# Sigmoid

In a binary liquid solution containing one noncondensable and one condensable component, it is customary to refer to the first as the solute and to the second as the solvent. Equation (13) is used for the normalization of the solvent s activity coefficient but Equation (14) is used for the solute. Since the normalizations for the two components are not the same, they are said to follow the unsymmetric convention. The standard-state fugacity of the solvent is the fugacity of the pure liquid. The standard-state fugacity of the solute is Henry s constant. [c.19]

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]

At moderate densities. Equation (3-lOb) provides a very good approximation. This approximation should be used only for densities less than (about) one half the critical density. As a rough rule, the virial equation truncated after the second term is valid for the present range [c.29]

Figure 1 shows second virial coefficients for four pure fluids as a function of temperature. Second virial coefficients for typical fluids are negative and increasingly so as the temperature falls only at the Boyle point, when the temperature is about 2.5 times the critical, does the second virial coefficient become positive. At a given temperature below the Boyle point, the magnitude of the second virial coefficient increases with [c.29]

Figure 3-1. Second virial coefficients for four fluids. |

Figure 3-2. Second virial coefficients for two binary systems. |

Bfi and 022- However, in the second binary, intermolecular forces between unlike molecules are much stronger than those between like molecules chloroform and ethyl acetate can strongly hydrogen bond with each other but only very weakly with them- [c.31]

The third edition of "Properties of Gases and Liquids" by Reid et al. (1977) lists useful group contribution methods for predicting critical properties. Contributions to the second [c.36]

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]

In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003 [c.68]

The enthalpy of a vapor mixture is obtained first, from zero-pressure heat capacities of the pure components and second, from corrections for the effects of mixing and pressure. [c.83]

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]

These systems are solved by a step-limited Newton-Raphson iteration, which, because of its second-order convergence characteristic, avoids the problem of "creeping" often encountered with first-order methods (Law and Bailey, 1967) [c.116]

In the generalized method of Hayden and O Connell (1975), the pure-component and cross second virial coefficients are given by the sum of two contributions [c.130]

Individual contributions to the second virial coefficient are calculated from temperature-dependent correlations [c.130]

As discussed in Chapter 3, the virial equation is suitable for describing vapor-phase nonidealities of nonassociating (or weakly associating) fluids at moderate densities. Equation (1) gives the second virial coefficient which is used directly in Equation (3-lOb) to calculate the fugacity coefficients. [c.133]

Equilibrium constants,, for all possible dimerization reactions are calculated from the metastable, bound, and chemical contributions to the second virial coefficients, B , as given by Equations (6) and (7). The equilibrium constants, K calculated using Equation (3-15). [c.133]

The total free contribution to the second virial coeffi-F [c.133]

The Lewis fugacity rule is used for calculating the fugacity coefficients of the true species, and (2) the second virial co- [c.134]

Judgment had to be exercised in data selection. For each fluid, all available data were first fit simultaneously and second, in groups of authors. Data that were obviously very old, data that were obviously in error, and data that were inconsistent with the rest of the data, were removed. [c.141]

OF SECOND CONST FUNC WRT INDE [c.240]

OF SECOND CONST FUNC WRT [c.240]

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]

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. [c.17]

The second term in Equation (15a) gives the enthalpy of mixing of the condensable components. It is difficult to estimate that enthalpy but fortunately it ma)ies only a small contri- [c.88]

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]

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. [c.211]

Subroutine BIJS2. This subroutine calculates the pure-component and cross second virial coefficients for binary mixtures according to the method of Hayden and O Connell (1975). [c.220]

Second card FORMAT(8F10.2), control variables for the regression. This program uses a Newton-Raphson type iteration which is susceptible to convergence problems with poor initial parameter estimates. Therefore, several features are implemented which help control oscillations, prevent divergence, and determine when convergence has been achieved. These features are controlled by the parameters on this card. The default values are the result of considerable experience and are adequate for the majority of situations. However, convergence may be enhanced in some cases with user supplied values. [c.222]

SECOND MEASURED DEPENDENT VARIABLES ERROR VARRIANCES OF SECOND MEASURED VAR IABLES [c.240]

See pages that mention the term

**Sigmoid**:

**[c.15] [c.16] [c.28] [c.28] [c.33] [c.34] [c.36] [c.41] [c.51] [c.84] [c.89] [c.118] [c.134] [c.136] [c.218] [c.219] [c.220] [c.224] [c.227]**

Advanced control engineering (2001) -- [ c.0 ]