Multicomponent Isotherms

Where both multicomponent and heat transfer effects appeared important, it was necessary to compare the results of selected runs via the MASC program under adiabatic conditions to the value obtained by multiplying the MTZ length prediction via SSMTZ (isothermal, multicomponent) by the ratio of MTZ lengths predicted by SSHTZ under adiabatic and isothermal conditions. If the values are essentially equivalent, the SSMTZ and SSHTZ programs could be used to analyze the MTZ data with a high degree of confidence. 

In this treatment only the ordinary and Knudsen diffusion mechanisms will be considered. Then, mass transport in isothermal, multicomponent gas phase systems is described by the following constitutive equation ... 

For diffusion in isothermal multicomponent systems the generalized driving force was written as a linear function of the relative velocities (m/ — My). In the general case, we must allow for coupling between the processes of heat and mass transfer and write constitutive relations for and q in terms of the (m — My) and V(l/r). With this allowance, the complete expression for the conductive heat flux is... 

Krishna, R. and Standart, G. L., Addition of Resistances for Non-Isothermal Multicomponent Mass Transfer, Letts. Heat Mass Transfer, 3, 41-48 (1976b). 

TABLE 9.1. Summary of Numerical Solutions for Isothermal Multicomponent Systems... 

Nagel, G., and Kluge, G., Non-isothermal multicomponent adsorption processes and their numerical treatment by means of integro-differential equations, Comput. Chem. Eng.. 13(9), 1025-1030(1989). 

Filippov, L.K., and Filippova, I.V.. Isothemial and non-isothermal multicomponent adsorption kinetics. Chem. Eng, Commun., 93, 147-166(1990). 

The calculation of single-stage equilibrium separations in multicomponent systems is implemented by a series of FORTRAN IV subroutines described in Chapter 7. These treat bubble and dewpoint calculations, isothermal and adiabatic equilibrium flash vaporizations, and liquid-liquid equilibrium "flash" separations. The treatment of multistage separation operations, which involves many additional considerations, is not considered in this monograph. 

Discussion of the concepts and procedures involved in designing packed gas absorption systems shall first be confined to simple gas absorption processes without compHcations isothermal absorption of a solute from a mixture containing an inert gas into a nonvolatile solvent without chemical reaction. Gas and Hquid are assumed to move through the packing in a plug-flow fashion. Deviations such as nonisotherma1 operation, multicomponent mass transfer effects, and departure from plug flow are treated in later sections. 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. 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 (7) as well as in the original paper (14). 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, ate not consistent with an ideal adsorbed... 

AH practical adsorbents have surfaces that are heterogeneous, both energetically and geometrically (not all pores are of uniform and constant dimensions). The degree of heterogeneity differs substantially from one adsorbent type to another. These heterogeneities are responsible for many nonlinearities, both in single component isotherms and in multicomponent adsorption selectivities. 

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. 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

Adsorbed-Solution Theoiy The common thennodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is... 

However, when complexation experiments are performed with both d- and L-enan-tiomers (Fig. 5-18), this leads to selectivity values between 1.4 and 1.9. It was shown that complexation by enantioselective micelles can effectively be described using straightforward multicomponent Langmuir isotherms [74]. 

Fig. 5-18. Selectivity for D,L-phenylalanine, fitted with a multicomponent Langmuir isotherm [73],...

Martinez-Ortiz, J. A., and D. B. Manley, Direct Solution of the Isothermal Gibbs-Duhem Equation for Multicomponent Systems, Ind. Eng. Chem. Process Des. Dev., 17, 3, (1978) p. 346. 

When the condensation process is not exactly isothermal but the temperature change is small such as where there is a significant change in pressure, or where a narrow boiling range multicomponent mixture is being condensed the logarithmic temperature difference can still be used but the temperature correction factor will be needed for multipass condensers. The appropriate terminal temperatures should be used in the calculation. 

For multicomponent condensation, the condensation will not be isothermal, leading to a nonlinear temperature-enthalpy profile for the condensation. If this is the case, then the exchanger can be divided into a number of zones with the temperature-enthalpy profiles linearized in each zone. Each zone is then modeled separately and zones summed to obtain the overall area requirement1. 

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. 

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