Solid-fluid equilibrium mixtures

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]. 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equilibrium, or the three roots (vapor, liquid, solid) characteristic of the triple point. 

Supercritical fluids are found in numerous applications thanks to their properties which vary with temperature and pressure. Supercritical fluids are put in contact with various compounds which also have physico-chemical properties dependant on temperature and pressure. Consequently, mixtures of these compounds with the supercritical solvent must be expected to behave in a complex way. For a binary mixture, for example, several types of phase equilibrium exist solid-fluid for low temperatures, solid-fluid-liquid when temperature rises, and liquid-fluid. 

In our research, we were led to characterise thermodynamically the mixtures composed of an organic compound and supercritical CO2 in a relatively wide range of temperatures, including several types of phase equilibrium. We looked for a single thermodynamic model which would be predictive (no parameters to adjust to the experimental data), valid for a wide range of temperatures and pressures, and also capable of representing solid-fluid and liquid-fluid equilibria. 

Competent design of chemical processes requires accurate knowledge of such process variables as the temperature, pressure, composition and phase of the process contents. Current predictive models for phase equilibria Involving supercritical fluids are limited due to the scarcity of data against which to test them. Phase equilibria data for solids In equilibrium with supercritical solvents are particularly sparse. The purpose of this work Is to expand the data base to facilitate the development of such models with emphasis on the melting point depressions encountered when solid mixtures are contacted with supercritical fluids. 

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. 

Type II (Solid-Fluid) System. In type II systems (when the solid and the SCF component are very dissimilar in molecular size, structure, and polarity), the S-L-V line is no longer continuous, and the critical (L = V) mixture curve also is not continuous. The branch of the three-phase S-L-V line starting with the triple point of the solid solute does not bend as much toward lower temperature with increasing pressure as it does in the case of type I system. This is because the SCF component is not very soluble in the heavy molten solute. The S-L-V line rises sharply with pressure and intersects the upper branch of the critical mixture (L = V) curve at the upper critical end point (LfCEP), and the lower temperature branch of the S-L-V line intersects the critical mixture curve at the lower critical end point (LCEP). Between the two branches of the S-L-V line there exists S-V equilibrium only (13). 

Equilibrium Treatment of Solidification. As an example of liquid-solid phase change in solid-fluid flow systems with the assumption of local thermal equilibrium imposed, consider the formulation of solid-fluid phase change (solidification/melting or sublimation/frosting) of a binary mixture. For this problem, the equilibrium condition extends to the local thermodynamic equilibrium where the local phasic temperature (thermal equilibrium), pressure (mechanical equilibrium), and chemical potential (chemical equilibrium) are assumed to be equal between the solid and the fluid phases. This is stated as... 

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. 

Fig. 2. Phase diagram describing lateral phase separations in the plane of bilayer membranes for binary mixtures of dielaidoylphosphatidylcholine (DEPC) and dipalmitoyl-phosphatidylcholine (DPPC). The two-phase region (F+S) represents an equilibrium between a homogeneous fluid solution F (La phase) and a solid solution phase S presumably having monoclinic symmetry (P(J. phase) in multilayers. This phase diagram is discussed in Refs. 19, 18, 4. The phase diagram was derived from studies of spin-label binding to the membranes.

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... 

By rapidly cooling the fluid mixture it is possible to minimise the readjustment of the equilibrium and to attain a solid condition in which tire original proportions of the mixture are approximately retained in the solid state tire allotropic change is so very slow as to allow careful and fairly prolonged examination of the mixture. It is then found that, the normal mobile liquid constituent (S ) has given rise to crystalline sulphur, soluble in carbon disulphide, whereas the dark-coloured viscous constituent (S ) has produced an amorphous solid, insoluble in this solvent4 (see also p. 10). A rough analysis of molten sulphur in... 

In a binary system more than two fluid phases are possible. For instance a mixture of pentanol and water can split into two liquid phases with a different composition a water-rich liquid phase and a pentanol-rich liquid-phase. If these two liquid phases are in equilibrium with a vapour phase we have a three-phase equilibrium. The existence of two pure solid phases is an often occuring case, but it is also possible that solid solutions or mixed crystals are formed and that solids exists in more than one crystal structure. 

There have been few studies reported in the literature in the area of multi-component adsorption and desorption rate modeling (1, 2,3., 4,5. These have generally employed simplified modeling approaches, and the model predictions have provided qualitative comparisons to the experimental data. The purpose of this study is to develop a comprehensive model for multi-component adsorption kinetics based on the following mechanistic process (1) film diffusion of each species from the fluid phase to the solid surface (2) adsorption on the surface from the solute mixture and (3) diffusion of the individual solute species into the interior of the particle. The model is general in that diffusion rates in both fluid and solid phases are considered, and no restrictions are made regarding adsorption equilibrium relationships. However, diffusional flows due to solute-solute interactions are assumed to be zero in both fluid and solid phases. 

Phenol and dodecyl benzene sulfonate are two solutes that have markedly different adsorption characteristics. The surface diffusion coefficient of phenol is about fourteen times greater than that for dodecyl benzene sulfonate. The equilibrium adsorption constants indicate that dodecyl benzene sulfonate has a much higher energy of adsorption than phenol (20,22). The adsorption rates from a mixture of these solutes can be predicted accurately, if (1) an adequate representation is obtained for the mixture equilibria, and (2) the diffusion rates in the solid and fluid phases are not affected by solute-solute interactions. 

A mathematical model has been developed to describe the kinetics of multicomponent adsorption. The model takes into account diffusional processes in both the solid and fluid phases, and nonlinear adsorption equilibrium. Comparison of model predictions with binary rate data indicates that the model predictions are in excellent for solutes with comparable diffusion rate characteristics. For solutes with markedly different diffusion rate constants, solute-solute interactions appear to affect the diffusional flows. In all cases, the total mixture concentration profiles predicted compares well with experimental data. 

If the temperature is too low, the solid cocoa butter formed this second phase, but with higher temperatures two fluid phases are in equilibrium one is rich in CO2 and the other is rich in cocoa butter. Kokot et al. [5] showed that at 313 K and 20 MPa, the mass fractions of CO2 for these two phases are 0.44 and 0.98. When working with two fluid phases and without stirring, the results of the experiments depend on the position of the outlet pipe of the first vessel top or bottom. If the fluid is extracted from the top of the vessel, the cocoa butter powder is produced from the richest phase in CO2. Conversely, it is the cocoa butter rich phase which is extracted from the bottom. If the mixture is homogenised by... 

The potential of supercritical extraction, a separation process in which a gas above its critical temperature is used as a solvent, has been widely recognized in the recent years. The first proposed applications have involved mainly compounds of low volatility, and processes that utilize supercritical fluids for the separation of solids from natural matrices (such as caffeine from coffee beans) are already in industrial operation. The use of supercritical fluids for separation of liquid mixtures, although of wider applicability, has been less well studied as the minimum number of components for any such separation is three (the solvent, and a binary mixture of components to be separated). The experimental study of phase equilibrium in ternary mixtures at high pressures is complicated and theoretical methods to correlate the observed phase behavior are lacking. 

Solubilities of meso-tetraphenylporphyrin (normal melting temperature 444°C) in pentane and in toluene have been measured at elevated temperatures and pressures. Three-phase, solid-liquid-gas equilibrium temperatures and pressures were also measured for these two binary mixtures at conditions near the critical point of the supercritical-fluid solvent. The solubility of the porphyrin in supercritical toluene is three orders of magnitude greater than that in supercritical pentane or in conventional liquid solvents at ambient temperatures and pressures. An analysis of the phase diagram for toluene-porphyrin mixtures shows that supercritical toluene is the preferred solvent for this porphyrin because (1) high solubilities are obtained at moderate pressures, and (2) the porphyrin can be easily recovered from solution by small reductions in pressure. 

However, for mixtures of TPP and toluene, a third (liquid) phase forms in the presence of the gas and the solid, at pressures well below the critical pressure of toluene. At higher pressures, gas-liquid and solid-liquid equilibria were observed, rather than gas-solid equilibrium. Thus, phase compositions for gas-liquid equilibrium were measured for this binary mixture to give TPP solubilities in each of the fluid phases. Pressures and temperatures for three-phase, solid-liquid-gas equilibrium were also measured for both binary mixtures. 

Measurements of solid solubilities in supercritical pentane were, in principle, identical to the bubble point or dew point measurements described above. The equilibrium pressure and corresponding solid solubility at a fixed temperature were determined from the measured pressure and known mixture composition in the cell when the last crystal of solid dissolved. These measurements were limited at low solubilities by the low porphyrin loadings, and at high solubilities by the dark purple color of the fluid phase which obscured observation of the solid phase. 

Adsorption is a process in which a species in a fluid (liquid or gas) mixture adheres to the surface of a solid with which the fluid is in contact. (This process should not be confused with absorption, in which a component of a gas mixture dissolves in a liquid solvent.) The solid is the adsorbent, and the species that adheres to the surface is the adsorbate. Good adsorbents such as activated carbon have extremely high specific surface areas (m surface/g solid), enabling small quantities of adsorbent to remove large quantities of adsorbate from fluid mixtures. An adsorption isotherm is a plot or equation that relates the equilibrium amount of an adsorbate held by a given mass of adsorbent to the adsorbate partial pressure or concentration in the surrounding gas at a specified temperature. 

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