Vapor-liquid equilibria data reduction

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. 

Vapor-Liquid Equilibrium Data Reduction for Acetone(1)-Methanol(2) System (Othmer, 1928)... 

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... 

Several techniques are available for measuring values of interaction second virial coefficients. The primary methods are reduction of mixture virial coefficients determined from PpT data reduction of vapor-liquid equilibrium data the differential pressure technique of Knobler et al.(1959) the Bumett-isochoric method of Hall and Eubank (1973) and reduction of gas chromatography data as originally proposed by Desty et al.(1962). The latter procedure is by far the most rapid, although it is probably the least accurate. 

The heat of mixing (excess enthalpy) and the excess Gibbs energy are also experimentally accessible, the heat of mixing by direct measurement and G (or In y) indirectly as a product of the reduction of vapor/liquid equilibrium data. Knowledge of and G allows calculation of S by Eq. (4-13) written for excess properties. 

When bienry vapor-liquid equilibrium data are reduced to yield binary parameters in UNlQUAC or in some other expression for g . it is usually not possible to obtain a unique set of bianty parameters that are in some significant sease "bsst for that bianry. Fora given binary mixture, bisary parameters are almost always at least partially correlated so that, when experimental uncertainties are taken into account, there are many sets of binary parameters dial can represent equally well the experimental data, When the gonl of data reduction is limited to representing the binary data, it does not matter which of these many sets of binary parameters is ured in the calculations. But when binary parameters are used lo predict ternary liquid-liquid equilibria (Type I), calculated results depend strongly on which set of binary parameters is used. 

Since all experimental data for vapor-liquid equilibrium have some experimental uncertainty, it follows that the parameters obtained from data reduction are not unique3. There are many sets of parameters that can represent the experimental data equally well, within experimental uncertainty. The experimental data used in data reduction are not sufficient to fix a unique set of best parameters. Realistic data reduction can determine only a region of parameters2. 

Other companies (e.g., Hoechst) have developed a slightly different process in which the water content is low in order to save CO feedstock. In the absence of water it turned out that the catalyst precipitates. Clearly, at low water concentrations the reduction of rhodium(III) back to rhodium(I) is much slower, but the formation of the trivalent rhodium species is reduced in the first place, because the HI content decreases with the water concentration. The water content is kept low by adding part of the methanol in the form of methyl acetate. Indeed, the shift reaction is now suppressed. Stabilization of the rhodium species and lowering of the HI content can be achieved by the addition of iodide salts. High reaction rates and low catalyst usage can be achieved at low reactor water concentration by the introduction of tertiary phosphine oxide additives.8 The kinetics of the title reaction with respect to [MeOH] change if H20 is used as a solvent instead of AcOH.9 Kinetic data for the Rh-catalyzed carbonylation of methanol have been critically analyzed. The discrepancy between the reaction rate constants is due to ignoring the effect of vapor-liquid equilibrium of the iodide promoter.10... 

Abbott, M. M. Van Ness, H. C. Vapor-liquid-equilibrium. 3. Data reduction with precise expressions for GE. AlChE J. 1975, 21, 62-71. 

EXPERIMENTAL METHODS, EQUIPMENT AND DATA REDUCTION 4.4.3.1 Vapor-liquid equilibrium (VLE) measurements... 

The data reduction of vapor-pressure osmometry (VPO) follows to some extent the same relations as outlined above. However, from its basic principles, it is not an equilibrium method, since one measures the (very) small difference between the boiling point temperatures of the pure solvent drop and the polymer solution drop in a dynamic regime. This temperature difference is the starting point for determining solvent activities. There is an analogy to the boiling point elevation in thermodynamic equilibrium. Therefore, in the steady state period of the experiment, the following relation can be applied if one assumes that the steady state is sufficiently near the vapor-liquid equilibrium and linear non-equilibrium thermodynamics is valid ... 

VAN1 Van Ness, H.C. Pedersen, F. Rasmussen, P. Vapor liquid equilibrium Part V. Data reduction by maximum likelihood. 7. 24(1978) 1055-1063. 

The data on boihng of concentrated polymeric solutions " demonstrate that in such sy stems thermodynamic, diffusional and rheological factors are of primary importance. The diagram of the hquid-vapor phase equilibrium is characterized by a decrease in the derivative dp/dT with the polymer concentration (dp/dT 0 at k 0). This leads to increase in both the nucleation energy and the detachment size of a bubble (see the equation [7.2.59]) and, consequently, to reduction of the bubbles generation frequency. Note that in reality the critical work, for a polymeric liquid may exceed the value predicted by the formula [7.2.59] because of manifestation of the elasticity of macromolecules. 

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