Multicomponent Phase Equilibria

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

In Chapter 5, we showed that the condition of phase equilibrium for multicomponent phases is that the chemical potential of each component must be the same in all the phases. That is... 

There are two types of multicomponent mixtures which occur In polymer phase equilibrium calculations solutions with multiple solvents or pol ers and solutions containing poly-disperse polymers. We will address these situations In turn. 

Mixture phase equilibrium calculations, types of, 24 680-681 Mixture-process design type, 8 399 commercial experimental design software compared, 8 398t Mixtures. See also Multicomponent mixtures Nonideal liquid mixtures acetylene containing, 2 186 adsorption, 2 593-594 adsorption isotherm models,... 

For the application of equation 212 to the case of multicomponent, two-phase equilibrium, the phases are denoted by a and p, and the following general expression is written for G ... 

The same reference (standard) state, f is chosen for the two phases, so that it cancels on both sides of equation 39. The products stffi and y" are referred to as activities. Because equation 39 holds for each component of a liquid—liquid system, it is possible to predict liquid—liquid phase splitting when the activity coefficients of the individual components in a multicomponent system are known. These values can come from vapor—liquid equilibrium experiments or from prediction methods developed for phase-equilibrium problems (4,5,10). Some binary systems can be modeled satisfactorily in this manner, but only rough estimations appear to be possible for multicomponent systems because activity coefficient models are not yet sufficiendy developed in this area. 

Solid-Phase Chemical Equilibrium. For the growth of multicomponent films, the solid film composition must be predicted from the gas-phase composition. In general, this prediction requires detailed information about transport rates and surface incorporation rates of individual species, but the necessary kinetics data are rarely available. On the other hand, the equilibrium analysis only requires thermodynamic data (e.g., phase equilibrium data), which often are available from liquid-phase-epitaxy studies, as discussed by Anderson in Chapter 3. 

When a liquid hydrocarbon mixture is present, the Lw-V-Lhc line in Figure 4.2b broadens to become an area, such as that labeled CFK in Figure 4.2c. This area is caused by the fact that a single hydrocarbon is no longer present, so a combination of hydrocarbon (and water) vapor pressures creates a broader phase equilibrium envelope. Consequently, the upper quadruple point (Q2) evolves into a line (KC) for the multicomponent hydrocarbon system. 

Stradi, B. A. Stadtherr, M. A. Brennecke, J. F. Multicomponent Phase Equilibrium Measurements and Modeling for the Allylic Epoxidation of traw.y-2-hexen-l-ol to (2R,3R)-( + )-3-Propyloxiranemethanol in High-Pressure Carbon Dioxide. J. Supercrit. Fluid 2001b, 20 (1), 1-13. 

The Gibbs phase rule is of use in phase equilibrium studies of multicomponent systems. 

Processes for supercritical extraction of oils have been described in numerous literature references, including Paulaitis et al. ( ), Ely and Baker (2), Gerard (3.), Stahl et al. (4K and Robey and Sunder (5). The literature lacks detailed phase equilibrium data on multicomponent essential oils with supercritical solvents in the proximity of the solvent critical temperature. 

Phase equilibrium is described by the multicomponent competitive Langmuir isotherm according to... 

There are several characteristics common to the describing equations of all types of multicomponent, vapor-liquid separation processes, both single- and multi-stage, that make it possible to exploit the inside-out concept in similar ways to solve them efficiently and reliably. The equations have as common members component and total mass balance, enthalpy balance, constitutive and phase equilibrium equations. In addition, all such processes require K-value or fugacity coefficient and vapor and liquid enthalpy models. In all cases the describing equations have similar forms, and depend on the primitive variables (temperature, pressure, phase rate and composition) in essentially the same ways. Before presenting the inside-out concept, it will be useful to identify two classes of conventional methods and discuss their main characteristics. 

On its way downwards, the liquid phase is of course depleted with respect to its more volatile component(s) and enriched in its heavier component(s). At the decisive locus, however, where both phases have their final contact (i.e., the top of the column), the composition of the liquid is obviously stationary. For a desired composition of the gas mixture, the appropriate values for the liquid phase composition and the saturator temperature must be found. This is best done in two successive steps, viz. by phase equilibrium calculations followed by experimental refinement of the calculated values. The multicomponent saturator showed an excellent performance, both in a unit for atmospheric pressure [18] and in a high-pressure apparatus [19, 20] So far, the discussion of methods for generating well defined feed mixtures in flow-type units has been restricted to gaseous streams. As a rule, liquid feed streams are much easier to prepare, simply by premixing the reactants in a reservoir and conveying this mixture to the reactor by means of a pump with a pulsation-free characteristic. 

The experimental data are correlated with equation of state models. The calculation of binary phase equilibrium data for FAEE is commonly based on the Peng-Robinson-equation-of-state, Yu et al. (1994). Up to now only the solubility of the oil components in the solvent has been subject of various studies. No attention was paid to a correlation of ternary data. The computation of ternary or multicomponent phase equilibrium is the basis to analyse and optimise the separation experiments. 

For binary diffusion, there is only one independent flow, force or concentration gradient, and diffusion coefficient. On the other hand, multicomponent diffusion differs from binary diffusion because of the possibility of interactions among the species in mixtures of three or more species. Some of the possible interactions are (1) a flow may be zero although its zero driving force vanishes, which is known as the diffusion barrier (2) the flow of a species may be in a direction opposite to that indicated by its driving force, which is called reverse flow and (3) the flow of a species may occur in the absence of a driving force, which may be called osmotic flow. The theory of nonequilibrium thermodynamics indicates that the chemical potential arises as the proper driving force for diffusion. This is also consistent with the condition of fluid phase equilibrium, which is satisfied when the chemical potentials of a species are equal in each phase. 

The design engineer dealing with polymer solutions must determine if a multicomponent mixture will separate into two or more phases and what the equilibrium compositions of these phases will be. Prausnitz et al. (1986) provides an excellent introduction to the field of phase equilibrium thermodynamics. 

The development of equations that successfully predict multicomponent phase equilibrium data from binary data with remarkable accuracy for engineering purposes not only improves the accuracy of tray-to-tray calculations but also lessens the amount of experimentation required to establish the phase equilibrium data. Such equations are the Wilson equation (13), the non-random two-liquid (NRTL) equation (14), and the local effective mole fractions (LEMF) equation (15, 16), a two-parameter version of the basically three-parameter NRTL equation. Larson and Tassios (17) showed that the Wilson and NRTL equations predict accurately ternary activity coefficients from binary data Hankin-son et al. (18) demonstrated that the Wilson equation predicts accurately... 

Cotterman, R. L., and Prausnitz, J. M., Continuous thermodynamics for phase-equilibrium calculations in chemical process design. In Kinetic and Thermodynamic Lumping of Multicomponent Mixtures (G. Astarita and R. I. Sandler, eds.). Elsevier, Amsterdam, 1991, p. 229. 

Hendricks, E. M., Simplified phase equilibrium equations for multicomponent systems. Fluid Phase Eq. 33,207 (1987). 

In Aspen Plus, solid components are identified as different types. Pure materials with measurable properties such as molecular weight, vapor pressure, and critical temperature and pressure are known as conventional solids and are present in the MIXED substream with other pure components. They can participate in any of the phase or reaction equilibria specified in any unit operation. If the solid phase participates only in reaction equilibrium but not in phase equilibrium (for example, when the solubility in the fluid phase is known to be very low), then it is called a conventional inert solid and is listed in a substream CISOLID. If a solid is not involved in either phase or reaction equilibrium, then it is a nonconventional solid and is assigned to substream NC. Nonconventional solids are defined by attributes rather than molecular properties and can be used for coal, cells, catalysts, bacteria, wood pulp, and other multicomponent solid materials. 

The N equations represented by Eq. (4-303) in conjunction with Eq. (4-305) may be solved for N unknown phase equilibrium variables. Eor a multicomponent system the calculation is formidable, but well suited to computer solution. 

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