Models liquid-vapor equilibrium

The procedure developed by Joris and Kalitventzeff (1987) aims to classify the variables and measurements involved in any type of plant model. The system of equations that represents plant operation involves state variables (temperature, pressure, partial molar flowrates of components, extents of reactions), measurements, and link variables (those that relate certain measurements to state variables). This system is made up of material and energy balances, liquid-vapor equilibrium relationships, pressure equality equations, link equations, etc. 

Substitution of 2H and H for 180 and lsO, respectively, in the above results in the identical relations for 2H/1H fractionation. The fractionation factor, a, is a function of the temperature at which condensation takes place and the phases involved. In the atmosphere, fractionation occurs between water vapor and liquid water or between water vapor and water ice. The temperature dependence of a(7j has been determined experimentally for liquid-vapor equilibrium (Majoube 1971b Horita and Wesolowski 1994), and for ice-vapor equilibrium (Merlivat and Nief 1967 Majoube 1971a). Existing experimental results are in quite close agreement and we use these relations in the model (Fig. 2). 

At this time, only a small number of nanoscale processes are characterized with transport phenomena equations. Therefore, if, for example, a chemical reaction takes place in a nanoscale process, we cannot couple the elementary chemical reaction act with the classical transport phenomena equations. However, researchers have found the keys to attaching the molecular process modelling to the chemical engineering requirements. For example in the liquid-vapor equilibrium, the solid surface adsorption and the properties of very fine porous ceramics computed earlier using molecular modelling have been successfully integrated in modelling based on transport phenomena [4.14]. In the same class of limits we can include the validity limits of the transfer phenomena equations which are based on parameters of the thermodynamic state. It is known [3.15] that the flow equations and, consequently, the heat and mass transport equations, are valid only for the... 

We have used a simple molecular model employed in simulations of oil/water/amphiphiles to investigate the self-assembly of amphiphiles in systems displaying liquid/vapor equilibrium. Studies of oil/water/amphiphile systems rarely concentrate on the role of the oil phase. Although it is rarely emphasized, many (if not most)... 

Essentially, the thermodynamic modeling consists in a set of equations that relate process parameters through mass and heat balances and liquid-vapor equilibrium equations. Below are the critical equations ... 

Now we need the energy of the lattice model liquid. At equilibrium, the exchange of particles between the liquid and the vapor phases involves little or no change in the internal quantum mechanical state of the particles—their rotations, vibrations, and electronic states do not change, to first approximation. So the energies that are relevant for vaporization are those between pairs of particles, not within each particle. In the gas phase, if it is ideal, the particles do not interact with each other. 

Liquid-vapor equilibrium of chain molecule fluids. Both analytic and numerical work has been recently done by Schweizer and co-workers. The compressibility route predictions of PRISM for this problem are extremely sensitive to closure approximation since the relevant fluid densities are very low and large-scale density fluctuations are present. The atomiclike MSA closure leads to qualitatively incorrect results as does the R-MMSA closure. However, the R-MPY/HTA approximation appears to be in excellent accord with the computer simulation studies of n-alkanes and model chain polymers, including a critical density that decreases weakly with N and a critical temperature that increases approximately logarithmically with N. 

Vapor-Liquid-Liquid Equilibrium. We have had limited experi-lence in rigorous three phase equilibrium calculations, vapor-liquid-liquid, primarily in single stage flash units. The implementation of such a three-phase equilibrium model in column calculation is scheduled in the future. Presently, a method also exists wherein complete immisclhility in the liquid phase can be specified between one component and all of the other components in the system e.g., between water and a set of hydrocarbons. The VLE ratios are normalized on an overall liquid basis so that the results can be used in conventional two-phase liquid-vapor equilibrium calculations. 

This model, which yields excellent results for polar and non-polar molecular liquids, is especially well suited for the study of liquid/ vapor equilibrium and the equilibrium between two liquids that are not completely miscible. Regardless of the number of components of the solution, the application of this model only requires the knowledge of two adjustment parameters per binary system, which can be deduced from the solution. The model is so widely applicable that it actually contains a number of previously classic models such as the models put forward by Van Laar, Wilson, Renon et al. (the NRTL - Non Random Two Liquids -model), Scatchard and Hildebrand, Flory and Huggins as special cases. In addition, it lends a physical meaning to the first three coefficients P, 5 and , in the Margules expansion (equation [2.1]). 

To build a molecular model of the equilibrium between a liquid and its vapor we first suppose that the liquid is introduced into an evacuated closed container. Vapor forms as molecules leave the surface of the liquid. Most evaporation takes place from the surface of the liquid because the molecules there are least strongly bound to their neighbors and can escape more easily than those in the bulk. Howevei as the number of molecules in the vapor increases, more of them become available to strike the surface of the liquid, stick to it, and become part of the liquid again. Eventually, the number of molecules returning to the liquid each second matches the number escaping (Fig. 8.2). The vapor is now condensing as fast as the liquid is vaporizing, and so the equilibrium is dynamic in the sense introduced in Section 7.11 ... 

Separation systems include in their mathematical models various vapor-liquid equilibrium (VLE) correlations that are specific to the binary or multicomponent system of interest. Such correlations are usually obtained by fitting VLE data by least squares. The nature of the data can depend on the level of sophistication of the experimental work. In some cases it is only feasible to measure the total pressure of a system as a function of the liquid phase mole fraction (no vapor phase mole fraction data are available). 

It is clear that as [A] approaches [A]sat, x approaches 1, and the surface-adsorbed layer thickness Eq. 11.60 goes to infinity that is, there is an infinite reservoir of liquid in equilibrium with the vapor. This is the desired limiting behavior for the model. 

The basic equations describing a single stage in a fractionator in which chemical reaction may occur include component material balances, vapor-liquid equilibrium relationships, and energy balance, and restrictions on the liquid vapor phase mol fractions. The model equations for stage j may be expressed as follows ... 

Contact angle — The contact angle is the angle of contact between a droplet of liquid and a flat rigid solid, measured within the liquid and perpendicular to the contact line where three phases (liquid, solid, vapor) meet. The simplest theoretical model of contact angle assumes thermodynamic equilibrium between three pure phases at constant temperature and pressure [i, ii]. Also, the droplet is assumed to be so small that the force of gravity does not distort its shape. If we denote the - interfacial tension of the solid-vapor interface as ysv. the interfacial tension of the solid-liquid interface as ySL and the interfacial tension of the liquid-vapor interface as yLV, then by a horizontal balance of mechanical forces (9 < 90°)... 

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