Vapor-liquid equilibria using the Gibbs energy and the common tangent plane criterion

Phase thermodynamics is often perceived as a difficult subject that many students never become fully comfortable with. The Gibbsian geometrical framework can help students to gain a better understanding of phase equilibria. An exercise to interpret the vapor-liquid equilibrium of a binary azeotropic mixture, using the equilibrium condition based on the common tangent plane criterion (the Gibbs stability test), is presented in this paper. From a T-composition phase diagram for the mixture, the temperature is set at different values: above, intermediate to, and below the boiling temperatures of the pure components, to intersect different regions of the system. Students prepare an Excel spreadsheet where the Gibbs energy of mixing of the vapor and liquid mixtures are calculated and represented over the whole range of compositions and then, apply the Gibbs stability test to ascertain the aggregation state of the system and to calculate the VL phase equilibrium compositions. Finally, Matlab is used to generate the 3D Gibbs energy of mixing surfaces for both phases over the whole range of temperatures which facilitates the geometrical interpretation of the vapor-liquid equilibrium.

GE models and algorithms for condensed phase equilibrium data regression in ternary systems: limitations and proposals

Phase equilibrium data regression is an unavoidable task necessary to obtain the appropriate values for any model to be used in separation equipment design for chemical process simulation and optimization. The accuracy of this process depends on different factors such as the experimental data quality, the selected model and the calculation algorithm. The present paper summarizes the results and conclusions achieved in our research on the capabilities and limitations of the existing GE models and about strategies that can be included in the correlation algorithms to improve the convergence and avoid inconsistencies. The NRTL model has been selected as a representative local composition model. New capabilities of this model, but also several relevant limitations, have been identified and some examples of the application of a modified NRTL equation have been discussed. Furthermore, a regression algorithm has been developed that allows for the advisable simultaneous regression of all the condensed phase equilibrium regions that are present in ternary systems at constant T and P. It includes specific strategies designed to avoid some of the pitfalls frequently found in commercial regression tools for phase equilibrium calculations. Most of the proposed strategies are based on the geometrical interpretation of the lowest common tangent plane equilibrium criterion, which allows an unambiguous comprehension of the behavior of the mixtures. The paper aims to show all the work as a whole in order to reveal the necessary efforts that must be devoted to overcome the difficulties that still exist in the phase equilibrium data regression problem.

Pitfalls on computing liquid-liquid phase equilibria using the k-value method

Poster presented in the 11th Mediterranean Congress of Chemical Engineering, Barcelona, October 21-24, 2008.

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.

Gibbs Energy of Mixing Function: Topological Analysis in Azeotropic Systems

ESAT 2014. 27th European Symposium on Applied Thermodynamics, Eindhoven University of Technology, July 6-9, 2014.

Una descomposición genética para el concepto de recta tangente

Presentamos una descomposición genética del concepto de recta tangente vista como una trayectoria hipotética de aprendizaje. Para generar esta descomposición genética se ha realizado un análisis histórico de la génesis del concepto, un análisis de libros de texto de Bachillerato, una síntesis de los resultados de las investigaciones sobre la comprensión de la recta tangente y hemos tenido en cuenta los resultados de un cuestionario respondido por alumnos de Bachillerato. La descomposición genética integra las perspectivas analítica local y geométrica como medio para favorecer la tematización del esquema de recta tangente.

Earth’s Rotation: A Challenging Problem in Mathematics and Physics

A suitable knowledge of the orientation and motion of the Earth in space is a common need in various fields. That knowledge has been ever necessary to carry out astronomical observations, but with the advent of the space age, it became essential for making observations of satellites and predicting and determining their orbits, and for observing the Earth from space as well. Given the relevant role it plays in Space Geodesy, Earth rotation is considered as one of the three pillars of Geodesy, the other two being geometry and gravity. Besides, research on Earth rotation has fostered advances in many fields, such as Mathematics, Astronomy and Geophysics, for centuries. One remarkable feature of the problem is in the extreme requirements of accuracy that must be fulfilled in the near future, about a millimetre on the tangent plane to the planet surface, roughly speaking. That challenges all of the theories that have been devised and used to-date; the paper makes a short review of some of the most relevant methods, which can be envisaged as milestones in Earth rotation research, emphasizing the Hamiltonian approach developed by the authors. Some contemporary problems are presented, as well as the main lines of future research prospected by the International Astronomical Union/International Association of Geodesy Joint Working Group on Theory of Earth Rotation, created in 2013.