Phase equilibria fluid-solid

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. 

Phase Equilibrium in Solid-Liquid-Supercritical Fluid Systems... 

This article is a review of some of the things that have been learned about the phase equilibrium between solids and liquids (SLE) or solids and fluids... 

Adsorption is a dynamic process in which some adsorbate molecules are transferring from the fluid phase onto the solid surface, while others are releasing from the surface back into the fluid. When the rate of these two processes becomes equal, adsorption equilibrium has been established. The equilibrium relationship between a speeific adsorbate and adsorbent is usually defined in terms of an adsorption isotherm, which expresses the amount of adsorbate adsorbed as a fimetion of the gas phase coneentration, at a eonstant temperature. 

In model equations, Uf denotes the linear velocity in the positive direction of z, z is the distance in flow direction with total length zr, C is concentration of fuel, s represents the void volume per unit volume of canister, and t is time. In addition to that, A, is the overall mass transfer coefficient, a, denotes the interfacial area for mass transfer ifom the fluid to the solid phase, ah denotes the interfacial area for heat transfer, p is density of each phase, Cp is heat capacity for a unit mass, hs is heat transfer coefficient, T is temperature, P is pressure, and AHi represents heat of adsorption. The subscript d refers bulk phase, s is solid phase of adsorbent, i is the component index. The superscript represents the equilibrium concentration. 

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

First-order adsorption kinetics model A simple first-order reaction model is based on a reversible reaction with equilibrium state being established between two phases (A— fluid, B—solid) ... 

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. 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.

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.]...

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. 

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. 

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. 

Weidner E, Wiesmet V, Knez Z et al (1997) Phase equilibrium (solid-liquid-gas) in polyethylene glycol-carbon dioxide systems. J Supercrit Fluids 10(3) 139-147... 

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. 

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. 

For the design of a process for formation of solid particles using supercritical fluids, data on solid - liquid and vapour - liquid phase equilibrium are essential. PGSS process is only possible for systems where enough gas is solubilized in the liquid. 

Several authors [3-9] studied the solubility of polymers in supercritical fluids due to research on fractionation of polymers. For solubility of SCF in polymers only limited number of experimental data are available till now [e.g. 4,5,10-12], Few data (for PEG S with molar mass up to 1000 g/mol) are available on the vapour-liquid phase equilibrium PEG -CO2 [13]. No data can be found on phase equilibrium solid-liquid for the binary PEG S -CO2. Experimental equipment and procedure for determination of phase equilibrium (vapour -liquid and solid -liquid) in the binary system PEG s -C02 are presented in [14]. It was found that the solubility of C02 in PEG is practically independent from the molecular mass of PEG and is influenced only by pressure and temperature of the system. 

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. 

In all cases in the thermodynamic analysis we considered partial pressures of H2O, CO2, and other volatiles to be independent variables, if they were not related to one another by reactions. In addition the general conclusion was drawn that in thermodynamic calculations of metamorphic reactions it is impossible to assume different isotropic pressures on the solid phases and fluid. Lithostatic (nonhydrostatic) pressure or loading pressure has practically no effect on equilibrium in elastic deformation of rocks. Isotropic pressure equal to fluid pressure in the case of an excess of volatiles should be considered an equilibrium factor in actual natural processes. 

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