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ArticlePITFALLS IN THE EVALUATION OF THE THERMODYNAMIC
CONSISTENCY OF EXPERIMENTAL VLE DATA SETSantonio marcilla, Maria del Mar Olaya, Mara Dolores Serrano, and MARA ANGELES GARRIDO
Ind. Eng. Chem. Res., Just Accepted Manuscript DOI: 10.1021/ie401646j Publication Date (Web): 29 Jul 2013Downloaded from http://pubs.acs.org on August 2, 2013
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PITFALLS IN THE EVALUATION OF THE THERMODYNAMIC
CONSISTENCY OF EXPERIMENTAL VLE DATA SETS
Antonio Marcilla*, Mara del Mar Olaya, Mara Dolores Serrano and
Mara Angeles Garrido
Chemical Engineering Department, University of Alicante, Apdo. 99, Alicante 03080,
Spain. Tel. (34) 965 903789 Fax (34) 965 903826, *e-mail: [email protected]
ABSTRACT
The thermodynamic consistency of almost ninety 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.
KEYWORDS
Thermodynamic consistency, Consistency tests, VLE data, NRTL, Gibbs energy of mixing.
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, Dortmunt
Data Bank DBB1 and NIST Source Data Archival System
2. Accurate VLE data are
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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: area or integral test3, point-to-point tests
4-6, L-W test
7, infinite
dilution test6,8
and differential test9. 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. propose the PAI test that is a combination of the
point-to-point, area and infinite dilution tests6,8,10
. Eubank and Lamonte11
advise about the
advantages of a two-step 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 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: area test3, point-
to-point test by van Ness4 and Kojima
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
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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 ten thousand 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 dynamically14
. 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.
However, existing TC tests possess many drawbacks, some of which have already been
discussed in the literature17,18
, and others that are partially the subject of the present paper.
It is known that the widely used area test can involve cancellation errors19
. Moreover, the
method is not sensitive at all to the measured total pressure data4,19
. 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
that the test results are highly sensitive to the model that is used17
. 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 ninety 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
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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 software16
.
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 values20
. This
is the context in which can be better understood the significance of the consistency
algorithm proposed by Kang et al.12
and the importance of the discussion presented in this
paper, 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.
2. THE AREA AND POINT-TO-POINT TC TESTS
In this section, a brief description of the area and the point-to-point tests is presented. For
more details about these and other TC tests several other reference works can be
consulted10,17,18
.
2.1. Area test
The general Gibbs-Duhem equation can be expressed as follows21
:
0dPRT
vdT
RT
Hlndx
i
E
2
E
ii =+
(1)
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where 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:
0dPRT
vdT
RT
Hdxln
o1
o2
o1
o2
P
P
ET
T 2
E
11
02
1 =+
(2)
where 0iT and 0iP
are the boiling points and the vapor pressures of pure component i,
respectively.
Under isothermal conditions, the second term in eq 2 vanishes22
and the third can be
neglected8. 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 1/ 2 versus x1
graph are equal3. The condition for passing this test is given by eq 4 using a deviation
parameter D:
0dxln 11
02
1 =
(3)
2BA
BA100D
+
= (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.
min
minmax
T
TT150J
= (5)
The derivation of this equation was examined by Wisniak18
, 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:
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( )min
minmax
Emax
Ea
T
TT
G
H34J
= (6)
where Tmax and Tmin are the maximum and minimum boiling temperatures over the entire
concentration range, EaH
is the average heat of excess and EmaxG
is the maximum Gibbs
energy of excess. Eq 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
Emax
Ea G/H has been proposed
18. The criterion for the VLE data set to pass the test is
.10JD
7
i
ii0ical
i P
xfy
= (7)
where 0if is the fugacity of pure component i as liquid and i 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 (three-parameter) equation
( )211221E
E xCxBxAxxxRT
Gg +=
= (8)
The activity coefficients are calculated from the gE function and its derivative as
1
E
2E
1dx
dgxgln += (9)
1
E
1E
2dx
dgxgln = (10)
The coefficients A, B and C in the gE function are calculated by fitting, using a comparison
between the experimental and calculated total pressure
220211
01
cal xpxpP += (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 y in order to
establish whether VLE data are consistent but, obviously, this quantity must be small. They
recommended inspecting the P and y versus x plots to verify that a random scatter about
zero occurs, enabling one to establish consistency of the experimental VLE data.
expi
calii yyy = (12)
2.2.2. The Fredenslund test
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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:
( )=
=
=n
1k
1kk21
EE xLaxx
RT
Gg (13)
where ak are the coefficients of order k and Lk(x1) are the Legendre polynomials (eq 14)
( ) ( ) ( ) ( ) ( ) ( )[ ]12k11k11k xL1kxL1x21k2k
1xL = (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:
01.0yyyexpi
calii = (15)
This is a widely used test, e.g., 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:
01.0n
yyn
1i
expi
cali
=
(16)
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 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 Series13
. 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 point-to-point5 tests have been applied to these systems. Results
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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
used13
, 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:
( ) ( )222111liq,M
xlnxxlnxRT
G+= (17)
where i 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 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 point-to-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
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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 100C and ethylene oxide + water at 20C (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 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 non-individual) 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 40C, 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 x1
II=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:
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1. The gM
vs. x curve reveals important information about the quality of the VLE data
that is neither apparent 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.
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 incorrect18,25
. 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 ago18
.
The VLE data of any system, for example water + 1,2-propanediol at 50 mmHg, can be
generated using the NRTL equation based on the parameter values published in
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
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( )2
EE
RT
H
T
RT/G=
(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 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
Wisniak18
. 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 five-parameter
NRTL equation as the required model. Jackson et al.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
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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 models 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.
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 paper28
,
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 is 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 298K with the NRTL
model. In what follows, a similar case is presented but, now, for a VLE data correlation.
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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
experimental point, specified below, belongs to the experimental data set of the water(1) +
1,2-propanediol(2) system at 25mmHg (no. 79 in Table S2 Supporting Information),
plotted in Figure 6a:
x1 = 0.030; y1 = 0.775; T = 83.5C
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= 145.85K; A21=
41.307K and =3.032. This yields the calculated point:
x1(cal) = 0.030; y1(cal) = 0.612; T(cal) = 83.5C
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.
+==i i
iioi
i
V,MV,M ylny
)T(p
Plny
RT
Gg (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:
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47.2dx
dg
expx1
L,M
=
(20)
65.0g
expx
L,M = (21)
The closest to the required gM,L
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 composition cited previously:
y1 = 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 non-isothermal
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.
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
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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 point-to-point test), a fact that is acknowledged by other
authors17
.
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 compiled in the DECHEMA data collection13
. 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
present29
. Chemical process simulation software packages obtain information from that
which is available in data banks, compilations, 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 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-to-point 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 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.
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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 test 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 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.
NOMENCLATURE
ak = Legendre polynomial coefficients
i,j = binary interaction parameters (K) for components i and j
A, B, C = Margules coefficients
fio = fugacity of pure component i
Fi = individual quality factor
GE (g
E) = 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
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pio = vapour 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 vapour phase
Greek letters
i,j = non-randomness NRTL factor
i = activity coefficient of component i
i = fugacity coefficient for i in the vapour phase
ASSOCIATED CONTENT
Supporting Information
Supplemental tables (S1 and S2) referenced in the text. This information is available free of
charge via the Internet at http://pubs.acs.org/.
ACKNOWLEDGEMENTS
We gratefully acknowledge financial support from the Vice-Presidency of Research
(University of Alicante, Spain).
LITERATURE
(1) Dortmund Data Bank Software Package (DDBSP). Dortmund Data Bank Software and
Separation Technology Gmbh. 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, 345348.
(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.
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(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, 269284.
(7) Wisniak, J. A New Test for the Thermodynamic Consistency of Vapor-Liquid
Equilibrium, Ind. Eng. Chem. Res. 1993, 32, 15311533.
(8) Kurihara, K.; Egawa, Y.; Ochi, K.; Kojima, K. Evaluation of thermodynamic
consistency of isobaric and isothermal binary vapor-liquid equilibrium data using the PAI
test, Fluid Phase Equilib. 2004, 219, 7585.
(9) Prausnitz, J.M.; Lichtenthaler, R.N.; Gomes de Azevedo, E. Molecular
Thermodynamics of fluid-phase equilibria, 3rd ed., Prentice Hall Ptr, New Jersey, 1999.
(10) Kurihara, K.; Egawa, Y.; Iino, S.; Ochi, K.; Kojima, K. Evaluation of thermodynamic
consistency of isobaric and isothermal binary vapor-liquid equilibrium data using the PAI
test II, alcohol+n-alkane, +aromatic, +cycloalkane systems, Fluid Phase Equilib. 2007,
257, 151162.
(11) Eubank, P.T.; Lamonte, B.G. 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 algorithm for vapor-liquid equilibrium data.
J. Chem. Eng. Data 2010, 55, 36313640.
(13) Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection.
Chemistry Data Series. DECHEMA: Frankfurt, 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.
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(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, 155197.
(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.; OConnell, 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, 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, 610611.
(24) Wang, H.; Xiao, J.; Shen, Y.; Ye, C.; Li, L.; Qiu, T. Experimental Measurements of
VaporLiquid 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,Volume 58, issue 6, 1827-1832.
(25) Wisniak, J. Comment on ref (12), J. Chem. Eng. Data, 2010, 55, 5394.
(26) Srensen, J.M.; Artl, W. Liquid-Liquid Equilibrium Data Collection. Vol V, Part 1 (p
419, 422), Chemistry Data Series, DECHEMA, Frankfurt.
(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, 1994.
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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)
DECHEMA [13] 1 152,5084 1866,3369 0,3747
2 444,3339 1997,5504 0,4850
3 412,0253 1735,4304 0,4465
4 152,5084 1866,3369 0,3747
5 294,7832 1893,5152 0,4276
6 619,3422 2708,5773 0,6185
7 -13,0045 1872,0758 0,2803
ChemCAD 6.4.0. 444.3322 1997.6031 0.4850
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a)
x1
RT
GM
b)
x1I x1
II
II
I
commontangent line
RT
GM
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).
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a)
b)
Figure 2. VLE experimental data for the binary system acetone (1) + water (2) at 100C
(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.
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a)
b)
Figure 3. VLE experimental data for the binary system ethylene oxide (1) + water (2) at
20C (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.
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a)
b)
Figure 4. VLE experimental data for the binary system water (1) + 1-hexanol (2) at 40C
(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.
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a)
b)
Figure 5. Schematic representation of the limitations of the NRTL model: a) systematic
study of the homogeneous gM
binary curves for x2min=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.
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a)
b)
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
(vapour) and gM
NRTL (liquid) functions for the selected VLE point
at T=83.5C (the non-isothermal gML
curve for all data set has been included).
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a)
b)
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 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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