Pitfalls in the Evaluation of the Thermodynamic Consistency of

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Pitfalls in the Evaluation of the Thermodynamic Consistency of Experimental VLE Data Sets Antonio Marcilla,* María del Mar Olaya, María Dolores Serrano, and María Angeles Garrido Chemical Engineering Department, University of Alicante, Apdo. 99, Alicante 03080, Spain S Supporting Information *

ABSTRACT: The thermodynamic consistency of almost 90 VLE data series, including isothermal and isobaric conditions for systems of both total and partial miscibility in the liquid phase, has been examined by means of the area and point-to-point tests. In addition, the Gibbs energy of mixing function calculated from these experimental data has been inspected, with some rather surprising results: certain data sets exhibiting high dispersion or leading to Gibbs energy of mixing curves inconsistent with the total or partial miscibility of the liquid phase, surprisingly, pass the tests. Several possible inconsistencies in the tests themselves or in their application are discussed. Related to this is a very interesting and ambitious initiative that arose within the NIST organization: the development of an algorithm to assess the quality of experimental VLE data. The present paper questions the applicability of two of the five tests that are combined in the algorithm. It further shows that the deviation of the experimental VLE data from the correlation obtained by a given model, the basis of some point-to-point tests, should not be used to evaluate the quality of these data.

1. INTRODUCTION Vapor−liquid equilibrium (VLE) data are essential for the simulation and design of many separation processes. These data are compiled in data banks such as, for example, the Dortmunt Data Bank DBB1 and NIST Source Data Archival System.2 Accurate VLE data are demanded for separation process design. VLE data are usually measured under isobaric or isothermal conditions and require the equilibrium vapor (y) or liquid (x) compositions as well as the temperature (T) or pressure (P) of the system, respectively. Accurate measurement of y is by far the most difficult, and therefore, many P − x or T − x data sets are frequently published. Only when a full set of measurements P − x, y or T − x, y (over determined system) is available is it possible to check whether they satisfy certain thermodynamic relationships (thermodynamic consistency tests or TC tests). In these cases, the VLE experimental data are declared thermodynamically consistent, but not necessarily correct. Conversely, if the experimental VLE data do not obey these conditions, then they will be inconsistent and can always be considered as such providing the thermodynamic consistency tests are applied rigorously. The fundamental Gibbs−Duhem (GD) equation is the most widely referenced condition for consistency of the experimental data. This equation can be handled in a number of ways, leading to a variety of consistency tests that can be broadly classified as an area or integral test,3 point-to-point tests,4−6 an L−W test,7 an infinite dilution test6,8 and a differential test.9 Experimental error propagates differently in each test, and therefore, some authors propose certain combinations of these tests as an overall check of the data. For example, Kojima et al. proposed the PAI test that is a combination of the point-to-point, area, and infinite dilution tests.6,8,10 Eubank et al.11 advise about the advantages of a twostep method to check the consistency of the VLE data via the GD equation. A recent and ambitious initiative is proposed in a paper by Kang et al.:12 the development of an algorithm to © 2013 American Chemical Society

assess the quality of experimental VLE data. This algorithm combines compliance of the data with the general Gibbs− Duhem equation, on the one hand, with consistency between the VLE data and the pure-compound vapor pressures, on the other. It employs four consistency tests based on the GD equation: the area test,3 point-to-point tests by van Ness et al.4 and Kojima et al.,6 and infinite dilution test.8 The results of these four tests plus consistency with pure-compound vapor pressures are represented numerically by their corresponding individual quality factors (Fi). These are then further combined to obtain a global quality factor (QVLE) for each one of the evaluated VLE data sets. Many efforts have been devoted to developing TC tests and applying them to large numbers of data series. One of the most extensive applications of TC tests and the results thereof is contained in the DECHEMA Chemistry Data Series13 compilation, where the area test3 (with the Herington approximation) is used in combination with the Fredenslund point-to-point test5 to check the consistency of about 10 000 VLE data sets. Another more recent example of the application of TC tests is the NIST Thermodata Engine (TDE) software package. It represents the first full-scale implementation of the dynamic data evaluation concept for thermophysical properties (including phase equilibria) and has led to the ability to produce critically evaluated data dynamically.14 For example, TDE 3.015 provides area test results for the VLE data sets of binary systems, and TDE 6.016 includes the algorithm proposed by Kang et al.12 to assess the quality of the experimental VLE data for binary and ternary mixtures. Received: Revised: Accepted: Published: 13198

May 23, 2013 July 12, 2013 July 29, 2013 July 29, 2013 dx.doi.org/10.1021/ie401646j | Ind. Eng. Chem. Res. 2013, 52, 13198−13208

Industrial & Engineering Chemistry Research

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However, existing TC tests possess many drawbacks, some of which have already been discussed in the literature,17,18 and others that are partially the subject of the present paper. It is known that the widely used area test can involve cancellation errors.19 Moreover, the method is not sensitive at all to the measured total pressure data.4,19 Besides, some of the data required by the tests are not available, and as a result, recourse is usually made to approximations. Jackson and Wilsak17 comment that “there has not yet been a thermodynamic consistency test rigorously applied to VLE data nor does there exist a set of data that is known a priori to be absolutely accurate.” The consequences of using some of the very popular approximations are not sufficiently known and will be discussed in the present paper. Finally, another very important question to be considered is the fact that some of the TC tests require models in order to be applied, e.g., excess Gibbs energy (gE) models. Several issues derived from this fact are already mentioned in the literature, such as the test results being highly sensitive to the model that is used.17 Here, we go further by attempting to present conclusive reasoning to demonstrate that TC tests, combined with the existing gE models (i.e., local composition models), are not suitable for the evaluation of experimental VLE data. Two of the most commonly used TC tests, the area and point-to-point (Fredenslund) tests, have been used to check the thermodynamic consistency of almost 90 VLE data sets as examples. These data sets include isothermal and isobaric conditions for both completely and partially miscible liquid phases. In this regard, the information supplied by a graphical representation of the Gibbs energy of mixing (gM) versus the liquid composition (x) is highly relevant. Some important inconsistencies found with the cited tests or in their application are illustrated using selected examples. This discussion is directly related to the initiative by Kang et al.,12 arising within the National Institute of Standards and Technology (NIST), to develop an algorithm to assess the quality of published experimental VLE data sets, which has already been implemented in the TDE 6.0 software.16 This initiative is not only valuable but also absolutely necessary and represents an advance toward the final objective of ensuring the quality of published experimental data. In this sense, some important steps have already been taken. An example of this is the joint statement by the editors of some journals and the Thermodynamics Research Center (TRC) of the NIST, which serves to facilitate the searching process when experimental data in submitted manuscripts must be compared with previously reported literature values.20 This is the context in which the significance of the consistency algorithm proposed by Kang et al.12 and the importance of the discussion presented in this paper can be better understood, which questions the applicability of two of the tests included in that algorithm because they can lead to a distorted picture of the quality of the experimental data, as conveniently illustrated by the examples below.

∑ xid ln γi + i

HE vE dT − dP = 0 2 RT RT

(1)

where γi is the activity coefficient of component i, xi is its molar fraction, HE is the excess enthalpy, and vE is the excess volume of the mixture, while R, T, and P retain their usual meaning. For a binary system, the integrated form of eq 1 becomes

∫0

1

ln

γ1 γ2

dx1 −

∫T

T1o o 2

HE dT + RT 2

∫P

P1o

o 2

vE dP = 0 RT

(2)

where Toi and Poi are the boiling point and the vapor pressure of pure component i, respectively. Under isothermal conditions, the second term in eq 2 vanishes,22 and the third can be neglected.8 Then, the area test can be performed according to the Redlich−Kister method (eq 3) that verifies whether the positive (A) and negative (B) areas in the ln γ1/γ2 versus x1 graph are equal.3 The condition for passing this test is given by eq 4 using a deviation parameter D:

∫0

1

ln

γ1 γ2

dx1 = 0

D = 100 ×

||A| − |B|| ≤2 |A | + |B |

(3)

(4)

Under isobaric conditions, the third term in eq 2 vanishes, but the second one now cannot be neglected. The evaluation of this term requires excess enthalpy HE data as a function of the temperature and composition. This information is scarce and rarely available and impedes rigorous calculation of eq 2. To overcome this problem, Herington23 proposed an empirical equation (eq 5) to approximately evaluate the integral term depending on HE. J = 150

Tmax − Tmin Tmin

(5)

The derivation of this equation was examined by Wisniak,18 who showed that it contained errors due to the very limited experimental information available to Herington at the time. Wisniak used an extensive database to show that the J parameter is better represented as indicated in eq 6: J = 34 ×

HaE (Tmax − Tmin) E Tmin Gmax

(6)

where Tmax and Tmin are the maximum and minimum boiling temperatures over the entire concentration range, HEa is the average heat of excess, and GEmax is the maximum Gibbs energy of excess. Equation 6 can be only used when heats of excess data (or their average values) are available. In the absence of this kind of data, a correlation to calculate the ratio HEa /GEmax has been proposed.18 The criterion for the VLE data set to pass the test is |D − J| < 10. In spite of the conclusions reached by Wisniak, at present many authors reporting experimental VLE data and even most commercial software packages including VLE data evaluation not only continue using the Herington equation12,15,16 but also do it incorrectly using (D − J) < 10 instead of |D − J| < 10 as a condition to pass the test.24 Since J is always positive, this mistake means that the difference (D − J) can be negative, and in these cases the test condition will be always satisfied, as has been previously discussed.25

2. AREA AND POINT-TO-POINT TC TESTS In this section, a brief description of the area and the point-topoint tests is presented. For more details about these and other TC tests, several other reference works can be consulted.10,17,18 2.1. Area Test. The general Gibbs−Duhem equation can be expressed as follows:21 13199

dx.doi.org/10.1021/ie401646j | Ind. Eng. Chem. Res. 2013, 52, 13198−13208


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2.2. Point-to-Point Tests. Various tests are referred to in the literature as point-to-point or point tests. Examples of these are the van Ness test,4 the Fredenslund test,5 and the Kojima test.6 A characteristic that they all have in common is that the consistency of every VLE data point is evaluated individually. This is also a characteristic of the L−W test,7 although it is usually classified separately. The van Ness test and a subsequent modification of it by Fredenslund et al. are briefly outlined below. 2.2.1. The van Ness Test. Van Ness et al.4 proposed to test the consistency of VLE data based on an ability to predict the composition of the vapor phase using eq 7: yical =

Lk (x1) =

− (k − 1) × Lk − 2 × (x1)]

Δyi = |yical − yiexp | ≤ 0.01

(7)

ΔGE = x1x 2(Ax 2 + Bx1 − Cx1x 2) RT

n

∑ i=1

dg E dx1

ln γ2 = g E − x1 ×

dg E dx1

(8)

(9)

(10) E

The coefficients A, B, and C in the g function are calculated by fitting, using a comparison between the experimental and calculated total pressure P cal = p1o × γ1 × x1 + p2o × γ2 × x 2

(11)

for which eqs 9 and 10 are used to obtain the activity coefficients. Next, the deviation between the calculated and experimental vapor composition is evaluated (eq 12). These authors did not place any numerical limit on Δy in order to establish whether VLE data are consistent, but obviously, this quantity must be small. They recommended inspecting the ΔP and Δy versus x plots to verify that a random scatter about zero occurs, enabling one to establish consistency of the experimental VLE data.

Δyi =

yical

−

yiexp

≤ 0.01 (16)

(17)

where γi is calculated from the experimental equilibrium data. It is important to remark that the gM versus x curve is not usually analyzed, in spite of the fact that it provides highly valuable information about the quality of the data, as we show in the present paper. A set of “good” VLE data should necessarily generate a gM curve possessing the following two characteristics: (a) a smooth tendency and (b) consistency with the total or partial miscibility of the liquid phase. That is to say, if the system presents a homogeneous liquid phase, the gM curve must be convex throughout the composition space (Figure 1a), but if the liquid phase is partially miscible, the gM curve must be concave in a region to allow for the existence of a

(12)

n

ΔGE = x1x 2 ∑ ak Lk (x1) RT k=1

n

GM,liq = x1 ln(x1 × γ1) + x 2 ln(x 2 × γ2) RT

2.2.2. The Fredenslund Test. Fredenslund et al.5 proposed several modifications to the van Ness test, such as the use of the highly flexible Legendre orthogonal polynomials (eq 13) to represent the Gibbs energy of excess: gE =

|yical − yiexp |

3. APPLICATION OF TC TESTS TO VLE DATA SETS The main goal of this work has been to apply several TC tests to VLE data sets for isothermal and isobaric systems that exhibit both total and partial miscibility in the liquid phase, with the aim of detecting problematic cases that allow identifying suspected limitations of the tests or in their application. The source of the experimental VLE data has been the book collection DECHEMA Chemistry Data Series.13 A total of 72 isothermal systems and 17 isobaric systems have been studied and are summarized in Tables S1 and S2, respectively, in the Supporting Information. The area and the Frendenslund pointto-point5 tests have been applied to these systems. Results are given as + (consistent) and − (inconsistent) in accordance with the corresponding test criteria. Blanks appear when the number of experimental points is too low or their distribution is not suitable. Results for these same TC tests are published in the data bank used,13 but the information for the point-to-point test results given there is for the overall data set, and our study required detailed information of the individual data points. In addition, the following relationships have been inspected graphically: vapor (y) versus liquid (x) compositions (equilibrium curve), P versus x and y, and also the Gibbs energy of mixing for the liquid phase (gM,liq or gM) versus x, with the aim of checking the trends and dispersion of the experimental points (i.e., smoothness, moderate or high dispersion). The following equation is used to obtain the gM,liq curve:

The activity coefficients are calculated from the gE function and its derivative as ln γ1 = g E + x 2 ×

(15)

This is a widely used test; e.g., the DECHEMA Chemistry Data Series13 applies this test together with the area test to check the consistency of all the VLE data sets it contains, but the evaluation of the data set is carried out globally by means of eq 16 instead of eq 15:

where f oi is the fugacity of pure component i as a liquid and φi is the fugacity coefficient for i in the vapor phase. In order to use eq 7, it is necessary to calculate the activity coefficients, and for this an expression for the excess Gibbs energy (gE) is required. These authors used the four-suffix Margules (threeparameter) equation gE =

(14)

In addition, for the VLE data to be considered consistent, the deviation between the experimental and calculated vapor compositions (eq 15) should not exceed a certain maximum established value:

f i0 × γi × xi P·φi

1 [(2k − 1) × (2x1 − 1) × Lk − 1(x1) k

(13)

where ak are the coefficients of order k and Lk(x1) are the Legendre polynomials (eq 14): 13200



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pass the point-to-point test and the former even the area test. In relation to the point-to-point test, the data set shown in Figure 2 passes the overall (but nonindividual) test. However, all the points plotted in Figure 3 are individually thermodynamically consistent according to this test. Figure 4 shows the water(1) + 1-hexanol(2) system at 40 °C, whose liquid phase is partially miscible. The data set selected for this example (no. 60 in Table S1 Supporting Information) generates a gM curve that is inconsistent because it reproduces a false LL splitting where the system is actually homogeneous. Conversely, in the composition interval where the system has a true LL equilibrium26 x1I = 0.305 and x1II = 0.998, a point with a homogeneous liquid phase is obtained. Despite all the inconsistencies in this data set, these experimental points still pass the (individual) point-to-point test. A possible gM curve that is consistent with the liquid behavior of the system has been included, just for the sake of illustration. The following partial conclusions are deduced from the study summarized in the present section: 1. The gM vs x curve reveals important information about the quality of the VLE data that is apparent neither in other typical representations nor in the test results. 2. Some inconsistencies in the TC tests or in their application should exist that justify the obtained results. It is especially important to clarify this last point because the area and point-to-point tests (by van Ness or Frendenslund) are used too often. For example, both are included in the algorithm proposed by Kang et al.12 whose end purpose is use in the quality evaluation of the main VLE data banks. In the next section, the area and van Ness point-to-point tests, such as they are used in the algorithm proposed by Kang et al., are analyzed and important inconsistencies discussed.

Figure 1. Gibbs energy of mixing (dimensionless) curves as a function of the molar fraction of the 1-component: (a) for a completely miscible binary system and (b) for a partially miscible binary system with LL splitting (I and II points).

common tangent line between the two liquid mixtures at equilibrium (points I and II in Figure 1b). The following represents some relevant data that summarize the information collected in Tables S1 and S2 (Supporting Information). Among the 89 data sets selected, 25 pass both tests, 20 data sets pass the area test but do not pass the pointto-point test, and 9 data sets pass the point-to-point test but not the area test. For 7 data sets, the results of the area test obtained in this study are not in agreement with those published in DECHEMA. This number increases to 11 for the point-to-point test. As regards the gM curve obtained from the experimental VLE data, 22 data sets show a smooth tendency, 28 data sets present a moderate dispersion and 39 exhibit a high dispersion or no trend at all. Besides, some of them (e.g., no. 46, 47, and 60 in Table S1, Supporting Information) correspond to the VLE data that are inconsistent with the total or partial miscibility of the liquid mixture, as is shown next. We reconciled the results of the TC tests with the aforementioned graphical representations for every one of the VLE data sets included in this study. This produced some rather unexpected results: some data sets exhibiting a smooth trend did not pass the tests, whereas others exhibiting high dispersion did. Furthermore, even data sets reproducing gM curves that are inconsistent with the total or partial miscibility of the liquid passed the tests. Obviously, a smooth trend in the data when represented graphically and consistency with the total or partial miscibility of the liquid phase do not guarantee thermodynamic consistency of the data. However, the opposite situation, high dispersion or inconsistent data that do pass the tests, is more difficult to justify. For example, the data sets for acetone + water at 100 °C and ethylene oxide + water at 20 °C (no. 24 and 7 in Table S1 Supporting Information), represented graphically in Figures 2 and 3, respectively, show dispersion in their gM curves. Nonetheless, both data sets do

4. SOME INCONSISTENCES IN THE APPLICATION OF TC TESTS 4.1. Area Test (with the Herington Approximation). As was explained in section 2, when the area test is applied to isobaric VLE data, experimental information about the excess enthalpy as a function of the temperature and composition is required. The approximation proposed by Herington to circumvent the necessity of these hard-to-come-by data is still widely used despite having been proved to be incorrect.18,25 A sign of its popularity is that the Herington equation is included in test 1 of the algorithm proposed by Kang et al.12,27 for the global numerical evaluation of VLE data sets. In this paper, we present a different approach that invalidates this equation and corroborates the conclusions reached by Wisniak many years ago.18 The VLE data of any system, for example water + 1,2propanediol at 50 mmHg, can be generated using the NRTL equation based on the parameter values published in the DECHEMA Chemistry Data Series (reference no. 80 in Table S2 Supporting Information). Because these equilibrium data are obtained by means of a thermodynamically consistent model, they are “totally” consistent according to the area test when it is applied rigorously using eq 2; the second term is evaluated by means of the NRTL equation using the relation ∂(GE /RT ) HE =− 2 ∂T RT

(18)

In contrast, when the Herington approximation is used in the application of the area test to these same VLE data (generated with the NRTL model), the result is negative: these data are 13201



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Figure 2. VLE experimental data for the binary system acetone (1) + water (2) at 100 °C (no. 24 in Table S1 Supporting Information): (a) Pressure vs x and y (molar fractions) and (b) gM as a function of the composition for the liquid phase.

parameter NRTL equation as the required model. Jackson and Wilsak.17 note that the required model may be empirical functions such as spline fits, although the traditional equations are generally preferred since proper limiting characteristics are already built into them. However, for far too many systems a satisfactory fitting is not obtained, and regrettably, not with any model either, at least not good enough to justify using the model as the standard of comparison for validating experimental data. In these cases, are the data inconsistent or is the model unable to represent the experimental phase behavior? The thermodynamic consistency tests, such as the van Ness point-to-point test, penalize the experimental data when the model is not capable of fitting them. Some authors17 have noticed that it is necessary to first find a thermodynamically consistent model that is capable of fitting the experimental data before the test can be applied, but this important observation is usually obviated. Furthermore, especially strict consideration should be given to a model’s ability to adequately fit a VLE data set when it is to be used as a standard of comparison; i.e., P and y residuals should be inspected to check that random distributions exist. Only in the paper by Jackson and Wilsak17 have we found some of these shortcomings of the test appropriately discussed. However, this excellent paper does not deduce anything regarding the very limited number of VLE data sets that could be evaluated if all these necessary requirements were taken into account.

now thermodynamically inconsistent! This demonstrates the complete unreliability of the results obtained when the Herington approximation is used to evaluate experimental VLE data. As a consequence, this equation should not be used in any procedure, algorithm, etc. whose purpose it is to evaluate such data sets. It may erroneously invalidate correctly obtained VLE data (such as the data obtained with the NRTL equation in the above example), or do the opposite of that, i.e. validate spurious data, as has been pointed out by Wisniak.18 If eq 6 is used instead of eq 5 to evaluate the J parameter, the VLE data set considered here produces the consistent result, showing that this equation is a better approximation of the rigorous one than the Herington equation. 4.2. Point-to-Point Test (van Ness). In this section, we present strong arguments to demonstrate that point-to-point tests based on models (i.e., local composition models such as NRTL) and used to verify the quality of experimental VLE data should no longer be used without due consideration to their limitations in representing the phase equilibria of many systems. The van Ness test (as well as the one by Fredenslund) is regarded as a modeling capability test. Kang et al.12 state literally that, “This test shows how a mathematical activity coefficient model can reproduce the experimental data accurately.” These authors include the van Ness test as part of their proposed algorithm and suggest using the five13202



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Figure 3. VLE experimental data for the binary system ethylene oxide (1) + water (2) at 20 °C (no. 7 in Table S1 Supporting Information): (a) Pressure vs x and y (molar fractions) and (b) gM as a function of the composition for the liquid phase.

experimental point, specified below, belongs to the experimental data set of the water(1) + 1,2-propanediol(2) system at 25 mmHg (no. 79 in Table S2 Supporting Information), plotted in Figure 6a:

The test is based on an excessive reliance on the existing excess Gibbs energy models, unfounded from our point of view as we have already demonstrated. In a previous paper,28 we carried out a systematic topological study of the Gibbs energy of mixing as a function of composition and demonstrated that the NRTL model exhibits “gaps” or regions where NRTL solutions for miscible binaries do not exist. In Figure 5a, an example of these gaps is shown for which the minimum value of gM (NRTL) is located at x1 = 0.35. Similar representations28 would show that the gap becomes progressively smaller as the minimum goes from 0.35 to 0.5. But what is more important is that the gaps themselves are responsible for the poor correlation of the LLE and VLE data of many systems. The above cited paper contains an example of the relationship between the “gaps” and the impossibility of fitting the experimental LLE data for a type I ternary system. This idea has been schematically represented in Figure 5b, where the fitting of the experimental tie-lines (LLE) requires that the gM binary curve of the 2−3 binary subsystem be exactly located where the model produces a gap, and as a consequence, no solution can be found using the model. This explains the poor LLE data correlation obtained for many systems using different models, e.g., methanol + diphenylamine + cyclohexane at 298 K with the NRTL model. In what follows, a similar case is presented but, now, for a VLE data correlation. The correlation of a unique experimental VLE data point, using the NRTL model as being representative of the local composition models, is considered in this example. This

x1 = 0.030, y1 = 0.775, T = 83.5 °C

The NRTL model is unable to fit this point because the vapor phase composition calculated using it deviates greatly from the experimental one. The best correlation that can be achieved uses the following NRTL binary interaction parameters A12 = 145.85 K, A21= 41.307 K, and α = 3.032. This yields the calculated point: x1(cal) = 0.030, y1(cal) = 0.612, T (cal) = 83.5 °C

The explanation for this poor correlation is, again, the existence of “gaps” in the NRTL model. This can be understood by taking into account that, for a vapor and liquid phase to be in equilibrium, a common tangent line to the respective vapor and liquid Gibbs energy of mixing functions (gM,V and gM,L) must exist at the vapor and liquid equilibrium compositions. The gM,V curve for the vapor phase has been calculated using eq 19 and is shown in Figure 6b. The reference state in this equation is the pure component as liquid at the same T and P of the system, and the vapor phase is considered to be ideal. 13203



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Figure 4. VLE experimental data for the binary system water (1) + 1-hexanol (2) at 40 °C (no. 60 in Table S1 Supporting Information): (a) Pressure vs x and y (molar fractions) and (b) gM as a function of the composition for the liquid phase.

g M,V =

GM,V = RT

∑ yi ln i

P pio (T )

+

∑ yi ln yi i

composition cited previously: Δy1 = 0.775 − 0.610 = 0.165. The gM,L curve obtained based on the NRTL parameters published in DECHEMA Chemistry Data Series (no. 79 in Table S2, Supporting Information), obtained from the global correlation of all the experimental VLE data included in that series, has also been plotted in Figure 6b. Obviously, this nonisothermal curve is somewhat different to that obtained by correlation of a single point. However, the solution for the specific point considered in this example is identical in both cases, as can be ascertained from the figure. At this point, we might wonder if we could resolve the problem by only increasing the number of parameters in the model, for example by taking into account the temperature or the composition dependence of the NRTL binary interaction parameters, as suggested by Kang et al.12 The reply would be no: the correlation of the above experimental VLE data set is not significantly improved when five, instead of three, interaction parameters are used; nor is it improved by incorporating temperature or composition dependencies into the model. The reason for this is that, although the additional parameters provide some additional flexibility, the gaps in the model are not filled in, and therefore, the capability of the model continues to be very limited.

(19)

Both the composition of the experimental vapor phase and the tangent line to the gM,V curve at the point in question are plotted in Figure 6b. For a perfect fitting of the specified VLE data point, the model should be able to generate a gM,L curve for the liquid phase, having the same tangent line in the experimental liquid composition as in the experimental vapor composition. A possible gM,L curve that satisfies this condition is also plotted in Figure 6b. However, the NRTL model is unable to generate a curve of these characteristics because it produces a “gap” in that region. In other words, the NRTL model cannot provide a solution for the following conditions: dg M,L dx1

= −2.47 (20)

xexp

g M,L |xexp = −0.65

(21) M,L

The closest to the required g curve that can be generated by the model is also shown in Figure 6b. The common tangent line between this curve for the liquid and the one for the vapor produces the large deviation in the calculated vapor 13204



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Figure 5. Schematic representation of the limitations of the NRTL model: (a) systematic study of the homogeneous gM binary curves for x2 min = 0.35 and (b) gM binary curves for a type 1 ternary system where the existence of “gaps” constrains the LL region calculated by the model.

compiled in the DECHEMA data collection.13 Table 1 shows the NRTL binary parameters obtained by fitting the available experimental data for every one of these sets. Moreover, only one of these data sets passes both area and point-to-point consistency tests (set number 2), but in this case the NRTL parameters reproduce data consistent with a type IV binary azeotrope (Figure 7), whereas the real behavior of the system corresponds to type I data, which means that there should be no LL miscibility gap present.29 Chemical process simulation software packages obtain information from that which is available in data banks, compilations, etc. and use the existing procedures to select data. Therefore, all the problems discussed here affect the results obtained when using chemical process simulation programs. Continuing with the previous example, the NRTL parameters in the database used by CHEMCAD 6.4.0 for this system coincide with those classified as set 2 shown in Table 1. They reproduce the inconsistent system plotted in Figure 7. Among the other parameter sets in Table 1, there are some that do not reproduce liquid−liquid splitting but

Given all these limitations of the existing gE models, such as NRTL, in correlating the experimental phase equilibrium data, i.e. LLE or VLE, it does not seem reasonable to penalize any experimental VLE data if it cannot be correlated by a given model or results in large deviations. Other different arguments can be used that reinforce this idea: • A comparison of the experimental and the calculated data is usually the procedure followed to check the capability of the models. Therefore, it does not seem logical to swap the role of every element in the comparison to the extent of turning the model into the standard of comparison. •The results can vary greatly when different gE models are used in the same TC tests (i.e., van Ness or Fredenslund pointto-point test), a fact that is acknowledged by other authors.17 The problems discussed in this paper have important practical repercussions. The following example illustrates how an excess of confidence in the existing consistency tests may lead to a selection of the wrong model parameters. Seven data sets for the water + 1-propanol binary system at 760 mmHg are 13205



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Figure 6. Experimental VLE data for the binary system water (1) + 1,2-propanediol (2) at 25 mmHg (no. 79 in Table S1 Supporting Information): (a) Temperature vs x and y (molar fractions) and (b) gM,V (vapor) and gM NRTL (liquid) functions for the selected VLE point at T = 83.5 °C (the nonisothermal gML curve for all data sets has been included).

arising within the NIST12 are absolutely necessary, but in order to avoid inconsistencies, the tests and their application to the data must be thoroughly revised, as discussed in the present paper.

Table 1. NRTL Parameters Values Obtained by Fitting Different Experimental VLE Data Sets for 1-Propanol (1) + Water (2) System at 760 mmHg data source

num.

A12 (cal/mol)

A21 (cal/mol)

α

DECHEMA [13]

1 2 3 4 5 6 7

152.5084 444.3339 412.0253 152.5084 294.7832 619.3422 −13.0045 444.3322

1866.3369 1997.5504 1735.4304 1866.3369 1893.5152 2708.5773 1872.0758 1997.6031

0.3747 0.4850 0.4465 0.3747 0.4276 0.6185 0.2803 0.4850

ChemCAD 6.4.0.

5. CONCLUSIONS The main conclusions of the present paper are the following: 1. Consistency tests not developed and/or applied with the required degree of rigor may erroneously invalidate correctly obtained VLE data or do the opposite, i.e., validate spurious data. 2. The deviation of the experimental VLE data with regard to correlation by means of a given model should not be used to assess the quality of these data until gE models capable of fitting all existing phase equilibrium behaviors are developed. The consistency tests based on this idea should not be applied. 3. Therefore, the applicability of two of the five tests that are combined in the algorithm propose by Kang et al.12 is questioned. A thorough revision of the strategies to develop and apply sound consistency tests is required that guarantees

rather type 1 binary data consistent with the behavior of this system, e.g., set 3. Most likely, the selection criterion to include set number 2, and no other parameters, is that this experimental data set is the only one that passes both the area and point-topoint consistency tests. An unfounded overconfidence in these consistency tests may have negative consequences, as just illustrated by the above example. Initiatives such as the one 13206



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ACKNOWLEDGMENTS We gratefully acknowledge financial support from the VicePresidency of Research (University of Alicante, Spain).

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NOMENCLATURE ak = Legendre polynomial coefficients Ai,j = binary interaction parameters (K) for components i and j A, B, C = Margules coefficients f io = fugacity of pure component i Fi = individual quality factor GE (gE) = Gibbs energy of excess (dimensionless) GM (gM) = Gibbs energy of mixing (dimensionless) HE = enthalpy of excess HaE = average enthalpy of excess Lk = Legendre polynomials P = pressure pio = vapor pressure of pure component i QVLE = global quality factor for VLE data T = temperature Tio = boiling point of pure component i vE = excess volume xi = mole fraction of component i in liquid phase yi = mole fraction of component i in vapor phase

Greek Letters

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utility and reliability, as well as the quality of the experimental equilibrium data. In the mean time, inspection of the Gibbs energy of mixing curve for the liquid (gM,L) versus the liquid composition, obtained from the experimental VLE data, can reveal important information about the quality of these data that should be taken into account. This curve must be both smooth and consistent with the partial or total miscibility behavior of the liquid mixture.

ASSOCIATED CONTENT

S Supporting Information *

Supporting tables (S1 and S2) referenced in the text. This information is available free of charge via the Internet at http:// pubs.acs.org/.

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REFERENCES

(1) Dortmund Data Bank Software Package (DDBSP); Dortmund Data Bank Software and Separation Technology Gmbh: Dortmund, Germany, 2001 (2) Frenkel, M.; Dong, Q.; Wilhoit, R. C.; Hall, K. R. TRC SOURCE Database: A Unique Tool for Automatic Production of Data Compilations. Int. J. Thermophys. 2001, 22, 215. (3) Redlich, O.; Kister, A. T. Algebraic representation of thermodynamic properties and the classification of solutions. Ind. Eng. Chem. 1948, 40, 345−348. (4) van Ness, H. C.; Byer, S. M.; Gibbs, R. E. Vapor-Liquid equilibrium: Part I. An appraisal of data reduction methods. AIChE J. 1973, 19, 238. (5) Fredenslund, A.; Gmehling, J.; Rasmunssen, P. Vapor-Liquid Equilibria Using UNIFAC; Elsevier: Amsterdam, 1977. (6) Kojima, K.; Moon, H. M.; Ochi, K. Thermodynamic Consistency Test of Vapor-Liquid Equilibrium Data. Fluid Phase Equilib. 1990, 56, 269−284. (7) Wisniak, J. A New Test for the Thermodynamic Consistency of Vapor-Liquid Equilibrium. Ind. Eng. Chem. Res. 1993, 32, 1531−1533. (8) Kurihara, K.; Egawa, Y.; Ochi, K.; Kojima, K. Evaluation of thermodynamic consistency of isobaric and isothermal binary vaporliquid equilibrium data using the PAI test. Fluid Phase Equilib. 2004, 219, 75−85. (9) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall Ptr: Upper Saddle River, NJ, 1999. (10) Kurihara, K.; Egawa, Y.; Iino, S.; Ochi, K.; Kojima, K. Evaluation of thermodynamic consistency of isobaric and isothermal binary vaporliquid equilibrium data using the PAI test II, alcohol+n-alkane, +aromatic, +cycloalkane systems. Fluid Phase Equilib. 2007, 257, 151− 162. (11) Eubank, P. T.; Lamonte, B. G.; Javier Alvarado, J. F. Consistency Tests for Binary VLE Data. J. Chem. Eng. Data 2000, 45, 1040−1048. (12) Kang, J. W.; Diky, V.; Chirico, R. D.; Magee, J. W.; Muzny, C. D.; Abdulagatov, I.; Kazakov, A. F.; Frenkel, M. Quality assessment

Figure 7. Equilibrium data for the 1-propanol (1) + water (2) binary system at 760 mmHg using the NRTL parameters in ChemCAD 6.4.0 (see Table 1): (a) y vs x and (b) temperature vs x and y (molar fractions). A false VLLE data point is generated leading to a type IV instead a type I system.

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αi,j = nonrandomness NRTL factor γi = activity coefficient of component i φi = fugacity coefficient for i in the vapor phase

AUTHOR INFORMATION

Corresponding Author

*Tel.: (34) 965 903789. Fax: (34) 965 903826. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 13207



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algorithm for vapor-liquid equilibrium data. J. Chem. Eng. Data 2010, 55, 3631−3640. (13) Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection. Chemistry Data Series; DECHEMA: Frankfurt, Germany, 1977−1984. (14) Frenkel, M.; Chirico, R. D.; Diky, V.; Yan, X.; Dong, Q.; Muzny, C. ThermoData Engine (TDE): Software Implementation of the Dynamic Data Evaluation Concept. J. Chem. Inf. Model. 2005, 45, 816−838. (15) Diky, V.; Chirico, R. D.; Kazakov, A. F.; Muzny, C. D.; Frenkel, M. ThermoData Engine (TDE): Software Implementation of the Dynamic Data Evaluation Concept. 3. Binary Mixtures. J. Chem. Inf. Model. 2009, 49, 503−517. (16) Diky, V.; Chirico, R. D.; Muzny, C. D.; Kazakov, A. F.; Kroenlein, K.; Magee, J. W.; Abdulagatov, I.; Kang, J. W.; Frenkel, M. ThermoData Engine (TDE): Software Implementation of the Dynamic Data Evaluation Concept. 7. Ternary Mixtures. J. Chem. Inf. Model. 2012, 52, 260−276. (17) Jackson, P. L.; Wilsak, R. A. Thermodynamic consistency tests based on the Gibbs-Duhem equation applied to isothermal, binary vapor-liquid equilibrium data: data evaluation and model testing. Fluid Phase Equilib. 1995, 103, 155−197. (18) Wisniak, J. The Herington Test for Thermodynamic Consistency. Ind. Eng. Chem. Res. 1994, 33, 177−180. (19) Van Ness, H. C. Thermodynamics in the treatment of vapor/ liquid equilibrium (VLE) data. Pure Appl. Chem. 1995, 67 (6), 859− 872. (20) Cummings, P. T.; de Loos, Th.W.; O’Connell, J. P. Joint Statement of Editors of Journals Publishing Thermophysical Property Data Process for Article Submission. Fluid Phase Equilib. 2009, 276, 165−166. (21) Van Ness, H. C. Classical Thermodynamics of Non-electrolyte Solutions; Pergamon: Oxford, 1964; p 79. (22) Tassios, D. Applied Chemical Engineering Thermodynamics; Springer: Berlin, 1993. (23) Herington, E. F. G. A Thermodynamic Consistency Test for the Internal Consistency of Experimental Data of Volatility Ratios. Nature 1947, 160, 610−611. (24) Wang, H.; Xiao, J.; Shen, Y.; Ye, C.; Li, L.; Qiu, T. Experimental Measurements of Vapor−Liquid Equilibrium Data for the Binary Systems of Methanol + 2-Butyl Acetate, 2-Butyl Alcohol + 2-Butyl Acetate, and Methyl Acetate + 2-Butyl Acetate at 101.33 kPa. J. Chem. Eng. Data 2013, 58 (issue 6), 1827−1832. (25) Wisniak, J. Comment on ref 12. J. Chem. Eng. Data 2010, 55, 5394. (26) Sørensen, J. M.; Artl, W. Liquid-Liquid Equilibrium Data Collection. Chemistry Data Series; DECHEMA: Frankfurt; Vol V, Part 1, pp 419, 422. (27) Kang, J. W.; Diky, V.; Chirico, R. D.; Magee, J. W.; Muzny, C. D.; Abdulagatov, I.; Kazakov, A. F.; Frenkel, M. Reply to Comments by J. Wisniak on ref (12). J. Chem. Eng. Data 2010, 55, 5395. (28) Marcilla, A.; Olaya, M. M.; Serrano, M. D.; Reyes-Labarta, J. A. Methods for improving models for condensed phase equilibrium calculations. Fluid Phase Equilib. 2010, 296, 15−24. (29) Gmehling, J.; Menke, J.; Krafczyk, J.; Fischer, K. Azeotropic data. Part I; VCH: Weinheim, Germany, 1994.

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Pitfalls in the Evaluation of the Thermodynamic Consistency of

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