Liquid-vapor equilibrium modeling

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 ... 

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]). 

Bowers and Mudawar (1994a) performed an experimental smdy of boiling flow within mini-channel (2.54 mm) and micro-channel d = 510 pm) heat sink and demonstrated that high values of heat flux can be achieved. Bowers and Mudawar (1994b) also modeled the pressure drop in the micro-channels and minichannels, using the Collier (1981) and Wallis (1969) homogenous equilibrium model, which assumes the liquid and vapor phases form a homogenous mixture with equal and uniform velocity, and properties were assumed to be uniform within each phase. 

Explosive boiling is certainly not the normal event to occur when liquids are heated. Thus, the very rapid vaporization process must be explained by theories other than standard equilibrium models. For example, if two liquids are brought into contact, and one is relatively nonvolatile but at a temperature significantly above the boiling point of the second liquid, an explosive rapid-phase transition sometimes results. Various models have been proposed to describe such transitions. None has been... 

D. THERMAL EQUILIBRIUM MODEL. The previous ease yields a model that is about as rigorous as one ean reasonably expeet. A final model, not quite as rigorous but usually quite adequate, is one in whieh thermal equihbrium between liquid and vapor is assumed to hold at all times. More simply, the vapor and. liquid temperatures are assumed equal to eaeh other T=T . This eliminates the need for an energy balanee for the vapor phase. It probably works pretty well because the sensible heat of the vapor is usually small compared with latent-heat effects. 

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. 

Figure 7 further shows that, as gaseous C02 moves up the absorber, phase equilibrium is achieved at the vapor-liquid interface. C02 then diffuses through the liquid film while reacting with the amines before it reaches the bulk liquid. Each reaction is constrained by chemical equilibrium but does not necessarily reach chemical equilibrium, depending primarily on the residence time (or liquid film thickness and liquid holdup for the bulk liquid) and temperature. Certainly kinetic rate expressions and the kinetic parameters need to be established for the kinetics-controlled reactions. While concentration-based kinetic rate expressions are often reported in the literature, activity-based kinetic rate expressions should be used in order to guarantee model consistency with the chemical equilibrium model for the aqueous phase solution chemistry. 

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 ... 

The application of Eq. (10.3) to specific phase-equilibrium problems requires use of models of solution behavior, which provide expressions for G or for the Hi as functions of temperature, pressure, and composition. The simplest of such expressions are for mixtures of ideal gases and for mixtures that form ideal solutions. These expressions, developed in this chapter, lead directly to Raoult s law, the simplest realistic relation between the compositions of phases coexisting in vapor/liquid equilibrium. Models of more general validity are treated in Chaps. 11 and 12. 

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