Correction: Maxwell-Stefan diffusion coefficient estimation for ternary systems: an ideal ternary alcohol system.

Correction for 'Maxwell-Stefan diffusion coefficient estimation for ternary systems: an ideal ternary alcohol system' by Tariq Allie-Ebrahim et al., Phys. Chem. Chem. Phys., 2017, 19, 16071-16077.

A predictive model for the diffusion of a highly non-ideal ternary system.

Diffusion plays a central part in many unit operations. The Maxwell-Stefan model is the dominant model for both gaseous and liquid diffusion. However, it was developed from the kinetic theory of gases, raising the question of whether it can be extended to non-ideal liquid systems. The dynamic fluctuation model is an alternative model based on the Cussler theory and predicts a smaller thermodynamic influence relative to the linear influence of the Maxwell-Stefan model due to dynamic concentration fluctuations. Since the dynamic fluctuation model, which uses the scaling factor α, had improved performance relative to the Maxwell-Stefan model for a wide range of binary systems, it is postulated that this improved performance should also be observed for a ternary system. In this work, the dynamic molecular fluctuation model was extended to a highly non-ideal ternary system, using the same scaling factor α, through matrix manipulation. Using self-diffusion data measured by NMR, mutual diffusion predictions of the developed model and the Maxwell-Stefan model were compared to experimental mutual diffusion data of the partially miscible system ethanol/toluene/n-decane. It is demonstrated that the dynamic fluctuation model gives improved predictions relative to the Maxwell-Stefan approach, consistent with previous observations on binary systems, showing that the reduced thermodynamic influence of the dynamic fluctuation model is an improvement. In addition, we show that the use of local mole fractions, to account for molecular association, in both the dynamic fluctuation and Maxwell-Stefan models, results in improved diffusion predictions for the ternary system. The results confirm that the dynamic fluctuation model improves predictions of mutual diffusion in liquid mixtures, suggesting a non-linear correction to the thermodynamic correction factor. The results also suggest that that the key assumptions in the Maxwell-Stefan model and its derivation, rooted in the kinetic theory of gases, are not entirely accurate for highly non-ideal liquid systems. The optimum α for the ternary system studied here is approximately 0.45, similarly to the optimum α of 0.40 to 0.80 for a range of binary systems previously studied, suggesting that the use of the α scaling factor, which is grounded in scaling laws theory, is of general validity.

Maxwell-Stefan diffusion coefficient estimation for ternary systems: an ideal ternary alcohol system.

The Maxwell-Stefan model is a popular diffusion model originally developed to model diffusion of gases, which can be considered thermodynamically ideal mixtures, although its application has been extended to model diffusion in non-ideal liquid mixtures as well. A drawback of the model is that it requires the Maxwell-Stefan diffusion coefficients, which are not based on measurable quantities but they have to be estimated. As a result, numerous estimation methods, such as the Darken model, have been proposed to estimate these diffusion coefficients. However, the Darken model was derived, and is only well defined, for binary systems. This model has been extended to ternary systems according to two proposed forms, one by R. Krishna and J. M. van Baten, Ind. Eng. Chem. Res., 2005, 44, 6939-6947 and the other by X. Liu, T. J. H. Vlugt and A. Bardow, Ind. Eng. Chem. Res., 2011, 50, 10350-10358. In this paper, the two forms have been analysed against the ideal ternary system of methanol/butan-1-ol/propan-1-ol and using experimental values of self-diffusion coefficients. In particular, using pulsed gradient stimulated echo nuclear magnetic resonance (PGSTE-NMR) we have measured the self-diffusion coefficients in various methanol/butan-1-ol/propan-1-ol mixtures. The experimental values of self-diffusion coefficients were then used as the input data required for the Darken model. The predictions of the two proposed multicomponent forms of this model were then compared to experimental values of mutual diffusion coefficients for the ideal alcohol ternary system. This experimental-based approach showed that the Liu's model gives better predictions compared to that of Krishna and van Baten, although it was only accurate to within 26%. Nonetheless, the multicomponent Darken model in conjunction with self-diffusion measurements from PGSTE-NMR represents an attractive method for a rapid estimation of mutual diffusion in multicomponent systems, especially when compared to exhaustive MD simulations.