Fluid systems, phase equilibrium consistency

A phase is the entirety of regions, where material properties either do not change or only change continually, but never change abmptly. However, it makes no difference whether the regions are spatially coherent or not (continuous or dispersed phase). A phase can consist of one or more chemically uniform substances, which are called components. A system can contain one phase (gas, liquid, solid), two phases (e.g., liquid/gas, fluid/sohd, fluid/fluid), or even more (in an evaporative crystallizer, e.g., there are a solid, a liquid and a gaseous phase) This chapter describes the thermo namic equilibrium between phases. 

Prediction of fluid phase behaviour at high pressures must be performed with an equation of state using the so called (p — 4> approach to attain a consistent result near the critical region of the mixture. In this sense, classic cubic equations of state are the more attractive to use due to their simplicity however, they present some limitations when applied to asymmetric systems. An alternative approach is provided by group contribution which is particularly attractive for reacting systems because it is often necessary to deal with mixtures for which no experimental phase equilibrium data are available. 

Let us describe the mathematical model of a three-phase non-isothermal compressible flow in porous media taking into account capillary effects. It is assumed that the movement of phases obeys the generalized Darcy s law. We assume that the phases are in the local thermal equilibrium, so that in any elementary volume the fluids saturating the porous medium and the rock have the same temperature. Furthermore, oil is assumed to be homogeneous non-evaporable fluid and oil reservoir consists of one type of rock. In this case, three-phase non-isothermal flow in a bounded domain 2 c M (d = 1, 2, 3) taking into account capillary forces and the phase transitions between the phases of water and heat transfer is described by the following system of equations ... 

Fluid dynamics (also called fluid mechanics) is the study of moving (deformable) matter, and includes liquids and gases, plasmas and, to some extent, plastic solids. From a fluid-mechanical point of view, matter can, in a broad sense, be considered to consist of fluid and solid, in a one-fluid system the difference between these two states being that a solid can resist shear stress by a static deformation, but a fluid can not. Notice also that thermodynamically a distinction between the gas and liquid states of matter carmot be made if temperature is above that of the so-called critical point, and below that temperature the only essential differences between these two phases are their differing equilibrium densities and compressibility. 

In pioneering research by Hailing and co-workers, it was demonstrated that the activity of water is a more representative and useful parameter than water concentration for describing enzymatic rates in nonaqueous enzymology. Water activity, or is defined as the fugacity of water contained in a mixture divided by the fugacity of pure water at the mixture s temperature. For a typical nonaqueous enzymatic reaction operated in a closed system, the medium will consist of a solvent (or fluid) phase, an enzyme-contaiifing solid phase, and air headspace above the solvent. As a first approximation, the water transport between the three phases is assumed to be at thermodynamic equilibrium. For such a situation, can be defined in terms of the air headspace properties ... 

To illustrate the use of this equilibrium criterion, consider the very simple, initially nonuniform system shown in Fig. 7.1-1. There a single-phase, single-component fluid in an adiabatic, constant-volume container has been divided into two subsystems by an imaginary-boundary. Each of these subsystems is assumed to contain the same chemical species of uniform thermodynamic properties. However, these subsystems are open to the flow of heat and mass across the imaginary internal boundary, and their temperature and-pressure need not be the same. For the composite system consisting of the two subsystems, the total mass (though, in fact, we will use number of moles), internal energy, volume, and entropy, all of which are extensive variables, are the sums of these respective quantities for the two subsystems, that is. 

Type III process consists of the repeated dissolution and precipitation of phase B from the Type II case. It is typically assumed in these calculations that (a) the system is closed to fluid, (b) the stoichiometry of the solid remains constant, (c) the isotopic fractionation factor between the freshly formed portion of the mineral and the fluid is constant (not necessarily equilibrium), and (d) isotopic exchange before dissolution and after precipitation is negligible. As a basis for their modeling, Dubinina and Lakshtanov... 

For simplicity, we consider here only a binary mixtures (A, B) and do not discuss the complications posed by extensions to multicomponent systems. Figure 1 shows a schematic phase diagram in the plane of variables temperature T and concentration c of species B. Kinetics of phase separation in bulk fluid mixtures is triggered by a rapid quench (at time t = 0) from the one-phase region into the miscibility gap. The initial equilibrium state (f < 0) is spatially homogeneous, apart from small-scale concentration inhomogeneities. The final equilibrium state towards which the system ultimately evolves (t oo) consists of... 

Remarks. Close inspection of the nonequilibrium model outputs reveals that assumption of nonequilibrium capillary pressure in the studied range of experimental conditions was not necessary and static equilibrium described by PcxPg-Pe was sufficient to account for the interfacial forces [54], However, recourse to empirical capillary relationships, such as the Leverett /-function, is unnecessary as the nonequilibrium two-phase flow model enables access to capillary pressure via entropy-consistent constitutive expressions for the macroscopic Helmholtz free energies. Also, the role of mass exchange between bulk fluid phase holdups and gas-liquid interfacial area was shown to play a nonnegligible role in the dynamics of trickle-bed reactor [ 54]. By accounting for the production/destruction of interfacial area, they prompted much briefer response times for the system to attain steady state compared to the case without inclusion of these mass exchange rates. 

We may evaluate thermodynamic stability under various conditions where a set of the independent thermodynamic variables is specified. Specification of the independent variables is equivalent to specification of ensemble. In any case, the number of water molecules is fixed to a constant value, and the temperature is also set to the constant one, T. The other mechanical conditions depend on the choice of the variables. The number of water molecules, irrespective of the choice of ensemble, is reserved for the extensive property to indicate the system size. Whatever the other properties are, they can be substituted for the (formal) intensive properties. According to the phase rule, the number of degrees of freedom for clathrate hydrate in equilibrium with the guest fluid consisting of a single component (A = 2,/jp = 2),//f = 2 + / c—r pis2. 

Ternary system consists of one volatile and two nonvolatile components, such phenomena as an azeotropy in liquid-gas equilibria and a formation of binary or ternary compounds are absent. Solid phases of volatile and each non-volatile components are completely immiscible and have the eutectic relations in equilibrium with fluid phases, whereas the solid phases of non-volatile components form a continuous solid solution. 

Monte Carlo methods offer a useful alternative to Molecular Dynamics techniques for the study of the equilibrium structure and properties, including phase behavior, of complex fluids. This is especially true of systems that exhibit a broad spectrum of characteristic relaxation times in such systems, the computational demands required to generate a long trajectory using Molecular Dynamics methods can be prohibitively large. In a fluid consisting of long chain molecules, for example, Monte Carlo techniques can now be used with confidence to determine thermodynamic properties, provided appropriate techniques are employed. 

Let us consider a multicomponent two-phase system with a plane interface of area A in complete equilibrium, and let us focus on the inhomogeneous interfacial region. Our approach is a point-thermodynamic approach [92-96], and our key assumption is that in an inhomogeneous system, it is possible to define, at least consistently, local values of the thermodynamic fields of pressure P, temperature T, chemical potential p, number density p, and Helmholtz free-energy density xg. At planar fluid-fluid interfaces, which are the interfaces of our interest here, the aforementioned fields and densities are functions only of the height z across the interface. 

Adsorption in a fixed bed is a complex system to model. Concentrations evidently vary with distance, and although they ultimately attain a steady form of distribution, they also vary with time. The model would consequently consist of two mass balances, one for the fluid phase and a second for the solid phase, and both of these would be partial differential equations in time and distance, which generally have to be solved numerically. To avoid this complication, a procediue has come into use in which mass transfer resistance is neglected and the two phases are everywhere assumed to be in equilibrium. The concentration then propagates in the shape of a rectangular front, shown in Figure 6.7 and denoted "Equilibrium." The... 

In a multicomponent system there can often be several solid phases or several liquid phases at equilibrium. For example, if one equilibrates mercury, a mineral oil, a methyl silicone oil, water, benzyl alcohol, and a perfluoro compound such as perfluoro (A-ethyl piperidine) at room temperature, one can obtain six coexisting liquid phases. Each of these phases consists of a large amount of one substance with small amounts of the other substances dissolved in it. Under ordinary conditions, a system can exhibit only a single gas phase. However, if certain gaseous mixtures are brought to supercritical temperatures and pressures, where the distinction between gas and liquid disappears, two fluid phases can form without first making a gas-liquid phase transition. 

Uno, S. Kurihara, K. Ochi, K. Kojima, K. Determination and correlation of vapor-liquid equilibrium for binary systems consisting of close-boiling components. Fluid Phase Equilib. 2007, 257, 139-146. 

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