A novel method for detecting and computing univolatility curves in ternary mixtures
Introduction
Separation of liquid mixtures is one of the most important tasks in the process industry where distillation is the most widely used technique. Remarkably, almost every product on the market contains chemicals that have undergone distillation (Kiss, 2014). Beyond conventional distillation of binary and multi-component mixtures, several additional distillation techniques are developed for breaking azeotropes or separating close boiling mixtures: pressure swing distillation, azeotropic and extractive distillation. These techniques are covered at length in several textbooks and reviews (Skiborowski et al., 2014, Gerbaud and Rodriguez-Donis, 2014, Arlt, 2014, Olujic, 2014).
Preliminary conceptual design of distillation processes is based on the knowledge of the mixture thermodynamics and on the analysis of the residue curve maps (RCMs). RCMs are useful to assess the feasibility of splits since they approximate the column composition profiles under total reflux (Doherty and Malone, 2001, Petlyuk, 2004). RCMs also display azeotropes and distillation boundaries, as well as unidistribution and univolatility manifolds. These geometrical concepts have been reviewed in several works. Particularly, the review paper of Kiva et al. (2003) provides a comprehensive historical review of RCMs mainly taking into account the Serafimov’s classification of 26 RCM diagrams of ternary systems (Hilmen et al.,2002). The important role of RCMs, unidistribution and univolatility manifolds has been well described by Widagdo and Seider, 1996, Ji and Liu, 2007, Skiborowski et al., 2014 for azeotropic distillation process design and by Rodriguez-Donis et al., 2009a, Rodriguez-Donis et al., 2009b, Rodriguez-Donis et al., 2012a, Rodriguez-Donis et al., 2012b, Luyben and Chien, 2010, Petlyuk et al., 2015 for extractive distillation design purposes. Noteworthy, the existence and the position of the univolatility curve, a particular type of isovolatility curve, determine the component to be drawn as distillate as well as the configuration of the extractive distillation column.
Isovolatility curves are curves along which the relative volatility of a pair of species is constant:
Along univolatility curves . As their properties are closely related to those of residue curves (Kiva et al., 2003), univolatility, isovolatility, and isodistribution curves are useful for studying the feasibility of distillation processes. For example, the most volatile component is likely to be recovered in the distillate stream with the so-called direct split whereas the least volatile is likely to be in the bottom stream with the so-called indirect split. In azeotropic and extractive distillation processes an entrainer is added to the liquid mixture to be separated, in order to enhance the relative volatility between the components. If the isovolatility rate increases towards the entrainer vertex, it is a good indicator of an easy separation (Laroche et al., 1991, Wahnschafft and Westerberg, 1993, Luyben and Chien, 2010). Furthermore in extractive distillation, the location of the univolatility curve and its intersection with the composition triangle edge determine the component to be withdrawn as a distillate product from the extractive column as well as the proper column configuration (Laroche et al., 1991, Lelkes et al., 1998, Gerbaud and Rodriguez-Donis, 2014).
As shown by Kiva and Serafimov (1973), univolatility curves divide the composition space, , into different K-order regions. Zhvanetskii et al. (1988) proposed the main principles describing all theoretically possible structures of univolatility curves for zeotropic ternary mixtures and their respective location according to the thermodynamic relationship between the distribution coefficients of the light component i, the intermediate j and the heavy component m. 33 possible structures of univolatility curves were reported under the assumption that for every pair of components there exists only one univolatility curve. In the succeeding paper from the same group (Reshetov et al., 1990), the classification was refined by introducing the following nomenclature:
- –
: an arc shape univolatility curve whose terminal points belong to the same binary side of the composition triangle;
- –
: the univolatility curve connecting two different binary sides of the composition triangle.
Later Reshetov and Kravchenko (2010) extended their earlier analysis to ternary mixtures having at least one binary azeotrope. Their main observations are:
- (a)
more than one univolatility curve having the same component index “” can appear in a ternary diagram;
- (b)
the univolatility curve that does not start at the binary azeotrope can be either or type. An curve can cross a separation boundary of the RCM;
- (c)
a ternary azeotrope can be crossed by any type of univolatility curve;
- (d)
if two univolatility curves intersect at some point, this point is a tangential binary azeotrope or a ternary azeotrope. In both cases there is a third univolatility curve of complementary type passing through this point;
- (e)
transitions from to (or vice versa) can occur as univolatility curves depend on pressure and temperature of vapor – liquid equilibrium (VLE).
Despite the increasing application of univolatility curves in conceptual design, there still lacks a method allowing:
- (1)
detection of the existence of univolatility curves independently of the presence of azeotropes, which is particularly important in the case of zeotropic mixtures;
- (2)
simple and efficient computation of univolatility curves.
In fact, numerical methods available in most chemical process simulators allow mainly the computation of the univolatility curves linked to azeotropic compositions. Missing univolatility curves not connected to azeotropes will result in improper design of the extractive distillation process. This problem can be solved by a fully iterative searching in the ternary composition space providing the composition values with equal relative volatility (Bogdanov and Kiva, 1977). Skiborowski et al. (2016) have recently proposed a method to detect the starting point of the univolatility curve. They locate all pinch branches that bifurcate when moving from the pure component vertex along the corresponding binary sides. The robustness of this approach to handle complex cases, such as biazeotropy when more than one univolatility curve ends on the same binary side, is well demonstrated. Their algorithm is based on MESH equations, including mass and energy balances, summation constraints, and equilibrium conditions.
A less tedious and less time-consuming method is proposed in this paper. It is based on the geometry of the boiling temperature surface constrained by the univolatility condition. This approach will also require the computation of starting binary compositions independently of the azeotrope condition. Such starting points can be easily computed with a vapor-liquid equilibrium model by using the intersections of the distribution coefficient curves on each binary side of the ternary diagram (Kiva et al., 2003).
This paper is organized as follows: First, we revisit the properties of the univolatility curves in RCMs, and prove that they are formed by critical points of the relative compositions. Then, we show that the topology of unidistribution and univolatility curves follows from both the global geometrical structure of the boiling temperature surface and the univolatility condition considered in the full three-dimensional composition - temperature state space. Such a consideration leads to a simple algorithm for the numerical computation of the univolatility curves and other similar objects by solving a system of ordinary differential equations. The starting points of the univolatility curves computation can be detected from the relationship between the distribution coefficients related to the binary pair “”: the binary distribution coefficients , and the ternary coefficient describing the ternary mixture with the third component “” at infinite dilution. Finally, we illustrate our approach by considering several topological configurations of RCM for two distinctive ternary mixtures (thermodynamic model parameters for both mixtures are available online as the supplementary material to this article). The ternary mixture diethylamine – chloroform – methanol at different pressures has two univolatility curves with the same component index “” and one univolatility curve of type . We also applied our method to the well-known (though uncommon) case of the binary mixture benzene – hexafluorobenzene exhibiting two azeotropes at atmospheric pressure. Considering hexane as the third component of the ternary mixture, we trace out the transformation of the type of the univolatility curve from to with the variation of pressure. The transformation of the topological structure of the univolatility curves is properly computed by using the new computational method.
Section snippets
Basic definitions and notations
Consider an open evaporation of a homogeneous ternary mixture at thermodynamic equilibrium of the vapor and liquid phases at constant pressure. Let denote the mole fractions in the liquid and in the vapor phases. T is the absolute temperature of the mixture. Since , only two mole fractions are independent. Selecting (arbitrarily) and , the possible compositions of the liquid belong to the Gibbs triangle , and we denote by its boundary.
Computation of univolatility curves in ternary mixtures. Case studies
Reshetov and Kravchenko, 2007a, Reshetov and Kravchenko, 2007b studied 6400 ternary mixtures including 1350 zeotropic mixtures, corresponding to Serafimov class 0.0–1. The structure of the univolatility curves was determined for 788 zeotropic systems, using the Wilson activity coefficient equation based on both, ternary experimental data and reconstructed data of binary mixtures. 15 of the possible 33 classes defined by Zhvanetskii et al. (1988) were found, and 28.4% of computed ternary
Conclusion
The feasibility of the separation and the design of distillation units can be assessed by the analysis of residue curve maps (RCMs) and univolatility and unidistribution curves. Kiva et al. (2003) reviewed their properties and interdependence. The topology of RCMs and of univolatility curve maps is not trivial even in the case of ternary mixtures, as is known from Serafimov’s and Zhvanetskii’s classifications (1988, 1990). These classifications are usually shown in the two dimensional
Acknowledgments
Ivonne Rodriguez Donis acknowledges financial support from the program SYNFERON (Optimised syngas fermentation for biofuels production; Project 4106-00035B) of the Innovation Fund Denmark.
References (31)
Azeotropic distillation
- et al.
Study on the feasibility of split crossing distillation compartment boundary
Chem. Eng. Process.
(2007) - et al.
Azeotropic phase equilibrium diagrams: a survey
Chem. Eng. Sci.
(2003) Vacuum and High-pressure distillation
- et al.
A novel method for the search and identification of feasible splits of extractive distillations in ternary mixtures
Chem. Eng. Res. Des.
(2015) - et al.
Conceptual design of azeotropic distillation processes
- et al.
Localization of single [Unity] K- and -lines in analysis of liquid-vapor phase diagrams
Russ. J. Phys. Chem.
(1977) Differential Geometry of Curves and Surfaces
(1976)- et al.
Conceptual Design of Distillation Systems
(2001) - et al.
Extractive distillation
Topology of ternary VLE diagrams: elementary cells
AIChE J.
Vapor-liquid equilibria at 101.3 kPa for diethylamine + chloroform
J. Chem. Eng. Data
Distillation technology – still young and full of breakthrough opportunities
J. Chem. Technol. Biotechnol.
Non-local rules of the movement of process lines for simple distillation in ternary systems
Russ. J. Phys. Chem.
Calculus of Several Variables
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