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VAPOR-LIQUID EQUILIBRIA Using the Gibbs Energy and the Common Tangent Plane Criterion

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VAPOR-LIQUID EQUILIBRIA Using the Gibbs Energy and the Common Tangent Plane Criterion ChE curriculum VAPOR-LIQUID EQUILIBRIA Using the Gibbs Energy and the Common Tangent Plane Criterion MARÍA DEL MAR OLAYA, JUAN A. REYES-LABARTA, MARÍA DOLORES SERRANO, ANTONIO MARCILLA University of Alicante • Apdo. 99, Alicante 03080, Spain hase thermodynamics is often perceived as a difficult overall composition. This is the case with the binary system subject with which many students never become fully in Figure 1(a); it is homogeneous for all compositions. The gM comfortable. It is our opinion that the Gibbsian geo- vs. composition curve is concave down, meaning that no split Pmetrical framework, which can be easily represented in Excel occurs in the global mixture composition to give two liquid spreadsheets, can help students to gain a better understanding phases. Geometrically, this implies that it is impossible to find of phase equilibria using only elementary concepts of high two different points on the gM curve sharing a common tangent school geometry. line. In contrast, the change of curvature in the gM function Phase equilibrium calculations are essential to the simula- as shown in Figure 1(b) permits the existence of two conju- tion and optimization of chemical processes. The task with gated points (I and II) that do share a common tangent line these calculations is to accurately predict the correct number and which, in turn, lead to the formation of two equilibrium of phases at equilibrium present in the system and their com- liquid phases (LL). Any initial mixture, as for example zi in positions. Methods for these calculations can be divided into Figure 1(b), located between the inflection points s on the M 2 M dx2 two main categories: the equation-solving approach (K-value g curve, is intrinsically unstable (d g / i <0) and splits method) and minimization of the Gibbs free energy. Isofugac- Antonio Marcilla is a professor of chemical engineering at Alicante Uni- ity conditions and mass balances form the set of equations in versity. He has presented courses in Unit Operations, Phase Equilibria, the equation-solving approach. Although the equation-solving and Chemical Reactor Laboratories. His research interests are pyrolisis, approach appears to be the most popular method of solution, it liquid-liquid extraction, polymers, and rheology. He is also currently involved with the study of polymer recycling via catalytic cracking and does not guarantee minimization of the global Gibbs energy, the problem of simultaneous correlation of fluid and condensed phase which is the thermodynamic requirement for equilibrium. equilibria. This is because isofugacity criterium is only a necessary but María del Mar Olaya completed her B.S. in chemistry in 1992 and Ph.D. in chemical engineering in 1996. She teaches a wide range of courses not a sufficient equilibrium condition. Minimization of the from freshman to senior level at the University of Alicante, Spain. Her global Gibbs free energy can be equivalently formulated as research interests include phase equilibria calculations and polymer the stability test or the common tangent test. structure, properties, and processing. Juan A. Reyes-Labarta received both his B.S. and Ph.D. in chemical The Gibbs stability condition is described and has been engineering in 1993 and 1998, respectively, at the University of Alicante used extensively in many references.[1, 2] It has been more (Spain). After post-doctoral stays at Carnegie Mellon University (USA) and the Institute of Polymer Science and Technology-CSIC (Spain), he is now frequently applied in liquid-liquid equilibrium rather than a full-time lecturer in Separation Processes and Molding Design. vapor-liquid equilibrium calculations. Gibbs showed that a María Dolores Serrano is a recent chemical engineering graduate of the necessary and sufficient condition for the absolute stability of University of Alicante. She is currently a post-graduate student, working a binary mixture at a fixed temperature, pressure, and overall on different aspects of phase equilibria. composition is that the Gibbs energy of mixing (gM) curve at no point be below the tangent line to the curve at the given © Copyright ChE Division of ASEE 2010 236 Chemical Engineering Education (a) (b) Figure 1. Dimensionless Gibbs energy of mixing (gM) for a binary liquid mixture as a function of the molar fraction of component i (xi): (a) for a homogeneous system, and (b) for a heteroge- neous system (LL). and gL. Obviously, a common reference state must be used for each of the components in the calculations, and for both of the phases involved (for example the pure component as liquid at the same P and T). VLE exists at a given T and P, whenever any initial mixture, such as zi in Figure 2, is thermodynami- cally unstable and splits into a vapor and liquid phases having compositions yi and xi, respectively, with a common tangent line to both gV and gL curves and a lower value for the overall Gibbs energy of mixing (M’’<M’<M). The geometrical perspective of the Gibbs energy minimiza- tion is not new and there are interesting papers dealing with this topic in depth. The paper of Jolls, et al.,[4] for instance, presents images of thermodynamic fundamental and state Figure 2. Dimensionless Gibbs energy curves (gV, gL) for a binary mixture with a VLE region at constant T and P as functions for pure binary and ternary systems. These authors a function of the molar fraction of component i. discuss the relationship between the model geometry and stability criteria. This geometrical perspective, however, is not usually considered in practice when dealing with VLE x I x II into two liquid phases having compositions i and i , and a using local composition models for the liquid phases, although lower value for the overall Gibbs energy of mixing (M’<M). it clearly illustrates the conditions for stable equilibrium vs. Global mixtures located between points I and s or s and II other possible unstable situations. The goal of the present 2 M dx2 are metastable or locally stable (d g / i >0). Therefore, the paper is to show an example of the visualization of the VLE inflection points separate the metastable equilibrium region from a Gibbs perspective for teaching purposes. from the definitely unstable region.[3] Of course, the Gibbs stability criteria can be extended to EDUCATIONAL ASPECTS ternary or multicomponent systems, where the gM curve is M M Our objective is to propose an exercise to analyze the replaced by the g surface or g hyper-surface and the com- VLE using the Gibbs common tangent plane criterion using mon tangent line by the common tangent plane or hyper-plane, Excel and Matlab. In the engineering education literature, a respectively. number of papers concerning the use of spreadsheets have The analytical expression for the Gibbs energy in LLE been described.[5, 6] Excel spreadsheets are used mainly due calculations is the same for both phases (let it be denoted by to their simplicity and built-in graphics capabilities. Matlab gL). This is not the case for equilibria involving different ag- software, which has more powerful graphical tools, can be gregation state phases such as vapor-liquid equilibria (VLE), used to represent 3-D diagrams that support the graphical where different expressions for the Gibbs energy must be used interpretation of equilibrium since plots can be easily rotated for each phase, yielding two possible Gibbs energy curves: gV and manipulated to facilitate their understanding. Vol. 44, No. 3, Summer 2010 237 This activity is framed in the subject of Principals of Gibbs Energy for the Liquid Phase Separation Processes in Chemical Engineering in a fourth- For liquid mixtures the ideal Gibbs energy of mixing (di- year course of a five-year program of chemical engineering. mensionless) is This task takes place during the first part of the course and consists of: gid,L x ln x ()5 =∑ i ⋅ i a) A lecture of the theoretical fundamentals of VLE, presen- i tation and analysis of T-x,y diagrams and sketch of Gibbs stability criterion. where the reference state for each component i is the pure b) A guided classroom solution of different VLE problems liquid at the temperature and pressure of the system. with increasing difficulty. Previous presentation of a simple Many equations have been proposed to model the excess non-azeotropic VLE binary system, and consequent solu- Gibbs energy and can be found in the literature. For example, tion of a binary homogenous azeotropic mixture, where the van Laar equation for binary systems: students work on their own computers using Excel and AA⋅ ⋅x⋅ x Matlab. gEL, = 12 21 1 2 ()6 c) Development of a project in groups of students where an A12 ⋅ x1 + A 21⋅ x 2 analogous analysis of VLE, but also of LLE, must be done for a heterogeneous azeotropic binary mixture. where A12 and A21 are the binary interaction parameters of d) As optional projects, VLE for ternary or more compli- the model that must be obtained by equilibrium data cor- cated mixtures could also be considered, but the visual- relation. ization becomes more complicated. THEORY Guess parameters The equilibrium condition for component i being simultane- ously present as a liquid and vapor equilibrium phase can be written as the isofugacity condition Guess Tb i v L fi = fi ()1 c P v 0 V oL oL i p , J i ϕi ⋅P ⋅ yi=p i ⋅ϕi ⋅exp dP⋅γi ⋅x i (2) i i ∫Po i RT v where ϕi is the fugacity coefficient for the vapor phase, P is y p 0 J x P i the total pressure, yi and xi are the molar fractions of compo- i i i oL i nent i in the vapor and liquid phases, respectively, pi is the oL vapor pressure of component i, ϕi is the fugacity coefficient po for the liquid phase at i , γi is the activity coefficient, defined vc by the selected model, and i is the molar volume of the no n ¦ yi 1 condensed phase as a function of pressure.
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