Ideal adsorbed phase model

Ideal Adsorbed Solution Theory. Perhaps the most successful general approach to the prediction of multicomponent equilibria from single-component isotherm data is ideal adsorbed solution theory. In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equilibrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equilibrium pressure for the pure component at Ike same spreading pressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption. Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, are not consistent with an ideal adsorbed phase and there is no way of knowing a priori whether or not a given system will show ideal behavior. 

The vacancy solution theory was developed by Suwanayuen and DanneE as a method of predicting multicomponent adsorption equilibria from singlecomponent isotherms without the assumption of an ideal adsorbed phase. A somewhat different analysis is given here although the essential features of the model are retained. 

The most common model for describing adsorption equilibrium in multi-component systems is the Ideal Adsorbed Solution (IAS) model, which was originally developed by Radke and Prausnitz [94]. This model relies on the assumption that the adsorbed phase forms an ideal solution and hence the name IAS model has been adopted. The following is a summary of the main equations and assumptions of this model (Eqs. 22-29). 

In order to determine the PSD of the micropores, Horvath-Kawazoe (H-K) method has been generally used. In 1983, Horvath and Kawazoe" developed a model for calculating the effective PSD of slit-shaped pores in molecular-sieve carbon from the adsorption isotherms. It is assumed that the micropores are either full or empty according to whether the adsorption pressure of the gas is greater or less than the characteristic value for particular micropore size. In H-K model, it is also assumed that the adsorbed phase thermodynamically behaves as a two-dimensional ideal gas. 

A multicomponent HSDM for acid cfye/carbon adsorption has been developed based on the ideal adsorbed solution theory (lAST) and the homogeneous surface diffusion model (H SDM) to predict the concentration versus time decay curves. The lAST with the Redlich-P eterson equation is used to determine the pair of liquid phase concentrations, Q and Qj, from the corresponding pair of solid phase concentrations, q j and q jy at fha surface of the carbon particle in the binary component. 

This study firstly aims at understanding adsorption properties of two HSZ towards three VOC (methyl ethyl ketone, toluene, and 1,4-dioxane), through single and binary adsorption equilibrium experiments. Secondly, the Ideal Adsorbed Solution Theory (IAST) established by Myers and Prausnitz [10], is applied to predict adsorption behaviour of binary systems on quasi homogeneous adsorbents, regarding the pure component isotherms fitting models [S]. Finally, extension of adsorbed phase to real behaviour is investigated [4]. 

The two-dimensional gas model assumes no mutual interaction of the adsorbed molecules. It is believed that the adsorbent creates a constant (across the surface) adsorption potential. Thus, in the framework of statistical thermodynamics, the model describes adsorption as the transition of a gas with three translational degrees of freedom into an adsorbed state with one vibrational and two translational degrees. Assuming ideal behavior and using molar quantities, one obtains the standard entropy in the adsorbed phase as the sum of the translational and vibrational entropies from Eqs. 5.28 and 5.29 ... 

Figure 4.6 illustrates the use of the IAS model to account for the competitive isotherm data of a ternary mixture of benzyl alcohol (BA), 2-phenylethanol (PE) and 2-methyl benzyl alcohol (MBA) in reversed phase liquid chromatography. The RAS model accounts for the nonideal behaviors in the mobile and the stationary phases through the variation of the activity coefficients with the concentrations. Figures 4.6d and 4.6e illustrate the variations of the activity coefficients in the stationary and the mobile phases, respectively. The solutes exhibit positive deviations from ideal behavior in the adsorbed phase and negative deviations from ideal behavior in the mobile phase. 

The IAS theory was later extended to account for the adsorption of gas mixtures on heterogenous surfaces [52,53]. It was also extended to calculate the competitive adsorption isotherms of components from hquid solutions [54]. At large solute loadings, the simplifying assumptions of the LAS theory must be relaxed in order to account for solute-solute interactions in the adsorbed phase. The IAS model is then replaced by the real adsorbed solution (RAS) model, in which the deviations of the adsorption equilibrium from ideal behavior are lumped into an activity coefficient [54,55]. Note that this deviation from ideal beha dor can also be due to the heterogeneity of the adsorbent surface rather than to adsorbate-adsorbate interactions, in which case the heterogeneous IAS model [55] should be used. 

IAS model for dilute liquid solution The IAS method was first proposed to accoimt for the adsorption of gas mixtures. It was later extended to multisolute adsorption from dilute liquid solutions [54]. Assuming that both the solution and the adsorbed phase are ideal, the following equation can be derived to calculate multi-solute equilibriirm composition [54]. 

The fifth approach is more a field than a concise method, since it embodies so many theoretical concepts and associated methods. All are grouped together as adsorbed mixture models. Basically, this involves treating the adsorbed mixture in the same manner that the liquid is treated when doing VLE calculations. The major distinction is that the adsorbed phase composition cannot be directly measnred (i.e., it can only be inferred) hence, it is difficult to pursue experimentally. A mixture model is nsed to account for interactions, which may be as simple as Raoult s law or as involved as Wilson s equation. These correspond roughly to the Ideal Adsorbed Solution theory and Vacancy Solution model, respectively. Pure component and mixture equilibrium data are required. The unfortunate aspect is that they require iterative root-finding procedures and integration, which complicates adsorber simnlation. They may be the only route to acceptably accurate answers, however. It would be nice if adsorbents could be selected to avoid both aspects, but adsorbate-adsorbate interactions may be nearly as important and as complicated as adsorbate-adsorbent interactions. 

In contrast to the binary Langmuir or SSTM models, the ideal adsorbed solution theory does not lead to a simple explicit relation for the adsorbed-phase composition and loading in terms of the partial pressures. Calculation of the equilibrium for a particular gas-phase composition therefore requires a trial and error procedure. 

Myers andPrausnitz [S] considered foe adsorbed phase to be an ideal solution, where Raoult s Law describes foe binary sorption equilibria. In foe IAS model foe partial pressure of a component in an id solution is equal to foe product of its mole fraction in that solution and foe pressure which it would exert as a pure sorbate, at foe same temperature and spreading pressure as that of foe mixture. No detailed physical model of sorption is involved anfi foe procedure is based upon rigorous thermodynamics. 

The multiphase ideal adsorbed solution theory (MIAST) is another model of the family of adsorbed solutions. Corrtraiy to HAST, an energy distribution function is assumed with differences of the local or molectrlar site energy. Therefore, every site has its own local adsorption isotherm arrd the adsorbate concentration differs from site to site. The adsorbate is not considered as a homogeneous phase but as a multiphase system. 

Equation (16) has also been applied to lAST [41]. When it is assumed that the gas mixture obeys the perfect gas law and that the adsorbed phase on each patch is ideal, the resulting model is termed HI AST (heterogeneous lAST). Since the HIAST requires the evaluation of the local equilibria on each energy patch with lAST, much more computational effort is required, and yet the increase in performance over the original lAST is only modest for most cases. Therefore, this theory has not been widely used in practical applications. 

It can be seen that lAST predictions seriously deviate from the experimental data for this system even though the adsorbed phase is known to be very close to ideal under the experimental conditions. This deviation therefore points to the surface heterogeneity. T o address this heterogeneity, new models continue to emerge in the study of multicomponent adsorption equilibria. 

Doong and Yang [61] proposed a simple way of predicting multicomponent equilibria by using the concept of TVFM. Their model is based on the idea that in the DA equation the total micropore volume of the adsorbent should be replaced by the "maximum available micropore volume" for each species. Other assumptions are that (1) the adsorbate-adsorbate interaction is negligible compared to the adsorbate-adsorbent interaction, (2) the parameters n and (3 o for any species are independent of the other species, and (3) the adsorbed phase is ideal. With these assumptions, the DA equation for each species in a binary system can be written as... 

It was demonstrated in the study of Gusev et al. [47] that the MSAM gives a good representation of both ideal and nonideal behavior of the adsorbed phase for a number of systems that cannot be described by the I AST. For example, the nonideal adsorption behavior of the ethane-methane system on BPL carbon (Fig. 1) is successfully simulated by the MSAM (dotted line in Fig. 1). It is noted,however, that the MSAM does introduce some arbitrary assumptions and unrealistic pictures in describing the adsorbed phase as the authors put it, "This model is intended for engineering application rather than for the fundamental mechanisms of multicomponent adsorption equilibria."... 

As a first-order deviation from ideal behavior one may assume that the adsorbed phase obeys the regular solution model ... 

Information concerning the relevant adsorption equilibria is generally an essential requirement for the analysis and design of an adsorption separation process. In Chapter 3 we considered adsorption equilibrium from the thermodynamic perspective and developed a number of simple idealized expressions for the equilibrium isotherm based on various assumptions concerning the nature of the adsorbed phase. The extent to which these models can provide a useful representation of the behavior of real systems was considered only superficially and is reviewed in this chapter. Since many practical adsorption systems involve the simultaneous adsorption of more than one component, the problems of correlating and predicting multicomponent equilibria from singlecomponent data are of particular importance and are therefore considered in some detail. 

If the adsorbed phase is thermodynamically ideal, it is possible to derive the equilibrium relationships for an adsorbed mixture directly from the pure-component isotherms using the methods outlined in Section 3.4 without postulating a specific model for the adsorbed phase. For an ideal binary systenr Eq. (3.49) becomes... 

At first sight the assumption of ideal behavior in the adsorbed phase seems highly improbable, but it has been shown that a number of systems conform closely to this model. Some examples arc listed in Table 4.5 but by no means all adsorbed mixtures show ideal behavior. Glcssner and Myers observed... 

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