Measured phase equilibrium data

The values of measured phase equilibrium data were plotted in P-T diagrams (fig.2) as lines of constant composition (shown for PEG 1500). On the abszisse a... 

One of the limitations of most phase-equilibrium data is that variances of experimental measurements are seldom known. 

The maximum-likelihood method is not limited to phase equilibrium data. It is applicable to any type of data for which a model can be postulated and for which there are known random measurement errors in the variables. P-V-T data, enthalpy data, solid-liquid adsorption data, etc., can all be reduced by this method. The advantages indicated here for vapor-liquid equilibrium data apply also to other data. 

In applying equation 33, Cpsl (the constant-pressure molar heat capacity of the stoichiometric liquid) is usually extrapolated from high-temperature measurements or assumed to be equal to Cpij of the compound, whereas the activity product, afXTjafXT), is estimated by interjection of a solution model with the parameters estimated from phase-equilibrium data involving the liquid phase (e.g., solid-liquid or vapor-liquid equilibrium systems). To relate equation 33 to an available data base, the activity product is expressed... 

The Si-As system has been reassessed based on the phase equilibrium data reported by Klemm and Pirscher [15], Ugay and Miroshnichenko [16], and Ugay et al. [17]. Arsenic solubility data measured by Trumbore [18], Sandhu and Reuter [19], Fair and Weber [20], Ohkawa et al. [21], Fair and Tsai [22], Miyamoto et al. [23] and the activity data given by Reuter [19], Ohkawa... 

A comprehensive collection of phase equilibrium data (including vapor-liquid, liquid-liquid, and solid-liquid data) is maintained by a group headed by Prof. Juergen Gmehling at the University of Oldenburg, Germany. This collection, known as the Dortmund Data Bank, includes LLE measurements as well as NRTL and UNIQUAC fitted parameters. The data bank also includes a compilation of infinite-dilution activity coefficients. The LLE collection is available as a series of... 

Phase equilibrium data have been measured for many binary mixtures with a special apparatus and are available in compiled form. These McCabe Thiele diagrams show the mole fraction of volatiles in the liquid phase in relation to the mole fraction of volatiles in the vapour phase in equilibrium at constant pressure [21, 22]. 

Activity coefficients, which play a central role in chemical thermodynamics, are usually obtained from the analysis of phase equilibrium measurements. However, with shifts in the chemical industry and the use of combinatorial chemistry, new chemicals are being introduced for which the needed phase equilibrium data may not be available. Therefore, predictive methods for estimating activity coefficients and phase behavior are needed. Group contribution methods, such as the ASOG [analytical solution of groups... 

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

Based on phase equilibrium data, a pilot plant for measuring the concentration profile along the column was built, and pilot production on a several-thousand-kg scale was undertaken (Fig. 13.4). 

Yet another type of internally consistent thermodynamic data base makes full use of phase equilibrium data as well as all other available thermochemical and physical measurements. The most complete example at present is the Berman, Brown and Greenwood (BBG) data base, described by Berman et al. (1984, 1985, 1986). Like the methods just described, the BBG approach analyzes all data simultaneously and produces an internally consistent set of thermodynamic data. However, the method differs significantly in using the mathematical technique called linear programming to deal with bracketed phase equilibrium data. 

This equation has units of moles it is important because T, P, N , V, and are all measurable. And although the excess chemical potentials cannot be measured directly, they can be extracted from phase-equilibrium data. (It is instructive to note that while absolute values for conceptuals, such as H, can never be measured, certain kinds of differences in conceptuals, such as and H ", can be.)... 

The ideal-gas and ideal-solution approaches also differ because they are based on different kinds of experimental data. The residual properties and fugadty coeffidents depend on volumetric data measurements of P, v, T, and x. But the excess properties and activity coeffidents depend on density measurements for calorimetric measurements for h, and phase-equilibrium data for and y,-. Modem modeling tends to rely on volumetric data (equations of state), and a prindpal feature of this chapter has been to establish how excess properties can be computed from residual properties and how activity coefficients can be computed from fugadty coeffidents. But note that such calculations can be performed in either direction that is, at least in principle. 

In the late 1970s when they looked over the existing literature, Hashizume et al. noticed that even for the simplest of quasi-binary solutions, i.e., ternary solutions containing two homologous polymers in a single solvent, systematic phase equilibrium data were virtually lacking. Since such data seemed essential for testing any proposed expression for g for quasi-binary solutions, they [22] undertook extensive measurements of cloud-point curves, binodals, and critical points on mixtures of two narrow-distribution polystyrenes f4 and flO dissolved... 

The FR method was introduced as a method for macroscopic measurement of diffusion coefficients, but it can also be used for measurement of equilibrium data. The diffusion coefficient is obtained from the locus of the maximum of the out-of-phase function, while the slope of the adsorption isotherm can be obtained from the low-frequency asymptote of the in-phase function [15]. Some applications of the FR methods to investigation of heterogeneous reaction systems have also been reported [46-51]. 

Figme 5 shows the phase equilibrium data of the system C02/methyl palmitate of both Inomata et al. [48] and Lockemarm [49]. Figure 5 hints towards significant differences between the data sets, yet insufficient information is available to determine which data set us superior. However, the data clearly shows that an increase in temperature results in an increase in phase transition pressure. Due to the scatter in the data and slight inconsistencies between the two sources the exact nature of the relationship between temperature and the phase transition pressure can not be determined. Additional measurements would be required therefore. Both sets of data do, however, indicate that total solubility can be achieved at moderate pressures (less than 25 MPa at temperatures below 343 K). 

Although distribution coefficients in metal extraction systems ate highly variable, they are theimody-namic quantities that can be measured and correlated as functions of the state of the system. Because of the multicomponent nature of metal extraction systems, one must take cate to identify a proper set of independent variables in constructing a correlation of phase equilibrium data. Another consideration in treating equilibrium extraction data is that one desires a systematic method for treating the effects of... 

As discussed in the chapters above reliable model parameters are most important. While mainly VLE data are used in the chemical industry, it is recommended to use all kinds of reliable data (phase equilibrium data (VLE, azeotropic data, SLE of eutectic systems, etc.), excess enthalpies) for fitting simultaneously -model parameters, which often have to be temperature dependent. To account only far the deviations from Raoult s law, it is recommended to use the pure component vapor pressures measured by the authors for every data set. This can be done by multiplying the vapor pressure with a correction factor, for the Antoine equation, this corresponds to changing the parameter A to A. Sometimes a large number of experimental data are available. 

The equilibrium constants, the UNIFAC interaction parameters, and the vapor pressure parameters of the components which do not exist in pure form have to be fitted simultaneously to phase equilibrium data and reaction equilibrium data obtained from spectroscopic measurements for the systems formaldehyde-water, formaldehyde-methanol, and the ternary system formaldehyde-water-methanol, for which large amounts of data are available. To keep the significance of the particular parameters, a number of simplifying assumptions are made. 

The predicted phase equilibrium is a strong function of the binary interaction parameters (BIPs). Process simulators have regression options to determine these parameters from experimental phase-equilibrium data. The fit gives a first-order approximation for the accuracy of the equation of state. This information should always be considered in estimating the accuracy of the simulatioa Additional simulations should be run with perturbed model parameters to get a feel for the uncertainty, and the user should realize that even this approach gives an optimistic approximation of the error introduced by the model. If BIPs are provided in the simulator and the user has no evidence that one equation of state is better than another, then a separate, conplete simulation should be performed for each of these equations of state. The difference between the simulations is a crude measure of the uncertainty introduced into the simulation by the uncertainty in the models. Again, the inferred uncertainty will be on the low side. 

Although the direct measurement of equilibrium data for mixtures at high pressures requires detailed experimental experience and expensive equipment, it is still an es sential and reliable way in order to obtain the data needed for the evaluation of high-pressure processes. Recently, Dohrn et al. (10] presented a classification of experimental methods for high-pressure phase equilibria. Figure 2.4 illustrates the two main groups analytical methods and synthetic methods. In case of analytical... 

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... 

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 (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

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