Multicomponent system composition vapor-liquid equilibria

Consider a multicomponent mixture in vapor-liquid equilibrium let x represent the set of mole fractions for the liquid and let y be the same for the vapor. In a closed system, the compositions x] and y will change, often drastically, with changes in T and P. However, in many systems the ratio y, /x,-, for each component i, is less sensitive to changes of state than is either x,- or y,- by itself. This observation is exploited by introducing two quantities the K-factor and the relative volatility ( 12.1.2). We have already encountered the K-factor in the Rachford-Rice method for flash calculations see (11.1.24) and Problem 11.7. 

For mixtures containing more than two species, an additional degree of freedom is available for each additional component. Thus, for a four-component system, the equihbrium vapor and liquid compositions are only fixed if the pressure, temperature, and mole fractious of two components are set. Representation of multicomponent vapor-hquid equihbrium data in tabular or graphical form of the type shown earlier for biuaiy systems is either difficult or impossible. Instead, such data, as well as biuaiy-system data, are commonly represented in terms of ivapor-liquid equilibrium ratios), which are defined by... 

In Chap. 6 we treated the thermodynamic properties of constant-composition fluids. However, many applications of chemical-engineering thermodynamics are to systems wherein multicomponent mixtures of gases or liquids undergo composition changes as the result of mixing or separation processes, the transfer of species from one phase to another, or chemical reaction. The properties of such systems depend on composition as well as on temperature and pressure. Our first task in this chapter is therefore to develop a fundamental property relation for homogeneous fluid mixtures of variable composition. We then derive equations applicable to mixtures of ideal gases and ideal solutions. Finally, we treat in detail a particularly simple description of multicomponent vapor/liquid equilibrium known as Raoult s law. 

A significant advantage of the Wilson equation is that it can be used to calculate the equilibrium compositions for multicomponent systems using only the Wilson coefficients obtained for the binary pairs that comprise the multicomponent mixture. The Wilson coefficients for several hundred binary systems are given in the DECHEMA vapor-liquid data collection (DECHEMA, 1977) and by Hirata (1975). Hirata gives methods for calculating the Wilson coefficients from vapor-liquid equilibrium experimental data. 

Using one of these activity coefficient equations it is possible to calculate liquid-liquid equihbrium (LLE) behavior of multicomponent hquid systems. Consider, for example, the ternary system of Figure 1. A system of overall composition A splits into two liquid phases B and C. The calculation of compositions of B and C is analogous to the flash ciculation of vapor-liquid equilibrium problems. By using the UNIQUAC equations to obtain the partition coefficients, Kj, this problem can be solved for any composition A of the overall system. The calculations are lengthy but computer programs for this purpose (2) have been published. In this paper simpler approximate methods for phase equilibrium problems of environmental interest is sought. For the moment it is sufficient to note that the activity coefficients provide the means of complete liquid-liquid equihbrium computations. 

Methods to estimate the thermal conductivity of liquid mixtures have been reviewed by Reid et al. (1977, 1987) and Rowley et al. (1988). Five methods are summarized by Reid et al. (1987), but three of these can be used only for binary mixtures. The two that can be extended to multicomponent mixtures are the Li method (Li 1976), and Rowley s method (Rowley et al. 1988). According to the latter the Li method does not accurately describe ternary behavior. Furthermore, it was indicated that the power law method (Reid et al. 1977 Rowley et al. 1988) successfully characterizes ternary mixture behavior when none of the pure component thermal conductivities differ by more than a factor of 2. But, the power law method should not be used when water is present in the mixture. Rowley s method is based on a local composition concept, and it uses NRTL parameters from vapor-liquid equilibrium data as part of the model. These parameters are available for a number of binary mixtures (Gmehling Onken 1977). When tested for 18 ternary systems, Rowley s method gave an average absolute deviation of 1.86%. 

In a multicomponent system of n components (i = 1,..., k,. .ri], to obtain the compositions of the vapor phase and the liquid phase leaving the separator under equilibrium, along with the total molar flow rates of the liquid product and the vapor product from the flash drum for a given feed condition, will require the solution of the appropriate governing equations. For a system of n components, there are n equations (6.3.53) describing vapor-liquid equilibrium, n equations (6.3.54) or (6.3.55), describing Xu in terms of Xif or in terms of Xif and one equation. 

Properties of multicomponent systems includes solubility, activity coefficients, diffusion coefficients, phase diagrams, vapor-liquid equilibrium data on mixtures, etc. These are generally functions of temperature, pressure, and composition. 

The UNIQUAC equation developed by Abrams and Prausnitz is usually preferred to the NRTL equation in the computer-aided design of separation processes. It is suitable for miscible and immiscible systems, and so can be used for vapor-liquid and liquid-liquid systems. As with the Wilson and NRTL equations, the equilibrium compositions for a multicomponent mixture can be predicted from experimental data for the binary pairs that comprise the mixture. Also, in the absence of experimental data for... 

For multicomponent systems which can be considered ideal, Raoult s law can be used to determine the composition of the vapor in equilibrium with the liquid. For example, for a system composed of four components. A, B, C, and D,... 

A column comprises individual separation stages in which the purification of the product is carried out by means of the effect that vapor and liquid have different compositions at equilibrium. Accordingly, the column design calls for knowledge of the phase equilibria of the systems [5]. Normally, phase equilibrium calculations are based on binary parameters describing the interactions of two different molecules. If multicomponent mixtures are considered, some of these interactions might be unknown. To obtain better simulation results, they should at least be estimated. This was the main reason for the development of the UNIFAC group contribution method twenty years ago. 

Starting from either side of the phase diagram, the situation is very much like a liquid-vapor phase change One component will preferentially change phase, and the other component will become more and more concentrated within the remaining liquid. Until, that is, a certain composition labeled is reached Then the two components will freeze simultaneously, and the solid that forms will have the same composition as the liquid. This composition is called the eutectic composition. At this composition, this liquid acts as if it were a pure component, so the solid and liquid phases have the same composition when in equilibrium at the eutectic temperature Tg. This pure component is called the eutectic. The eutectic is similar to the azeotrope in liquid-vapor phase diagrams. Not all systems will have eutectics some systems may have more than one, and the composition of the eutectic(s) of a multicomponent system is characteristic of the components. That is, you cannot predict a eutectic for any given system. 

One of the more widely used methods for calculating the number of theoretical stages in multicomponent systems was developed by Lewis and Matheson [6]. Again, the molar flow rates of vapor and liquid in each section are assumed to be constant. On each equilibrium stage, the summation of the concentrations of all vapor components must equal unity. The same is true for the summation of the concentrations of all liquid components. Further, the vapor and liquid compositions of each component are related by the K value for that component. 

If you apply the Gibbs phase rule to a multicomponent gas-liquid system at equilibrium, you will discover that the compositions of the two phases at a given temperature and pressure are not independent. Once the composition of one of the phases is specified (in terms of mole fractions. mass fractions, concentrations, or. for the vapor phase, partial pressures), the composition of the other phase is fixed and, in principle, can be determined from physical properties of the system components. 

---