Comparative analysis of the performance of mixing rules for density prediction of simple chemical mixtures

Four different mixing rules (MRs) in three equations of state (EOSs) have been used to account for the intermolecular forces of attraction between dissimilar molecules of different substances that form simple mixtures. The combined effects of the co-volumes of all constituent species of the mixtures were also considered, and the densities of these simple mixtures were predicted. Thereafter, the density results obtained were compared with accurately simulated experimental density values, and the effectiveness of these MRs was determined and compared. The four MRs compared are geometric mean average (GMA), whole square root average (SRA), Expanded geometric average (EGA), and simple average (SA) of attractive force parameter. They were all used in Van der Waals, Redlich Kwong, and Peng Robinson EOSs for two simple mixtures: a binary system (Ammonia – Water system) and a ternary mixture (methyl acetate – water – toluene system). It was found that GMA and EGA gave reasonably accurate estimates of the mixture attractive force parameter (am) and hence good density prediction for both Ammonia – Water and Methyl acetate – Water – Toluene systems. SRA gave unrealistic values of mixture densities for both systems and was discarded. SA gave a somewhat good result with Peng Robinson EOS for the ammonia-water system, but not that good in Redlich Kwong EOS and very poor in Van der Waals EOS. SA does not give reasonable estimates of the mixture densities with the three EOSs considered for the methyl acetate – water – toluene system.

Induced Dualistic Geometry of Finitely Parametrized Probability Densities on Manifolds

This paper aims to describe the geometrical structure and explicit expressions of family of finitely parametrized probability densities over smooth manifold $M$. The geometry of family of probability densities on $M$ are inherited from probability densities on Euclidean spaces $\left\{U_\alpha \right\}$ via bundle morphisms, induced by an orientation-preserving diffeomorphisms $\rho_\alpha:U_\alpha \rightarrow M$. Current literature inherits densities on $M$ from tangent spaces via Riemannian exponential map $\exp: T_x M \rightarrow M$; densities on $M$ are defined locally on region where the exponential map is a diffeomorphism. We generalize this approach with an arbitrary orientation-preserving bundle morphism; we show that the dualistic geometry of family of densities on $U_\alpha$ can be inherited to family of densities on $M$. Furthermore, we provide explicit expressions for parametrized probability densities on $\rho_\alpha(U_\alpha) \subset M$. Finally, using the component densities on $\rho_\alpha(U_\alpha)$, we construct parametrized mixture densities on totally bounded subsets of $M$. We provide a description of inherited mixture product dualistic geometry of the family of mixture densities.