Pure-component Parameter Estimation

SAFT intermolecular-potential model parameters are traditionally determined by fitting to experimental data, as other equations of state for pure compound parameters vapour pressure and saturated liquid densities are typically used. These properties are chosen as the vapour pressure is of key interest in practical applications and depends strongly on the energy parameters, while the use of liquid density data is important in determining size related parameters. 

A least-squares objective function with the chosen residuals, given by 

A different approach is to reduce the number of parameters that need to be determined by proposing transferable models. Towards this end, the use of 

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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. 

To estimate the pure component parameters, we used the technique of Panagiotopoulos and Kumar (11). The technique provides parameters that exactly reproduce the vapor pressure and liquid density of a subcritical component. Table II presents the pure component parameters that were used. For the supercritical components, the usual acentric factor correlation was utilized. 

In addition to pure component parameters, mixture calculations require the estimation of the unlike-pair interaction parameters. These were obtained in this study using the Lorenz-Berthelot rules ... 

For biomaterials that are thermally unstable and decompose before reaching the critical temperature, several estimation techniques are available. We have used the Lydersen group contributions method ( ). Other techniques available for predicting critical properties have been reviewed and evaluated by Spencer and Daubert ( ) and Brunner and Hederer Qfi). It is also possible to determine the EOS parameters from readily measurable data such as vapor pressure, and liquid molar volume instead of critical properties (11). We used the Lydersen method to get pure component parameters because the vapor compositions we obtained were in closer agreement with experiment than those we got from pure component parameters derived by Brunner s method. The critical properties we used for the systems we studied are summarized in Table II. 

Mixtures Both liquid and vapor densities can be estimated using pure-component CS and EoS methods by treating the fluid as a pseudo-pure component with effective parameters calculated from the pure-component parameters and using ad hoc mixing rules. 

A number of methods based on regular solution theory also are available. Only pure-component parameters are needed to make estimates, so they may be applied when UNIFAC group-interaction parameters are not available. The Hansen solubility parameter model divides the Hildebrand solubility parameter into three parts to obtain parameters 8d, 5p, and 5 accounting for nonpolar (dispersion), polar, and hydrogenbonding effects [Hansen,/. Paint Technot, 39, pp. 104-117 (1967)]) An activity coefficient may be estimated by using an equation of the form... 

When volumetric data are available for a polar gas or for hydrocarbon gas mixtures containing polar species, the Won modification of the Soave equation of state can be used to determine the two pure-component parameters Ap( Tc) and a for the polar compound. With these parameters, the Won modified equation of state provides good estimates of VLE K-ratios of trace polar compounds in hydrocarbon-rich mixtures. 

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... 

SolventPro also has the option to use the original UNIFAC model as one of the GE-model or the PC-S AFT equation of state [9,10] to estimate the solid solnbility as shown in Fignre 10.13. Note that the PC-SAFT pure-component parameters are predicted throngh a GC method [9, 10]. 

FIGURE 10.13 SLE for binary mixture of aspirin (1) + 2-proponal (2) using group contoibu-tion method for estimating PC-SAFT pure component parameters, a Experimental data [53] Eutectic point. 

Using the pure-component parameters given in Table 10.5 in combination with the mixing rules (Eq. (10.43)), the depth of the potential ( u/k) and the temperature-independent segment diameter of the mixture can be estimated ... 

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]. 

Kouskoumvekaki, LA., von Solms, N., Lindvig, T, Michelsen, M.L., and Kontogeorgis, G.M., 2004. Novel method for estimating pure-component parameters for polymers Application to the PC-SAFT equation of state. Ind. Eng. Chem. Res., 43 2830-2838. 

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. 

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. ... 

In this equation, all molecules are divided into four groups paraffins (P), olefins (O), naphthenics (N), and aromatics (A). The v values represent the volume fractions of each component used, while the fa values are the blending values, which were calculated for each of the molecular lumps shown in Table 2. Pure component octane numbers used are designated as ON/, but one should note that in the development of the model, 57 molecular lumps were made based on GC analysis, and pure component ONs were assigned to each lump, and not necessarily each pure component. The kt values are calculated interaction parameters between paraffins, olefins, and naphthenics, and are also shown in Table 2. Based on this equation, and knowing the composition and pure octane numbers of a fuel mixture, an estimation of the blending ON may then be made. 

If the potential parameters for the pure components are not found in the tables given in (Hll) and (Bll), and if viscosity and second virial data are not available for their determination, then for the Lennard-Jones (6-12) potential it is possible as a last resort to estimate these parameters from the properties of the substance at its critical point c, its melting point m, or its boiling point b these relations give /k in °K. and a in Angstrom units (1 A. = 10-3 cm.) ... 

The estimated pure component spectra K (or the measured pure component spectra, K, if they are sufficiently relevant) are the parameters for the CLS model. Once they are determined, the concentrations for all analytes in a future sample (cp) can be estimated from the spectrum of that sample (xp) ... 

Semikinetic modeling is illustrated by a generalized model for naphtha pyrolysis. The empiricism associated with the semikinetic model dictates the need for an extensive data base for parameter estimates. The naphtha data base consists of about 400 tests covering pure components and their mixtures and 17 naphthas (25). The pure components studied were normal and isopentanes, cyclohexane, and n-heptane. The wide range of naphtha feed properties is summarized in Table III. 

Graessley et al, (1994,1995) have proposed an alternative scheme that correlates the x values for polyolefin blends with differences in inferred values of the solubility parameters of the pure components [see Eq. (2-31)]. For the series of polyolefins they studied, values of for the pure components can be assigned in such a way that the x parameters of the mixtures computed from Eq. (2-31) are in reasonable qualitative agreement with measured values. Direct estimates of the (5 s can be obtained from the following formula derived from the van der Waals cohesive energy density (Hildebrand and Scott 1950 Krishnamoorti et al. 1996) ... 

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