One of the essential ingredients for rational design of such separation operations is a knowledge of the required phase equilibria. The purpose of the present monograph is to present a technique, implemented for digital computers, to estimate these equilibria from a minimum of experimental information.  [c.1]

While much attention has been given to the development of computer techniques for design of distillation and absorption columns, much less attention has been devoted to the development of such techniques for equipment using liquid-liquid extraction. However, regardless of the nature of the operation, few systematic attempts have been made to organize phase-equilibrium information for direct use in chemical process design. This monograph presents a systematic procedure for calculating multi-component vapor-liquid and liquid-liquid equilibria for mixtures commonly encountered in the chemical process industries. Attention is limited to systems at low or moderate pressures. Pertinent references to previous work are given at the end of this chapter.  [c.1]

The possible number of liquid and vapor mixtures in technological processes is incredibly large, and it is unreasonable to expect that experimental vapor-liquid and liquid-liquid equilibria will ever be available for a significant fraction of this number. Further, obtaining good experimental data requires appreciable experimental skill, experience, and patience. It is, therefore, an economic necessity to consider techniques for calculating phase equilibria for multicomponent mixtures from few experimental data. Such techniques should require only a  [c.1]

Compilation of data for binary mixtures reports some vapor-liquid equilibrium data as well as other properties such as density and viscosity.  [c.12]

In Equation (11), V is the total volume containing n moles of component i, n moles of component j, etc. The differentiation is carried out such that, in addition to temperature and pressure, all mole numbers (except n ) are held constant.  [c.16]

Equation (12), applicable at low or moderate pressures, is used in this monograph for typical vapor mixtures. However, when the vapor phase contains a strongly dimerizing component such as carboxylic acid. Equation (7) is not applicable and  [c.16]

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention.  [c.18]

The standard-state fugacity of any component must be evaluated at the same temperature as that of the solution, regardless of whether the symmetric or unsymmetric convention is used for activity-coefficient normalization. But what about the pressure At low pressures, the effect of pressure on the thermodynamic properties of condensed phases is negligible and under such con-  [c.19]

When the pressure is low and mixture conditions are far from critical, activity coefficients are essentially independent of pressure. For such conditions it is common practice to set P = P in Equations (18) and (19). Coupled with the assumption that v = v, substitution gives the familiar equation  [c.22]

This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture.  [c.26]

The fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure  [c.26]

While vapor-phase corrections may be small for nonpolar molecules at low pressure, such corrections are usually not negligible for mixtures containing polar molecules. Vapor-phase corrections are extremely important for mixtures containing one or more carboxylic acids.  [c.38]

The parameters r, q, and q are pure-component molecular-structure constants depending on molecular size and external surface areas. In the original formulation (Abrams and Prausnitz, 1975), q = q. To obtain better agreement for mixtures containing water or alcohols, q for water and alcohols has here been obtained empirically to give an optimum fit to a variety of systems containing these components. For alcohols, the surface of interaction q is smaller than the geometric external surface q, indicating that for alcohols, intermolecular attraction is determined primarily by the OH group. Appendix C presents values of these structural parameters.  [c.42]

Since the accuracy of experimental data is frequently not high, and since experimental data are hardly ever plentiful, it is important to reduce the available data with care using a suitable statistical method and using a model for the excess Gibbs energy which contains only a minimum of binary parameters. Rarely are experimental data of sufficient quality and quantity to justify more than three binary parameters and, all too often, the data justify no more than two such parameters. When data sources (5) or (6) or (7) are used alone, it is not possible to use a three- (or more)-parameter model without making additional arbitrary assumptions. For typical engineering calculations, therefore, it is desirable to use a two-parameter model such as UNIQUAC.  [c.43]

Unfortunately, good binary data are often not available, and no model, including the modified UNIQUAC equation, is entirely adequate. Therefore, we require a calculation method which allows utilization of some ternary data in the parameter estimation such that the ternary system is well represented. A method toward that end is described in the next section.  [c.66]

For an ideal vapor mixture of m components, there is no enthalpy of mixing. The enthalpy of such a mixture is then  [c.84]

For such associating vapor mixtures, the "chemical" theory is used, as discussed in Chapter 3. The derivation of Ah  [c.85]

Two generally accepted models for the vapor phase were discussed in Chapter 3 and one particular model for the liquid phase (UNIQUAC) was discussed in Chapter 4. Unfortunately, these, and all other presently available models, are only approximate when used to calculate equilibrium properties of dense fluid mixtures. Therefore, any such model must contain a number of adjustable parameters, which can only be obtained from experimental measurements. The predictions of the model may be sensitive to the values selected for model parameters, and the data available may contain significant measurement errors. Thus, it is of major importance that serious consideration be given to the proper treatment of experimental measurements for mixtures to obtain the most appropriate values for parameters in models such as UNIQUAC.  [c.96]

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization.  [c.96]

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction.  [c.97]

The sum of the squared differences between calculated and measures pressures is minimized as a function of model parameters. This method, often called Barker s method (Barker, 1953), ignores information contained in vapor-phase mole fraction measurements such information is normally only used for consistency tests, as discussed by Van Ness et al. (1973). Nevertheless, when high-quality experimental data are available. Barker s method often gives excellent results (Abbott and Van Ness, 1975).  [c.97]

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation.  [c.105]

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements.  [c.106]

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965).  [c.108]

The product streams can be a vapor and a liquid or two immiscible liquids. The process may consist of a single equilibrium stage, as with a flash drum or single mixer-settler, or of a cascade of equilibrium stages, as with staged distillation or extraction. Multistage separations are usually arranged with countercurrent flow of the phases between the stages. The calculation of all such equilibrium separations is based on enthalpy and component material balances over the separation stage, in combination with the requirements that, for each component, the fugacities be equal in the two streams exiting the stage.  [c.110]

In this chapter we present efficient calculation procedures for single-stage equilibrium separations subroutines implementing these procedures are given in Appendices F and G. While we recognize the great importance of multistage separations, it must be realized that the efficient computation of such processes requires very careful resolution of the large number of simultaneous equilibrium stages involved in a countercurrent cascade. The dominant consideration in such multistage computation procedures is usually the technique used to achieve this simultaneous solution rather than the equilibrium treatment of the stages themselves. (Goldstein and Stanfield, 1970 Holland,  [c.110]

The single-stage separations for which we present computational procedures are the incipient separations (one product phase present in very small amount) represented by bubble and dew-point calculations, vapor-liquid equilibrium separations at fixed pressure under isothermal or adiabatic conditions, and liquid-liquid equilibrium separations at fixed pressure and temperature. These calculations are implemented by FORTRAN IV subroutines designed to minimize the number of vapor and liquid-phase fugacity evaluations necessary to achieve satisfactory solutions. This criterion for efficiency of the algorithms is based on the recognition that, with relatively rigorous thermodynamic methods such as those used here, most of the computation effort in any separation calculation is devoted to evaluation of thermodynamic equilibrium functions. It is important to avoid unnecessary calculations of fugacities or fugacity (activity) coefficients in computer programs used in chemical engineering practice.  [c.111]

If any iteration yields a such that  [c.116]

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate.  [c.124]

Appendix C presents the best set of constants for Equation (2). Also shown are the temperature limits of the real experimental data. Users must exercise caution when using the correlation outside the range of real data such use should, in general, be avoided.  [c.140]

Methane Propane 311-344 -2.01 604 Sage (1950)  [c.209]

Vapor-liquid and liquid-liquid equilibria depend on the nature of the components present, on their concentrations in both phases, and on the temperature and pressure of the system. Because of the large number of variables which determine multi-component equilibria, it is essential to utilize an efficient organizational tool which reduces available experimental data to a small number of theoretically significant functions and parameters these functions and parcimeters may then be called upon to form the building blocks upon which to construct the desired equilibria. Such an organizational tool is provided by thermodynamic analysis and synthesis. First, limited pure-component and binary data are analyzed to yield fundamental thermodynamic quantities. Second, these quantities are reduced to obtain parameters in a molecular model. That model, by synthesis, may be used to calculate the phase behavior of multicomponent liquids and vapors. In this way, it is possible to "scale up" data on binary and pure-component systems to obtain good estimates of the properties of multicomponent mixtures of a large variety of components including water, polar organic solvents such as ketones, alcohols, nitriles, etc., and paraffinic, naphthenic, and aromatic hydrocarbons.  [c.2]

Activity-coefficient data at infinite dilution often provide an excellent method for obtaining binary parameters as shown, for example, by Eclcert and Schreiber (1971) and by Nicolaides and Eckert (1978). Unfortunately, such data are rare.  [c.43]

In some cases, the temperature of the system may be larger than the critical temperature of one (or more) of the components, i.e., system temperature T may exceed T. . In that event, component i is a supercritical component, one that cannot exist as a pure liquid at temperature T. For this component, it is still possible to use symmetric normalization of the activity coefficient (y - 1 as x - 1) provided that some method of extrapolation is used to evaluate the standard-state fugacity which, in this case, is the fugacity of pure liquid i at system temperature T. For highly supercritical components (T Tj,.), such extrapolation is extremely arbitrary as a result, we have no assurance that when experimental data are reduced, the activity coefficient tends to obey the necessary boundary condition 1  [c.58]

The enthalpy of mixing for a liquid solution cannot be calculated accurately without specific data for that solution. Although it is always possible to differentiate an expression for the excess Gibbs energy with respect to temperature to find the enthalpy of mixing, such a procedure is often not reliable because the temperature dependence of the binary-mixture parameter of the liquid-phase model is usually not known. Far better results are achieved if vapor-liquid-equilibrium data are available over a range of temperatures or if some calorimetric mixing data can be used to fix temperature-dependent parameters." Fortunately, the enthalpy of mixing provides only a minor contribution to the total enthalpy of nonelectrolyte liquid mixtures. For engineering purposes it is therefore usually sufficient to estimate the enthalpy of mixing crudely or to neglect it entirely.  [c.83]

Such step-limiting is often helpful because the direction of correction provided by the Newton-Raphson procedure, that is, the relative magnitudes of the elements of the vector J G, is very frequently more reliable than the magnitude of the correction (Naphtali, 1964). In application, t is initially set to 1, and remains at this value as long as the Newton-Raphson correotions serve to decrease the norm (magnitude) of G, that is, for  [c.116]

If the objective function is considered two-dimensional, consisting of Equations (7-13) and (7-14) and the vector X includes only T and a, then the only change in the iteration is that the derivatives of with respect to composition are ignored in establishing the Newton-Raphson corrections to T and a. The new compositions can then be determined from Equations (7-8) and (7-9). Such a simplified procedure sacrifices little in convergence rate for vapor-liquid systems, where the contributions of compfosition-derivatives to changes in T and a are almost always smad 1. This approach requires only two evaluations of per iteration and still avoids creeping since it is essentially second-order in the limit as convergence is approached.  [c.117]

The subroutine is well suited to the typical problems of liquid-liquid separation calculations wehre good estimates of equilibrium phase compositions are not available. However, if very good initial estimates of conjugate-phase compositions are available h. priori, more effective procedures, with second-order convergence, can probably be developed for special applications such as tracing the entire boundary of a two-phase region.  [c.128]



See pages that mention the term Szyszkowski : [c.18]    [c.29]    [c.29]    [c.31]    [c.114]    [c.131]    [c.141]    [c.157]    [c.209]    [c.209]    [c.209]   
Physical chemistry of surfaces (0) -- [ c.67 ]