# Myrrh

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. [c.4]

In Chapter 2 we discuss briefly the thermodynamic functions whereby the abstract fugacities are related to the measurable, real quantities temperature, pressure, and composition. This formulation is then given more completely in Chapters 3 and 4, which present detailed material on vapor-phase and liquid-phase fugacities, respectively. [c.5]

Comprehensive data collection for more than 6000 binary and multicomponent mixtures at moderate pressures. Data correlation and consistency tests are given for each data set. [c.8]

The thermodynamic treatment of multicomponent phase equilibria, introduced by J. W. Gibbs, is based on the concept of the chemical potential. Two phases are in thermodynamic equilibrium when the temperature of one phase is equal to that of the other and when the chemical potential of each component present is the same in both phases. For engineering purposes, the chemical potential is an awkward quantity, devoid of any immediate sense of physical reality. G. N. Lewis showed that a physically more meaningful quantity, equivalent to the chemical potential, could be obtained by a simple transformation the result of this transformation is a quantity called the fugacity, which has units of pressure. Physically, it is convenient to think of the fugacity as a thermodynamic pressure since, in a mixture of ideal gases, the fugacity of each component is equal to its partial pressure. In real mixtures, the fugacity can be considered as a partial pressure, corrected for nonideal behavior. [c.14]

Equation (10a) is somewhat inconvenient first, because we prefer to use pressure rather than volume as our independent variable, and second, because little is known about third virial coefficients It is therefore more practical to substitute [c.28]

Equation (10b) is used in this work whenever the vapor mixture does not contain one or more carboxylic acids. [c.28]

Next, and more difficult, is the calculation of the true mole fraction This calculation is achieved by simultaneous [c.34]

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]

Equations (2) and (3) are physically meaningful only in the temperature range bounded by the triple-point temperature and the critical temperature. Nevertheless, it is often useful to extrapolate these equations either to lower or, more often, to higher temperatures. In this monograph we have extrapolated the function F [Equation (3)] to a reduced temperature of nearly 2. We do not recommend further extrapolation. For highly supercritical components it is better to use the unsymmetric normalization for activity coefficients as indicated in Chapter 2 and as discussed further in a later section of this chapter. [c.40]

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]

Vapor-Liquid Equilibria for Mixtures Containing One or More Noncondensable Components [c.58]

Our experience with multicomponent vapor-liquid equilibria suggests that for system temperatures well below the critical of every component, good multicomponent results are usually obtained, especially where binary parameters are chosen with care. However, when the system temperature is near or above the critical of one (or more) of the components, multicomponent predictions may be in error, even though all binary pairs are fit well. [c.61]

Liquid-liquid equilibria are much more sensitive than vapor-liquid equilibria to small changes in the effect of composition on activity coefficients. Therefore, calculations for liquid-liquid equilibria should be based, whenever possible, at least in part, on experimental liquid-liquid data. [c.63]

In the next three sections we discuss calculation of liquid-liquid equilibria (LLE) for ternary systems and then conclude the chapter with a discussion of LLE for systems containing more than three components. [c.63]

For systems of type II, if the mutual binary solubility (LLE) data are known for the two partially miscible pairs, and if reasonable vapor-liquid equilibrium (VLE) data are known for the miscible pair, it is relatively simple to predict the ternary equilibria. For systems of type I, which has a plait point, reliable calculations are much more difficult. However, sometimes useful quantitative predictions can be obtained for type I systems with binary data alone provided that [c.63]

The continuous line in Figure 16 shows results from fitting a single tie line in addition to the binary data. Only slight improvement is obtained in prediction of the two-phase region more important, however, prediction of solute distribution is improved. Incorporation of the single ternary tie line into the method of data reduction produces only a small loss of accuracy in the representation of VLE for the two binary systems. [c.69]

On triangular diagrams, comparisons of calculated and experimental results can be deceiving. A more realistic representation is provided by Figure 18, comparing experimental solute distributions with those calculated from the UNIQUAC equation for four ternary systems. For three of these systems, calculations were made using the parameters determined from binary data plus one ternary tie line however, for the 2,2,4-trimethylpen-tane-furfural-cyclohexane system, parameters were obtained from binary data alone. With the exception of the region very near the plait point, calculated distributions are good. Fortunately, commercial extractions are almost never conducted near the plait point since the small density difference in the plait-point region causes hydrodynamic difficulties (flooding). [c.71]

Two further examples of type I ternary systems are shown in Figure 19 which presents calculated and observed selectivities. For successful extraction, selectivity is often a more important index than the distribution coefficient. Calculations are shown for the case where binary data alone are used and where binary data are used together with a single ternary tie line. It is evident that calculated selectivities are substantially improved by including limited ternary tie-line data in data reduction. [c.71]

Liquid-Liquid Equilibria for Four (or More) Components [c.71]

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. [c.76]

In Equation (15), the third term is much more important than the second term. The third term gives the enthalpy of the ideal liquid mixture (corrected to zero pressure) relative to that of the ideal vapor at the same temperature and composition. The second term gives the excess enthalpy, i.e. the liquid-phase enthalpy of mixing often little basis exists for evaluation of this term, but fortunately its contribution to total liquid enthalpy is usually not large. [c.86]

While many methods for parameter estimation have been proposed, experience has shown some to be more effective than others. Since most phenomenological models are nonlinear in their adjustable parameters, the best estimates of these parameters can be obtained from a formalized method which properly treats the statistical behavior of the errors associated with all experimental observations. For reliable process-design calculations, we require not only estimates of the parameters but also a measure of the errors in the parameters and an indication of the accuracy of the data. [c.96]

For each experiment, the true values of the measured variables are related by one or more constraints. Because the number of data points exceeds the number of parameters to be estimated, all constraint equations are not exactly satisfied for all experimental measurements. Exact agreement between theory and experiment is not achieved due to random and systematic errors in the data and to "lack of fit" of the model to the data. Optimum parameters and true values corresponding to the experimental measurements must be found by satisfaction of an appropriate statistical criterion. [c.98]

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). [c.107]

There is justification for allowing t to increase beyond 1, and in many particular applications this may be desirable. Here a more conservative approach is used to reduce the chance of unstable iterations. [c.116]

The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. [c.117]

Calculations for wide-boiling mixtures are a little more difficult to converge, especially for mixtures having very light or noncondensable components together with relatively nonvolatile components and lacking components of intermediate volatility. [c.124]

Flash calculations for these mixtures usually require four to eight iterations. Cases 5 and 6 in Table 1 have feeds of this type, including noncondensable components in Case 6. Within the limits of the thermodynamic framework used here, no case has been encountered where FLASH has required more than 12 iterations for satisfactory convergence. [c.124]

Liquid-liquid equilibrium separation calculations are superficially similar to isothermal vapor-liquid flash calculations. They also use the objective function. Equation (7-13), in a step-limited Newton-Raphson iteration for a, which is here E/F. However, because of the very strong dependence of equilibrium ratios on phase compositions, a computation as described for isothermal flash processes can converge very slowly, especially near the plait point. (Sometimes 50 or more iterations are required. ) [c.124]

Appendix C-5 lists selected UNIQUAC binary parameters and characteristic binary parameters for noncondensable-condensable interactions for 150 binary pairs. For any binary pair, the parameters shown are believed to be the best now available. Parameters listed here were chosen from the more extensive lists in Appendix C-6 and C-7. A12 and A21 correspond to the UNIQUAC [c.144]

More general forms of the Gibbs-Duhem equation have been derived to allow for variations in temperature or pressure (or both) but these are not useful for our purposes since they are not easily integrated. Equation (16) is satisfied by various simple algebraic forms relating an y to x well-ltnown examples are the Margules and van Laar equations but many others exist. The particular relation used in this work, the UNIQUAC equation, while significantly different from the equations of Margules and van Laar, is also a solution to the Gibbs-Duhem differential equation. [c.20]

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]

As illustrated here with the UNIQDAC equation, an optimum set of binary parameters can be obtained using simultaneously binary VLE data, binary LLE data, and one (or more) ternary tieline data. The maximum-likelihood principle described in Chapter 6 provides the basis for parameter estimation. The parameters obtained give good representation of ternary data for a wide variety of systems. More important, however, as outlined here, calculations based on a model for the excess Gibbs energy provide a systematic procedure for predicting VLE and LLE for systems containing more than three components. [c.79]

The off-diagonal elements of the variance-covariance matrix represent the covariances between different parameters. From the covariances and variances, correlation coefficients between parameters can be calculated. When the parameters are completely independent, the correlation coefficient is zero. As the parameters become more correlated, the correlation coefficient approaches a value of +1 or -1. [c.102]

Equation (7-8). However, for liquid-liquid equilibria, the equilibrium ratios are strong functions of both phase compositions. The system is thus far more difficult to solve than the superficially similar system of equations for the isothermal vapor-liquid flash. In fact, some of the arguments leading to the selection of the Rachford-Rice form for Equation (7-17) do not apply strictly in the case of two liquid phases. Nevertheless, this form does avoid spurious roots at a = 0 or 1 and has been shown, by extensive experience, to be marltedly superior to alternatives. [c.115]

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]

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. [c.120]

As the feed composition approaches a plait point, the rate of convergence of the calculation procedure is markedly reduced. Typically, 10 to 20 iterations are required, as shown in Cases 2 and 6 for ternary type-I systems. Very near a plait point, convergence can be extremely slow, requiring 50 iterations or more. ELIPS checks for these situations, terminates without a solution, and returns an error flag (ERR=7) to avoid unwarranted computational effort. This is not a significant disadvantage since liquid-liquid separations are not intentionally conducted near plait points. [c.127]

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]

At temperatures above those corresponding to the highest experimental pressures, data were generated using the Lyckman correlation all of these were assigned an uncertainty of 5% of the standard-state fugacity at zero pressure. Frequently, this uncertainty amounts to one half or more atmosphere for the lowest point, and to 1 to 5 atmospheres for the highest point. [c.142]

NOTE - r NG GIl ES THE TENPERArURE RANGE tKl OF THE EXPERIMENTAL DATA USED TO FIT THE CONSTANTS CONSTANTS FOR NCNCONDENSABLES CCOMPONENTS 1-B) MERE DETERMINED FROM A GENERALIZED CORRELATION FOR THE HYPOTHETICAL REFERENCE FUGACITY. [c.154]

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