Further Advance to a Practical Methodology To Assess VaporâLiquid

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Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Further Advance to a Practical Methodology To Assess Vapor− Liquid Equilibrium Data: Influence on Binaries Rectification Adriel Sosa,† Juan Ortega,*,† Luis Fernań dez,† Jose ́ M. Pacheco,† Jaime Wisniak,‡ and Arturo Romero§ †

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Grupo de Ingeniería Térmica (IDeTIC), Parque Científico-Tecnológico, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Canary Islands 35017, Spain ‡ Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel § Departamento de Ingeniería Química, Universidad Complutense de Madrid, Madrid 28040, Spain S Supporting Information *

ABSTRACT: Thermodynamic consistency tests are employed to verify the quality of data obtained from experiments on phase equilibria. Validated data sets guarantee their subsequent use in computations, leading to process design. In the present paper, three data sets obtained from binary systems, chosen as significant examples, are studied by applying several thermodynamic validation methods for vapor−liquid equilibrium (VLE) data to them. Another previous task is recommended: to inspect the graphs with the different values that characterize the VLE data, looking for the coherence between those two tasks. Different choices point to variations in the design of separation equipment for the selected solutions. In all instances, a biunivocal relationship is established between the data degree of inconsistency and the parameters of the equipment designed by simulation. It is shown that, when the system has an azeotrope or when the boiling temperatures of the pure participating compounds are close, the influence of the data check/test in the design/simulation process is more important than for the case of nonazeotropic systems.

1. INTRODUCTION Thermodynamic consistency tests are mathematical procedures used to verify that phase equilibria experimental data comply with the scientific formalism imposed by thermodynamics. These tests are necessary because the number of variables measured in phase equilibria experiments is often larger than the degrees of freedom of the system, being therefore overdetermined and, in many cases, the published data is incomplete. Consequently, application of the tests is mandatory to check that the generated data fulfill the basic relationships of thermodynamics. After their application, engineers and researchers can discriminate the validity of all data to guarantee the reliability of subsequent calculations (i.e., process simulation, equipment design, etc.). Nevertheless, data quality verification does not ensure that experimental measurements (considered individually or forming part of a series) represent the physical problem but only the fulfillment of the relations imposed by the scientific theory. Essentially, these tests define: (a) the discrepancy between the observed variables that characterize the equilibria and their real values, and (b) the limits to establish acceptable values for discrepancies, defining indices to evaluate the consistency/ inconsistency of the values obtained and, by extension, of the system studied. This is relevant since some cases can be presented to comply with the second rule (falsified data, values obtained for only one of the vapor−liquid equilibrium (VLE) © XXXX American Chemical Society

phases, i.e., liquid phase, while the vapor composition is obtained by calculation, etc.) but not the first or vice versa. The methodology used by most of the existing methods is based on the resolution of the Gibbs−Duhem equation,1 expressed in differential form as n

∑ xi dln γi = i=1

vE hE dp − dT RT RT 2

(1)

Different resolution methods of this equation have been proposed, resulting in different approaches for data verification. The mathematical resolution of eq 1 is not trivial, and it is not reasonable to simplify it by elimination of both terms on the right-hand side. The most common approach is to define different metrics to quantify the discrepancies indicated above (a), constituting the differences of the criteria of consistency/ inconsistency of experimental values obtained by a researcher. The randomness of these criteria leads to the generation of many procedures to define the quality of the phase equilibrium data, especially for vapor−liquid equilibria due to the inherent value of these data on their use in distillation, one of the most common unit operations in chemical engineering. Many Received: April 18, 2019 Accepted: August 12, 2019

A

DOI: 10.1021/acs.jced.9b00344 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Figure 1. Scheme indicating the ways for the treatment of experimental vapor−liquid equilibrium data and the subsequent operations.

quality. The metrics used in each method were given, and a discussion on the consistency of each of the selected systems was presented. The authors of the present work14,15 have also recently introduced a rigorous methodology to integrate eq 1, with the intention of generalizing the formalism to all cases of phase equilibria (vapor−liquid, liquid−liquid, solid−liquid, etc.) in both binary and multicomponent systems. The main goal of the work is to present the practical application and the consequences of some existing methods to different data sets taken from the literature. Another aim is to show the influences of the evaluation tests (consistency or inconsistency) of VLE data in subsequent uses such as modeling, simulation, and/or design of separation devices. The scheme depicted in Figure 1 shows the sequence of tasks that should be fulfilled to obtain an optimal design of a separation process starting from laboratory experimentation.

publications have appeared (and continue to appear) in the literature of the subject in the last years, as there is no global consensus in the scientific community about the best strategy to validate phase equilibria data. The following items must be highlighted: (1) The existence of numerous works dealing with this topic2−15 is due to the incomplete validations provided by existing tests in the current literature. They rely on subjective metrics to approve or reject a certain data series. In other words, there are cases where different tests applied to a same data set yield contradictory results, so their scientific rigor should be questioned. (2) In other cases, some of these tools, trying to simplify the designed procedure (based on the conjunction of thermodynamic formalism and mathematical rigor), lead to serious errors in the results of the analysis.10−12 (3) Some authors6,7 have combined different methods to generate a single evaluation procedure, in order to achieve the necessary rigor; however, these “mixtures” do not always provide good results. The exclusive selection of some of the existing methods is inappropriate when their application depends on the characteristics of the variables related to the equilibrium, since some metrics fit better for some cases than others, as a function of the shape of the equilibrium diagrams.

2. WORK DESCRIPTION The procedure shown in the scheme of Figure 1 seems to link the starting point (E) with a final goal, such as the simulation (S) and the process design. However, none of the blocks are independent; hence, there are routes of two or more tasks, such as E-T, E-T-M, etc., or minicycles of tasks, i.e., E-T-M-E and E-M-S-E, which depend on the final purpose. The methodology proposed in this work is general, and with it, a single address is sought: E-T-M-S, assessing especially how the quality of the experimental data (the first three tasks) affects the simulation of separation process. To provide an adequate methodology, useful for other researchers, several cases are implemented. For each of them, two data sets have been chosen using different procedures,3−9 according to a previously established procedure13 based mainly on two actions:

A previous work13 described some of the most used methods in the evaluation of the consistency of VLE data of volatile compounds. The first recommendation was to carry out a visual analysis of the graphs containing the experimental information obtained, before proceeding to a mathematical− thermodynamic analysis in order to assess the VLE data B



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minimum boiling point in the representation of VLE data of the mixture. The y1 vs x1 diagram below shows the presence of a singular point at intermediate composition. To achieve the separation of this type of system, several alternatives have been proposed, such as azeotropic distillation, extractive distillation, membrane separation processes, and others.

(a) A visualization of the different graphs can be drawn with the experimental information directly obtained (p,T,x,y) and the calculated one to characterize the VLE. The coherence of the data from this information should be evident, offering a first selection of data. (b) Different methods are applied simultaneously to check the thermodynamic quality of the data, and the results are evaluated to indicate the nature of the errors. The following were used: the area test,3 the tests of Fredenslund et al.,6 Wisniak,7 and Kojima et al.,8 and the direct Van Ness test.9 Additionally, a new method recently published,14 based on the binomial {model + test}, has been included here; see the Supporting Information (S1 and S2). The selection of the systems used in this work was based on two main criteria: their importance for chemical engineering and the presence of some singularity. The binary methanol(1) + water(2), zeotropic, with a representation of y1 vs x1 is shown below. With the exception of a fold in the methanol-rich zone, the equilibrium diagram reflects clearly a favorable separation. It is a binary considered as “standard”, and the available experimental information is even used to test ebulliometers. In this case, the simple distillation is the classical process of methanol purification, whose use as a solvent, fuel additive, etc., is well-known.

The most significant and complex case of the three is the last one, for which a distillation technique other than simple rectification is proposed, such as Pressure Swing Distillation (PSD), carried out with two distillation columns, one operating at low pressure and the other at high pressure. This method, used to separate homogeneous azeotropic solutions, is successful when the position of the azeotropic point is sensitive to pressure, that is, when the azeotropic compositions at the chosen pressures, low and high, are different enough to change the VLE and the coordinates of the azeotrope. Application of this procedure, in addition to allowing the aforementioned separation, underlines the importance of using consistent data to write down the models later used for calculations outside of the measuring region. The simulation of the three cases studied is made by a RadFrac block of Aspen Plus 8.8.17 Although the mathematical−thermodynamic treatment applied to the data of each of the systems chosen is similar to standardize the applicability criteria, each of the proposed cases generates different information.

The second binary selected in the existing databases is that formed by 1-hexene(1) + hexane(2), which exhibits a quasiideal behavior, with very small differences between the compositions of both phases, as shown in the figure (y1 vs x1) below. This behavior makes the separation of solutions of this nature difficult. In general, the separation of the alkene(1) + alkane(2) mixtures is a relevant challenge in chemical engineering,16 whose resolution should not be postponed in time.

3. VLE-DATA CHECKING: INCIDENCE ON THE SIMULATION OF RECTIFICATION PROCESSES 3.1. Separation of the Binary Methanol−Water. The VLE of zeotropic solutions having relative high volatility values, α12, are easily separated by distillation. In these cases, the rectification operation of the solutions satisfying these criteria is performed in individual columns with a relatively small number of stages and a low cost. However, there are certain connotations to using systems of this type: they are composed of substances having very distinct vapor pressures and deviating moderately from ideality. This group is well represented by the binary system methanol−water. Some of the equilibrium data reported in the literature present an inflection (fold) at high methanol compositions, that is, at x1 → 1, where α12 → 1, making the separation difficult. Other data series do not show this singularity or describe it incorrectly, even with values α12 < 1, pointing to the presence of a pseudoazeotrope. Therefore, one could expect that the selection of a precise data series may significantly affect the simulation of the separation process of the solution chosen. The literature18 contains a large amount of experimental isop VLE data for the binary methanol−water. Two data series have been chosen here: those published by Dalager,19 and the

The separation of the ethyl ethanoate(1) + ethanol(2) azeotropic solution is more complex, and it cannot be performed by simple distillation due to the presence of a C



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one reported by Dunlop.20 Figure 2 shows a comparison of both sets of data. The thermodynamic consistency of both

does show the systematic error, especially in the two last values of γ2, in the region where x1 → 1. The visual analysis of these representations is a first step to contrast and complement the information provided by the consistency tests in relation to the anomalies observed. Although the numerical values obtained for the consistency by each of the tests are reported in the Table S2, some details of the results obtained are discussed for the first case. Figure S1 shows the pertinent results. Figure S1(a) presents the results of applying the area test to Dalager’s19 data, clearly illustrating the differences in the values of the positive A+ and negative A− areas. However, this result is strongly conditioned by the equation used to interpolate the values of ln(γ1/γ2) which, in this case, corresponds to a fifth-degree polynomial that adequately reproduces the function ln(γ1/γ2) = φ(x1). The parameter of the consistency test, DA = 21%, is ten times larger than the accepted limit for this method. Therefore, the area test rules out this data series. Figure S1(b) shows the results of the Wisniak test,7 fulfilling the condition 0.92 0.01, proving that the pertinent data series is inconsistent. The values of δp show important differences, close to 1 kPa in some points, presenting a direct relation with nonvalidated points. The direct Van Ness test9 produces the point-to-point results shown in Figure S1(e), with a nonrandom distribution of data, especially in the regions of x1 < 0.2 and x1 > 0.9, due to systematic errors. The average value of δ̅ ln(γ1/γ2) = 0.5 is higher than the limit established for this test, so the data series is also invalidated by this method. The systematic error shown by the previous methods, Figure S1(f), is reflected by the Kojima et al. test,8 Figure S1(f), which also rejects this data series. The results delivered by the proposed new test,14,15 Figure S1(g,h), show that about half of the data are not accepted by the integral form and only one point passes the dif ferential form. Probably, this is due to the existence of a large random error in the compositions measurements of both phases and to a certain systematic error in the determination of the vapor phase compositions. Therefore, this method also rejects the data series. In summary, four of the tests reveal inconsistencies in the data series, and only the Wisniak test7 considers the Dalager19 data valid. However, the global assessment declares the thermodynamic inconsistency of the experimental data. A second example to analyze the methanol(1) + water(2) systems is the isobaric data of Dunlop,20 Figure 4(a). The published information was used to calculate the values of the activity coefficients γi and the adimensional Gibbs function, Figure 4(b). The experimental values T−x1,y1 show an acceptable behavior for the iso-p VLE; however, the distribution of the calculated quantities, Figure 4(b), reflects some irregularities, especially in the water-rich region where x1 → 0. Although some values distort the representation of gE/RT

Figure 2. Plot of experimental values (y1 − x1) vs x1 of the binary methanol(1) + water(2) measured by (- - ● - -, red) Dalager19 and (- ● - -, black) Dunlop.20

series is analyzed using the methodology performed by the authors in ref 13, which uses several tests with a prior qualitative analysis based on certain graphical representations. When a chemical engineer uses data of the methanol−water system to model ↔ design the separation units of equipment, important differences are evidenced between the two data sets and a natural question arises: what should be done? Especially relevant is the presence of an azeotrope in Dalager’s19 data that does not appear in those of Dunlop.20 The study presented in the following subsections analyses the effects of these differences on the calculations in a separation process. 3.1.1. Thermodynamic Consistency of Dalager’s19 and Dunlop’s20 Data. Figure 3(a) shows the data T and x1, y1 for

Figure 3. Plot of data by Dalager19 for the binary methanol(1) + water(2). (a) Representation of T−x1, y1 (●) and (b) activity coefficients γi (▲) and gE/RT (×).

the solution x1 methanol + x2 water at 101.32 kPa published by Dalager.19 A glance at the graph reveals the presence of a random error in the distribution of points (y1, T) and some anomalous inflections in the representation of the two equilibrium curves T = T(x1) and T = T(y1), which can be associated with systematic errors. One example of these irregularities is the presence of an inflection point or “bulge” in the function T = T(x1) for values of x1 ≈ 0.3. The activity coefficients and the values of the excess Gibbs energy gE/RT vs x1, shown in Figure 3(b), reflect the above random error, especially in the values of the adimensional quantity gE/RT, with an irregular behavior. The graph of the activity coefficients γi does not clearly reflect this error, although it D



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using a model proposed in previous works21 (see the Supporting Information), which gives excellent results in its different applications.22,23 Table1 shows the composition and temperature profiles for this design and the energy consumptions in the condenser and the boiler. Table 1. Simulation of the Distillation Process for the Methanol(1)−Water(2) System Using the Model with Data by Dalager19 and Dunlop20,a

Figure 4. Plot of data by Dunlop20 for the binary methanol(1) + water(2). (a) Representation of T−x1, y1 (●) and (b) activity coefficients γi (▲) and gE/RT (×).

in the region of x1 < 0.4, the overall evaluation of the sample can be considered to be positive, especially if these data are compared with those of Dalager;19 see Figure 2 with a more irregular distribution of points. Figure S2(a−f) shows the analytical evaluation obtained after applying the different consistency tests to the Dunlop20 data series. The results lead to the following comments. The area test gives a favorable evaluation, with DA = 0.6% and Figure S2(a). The Wisniak test gives a value of DW = 1.6 (the limit xaze. Therefore, this series is accepted globally by the proposed test, but the indicated region is rejected by a point-to-point criterion. The second series of data analyzed for this system was taken from the publication by Orchillés et al.,27 and the isobaric VLE is presented in Figures 10 and 12(a,b). This series shows a more coherent behavior with a distribution of points in the two graphs that offers some quality guarantee, although the qualitative description of the VLE is almost identical to that of the previous case. The good quality of the data in this series is evident and can be observed in the equilibrium diagrams and in the representation of the quantities calculated to characterize the situation of each stage, Figure 12(b). The quality of these values translates into obtaining positive results with all H



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stage 10 with a reflux ratio of 5. The difference in pressures between the high and low streams is not great, which limits the maximum purities that can be achieved in the process. It would be inappropriate to choose a higher pressure difference as the experimental data were measured at 101.32 kPa and the extrapolation performed to reach the simulation should not go far beyond the fixed values. For the initial feed to the process, a flow rate of 30% v/v in ethyl ethanoate is established, at 298 K and 101.32 kPa, corresponding to flow A1 of the diagram shown in Figure 13, where A3 and B3 are the streams corresponding to the final products. Tables 3 and 4 show the temperature and composition profiles of columns A and B, respectively, obtained in the

Figure 12. Representation of data published by Orchillés et al.27 for the binary ethyl ethanoate(1) + ethanol(2). (a) Plot of T−x1, y1 (●) and (b) activity coefficients γi (▲) and and gE/RT (×). (−) Modeling results.

Table 3. Simulation of Column A (p = 30 kPa) of the Binary Ethyl Ethanoate(1)−Ethanol(2) Using the Model Obtained with Data by Furnas and Leighton26 and Orchillés et al.27,a

the consistency tests used to evaluate the experimental data. Figure S6(a−f) shows the results of the graphical evaluation of this series by the different methods. In summary, in spite of the differences found in the two data series (Figure 10) and the problems reported for the data of Furnas and Leighton,26 validation with the traditional tests does not clearly distinguish the preference for one series or the other. However, the proposed method indicates a better quality in the data of Orchillés et al.,27 which is selected for the modeling and simulation of this system. 3.3.2. Simulation of a Separation Process for the Ethyl Ethanoate−Ethanol System. To resolve the separation of this azeotropic solution a “Pressure Swing” process is proposed, operating between 30 kPa (column A) and 150 kPa (column B), as shown in Figure 13. The column conditions are established previously, both with 20 stages. In column A, the main feed is located in stage 8, and the reflux enters in stage 5 with a reflux ratio of 10. Column B (high pressure) is fed in

a

Shading: feeding stage; red: output compositions.

simulations. Both cases used the modeling applied (see the Supporting Information) with the two data series analyzed in the previous section. The temperature profiles of both columns are shown in Figures 14 and 15. Of these data, the values emitted by the simulator merit some discussion, at least the most significant ones, in relation to the separation process defined. Differences are observed in the output compositions generated by the two simulations on the data series under study. Specifically, the data of Orchillés et al.27 produce an effluent at the outlet of column B (B3) with a composition of almost 99% ethyl ethanoate. However, in the same column, the simulation obtained with the model representing the data published by Furnas and Leighton26 produces an effluent with a purity of only 83%. There is also a difference in the purity of the products from column A (A3): ≈83% in alcohol when using the model with the data from Furnas and Leighton26 and 87% when the operation is simulated with the model representing the data of Orchillés et al.27 In this case, the

Figure 13. Pressure Swing process diagram designed to separate ethyl ethanoate−ethanol. Low pressure line (30 kPa): A1: feed to column A. A3: output of purified ethanol. A2: azeotropic stream at low pressure. A4: recirculation at low pressure. High pressure line (150 kPa): B1: feed to column B. B3: output of purified ethyl ethanoate. B2: low pressure recirculation. Equipment: A: column A (low pressure). B: column B (high pressure). V1: pump. V2: expansion valve. I



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Table 4. Simulation of Column B (p = 150 kPa) of the Binary Ethyl Ethanoate(1)−Ethanol(2) Using the Model Obtained with Data by Furnas and Leighton26 and Orchillés et al.27,a

Figure 15. Plots of liquid composition x1 and temperature profiles, estimated for the equilibrium stages, obtained in the simulation of the distillation column B for the binary ethyl ethanoate (1) + ethanol(2). (−, red) In the simulation of the column B, the modeling was made with data by Furnas and Leighton.26 (−, blue) The simulation of column B was made using the model with data by Orchillés et al.27 Dashed lines correspond to the projections on the plane stage vs x1 and T vs x1.

a

Shading: feeding stage; red: output compositions.

by each one. These differences should be taken into account when choosing a technically and economically optimum design and result from the difference in purities obtained and the flows in the column as well as the estimations of the mixing enthalpies in both models.

4. CONCLUSIONS The results obtained in the simulation of a given process when using different experimental data series reveal important aspects that should be highlighted. This is particularly relevant in the treatment and validation of phase equilibrium data, owing to the repercussions of the quality of the data in the distillation operations design. The problems are discussed through a complete analysis of the thermodynamic consistency of several data series that belong to three examples proposed for the VLE binary systems: zeotropic and azeotropic systems and one system with a quasi-ideal behavior. A new procedure described in a previous work13 has been applied, highlighting the visual check of the data before applying several tests. Some particular comments on the application of the tests give rise to different situations that are discussed. In zeotropic systems, one of the analyzed cases has a remarkable difference between the boiling points of the pure compounds, so that the complete separation of the mixture takes place in a few equilibrium stages. In these cases, an inconsistency, even of high magnitude, such as that presented by Dalager’s19 data, has little effect on the simulation of the rectification column. However, in systems exhibiting a quasiideal behavior (with minimal differences in the VLE phase compositions) for which the separation is more complex, as the case of the binary 1-hexene + hexane, the inconsistency assessment of the data significantly modifies the optimal column design. This effect is more significant in complex

Figure 14. Plots of liquid composition x1 and temperature profiles, estimated for the equilibrium stages, obtained in the simulation of the distillation column A for the binary ethyl ethanoate(1) + ethanol(2). (−, red) In the simulation of column A, the modeling was made with data by Furnas and Leighton.26 (−, blue) In the simulation of column A, the modeling was made with data by Orchillés et al.27 Dashed lines correspond to the projections on the plane stage vs x1 and T vs x1.

purification of ethanol at low pressure in column A does not produce estimations so different. The simulation shows relevant information about the energy consumptions of each column, depending on the models used for each series. The values obtained are recorded in Tables 3 and 4 and show a significant difference in the energy consumed J



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operations that require the presence of more than one column, especially when operating outside the conditions in which the experimental VLE data were measured. Regarding the global data treatment, with the sequence of operations in the E → T → M calculation, it is important to note that the analysis carried out with the different tests chosen constitutes a necessary verification but is not in itself suf f icient. This was observed in the third case studied, with the data published by Furnas and Leighton26 for the ethyl ethanoate + ethanol solution. In this case, the consistency tests do not detect the existence of specific problems in the experimental data, especially in the region close to the azeotrope. The incidence of the study carried out on the simulation/ design of the rectification operations for the selected systems was compared with that predicted by the commercial simulator using the UNIFAC28 modeling. The results obtained for methanol + water and 1-hexene + hexane systems are similar to those obtained with the direct procedure proposed in this work. This is because the modeling with UNIFAC of the first system is adequate (a large amount of information is available in the literature) and of the second has high ideality. However, the simulation of the separation process of the azeotropic mixture ethyl ethanoate + ethanol, by means of PSD, generates very different results, estimating even a much higher energy consumption (50%) than that estimated by the direct procedure used. This is because the azeotrope estimated by UNIFAC shifts in relation to the experiment, and it is necessary to vary the process flows. The authors consider that the present study is of interest for researchers working in the phase equilibrium field. Therefore, the work carried out can be summarized in two main recommendations: (1) the experimental VLE data, both direct and indirect (such as thermodynamic functions calculated), should be graphed and analyzed for visual coherence of the available information. (2) Thermodynamic verification of the quality of the data (one by one and together) should be done using several tests, not only one chosen by the researcher’s interest. The information obtained should be included in the Supporting Information of the publications for its pertinent exam.

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ACKNOWLEDGMENTS

The authors are grateful for financial support from the Spanish Ministry MINECO (project CTQ2015-68428-P). A.S. is grateful for the support received from the ACIISI (Canary Government, 2015010110); L.F. thanks MINECO for the postdoctoral contract received under the Juan de la Cierva program (FJCI-2017-31784).

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NOMENCLATURE DA parameter characterizing the Area test DW parameter characterizing the Wisniak test OF objective function, eq S3 gE/RT adimensional excess Gibbs function gi coefficients of eq S1 Gij coefficients of eq S2 hE excess molar enthalpy, J·mol−1 Li/Wi parameter characterizing the Wisniak test m number of experimental points p pressure, kPa R gas constant, J·mol−1·K−1 s(Y) mean quadratic deviation for Y, eq S4 T temperature, K x1 liquid phase composition of compound 1 y1 vapor phase composition of compound 1 zi active fraction of compound i, eq S1 Greek Letters

α relative volatility δ deviation or difference between two values γi activity coefficient of compound i

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REFERENCES

(1) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 7th ed.; McGraw-Hill, Inc.: 2005. (2) Herington, E. F. G. A Thermodynamic Test for the Internal Consistency of Experimental Data on Volatility Ratios. Nature 1947, 160, 610−611. (3) Redlich, O.; Kister, T. Thermodynamics of Nonelectrolyte solutions. Ind. Eng. Chem. 1948, 40, 341−345. (4) Van Ness, H. C. Precise testing of binary vapour-liquid equilibrium data by Gibbs-Duhem equation. Chem. Eng. Sci. 1959, 11, 118−124. (5) Van Ness, H. C.; Byer, S. M.; Gibbs, R. E. Vapor-Liquid Equilibrium: Part 1. An appraisal of data reduction methods. AIChE J. 1973, 19, 238−244. (6) Fredenslund, A.; Gmehling, J.; Rasmussen, P. Vapor-liquid equilibria using UNIFAC a group contribution method; Elsevier: Amsterdam, 1977. (7) Wisniak, J. A new test for the thermodynamic consistency of vapor-liquid equilibrium. Ind. Eng. Chem. Res. 1993, 32, 1531−1533. (8) Kojima, K.; Moon, H. M.; Ochi, K. Thermodynamic consistency test of vapor-liquid equilibrium data. Fluid Phase Equilib. 1990, 56, 269−284. (9) Van Ness, H. C. Thermodynamics in the treatment of vapor/ liquid equilibrium (VLE) data. Pure Appl. Chem. 1995, 67, 859−872. (10) Marcilla, A.; Olaya, M. M.; Serrano, M. D.; Garrido, A. A. Pitfalls in the evaluation of the thermodynamic consistency of experimental VLE data sets. Ind. Eng. Chem. Res. 2013, 52, 13198− 13208. (11) Wisniak, J.; Apelblat, A.; Segura, H. An assessment of thermodynamic consistency tests for vapour-liquid equilibrium data. Phys. Chem. Liq. 1997, 35, 1−58. (12) Wisniak, J. The Herington test for thermodynamic consistency. Ind. Eng. Chem. Res. 1994, 33, 177−180.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.9b00344.

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Description of the model used (S1) and its parametrization for each of the selected systems (Table S1); description of the proposed consistency method (S2); results obtained by applying the different consistency tests to the selected systems in this work (Table S2); graphical results obtained by applying the different consistency tests to the selected systems in this work (Figures S1 to S6) (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Juan Ortega: 0000-0002-8304-2171 Notes

The authors declare no competing financial interest. K



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