Fluid-Solid Equilibrium

Crystallization from the gas phase, and in particular fi om a gas phase above the critical temperature and pressure of the solution, has attracted much interest in the literature for making particles of uniform size via a process known as supercritical 

The thermodynamic relationships for a pure solid solute A (assuming that the gas does not dissolve in the solid) in equilibrium with a supercritical solution is given by 

The fugacity coefficient 0 is generally calculated from an equation of state. However, many equations of state require a knowledge of the critical parameters of the solute, which may not always be available. Nevertheless, solubilities can be correlated, and sometimes extrapolated, using this approach. The addition of more solutes poses few problems from a thermodynamic viewpoint, as long as the appropriate solid-state fugacity is used in the calculations. This type of approach may also be used to study 

---

Solid-Fluid Equilibria The phase diagrams of binai y mixtures in which the heavier component (tne solute) is normally a solid at the critical temperature of the light component (the solvent) include solid-liquid-vapor (SLV) cui ves which may or may not intersect the LV critical cui ve. The solubility of the solid is vei y sensitive to pressure and temperature in compressible regions where the solvent s density and solubility parameter are highly variable. In contrast, plots of the log of the solubility versus density at constant temperature exhibit fairly simple linear behavior. 

Crystalline sodium metasilicates, manufacture of, 22 464 Crystalline solids, solid-fluid equilibria for, 24 10-11... 

Solid-fluid equilibria are important for supercritical fluid processing design (Bush and Eckert, 1998). 

One of the simplest cases of phase behavior modeling is that of solid—fluid equilibria for crystalline solids, in which the solubility of the fluid in the solid phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component iyf y are equal for all phases at equilibrium under constant temperature and pressure (51). The solid-phase fugacity, ff, can be represented by the following expression at temperature T ... 

The aim of this paper is to explore the feasibility of the three nixing rules Wong-Sandler (4), Heidemann-Kokal (5) and MF1V2 (Dahl-Michelsen (6)) to the solid-fluid equilibria occurring in the binary systems composed of a organic solid and four supercritical fluids, with the calculations of solubility and density of the mixture. 

A study of the representation of solid-fluid equilibria with the "Excess Function-Equation of State" model. The results obtained with [5] and [6] show that this model is as efficient as the HSVDW [7], AVDW-DDLC [8], or KBST [9] models. 

The interaction parameter Eg was first adjusted to the experimental data of the liquid-fluid and solid-fluid equilibria. The representation of both types of equilibria was very satisfactory, which proves that the "Excess Function-Equation of State" model with an adjusted parameter is well adapted to both cases ([4-6], [13]). 

The interaction parameters of the form (10) are appropriate for the liquid-fluid equilibria for a large number of compounds [4], A study of the "continuity" of the model in the field of solid-fluid equilibria is a pre-requisite for the development of a single all-purpose group -contribution method. 

The temperature function appearing in the relation (10) is indeed specific for a couple of given molecules i and j. We analysed the correlation existing between lnE12 and ln(T°/T) for hydrocarbon-CC>2 couples with experimentally established liquid-fluid and solid-fluid equilibria (Figure 1). The values of En are those adjusted to experimental data to be found in the literature. 

The interaction parameters adjusted to the experimental data of liquid-fluid and solid-fluid equilibria seem to vary continuously with the temperature. A common definition of the groups... 

In relations (11) and (12), i is a parameter which indicates the length of the chain of normal alkanes it is equal to the number of carbon atoms in the molecule, a and Pi are constants, C2, C3 and Cp are the site fractions occupied in the molecule by the condensed aromatic carbon atoms common to two cycles, three cycles and condensed atoms in angular position. The Cas parameter represents the fraction of the sites for the substituted aromatic carbons belonging to condensed cycles, which must be taken into account for the substituted derivatives of naphthalene. Table 1 shows the values of the parameters adjusted to the experimental data of the liquid-fluid and solid-fluid equilibria as a whole. 

The calculation of the liquid-fluid equilibria with the group contribution method has been presented elsewhere [4], The matrix of the parameters of group interaction (Table 1) contains values readjusted relative to the matrix obtained considering only liquid-fluid equilibria [4], These parameters are Ai5, A35, A45 and A59. The introduction of supplementary increments for the form of the molecules (a3 and P3.5) enables good results to be obtained for the calculation of solid-fluid equilibria and does not essentially modify the representation of liquid-fluid equilibria. The average relative deviation 5r(x) for the experimental data as a whole is 16 7% for the group contribution method and 13.3% for the model with E12 adjusted. The experimental data concern 40 isotherms (P,x) for 11 binary mixtures of solid aromatic hydrocarbon with supercritical C02. 

Kurnik, R. T. 1981. Supercritical fluid extraction A study of binary and multicomponent solid-fluid equilibria. Ph.D. diss., Massachusetts Institute of Technology, Cambridge, MA. 

Adsorption is a separation process in which certain components of a fluid phase are transferred to the surface of a solid adsorbent. Usually the small particles of adsorbent are held in a fixed bed, and fluid is passed continuously through the bed until the solid is nearly saturated and the desired separation can no longer be achieved. The flow is then switched to a second bed until the saturated adsorbent can be replaced or regenerated. Ion exhange is another process that is usually carried out in this semibatch fashion in a fixed bed. Water that is to be softened or deionized is passed over beads of ion-exchange resin in a column until the resin becomes nearly saturated. The removal of trace impurities by reaction with solids can also be carried out in fixed beds, and the removal of H2S from synthesis gas with ZnO pellets is a well-known example. For all these processes, the performance depends on solid-fluid equilibria and on mass-transfer rates. In this chapter the emphasis is on adsorption, but the general methods of analysis and design are applicable to other fixed-bed processes. 

The approach now summarized to represent solid-fluid equilibria can be profitably used in the development of both RESS and CSS processes. 

The criteria for equilibria involving solid phases are exactly those given in 7.3.5 for any phase-equilibrium situation phases in equilibrium have the same temperatures, pressures, and fugacities. Moreover, pure-component solid-fluid equilibria obey the Clapeyron equation (8.2.27). This means the latent heat of melting is proportional to the slope of the melting curve on a PT diagram and the latent heat of sublimation is proportional to the slope of the sublimation curve. In the case of solid-gas equilibria, the Clausius-Clapeyron equation (8.2.30) often provides a reliable relation between temperature and sublimation pressures, analogous to that for vapor-liquid equilibria. 

Besides solid-fluid equilibria, some pure materials can exist in more than one stable solid structure, giving rise to solid-solid equilibria. Examples include equilibria between the fee and bcc forms of iron, equilibria between rhombic and monoclinic sulfur, and equilibria among the many different phases of ice. Such solid-solid phase transitions are accompanied by a volume change and a latent heat, and these two quantities are related through the Clapeyron equation (8.2.27). When a pure material can undergo solid-solid phase transitions, then the substance usually exhibits multiple triple points. Besides the usual solid-vapor-liquid point, the pure substance might also exist in solid-solid-liquid or solid-solid-solid equilibria. Several such triple points occur in water, caused by equilibria involving various forms of ice [13]. 

We have shown how models for volumetric equations of state can be used with stability criteria to predict vapor-liquid phase separations. However, not all phase equilibria are conveniently described by volumetric equations of state for example, liquid-liquid, solid-solid, and solid-fluid equilibria are usually correlated using models for the excess Gibbs energy g. When solid phases are present, one motivation for not using a PvT equation is to avoid the introduction of spurious fluid-solid critical points, as discussed in 8.2.5. A second motivation is that properties of liquids and solids are little affected by moderate changes in pressure, so PvT equations can be unnecessarily complicated when applied to condensed phases. In contrast, g -models often do not contain pressure or density instead, they attempt to account only for the effects of temperature and composition. Such models are thereby limited to descriptions of phase separations that are driven by diffusional instabilities, and the stability behavior must be of class I (see 8.4.2). In this section we show how a g -model can describe liquid-liquid and solid-solid equilibria. 

Sections 9.3-9.S present the common phase behavior of binary mixtures 9.3 describes vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria at low pressures 9.4 considers solid-fluid equilibria and 9.5 discusses common high-pressure fluid-phase equilibria. Then 9.6 briefly describes the basic vapor-liquid and liquid-liquid equilibria that can occur in ternary mixtures. This chapter describes many apparently different phase behaviors, and so we try to show when those differences are more apparent than real. The organization is intended to bring out underlying similarities, thereby reducing the number of different things to be learned. 

We now describe the phase behavior exhibited by binary mixtures at modest pressures. The kinds of behavior observed in Nature include vapor-liquid equilibria (VLE, 9.3.1-9.3.3), azeotropes ( 9.3.4), critical points ( 9.3.5), liquid-liquid equilibria (LLE, 9.3.6), and vapor-liquid-liquid equilibria (VLLE, 9.3.7). When solid-fluid equilibria occur ( 9.4), many (but not all) of the resulting phase diagrams are analogous to their counterparts in fluid-fluid equilibria for example, many liquid-solid diagrams are analogous to vapor-liquid diagrams. 

Freeze-out of a Solid from a Liquid Solution. Thermodynamic textbooks and journal articles, for the most part, give greatest coverage to the phase equilibria of fluid phases but, with few exceptions, little to solids. However, prediction of solid-fluid equilibria is an important industrial problem. This is especially true for process designs related to natural gas and cryogenic plants, where the solubility limits at the incipient solid phase separation of, for example, water, CO2 and heavy -hydrocarbons must be accurately known to prevent costly plant shutdowns with... 

---