Fugacity equilibrium calculations

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

In apphcatious to equilibrium calculations, the fugacity coefficients of species iu a mixture are required. Given au expression for G /RT as aetermiued from Eq. (4-158) for a coustaut-compositiou mixture, the corresponding recipe for In is found through the partial-property relation... 

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

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 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. 

For every initial H20/C0 ratio, the mixture mole fractions, hence the critical temperature and volume, are determined by the reaction extent e. The equilibrium constant is calculated at the critical temperature. The fugacities are calculated also at the critical condition for the given e. The function F, defined as... 

Use of equation 247 for actual calculations requires explicit introduction of composition variables. As in phase-equilibrium calculations, this is normally done for gas phases through the fugacity coefficient and for liquid phases through the activity coefficient. Thus, either... 

Chemical Potential. Equilibrium calculations are based on the equality of individual chemical potentials (and fugacities) between phases in contact (10). In gas—solid adsorption, the equilibrium state can be defined in terms of an adsorption potential, which is an extension of the chemical potential concept to pore-filling (physisorption) onto microporous solids (11—16). 

This assumption is a good one at present day pressures of 500-800 psia for solids however, proposed operating pressures of 2000-3000 psia even at the temperatures of concern, may require some correction to the perfect gas law. Under these conditions, one should use the Beattie-Bridgeman or van der Waal s equation for the state equation and fugacity coefficients in the equilibrium calculations. 

The advantage of using fugacity to calculate the equilibrium distribution coefficients becomes apparent when one compares the fugacity capacities of a HOP for several different phases. For example, consider a region of the unsaturated zone just below the ground surface where naphthalene is distributed between air, water, pure phase octanol, and soil at equilibrium. The fugacity capacities for these phases are repeated below in Eqs. (46)-(49) ... 

The comparison highlights the difference between the nonideal hydrogen/steam/water case and the ideal carbonmonox-ide/carbondioxide case. The difference can be detected only if fugacity-based calculations as displayed in the introduction to this book are made using the JANAF tables, (Chase etah, 1998). The equilibrium concentrations, the equilibrium constant and the Nernst potential difference V, in the hydrogen case, are a function of both pressure and temperature. declines with pressure. In the carbon monoxide perfect gas case, the same variables are a function of temperature only. The pressure coefficient is zero. 

Liquid-liquid equilibrium calculations remain a problem area especially for systems containing non-volatile species such as strong electrolytes or high polymers. These species have negligible or no fugacities, and, as a result, many flash algorithms cannot properly account for them. 

For the vapor-liquid equilibrium calculations, at the equilibrium state the fugacities for all species i must be the same in all phases, namely... 

Vapor-liquid phase equilibrium calculations have to be conducted for the estimation of solubility in the vapor phase (16,17). Alternatively, a cubic EOS can be applied for the estimation of properties of the liquid phase. The equality of fugacity in the two phases can be written as... 

This result is hardly different from that based on the ideal gas assumption. The fugacity coefficients in the equilibrium equation clearly cancel one another. This is not uncommon in reaction equilibrium calculations, as there are always products and reactions, making the ideal gas assumption far more useful than might be expected. 

The theory and conditions for phase equilibrium are well established. If more than one phase is present, then the chemical potential of a component is the same in all phases present. As chemical potential is linked functionally to the concepts of fugacity and activity, models for phase behavior prediction and correlation based on chemical potentials, fugacities, and activities have been developed. Historically, phase equilibrium calculations for hydrocarbon mixtures have been fragmented with liquid-vapor, liquid-liquid, and other phase equilibrium calculations, subject to separate and diverse treatments depending on the temperature, pressure, and component properties. Many of these methods and approaches arose to meet specific needs in the chemical process industries. Poling, Prausnitz,... 

This expression of the condition for equilibrium is used in phase equilibrium calculations more frequently than the equality of the chemical potentials because the fugacity is more closely related to observable properties than the chemical potential. 

Equilibrium compositions of liquid phases at equilibrium are calculated by equating the component fugacities, similar to vapor-liquid equilibrium calculations, described in more detail in Chapter 2. The activity coefficients may be calculated by equations presented in Section 1.3.3, in particular the UNIQUAC and NRTL equations. The composition dependence of these equations is developed to the point where the same equation with the same constants can predict activity coefficients over wide ranges of composition, thus allowing it to predict two immiscible liquid phases at equilibrium. 

The principles and algorithms for calculating fluid-phase equilibria are discussed in many textbooks [36 0]. Here, we focus on methods and data requirements for calculating the component fugacities in a phase as a function of temperature, pressure, and composition this is the key element in all phase-equilibrium calculations. 

An equation of state, applicable to all fluid phases, is paitiodariy useful for phase-equilibrium calculations where a liquid phase and a vapor phase coexist at high pressures. At such conditions, conventional activity coefficients are not useful because, with rare exceptions, at least one of the mixture s components is supercritical that is, (he system temperature is above (hat component s critical temperature. In that event, one must employ special standard states for the activity coefficients of the supercritical components (see Section 1.5-2). That complication is avoided when ail fugacities are calculated front en equation of state. 

Other approaches to the computation of solid-liquid equilibria are shown in Table 11.2-3. The Soave-Redlich-Kwong equation of state evaluates fugacities to calculate solid-liquid equilibria,7 while Wenzel and Schmidt developed a modified van der Waals equation of state forthe representation ofphase equilibria. The Wenzel-Scbmidt approach generates fugacities, from which the authors developed a trial-and-error approach to compute solid-liquid equilibrium. Unno et a .9 recently presented a simplification of the solution of groups model (ASOG) that allows prediction of solution equilibrium from limited vapor-liquid equilibrium data. 

The fugacity function has been introduced because its relation to the Gibbs energy makes it useful in phase equilibrium calculations. The present criterion for equilibrium between two phases, (Eq. 7.1-9c) is C = C , with the restriction that the temperature and pressure be constant and equal in the two phases. Using this equality and the definition of the fugacity (Eq. 7.4-6) gives... 

Since the fugacity function is of central importance in phase equilibrium calculations, we consider here the computation of the fugacity for pure gases, liquids, and solids. 

Therefore, at equilibrium, the fugacity of each species must be the same in the two phases. Since this result follows directly from Eq. 8.7-10, it may be substituted for it. Furthermore, since we can make estimates for the fugacity of a species in a mixture in a more direct fashion than for partial molar Gibbs energies, it is more convenient to use Eq. 9.2-11 as the basis for phase equilibrium calculations. 

Note that the fugacity of the pure liquid, P), in Eq. 9.3-11 can be found from the methods of Sec. 7.4b.] As will be seen in Chapters 10 to 12, the calculation of the activity coefficient for each species in a mixture is an important step in many phase equilibrium calculations. Therefore, much of this chapter deals with models (equations) for G and activity coefficients. 

The objective of this chapter has been to develop methods of estimating species fugac-ities in mixtures. These methods are very important in phase equilibrium calculations, as will be seen in the following chapters. Because of the variety of methods discussed, there may be some confusion as to which fugacity estimation technique applies in a given situation. The comments that follow may be helpful in choosing among the three main methods discussed in this chapter ... 

The starting point for any phase equilibrium calculation is, of course, the equality of fugacities of each species in each phase, that is. 

We consider only one additional type of phase equilibrium calculation here, the isothermal flash calculation discussed in Sec. 10.1. In this calculation one needs to satisfy the equality of species fugacities relation (Eq.-10.3-1) as in other phase equilibrium calculations and also the mass balances (based on 1 mole of feed of mole fractions Zi f) discussed earlier,... 

In this chapter we consider several other types of phase equilibria, mostly involving a fluid and a solid. This includes the solubility of a solid in a liquid, gas, and a supercritical fluid the partitioning of a solid (or a liquid) between two partially soluble liquids the freezing point of a solid from a liquid mixture and the behavior of solid mixtures. Also considered is the environmental problem of how a chemical partitions between different parts of the environment. Although these areas of application appear to be quite different, they are connected by the same starting point as for all phase equilibrium calculations, which is the equality of fugacities of each species in each phase ... 

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

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