On the Riemannian structure of the residue curves maps

Abstract

a b s t r a c tIn this paper, we revise the structure of the residue curve maps (RCM) theory of simple evaporation from the point of view of Differential Geometry. RCM are broadly used for the qualitative analysis of distillation of multicomponent mixtures within the thermodynamic equilibrium model. Nevertheless, some of their basic properties are still a matter of discussion. For instance, this concerns the connection between RCM and the associated boiling temperature surface and the topological characterization of the distillation boundaries. In this paper we put in evidence the Riemannian metric hidden behind the thermodynamic equilibrium condition written in the form of the van der Waals-Storonkin equation, and we show that the differential equations of residue curves have formal gradient structure. We discuss the first non-trivial consequences of this fact for the RCM theory of ternary mixtures.

Classification for ternary flash point mixtures diagrams regarding miscible flammable compounds

“…As it was recently shown in Ref. [20], eq. (8) implies that the equilibrium vector v restricted to the boiling temperature surface is equal to the generalized gradient of the boiling temperature function T b :…”

“…The only possible types of the RCMs singular points are imposed by the gradient type of the dynamical system in eq. (6) [20]. Moreover, according to eq.…”

“…Below we will mark by (*) the parameters related to the saturated pressure reference state. (20) Recall now that system S' and the closed-cup flash point system S have the same activity coefficients for given pressure, temperature and liquid composition. Therefore g 0 i ≡g i .…”

Review of extractive distillation. Process design, operation, optimization and control

“…The stationary points of the set of differential equations are pure components and azeotropes (Kiva et al, 2003). Since it is proven that residue curves are solutions to the gradient system with respect to the boiling temperature (Shcherbakova et al, 2015), the stable node is the highest boiling point of the distillation region, the unstable node is the lowest boiling point and saddle points are intermediate boiling points. Supplanting less compact classifications (Matsuyama and Nishimura, 1977;Doherty and Caldarola, 1985;Peterson and Partin, 1997), Serafimov's topological classification of 26 ternary diagrams is now used (Serafimov, 1970(Serafimov, , 1996 and is completed for zeotropic ternary diagrams by Reshetov and Kravchenko (2007) with 33 zeotropic classes, which are distinguished by different volatility order regions.…”