Chrastil Correlation Discontinuity Examples

Different Density Sources:  

Fitting solubility data to Chrastil’s correlation requires a PvT source for density as a function of temperature and pressure.  The resulting parameters are tied to that PvT source.  Using the parameters with a different PvT source for density creates a discontinuity from the original solubility data.  

To illustrate consider the fitting of naphthalene solubility data (Schmitt and Reid, J Chem Eng Data, 1986) to Chrastil’s Correlation using the Fundamental EOS (FEOS, Span and Wagner, 1996) as the source for density.  The resulting Chrastil parameters are -4279.082, -1.1321, 1.5146 for α, β, and γ (Greek characters are used for Chrastil’s A, B and C to signify using solubility data in mol/mol rather than g/L), respectively.  The resulting Absolute Average Relative Deviation (AARD) is 18.4%.  At 310 K and 100 bar, the FEOS provides a carbon dioxide density of 685.77 kg/m3 and this density with the Chrastil parameters leads to a calculated solubility of 6.45E-03 mol/mol.  However, using the Peng-Robinson EOS (PR-EOS) provides a density of 622.25 kg/m3 and a calculated solubility of 5.56E-03 mol/mol.  A 14% discontinuity in the solubility result is generated by using a different density source.  This discontinuity will not be consistent for all solutes and all conditions.  At different conditions, the difference between the FEOS and PR-EOS density values will change.  For different solutes, the dependence on density varies (evident with different values of γ).  For example, caffeine solubility data (Johannsen and Brunner, Fluid Phase Equilibria, 1994) leads to a γ value of 5.413541 (α of -4388.9373, β of -30.555812 and an AARD of 4.86%).  Using the fitted Chrastil parameters based on FEOS density leads to a 41% discontinuity in the solubility calculated (at 310 K and 100 bar) with the PR-EOS and FEOS density.  Thus, the Chrastil fitted parameters must only be used with the same density source as was used to fit the original solubility data.  

Different Fitting Techniques:

Fitting the log of the solubility data (to linearize the model) and then conducting the fit by minimizing the least squares error is a common practice.  This was broadly suggested in Chrastil’s original work and is readily available in tools such as Excel’s®️.   The resulting fitted parameters are then used to calculate and report the AARD performance of the fit.  However, the resulting parameters do not lead to a true minimum AARD.  The log scale and minimizing the least squared error both create issues.  

To illustrate (with Johannsen and Brunner’s caffeine data), Excel’s®️ multivariable regression tool (linearizing Chrastil’s equation by taking the natural logarithms and fitting ln(y) against 1/T and ln(ρ) ) provides Chrastil coefficients of -4170.0989, -26.816010, and 4.758303 for α, β, and γ, respectively.  The resulting AARD is 5.02% which is slightly higher than the true minimum AARD achieved through minimizing the AARD value in the fitting process.  In this illustration, the difference is minor and is likely associated with the caffeine data used only spanned a factor of 4 difference between the minimum and maximum measured solubility.  Bartle et al.’s pyrene data (J Chem Eng Data, 1990) spanned a factor of 27 (max to min measured solubility) and lead to a difference in the AARD of 4.95% to 5.54% in going from minimizing the AARD to minimizing the least squared error on a log-scale.  This fitting discontinuity is not nearly as significant as that created by using different density values but it is still a discontinuity that is unnecessary to introduce.   

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