Phase Equilibria Models

A very fine example was provided by the extensive use of Professor Pitzer s electrolyte activity coefficient theory within several acid gas phase equilibrium models. 

The present paper deals with one aspect of this problem the calculation of phase separation critical points in reacting mixtures. The model employed is the Soave-Redlich-Kwong equation of state (1 ), which is typical of several equations of state (2, 5) which have relatively recently come into wide use as phase equilibrium models for light gas mixtures, sometimes including water and the acid gases as components (4, . 5, 6). If the critical point contained in the equation of state (perhaps even for the mixture at reaction equilibrium) can be found directly, the result will aid in other equilibrium computations. 

Following this, the thermodynamic arguments needed for determining CMC are discussed (Section 8.5). Here, we describe two approaches, namely, the mass action model (based on treating micellization as a chemical reaction ) and the phase equilibrium model (which treats micellization as a phase separation phenomenon). The entropy change due to micellization and the concept of hydrophobic effect are also described, along with the definition of thermodynamic standard states. 

In summary, whether a reaction equilibrium or a phase equilibrium approach is adopted depends on the size of the micelles formed. In aqueous systems the phase equilibrium model is generally used. In Section 8.5 we see that thermodynamic analyses based on either model merge as n increases. Since a degree of approximation is introduced by using the phase equilibrium model to describe micellization, micelles are sometimes called pseudophases. 

An illustration of how both the reaction equilibrium and phase equilibrium models can be applied to micellization is provided by Example 8.1. 

EXAMPLE 8.1 Reaction Equilibrium and Phase Equilibrium Models of Micellization. Research in which the CMC of an ionic surfactant M+ S is studied as a function of added salt, say M+X, ... 

Under what conditions can the formation of micelles be described in terms of reaction-equilibrium formalism When is the phase equilibrium model appropriate ... 

Reaction equilibrium and phase equilibrium models of micellization 361... 

Equations of State. An equation of state can be an exceptional tool for property prediction and phase equilibrium modeling. The term equation of state refers to the equilibrium relation among pressure, volume, temperature, and composition of a substance (2). This substance can be a pure chemical or a uniform mixture of chemicals in gaseous or liquid form. 

Let us test our two-phase model numerically. The model is captured in the two equations (6.99) and (6.102) that account for the gas and liquid molar balances. We shall compare these results for nonequilibrium stages with those of the one-phase equilibrium model given in equation (6.85) earlier. 

Mujtaba (1989) simulated the same example for the first product cut using a reflux ratio profile very close to that used by Nad and Spiegel in their own simulation and a nonideal phase equilibrium model (SRK). The purpose of this was to show that a better model (model type IV) and better integration method achieves even a better fit to their experimental data. Also the problem was simulated using an ideal phase equilibrium model (Antoine s equation) and the computational details were presented. The input data to the problem are given in Table 4.7. 

Vapour phase enthalpies were calculated using ideal gas heat capacity values and the liquid phase enthalpies were calculated by subtracting heat of vaporisation from the vapour enthalpies. The input data required to evaluate these thermodynamic properties were taken from Reid et al. (1977). Initialisation of the plate and condenser compositions (differential variables) was done using the fresh feed composition according to the policy described in section 4.1.1.(a). The simulation results are presented in Table 4.8. It shows that the product composition obtained by both ideal and nonideal phase equilibrium models are very close those obtained experimentally. However, the computation times for the two cases are considerably different. As can be seen from Table 4.8 about 67% time saving (compared to nonideal case) is possible when ideal equilibrium is used. 

Figure 4.7 shows the simulated instant distillate composition profiles by Mujtaba and by that of Nad and Spiegel using nonideal phase equilibrium models. The figure also includes experimentally obtained instant distillate composition data and the adjusted reflux ratio profiles used by Nad and Spiegel and Mujtaba. 

Keywords Hydrogen, metal hydrides, intermetallic compounds, phase equilibriums, model of non-ideal lattice gas. 

An improved local composition expression and its implications for phase equilibrium models. 

An Improved Local Composition Expression and Its Implication for Phase Equilibrium Models... 

A modified local composition (LC) expression is suggested, which accounts for the recent finding that the LC in an ideal binary mixture should be equal to the bulk composition only when the molar volumes of the two pure components are equal. However, the expressions available in the literature for the LCs in binary mixtures do not satisfy this requirement. Some LCs are examined including the popular LC-based NRTL model, to show how the above inconsistency can be eliminated. Further, the emphasis is on the modified NRTL model. The newly derived activity coefficient expressions have three adjustable parameters as the NRTL equations do, but contain, in addition, the ratio of the molar volumes of the pure components, a quantity that is usually available. The correlation capability of the modified activity coefficients was compared to the traditional NRTL equations for 42 vapor—liquid equilibrium data sets from two different kinds of binary mixtures (i) highly nonideal alcohol/water mixtures (33 sets), and (ii) mixtures formed of weakly interacting components, such as benzene, hexafiuorobenzene, toluene, and cyclohexane (9 sets). The new equations provided better performances in correlating the vapor pressure than the NRTL for 36 data sets, less well for 4 data sets, and equal performances for 2 data sets. Similar modifications can be applied to any phase equilibrium model based on the LC concept. 

When inorganic solids and water are present, an electrolyte phase equilibrium model must be selected for the aqueous phase, to properly account for the dissolution of the solid and formation of ions in solution. 

It is often necessary to add user components to complete a simulation model. The design engineer should always be cautious when interpreting simulation results for models that include user components. Phase equilibrium predictions for flashes, decanters, extraction, distillation, and crystallization operations should be carefully checked against laboratory data to ensure that the model is correctly predicting the component distribution between the phases. If the fit is poor, the binary interaction parameters in the phase equilibrium model can be tuned to improve the prediction. 

Choice of Phase Equilibrium Model for Design Calculations... 

Phase Equilibrium Modeling for Nylon-6 Process Simulation... 

Cubic equations of state (EOS) such as the Redlich-Kwong (RK), Soave-Redlich-Kwong and Peng>Robinson equations of state have become important tools in the area of phase equilibrium modeling, especially for systems at pressures close to or above the critical pressure of one or more of these system components. The functional form of the Soave-Redlich-Kwong and Peng-Robinson equations of state can be represented in a general manner as shown in Equation 2 ... 

The goal of predictive phase equilibrium models is to provide reliable and accurate predictions of the phase behavior of mixtures in the absence of experimental data. For low and moderate pressures, this has been accomplished to a considerable extent by using the group contribution activity coefficient methods, such as the UNIFAC or ASOG models, for the activity coefficient term in eqn. (2.3.8). The combination of such group contribution methods with equations of state is very attractive because it makes the EOS approach completely predictive and the group contribution method... 

Gros HP, Bottini SB, Brignole EA. High pressure phase equilibrium modeling of mixtures containing associated compounds and gases. Fluid Phase Equilibria 1997 139 75-87. 

Amine-extraction equilibria can also be modeled by chemical-reaction equilibrium constants. Figure 8.3-3 indicates that cations such as iron(IIl), zinc, cobelt(ll) and coppeifU) exhibit high distribution coefficients with chloride solutions, wherese nickel. iron(II), and manganese are not extracted to any great extent. The besis for the differences in distribution coefficients lies mainly in the tendency for the former group of cations to fonn chloride complexes. Stability constants for these complexes are available in the literature,11 and they can be used to develop quantitative phase-equilibrium models. 

With both staged equipment and differential contactors, availability oradequate phase-equilibrium models and rate expressions would allow application of existing correlations and simulation algorithm). For example. knowledge of metal-extraction kinetics in terms of interfacial species concentrations conld be combined with correlations of film mass transfer coefficients in a particular type of equipment to obtain the inlerfacial flux as a fuuction of bulk concentrations. Correlations or separate measurements of inierfacial area and an estimate of dispersion characteristics would allow calculation of extraction performance as a... 

The PCB concentration of eels in this river is a factor of 26 300 higher than the PCB concentration in the river water. The very simple phase equilibrium model for the distribution of the chemical used here is able to predict this very large bioconcentration effect to within a factor of2.. ... 

A simple dynamic model. The maximum magnitude of noble gas fractionation that can occur when two phases have been equilibrated is summarized for Ne/Ar in Figure 7. Although the phase equilibrium model demonstrates the effect of the physical conditions in a system on the limits of noble gas fractionation, the phase equilibrium model represents only one end-member of the processes that may be occurring in a dynamic subsurface fluid environment. To convey some sense of the relevance of the phase equilibrium model in a dynamic system it is useful to consider the extent to which noble gases partition and fractionate between phases when a gas bubble passes through a column of liquid (Ballentine 1991 Fig. 8). 

Figure 8. (A) A water column is divided into fifty equal unit cells and it is assumed there is no liquid or dissolved gas between cells. Each cell originally has the noble gas content of air-equilibrated water and all calculated Ne/Ar ratios are normalized to this value to obtain a fractionation factor F. The column temperature is taken to be 325 K, which for pure water gives Knc = 133245 atm and Kaf= 55389 atm. A gas bubble of constant volume is passed sequentially through the column, equilibrium assumed to occur in each water cell and the Ne and Ar partitioned into the respective gas and water phases (Eqn. 16). The evolution of the Ne/Ar ratio in the gas bubble (bold) and each water phase increment (Faint) is shown for different gas/water volume ratios, Vg/Vi. The gas bubble Ne/Ar ratio approaches the maximum fractionation value predicted for a gas/water phase equilibrium where as Vg/Vi -> 0, F Knc/Kat. The cell Vg/Vi ratio only determines the rate at which this hmit is approached. (B) The same water column with a fixed cell Vg/Vi ratio of 0.01. n subsequent bubbles are passed through the column and the He/Ne distribution between phases calculated at each stage. The gas bubble Ne/Ar ratio evolution for n = 1, 10, 20 and 30 is shown in bold, together with the residual Ne/Ar in the water colunm cells (faint lines). All gas bubbles approach the limit imposed by the phase equilibrium model. The water phase is fractioned in the opposite sense and is fractionated in proportion to the magnitude of gas loss following the Rayleigh fractionation law (Eqn. 24).

The phase equilibrium model limits the maximum noble gas fractionation that will occur in a gas phase migrating though groundwater. 

By direct analogy, noble gas fractionation in an oil phase migrating through groundwater will be similarly limited by the phase equilibrium model. 

A three-phase equilibrium model for partitioning of solute, E, between a bulk aqueous phase and dissolved cyclodextrin, and between the bulk aqueous phase and the stationary phase, L3, allows one to derive equations relating capacity factor, k, to the molar concentration of CD. The equation given below is similar to that derived for micellar chromatography which also assumes a three-phase model (12. 131,... 

Depending on the system at hand, the equilibrium ratio AT, may be either constant (as in Henry s law), or a function of temperature, pressure, and/or composition. In this book, the following phase equilibrium models are primarily models dealt with (1) constant relative volatilities, (2) ideal solutions using Raoult s law, and (3) nonideal solutions using a modified Raoult s law and the NRTL activity coefficient model, although other activity coefficient models are also applicable. Each of these three models is briefly discussed here. 

As discussed in Section 2.4, the residue curve equation is merely a mass balance, and can be used for any relationship between x and y. In this scenario, however, it is logical and common practice to assume that the streams emerging from a tray, or packing segment, are in equilibrium with each other. This then allows y to be defined by the appropriate thermodynamic phase equilibrium model. 

FIGURE 4.11 Superimposed RCMs for the ethyl acetate/water/ethanol stem using the NRTL and UNIQUAC phase equilibrium models at a pressure of 0.825 atm. Shaded LLE region also shown. 

FIGURE 4.12 Superimposed CPMs for the ethyl acetateAvater/ethanol system at a = 1 and X/7=Xai>=[0.8 0.1] using the NRTL and UNIQUAC phase equilibrium models at P=0.825 atm. Experimental data points have also been plotted. 

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