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Thermodynamics, Transport, and Fluid Mechanics
Improving the Predictability of Chemical Equilibrium Software Qi LIU, Christophe Proust, Francois Gomez, Denis Luart, and Christophe Len Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b03571 • Publication Date (Web): 10 Dec 2018 Downloaded from http://pubs.acs.org on December 13, 2018
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Industrial & Engineering Chemistry Research
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Improving the Predictability of Chemical
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Equilibrium Software
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Qi Liua, Christophe Prousta,d,*, François Gomeza, Denis Luartb, Christophe Lena,c
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a Sorbonne Universités, Universite de Technologie de Compiegne, Centre de Recherche
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Royallieu, CS60319, F-60203 Compiegne, France
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b Ecole Superieure de Chimie Organique et Minerale, 1, rue du reseau Jean-Marie
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Buckmaster, F-60200 Compiègne, France
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c PSL University, UMR 8247 CNRS Chimie ParisTech, Institut de Recherche de Chimie, 11
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rue Pierre et Marie Curie, F-75005 Paris, France
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d INERIS, dept DRA, parc Technologique Alata P.O box. 2, F-60550 Verneuil-en-Halatte,
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France
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Abstract:
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Over the past few decades, researchers have been developing tools to predict chemical
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reactions to aid the growing field of industrial chemistry. Currently, a large variety of
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numerical tools are used to predict the final chemical equilibrium based on the minimization
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of the Gibbs free energy. Due to the mathematical complexity of the problem, numerical
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methods were developed to solve this problem. These methods were reviewed in another
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study1, 2 exhibiting their limitations and proposing an alternative. In this study, the sensitivity
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of the prediction as function of the thermochemical (input) parameters is discussed showing
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that significant deviations are possible when the relative uncertainty between the enthalpies
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of formation is larger than a few kJ/mole. Often the scatter between various data sources is
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much larger than this. To solve this difficulty, it is attempted to derive all the required
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thermodynamical parameters from a base of molecular descriptors common to the chemistry
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targeted in this work (organic). The group contribution theory is implemented and in
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particular the UNIFAC descriptors and is shown to give very satisfactory results.
26 27 28
Keywords
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Gibbs free energy minimization, thermodynamic equilibrium calculations, group contribution
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theory.
31 32
Introduction
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The development of computers made it possible to predict the final equilibrium
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composition of a multiphase and reactive mixture by minimizing the total Gibbs energy of the
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products. The method is potentially powerful but suffers from some practical limitations. The
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first limitation is to be able to find the absolute minimum of the Gibbs energy while
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preserving the conservation of the atoms. Very innovative step by step mathematical
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techniques were proposed which may not converge to the absolute minimum (may converge
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at some local “constrained” minimum) depending on the choice of the starting point. These
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aspects were reviewed in another paper,1, 2 and an alternative minimization technique was
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proposed to avoid this. The second limitation is undoubtedly the accuracy of the
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thermochemical data incorporated in the database of the software and is addressed below.
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Lastly, the third limitation is the choice of the product list, which strongly orientate towards
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the final result. The method of selection of the product will be addressed in a forthcoming 2 ACS Paragon Plus Environment
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Industrial & Engineering Chemistry Research
paper.
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The following work deals with the influence of the uncertainties of the thermochemical
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data on the prediction of thermodynamic equilibria. This influence is first quantified, which
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was not openly done in the literature before. Second, it is shown that the group contribution
49
theory can be used to calculate all the required thermodynamical parameters in order to bring
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more coherence and reduce the influence of scattering of the thermodynamical data sources.
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These developments were implemented in CIRCE code, a homemade software prepared by
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the present authors, and the performances of this technique are illustrated by comparison to
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tricky experimental configurations.
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Influence on the accuracy of the thermochemical data on the chemical equilibrium
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Fundamentals of chemical equilibrium
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Only the leading aspects of the underlying thermochemistry are recalled below.1, 2 It is
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assumed that a chemical reaction occurs between nSp molecules (They are indexed by
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i=1…nSp) composed of nEl atoms (They are indexed by j=1…nEl). If aij is the number of
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atom j in the molecule i (available in ni moles in the mixture), then the conservation of mass
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can be expressed as (bj is the total number of atoms j in the initial mixture):
62
𝜖𝑗 =
𝑁 𝑖=1 𝑎𝑖𝑗
∙ 𝑛𝑖 − 𝑏𝑗 = 0 𝑓𝑜𝑟 1 < 𝑗 < 𝑛𝐸𝑙
(1)
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In the most general situation, the chemical potential μi of the molecule i in a mixture is μi(P,
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T)= μio(T)+RT.lnai where μio(T) is the standard Gibbs energy of formation of the species and
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ai stands for its “activity” in the mixture. For ideal mixtures, ai is simply the molecular
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fraction xi. Often ai is expressed as a function of the molar fraction xi, such as, ai=γi.xi where
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γi is the coefficient of activity. This represents the intermolecular interactions which may
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favor or limit the mobility of the species as compared to an ideal mixture where the
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intermolecular forces are zero. Note that for solids, it is often assumed that the chemical 3 ACS Paragon Plus Environment
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potential mostly depends on the temperature and does not depend on other compounds.
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At constant temperature and pressure, the chemical equilibrium is reached when the Gibbs
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energy of the mixture is minimum. The Gibbs energy of the mixture is most often expressed
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as the sum of the contributions of the Gibbs energies of formation of each pure species G0, of
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the mixing of the species Gmix (the increase of the entropy) and of the non-ideality GE
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(“excess Gibbs energy”) caused by the intermolecular forces in the real mixture:
76
𝐺0 =
77
𝐺𝑚𝑖𝑥 =
78
𝐺𝐸 =
79
𝐺 = 𝐺 0 + 𝐺𝑚𝑖𝑥 + 𝐺 𝐸
𝑁𝑆 𝑖=1 𝑛𝑖
∙ 𝜇𝑖0 +
𝑁𝐿 𝑖=1 𝑛𝑖 𝑁𝐿 𝑖=1 𝑛𝑖
𝑁𝐿 𝑖=1 𝑛𝑖
∙ 𝑅 ∙ 𝑇 ∙ 𝑙𝑛
∙ 𝜇𝑖0 + 𝑛𝑖 𝑁𝐿 𝑛 𝑖 𝑖
∙ 𝑅 ∙ 𝑇 ∙ 𝑙𝑛 𝛾𝑖 +
𝑁𝐺 𝑖=1 𝑛𝑖
+
𝑁𝐺 𝑖=1 𝑛𝑖
(2)
∙ 𝜇𝑖0
𝑁𝐺 𝑖=1 𝑛𝑖
∙ 𝑅 ∙ 𝑇 ∙ 𝑙𝑛
𝑛𝑖 𝑁𝐺 𝑛 𝑖 𝑖
∙ 𝑅 ∙ 𝑇 ∙ 𝑙𝑛 𝛾𝑖
(3) (4) (5)
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Where μi0(T) is the Gibbs energy of formation of component i (considered pure) at
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temperature T and pressure P of the reaction, γi the activity coefficient of component i in the
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real mixture at T and P, NS, NL and NG stand respectively for the number of solid, liquid and
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gaseous species.
84
Uncertainties related to the accuracy of the thermochemical data
85 86
The required thermochemical data are the chemical potentials of the pure components and the activity coefficients.
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Considering a three components mixture (n1, n2, and n3) with one product (n3) being
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diluents (no variations of its number of moles). The atom balance (1) between n1 and n2,
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imposes a relationship which can be written as below:
90
𝑛2 = 𝐶 − 𝐴 ∙ 𝑛1
(6)
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Where C and A are constants. In the present context where n1 turns into n2, typically for a
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phase change which is considered here, A is unity. Equation (6) can be inserted into (2) to
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(5),and the only dependent variable is n1 since n3 is a constant. The minimum of G is 4 ACS Paragon Plus Environment
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obtained when 𝑑𝐺 𝑑𝑛1 = 0. If we consider that n1 is a liquid and n2 and n3 are gases, after
95
some straightforward calculations (Appendix S1), the final expression can be expressed as: 𝑑𝛾
1−𝑥 2 ∙𝑥 2
𝑑𝛾
0 = 𝜇10 − 𝐴 ∙ 𝜇20 − 𝐴 ∙ 𝑅 ∙ 𝑇 ∙ 𝑙𝑛 𝑥2 − 𝐴 ∙ 𝑅 ∙ 𝑇 ∙ 𝑙𝑛𝛾2 − 𝑑𝑥2 ∙
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Where x2 is the molar fraction of component 2 in the gaseous phase (containing only
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components 2 and 3) and is the eigenvalue parameter of this problem. The contribution of the
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formation energies of the components and the molecular interactions are now rather explicit.
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2
𝐴∙𝛾2
+ 𝑑𝑥3 ∙
1−𝑥 2 2
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2
𝛾3
(7)
The influence of the chemical potential of the pure components is illustrated first setting the
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fugacity coefficients equal to 1 and constant (ideal mixtures). Then (7) reduces to:
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ln 𝑥2 =
𝜇 10 −𝐴∙𝜇 20
(8)
𝐴∙𝑅∙𝑇
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This equation suggests that the accuracy of the determination of the chemical potentials
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should be smaller than R.T or, more accurately that the method used to determine the
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chemical potentials should not generate deviations from one component to another
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component comparable to R.T, and thus it should typically be smaller than 1 kJ/mole.
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A more practical illustration is proposed for a 5 components chemical reaction consisting
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of a water-gas shift simulation (C, CO, CO2, H2O, H2). In this kind of reactions, the amount
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of produced CO can be seen as a sensitive indicator of the global reaction since it is produced
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by the C/H2O reaction and consumed by the CO/H2O reaction. The evolution of the CO
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molar fraction is shown in Figure 1 as a function of pressure and temperature of the reactions.
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All components except C are gases obeying the perfect gas law. In this simulation, the
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standard enthalpy of formation for CO is 110 kJ/mole.
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0.6
0.5
XCO
0.4
0.3
0.2
0.1
0.0 0
200
400
600
800
1000
1200
1400
1600
1800
2000
T/K
114 115
Figure 1. CO volumetric fraction in a 5 components water shift reaction as a function of
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pressure and temperature of the reaction under the assumption of perfect gas reference cases
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(CIRCE code).10kPa, ■; 100kPa, ●; 1MPa, ▲.
118 119
In Figure 2, the changes in the CO volumetric fraction is shown when the standard enthalpy
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of formation of CO is changed by ±1%, ±5%, and ±10%, respectively.
0.6
a)
0.5
0.4
xCO
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0.3
0.2
0.1
0.0 0
121
200
400
600
800
1000 1200 1400 1600 1800 2000 T/K
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0.6
b) 0.5
xCO
0.4
0.3
0.2
0.1
0.0 0
200
400
600
800
1000 1200 1400 1600 1800 2000 T/K
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0.6
c) 0.5
0.4
xCO
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Industrial & Engineering Chemistry Research
0.3
0.2
0.1
0.0 0
200
400
600
800
1000 1200 1400 1600 1800 2000 T/K
123 124
Figure 2. CO volumetric fraction in a 5 components water shift reaction as a function of
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different pressure a) 100kPa, b) 10kPa and c) 1MPa and temperature of the reaction under the
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assumption of a perfect gas influence of the accuracy of thermochemical data (variation of
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the standard enthalpy of formation of CO). Reference case at 100kPa, ■; HfCO+0.01HfCO, ●;
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HfCO+0.05HfCO, ▲; HfCO+0.1HfCO, ▼; HfCO-0.01HfCO, ◆; HfCO-0.05HfCO, ◄; HfCO-
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0.1HfCO, ►.
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This example shows an evident influence on the values of the thermochemical data 7 ACS Paragon Plus Environment
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particularly concerning the Gibbs energy of formation of the components (in which the
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standard enthalpy of formation is the main aspect). The pressure influences the evolution of
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the entropy. It can be observed that a 10% change in the standard enthalpy of formation of
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CO (i.e., ±10 kJ/mole) produces a variation into the CO concentration within a factor of two
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at 1 MPa and 1000 K for instance. A ±1% variations (±1 kJ/mole), would produce, in the
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same configuration, only a relative scatter of 10% of the CO concentration which might be
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judged acceptable in chemical engineering.
138
This example should be considered as an illustration. But together with the simple
139
theoretical approach proposed in Equation (8), it shows that even if the minimization
140
technique is robust enough and the product list is well selected, erroneous results can be
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obtained if the relative accuracy of the thermochemical data is lower than, typically 1kJ/mole.
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This last point questions the used of the published database where the uncertainties about the
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energies of formations may amount ±10kJ/mole.3
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Now, the question of the accuracy of the activity coefficients is addressed. From equation
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(7), the deviation of the molar fraction of component 2 from the ideal mixture case (given by
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equation (8)) reads: 𝑥2
𝑑𝛾
= − 𝑙𝑛𝛾2 − 𝑑𝑥2 ∙
1−𝑥 2 ∙𝑥 2 𝐴∙𝛾2
𝑑𝛾
+ 𝑑𝑥3 ∙
1−𝑥 2 2
(9)
147
𝑙𝑛
148
A graphical representation is proposed on Figure 3 in which the ideal mixture case is
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compared to a constant (γ2=2) and also to a variable activity coefficient (γ2=2 and
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1/γ2∙dγ2/dx2=1), which is more realistic. Note that both the chosen values of coefficient of
151
activity and their variations are typically in the orders of magnitude encountered in real
152
systems.4
𝑥 2_𝑖𝑑
2
2
𝛾3
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2.0
1.5
x2/x2id
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Industrial & Engineering Chemistry Research
1.0
0.5
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x2
153 154
Figure 3. ideal mixture approximation as compared a constant (γ2=2) and to a variable of the
155
coefficient of activity (γ2=2 and 1/γ2∙dγ2/dx2=1). The latter case is labeled “real mixture”.
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ideal mixture,■; constant activity coefficient,●; real mixture,▲.
157 158
It appears that a significant deviation is possible, within the ratios of 2 to 4, when the
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details of the variations of the coefficient of activity are ignored. Even a quasi-constant
160
approximation (with dγ2/dx2=0) may not be satisfactory.
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An illustration is presented below about the distillation of ethanol-water mixtures at 100
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kPa pressure (Figure 4). Assuming ideal mixtures (Figure 4-b) provides results rather far
163
from reality. In textbooks, the activity coefficients of ethanol and water are respectively on
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the order of two so that the real concentrations in the gaseous phase should be about half of
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the concentration of the ideal mixture (see Figure 3). However, even after this correction (not
166
shown in Figure 4), the agreement would remain poor. The variations of the activity
167
coefficient with the compositions should be included. This is partly done in the RAND
168
method5 for instance (Figure 4-a)but it can be realized that the simulated values deviate by a
169
few % from the experiments. This is presumably due to the quadratic Taylor approximation 9 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research
170
of G which is done in the RAND method to run the step by step algorithm.
375
370
T/K
365
360
355
350 0.0
0.2
0.4
0.6
0.8
1.0
xEthanol
171 (a)
172
375
370
365
T/K
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360
355
350 0.0
0.2
0.4
0.6
0.8
1.0
xEthanol
173 174
(b)
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Figure 4. Simulation of vapor-liquid equilibria of the water-ethanol mixture at 100kPa
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pressure using the RAND method (ASPEN Plus®-RGIBBS code) (a) and the ideal mixture
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model approximation (CIRCE code) (b). Simulations -vapor, ○; experiments -vapor, ●;
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simulations-liquid,□; experiments-liquid, ■. 10 ACS Paragon Plus Environment
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Industrial & Engineering Chemistry Research
179 180
So there is a need to find a way to estimate all the thermochemical properties for the pure
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components and the intermolecular forces in a “coherent” way. A systematic bias may not be
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so problematic since the absolute value of G is not looked for but only its minimum, but
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scattering/inconsistencies between the data sources for the components would easily produce
184
large uncertainties. A common route/model for estimating the chemical potentials should be
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looked for, or in other words, based on the same physical principles. Possibilities are
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presented hereafter.
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Thermochemical properties and interaction parameters
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Group contribution theory
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Equation (5) contains both a historical and conceptual vision of thermodynamics.
191
Historically, pure and ideal mixtures where considered at first (G0 and Gmix) and conceptually
192
since real fluids and mixtures (GE) are approached as deviations from the ideal cases, which
193
can be calculated via some virtual transformation.
194
So what is needed is a common set of descriptors enabling the estimation of the enthalpy of
195
formation of the pure ideal components, of the real fluid “transformation” (using the equation
196
of state of the fluid) and of the activity coefficients.
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As explained in textbooks, many attempts were made over the last century to connect
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molecular descriptors to the thermodynamic properties. The simplest and perhaps the first
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attempt is to use the property and number of each atom and to add up each atomic
200
contribution to the searched property for the molecule (additivity rule), just as what is done to
201
calculate the molar mass of a molecule. But even for this simple property, careful verification
202
shows that the contribution of a given type of atom depends on the neighboring atoms. The
203
extent to this effect depends upon the importance of outer valence electrons. A refinement is 11 ACS Paragon Plus Environment
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to consider on contributions in which specific differences between various types of bonds (ex
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for carbon: carbon-carbon, carbon-oxygen, carbon-nitrogen...) could be integrated. But this
206
adds many complications. A medium way is to consider groups of atoms from which most
207
molecules are made of and apply the additivity to these groups. To some extent, such groups
208
have built-in information on the valence structure associated with a significant proportion of
209
the atoms present. The “group contribution” theory is hence widely and successfully used at
210
least in organic chemistry.
211
Perhaps the most developed side of the group contribution theory appears in the
212
determination of the activity coefficient via the well-known UNIFAC method.6 Today, in
213
organic chemistry, UNIFAC incorporates about 54 main groups and about 113 subgroups
214
from the DDBST7 database. It is recalled that “main groups” are groups of atoms
215
(“subgroups”) sharing the same interaction effects with the other groups of atoms. The
216
“subgroups” describe the interactions of the molecule into the mixture. Using UNIFAC the
217
activity coefficients in the standard conditions can be calculated. But the group contribution
218
theory can also be used very satisfactorily to derive most of the important thermodynamic
219
parameters like those for the equation of state of each component from which the real fluid
220
behavior can be calculated.6 The latter method provides the critical parameters, temperature
221
Tc, pressure Pc and volume Vc from the group contribution theory enabling the derivation of
222
the “co-volume” and “energy” parameters of the equation of state. An extension was
223
proposed by Gani-Constantinou8 about the “acentric factor” of the equation of state.
224
Joback’s9 method also uses the group contribution theory to derive the standard enthalpy,
225
entropy of formation,… of the ideal gas form of the components. Further refinements were
226
proposed by Klincewicz,10 Ambrose…11
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Thus, the group contribution theory was implemented (in CIRCE). The molecular
228
descriptors are those from UNIFAC. The ideal gas thermodynamical data (standard enthalpy 12 ACS Paragon Plus Environment
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229
and entropy of formation, heat capacity,…) and real fluid parameters (acentric factor, critical
230
parameters,…) were extracted from the works of Joback.9 The activity coefficient is derived
231
from the UNIFAC theory.
232
A “cubic” equation of state was chosen to account for the real fluid “transformation”
233
because of its ease of implementation in numerical codes. However, since the activity
234
coefficients in a mixture can also be extracted from the equation of state of the mixture
235
(according to the mixing rule associated with the equation of state), precautions were taken so
236
that the activity coefficients, appearing in GE, obtained through the equation of state (EoS)
237
were identical to those obtained using the UNIFAC theory for the standard conditions. This
238
important aspect is clarified later.
239 240
The final organization of the thermodynamical properties generator of CIRCE software is represented schematically in Figure 5.
Molecular descriptors from UNIFAC
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Ideal fluid properties
Real fluid properties (EoS parameters & mixing rules)
Fitting Activity coefficients
241 242
Figure 5. Organization of the thermodynamical properties generator of CIRCE software.
243 244
When running a simulation, the properties of each component considered as an ideal gas at 13 ACS Paragon Plus Environment
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245
the temperature and the pressure of the reaction are first calculated. The non-ideality of the
246
component is added under the form of the “departure function” detailed later (integration of
247
the equation of state from the ideal gas state to the real fluid state – liquid or gaseous). This
248
calculation provides G0. Then the activity coefficients are calculated using UNIFAC (thus in
249
the standard conditions) and incorporated in the EoS. Using the EoS, the activity coefficients
250
are computed at T and P of the reaction. Note that if liquids, gases, and even supercritical
251
fluids can be handled this way, provided the EoS is well chosen, this does not hold for solids,
252
for which a classical method is used.
253 254
The details are presented in the next paragraphs. Ideal (and some real) fluid properties
255
The following parameters are needed to run a calculation: standard formation enthalpy and
256
entropy, heat capacities at specific temperatures for extrapolation purposes (typically at 290K,
257
1000K, 2500K, 3500K, and 5000K), the critical parameters (Vc, Tc, Pc) and the acentric
258
factor.
259
The Joback method uses the group contribution theory as an extension of the pioneering
260
work of Parks and Huffman.12 Note that Joback assumed no interactions between the groups
261
so that only linear combinations are considered. These data used in the development of all the
262
methods were obtained from the literature. The experimental value of Critical property values:
263
Tc, Pc, and Vc from Ambrose13 and Reid et al.14 have been used. The data from Reid et al.14
264
0 0 and Stull et al15 has been used for the thermodynamic properties(∆𝐻𝑓,298 , ∆𝐺𝑓,298 and 𝐶𝑝0 ).
265
Multiple linear regression techniques were employed to determine the group contributions for
266
each structurally-dependent parameter.7 The acentric factor is given by Lydersen’s work.16
267 268
Note however that the “groups” considered by Joback are not those from UNIFAC but,
269
fortunately, the latter can be globally deduced from the former by additions which enables a 14 ACS Paragon Plus Environment
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unique type of “groups” (those from UNIFAC) to be considered. After having redefined the
271
“groups” in the Joback technique identical to those from UNIFAC a verification was
272
performed on typical molecules (Table 1). The analysis was carried out using methanol,
273
ethanol, formic acid acetaldehyde molecules which are simple and common in the chemistry.
274
The results are in agreement with those obtained by Joback, except may be a better accuracy
275
on the enthalpy of formation.
276 277
Table 1. Comparison of the modified JOBACK method with experimental data.9
Simulated C2H5OH Measured C2H5OH Simulated CH3COOH Measured CH3COOH Simulated CH3OH Measured CH3OH Simulated CH3CHO Measured CH3CHO Standard deviation
0 Tb(K) Tc(K) Pc(100kPa) Vc(cm3/mole) ∆𝐻𝑓,298 (kJ/mole) ΔHv,b(kJ/mole) 337.34 499.11 57.57 166.50 -236.84 36.64
351.5
514
63
168
-234
42.3
390.67 587.25
57.31
171.50
-434.88
40.67
391.2
57.81
170.34
-433
50.3
66.97
110.50
-216.20
33.91
81
117
-205
35.21
293.82 465.29
56.03
164.50
-170.19
26.92
293.9
466.0
55.7
151.38
-170.7
26.12
20
20
8
5
2
6
593
314.46 475.49 337.8
513
278
a
Tb is boiling temperature. bTc is the critical temperature. c Pc is the critical pressure. dVc is the
279
critical volume. eΔH0f,298 is the enthalpy of formation at 298K. fΔH0v,b is the enthalpy of
280
formation at 298K.
281 282 283 284 285
Equation of state
The ideal gas equation of state was established empirically and theoretically mainly during the 18th and 19th century. It is assumed that molecules are infinitely small and do not interact. In 1893, Van der Waals concluded that in real gases the volume occupied by the molecules 15 ACS Paragon Plus Environment
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Page 16 of 27
286
is not zero and should be subtracted from the total volume to retrieve the real free volume and
287
that molecules interact with various forces which change their velocities, and thus the
288
pressure and a correction term should be added.17
289
The ideal gas can be considered as a limiting case of the van der Waals equation of state
290
when P tends to zero since in such conditions Vm tends to infinity in such a way that the
291
correction terms are negligible. This remark is used to calculate the “departure function”.
292
It was recognized that although being significant progress, the van der Waals equation is
293
deficient on many aspects and alternative EoS, based on a very similar equation, were
294
proposed later like the RK equation,18 the SRK19 EoS, and the well-known Peng-Robinson
295
EoS.20 The latter, detailed below, is particularly efficient in predicting liquid densities and
296
vapor pressures even on the saturation line.
297
As it stands, however, the Peng Robinson EoS has deficiencies mostly when mixtures are
298
concerned for which the “binary interaction” parameters are to be fitted from experimental
299
data.
300
To overcome this deficiency, Vidal in 197820 coupled the Peng-Robinson equation to the
301
GE parameter, incorporating this correction into the energy parameter (α). The criterion is that
302
GE derived from the EoS should be equal to that obtained from an explicit formulation (like
303
UNIFAC…) of the activity coefficients at some reference pressure.
304
As a reminder, the activity coefficients are associated with the fugacities (or more precisely
305
to the fugacity coefficients) which can be derived from the EoS (Appendix S1). If 𝜑𝑖 is the
306
fugacity coefficient of component i in the mixture, and 𝜑𝑖 is the pure compound i, the excess
307
Gibbs energy, GE is calculated as:
308 309 310
𝐺𝐸 𝑅𝑇
=
𝑖 𝑥𝑖
𝜑
ln 𝜑 𝑖 = 𝑖
𝑖
(10)
𝑥𝑖 ln 𝛾𝑖
Since the fugacities can be derived from the EoS so that the following relationship is found between the activity coefficients obtained from UNIFAC and those derived from the EoS: 16 ACS Paragon Plus Environment
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311 312 313 314
Industrial & Engineering Chemistry Research
𝜑
ln 𝛾𝑖 = ln 𝜑 𝑖 =
𝜕𝐺 𝐸
(11)
𝜕𝑛 𝑖 𝑇,𝑃,𝑛 𝑗 ≠𝑖
𝑖
In Huron-Vidal, Wong-Sandler approaches the reference pressure to which this equality is applied in infinity whereas for PSRK model the reference pressure is zero. In the present case, the LCVM method21 is selected which does not specify a specific
315
pressure.22 For this the Peng-Robinson EoS is used as:
316
𝑃 = 𝑉−𝑏 − 𝑉
317
𝑅𝑇
𝑎
The pure component parameters a and b are:
318
𝑎 = 0.45724
319
𝑏=
320 321 322 323 324
(12)
𝑉+𝑏 +𝑏 𝑉−𝑏
𝑖
𝑅 2 𝑇𝑐2 𝑃𝑐
(13)
∙ 𝑓 𝑇𝑟
(14)
𝑥𝑖 𝑏𝑖
If (𝑇𝑟 ≤ 1): 𝑓 𝑇𝑟 = 1 + 𝑐1 1 − 𝑇𝑟 + 𝑐2 1 − 𝑇𝑟
2
+ 𝑐3 1 − 𝑇𝑟
3 3
(15)
If (𝑇𝑟 ≥ 1): 𝑓 𝑇𝑟 = 1 + 𝑐1 1 − 𝑇𝑟
2
(16)
Coquelet et al.23 proposed a method to generate the parameter of c1, c2, and c3 automatically:
325
𝑐1 = 0.1316𝜔2 + 1.4031𝜔 + 0.3906
(17)
326
𝑐2 = −1.3127𝜔2 + 0.3015𝜔 ± 0.1213
(18)
327
𝑐3 = 0.7661𝜔 + 0.3041
(19)
328
A linear combination of the mixing rules of Huron and Vidal24 and Michelsen25 is
329
implemented, which does need a reference pressure.21 And it has been proved26,
27
330
superior to the other EOS/GE models. Clearly, other categories of EoS existed, using the
331
group contribution theory23 and based on the Statistical Associating Fluid Theory24, but a
332
“cubic” EoS was preferred in this work. The pressure/temperature domain of the LCVM EoS
333
is also well adapted to the practical applications targeted by our research group.
to be
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334 335 336
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The co-volume parameter is unmodified as compared to the standard Peng Robinson EoS (12). But the innovation is in the formulation of the energy parameter a. 𝑎 𝑏∙𝑅∙𝑇
=𝜆∙
1 𝐴𝑉
𝐺𝐸
∙ 𝑅𝑇 +
𝑎
𝑖 𝑥𝑖 𝑏 ∙𝑅∙𝑇 + 1−𝜆 ∙ 𝑖
1 𝐴𝑀
∙
𝐺𝐸 𝑅𝑇
×
𝑥𝑖 ln
𝑏 𝑏𝑖
+
𝑎
(20)
𝑖 𝑥𝑖 𝑏 ∙𝑅∙𝑇 𝑖
𝑏
337
Where the terms 𝐺 𝐸 𝑅𝑇 is calculated using the UNIFAC model, the terms
338
𝑖 𝑥𝑖 𝑏 ∙𝑅∙𝑇 come from the Peng Robinson Equation of state, and AM, AV are constant
339
coefficients proposed respectively by Michelsen25 and Vidal24 ( 𝐴𝑀 = −0.52 , and 𝐴𝑉 =
340
−0.623 ). Coefficient λ is a sort of relaxation parameter, and its value was selected
341
empirically at 0.36.
342
The “departure function”
𝑥𝑖 ln
𝑏𝑖
and
𝑎
𝑖
343
By definition, the “departure functions” are used to calculate the difference between a real
344
fluid and the ideal fluid. The principle of the “virtual” transformation of an ideal gas to a real
345
fluid is outlined in the appendix S2. The resolution uses the expression of the thermodynamic
346
function as a function of the state variables (P, V, and T). For the Gibbs energy:
347
G 𝑇, 𝑃
348
𝑟𝑒𝑎𝑙
− G 𝑇, 𝑃
𝑖𝑑𝑒𝑎𝑙
=
𝑟𝑒𝑎𝑙 𝑖𝑑𝑒𝑎𝑙
𝑉 𝑑𝑃 =
𝑟𝑒𝑎𝑙 𝑖𝑑𝑒𝑎𝑙
𝑑 𝑃𝑉 −
𝑉 𝑃 𝑑𝑣 𝑉=∞
= 𝑃𝑉 − 𝑅𝑇 −
𝑉 𝑃𝑑𝑉 𝑉=∞
(21)
The departure function is used to calculate the chemical potentials of the pure components
349
(to calculate G0) for which the LCVM EoS reduces to the (modified) Peng Robinson EoS.
350
Performances
351
Some simulations using the above methods (Joback, UNIFAC, LCVM, departure
352
functions...) are presented below for some critical systems. Note that the minimization
353
technique is the MGCE method presented elsewhere1. All these are implemented in CIRCE
354
software. The first is the vapor-liquid equilibria of the ethanol-water mixture which was
355
shown to exhibit specific difficulties and illustrates the influence of significant molecular
356
interactions. The second and third systems involve the investigation of the impact of the
357
pressure on the prediction of a vapor/liquid multicomponent equilibrium and the last one deal 18 ACS Paragon Plus Environment
Page 19 of 27
358
with a supercritical dissolution case traditionally challenging to handle.
359
Vapor-liquid equilibrium of a significantly polar system
360
The distillation at ambient pressure as the function of the temperature of an equimolar
361
ethanol-water mixture is considered identical to that presented in Figure 4. The results are
362
shown in Figure 6. The simulation results are entirely consistent with the experimental data,
363
suggesting first that the thermodynamical data are calculated relevantly and that the
364
minimization techniques can conveniently follow the variations of the activity coefficients.
375
370
365
T/K
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
360
355
350 0.0
0.2
0.4
0.6
0.8
1.0
XEthanol
365 366
Figure 6. Simulation of vapor-liquid equilibria of the water-ethanol mixture (100kPa
367
pressure) using the LCVM/departure function/MGCE methods (CIRCE code). Simulations:
368
vapor, ○; experiments: vapor,●; simulations: liquid,□; experiments: liquid,■.
369 370 371 372
High-pressure distillation cases
The distillation of n-pentane in propane at 344 K and various pressures (non-polar system) is shown in Figure 7. The simulation result agrees well with available experimental data.
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100
80
P/100kPa
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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60
40
20
0 0.0
0.2
xC3H8
373 374
Figure 7. Prediction of the P-x-y diagram for the distillation propane/n-pentane at 344.26 K
375
using the LCVM/departure function/MGCE methods (CIRCE code), Experimental data from
376
Knapp et al. (1982).28Simulations: vapor, ○; experiments: vapor, ●; simulations: liquid,□;
377
experiments: liquid,■.
378 379 380
The distillation of methanol in water at 423 K and various pressures (polar system) is shown in Figure 8 again showing a satisfactory agreement with available experimental data.
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14
12
P/100kPa
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10
8
6
4 0
20
40
60
80
100
xmethanol
381 382
Figure 8. Prediction of the P-x-y diagram for the distillation methanol-water at 423 K using
383
the LCVM/departure function/MGCE methods (CIRCE code), Experimental data from
384
Griswold and Wong.29 Simulations: vapor, ○; experiments: vapor, ●; simulations: liquid,□;
385
experiments: liquid,■.
386 387
Solubility of supercritical CO2 in methanol
388
Solubility is the property of gases for instance (CO2 here) to dissolve in liquids (methanol).
389
The solvent is assumed to have negligible vapor pressure and to remain entirely in the liquid
390
phase. The specific difficulty is linked to the possibility of the mixture to become
391
supercritical. But the general character of the LCVM EoS offers in principle the options to
392
deal with the supercritical state since no reference pressure is used to couple the EoS with GE.
393
Some example of the solubility of CO2 have been inverstigated.30, 31A typical example of
394
the solubility of methanol in CO2 is shown in Figure 9. In this particular example, CO2
395
reaches supercritical state above 7 MPa whereas methanol is not. The mixture is neither
396
liquid nor gas.32 The simulation strategy enables to mimic this behavior although with a
397
reduced accuracy than in the example above. A possible reason for this may lie in the fact 21 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research
398
that the parameters of the EoS are only approximate using the Constantinou approaches 7 and
399
that the deviations become more influenced when P and T are not far from the critical points.
80
60
P/100kPa
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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40
20
0
0
20
40
60
80
100
xCO2
400 401
Figure 9. Prediction of the solubility of methanol in CO2 at 313K using the LCVM/departure
402
function/MGCE methods (CIRCE code), Experimental data by Suzuki et al.32 at
403
313K.simulations: vapor, ○; experiments: vapor, ●; simulations: liquid, □; experiments:
404
liquid, ■.
405 406
Conclusions
407
This work is a contribution to the simulation of the thermodynamic equilibrium of reactive
408
multiphase mixtures. Besides the specific difficulties in minimizing the very complex Gibbs
409
energy of the mixture, addressed in another article, the attention is focused in the present
410
work on the incidence of the inputted thermodynamical data on the final result. It is shown
411
that inaccuracies more significant than 1 kJ/mole on the Gibbs formation energy of the
412
component have a visible impact. Similarly, the evolution of the coefficient of activities as a
413
function of the composition of the mixture is important. Thus using data from different
414
sources and (too) rough models for activity coefficients may lead to uncontrolled deviations 22 ACS Paragon Plus Environment
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415
in the final results. To limit such occurrence, it is proposed to use the same molecular
416
descriptors to calculate the thermodynamical data and activity coefficients. The group
417
contribution theory is implemented not only to calculate the activity coefficients (UNIFAC)
418
but also to estimate the standard energies of formation of the species (Joback’s approach) and
419
to introduce the effect of pressure and temperature via the LCVM equation of state.
420
Some examples suggest that this approach can give very satisfactory results.
421 422
NOTATION A,C aij ai AV,AM ai,aj a,b bi bj C1,C2,C3 Cp0 fi G0 Gmix GE
Constant number of atoms of element j in compound i activity of compound i constant coefficients proposed respectively by Michelsen and Vidal interaction parameter the parameter of PENG Robinson volume parameter total number moles of atoms j in the mixture the parameter for the LCVM Heat capacity (J/K) fugacity of component i (Pa) standard Gibbs energy (J) Gibbs energy of mixture (J) Excess Gibbs energy (J)
ΔG0f,298 Hf ∆Hv,b ni nSp nEl NG,NL,NS P R Tr Tf Tb T Vm Vc, Tc, Pc
Gibbs formation energy (J) enthalpie of formation (J) enthalpie of evaporation (J) mole number of i number of species number of elements number of Gas, liquid and solid Pressure (Pa) gas constant (J mol−1K−1) relative temperature temperature of fusion (K) boiling point (K) Temperature (K) molar volume (m3) the critical parameter 23 ACS Paragon Plus Environment
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Z
Page 24 of 27
compressibility factor
423 424
Greek Symbols: 𝜖𝑖 μio(T) μi γi Φi δij 𝜑𝑖 𝜑𝑖 Λ Ω Κ
function of the conservation of mass chemical potential (J) chemical potential of component I (J) Activity coeffiient of component i fugacity coefficient i empirically determined binary interaction coefficient Fugacity coefficient of component i in the mixture Fugacity coefficient of pure component i a sort of relaxation parameter acentric factor constant characteristic of each substance
425 426 427
Supporting Information
428
Supporting Information Available: Appendix S1: analytical derivation of equation (7).
429
Appendix S2: Estimate the departure function.
430 431
Corresponding author:
432
Christophe Proust,
433
*E-mail: christophe.proust@ineris.fr
434
ORCID
435
Christophe. PROUST: 0000-0002-9097-4292
436 437
Funding Sources
438
None.
439
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References:
441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488
1. Liu, Q.; Proust, C.; Gomez, F.; Luart, D.; Len, C., Predicting multi-phase chemical equilibria using a Monte Carlo technique. Comput. Chem. Eng. submitted in 2018. 2. Liu, Q. CIRCE a new software to predict the steady state equilibrium of chemical reactions Universite de Technologie de Compiegne, 2018. 3. Yaws, C. L., Physical properties. In Handbook of Chemical Compound Data for Process Safety, Elsevier: 1997; pp 1-26. 4. Hino, T.; Song, Y.; Prausnitz, J. M., Liquid‐liquid equilibria and theta temperatures in homopolymer‐solvent solutions from a perturbed hard‐sphere‐chain equation of state. J. Polym. Sci., Part B: Polym. Phys. 2015, 34, (12), 1961-1976. 5. Gautam, R.; Seider, W. D., Computation of phase and chemical equilibrium: Part I. Local and constrained minima in Gibbs free energy. AlChE J. 1979, 25, (6), 991-999. 6. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M., Group‐contribution estimation of activity coefficients in nonideal liquid mixtures. AlChE J. 1975, 21, (6), 1086-1099. 7. Gmehling, J.; Rarey, J.; Menke, J., Dortmund Data Bank. Oldenburg (29/08/2013) http://www.ddbst.com 2008. 8. Constantinou, L.; Gani, R., New group contribution method for estimating properties of pure compounds. AlChE J. 1994, 40, (10), 1697-1710. 9. Joback, K. G.; Reid, R. C., Estimation of pure-component properties from groupcontributions. Chem. Eng. Commun. 1987, 57, (1-6), 233-243. 10. Klincewicz, K.; Reid, R., Estimation of critical properties with group contribution methods. AlChE J. 1984, 30, (1), 137-142. Ambrose, D., Correlation and Estimation of Vapour-liquid Critical Properties: I, 11. Critical Temperatures of Organic Compounds. National Physical Laboratory: 1978. 12. Parks, G. S.; Huffman, H. M., Free energies of some organic compounds. 1932. 13. Ambrose, D., Correlation and estimation of vapour-liquid critical properties. Part 1: Critical temperatures of organic compounds. 1978. Poling, B. E.; Prausnitz, J. M.; John Paul, O. C.; Reid, R. C., The properties of gases 14. and liquids. Mcgraw-hill New York: 2001; Vol. 5. 15. Stull, D. R.; Westrum, E. F.; Sinke, G. C., The chemical thermodynamics of organic compounds. 1969. 16. Lydersen, A. L.; Station, M. E. E., Estimation of critical properties of organic compounds by the method of group contributions. University of Wisconsin: 1955. 17. van der Waals, J. D., The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 1979, 20, (2), 200-244. 18. Redlich, O.; Kwong, J. N., On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chem. Rev. 1949, 44, (1), 233-244. 19. Soave, G., Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 1972, 27, (6), 1197-1203. 20. Peng, D.-Y.; Robinson, D. B., A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, (1), 59-64. 21. Boukouvalas, C.; Spiliotis, N.; Coutsikos, P.; Tzouvaras, N.; Tassios, D., Prediction of vapor-liquid equilibrium with the LCVM model: a linear combination of the Vidal and Michelsen mixing rules coupled with the original UNIF. Fluid Phase Equilib. 1994, 92, 75106. 22. Kontogeorgis, G. M.; Coutsikos, P., Thirty Years with EoS/GE Models-What Have We Learned? Ind. Eng. Chem. Res. 2012, 51, (11), 4119-4142. 23. Coquelet, C.; Chapoy, A.; Richon, D., Development of a new alpha function for the Peng–Robinson equation of state: comparative study of alpha function models for pure gases 25 ACS Paragon Plus Environment
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(natural gas components) and water-gas systems. Int. J. Thermophys. 2004, 25, (1), 133-158. 24. Huron, M.-J.; Vidal, J., New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilib. 1979, 3, (4), 255-271. 25. Michelsen, M. L., A method for incorporating excess Gibbs energy models in equations of state. Fluid Phase Equilib. 1990, 60, (1-2), 47-58. 26. Spiliotis, N.; Boukouvalas, C.; Tzouvaras, N.; Tassios, D., Application of the LCVM model to multicomponent systems: Extension of the UNIFAC interaction parameter table and prediction of the phase behavior of synthetic gas condensate and oil systems. Fluid Phase Equilib. 1994, 101, 187-210. 27. Voutsas, E. C.; Spiliotis, N.; Kalospiros, N. S.; Tassios, D., Prediction of vapor-liquid equilibria at low and high pressures using UNIFAC-based models. Ind. Eng. Chem. Res. 1995, 34, (2), 681-687. 28. Knapp, H., " Vapor-Liguid Equilibria for Mixtures of Low Boiling Substances" Part14. DECHEMA Chemistry Data Series 1982. 29. Griswold, J.; Wong, S. In Phase-equilibria of the acetone-methanol-water system from 100 degrees C into the critical region, Chemical Engineering Progress Symposium Series, 1952; 1952; pp 18-34. 30. Zirrahi, M.; Azinfar, B.; Hassanzadeh, H.; Abedi, J., Measuring and modeling the solubility and density for CO2–toluene and C2H6–toluene systems. J. Chem. Eng. Data 2015, 60, (6), 1592-1599. 31. Chen, S.; Chen, S.; Fei, X.; Zhang, Y.; Qin, L., Solubility and characterization of CO2 in 40 mass% N-ethylmonoethanolamine solutions: explorations for an efficient nonaqueous solution. Ind. Eng. Chem. Res. 2015, 54, (29), 7212-7218. 32. Suzuki, K.; Sue, H.; Itou, M.; Smith, R. L.; Inomata, H.; Arai, K.; Saito, S., Isothermal vapor-liquid equilibrium data for binary systems at high pressures: carbon dioxide-methanol, carbon dioxide-ethanol, carbon dioxide-1-propanol, methane-ethanol, methane-1-propanol, ethane-ethanol, and ethane-1-propanol systems. J. Chem. Eng. Data 1990, 35, (1), 63-66.
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Molecular descriptors from UNIFAC
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Industrial & Engineering Chemistry Research
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Real fluid properties (EoS parameters & mixing rules)
Activity coefficients
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The Software architecture.
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