Thermodynamic Modeling of Mixed Solvent Electrolyte Solutions: Challenges and Practical Guide
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Thermodynamic Modeling of Mixed Solvent Electrolyte Solutions: Challenges and Practical Guide
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Journal of Chemical & Engineering Data

Cite this: J. Chem. Eng. Data 2025, 70, 11, 4331–4350
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https://doi.org/10.1021/acs.jced.5c00412
Published October 19, 2025

Copyright © 2025 The Authors. Published by American Chemical Society. This publication is licensed under these Terms of Use.

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Abstract

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Electrolyte systems are characterized by their (i) strong nonideal behavior requiring advanced thermodynamic models, (ii) reactive components necessitating knowledge of formation properties, and (iii) multiphase nature, which demands robust reactive algorithms. These complexities are crucial in industrial applications, leading to the creation of the Joint Industry Project (JIP) EleTher, which aims to promote further academic exploration and provide recommendations for parametrizing such systems. To better understand interactions in electrolyte systems, the study focuses on quaternary systems comprising water, an acid, a base, and a cosolvent. This approach isolates pH effects from the dielectric constant influences. Due to limited data, only ternary subsystems were analyzed. The study evaluated the accuracy of three approaches: full dissociation (FD), no dissociation (ND), and partial dissociation (PD). While FD and ND approaches performed adequately for vapor–liquid equilibrium, they failed when speciation became significant. The PD approach, though more accurate, revealed challenges with parameter optimization as multiple local minima resulted in varying species distributions. A hybrid approach combining FD, ND, and PD models is recommended for achieving the most physically meaningful results. The study utilized the eNRTL method CPA equation of state within in-house tools (Carnot and ATOUT) to conduct the analysis.

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Copyright © 2025 The Authors. Published by American Chemical Society

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Published as part of Journal of Chemical & Engineering Data special issue “In Honor of Frederico W. Tavares”.

Introduction

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A wide range of applications in the process industry involves fluids with electrolytic components. One may mention, without being exhaustive, distillation, (1) corrosion, (2) hydrometallurgy, (3) carbon dioxide capture processes involving reactive solvents, (4) geothermal (5) or geochemical engineering, (6) biomass treatment, (7) batteries, (8) etc. Several industrial surveys, (9,10) as well as a recent opinion paper (11) proposed in the frame of the European Working Party on Thermodynamics and Transport Properties (https://efce.info/WP_TTP.html), systematically emphasize the importance of investigating the thermodynamics of electrolytic systems in more depth.
Although the thermodynamic behavior of most molecular species is now accurately reproduced, that of ions within electrolytic systems still represents significant challenges and unanswered questions. (10−12) Several key challenges arise in the modeling of industrial electrolytic systems:

  • no agreement in the scientific community about the best model to use for combined long-range and short-range interactions. The industrial models used to date often have a very large number of parameters. (13,14)

  • The presence of chemical equilibrium implies the need to work with reactive algorithms searching for a global minimum (phase and chemical). (15)

  • Depending on the system as well as pressure and temperature ranges, different types of phase equilibria can occur (a vapor, one or several liquid phases, or one or several solid phases), which obviously modify the way properties of interest are addressed. As a result, a robust reactive flash algorithm is required for phase stability analysis to take all possible changes in phase equilibria into consideration. (16)

  • Given the complexity of the industrial systems, the reference experimental data are often insufficient for accurate process design, meaning that data extrapolation are predictive methods may be needed.

Furthermore, the Industrial Use of Thermodynamics (IUT) symposium, which focused on electrolyte thermodynamics, (12) emphasized that the importance of the various phenomena may vary depending on the properties relevant for a specific industrial application. Hence, although a given set of simplifications is acceptable in one case, it may no longer be the case for other applications. Therefore, extrapolations only based on global reasoning can lead to spurious estimations and extremely dangerous situations.
A Joint Industrial Project (JIP) called EleTher (for Electrolyte Thermodynamics) was created, bringing together industrial companies, their know-how, and their tools, within a collaborative community. The basic idea of the JIP EleTher is to address the need of assessing best practices in complex electrolyte system parametrization as well as communicating challenges of thermodynamics for electrolyte solutions. More precisely, the objective is not to solve the many complex issues that exist but rather to raise attention to the topic and promote collaborative efforts. The first edition of the JIP EleTher (17) was constructed on a three-step workflow, as follows: (i) data analysis; (ii) data extrapolation; and (iii) model parametrization. The initial paper described the first step in detail, investigating binary systems consisting of alkali halide salts in water. (17) Since then, the work has been extended to mixed solvent data. (18) The need to extrapolate from available data and their conditions to those generally encountered in industrial applications requires the identification of precise physical models, or even, if necessary, their reparameterization. (19) This latter point led to the second edition of the JIP EleTher which mainly focuses on parametrization issues. As the industrial users require robust models integrated into software, the industrial community has been extended to incorporate both software providers and end-users of thermodynamic models.
To illustrate the industrial complexity while remaining tractable in terms of system complexity, it was decided to use case studies with quaternary mixtures. Water is necessarily present, and a salt that is considered as a combination of an acid and a base (this way, the impact of the pH is made visible). A cosolvent is a fourth component. The presence of this last compound makes it possible to investigate the impact of the change in the dielectric constant on the system properties. As a consequence of the presence of the cosolvent, no salt can be considered as “strong”: the chemical equilibrium that rules the formation of ion pairs will come into the picture. The quaternary system then becomes much more complex as it potentially forms a nine-component system, which implies a complex parametrization between all pairs (Figure 1). However, in practice, simplifying assumptions is often considered. In this work, we tried to investigate the phase behavior of electrolyte solutions based on dissociation assumptions regarding the speciation as follows:

  • no dissociation (ND) calculations where all species in the system are in molecular form.

  • FD (Full dissociation) calculations considering only the ionic form of salt species.

  • PD (Partial dissociation) calculations that consider reaction equilibrium between molecules and ions.

Figure 1

Figure 1. Schematic representation of the selected quaternary system, which turns out to be a nine-component system due to the reactive equilibria. HA and BOH stand for acid and base, respectively.

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The targeted industrial applications serve as the foundation for this study. These include, for example, wastewater treatment, the production of specialty chemicals, and the biomass-to-alcohols process, where fermentation generates carboxylic acids as byproducts, necessitating pH control through the use of strong bases. To explore these applications further, we will analyze a quaternary system comprising a weak acid, acetic acid (HAc), a strong base, potassium hydroxide (KOH), and a cosolvent, methanol. The practical choice of this case study is based on the availability of experimental data in the open literature. Our primary focus will be on vapor–liquid equilibrium calculations while also considering other properties such as ionic activity coefficients and speciation. This system, commonly encountered in industrial contexts, provides valuable insights into improving processes that depend on electrolyte solutions.
In practice, a salt may precipitate (potassium acetate, KAc) from an acid–base reaction. Many questions have been raised during the thermodynamic modeling: What is the optimal parametrization strategy (e.g., which parameters are fitted on which data types)? How to deal with parameter degeneracy? What is the sensitivity of the model’s outputs with respect to their parameters? These questions will be addressed in this paper in a systematic analysis. Since there are no available experimental data in the open literature for the quaternary system, only the binary and ternary systems will be investigated. The conclusions drawn in this study are the fruit of collaborations and discussions with the industrial partners of the JIP.
In what follows, the experimental data available in the open literature for the systems of interest are summarized, including both chemical reaction and phase equilibrium data. Next, the models used for the thermodynamic modeling are discussed. The section dedicated to results and discussions will provide a focus on the parametrization, presenting difficulties and the solutions proposed to overcome them. The article ends with conclusions and perspectives.

Experimental Data

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Considering a ternary system containing water, a cosolvent, and a salt (and excluding the case of liquid–liquid equilibrium), one may distinguish three types of activity coefficients as summarized in Figure 2. The first, called γs is the usual definition of a solvent activity coefficient: it describes the difference between the chemical potentials of a solvent molecule (water or cosolvent) in the mixture and in their respective reference state (which is the pure liquid component). The impact of the presence of salt on this activity coefficient will yield information regarding the salting-in or salting-out behavior of the solvent. The second is the Gibbs energy of transfer (γit), (20,21) expressing the difference between ionic chemical potential (μis) in the mixed solvent reference state and (μiw) in the water reference state. As we will point out, this difference will provide information regarding the modification of the speciation behavior (equilibrium constant) when comparing an aqueous and a nonaqueous solvent. The third is what is conventionally called the ionic activity coefficient (γi), related to the difference between the ion chemical potential in the mixture and the infinitely diluted ion in the mixed solvent reference state. This property is often obtained from potentiometry measurements (22) and is important for describing solid–liquid equilibria. Note that the reference states for ions may be taken based on the molality or mole fraction, which will modify the numerical value of the activity coefficients. The transformation from one to the other is provided in textbooks. (23)

Figure 2

Figure 2. Three types of activity coefficients for a ternary system containing water, a cosolvent, and a salt: (i) blue, solvent activity coefficients; (ii) red, transfer activity coefficient of the ions between pure water and mixed solvent; (iii) green, conventional ionic activity coefficient. The orientation of the arrows shows the type of derivative of the Gibbs energy that they correspond to. SLE stands for solid–liquid equilibrium. The definitions of the variables can be found in the text.

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Physical Phase Equilibrium

Existing Data

Regarding our study case, the ternary system containing water (W; CAS 7732-18-5) was chosen considering the data available in the literature. We have selected methanol as a cosolvent (MeOH, CAS 67-56-1), acetic acid (HAc, CAS 64-19-7) as a representative of an acid “HA” and potassium hydroxide (KOH, CAS 1310-58-3) as a base “BOH”. As aforementioned, a salt “AB” may form in the reactive system (Figure 1). In our case, this corresponds to potassium acetate (KAc, CAS 127-08-2), which should therefore also be considered. An overview of the available experimental data for binary and ternary systems is proposed in Table 1.
Table 1. Overview of the Experimental Data of Binary and Ternary Systems of Interestc
systemtypeareferenceT/KP/barsalt molality/molalNPb
binaryW + MeOHVLE-isothermxyP (24−26)298.15–413.15 N/A24, 11, 22
 W + HAcVLE-isothermxyP (27−29),b298.15–333.15 N/A13, 28, 9
 MeOH + HAcVLE - isothermxyP (30–32)298.15–361.490.02–1.79N/A70
 W + KAcMIAC  (33,34)298.150.1–3.511, 2435
  VLE-isobarxT (35)373.15–384.1510.358–823
  VLE-isothermxP (36)358.15–378.150.462–1.170.756–5.2626
   xP (37)373.15–373.150.6–1.013250–1010
   xP (38)278.15–308.150.0022–0.0541.133–24.1226127
 W + KOHMIAC  (34,39–42)298.150.001–2043, 32, 11, 13, 8159
  VLE isobarxT (43)373.15–613.151.013–1.0130–33.44345
  VLE isothermxP (44)573.15–693.154.903–234.41.717–60.13441
   xP (37)373.15–373.150.496–1.013250–1010
   xP (45)273.15–273.150.0008–0.0060.995–31.2348
   xP (46)293.15–298.150.002–0.031.535–23.80942
   xP (47)298.15–623.150.0078–157.60.8–14.26176
 HAc + KAcVLE isobarxT (35)390.55–400.7510.05–1.3533
 MeOH + KAcVLE isobarxT (48)338.95–346.951.0133–1.01331.039–6.234
  VLE isothermxP (49)298.15–313.150.1482–0.35740–2.51124
TernaryW + MeOH + HAcVLE isobarxT (50)341.08–381.641.01325N/A34
  VLE isobarxyT (51)350.55–377.751.01325N/A28
 W + MeOH + KAcVLE isobarxyT (48)338.95–381.951.013251.04–6.2376
 W + MeOH + KOHVLE isothermxyP (52)298–3080.923–0.9831.47–718
 W + HAc + KAcVLE isobarxyT (35)373.55–398.7510.18–6.7796
   xyT (53)373.15–393.151.013251–3.0615
a

For isobars, xyT means that both phase compositions are available; xT means that only bubble points are given. For isotherms, xyP means that both phase compositions are available; xP means that only bubble points are given.

b

NP: number of points available in each respective reference.

c

Vapor-Liquid equilibrium (VLE) data and Mean Ionic Activity Coefficients (MIAC) are used.

Experimental Activity Coefficients Illustrating the Salting In/Out Behavior

Salting effect can be investigated without any model by simply transforming the PTxy data and calculating the “experimental” activity coefficient of the solvents using
γi=yiPxiPiσ
(1)
where xi and yi are the molar fractions of solvent i in the liquid and vapor phases, respectively; P is the pressure, and Piσ is the saturation pressure of solvent i. In this expression, the Poynting correction and the nonideality in the vapor phase are neglected. When the activity coefficient decreases with salt content, the compound is increasingly soluble with salinity, or salted in; when the activity coefficient increases, then the component is salted out.
Figure 3 compares activity coefficients of methanol and water calculated by eq 1 using the experimental data of water/methanol/potassium acetate. It can be clearly seen that the introduction of potassium acetate into the methanol–water system significantly modifies the behavior of the components by influencing their activity coefficients through salting-out and salting-in mechanisms. Methanol experiences a salting-out effect, where its activity coefficient rises, leading to a decrease in its solubility within the aqueous phase. This happens because the salt ions predominantly interact with water molecules, reducing their capacity to hydrate methanol and consequently pushing methanol out of the solution. In contrast, water undergoes a salting-in effect, which lowers its activity coefficient. This effect stems from the strong bonds formed between salt ions and water molecules, which stabilize the water structure and increase the effective concentration in the mixture.

Figure 3

Figure 3. Solvent activity coefficient (computed by eq 1) in the water/methanol/KAc ternary system: (a) methanol and (b) water. Data were extracted from ref (48).

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Chemical Equilibrium

To consider partial dissociation in reactive equilibrium computations, the equilibrium constant is required. Hence, this section is planned with the intent to fill in, classify, and offer an overview of experimentally derived data on equilibrium constants of electrolytes relevant to our work in the literature. The constants are often measured using potentiometric or spectroscopic methods. (54) The potentiometric methods are essentially titration methods, where the pH of the solution is tracked upon addition of a strong acid (for the bases). Spectroscopically, actual speciation is tracked. This latter approach brings a lot of questions regarding the nanostructure of the species considered: in reality, many different hydrated forms may coexist, which complicates the analysis substantially. It should be noted that the equilibrium constants reviewed in this work are reported on a molality basis with respect to pure water reference state.

Hydroxide Salts

In this section, experimental data on the dissociation constant of three hydroxide salts in water; LiOH(aq), NaOH(aq), and KOH(aq) are reviewed and presented in Figure 4 (a global picture of all hydroxides is also provided). In addition, for each salt, a predictive approach proposed by Akinfiev (55) is tested. The predictive approach is based on the AD EoS (Akinfiev-Diamond Equation of State) (56) coupled with quantum chemical estimations. A large value of K means that the ionic form is overriding, while a small value indicates that the molecular (ion pair) form is more important. As aforementioned, the standard state for aqueous species in these equilibrium constant data is defined as a hypothetical 1 mol solution, with unit activity referenced to infinite dilution.

Figure 4

Figure 4. Molality-based dissociation constants of (a) lithium, (b) sodium, and (c) potassium hydroxides as a function of the inverse of the temperature. (55−65) The (d) diagram provided a comparison of (a–c) using Akinfiev-calculated data.

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Acetate Salts

In this work, we reviewed dissociation constant data (on a molality basis and with respect to an aqueous-phase reference state) for a series of acetate salts, i.e., LiAc(aq), NaAc(aq), and KAc(aq). Figure 5 illustrates the behavior of dissociation constants of acetate salts as a function of the inverse of temperature. In addition, for each acetate salt, predictions are proposed by using the Shock and Koretsky approach based on the revised-HKF equations of states. (65)

Figure 5

Figure 5. Dissociation constants of acetate salts containing (a) lithium, (b) sodium, and (c) potassium as a function of the inverse of temperature. (54,64−70) The diagram labeled (d) provides a comparison of (a–c) data.

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Computational Method

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Model and Adjusted Parameters

Physical Models

In this study, the in-house IFPEN thermodynamics library Carnot (71) is employed for thermodynamic calculations based on a heterogeneous γ – φ approach. The activity coefficients of mixture components in the liquid and aqueous phases are modeled by the Electrolyte Non-Random Two-Liquid (eNRTL) model, (72,73) whereas the Cubic Plus Association (CPA) equation of state (74) is used to describe the nonideality of the vapor phase instead of the Haydon O’Connell model, (75) as recommended in the Aspen Plus software. Details about these models are provided in Supporting Information 1. The eNRTL models in the Carnot Software have been validated by comparing the results with the commercial tools AspenPlus (76) and Simulis Thermodynamics. The use of CPA was motivated by the absence of the HOC model in Carnot. Yet, we compared the results of both models for the fugacity coefficient of acetic acid in the vapor phase and found them to be in good agreement. Acetic acid was chosen for this comparison due to its known tendency to dimerize in the vapor phase, which presents a more challenging test case for evaluating model performance. As shown in Figure 6, the results follow a similar trend, indicating the consistency of both models in representing the vapor-phase behavior of acetic acid.

Figure 6

Figure 6. Comparison of fugacity coefficient of acetic acid in the vapor phase as predicted by HOC and CPA equations of state.

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The eNRTL model extends the traditional NRTL model by incorporating terms that account for both long-range electrostatic interactions between ions (originating from the Pitzer-Debye–Hückel (PDH) equation) and short-range molecular interactions between species (Born term). The eNRTL parameters are defined for pairs, for each binary, as follows: i–j, αij = αji; τij(T) and τji(T). Therein, αij is temperature-independent, while τji(T) is expressed according to eq 2.
τji(T)=τji0+τjiT(1T−1T0)
(2)
where T0 is the reference temperature, which is set to 298.15 K, and τ0 and τT are binary interaction parameters. This makes a total of five parameters per binary. Unless otherwise noted, only the parameters at the reference temperature were regressed, keeping all τT = 0.
This short paragraph is intended to both remind and provide generic abbreviations that are used here below: W stands for water, A stands for acid, M or MeOH for methanol, S for salt (i.e., neutral molecule or ion pair), and I for ions. For notation of parameters, the same symbols but in lower case are used (e.g., αwi then denotes the α parameter between water and salt ions). In other words, τws and τsw refer to the two water and molecular salt parameters, while τwi and τiw, refer to the water + ions parameters.

Chemical Models

For our quaternary system of interest, the following chemical reactions (eqs 3–6) can occur
2H2O⇔K1H3O++OH−
(3)
HAc+H2O⇔K2H3O++Ac−
(4)
KAc⇔K3K++Ac−
(5)
KOH⇔K4K++OH−
(6)
Finally, the quaternary system (water, cosolvent, acid, and strong base) results in a 9-component system when all chemical reactions and potential species are considered. More precisely, the nine possible components that may occur are H2O, MeOH, HAc, KOH, KAc, K+, Ac–, OH–, and H3O+. In practice, because the selected base is strong and the acid is weak, we will see that H3O+ is required for only pH-type calculations.
Considering the limited temperature range (up to 400 K) that is investigated, the equilibrium constants used in this work (on a molality basis and with respect to an aqueous-phase reference state) are given by the following linear relationship
ln(K)=(a+bT)RT
(7)
Values of the parameters a and b in eq 7 for water and the different considered electrolytes are summarized in Table 2, and the variation with the inverse of temperature, compared to experimental data, is plotted in Figure 7. Clearly, it is observed that water and acetic acid are very stable in their molecular form (small dissociation constant), as compared to KOH and KAc (in that order), which dissociate easily into ions, at least at low temperatures. The water dissociation constant has been validated based on the revised release on the ionization constant of H2O (77)
Table 2. Coefficients of eq 7 for Studied Compounds (Molality Base)
electrolytereaction constantabtemperature range
H2OK1–26061.1–46.9191298–473 K
HAcK228183.4–51.0388298–373 K
KAcK38052.0–22.6331298–363 K
KOHK423684.6–57.7287373–573 K

Figure 7

Figure 7. Dissociation constant as a function of temperature using eq 7, together with experimental data from ref (78) for H2O, from ref (79) for HAc, from refs (66,and67) for KAc, and from ref (57) for KOH.

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Properties

Phase Equilibrium Calculation

The equilibrium is based on the partition coefficient Ki, which is computed using the heterogeneous γ – φ approach
Ki=φiσγiLPiPiσPφiV
(8)
where P is the pressure of the system, Pi the Poynting factor, Piσ the saturation pressure, φiσ(Piσ) and φiV(P) the vapor phase fugacity coefficient calculated at pressure Piσ and P, respectively, and γiL stands for the activity coefficient of species in the liquid phase. As mentioned, the CPA equation of state is used to compute both fugacity coefficients: φiV(P) and φiσ(Piσ). It is important to note that these two terms are not computed at the same pressure, nor composition, but it is mandatory that the model used is the same so that in the limit of the pure component (where pressure equals the pure component vapor pressure) φiV(P) = φiσ(Piσ). We shall call φiV(P)φiσ(Piσ)=γiV the vapor phase activity coefficient, as shown in Figure 6.

Mean Ionic Activity Coefficient (MIAC)

The ionic activity coefficients provided by the eNRTL model cannot be directly used to obtain the mean ionic activity coefficient (MIAC) that is compared to experimental data; additional treatment is needed because the model provides the mole fraction-based (or rational) activity coefficient. The relation between the rational (noted with exponent x) to molality-based (noted with exponent m) asymetric activity coefficients is as follows
γimxs=γi*,x
(9)
where xs is the solvent mole fraction. Moreover, when the computation is performed using partial dissociation (PD), the MIAC is computed from ref (80)
MIAC=αγAmγCm
(10)
where α is the fraction of free ions (that equals one for Full Dissociation, FD), and A and C stand for anion and cation, respectively. The fraction of free ions can be obtained from the speciation.

pH

The pH is defined in terms of the activity on a molality base of H+ ions in solution. (81) However, in practice, it is not the proton but the hydronium ion that is present in the solution, and the equilibrium between H3O+ and H+ (eq 11) has to be considered.
H++H2O⇔H3O+
(11)
Kosinski et al. (82) suggested writing the equilibrium constant of the reaction (eq 11) as
aH3O+aH+aH2O=exp(μH3O+ref−μH+ref−μH2OrefRT)
where a is activity, μ is chemical potential, R is the universal gas constant, and T is temperature. The authors proposed to set the equilibrium constant to unity, which is equivalent to stating that
μH3O+ref=μH+ref+μH2Oref
where per convention, μH+ref=0, resulting in
aH3O+=aH2OaH+
The pH is thus defined as follows
pH=−log10(aH+)=−log10(aH3O+)+log10(aH2O)
It is noteworthy that the water activity is close to one, but in mixed solvents, this may no longer be true. The pH is then calculated as follows
pH=−log10(mH3O+γH3O+*,mm0)+log10(xH2OγH2Ox)=−log10(xH3O+γH3O+*,xMwm0)+log10(xH2OγH2Ox)
(12)
With xH3O+ and γH3O+*,x the mole fraction and the asymmetric activity coefficient of hydronium ion, Mw the molecular weight of water, and the reference molality m0 = 1 mol/kg. Note that this pH definition is a water-based reference state. When the experimental data are reported on a solvent-specific basis, the pH should be recalculated to the corresponding reference state. (82) In this work, the formalism defined on a water-based scale is used to evaluate the pH.

Parameterization of Binary and Ternary Systems

The molecular interaction parameters are taken directly from the literature if available and presented along with the regressed parameters in Supporting Information 1. Several issues are raised in this essential step of the work: considering a given set of assumptions (and therefore of parameters to be regressed), one must first define which parameters have to be fitted and the data available for this task; second, one must be able to deal with parameter degeneracy and computation failure (i.e., numerical issues). These latter points are discussed in the two following subsections.

Data Selection for Parameter Regression

While the use of binary systems for fitting binary parameters is generally recommended, we noticed that this approach is not applicable in the case of reactive mixtures. For example, parameters between water and ion pairs (also called molecular salt) will not have a significant impact on the water-salt binary systems. Indeed, salts dissociate in water, and therefore, their molecular form (ion pair) is almost absent in the solution. Similarly, ionic species are almost absent from the nonaqueous solvent, meaning that the ion–solvent parameters have no impact on the simulation of these binaries. This is why on several occasions we will recommend fitting binary data on ternaries, using the PD approach. The drawback stands in the strong parameter degeneracy. In all cases, we try defining parameters independently of the computational approach (PD, ND, or FD).
In this work, we have focused only on less well-identified parameters of the eNRTL model: although the dissociation constants (Table 2) carry some uncertainty and have an impact on the result, they have not been modified. Among the eNRTL parameters, those that concern only molecular species were directly taken from the Aspen Plus software and are available and reported in Supporting Information 1. Additionally, Chen et al. (72) proposed nonzero default parameters for water–ion and solvent–ion systems (τwi = 8; τiw = −4 and τmi = 10, τim = −2), which will be considered further.
The reference data used during the parameter regression may be MIAC, VLE (and osmotic coefficient if available), or both, as indicated hereafter. The other available data are used for the validation of the approach. In addition, for the PD approach, we will investigate the speciation trends that result from the developed models to make sure that they correspond to the expected behavior.

Numerical Issues

Parameter regression is performed by minimizing an objective function, defined as the sum of the square of the relative deviations between experimental and calculated values, as follows
OF=∑j=1nds∑i=1ndp,vj(Δi)2ndpj
(13)
where nds is the number of data sets, ndpj is the number of data points in the data set j, and relative deviations (Δi) are computed according to the following scheme
{Δi=Xcali,j−Xexpi,j⁣ifXexpi,j=0Δi=Xcali,j−Xexpi,jXexpi,j⁣otherwise
with X referring as to any property j. The subscript cal denotes calculated values, and the subscript exp denotes experimental values.
The IFPEN regression tool ATOUT, (83) was used for parameter regression. Two challenges have been observed: (1) the presence of deep valleys pointing to parameter degeneracy, and (2) parameter sets that are evaluated during the regression sometimes yield nonconvergence issues. The first can be observed in an example plot provided in Figure 8. In the example shown (methanol/K+Ac– binary system), one notices that the recommended parameters [τmi, τim] = [10, −2] (blue dot) yield an unmanageable objective function. While other parameter sets [τmi, τim] = [8, −4], that is recommended for the water/ion pair, and [τmi, τim] = [0, 0] are within the valley. The optimization solver is somehow “blind” outside the valley and often terminates prematurely before obtaining a global minimum. In this example, the parameter intervals (minimum and maximum values) have been deliberately chosen to clearly outline the response surface of the objective function.

Figure 8

Figure 8. Example of response surfaces for the methanol─KAc parameter regression. The points indicate the location of different default parameter sets: [τmi, τim] = [10, −2] (blue dot), [τmi, τim] = [8, −4] (brown dot), and [τmi, τim] = [0, 0] (green dot).

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To deal with nonconvergence issues during the regression, the objective function was modified as follows
OF′=OF+∑j=1nds(ndpj−ndp,vj)∑j=1ndsndpj
(14)
where ndp,vj with the subscript v is the number of valid (or converged) calculations in data set j. The second term of the objective function is the rate of nonconvergence calculations, which allows one to penalize poor parameter sets leading to a lot of nonconvergence issues due to numerical problems but provides low deviations from experimental data in a few converged simulations.
During the parametrization procedure, it was found to be useful to follow a two-stage regression approach:

  • 1st stage: a “coarse” regression using an evolutionary algorithm solver allowing to generate potential solution candidates.

  • 2nd stage: a “fine” regression using a solver with a quadratic approximated model to obtain the best and final parameters.

The regression process in this work employs two optimization algorithms: the CMA evolutionary solver (84,85) and the local DFO (Derivative-Free Optimization) method SQA (Sequential Quadratic Approximation). (86) Both algorithms are available in ATOUT software.

Results and Discussion

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The four ternary systems mentioned in Table 1 are discussed hereafter, always keeping in mind that there are three dissociation-related modeling scenarios, called PD, ND, and FD. At first, the aqueous binary systems containing ionic species will be analyzed, then we focus on ternary systems containing methanol, subsequently systems involving acetic acid, and finally the system that is expected to contain only very small amounts of ions, yet for which properties other than phase equilibrium may be important.

Binaries with Water

Water/KAc

The parameters extracted from Aspen Plus software were used to describe MIAC and VLE data at 298 K, using the three approaches (PD, FD, and ND). The predicted values are presented in Figure 9 where they are compared to experimental values when available. Figure 9a demonstrates that on the considered range of molality values─up to about 4 M (the highest concentration of the available data)─the MIAC data are well predicted using the default parameters, with both PD and FD. Figure 9b compares experimental and predicted values resulting from the vapor pressure calculations. The bad performance of the ND approach can be related to the fact that the pressure is almost proportional to the mole fraction of water. This mole fraction is clearly not the same whether only ion pairs are considered or both ions are counted separately. FD and PD show almost identical results with a slight overestimation of experimental reference values. It is noteworthy that discrepancies arise when reaching a salt molality of about 5 M. This shows that beyond that concentration, nonidealities become important and must be properly accounted for. Figure 9c,d shows the predicted speciation of the mixture W/KAc, using the PD approach. The diagrams in Figure 9 indicate that the default parameters are reliable up to 5 M. Beyond this value, the predictive approaches show that the deviation increases with increasing salt molality values. If these parameters are used together with the PD approach, it is then possible to obtain speciation.

Figure 9

Figure 9. Mean activity coefficient (a) and vapor pressure (b) of Water/KAc mixtures at 25 °C with the default Aspen Plus parameters using Carnot. Calculation of MIAC with ND option is impossible because it requires having ions in the system. Corresponding speciations for MIAC (c) and vapor pressure (d) calculations using the PD approach. MIAC Data are extracted from ref (34) and vapor pressure data from ref (38).

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In order to improve the prediction accuracy of the vapor pressure for molality beyond 5 M, we have further optimized the two binary pair parameters between water and K+/Ac– (keeping the water─KAc ion pair parameters extracted from the Aspen Plus software). The regression was performed on two properties: MIAC (γ±) at 298.15 K and VLE data (p) at different temperatures ranging from 278.15 to 378.15 K. Full details about the two regressed parameter sets obtained can be found in Supporting Information 1. Note that a sensitivity analysis was performed on water-KOH (see response surface in Figure S1 of Supporting Information), which showed, not surprisingly, that the water–ion parameter is more influential on the selected data than the water─molecular KAc parameters. Note also that no temperature-dependent parameters were used. Figure 10c,d shows the deviations for these two properties, using the three calculation approaches (except for the MIAC property that cannot be calculated by ND in the absence of ions). It appears that ND performs poorly on the description of VLE (other W/KAc parameters probably need to be fitted), but that PD and FD lead to similar results, with a slightly better performance for PD on VLE. In contrast, FD performs slightly better than PD on MIAC. This might be due to the fact that the default molecular parameters used were not optimal, since in the PD approach, a small amount of salt remains in the molecular form.

Figure 10

Figure 10. Mean activity coefficient (a) and vapor pressure (b) of Water/KAc mixtures at 25 °C and deviations on MIAC (c) and vapor pressure (d). Experimental data (34,36,38) and modeling results with regressed parameters.

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For the water/KAc binary system, the MIAC data do not cover an extensive composition range (only up to 4 M). VLE is easy to represent (except with ND, which is clearly the wrong choice). The default parameter set is, however, not sufficient, even for VLE, beyond 5 M. Acceptable regressed parameter sets for both FD and PD can be found in Supporting Information 1.

Water/KOH

Potassium hydroxide is a strong base, meaning that it could be reasonably assumed to be totally dissociated into K+ and OH– species in aqueous solution (in the limit of its solubility). This is also the usual procedure (FD) recommended in the AspenTech and Fives-ProSim documentation, which leads to satisfactory results for VLE calculations. When considering the MIAC data as well, data are available up to high molality values, and the ionic activity coefficient reaches high values (up to 50), as shown in Figure 11. This figure also provides the calculation results using default parameters from Aspen Plus software for τwi and τws = [0, 0], considering both FD and PD. Using full dissociation and default parameters of Aspen Plus (labeled as Default FD) yields satisfactory results up to 10 M. However, large deviations can be observed at higher concentrations, as shown in Figure 11a.

Figure 11

Figure 11. KOH mean activity coefficient (a) and vapor pressure (b) with different methods and parameters (experimental data are from ref (34)). Reg stands for regressed. Speciation of Psat PD calculations as a function of salt molality using default (c) and regressed (d) parameters.

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Regarding the PD approach, one may expect that the mole fraction of the KOH ion pair remains much smaller than that of ions K+/OH–. However, it is found that it depends strongly on the parameters. Water–ions parameters from Aspen Plus and [τws, τsw] = [0, 0] are first used to evaluate the PD approach, which led to the results of the MIAC and VLE calculations presented in Figure 11a,b. It is noticeable that the simulated curves deviate greatly from the experimental data. Additionally, salt dissociation decreases significantly, as illustrated by a large amount of KOH remaining in its molecular form in Figure 11c. It requires, therefore, model parametrization for the PD approach to enhance predictive accuracy.
In the first attempt, both w-KOH and w-K+/OH– interaction parameters were regressed at the same time. It was found that the α parameter between water and ions should also be taken into consideration during regression in order to obtain satisfying results: the default value (αij = 0.2) did not seem to be optimal. Only binary MIAC (γ ±) is used for this regression, and the VLE data are then used for validation. The number of parameters (5 in total) over one data series seems to be high.
The obtained values for the regressed parameters are given in Supporting Information 1, and the numerical results are illustrated in Figure 11. The adjusted parameters provide very good results for both PD and FD calculations (PD and FD curves almost overlap; therefore, only the PD curve is shown). The similar results obtained with the PD and FD approaches mean that the PD approach becomes equivalent to an FD description. Only a slight deviation between experimental and numerical results is noticed over a wide range of salt molality (up to 20 M).
The vapor pressure data were used for validation. Due to Gibbs–Duhem consistency, it is expected for such a binary mixture that with good ionic activity coefficients the solvent activity would also be well represented. The modeling results (Figure 11b) show that the model predicts correctly the vapor pressure data even at high molalities. The speciation plot with the regressed parameters (Figure 11d) shows that the dissociation remains strong even at very large KOH concentrations.
The experimental data as well as simulation results of the saturation pressure at different temperatures are illustrated in Figure 12. Although the parameters have been fitted only on ambient temperature data, it provides predictions in good agreement with reference data, even at very high temperatures. Without implementing temperature-dependent coefficients, this was also reported by Valverde et al. (87) Osmotic coefficient data also exist and have been compared with the model. They fit nicely at 298.15 K, which is expected considering the interrelationship between these and the MIAC data. However, at higher temperatures, the eNRTL with temperature-independent parameters can no longer follow the trend. This was not further investigated in this work because the impact of temperature was outside its scope.

Figure 12

Figure 12. Vapor pressures of the water + KOH system with regressed parameters. The data are from refs (46,and47).

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One may conclude for the water/KOH binary system that binary interaction parameters are needed for concentrations higher than 5 M, but that the MIAC data that cover a very large concentration range and reach values up to 50 are impossible to describe with the default parameters. Both FD and PD parameters can be obtained, but in all cases, the difficulty is such that the nonrandomness parameter, α, needs to be tuned along with τij0 and τji0. To reduce the number of adjusted parameters and address concerns of parameter degeneracy, we believe that fixing [tws, tsw] at [0, 0] and then regressing the water–ion interactions may also be a relevant approach.

Systems with Methanol Cosolvent

Water/Methanol/KAc

Four types of parameters will be discussed in this section, each time with three computational approaches. For the first type, it consists of analyzing the results obtained with the Aspen Plus parameters (MSI_0) along with water/K+Ac– parameters from Aspen Plus. It should be noted that using the MSI_0 parameters with water/K+Ac– regressed in this work (labeled MSI_0b) leads to nonconvergence issues in PD calculations in the presence of ion pairs. Therefore, the methanol/K+Ac– interaction should also be adjusted afterward. The second type of parameter refers to the regressed parameters (MSI_1), which are obtained by regressing binary parameters on the binary data (and then applying them for predicting ternary properties). The two last parameter types (MSI_2 and MSI_3) represent two alternatives that include ternary data during the parametrization. The difference between the MSI_2 and MSI_3 parameter sets will be discussed later. It is noteworthy that the water/K+Ac– parameters previously regressed are used for these three parameter sets.
The results obtained using MSI_0 are shown in Figure 13 where they are compared to experimental values at 298 and 313 K. Clearly, the Aspen Plus parameters are designed to consider KAc as a molecular species, which is reasonable in this context. The FD calculations, performed for comparison purposes, show an opposite trend compared to the experimental data. On the contrary, the ND and PD lead to results in agreement with experimental data. This can be explained by the strongly positive activity coefficient of methanol in the presence of ions. Indeed, methanol, as any organic compound, is known for showing a salting-out behavior in the presence of ions. Yet, there are in fact no ions, so that in practice the methanol activity coefficient is much closer to one.

Figure 13

Figure 13. VLE calculation versus experimental data from refs (49and50) (a) Comparison of the methanol-potassium acetate vapor pressure (isotherm data) using the fully dissociated (FD) approach, partially dissociated (PD), or nondissociated (ND) assumptions with Aspen Plus NRTL parameters for both water/ion pair and methanol/ion pair interactions using MSI_0 set; (b) liquid speciation with PD calculation using MSI_0 set; (c) vapor pressure deviation on isobar data (1 bar) using MSI_0 and MSI_1 sets.

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Considering the reaction equilibrium of KAc (eq 5), the PD calculation results are similar to those generated with the ND case. This can be explained by the domination of molecular KAc compared to ionic species in the system, as observed in the liquid speciation in Figure 13b. The deviation of the model for isobaric data (1 bar) is presented in Figure 13c. The situation is the same as for the isothermal data, except that the deviation is higher, with a maximal deviation of about 160% for the FD calculation and about 15% for the ND and PD calculations. It is noticed that the deviation increases with the salt concentration. This could be explained by the nonoptimal parameters between methanol and the salt ionic species. At high salt molality values, the ion concentrations in the system are expected to increase, resulting in the need for more accurate ion parameters. In the presence of water (for the ternary mixture), accurate ion–methanol parameters will be required.
In a second stage, the MSI_1 approach is attempted, which consists of further improving the modeling of the binary system by fitting the two methanol-salt (τms and τsm) parameters on the binary isothermal VLE data (49) using the ND approach (the τmi and τim parameters are kept at their default values). The two last approaches consider simultaneously binary (MeOH/KAc) and ternary (W/MeOH/KAc) data in the objective function, with four adjustable parameters (τmi, τim, τms, and τsm). A PD approach is necessary because we assume that the speciation will be very different in aqueous and in nonaqueous solvents. Using the same database, two numerically distinct parameter sets have been obtained (labeled MSI_2 and MSI_3), pointing to a parameter degeneracy problem.
Figure 14 summarizes the performance of the different parameter sets on binary (MeOH/KAc) and ternary (W/MeOH/KAc) experimental data, with ND, FD, and PD calculations. Figure 15 provides the detailed results for MSI_2 and MSI_3. It is observed that

  • It is possible to use exclusively parameters from Aspen Plus (MSI_0, blue bars) for ND or PD on binary systems. Regarding the results for ternary systems, predictions deviate significantly from the experimental data regardless of the used approach (deviation of ∼80%).

  • The MSI_0b set with methanol/KAc and methanol/K+Ac– parameters from Aspen Plus and for water/K+Ac– regressed parameters is not optimal with very large deviations on binary with PD approach (Bin Data─PD) and all ternary calculations.

  • The regressed parameters set MSI_1 obtained from regression on the binary data set provides reliable results only in the case of ND calculation. Once salt dissociation is considered, the default parameter set (τmi, τim) = (10, −2) is no longer optimal as shown in Figure 13c.

  • The two other parameter sets, MSI_2 and MSI_3, provide almost the same final objective function (average relative deviation) of about 18%–20%, on the ternary VLE but yield different speciation in the mixed solvent, as illustrated in Figure 15. The speciation resulting from the MSI_3 parameter set shows a strong salt dissociation in methanol, which is inconsistent within a nonaqueous solvent. In contrast, the MSI_2 parameters yield more ion pairs, which is more consistent with the expectation. This explains why MSI_2 performs better on the binary data than MSI_3 (Figure 14). Interestingly, both sets provide reasonable results for all VLE calculations. However, it should be noted that MSI_2 and MSI_3 lead to different pH values (Figure 16), especially at high methanol fraction.

Figure 14

Figure 14. Performance of different parameter sets on binary (MeOH/KAc) and ternary (W/MeOH/KAc) experimental data with ND, FD, and PD calculations. Bin and Ter stand for binary and ternary, respectively.

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Figure 15

Figure 15. Comparison of the ternary W/MeOH/KAc performance on partial pressures calculations and corresponding speciation with MSI_2 (top) and MSI_3 (bottom) parameter sets at 1, 2, 4, and 6 M potassium acetate. Data are extracted from ref (48).

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Figure 16

Figure 16. pH of 9% KAc in water–methanol mixed solvent at 298.15 K computed with MSI_2 and MSI_3 parameter sets.

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As a consequence of these observations, it is proposed to reduce the parameter degeneracy (which was already made visible in Figure 8) by imposing an additional constraint on the objective function. This constraint requires that the ND approach, when applied to the binary methanol/KAc mixture using the same set of parameters, achieves a low deviation. This condition should be explicitly incorporated into the objective function to enhance its effectiveness.

Systems with Acetic Acid Solvent

Water/HAc/KAc

Acetic acid as a solvent has a small dielectric constant. It is therefore expected that potassium acetate remains in ion-pair form. This is clearly also what is recommended by Aspen Plus, as illustrated in Figure 17. The FD calculation shows a decreasing bubble temperature (corresponding to an increasing bubble pressure) upon adding KAc. Note that the eNRTL model parameters are zeroed out for τai and τia, meaning that the contribution of the physical part of the model (NRTL) is null, and only the PDH contribution is nonzero. Therefore, the FD approach makes clearly no physical sense because not only does it modify the mole fraction of water by considering the salt as two species, but also it wrongly yields a large activity coefficient to the HAc solvent.

Figure 17

Figure 17. Comparison of the HAc/KAc bubble temperature at 1 bar, using two options: fully dissociated KAc or molecular KAc with eNRTL using ASI_0 parameter sets. Experimental data were extracted from ref (35).

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As in the case of methanol/KAc, four parameter sets are considered. At first, the Aspen Plus default parameters are tested (ASI_0): (τas, τsa) = (0, 0) and (τai, τia) = (0, 0), along with the water/K+Ac– (wi) parameters from Aspen Plus. These parameters can also be combined with the water/K+Ac– parameters regressed in this work, making the labeled ASI_0b set. Then, two regression approaches have been performed. The first regression leading to the set labeled ASI_1 uses the binary experimental data that are very limited, with only 33 pressure data points over a narrow temperature range from 390.55 to 400.75 K. (35) In that case, only the solvent-molecular salt parameters are determined (two: tas and tsa), using the ND approach. The second regression considers a combined binary and ternary (regression labeled Bin + Ter Data -PD, i.e., W/HAc/KAc (35)) database. Here, the PD approach is needed with four adjustable parameters (tas, tsa, tai, and tia). The considered properties in these regressions are the bubble pressure of the mixture for binary data and partial pressures of water and acetic acid for ternary data. For “Bin + Ter Data PD regression”, two solutions have been investigated, labeled ASI_2 and ASI_3, respectively, which depend on the initialization only. One can notice that the parameters of ASI_2 and ASI_3 are quite different from each other, which again points to parameter degeneracy.
Figure 18 summarizes the performance of the different parameter sets on the binary (HAc/KAc) and ternary (W/HAc/KAc) experimental data with ND, FD, and PD calculations. Note that the vertical axis is on a logarithmic scale.

  • ASI_0 (Aspen Plus parameters), ASI_0b (methanol/salt and methanol/ions parameters from Aspen Plus with water/ions parameters regressed in this work), and the ASI_1 parameter set regressed on the binary data set provide adequate results only in the case of ND and PD calculations on binary data. While considering salt dissociation, the default parameter set (τai, τia) = (0,0) is not optimal.

  • Both ASI_2 and ASI_3 parameter sets are obtained from regression on combined binary and ternary data using the PD assumption. We can notice that both sets provide acceptable results with an objective function (average relative deviation) output value in the same range of about 35–45%. It was difficult to obtain better results because of some infinite dilution data (at very low concentrations of water or acetic acid of less than 1%), leading to high deviations in partial pressures and, consequently, to high objective function values. Figure 19 shows the detailed deviations and speciations for the binary system (HAc/KAc). Simulation results with ASI_2 seem to better fit experimental pressure binary data. From the right-hand part of Figure 19, it is also observed that the ASI_2 PD speciation is dominated by the molecular KAc compound, while ASI_3 results in high ionic concentration in the system. We know that ASI_2 results are more realistic than those of ASI_3, but to the best of our knowledge there is no speciation data to compare our predictions with. Yet, we can notice that deviations resulting from applying the “ND” assumption using the ASI_3 parameters on the binary data are much larger than those obtained with the ASI_2 parameters. Note that the “ND” calculations with both ASI_2 and ASI_3 parameters are performed a posteriori of the regression. This observation should then be used as a guide to determine which set is the most physically consistent.

Figure 18

Figure 18. Performance of different parameter sets on binary (HAc/KAc) and ternary (W/HAc/KAc) experimental data with ND, FD, and PD calculations. It is observed that.

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Figure 19

Figure 19. Comparison of the binary acetic acid-potassium acetate bubble pressure (Psat) PD calculations and corresponding speciation with ASI_2 and ASI_3 parameter sets on isobar data (1 bar) from ref (35) with corresponding speciation.

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Figure 20 shows the deviation in the partial pressure of volatile compounds (water and acetic acid) as well as the corresponding speciation of the system. It shows that numerical simulations performed with ASI_2 (top) and ASI_3 (bottom) parameter sets lead to almost the same performance on VLE data. As compared to ASI_3, ASI_2 is slightly better for water but worse for acetic acid. No parameter set can predict correctly the behavior of water at infinite dilution where the water salt-free mole fraction is lower than 5 × 10–3. The right-hand side plots show the speciation using the two different parameter sets. One notices that the results are extremely different between ASI_2 or ASI_3 parameters: the concentration of ions increases with the water content for ASI_2, while for ASI_3, ionic species are dominant throughout the mixed solvent domain. The salt dissociation increases with the H2O mole fraction for ASI_2 and decreases for ASI_3. Given these observations, ASI_2 seems to be more reasonable. But once again, it requires complementary data to validate.

Figure 20

Figure 20. Comparison of the ternary W/HAc/KAc deviations on partial pressures with PD calculations and corresponding speciation with ASI_2 (top) and ASI_3 (bottom) parameter sets on isobaric data (1 bar) from ref (35). The molar ratio of acetic acid to water is 0.01, 0.03, 0.05, 0.07, 0.09, and 0.11.

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Water/Methanol/HAc

The vapor–liquid equilibrium of this system is perfectly well represented by molecular parameters from Aspen Plus software (ND approach; see Figure 21). It is observed that in most cases the average deviation is about 20%, with the maximal deviations located at infinite dilution.

Figure 21

Figure 21. Deviations in partial pressure calculations using Carnot. Isobar data are taken at atmospheric pressure from ref (51) with temperature ranging from 350.55 to 377.75 K.

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The conclusion for this system is that the molecular parameters from the literature are sufficient for describing VLE. However, if one wants to have another property that is related to speciation, as for example pH (for instance, crucial for, e.g., corrosion issues), then it is essential to work with PD and an adequate parameter set.

Conclusions

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In this study, we explored the complexities of thermodynamic modeling in electrolyte solutions, with a focus on the challenges and strategies involved in parametrizing multicomponent systems. Conducted under the JIP EleTher initiative, this work aims to develop a “best practices” guide for setting up thermodynamic models that address both molecular and ionic interactions in electrolyte solutions. Although the project initially aimed to target quaternary systems (water, cosolvent, acid, and strong base), our focus shifted to ternary systems due to the current scarcity of experimental data.
We employed a heterogeneous γ – φ approach for simulating the thermodynamics of the electrolyte solutions. The eNRTL method was applied to model the activity coefficients of components in the liquid and aqueous phases, while the CPA equation of state accounted for the nonideal behavior of the vapor phase. The thermodynamic model and flash calculation framework were implemented in our in-house library and employed throughout this study.
A first observation concerns the difficulty of initializing the parametrization properly: the model may, in some cases, yield very strong deviations. It is then suggested to add a correction to the objective function that accounts for the nonconvergence calculations. In addition, it was also found useful to start with an evolutionary algorithm that generates reasonable initial sets that can then be solved with a more conventional minimizer.
Regarding the acceptable simplifying assumptions, we clearly need to distinguish single-solvent and mixed-solvent systems. In the former, the assumptions of either no dissociation (ND, for nonaqueous solvents) or full dissociation (FD, for aqueous solvents) can be used to adequately fit the data. We sometimes had to refit parameters at a high salt concentration. Indeed, the default parameters are generally adequate until 4 M solutions and for vapor–liquid equilibria. When including the MIAC, the parametrization is generally much more stringent, and it may be required to refit the third eNRTL parameter, alpha. Note that no MIAC value can be computed using the “ND” approach because no ions are then present in solution.
When dealing with ternary systems, it is recommended to use the partial dissociation (PD) approach. We then need to consider more compounds and therefore many more binary interaction parameters, resulting in parameter degeneracy (multiple parameter sets yield the same result). In order to further constrain the regression, we then recommend using an objective function (OF) that combines several calculation modes at once, always with the same parameter set
OF=OFPD(alldata)+OFND(non−aqueousbinaries)+OFFD(aqueousbinaries)
(15)
where the individual OFs are defined as in eq 14, i.e., including a correction for nonconverged situations. Such an objective function can yield a set of parameters that perform robustly across all scenarios, regardless of the underlying assumption about the chemical reactions
Obviously, if speciation data were available, this would equally improve the results.
A last investigation was performed to evaluate the pH of acetic acid in a mixed solvent. While in other cases it was not needed to consider the H3O+, the computation of the acidity level requires an accurate level of information about the H3O+ activity. The approach proposed by Kosinski et al. (82) was then applied, but it requires refitting of cosolvent–ion parameters on mixed solvent acid titration data.
Finally, as a best practice, the model parameters should always be refitted, since their values depend on the assumed reaction framework─fully dissociated (FD), partially dissociated (PD), or nondissociated (ND). Out-of-box Aspen parameters need to be validated with these different assumptions. For example, a similar study has been conducted using Aspen Properties, and the results with refitted parameters gave good matching of the experimental data of the same multicomponent electrolyte system.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jced.5c00412.

  • Contains the description of the model and all the parameters that have been used in this work (PDF)

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Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Authors
    • Saheb Maghsoodloo - IFP Energies Nouvelles, 1 et 4 Avenue de Bois-Préau, 92852 Rueil-Malmaison, Cedex, France;  Email: [email protected]
    • Tri Dat Ngo - IFP Energies Nouvelles, 1 et 4 Avenue de Bois-Préau, 92852 Rueil-Malmaison, Cedex, France;  Email: [email protected]
    • Jean-Charles de Hemptinne - IFP Energies Nouvelles, 1 et 4 Avenue de Bois-Préau, 92852 Rueil-Malmaison, Cedex, France;  Orcidhttps://orcid.org/0000-0003-1607-3960;  Email: [email protected]
  • Authors
    • Edouard Moine - Fives ProSim, 51 Rue Ampère, Immeuble Stratège A, 31670 Labège, France
    • Shu Wang - AspenTech, 20 Crosby Dr, Bedford, Massachusetts 01730, United States;  Orcidhttps://orcid.org/0000-0002-7070-5095
    • Bjorn Maribo-Mogensen - Hafnium Laboratories, Vestergade 16, Third Fl., 1456 Copenhagen, Denmark;  Orcidhttps://orcid.org/0000-0001-5447-412X
    • Emrah Altuntepe - Covestro, Kaiser Wilhelm-Allee 60, 51373 Leverkusen, Germany
    • Salvador Asensio-Delgado - Syensqo, 85 Av des Frères Perret, 69190 Saint Fons, France;  Orcidhttps://orcid.org/0000-0002-0722-6139
    • Stephanie Peper - Bayer, Kaiser Wilhelm-Allee 3, 51373 Leverkusen, Germany;  Orcidhttps://orcid.org/0000-0002-9660-1745
    • Andrés González de Castilla - Bayer, Kaiser Wilhelm-Allee 3, 51373 Leverkusen, Germany;  Orcidhttps://orcid.org/0000-0003-0536-7402
    • Pascal Ferrari - Orano, 23 Pl. de Wicklow, 78180 Montigny-le-Bretonneux, France
    • Ellen Steimers - BASF SE, Carl-Bosch Strasse 38, 67063 Ludwigshafen, Germany
    • Susanna Kuitunen - NESTE, Länsitie 6, 06400 Porvoo, Finland
  • Author Contributions

    Corresponding Authors: Edouard Moine: review and validation; Shu Wang: review, editing, and validation; Bjorn Maribo-Mogensen: review; Emrah Altuntepe: review and funding; Salvador Asensio-Delgado:: review and funding; Stephanie Peper:: review and funding; Andrés González de Castilla:: review and funding; Pascal Ferrari:: review and funding; Ellen Steimers: review, editing, and funding; and Susanna Kuitunen: review, funding.

  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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The authors acknowledge the EleTher Joint Industry Project that has financed the work. The corresponding authors thank the personal input of all industry participants.

References

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  • Abstract

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    Figure 1

    Figure 1. Schematic representation of the selected quaternary system, which turns out to be a nine-component system due to the reactive equilibria. HA and BOH stand for acid and base, respectively.

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    Figure 2

    Figure 2. Three types of activity coefficients for a ternary system containing water, a cosolvent, and a salt: (i) blue, solvent activity coefficients; (ii) red, transfer activity coefficient of the ions between pure water and mixed solvent; (iii) green, conventional ionic activity coefficient. The orientation of the arrows shows the type of derivative of the Gibbs energy that they correspond to. SLE stands for solid–liquid equilibrium. The definitions of the variables can be found in the text.

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    Figure 3

    Figure 3. Solvent activity coefficient (computed by eq 1) in the water/methanol/KAc ternary system: (a) methanol and (b) water. Data were extracted from ref (48).

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    Figure 4

    Figure 4. Molality-based dissociation constants of (a) lithium, (b) sodium, and (c) potassium hydroxides as a function of the inverse of the temperature. (55−65) The (d) diagram provided a comparison of (a–c) using Akinfiev-calculated data.

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    Figure 5

    Figure 5. Dissociation constants of acetate salts containing (a) lithium, (b) sodium, and (c) potassium as a function of the inverse of temperature. (54,64−70) The diagram labeled (d) provides a comparison of (a–c) data.

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    Figure 6

    Figure 6. Comparison of fugacity coefficient of acetic acid in the vapor phase as predicted by HOC and CPA equations of state.

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    Figure 7

    Figure 7. Dissociation constant as a function of temperature using eq 7, together with experimental data from ref (78) for H2O, from ref (79) for HAc, from refs (66,and67) for KAc, and from ref (57) for KOH.

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    Figure 8

    Figure 8. Example of response surfaces for the methanol─KAc parameter regression. The points indicate the location of different default parameter sets: [τmi, τim] = [10, −2] (blue dot), [τmi, τim] = [8, −4] (brown dot), and [τmi, τim] = [0, 0] (green dot).

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    Figure 9

    Figure 9. Mean activity coefficient (a) and vapor pressure (b) of Water/KAc mixtures at 25 °C with the default Aspen Plus parameters using Carnot. Calculation of MIAC with ND option is impossible because it requires having ions in the system. Corresponding speciations for MIAC (c) and vapor pressure (d) calculations using the PD approach. MIAC Data are extracted from ref (34) and vapor pressure data from ref (38).

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    Figure 10

    Figure 10. Mean activity coefficient (a) and vapor pressure (b) of Water/KAc mixtures at 25 °C and deviations on MIAC (c) and vapor pressure (d). Experimental data (34,36,38) and modeling results with regressed parameters.

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    Figure 11

    Figure 11. KOH mean activity coefficient (a) and vapor pressure (b) with different methods and parameters (experimental data are from ref (34)). Reg stands for regressed. Speciation of Psat PD calculations as a function of salt molality using default (c) and regressed (d) parameters.

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    Figure 12

    Figure 12. Vapor pressures of the water + KOH system with regressed parameters. The data are from refs (46,and47).

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    Figure 13

    Figure 13. VLE calculation versus experimental data from refs (49and50) (a) Comparison of the methanol-potassium acetate vapor pressure (isotherm data) using the fully dissociated (FD) approach, partially dissociated (PD), or nondissociated (ND) assumptions with Aspen Plus NRTL parameters for both water/ion pair and methanol/ion pair interactions using MSI_0 set; (b) liquid speciation with PD calculation using MSI_0 set; (c) vapor pressure deviation on isobar data (1 bar) using MSI_0 and MSI_1 sets.

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    Figure 14

    Figure 14. Performance of different parameter sets on binary (MeOH/KAc) and ternary (W/MeOH/KAc) experimental data with ND, FD, and PD calculations. Bin and Ter stand for binary and ternary, respectively.

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    Figure 15

    Figure 15. Comparison of the ternary W/MeOH/KAc performance on partial pressures calculations and corresponding speciation with MSI_2 (top) and MSI_3 (bottom) parameter sets at 1, 2, 4, and 6 M potassium acetate. Data are extracted from ref (48).

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    Figure 16

    Figure 16. pH of 9% KAc in water–methanol mixed solvent at 298.15 K computed with MSI_2 and MSI_3 parameter sets.

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    Figure 17

    Figure 17. Comparison of the HAc/KAc bubble temperature at 1 bar, using two options: fully dissociated KAc or molecular KAc with eNRTL using ASI_0 parameter sets. Experimental data were extracted from ref (35).

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    Figure 18

    Figure 18. Performance of different parameter sets on binary (HAc/KAc) and ternary (W/HAc/KAc) experimental data with ND, FD, and PD calculations. It is observed that.

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    Figure 19

    Figure 19. Comparison of the binary acetic acid-potassium acetate bubble pressure (Psat) PD calculations and corresponding speciation with ASI_2 and ASI_3 parameter sets on isobar data (1 bar) from ref (35) with corresponding speciation.

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    Figure 20

    Figure 20. Comparison of the ternary W/HAc/KAc deviations on partial pressures with PD calculations and corresponding speciation with ASI_2 (top) and ASI_3 (bottom) parameter sets on isobaric data (1 bar) from ref (35). The molar ratio of acetic acid to water is 0.01, 0.03, 0.05, 0.07, 0.09, and 0.11.

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    Figure 21

    Figure 21. Deviations in partial pressure calculations using Carnot. Isobar data are taken at atmospheric pressure from ref (51) with temperature ranging from 350.55 to 377.75 K.

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  • Supporting Information

    Supporting Information

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