Fluid systems, phase equilibrium state

In practice, colloidal systems do not always reach tlie predicted equilibrium state, which is observed here for tlie case of narrow attractions. On increasing tlie polymer concentration, a fluid-crystal phase separation may be induced, but at higher concentration crystallization is arrested and amorjihous gels have been found to fonn instead [101, 102]. Close to the phase boundary, transient gels were observed, in which phase separation proceeded after a lag time. 

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

There are numerous sources of phase equilibrium data available that serve as a database to those developing or improving equations of state. References to these databases are widely available. In addition, new data are added mainly through the Journal of Chemical and Engineering Data and the journal Fluid Phase Equilibria. Next, we give data for two systems so that the reader may practice the estimation methods discussed in this chapter. 

The extra pieces of information describe the extent of the system - the amounts of fluid and minerals that are present. It is not necessary to know the system s extent to determine its equilibrium state, but in reaction modeling (see Chapter 13) we generally want to track the masses of solution and minerals in the system we also must know these masses to search for the system s stable phase assemblage (as described in Section 4.4). 

Two additional points about Equation (8) need to be discussed here. Equation (8) contains mj in the denominator. Thus the solution concentrations must be known before the first increment dE, is taken and none of them can be zero. In practice this means that the set of nonlinear equations (mass action and balance equations) describing the fluid phase in its initial unperturbed equilibrium state must be solved once. Further, Equation (8) does not completely describe a heterogeneous system at partial equilibrium. 

Optimizing solvents and solvent mixtures can be done empirically or through modeling. An example of the latter involves a single Sanchez-Lacombe lattice fluid equation of state, used to model both phases for a polymer-supercritical fluid-cosolvent system. This method works well over a wide pressure range both volumetric and phase equilibrium properties for a cross-linked poly(dimethyl siloxane) phase in contact with CO2 modified by a number of cosolvents (West et al., 1998). 

In summary, we refer to Figure 5.5, which may be considered as the projection of the entire equilibrium surface on the entropy-volume plane. All of the equilibrium states of the system when it exists in the single-phase fluid state lie in the area above the curves alevd. All of the equilibrium states of the system when it exists in the single-phase solid state lie in the area bounded by the lines bs and sc. These areas are the projections of the primary surfaces. The two-phase systems are represented by the shaded areas alsb, lev, and csvd. These areas are the projections of the derived surfaces for these states. Finally, the triangular area slv represents the projection of the tangent plane at the triple point, and represents all possible states of the system at the triple point. This area also is a projection of a derived surface. 

A two-phase flowing fluid is composed of both liquid and gas, generally in an equilibrium state when the pressure decreases because of flow through the pipe, more of the liquid will flash to vapor. This will lower the temperature, since the removal of latent heat energy taken by the flashed vapor cools the system. 

The calculations reported in this paper and a related series of publications indicate that it is now quite feasible to obtain reasonably accurate results for phase equilibria in simple fluid mixtures directly from molecular simulation. What is the possible value of such results Clearly, because of the lack of accurate intermolecular potentials optimized for phase equilibrium calculations for most systems of practical interest, the immediate application of molecular simulation techniques as a replacement of the established modelling methods is not possible (or even desirable). For obtaining accurate results, the intermolecular potential parameters must be fitted to experimental results, in much the same way as parameters for equation-of-state or activity coefficient models. This conclusion is supported by other molecular-simulation based predictions of phase equilibria in similar systems (6). However, there is an important difference between the potential parameters in molecular simulation methods and fitted parameters of thermodynamic models. Molecular simulation calculations, such as the ones reported here, involve no approximations beyond those inherent in the potential models. The calculated behavior of a system with assumed intermolecular potentials is exact for any conditions of pressure, temperature or composition. Thus, if a good potential model for a component can be developed, it can be reliably used for predictions in the absence of experimental information. 

Browarzik, D. and Kowalewski, M., Calculation of the stability and phase equilibrium in the system polystyrene -1- cyclohexane + carbon dioxide based on equations of state. Fluid Phase Equilibria, 163, 43-60, 1999. 

Most of the common separation methods used in the chemical industry rely on a well-known observation when a multicomponent two-phase system is given sufficient ttmu to attain a statioenry state called equilibrium, the composition of one phase is different from thet of the other. It is this property of nature which eenbles separation of fluid mixtures by distillation, extraction, and other diffusions operations. For rational design of such operations it is necessary to heve a quantitative description of how a component distributes itself between two contacting phases. Phase-equilibrium thermodynamics, summarized here, provides a framework for establishing that description. 

An equation of state, applicable to all fluid phases, is paitiodariy useful for phase-equilibrium calculations where a liquid phase and a vapor phase coexist at high pressures. At such conditions, conventional activity coefficients are not useful because, with rare exceptions, at least one of the mixture s components is supercritical that is, (he system temperature is above (hat component s critical temperature. In that event, one must employ special standard states for the activity coefficients of the supercritical components (see Section 1.5-2). That complication is avoided when ail fugacities are calculated front en equation of state. 

It is useful to mention another class of problems related to those referred to in the previous paragraphs, but that is not considered here. We do not try to answer the ques- tion of how fast a system will respond to a change in constraints that is, we do not try to study system dynamics. The answers to such problems, depending on the system and its constraints, may involve chemical kinetics, heat or mass transfer, and fluid mechanics, all of which are studied elsewhere. Thus, in the example above, we are interested in the final state of the gas in each cylinder, but not in computing how long a valve of given size must be held open to allow the necessary amount of gas to pass from one cylinder to the other. Similarly, when, in Chapters 10, 11, and 12, we study phase equilibrium and, in Chapter 13, chemical equilibrium, our interest is in the prediction of the equilibrium state, not in how long it will take to achieve this equilibrium state. —... 

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