langmuir isotherm. 

Recently, Praharso et al also developed a Langmuir-Hinshelwood type of kinetic model for the SR kinetics of i-Cg over a Ni-based catalyst. In their model, it was assumed that both the hydrocarbon and steam dissociatively chemisorb on two different dual sites on the catalyst surface. The bimolecular surface reaction between dissociated adsorbed species was proposed as the ratedetermining step. The following generalized rate expression was proposed ... 

The sorption and diffusion behaviour of gas mixtures is of particular interest from the point of view of membrane gas separation, which is steadily gaining in importance by virtue of its low energy requirements. On the basis of the dual mode sorption model, one may reasonably expect that sorption of a binary gas mixture A, B in the polymer matrix will exhibit little gas-gas interaction and hence will tend to occur essentially additively. In the Langmuir-like mode of sorption, on the other hand, there will be competition between A and B for the limited number of available sites. These considerations led 67) to the following reformulation of Eqs. (8) and (9)... 

Pignet and Schmidt have fitted a Langmuir-Hinshelwood model for non-competitive dual-site adsorption and with dissociative oxygen adsorption ... 

Fredrickson and Helfand incorporated coupling terms in the transport equation and showed that "these terms provide mobility to molecules that are absorbed into microvoids, even if the molecules have an intrinsic diffusion coefficient in the hole phase that vanishes." These investigators proposed experiments that could provide information on the size and topological connectivity of microcavities (51). Related studies have also been made by Chern, Koros, et al. (52) and by Barrer (53). However, Petropoulos (54) has expressed the opinion that the treatment of Fredrickson and Helfand (51 ) as well as that of Barrer (53) "introduces more diffusion parameters than can reasonably be expected to be measurable on the basis of past experience". Petropoulos also showed that these treatments impose certain limitations on the physical meaning of the diffusion parameters. In the same study, Petropoulos has examined the modification of his dual-mode sorption model ( 5) which are necessary if the Langmuir domains are sufficiently extensive to constitute a macroscopically recognizable phase rather than scattered individual sites (or microcavities) (54). ... 

In their pioneering work Dodd and Watson (3) correlated the dehydrogenation data by Langmuir-Hinshelwood rate models and found that a dual-site surface rate-controlling model is most plausible. Noda and co-workers (21) and more recently Carra and his colleagues (2) obtained essentially the same results as Dodd and Watson (3) for the chromia-alumina catalyzed dehydrogenation of n-butane. As Table I shows, somewhat different values for the orders of power function models are quoted in the references using this method of correlation. 

The basic problem can be stated as follows minimize an objective function fjx), where x is a vector variable comprising the parameters to be estimated, andXx) is the sum of squares to be minimized. For instance, in the oxidation of carbon monoxide, already mentioned in Chapter 4 and discussed in more detail in Chapter 11 (and in [Asprey, 1997]), the following Langmuir-Hinshelwood Dual Site Dissociative Adsorption Model is used to fit experimental data ... 

At elevated pressures, the dependence of the solute concentration on the ideal gas pressure can be represented as a sum of linear terms, bjp (for sites j where bjp 1), and Langmuir-like terms, bjP/(l -i- bjp) (for sites] where the term bjp is at least comparable to unity). If a system consists only of linear sites and a set of identical Langmuir sites, the well-known dual-mode-sorptiorr model [2] is obtained [54],... 

However, as the reaction proceeds, the add adsorption term might have to be introduced and in such an event the reaction kinetics would be represented by a dual site mechanistic, such as Langmuir-Hinshelwood (LH) model. 

The extension of this model to mixed-gases is considerably more difficult than the extension of the simple dual-mode transport models. To include gas mixtures, the Barrer model must be modified to accurately describe the fraction of filled Langmuir sorption sites when more than one gas species is present in the pol5mier. To account for the presence of a second gas, the flux expressions can be rewritten using the following definition for the fraction of filled holes ... 

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