Pure-component parameters values

Determination of pure component parameters. In order to use the EOS to model real substances one needs to obtain pure component below its critical point, a technique suggested by Joffe et al. (18) was used. This involves the matching of chemical potentials of each component in the liquid and the vapour phases at the vapour pressure of the substance. Also, the actual and predicted saturated liquid densities were matched. The set of equations so obtained was solved by the use of a standard Newton s method to yield the pure component parameters. Values of exl and v for ethanol and water at several temperatures are shown in Table 1. In this calculation vH and z were set to 9.75 x 10"6 m3 mole"1 and 10, respectively (1 ). The capability of the lattice EOS to fit pure component VLE was found to be quite insensitive to variations in z (6[Pg.90]

For nonpolar, nonhydrocarbon vapor mixtures at high pressures, the method of Dean and Stiel [Eq- (2-102)] discussed earlier can be used. The accuracy of the method is excellent and dependent on the pure component viscosity values used as input parameters. 

The evaluation of the sublimation pressure is a problem since most of the compounds to be extracted with the supercritical fluids exhibit sublimation pressures of the order of 10 14 bar, and as a consequence these data cannot be determined experimentally. The sublimation pressure is thus usually estimated by empirical correlations, which are often developed only for hydrocarbon compounds. In the correlation of solubility data this problem can be solved empirically by considering the pure component parameters as fitting-parameters. Better results are obviously obtained [61], but the physical significance of the numerical values of the parameters obtained is doubtful. For example, different pure component properties can be obtained for the same solute using solubility data for different binary mixtures. 

In order to test the applicability of the model to polymer-SCF systems, a hypothetical system of CC>2 and a monodisperse -mer with a monomeric unit molecular weight of 100 was simulated. Pure component parameters for the polymer, polystyrene, were obtained from Panayiotou and Vera (16). Constant values of kj< were used for the polymer system, where the degree of polymerization, , varied between 1 and 7. It was assumed that all chains had the same e, and v scaled as the molecular weight of the chain. Figure 5 shows the results of the predicted mole fraction of the -mer in the SCF phase. 

Simulations of ternary systems were performed using the pure component parameters in Table I and the cross parameters for the systems acetone/ CO2 and water/C02 determined previously (fi j - 1 and 0.81 respectively). Because of expected difficulties similar to the ones mentioned for the water/C02 system, no attempt was made to simulate the system acetone/water near room temperature. Thus, we set the acetone/water interaction parameters to the values from the Lorenz-Berthelot rules with fi j-l. Direct simulations of ternary phase equilibria have not been previously reported to the best of our knowledge. 

These interaction parameters are used in place of the corresponding pure-component parameters to determine the B,j values. 

Numerical values for the pure-component parameters 9 and ate detemiined from Eqs. (1.3-26)-(1.3-28) ... 

Here, irm is the value of parameter tt for a mixture, and the quantity characterizes interactions between species i and /. If j = i, then tt == iru( = TTf), the parameter for pure i. Once a scheme is devised for assignment of numbers to pure-component parameters, the value of 7r< is unambiguous. 

The difficulty in the application of Equation 29 is in the assignment of values to tt when i /. While such a parameter is said to represent the effects of unlike-pair interactions, it is often difficult to give this idea a clear physical or mathematical statement. Thus one usually resorts to combination rules, which relate nto the pure-component parameters and (possibly) to an empirical interaction parameter, a number (by optimistic definition) usually either of order zero or of order unity. 

Methanol in Hydrocarbon-Rich Vapor and Liquid. The volumetric properties of methanol gas (12) and the second virial cross coefficients of methanol and light gases (13) were used to determine the pure-component parameters AP(TC) and a for methanol. Table II shows the enthalpy departure of gaseous methanol from ideal gas at three temperatures and several pressures. For comparison, the experimental values (14) and the values calculated by the Soave equation (1) are also shown. Table II indicates that the Won modified equation of state predicts the enthalpy departure of methanol very well at low temperatures and fairly well at high temperatures, but that the original Soave equation considerably underestimates the enthalpy departure at all temperatures and pressures. Since the original Soave equation was meant to be applied only to hydrocarbons, we are not surprised at this result. Comparison of calculated and experimental second virial cross coefficients between methanol and methane (and also C02) is presented elsewhere (15).)... 

Once procedures for calculating pure-component parameters and mixing rules are established, the calculation of component fugacity coefficients 4>i for both vapor and liquid phases follows standard procedures (see e.g. (4)). For VLE calculations, the distribution of components between phases is expressed generally as the K-value—the vapor mole fraction divided by the liquid mole fraction—related to fugacity coefficients for each component by ... 

For rubbery phases, the models are used in their original equilibrium formulations, which requires knowledge of the pure component parameters and the binary interaction parameters entering the mixing rules associated with the models. The former can be retrieved from pure component volumetric data at different temperatures and pressures and, when applicable, from vapor pressure data for each pair of substances the binary parameter is either retrieved from volumetric data or adjusted to the solubility data. In several cases, the default value offers a reasonable estimation of the solubility isotherms. 

It is conventional to estimate values for unlike parameters (such as a,y) by combining the pure-component parameters (a,-,- = Apu i = pure j)/ such prescriptions... 

The ratio Uo/k or u/k is analogous to the characteristic parameter T in the equations above. There are two additional volume and energy parameters if association is taken into account. In its essence, the SAFT equation of state needs three pure component parameters which have to be fitted to equilibrium data V , Uj/k and r. Fitting of the segment number looks somewhat curious to a polymer scientist, but it is simply a model parameter, like the c-parameter in the equations above, which is also proportional to r. One may note additionally that fitting to specific volume PVT-data leads to a characteristic ratio r/M (which is a specific r-value), as in the equations above, with a specific c-parameter. Several modifications and approximations within the SAFT-framework have been developed in the literature. Banaszak et or Shukla and Chapman extended the concept to copolymers. 

The molecular parameters characterizing the pure components and the mixtures in the S-S theory, are taken from reference [6], The pure component parameters were estimated from equation of state data [13,14]. Values for the mixing parameters e i2 and v i2 were adjusted to give quantitative agreement between the computed and experimental critical conditions. Since all the model parameters are available, we are in a position to predict other thermodynamic properties. As an example, spinodal conditions are considered. Details concerning the computational methods have been presented elsewhere [5]. It can be observed in Figure 1 that, in comparison to the experimental spinodals, the predicted spinodals become too narrow with decreasing molar mass. If the flexibility parameter c is allowed to vary with molar mass in a manner dictated by the experimental spinodal data, a quantitative description of these data can be obtained [6]. 

According to the regular solution theory of Hildebrand the f-parameter can be approximated by Eq. (41) [8], where Vs is the molar volume of the solvent and 4 and Sp are the solubility parameters of solvent and polymer, respectively. Since these solubility parameters are pure component parameters, Eq. (41) combined with Eq. (27) results in a predictive model. However, since many simplifications are involved, the results of this model can be considered as only a rough estimate. Following the slogan like dissolves like , a good solvent for a polymer is a solvent for which Ss and Sp have similar values. 

Both SAFT and PC-SAFT contain pure component parameters the energy parameter or u, the hard-sphere diameter a, or the hard-sphere volume and the number of segments m per molecule. For small (solvent) molecules these parameters are obtained from a fit of vapor pressure data and saturated liquid volume data. Since they do not have a vapor pressure, this fit is not possible for polymers, and the pure component polymer parameters are obtained from a fit to PVT data of the molten polymer or from a fit to PVT data and binary phase equilibrium data. For the description of a mixture one needs one binary interaction parameter ky per binary, which has to be fitted to phase equilibrium data. If necessary, ky can be made temperature-dependent. In general, phase equilibria are very sensitive to the kij value. 

The BWRS equation is Starling s modification of the Benedlct-Webb-Rubin equation of state. It contains eleven adjustable pure component parameters plus a binary interaction parameter for each component pair. Thus, a typical 20 component mixture would be characterized by 220 pure component parameters and 180 different binary interaction parameters—a total of AOO constants. Exxon s set of constants were determined by multi-property regression, a procedure in which parameters are adjusted until available data on density, enthalpy, vapor pressure, K-values, sonic velocity, and specific heats are all matched simultaneously. The large number of constants to be determined requires that these data be accurate and that they cover a wide range of conditions. Nearly 20,000 data points were used to determine our set of constants. The large amount of data required limits the components that can be handled to the relatively few for which such data exist and may also place a practical limit on the number of parameters desirable in an equation of state. 

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. ... 

PARIN loads values of pure component and binary parameters from formatted card images into labeled common blocks /PURE/ and /BINARY/ for a maximum of 100 components. 

A key feature of this model is that no data for mixtures are required to apply the regular-solution equations because the solubiHty parameters are evaluated from pure-component data. Results based on these equations should be treated as only quaHtative. However, mixtures of nonpolar or slightly polar, nonassociating chemicals, can sometimes be modeled adequately (1,3,18). AppHcations of this model have been limited to hydrocarbons (qv) and a few gases associated with petroleum (qv) and natural gas (see Gas, natural) processiag, such as N2, H2, CO2, and H2S. Values for 5 and H can be found ia many references (1—3,7). 

The terms p, T, and v are characteristic reducing parameters which may be obtained by fitting pressure-volume-temperature data (density, thermal expansion coefficient, and thermal pressure coefficient) for each pure component in the mixture (3,12). Values of p, v, and T are given in Tables I and II. 

This introduces the gas-phase residence time VgjQg as a new parameter. It also introduces an ambiguity regarding the term kgAi(a — Ug). There is no resistance to mass transfer within a pure component so kgAj oo and a — Ug 0. Thus, kgAi(a — Ug) is an indeterminate form of the oo x 0 variety. Its value must continue to equal the rate at which oxygen is transferred into the liquid phase. Equation (11.5) remains true and the pair of simultaneous ODEs become... 

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