Phase equilibrium calculations, solid-fluid

Pure solid + fluid phase equilibrium calculations are challenging but can, in principle, be modeled if the triple point of the pure solid and the enthalpy of fusion are known, the physical state of the solid does not change with temperature and pressure, and a chemical potential model (or equivalent), with known coefficients, for solid constituents is available. These conditions are rarely met even for simple mixtures and it is difficult to generalize multiphase behavior prediction results involving even well-defined solids. The presence of polymorphs, solid-solid transitions, and solid compounds provide additional modeling challenges, for example, ice, gas hydrates, and solid hydrocarbons all have multiple forms. 

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 ... 

Solubility calculations are merely phase-equilibrium calculations applied to supercritical gases in liquids, solids in liquids, and solutes in near-critical fluids. The last application has drawn substantial attention, for near-critical extraction processes are being applied, not only in the chemical and energy industries, but also in food processing, purification of biological products, and clean-up of hazardous wastes. 

Nichita et al calculated the wax precipitation from hydrocarbon mixtures using a cubic equation of state (see Chapter 4) to describe the vapour and the liquid lumping into pseudo-components to simplify the phase equilibrium calculation. However, the information lost in this procedure effected the location of the predicted solid phase transition. This issue was avoided by an inverse lumping procedure, in which the equilibrium constants of the original system are related to some quantities evaluated from lumped fluid flash results. The method was tested for two synthetic and one real mixture yielding good agreement between calculated and experimental results. 

Advances continue in the treatment of detonation mixtures that include explicit polar and ionic contributions. The new formalism places on a solid footing the modeling of polar species, opens the possibility of realistic multiple fluid phase chemical equilibrium calculations (polar—nonpolar phase segregation), extends the validity domain of the EXP6 library,40 and opens the possibility of applications in a wider regime of pressures and temperatures. 

Using this equation, the ratio of the fluid volume to the solid mass needed to achieve a desired fluid-phase equilibrium concentration (X) can be calculated. We can achieve a lower liquid-phase concentration level by using a lower Vim ratio, using, for example, a higher amount of solid. Thus, equilibrium calculations result in the maximum V/m ratio that should be used to achieve the desired equilibrium (final) concentration. But, how much time do we need to achieve our goal in a batch reactor This is a question to be answered by kinetics. 

Fig. 6. Schematic model of two-phase system, with solid domain S in equilibrium with fluid domain F. The relative proportions, and compositions of F and S are determined from, for example, the phase diagram in Fig. 2. The 13C nuclear resonance spectra of lipids in the solid phase S are broader than the nuclear resonance spectra of lipids in the fluid phase F the spectra of lipids in the fluid phase F can be broadened if they diffuse to, and stick to, the solid phase. For a detailed theoretical calculation of line widths expected for this geometry, see Ref. 4. The bilayer length is L, and the domain boundary is b - b. [Reprinted with permission from P. Brulet and H. M. McConnell, J. Am. Chem. Soc., 98, 1314 (1977). Copyright by American Chemical Society.]...

Let us assume some small number n of lipid molecules can form a relatively stable solid phase cluster when the temperature and composition of the lipid mixture is such that, according to the phase diagram, solid phase can exist in equilibrium with the fluid phase. (For example, we later assume that n 10.) Let us further assume that (1) the temperature and composition of the lipid mixture is such that X is small, X 1, and (2) all the solid phase present is in the form of clusters of n molecules each. If the clusters are randomly distributed in the plane of the membrane, then each cluster will be surrounded by a number of fluid molecules of the order of magnitude of N n/X. The area occupied by the surrounding fluid phase molecules is then NA0 where, A0 60A2. Let us now calculate lower limit on X, Xmin, such that each molecule in... 

Molecular dynamics calculations of Hoover and Ree (25) have indicated that a fluid-solid transition occurs in a system of hard spheres even in the absence of attractive forces. The fluid exists for particle volume fractions up to a value rj = 0.49 and at this point, a solid phase with ij = 0.55 is predicted to coexist in equilibrium with the fluid phase. When the particle volume fraction lies in the range 0.55 < jj < 0.74, the solid phase is stable. The upper limit for ij corresponds to the density at closest packing for a face-centered-cubic (fee) arrangement of the particles. 

Clarke, M.A. Bishnoi, P.R. Development of a new equation of state for mixed salt and mixed solvent systems, and application to vapour liquid equilibrium and solid (hydrate) vapour liquid equilibrium calculations. Fluid Phase Equilibria 2004, 220, 21-35. 

In pursuing an accurate thermodynamic description of the three-phase, three-component system, the phase equilibrium compositions can be calculated after pressure and temperature have been fixed, since it is known from the Gibbs phase rule that there are only 2 degrees of freedom. There are five unknown compositions, assuming that the solid is crystalline and pure and that its solubility in the vapor/fluid phase is negligible. Two of these unknown mole fractions are eliminated by the constraints that the mole fractions in each phase sum up to unity. To find these three unknown mole fractions, namely, xi, X3, and y2, only three equilibrium relations are required. 

FIGURE 16.7 Average deviations in solid-liquid equilibrium calculations, as a function of the solute carbon number, for alkane systems using various FV models. E-FV is the Entropic-FV model. F-FVl.l is the Flory-FV model using c= 1.1. GCFLORY EoS is the GC-Flory equation of state. (From Coutinho, J.A.P. et al.. Fluid Phase Equilibria, 103, 23, 1995. With permission.)... 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

The fugacity function is central to the calculation of phase equilibrium. This should be apparent from the earlier discussion of this chapter and from the calculations of Sec. 7.5, which established that once we had the pure fluid fugacity, phase behavior in a pure fluid could be predicted. Consequently, for the remainder of this chapter we will be concerned with estimating the fugacity of species in gaseous, liquid, and solid mixtures. 

From the outset the relationships between the fugacity and the state variables are highly nonlinear. To determine the composition of each phase for a SLV system such that Equations 1 and 2 are satisfied requires that an iterative method be used. Because of the constraints imposed on the system by the phase rule somewhat different procedures were used in this study to compute the SLV equilibrium condition for multicomponent systems and for binary systems, respectively. Both procedures calculate the fluid-phase compositions of a given mixture at the incipient solid-formation condition. 

Figure 4. The equilibrium fraction of monomers in the coexisting liquid-vapor phase of an associating fluid with one square-well bonding site. The liquid-phase fractions of monomers are on the left-hand side of the figure. The circles are data from RCMC-Gibbs ensemble simulations, and the lines are calculations from three different implementations of a theory for associating fluids. The solid line uses exact values of the reference fluid radial distribution function the dashed and long dashed-short dashed lines use the WCA and modified WCA approximations to the radial distribution function, respectively. (Reprinted with permission from Muller et al. [43]. Copyright 1995 American Institute of Physics.)...

The fugacity of the pure solid compound 2 can be described by the sublimation pressure, the fugacity coefficient in the saturation state and the Poyntin.g factor, so that the following phase equilibrium relation is obtained for the calculation of the solubility of the solid 2 in the supercritical fluid ... 

A semi-grand canonical treatment for the phase behaviour of colloidal spheres plus non-adsorbing polymers was proposed by Lekkerkerker [141], who developed free volume theory (also called osmotic equilibrium theory ), see Chap. 3. The main difference with TPT [115] is that free volume theory (FVT) accounts for polymer partitioning between the phases and corrects for multiple overlap of depletion layers, hence avoids the assumption of pair-wise additivity which becomes inaccurate for relatively thick depletion layers. These effects are incorporated through scaled particle theory (see for instance [136] and references therein). The resulting free volume theory (FVT) phase diagrams calculated by Lekkerkerker et al. [142] revealed that for <0.3 coexisting fluid-solid phases are predicted, whereas at low colloid volume fractions a gas-hquid coexistence is found for q > 0.3, as was predicted by TPT. 

Fig. 2.37. Phase diagram for Ca0-Na20 Si02-(Al203)-H20 system in equilibrium with quartz at 400°C and 400 bars. Plagioclase solid solution can be represented by the albite and anorthite fields, whereas epidote is represented by clinozoisite. Note that the clinozoisite field is adjacent to the anorthite field, suggesting that fluids with high Ca/(H+) might equilibrate with excess anorthite by replacing it with epidote. The location of the albite-anorthite-epidote equilibrium point is a function of epidote and plagioclase composition and depends on the model used for calculation of the thermodynamic properties of aqueous cations (Berndt et al., 1989).

Detailed speciation calculations for the fluid coupled with mineralogical investigation of the solid paragenesis generally allow a sufficiently precise estimate of the T of equilibrium. As figure 8.30A shows, the chemistry of the fluid is buffered by wall-rock minerals, so that the saturation curves of other phases not pertaining to the system of interest are scattered (figure 8.28B). 

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