Adsorbents multicomponent isotherm equations

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

The pure component adsorption equilibrium of ethane and propane are measured on Norit AC at three temperatures (30, 60 and 90 °C). All experimental data of two species at three temperatures are employed simultaneously to fit the isotherm equation to extract the isothermal parameters. Since an extended Langmuir equation is used to describe the local multicomponent isotherm, the maximum adsorbed capacity is forced to be the same for ethane and propane in order to satisfy the thermodynamic consistency. The saturation capacity was assumed to be temperature dependent while the other parameters, bo and u], are temperature independent but species dependent. The derived isotherm parameters for ethane and propane are tabulated in Table 1. The experimental data (symbols) and the model fittings (solid lines)... 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-... 

Myers and Prausnitz [49] developed the ideal adsorbed solution (IAS) model in order to predict thermodjmamically consistent multicomponent isotherms of gas mixtures, using only experimental data acquired for single solute adsorption. The initial equation of the IAS theory for gases is... 

The prediction of multicomponent equilibria based on the information derived from the analysis of single component adsorption data is an important issue particularly in the domain of liquid chromatography. To solve the general adsorption isotherm, Equation (27.2), Quinones et al. [156] have proposed an extension of the Jovanovic-Freundlich isotherm for each component of the mixture as local adsorption isotherms. They tested the model with experimental data on the system 2-phenylethanol and 3-phenylpropanol mixtures adsorbed on silica. The experimental data was published elsewhere [157]. The local isotherm employed to solve Equation (27.2) includes lateral interactions, which means a step forward with respect to, that is, Langmuir equation. The results obtained account better for competitive data. One drawback of the model concerns the computational time needed to invert Equation (27.2) nevertheless the authors proposed a method to minimize it. The success of this model compared to other resides in that it takes into account the two main sources of nonideal behavior surface heterogeneity and adsorbate-adsorbate interactions. The authors pointed out that there is some degree of thermodynamic inconsistency in this and other models based on similar -assumptions. These inconsistencies could arise from the simplihcations included in their derivation and the main one is related to the monolayer capacity of each component [156]. 

A number of fundamental approaches have been taken to derive the necessary adsorption isotherm. If the adsorbed fluid is assumed to behave like a two dimensional nonideal fluid, then the Equation of State developed for three dimensional fluids can be applied to two dimensional fluids with a proper change of variables. The 2D-EOS adsorption isotherm equations are not popularly used in the description of data, but they have an advantage of easily extending to multicomponent mixtures by using a proper mixing rule for the adsorption parameters. 

This equation is known in the literature as the extended Langmuir isotherm equation, which gives the adsorbed concentration of the species i in the multicomponent system. For a binary system. Figure 5.2-1 shows plots of the fractional coverage of the component 1 versus biPj with the parameter b2P2 as the varying parameter. We see that the presence of the additional component 2 causes a decrease in the surface concentration of the component 1 and vice versa due to the competition of the two species. The reduction in the adsorbed concentration of the species 1 is... 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

A new molecular simulation technique is developed to solve the perturbation equations for a multicomponent, isothermal stured-tank adsorber under equilibrium controlled conditions. The method is a hybrid between die Gibbs ensemble and Grand Canonical Monte Carlo methods, coupled to macroscopic material balances. The bulk and adsorbed phases are simulated as two separate boxes, but the former is not actually modelled. To the best of our knowledge, this is the first attempt to predict the macroscopic behavior of an adsorption process from knowledge of the intermolecular forces by combining atomistic and continuum modelling into a single computational tool. 

Eijuillbrium. Among the aspects of adsorption, equiUbtium is the most studied and pubUshed. Many different adsorption equiUbtium equations are used for the gas phase the more important have been presented (see section on Isotherm Models). Equally important is the adsorbed phase mixing rule that is used with these other models to predict multicomponent behavior. 

For constant-separation factor systems, the /(-I rails formal ion of Helfferich and Klein (gen. refs.) or the method of Rhee et al. [AlChE J., 28, 423 (1982)] can be used [see also Helfferich, Chem. Eng. Sci., 46, 3320 (1991)]. The equations that follow are adapted from Frenz and Horvath [AlChE ]., 31, 400 (1985)] and are based on the h I ransiomialion. They refer to the separation of a mixture of M — 1 components with a displacer (component 1) that is more strongly adsorbed than any of the feed solutes. The multicomponent Langmuir isotherm [Eq. (16-39)] is assumed valid with equal monolayer capacities, and components are ranked numerically in order of decreasing affinity for the stationary phase (i.e., Ki > K2 > Km). 

The problem of predicting multicomponent adsorption equilibria from single-component isotherm data has attracted considerable attention, and several more sophisticated approaches have been developed, including the ideal adsorbed solution theory and the vacancy solution theory. These theories provide useful quantitative correlations for a number of binary and ternary systems, although available experimental data are somewhat limited. A simpler but purely empirical approach is to use a modified form of isotherm expression based on Langmuir-Freundlich or loading ratio correlation equations ... 

The adsorption of gas mixtures has been extensively studied. For example, Wendland et al. [64] applied the Bom—Green—Yvon approach using a coarse grained density to study the adsorption of subcritical Lennard-Jones fluids. In a subsequent paper, they tested their equations with simulated adsorption isotherms of several model mixtures [65]. They compared the adsorption of model gases with an equal molecular size but different adsorption potentials. They discussed the stmcture of the adsorbed phase, adsorption isotherms, and selectivity curves. Based on the vacancy solution theory [66], Nguyen and Do [67] developed a new technique for predicting the multicomponent adsorption equihbria of supercritical fluids in microporous carbons. They concluded that the degree of adsorption enhancement, due to the proximity of the pore... 

Equations of State If an equation of state is specified for a multicomponent adsorbed phase of the form itA/iSiT) = fitii,. . . ), then tne isotherms are cietermined using [Van Ness, Ind. Eng. Chem. Fundam., 8, 464-473 (1969)]... 

A multicomponent model of adsorption can be reduced to three main components first, H2S, second, VOC with average properties of found species, and third, H2O. Moreover, H2O adsorption can be described as quasi-equilibrium due to the high concenbation of H2O and the long operation time of the carbon bed. Rectangular type isotherms are used to model the equilibrium of VOC and H2S. The amount of H2S adsorbed depends on available space left after VOC adsorption and it can be described bj equation ... 

We have considered the case of multicomponent adsorption under isothermal conditions in the last section. Such an isothermal condition occurs when the particle is very small or when the environment is well stirred or when the heat of adsorption is low. If these criteria are not met, the particle temperature will vary. Heat is released during adsorption while it is absorbed by the particle when desorption occurs, leading to particle temperature rise in adsorption and temperature drop in desorption. The particle temperature variation depends on the rate of heat released and the dissipation rate of energy to the surrounding. In the displacement situation, that is one or more adsorbates are displacing the others, the particle temperature variation depends also on the relative heats of adsorption of displacing adsorbates and displaced adsorbates. Details of this can only be seen from the solution of coupled mass and heat balance equations. 

Kapoor et al. [79] proposed a heterogeneous extended Langmuir (HEL) model for the description of multicomponent equilibria on heterogeneous adsorbents. With the integral equation approach of Eq. (16), the general isotherm for a pure component system can be simplified as... 

Minka and Myers (8) have extended the concept of surface excess and selectivity to multicomponent mixtures. They applied a theory of an ideal adsorbed phase to predict the adsorption behavior of ternary mixtures from adsorption measurements in binary systems. Having binary data in the form of Equation (10) a ternary isotherm is calculated as follows ... 

The analysis of macropore diffusion in binary or multicomponent systenis presents no particular problems since the transport properties of one compos nent are not directly affected by changes ini the concentration of the bther components. In an adsorbed phase the situation is more complex since ih addition to any possible direct effect on thei mobility, the driving force for each component (chemical potential gradient is modified, through the multi-component equilibrium isotherm, by the coiicentration levels of all components in the system. The diffusion equations for each component are therefore directly coupled through the equilibrium relationship. Because of the complexity of the problem, diffusion in a mixed adscjrbed phase has been studied tjs only a limited extent. 

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