Multiphase equilibrium

The behavior of multiphase systems is described by the use of phase diagrams. The following is a review of important terminology. 

Phase diagram - a map that indicates which phases are stable as a function of the relevant thermodynamic variables (e.g. T, P, composition, etc.) for a given system. 

Phase - a physically distinct part of a system whose intensive properties are homogeneous and which has definite bounding surfaces. 

Homogeneous system - a system that consists of one phase, e.g. water in a glass. 

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Gibb s Phase Rule. The phase rule derived by W. J. Gibbs applies to multiphase equilibria in multicomponent systems, in the absence of chemical reactions. It is written as... 

Lopes, J. N. C. Tildesley, D. J., Multiphase equilibria using the Gibbs ensemble Monte Carlo method, Mol. Phys. 1997, 92, 187-196... 

A little more elaborate was the multiphase equilibria approach, adopted earlier in Ref.64 Equations (56) and (73) are still used, but Eq. (1) is not because direct observation proved that the drop (e.gof liquid tin) raised a ridge on the solid (e.g., molybdenum) so that the vertical cross section of the system looked like Fig. 24, see also Fig. 20. Let the angles formed by the tension y and the hypothetical tensions 7S and 7si with the horizontal plane be, respectively, 0, a, and 5. As a correction, symbol 7SV was substituted for 7S because the energy of the vapor — solid interface presumably was influenced by the vapor of the liquid (i.e., molten tin). Of course, the y of tin melts presumably also was altered by dissolution of molybdenum, but this possibility was disregarded. For equilibrium of all horizontal forces, the equality... 

For a multiphase equilibria containing a number of components distributed among them, the chemical potential of any component is the same in all the phases. 

Ballard, A.L., A Non-Ideal Hydrate Solid Solution Model for a Multiphase Equilibria Program, Ph.D. Thesis, Colorado School of Mines, Golden, CO (2002). 

Asher, GB., Development of a Computerized Thermal Conductivity Measurement System Utilizing the Transient Needle Probe Technique An Application to Hydrates in Porous Media, Dissertation, Colorado School of Mines, Golden, CO (1987). Ashworth, T., Johnson, L.R., Lai, L.P., High Temperatures-HighPressures, 17,413 (1985). Avlonitis, D., Multiphase Equilibria in Oil-Water Hydrate Forming Systems, M.Sc. Thesis, Heriot-Watt University, Edinburgh, Scotland (1988). 

High Pressure Multiphase Equilibria in the Ternary System Carbon Dioxide-Water-1-Propanol... 

The present paper gives an overview of results on high-pressure phase equilibria in the ternary system carbon dioxide-water-1-propanol, which has been investigated at temperatures between 288 and 333 K and pressures up to 16 MPa. Furthermore, pressure-temperature data on critical lines, which bound the region where multiphase equilibria are oberserved were taken. This study continues the series of previous investigations on ternary systems with the polar solvents acetone [2], isopropanol [3] and propionic add [4], A classification of the different types of phase behaviour and thermodynamic methods to model the complex phase behaviour with cubic equations of state are discussed. 

The weathering of minerals can be understood as a continuous dissolution-precipitation process involving them. The process can be very complicated, leading to multiphase equilibria, in which more than one solid phase, solution, and gas phase may be present. For example, primary silicates transform to secondary silicate minerals via such weathering reactions (Stumm and Wollast, 1990), as in the formation of kaolinite (AljSijC OH) from anorthite (CaAl2Si208) ... 

Multiphase Equilibria for Water—Carbon Dioxide—2-Propanol Mixtures at Elevated Pressures... 

Miller, M.M. Luks, K.D. Observations on the multiphase equilibria behavior of C02-rich and... 

Peters, C.J. Multiphase equilibria in near-critical solvent. In Supercritical Fluids Fundamentals for Application, Kiran, E., Levelt Sengers, J.M.H., Eds. NATO ASI Series, Series E Applied Sciences Vol. 273 Kluwer Academic Publishers, 1994 117-145. 

Peters, C. J., Florusse, L. J., Hahre, S., and de Swaan Arons, J., 1995. Fluid multiphase equilibria and critical phenomena in binary and ternary mixtures of carbon dioxide, certain -alkanols and tetradecane. Fluid Phase Eq., 110 157-173. 

Hong SP, Luks KD. Multiphase equilibria of the mixture carbon dioxide-naphthalene-toluene. Fluid Phase Equilibria 1992 74 133. 

The SAFT equation of state was proposed by Radosz, Gubbins, Jackson, and Chapman and is a model derived based on the perturbation theory of Weirtheim. SAFT is a noncubic equation with separate terms for the various effects (dispersion, polar, chain, hydrogen bonding). SAFT has already found extensive application in both polymer and oil industry, where different capabilities of the model have been exploited. In the oil industry, it is used for describing the complex multiphase equilibria of hydrogen bonding multicomponent systems, e.g., water-oil-alcohols (glycols). Several recent reviews of the SAFT equation of state are available, all of which present results for polymer solutions. 

Luks, K. D., and J. P. Kohn. 1984. The topography of multiphase equilibria behavior What can it tell the design engineer. Proc. 63rd Annu. Convention, Gas Processors Assoc., 181. 

White, G. L., and C. T. Lira. 1991a. High pressure multiphase equilibria of CO2 with polycyclic aromatic hydrocarbons. Paper presented at the 2d International Symposium on Supercritical Fluids. Boston, MA, May. 

The present contributions show how carefully chosen thermochemical data, in conjunction with the computer program ChemSage, can assist in the understanding of processes in steel metallurgy. Furthermore, we have shown that the same software can also be employed in combustion calculations or for the understanding of phase relations in hard-metal alloys. In all cases rapid and reliable answers for multicomponent, multiphase equilibria are obtained. 

Ultimately, we want to develop a computational procedure for determining the observability of a state proposed for a binary fluid. The motivation is that we want to avoid trying to solve phase-equilibrium problems that do not exist. Therefore we first test for observability, and if multiphase situations are observable, then we solve for phase compositions, if they are required. In this section we consider situations in which the proposed state is identified by specifying values for T, P, and Xp Such a state could be in any one of three observable conditions (a) a stable single phase, (b) a stable multiphase equilibrium, or (c) a metastable single phase. Some metastable phases can only relax to a stable single phase, but other metastable phases can split into multiple phases. Multiphase equilibria in binaries are predominantly two-phase situations, so we will restrict our attention to those possibilities here however, three and four-phase binaries are also possible. 

C. J. Peters, "Multiphase Equilibria in Near-Critical Solvents," in Supercritical Fluids Fundamentals for Applications, E. Kiran and J. M. H. Levelt Sengers (eds.), NATO ASI Series E, vol. 273, Kluwer Academic Publishers, Dordrecht, 1994, p. 117. 

W. D. Seider and S. Widagdo, "Multiphase Equilibria of Reactive Systems," Fluid Phase Equil, 123, 283 (1996). 

The various identified phases—Li, L3, L4, and —are separated by several complex multiphase equilibria including several three-phase regions. The tie-lines involving the L4 phase have not been measured precisely. However, X-ray data obtained for octanol suggest that the tie-lines be drawn around the L4 phase as shown in Fig. 7. Note that in region dj between the Li and L4 phases, the macroscopic phase separation between an isotropic phase and the lamellar phase occurs after a few weeks. 

For the copolymer B, Eq. (35b) is valid for all species identified by segment number rB and chemical composition Yb and considered to be continuously distributed within the total domains of these variables. Multiphase equilibria, though not considered here, may be treated analogously by extending Eqs. (35) to more phases and calculating the properties of more than one unknown phase by additional equations of the type developed below for the equiUbrium of two phases. In many cases, multiphase equilibria are obtained as multiple solutions of Eqs. (35). 

Two copolymers are named B and C (to keep A for the solvents. Sect. 3). The phase equilibrium conditions for two coexisting phases (multiphase equilibria are again neglected) in continuous thermodynamics [35,40] read... 

Peters, CJ., Florusse, LJ. and HShre, S. (1996) Measurements on fluid multiphase equilibria in ternary mixtures of carbon dioxide, tetradecane and certain n-alkanols. The Journal of Supercritical Fluids 9, 135-140. 

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