# Boyd

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]

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]

When the same standard-state fugacity is used in both phases. Equation (5) can be rewritten [c.15]

We make the simplifying assumption that both and are functions only of temperature, not of pressure and composition. For a condensable component it follows that at the same tempera-ture, . [c.22]

Figure 5 shows fugacity coefficients for the system acetaldehyde-acetic acid at 90°C and 0.25 atm. Calculations are based on the "chemical" theory of vapor imperfection. Although the pressure is far below atmospheric, fugacity coefficients for both components are well removed from unity. Because of strong dimerization between acetic acid molecules and weak dimerization between the other possible pairs, deviations from ideality are large, much larger than one might expect at this low pressure. [c.34]

In the first, both components strongly associate with themselves and with each other. In the second, only one of the components associates strongly. For both systems, representation of the data is very good. However, the interesting quality of these systems is that whereas the fugacity coefficients are significantly remote from unity, the activity coefficients show only minor deviations from ideal-solution behavior. Figures 6 and 7 in Chapter 3 indicate that the fugacity coefficients show marked departure from ideality. In these systems, the major contribution to nonideality occurs in the vapor phase. Failure to take into account these strong vapor-phase nonidealities would result in erroneous activity-coefficient parameters, a 2 21 [c.51]

Figure 4-8. Vapor-liquid equilibria for a binary system where both components solvate and associate strongly in the vapor phase. |

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the [c.61]

Equation (23) holds only when, for every component i, the same standard-state fugacity is used in both liquid phases. [c.63]

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. [c.67]

The method described here is based on the high degree of correlation of model parameters, in this case, UNIQUAC parameters. Thus, although a certain set of binary parameters may be best for VLE data, we are able to find other sets of binary parameters for the miscible binaries which significantly improve ternary LLE prediction while only slightly decreasing accuracy of representation of the binary VLE. Fitting ternary LLE data only, may yield unrealistic parameters that predict grossly erroneous results when used in regions not identical to those employed in data reduction. By contrast, fitting ternary LLE data simultaneously with binary VLE data, effectively provides constraints on the binary parameters, preventing them from attaining arbitrary values of little physical significance. Determination of a single set of parameters which can adequately represent both VLE and LLE is particularly important in three-phase distillation. [c.69]

Type B. Components 1 and 2 are only partly miscible with each other. Both 1 and 2 are completely miscible with all other components in the system (3 through m). Components 3 through m are also miscible in all proportions. Both binary and ternary data are needed for a reliable description of the multicomponent LLE [c.74]

Type C. Component 1 is only partially miscible with coin -ponents 3 through m, but it is totally miscible with component 2. Components 2 through m are miscible with each other in all proportions. Again, both binary data and ternary tie-line data are needed [c.74]

[c.88]

Figure 1 gives an enthalpy-concentration diagram for ethanol(1)-water(2) at 1 atm. (The reference enthalpy is defined as that of the pure liquid at 0°C and 1 atm.) In this case both components are condensables. The liquid-phase enthalpy of mixing [c.89]

Finally, Table 2 shows enthalpy calculations for the system nitrogen-water at 100 atm. in the range 313.5-584.7°K. [See also Figure (4-13).] The mole fraction of nitrogen in the liquid phase is small throughout, but that in the vapor phase varies from essentially unity at the low-temperature end to zero at the high-temperature end. In the liquid phase, the enthalpy is determined primarily by the temperature, but in the vapor phase it is determined by both temperature and composition. [c.93]

This chapter presents quantitative methods for calculation of enthalpies of vapor-phase and liquid-phase mixtures. These methods rely primarily on pure-component data, in particular ideal-vapor heat capacities and vapor-pressure data, both as functions of temperature. Vapor-phase corrections for nonideality are usually relatively small. Liquid-phase excess enthalpies are also usually not important. As indicated in Chapter 4, for mixtures containing noncondensable components, we restrict attention to liquid solutions which are dilute with respect to all noncondensable components. [c.93]

An apparent systematic error may be due to an erroneous value of one or both of the pure-component vapor pressures as discussed by several authors (Van Ness et al., 1973 Fabries and Renon, 1975 Abbott and Van Ness, 1977). In some cases, highly inaccurate estimates of binary parameters may occur. Fabries and Renon recommend that when no pure-component vapor-pressure data are given, or if the given values appear to be of doubtful validity, then the unknown vapor pressure should be included as one of the adjustable parameters. If, after making these corrections, the residuals again display a nonrandom pattern, then it is likely that there is systematic error present in the measurements. [c.107]

In both of these cases. Equation (7-14) becomes trivial as h or [c.114]

Figure 7-1. Incipient equilibrium vapor-phase compositions calculated with subroutine BUDET. |

Both vapor-liquid flash calculations are implemented by the FORTRAN IV subroutine FLASH, which is described and listed in Appendix F. This subroutine can accept vapor and liquid feed streams simultaneously. It provides for input of estimates of vaporization, vapor and liquid compositions, and, for the adiabatic calculation, temperature, but makes its own initial estimates as specified above in the absence (0 values) of the external estimates. No cases have been encountered in which convergence is not achieved from internal initial estimates. [c.122]

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function [c.125]

The data, both real and generated, were then fit to a function of the form [c.139]

BETA cols 11-20 oscillation control parameter default value is set equal to 0.25. To help prevent oscillations (thus slowing convergence) we not only require that the sum of squares, SSQ, decreases [c.222]

ALST -SSTL -I TMX BETA -RP [c.240]

ONE OR BOTH COMP ARE ORGANIC ACIDS. [c.266]

Standard-state fugacities at zero pressure are evaluated using the Equation (A-2) for both condensable and noncondensable components. The Rackett Equation (B-2) is evaluated to determine the liquid molar volumes as a function of temperature. Standard-state fugacities at system temperature and pressure are given by the product of the standard-state fugacity at zero pressure and the Poynting correction shown in Equation (4-1). Double precision is advisable. [c.308]

A step-limited Newton-Raphson iteration, applied to the Rachford-Rice objective function, is used to solve for A, the vapor to feed mole ratio, for an isothermal flash. For an adiabatic flash, an enthalpy balance is included in a two-dimensional Newton-Raphson iteration to yield both A and T. Details are given in Chapter 7. [c.319]

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]

Equation (1) is of little practical use unless the fuga-cities can be related to the experimentally accessible quantities X, y, T, and P, where x stands for the composition (expressed in mole fraction) of the liquid phase, y for the composition (also expressed in mole fraction) of the vapor phase, T for the absolute temperature, and P for the total pressure, assumed to be the same for both phases. The desired relationship between fugacities and experimentally accessible quantities is facilitated by two auxiliary functions which are given the symbols (f [c.14]

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]

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]

The bubble and dew-point temperature calculations have been implemented by the FORTRAN IV subroutine BUDET and the pressure calculations by subroutine BUDEP, which are described and listed in Appendix F. These subroutines calculate the unknown temperature or pressure, given feed composition and the fixed pressure or temperature. They provide for input of initial estimates of the temperature or pressure sought, but converge quickly from any estimates within the range of validity of the thermodynamic framework. Standard initial estimates are provided by the subroutines. [c.119]

See pages that mention the term

**Boyd**:

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