Phase equilibrium components

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

The amounts of each phase and their compositions are calculated by resolving the equations of phase equilibrium and material balance for each component. For this, the partial fugacities of each constituent are determined ... 

The criterion for phase equilibrium is given by Eq. (8.14) to be the equality of chemical potential in the phases in question for each of the components in the mixture. In Sec. 8.8 we shall use this idea to discuss the osmotic pressure of a... 

For each stage J, the following 2C -1- 3 component material-balance (M), phase-equilibrium (E), mole-fraction-summation (S), and energy-balance (H) equations apply, where C is the number of chemical species ... 

The activity coefficient y can be defined as the escaping tendency of a component relative to Raonlt s law in vapor-liqnid eqnihbrinm (see Sec. 4 in this handbook or Null, Phase Equilibrium in Process Design, Wiley-Interscience, 1970). 

Note that this equation holds for any component in a multi-component mixture. The integral on the right-hand side can only be evaluated if the vapor mole fraction y is known as a function of the mole fraction Xr in the still. Assuming phase equilibrium between liquid and vapor in the still, the vapor mole fraction y x ) is defined by the equilibrium curve. Agitation of the liquid in tire still and low boilup rates tend to improve the validity of this assumption. 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

In words, for phase equilibrium to occur in a constant temperature and pressure process, the chemical potential of each component must have the same value in all of the phases that are in equilibrium. 

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... 

If we could prevent the mixture from separating into two phases at temperatures below Tc, we would expect the point of inflection to develop into curves similar to those shown in Figure 8.17 as the dotted line for /2, with a maximum and minimum in the fugacity curve. This behavior would require that the fugacity of a component decreases with increasing mole fraction. In reality, this does not happen, except for the possibility of a small amount of supersaturation that may persist briefly. Instead, the mixture separates into two phases. These phases are in equilibrium so that the chemical potential and vapor fugacity of each component is the same in both phases, That is, if we represent the phase equilibrium as... 

In processing, it is frequently necessary to separate a mixture into its components and, in a physical process, differences in a particular property are exploited as the basis for the separation process. Thus, fractional distillation depends on differences in volatility. gas absorption on differences in solubility of the gases in a selective absorbent and, similarly, liquid-liquid extraction is based on on the selectivity of an immiscible liquid solvent for one of the constituents. The rate at which the process takes place is dependent both on the driving force (concentration difference) and on the mass transfer resistance. In most of these applications, mass transfer takes place across a phase boundary where the concentrations on either side of the interface are related by the phase equilibrium relationship. Where a chemical reaction takes place during the course of the mass transfer process, the overall transfer rate depends on both the chemical kinetics of the reaction and on the mass transfer resistance, and it is important to understand the relative significance of these two factors in any practical application. 

The computational problem of polymer phase equilibrium is to provide an adequate representation of the chemical potentials of each component in solution as a function of temperature, pressure, and composition. 

B bulk property d deactivation e effective property G gas phase i component index i reaction index L liquid phase p catalyst particle property equilibrium conditions... 

When the original compositions of the outer phases are different, the permselective membrane will prevent the complete leveling of these compositions. Some equilibrium component distribution between phases (a) and (p) will be established, and between points A and B a potential difference called the membrane potential (or transmembrane potential) (p will develop. This potential difference is determined by... 

For the purpose of deciding which phase equilibrium method to use, it is convenient to classify components into the classes shown in Table 8.10. 

The fraction of benzene, toluene, styrene and ethylbenzene condensed can be determined from phase equilibrium calculations. The percent of the various components entering... 

The phase equilibrium for pure components is illustrated in Figure 4.1. At low temperatures, the component forms a solid phase. At high temperatures and low pressures, the component forms a vapor phase. At high pressures and high temperatures, the component forms a liquid phase. The phase equilibrium boundaries between each of the phases are illustrated in Figure 4.1. The point where the three phase equilibrium boundaries meet is the triple point, where solid, liquid and vapor coexist. The phase equilibrium boundary between liquid and vapor terminates at the critical point. Above the critical temperature, no liquid forms, no matter how high the pressure. The phase equilibrium boundary between liquid and vapor connects the triple point and the... 

Thus, equilibrium is achieved when the escaping tendency from the vapor and liquid phases for Component i are equal. The vapor-phase fugacity coefficient, fj, can be defined by the expression ... 

In the case of vapor-liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. 

Characterization of Equilibria Phase equilibrium between fluid and sorbed phases for one or many components in adsorption or two or more species in ion exchange is usually the single most... 

Phase solubility analysis is a technique to determine the purity of a substance based on a careful study of its solubility behavior [38,39]. The method has its theoretical basis in the phase mle, developed by Gibbs, in which the equilibrium existing in a system is defined by the relation between the number of coexisting phases and components. The equilibrium solubility of a material in a particular solvent, although a function of temperature and pressure, is nevertheless an intrinsic property of that material. Any deviation from the solubility exhibited by a pure sample arises from the presence of impurities and/or crystal defects, and so accurate solubility measurements can be used to deduce the purity of the sample. 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

One-bath, one-stage dyeing process, 9 198 One-component system, phase equilibrium in, 24 661-665... 

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