Calculation of Phase Equilibrium

Design of extraction processes and equipment is based on mass transfer and thermodynamic data. Among such thermodynamic data, phase equilibrium data for mixtures, that is, the distribution of components between different phases, are among the most important. Equations for the calculations of phase equilibria can be used in process simulation programs like PROCESS and ASPEN. 

The adjustable interaction constants Q can be evaluated from the experimental data for three-component systems these constants can then be employed for concentration of temperature interpolations and also for calculation of phase equilibria in multicomponent systems. Moreover, the constants Q usually depend very little on temperature, as the relative molalities, related to the solubility of the substance in the pure solvent, are employed hence calculations of other isotherms can be carried out easily. 

Another area which saw increasing attention was semiconductor materials. Ishida et al. (1989) constructed a database which combined the calculation of phase equilibria in III-V compounds with the calculation of band gaps in the same systems. This was to be the forerunner of future attempts to expand the database... 

This section will give examples of how CALPHAD calculations have been used for materials which are in practical use and is concerned with calculations of critical temperatures and the amoimt and composition of phases in duplex and multi-phase types of alloy. These cases provide an excellent opportunity to compare predicted calculations of phase equilibria against an extensive literature of experimental measurements. This can be used to show that the CALPHAD route provides results whose accuracy lies close to what would be expected from experimental measurements. The ability to statistically validate databases is a key factor in seeing the CALPHAD methodology become increasingly used in practical applications. 

Saunders, N. and Sundman, B. (1996) Fe-DATA, a database for calculation of phase equilibria in Fe-based alloys. 

The last term is linear in px and can be disregarded for the calculation of phase equilibria. This then gives the following simple result for the moment free energy ... 

Weiss J, Lukas HL, Petzow G (1983) Calculation of Phase Equilibria in Systems based on Si3N4. In Riley FL (ed) Progress in Nitrogen Ceramics. NATO ASI Series E. 65 77... 

Calculation of phase equilibria from the chemical potentials... 

Molecular simulation methods for calculation of phase equilibria... 

The binary mixture parameter has been fitted to VLE data for 29 systems its values are in Table 1. It should be noted that is independent of temperature and always very close to unity. The calculation of phase equilibria was performed by means of the algorithm of Deiters [8, 9], The reproduction of VLE data and the predictions of LLE data, of excess volumes, of virial coefficients are very good for all 29 binary mixtures investigated [3]. 

Wang,S.H., W. B. Whiting A Comparison of Distrbution Functions for Calculation of Phase Equilibria of Continuous Mixtures Chem. Engng. Comm. 71,127-143(1988). 

In calculations of phase equilibria that appear to be H2O undersaturated, it is commonly assumed that a fluid phase is present and H2O activity is lowered by CO2 in the fluid. The latter is not necessarily true, and this section examines the differences between the limited availability... 

The chemical potential provides the fundamental criteria for determining phase equilibria. Like many thermodynamic functions, there is no absolute value for chemical potential. The Gibbs free energy function is related to both the enthalpy and entropy for which there is no absolute value. Moreover, there are some other undesirable properties of the chemical potential that make it less than suitable for practical calculations of phase equilibria. Thus, G.N. Lewis introduced the concept of fugacity, which can be related to the chemical potential and has a relationship closer to real world intensive properties. With Lewis s definition, there still remains the problem of absolute value for the function. Thus,... 

Fig. 10. Schematic of the Gibbs ensemble Monte Carlo simulation method for calculation of phase equilibria of confined fluids [22].

Pfohl, O., Dohrn, R., and Riebesell, C., Measurement and calculation of phase equilibria in the system n-pentane + poly(dimethylsiloxane) at 35°C and at 150°C, Fluid Phase Equilibria, submitted for publication, 2001. 

Peters, C. J., J. L. de Roo, and R. N. Lichtenthaler. 1987. Measurements and calculations of phase equilibria of binary mixtures of ethane + eicosane. Part I Vapour + liquid equilibria. J. Fluid Phase Equil. 34 287-308. 

Numerical calculations of phase equilibria require thermodynamic data or cotestations of data. For pure componeats, the requisite data may include saturation pressures (or temperatures), hem capacities, latent hems, and volumetric properties. For mixtures, one requires a PVTx equation of state (for determitiation of 4>j), and/or en expression for the molar excess Gibbs eenrgy g (for deiermiention of y,). We have disoussed in Sections 1.3 and 1.4 the correlating capabilities of selected equations or mite and expressions for gR, and the behavior of the fogacily cnefficients and activity coefficients derived from them. 

The Soave-Redlich-Kwong equation is rapidly gaining acceptance by the hydrocarbon processing industry. Further developments, such as that of Peng and Robinson,are likely to improve predictions of liquid density and phase equilibria in the critical region. In general however, use of such equations appears to be limited to relatively small, nonpolar molecules. Calculations of phase equilibria with the S-R-K equations require initial estimates of the phase compositions. 

In the last few years many contributions dealing with the calculation of phase equilibria have been published. It would lead us too far to mention all of these, particularly since this subject is exhaustively treated by Schuberth [17] and Hala et al. [78]. Their books include extensive bibliographies. Of particular interest in this context is a remark made by Spath (cf. chap. 3, ref. [17]) about a simplification in representing and evaluating vapour-liquid equihbria. 

A Fortran programme has been elaborated by Williams and Henley [89] for tlie computation of multicomponent vapour-liquid equilibria. To take into account real behaviours a number of subprogrammes are available which enable fugacities to be calculated by means of the virial equation, the Redlich-Kwong relation or according to Chao-Seader. Activity coefficients may be considered following Wilson, van baar or Hildebrand. The state of the art of precalculating vapour-liquid equilibria in multicomponent mixtures was surveyed by Hala [89a]. Lu and Polak [89b] discussed the special requirements for the calculation of phase equilibria at low temperatures (20 K to room temperature). 

Numerous thermodynamic calculations of phase equilibria in the Si-C-N system have been published but only few experimental investigations are documented. Calculated isothermal sections [117, 234-237], isopleths [117, 234], different types of potential phase diagrams [234,235,238-244] and phase fraction diagrams [234, 239] were presented. Additional information is provided by [245]. No or very low solid solubilities between SiC and Si3N4 could be detected by X-ray diffraction up to 2500 K [246, 247]. Also the nitrogen solubility in SiC is low. For more experimental information see [248, 249]. 

In conclusion, it is possible to concentrate the flavor fraction of cold-pressed citrus oils with supercritical fluid technology by selectively extracting the terpenes from the oil. During continuous extractions, the amount of extract followed a linear trend with time over the first 5 hours of extraction and it increased five times when the flow rate was increased ten times. Since the design of supercritical fluid extraction and solvent regeneration processes for the concentration of citrus oils require accurate calculation of phase equilibria, more research must be done to determine the equilibrium solubility data, the thermod3mamic model to represent the system, and the economic feasibility of the process. 

In this chapter we describe the kinds of phase behavior that are commonly observed in pure fluids, binary mixtures, and some ternary mixtures. The descriptions typically take the form of phase diagrams, and we show how studies of phase behavior can be made systematic by identifying classes of diagrams. Since we are interested in describing what is actually seen, the mixture diagrams presented in this chapter are plotted in terms of measurables usually temperature, pressure, composition, or a subset of those. Calculations of phase equilibria necessarily involves conceptuals, and such calculations are discussed in Chapter 10. Here we only describe phenomena. 

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