Equilibrium multicomponent

The condition for boiling, of a liquid mixture, is that the sum of the partial pressures of the components of the liquid phase mixture is equal to the total pressure of the system 

As discussed by Franks (1972), in order to solve this system of equations, a value of temperature T must be found to satisfy the condition that the difference term 6 = P - Zpj is very small, i.e., that the equilibrium condition is satisfied. This is known as a bubble point calculation. The above system of defining equations, however represent, an implicit algebraic loop and the trial and error solution procedure can be very time consuming, especially when incorporated into a dynamic simulation program. 

An information flow diagram showing the iteration for 6 is shown in Fig. 3.63. Examples of multicomponent equilibria with activity coefficients are given in the simulation examples BUBBLE and STEAM. 

---

Kofke D A and Glandt E D 1988 Monte Carlo simulation of multicomponent equilibria in a semigrand canonical ensemble/Wo/. Phys. 64 1105-31... 

Multicomponent equilibria combined with distillation heat effects are discussed in more detail in Section 3.3.4 below. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

They have been found useful as an empirical correlation method for adsorption on molecular sieves [Maurer, Am. Chem. Soc. Symp. Ser. 135, 73 (1980)]. Other attempts at prediction or correlation of multicomponent adsorption data are reviewed by Ruthven (1984). In general, however, multicomponent equilibria are not well correlatable in general form so that design of equipment is best based on direct laboratory data with the exact mixture and the exact adsorbent at anticipated pressure and temperature. 

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. 

Multi-Component. Only a representative selection of the many possible combinations of multicomponent equilibria involving the 4 gases used in the current program is presented in this paper. 

The fundamental adsorptive properties governing the performance of the separation processes are the multicomponent equilibria, kinetics, and heat. A large volume of data, as well as models to describe them, exist in the published literature only for adsorption of pure gases and binary liquid mixtures. Binary gas adsorption data are sporadic. Multicomponent data are rare. Existence of adsorbent heterogeneity can introduce severe complexity in the multicomponent adsorption behavior. 

In addition to modeling polypropylene formation, scientists at BP Amoco Chemicals have successfully used computations to model desulfurization of light naphtha, flue gas multicomponent equilibria, methane-to-methanol conversion, and oxygen scavenging films. 

Selectivity is a means by which mixture equilibria can be grasped. Several different definitions exist, but all basically are ratios of what is adsorbed to what remains in the fluid phase at equilibrium. Selectivity, therefore, provides a simple description of the nature of multicomponent equilibria, although the values are seldom employed in mathematical models. Some common definitions are... 

A significant amount of work has been done on determining sorption capacity and the kinetics of sorption in zeolites because of their applications as adsorbents and as catalysts in the chemical process industry. However, most of this work has been done with single components, whereas all practical applications involve multicomponent mixtures. Hence, measurement of binary or multicomponent equilibria on zeolites is of considerable importance. 

Already its dissolution in protic solvents initiated changes in the structure of secologanin and resulted multicomponent equilibria, therefore the NMR spectra of secologanin and some of its derivatives should be recorded in aprotic solvents (chloroform, benzene, acetonitril, etc). The... 

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]. 

The last three chapters deal with the fundamental and empirical approaches of adsorption isotherm for pure components. They provide the foundation for the investigation of adsorption systems. Most, if not all, adsorption systems usually involve more than one component, and therefore adsorption equilibria involving competition between molecules of different type is needed for the understanding of the system as well as for the design purposes. In this chapter, we will discuss adsorption equilibria for multicomponent system, and we start with the simplest theory for describing multicomponent equilibria, the extended Langmuir isotherm equation. This is then followed by a very popularly used IAS theory. Since this theory is based on the solution thermodynamics, it is independent of the actual model of adsorption. Various versions of the IAS theory are presented, starting with the Myers and Prausnitz theory, followed by the LeVan and Vermeulen approach for binary systems, and then other versions, such as the Fast IAS theory which is developed to speed up the computation. Other multicomponent equilibria theories, such as the Real Adsorption Solution Theory (RAST), the Nitta et al. s theory, the potential theory, etc. are also discussed in this chapter. 

A programming code FastlAS is provided with this book, and readers are encouranged to use the code to perform calculation of multicomponent equilibria. We illustrate this with the following example. 

For the case where the extended Langmuir isotherm can describe the multicomponent equilibria, the non-dimensional diffusivity matrix is given by ... 

We first present the multicomponent equilibria and then deal with the multicomponent kinetics next. 

Equation (16) is the general integral equation from which many multicomponent equilibria models are derived [38,39]. Sircar [40] also used the binary selectivity S instead of energy e in Eq. (16) to study the role of adsorbent heterogeneity in adsorption from a binary liquid mixture. 

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... 

---