The Vapor-Liquid Equilibrium Problem

H0W many degrees of freedom (F) are there, i.e. how many of these variables must be specified so that the intensive state of this system is determined  

According to the Phase Rule (Section 12.7.1) F = + 2- 2 = k i.e. specification of k of these 2k variables suffices for the complete description of the intensive state of the system. 

The remaining k variables would thus have to be determined experimentally. This approach is time-consuming and costly, however, and our objective is the evaluation of these k variables using the minimum amount of experimental information possible. This represents the general statement of the vapor-liquid equilibrium problem. The typical cases encountered in practice, are presented in Table 13.1. 

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The vapor-liquid equilibrium was computed from the EOS model using the reliable and robust method of Hua et al 14-16) based on interval analysis. Their method can find the correct thermodynamically stable solution to the vapor-liquid equilibrium problem with mathematical and computational certainty. Additionally, the tangent plane distance method 17,18) was used to test the predicted liquid and vapor phase compositions for global thermodynamic phase stability. 

The methodology for solving the vapor-liquid equilibrium problem using an equation of state has been outlined in Section 13.4 and involves expressing the fugacity coefficients of the mixture components in the liquid (/) and vapor (v) phases through an equation of state (EoS). The equilibrium ratio Kj is then calculated from ... 

In general, the formulation of the problem of vapor-liquid equilibria in these systems is not difficult. One has the mass balances, dissociation equilibria in the solution, the equation of electroneutrality and the expressions for the vapor-liquid equilibrium of each molecular species (equality of activities). The result is a system of non-linear equations which must be solved. The main thermodynamic problem is the relation of the activities of the species to be measurable properties, such as pressure and composition. In order to do this a model is needed and the parameters in the model are usually obtained from experimental data on the mixtures involved. Calculations of this type are well-known in geological systems O) where the vapor-liquid equilibria are usually neglected. 

At system pressures up to several tens of MPa, the fugacity coefficients, < > and (j), and the Poynting factor, 7Zp are usually near unity. A simplified version of equation 19 can therefore be used for the majority of vapor—liquid equilibrium problems ... 

Having identified discrete components as our criterion, the next problem is how to define the equilibrium of three components. In Chap. 2, for example, the vapor-liquid equilibrium of a flash—a single component transfer between two phases—was derived. A certain quantity of a component, called the y fraction, is vaporized into the vapor phase as an equal amount of the same component, called the x fraction, is dissolved in the liquid phase. The K value (the equilibrium constant of K=y/x) is used to determine the component distribution results. This same logic may be used here. Although we cannot use these same K values, as they do not apply, we can apply another database. 

One further vapor/liquid equilibrium problem is the flash calculation. origin of the name is in the change that occurs when a liquid under press passes through a valve to a pressure low enough that some of the liquid vapori or flashes, producing a two-phase stream of vapor and liquid in equilibri We consider here only the P.f -flash, which refers to any calculation of quantities and compositions of the vapor and liquid phases making up a two-ph system in equilibrium at known P, T, and overall composition. 

Since Pf is a function of temperature only, Raoult s law is a set of N equations in the variables T, P, y,, and, . There are, in fact, N - 1 independent vapor-phase mole fractions (the y,- s), N - 1 independent liquid-phase mole fractions (the x, s), and T and P. This makes a total of 21V independent variables related by N equations. The specification of N of these variables in the formulation of a vapor/liquid equilibrium problem allows the remaining N variables to be determined by the simultaneous solution of the N equilibrium relations given here by Raoult s law. In practice, one usually specifies either T or P and either the liquid-phase or the vapor-phase composition, fixing 1 + N - 1) = N variables. 

Your task in this problem will be to use a spreadsheet to generate a Txy diagram for a two-component system, using Raoult s law to express the vapor-liquid equilibrium distribution of each species. The spreadsheet will be constructed for the chloroform-benzene system at 1 atm (for which Raouit s law is not a very good approximation), but it can then be used for any other system by substituting different Antoine equation constants. 

Water in THF is the second most important problem as it makes polymerization impossible. The vapor/liquid equilibrium diagram of the THF/water system at atmospheric pressure is shown in Figure 95. 

Take a mixture of two or more chemicals in a temperature regime where both have a significant vapor pressure. The composition of the mixture in the vapor is different from that in the liquid. By harnessing this difference, you can separate two chemicals, which is the basis of distillation. To calculate this phenomenon, though, you need to predict thermodynamic quantities such as fugacity, and then perform mass and energy balances over the system. This chapter explains how to predict the thermodynamic properties and then how to solve equations for a phase separation. While phase separation is only one part of the distillation process, it is the basis for the entire process. In this chapter you will learn to solve vapor-liquid equilibrium problems, and these principles are employed in calculations for distillation towers in Chapters 6 and 7. Vapor-liquid equilibria problems are expressed as algebraic equations, and the methods used are the same ones as introduced in Chapter 2. 

Using one of these activity coefficient equations it is possible to calculate liquid-liquid equihbrium (LLE) behavior of multicomponent hquid systems. Consider, for example, the ternary system of Figure 1. A system of overall composition A splits into two liquid phases B and C. The calculation of compositions of B and C is analogous to the flash ciculation of vapor-liquid equilibrium problems. By using the UNIQUAC equations to obtain the partition coefficients, Kj, this problem can be solved for any composition A of the overall system. The calculations are lengthy but computer programs for this purpose (2) have been published. In this paper simpler approximate methods for phase equilibrium problems of environmental interest is sought. For the moment it is sufficient to note that the activity coefficients provide the means of complete liquid-liquid equihbrium computations. 

Another type of vapor-liquid equilibrium problem, and one that is more important for designing separation equipment, is computing the two-phase equilibrium state when either a liquid of known composition is partially vaporized or a vapor is partially condensed as a result of a change in temperature and/or pressure. Such a problem is gener-... 

The A"-factor formulation introduced in this calculation is frequently useful in solving vapor-liquid equilibrium problems. The procedure is easily generalized to nonideal liquid and vapor phases as follows ... 

BCnowledge of the vapor-liquid equilibrium (VLB) behavior in these mixtures is necessary to design and to optimize the separation in the flash vessel, which is part of the extractive process considered in this work, described in the next section. The problem of phase equilibrium consists on the calculation of some variables of the set T, P, x, and y) when some of them are known. 

Prausnitz (1,2) has discussed this problem extensively, but the most successful techniques, which are based on either closed equations of state, such as discussed in this symposium, or on dilute liquid solution reference states such as in Prausnitz and Chueh (3), are limited to systems containing nonpolar species or dilute quantities of weakly polar substances. The purpose of this chapter is to describe a novel method for calculating the properties of liquids containing supercritical components which requires relatively few data and is of general applicability. Used with a vapor equation of state, the vapor-liquid equilibrium for these systems can be predicted to a high degree of accuracy even though the liquid may be 30 mol % or more of the supercritical species and the pressure more than 1000 bar. 

Vapor permeation and pervaporation are membrane separation processes that employ dense, non-porous membranes for the selective separation of dilute solutes from a vapor or liquid bulk, respectively, into a solute-enriched vapor phase. The separation concept of vapor permeation and pervaporation is based on the molecular interaction between the feed components and the dense membrane, unlike some pressure-driven membrane processes such as microfiltration, whose general separation mechanism is primarily based on size-exclusion. Hence, the membrane serves as a selective transport barrier during the permeation of solutes from the feed (upstream) phase to the downstream phase and, in this way, possesses an additional selectivity (permselectivity) compared to evaporative techniques, such as distillation (see Chapter 3.1). This is an advantage when, for example, a feed stream consists of an azeotrope that, by definition, caimot be further separated by distillation. Introducing a permselective membrane barrier through which separation is controlled by solute-membrane interactions rather than those dominating the vapor-liquid equilibrium, such an evaporative separation problem can be overcome without the need for external aids such as entrainers. The most common example for such an application is the dehydration of ethanol. 

A rigorous simulation and optimization of reactive distillation processes usually is based on nonlinear fimctions for a realistic description of the reaction kinetics and the vapor-liquid-equilibrium. Within GAMS models, this description leads to very complex models that often face convergence problems. By using the new so-called external functions, the situation can be improved by transferring calculation procedures to an external module. 

As demonstrated by former publications, the GAMS modeling system has been successfully used for the MINLP-optimization of single reactive distillation columns (Poth et al., 2001 Jackson and Grossmann 2001). The strong nonlinear functions required for a realistic description of the reaction kinetics and the vapor-liquid-equilibrium in these cases lead to very complex GAMS models that may face convergence problems. 

The problem becomes even more complex when chemical reactions occur in the liquid phase. The reactions affect both the vapor-liquid equilibrium and the rate of mass transfer. However, correlations have been developed to predict vapor-liquid equilibria for amine-... 

If we know the system temperature we can use Raouh s law (and the assumption that the vapor is an ideal gas) to solve some simple vapor-liquid equilibrium problems. 

We return then to the general vapor-liquid equilibrium problem and discuss how the activity coefficient is used in solving it. We will see that this is accomplished through the so-called bubble and dew point calculations. 

We have concluded, thus, our discussion of the activity coefficient and are ready to consider its use in solving vapor-liquid equilibrium problems. 

The typical vapor-liquid equilibrium problem is solved through bubble or dew point calculations. 

Vapor/liquid equilibrium (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

Step 1. Assume the composition of the liquid in the evaporator at equilibrium with its vapor to be 75 mol% propane and 25 mol% butane. This is the initial assumption. If it is correct, the composition of the initial charge can be checked. If it is not correct, the problem must be reworked with a new equilibrium assumption. The composition of the vapor in equilibrium with this liquid is determined from the following equation. 

We are interested in comparing the effectiveness of the various equations of state in predicting the (p. V. T) properties. We will limit our comparisons to Tr > 1 since for Tr < 1 condensations to the liquid phase occur. Prediction of (vapor + liquid) equilibrium would be of interest, but these predictions present serious problems, since in some instances the equations of state do not converge for Tr< 1. 

This illustrates the statement made earlier that the most convenient choice of standard state may depend on the problem. For gas-phase problems involving A, it is convenient to choose the standard state for A as an ideal gas at 1 atm pressure. But, where the vapor of A is in equilibrium with a solution, it is sometimes convenient to choose the standard state as the pure liquid. Since /a is the same for the pure liquid and the vapor in equilibrium... 

None of the experimental techniques described by Bonner, however, has been capable of providing reliable vapor-liquid equilibrium data at the combined extremes of elevated temperature and reduced pressure, conditions applicable to most commercial polymer-stripping operations. This problem has been addressed by Meyer and Blanks (1982), who developed a modified isopiestic technique that could be used when solubilities are low. Although the success of this new technique was demonstrated using just polyethylene with isobutane and propane, the idea shows considerable promise for obtaining data at unusual conditions of temperature and pressure. 

The problem is best understood by considering an example. One of the most common iterative calculations is a vapor-liquid equilibrium bubblepoint calculation. 

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