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## Mean spherical approximation theory (MSA)

The MSA theory is based on the perturbation of polar fluids where the reference system is in the Percus-Yevick approximation using the Ornstein-Zernike equation as a specific closure. MSA theory has both an implicit and an explicit version, the explicit (restricted) version is simple and uses a common ion diameter whereas implicit (non-restricted) version is solved iteratively [36]. According to some findings, it is said that MSA theory (section 2.3.4.2) has an edge over DH theory (section 2.3.4.1). Galindo et al. [71] compared the MSA and DH theories for describing short-range interactions and reported that at a higher salt concentration of NaCl densities were more accurately represented by MSA than by DH. However, for the representation of vapor pressure, the performance was nearly the same. A Taylor series expansion and the comparison of the mathematical form of both these theories by Lin et al. [47] showed that there are very little differences when assuming the same ion diameters. A recent comparison made by MariboMogensenet al. [81] showed that the two theories (DH vs NP-MSA) when compared numerically in terms of screening length gave similar results. Since we are adapting a non-restricted primitive version of MSA we present the final expression in terms of Helmholtz energy.

### Review of the existing electrolyte and mixedsolvent electrolyte EoS

A good review of the electrolyte thermodynamics is made by Loehe and Donohue [83], that included models from 1985-1997. Another review by Prausnitz [84] presented a brief account of electrolyte thermodynamics including their applicability in the biotechnology industry. A short review by Pinsky and Takano [85] presents some local composition models emphasizing computational details of activity coefficient models. Lin et al. [47] and Tan et al. [86] presented an account on electrolyte equation of state in conjunction with SAFT and electrolytic theories. In addition to these, Michelsen and Mollerup [43] presented a thorough discussion including a derivation of modern Debye-Hückel theory and the theories of dipolar ions.

Similarly, many authors have investigated equations of state and also several electrolyte thermodynamic equations of state have been developed. An extensive review is already presented by Kontogeorgis [36] and MariboMogensen [87]. An extension of those reviews including some newer ones is presented herein tables 2.2 and 2.3. These tables present the models with a mention of their parent/base model, the type of Coulombic term employed, whether a Born term is used, and the functional form of the dielectric constant if needed (i.e. density-dependent –indicated with V; salt concentration-dependent – indicated with i) used for the electrolyte terms, the presence of ion-ion or ion-solvent association. This table shows that the models differ on the incorporation of short-range [7, 88–90] and long-range forces as Debye-Hückel (DH) or mean spherical approximation(MSA).

#### Statistical associating fluid theory (SAFT)

In the current section, we present details about SAFT. The reasons for choosing SAFT as a basis for this work is due to its success in describing non-electrolyte systems. This section will provide a theoretical background followed by various developments made in SAFT.

**Perturbation theory**

Perturbation theory is a mathematical theory which provides an approximate solution to a problem. The solution is obtained by adding values obtained by approximating perturbation terms to the solution of exact terms as in a Taylor series expansion.

The mathematical expression describing the residual Helmholtz energy is given as in equation 2.19. Ares = Aref + Aperturbation (2.19).

Where Aref refers to the Helmholtz energy of reference state is often the hard sphere fluid. The perturbation term accounts for various interactions such as repulsion, chain, dispersion, association, polar (figure 2.4) and in the current work for electrolytic contributions.

**Mathematical description of PC-SAFT**

In SAFT, the interactions between the molecules are accounted as perturbations. The physical picture of SAFT is based on the statistical perturbation theory. The reference is most often the hard sphere, except for PC-SAFT where it is the hard chain. The perturbation applied to this reference is based on some picture of the pair potential (Lennard-Jones, square well or other) and is called the dispersive contribution. The hard spheres are assumed to contain association sites, making it possible, using Wertheim’s theory, to construct a perturbation related to association interactions. Considering infinite association strengths between some selected sites, it is possible to construct a chain with its specific perturbation term. The representative equation 2.20 for final residual Helmholtz energy is given as a follows. Figure 2.4 illustrates the interactions and construction of GC-PPC-SAFT.

**Previous Descriptions of Water with the PC-SAFT EoS**

Water is a polar solvent which has strong association tendency. It readily forms hydrogen bonds to itself and to other molecules capable of forming such bond. In current GCPPC- SAFT prior to modification, water is viewed as polar associating component, however, there has been significant deviations in predicting liquid densities of pure water.

NguyenHuynh [152] has described pure water according to a 4C association scheme and a dipolar character. The deviations in liquid densities of pure water were 4.74% for liquid volume and 3.36% for vapor pressure. Moreover, at room temperature, these deviations remained significantly high, around 10% (Figure 3.1).

**A new Temperature Dependence of Water Diameter**

After an exhaustive investigation, it was clear that accurate modeling of liquid densities in water cannot be obtained without modification of its temperature independent diameter σw (equation 3.3), which was also reported by Held and Sadowski [60]. However, the expression proposed by Held is corrective in nature and contains 4 regressed parameters valid on a reduced temperature range. It appears that a very small change in the value of this parameter may change the resulting densities significantly. This is coherent with observations made by NguyenHuynh et al. [152]. Thus, it was important to identify the exact behavior of the optimal water diameter with our model, on a larger temperature range and then systematically incorporate that behavior in the model. To that end, the optimal diameter was fitted on both vapor pressure and saturated liquid molar volume (data originating from DIPPR correlation). The segment diameter σ was regressed at short temperature intervals (∼10K) of vapor pressure and saturated liquid volume data. This process was repeated over entire range of temperatures from 273.15 to 550 K and the regressed values were plotted as a function of temperature (average of the temperature intervals on which it was regressed). As shown in figure 3.2, this clearly exhibits that the segment diameter of water should increase with temperature if we want it to match accurately and simultaneously liquid densities and vapour pressures. The functional form that was chosen to fit the observed behaviour is as follows: σT,W = σW + Tdep,1.exp(Tdep,2 × T) + Tdep,3.

**Table of contents :**

List of figures

List of tables

Nomenclature

**1 Setting up stage **

1.1 Introduction

1.2 A very brief history of thermodynamics

1.3 Motivation for this work

1.4 Roles of Electrolytes in Various Industries: Scope of this work

1.4.1 Biorefining industry

1.4.2 Oil and Gas industry

1.4.3 Carbon capture and sequestration (CCS)

1.4.4 Acid gas injection

1.4.5 Pharmaceutical industry

1.4.6 Other uses of electrolytes

1.5 Aqueous two-phase systems

1.6 Objectives of this research

**2 Electrolyte thermodynamic model and the State of the art **

2.1 Introduction

2.2 Activity coefficient models vs EoS

2.3 Interactions in electrolyte systems and theories

2.3.1 Discharge

2.3.2 Repulsion and dispersion

2.3.3 The Structure-forming step

2.3.4 Electrolyte terms

2.4 Review of the existing electrolyte and mixed-solvent electrolyte EoS

2.4.1 State of the art

2.4.2 Choice of thermodynamic model

2.5 Statistical associating fluid theory (SAFT)

2.5.1 Perturbation theory

2.5.2 History of Statistical Associating Fluid Theory (SAFT)

2.5.3 Mathematical description of PC-SAFT

2.5.4 Various terms of ePPC-SAFT

2.5.5 Group contribution approach

2.5.6 Cross association parameters

**3 Modified model of PC-SAFT for water **

3.1 Abstract

3.2 Introduction

3.3 Model

3.3.1 Previous Descriptions of Water with the PC-SAFT EoS

3.3.2 A new Temperature Dependence of Water Diameter

3.4 Results for binary mixtures

3.4.1 Mutual solubilities

3.4.2 Octanol/Water Partition coefficient

3.4.3 Gibbs energy of Hydrogen Bonding

3.4.4 Conclusion

4 Modeling of strong electrolytes

4.1 Abstract

4.2 Introduction

4.3 Some thoughts and arguments related to the choices made in this work .

4.3.1 What is solvation?

4.3.2 Specificities related to the GC-ePPC-SAFT model

4.3.3 Parameters from previous work

4.4 Ion parameterization procedure

4.5 Regression Results

4.5.1 Correlation results for Alkali halide brines

4.5.2 Solvation Gibbs energies for Alkali halide brines

4.6 Alkanes and acid gases with brines: salting out effect in presence of organic compounds

4.7 Mixed solvent electrolytes

4.8 Conclusion

**5 Modeling LLE of mixed-solvent electrolytes **

5.1 Introduction

5.2 Algorithmic issue: Electroneutrality

5.2.1 Generalities

5.2.2 Current state: without electroneutrality

5.2.3 A new method for electroneutrality: Modifying the fugacity coefficient104

5.3 The machinery

5.3.1 Prediction of the LLE by the non-parameterized model

5.3.2 Partition coefficient of salts

5.4 Parameterizing mixed solvent salt systems using available data

5.4.1 Dielectric constant of mixed-solvents

5.4.2 MIAC of mixed solvent electrolyte systems: Analysis

5.4.3 Approach for parameterization of mixed-solvent salt systems

5.4.4 Result of MIAC for mixed solvent electrolytes

5.4.5 Parameter for 1-Butanol-water-salt systems from LLE data .

5.5 Final results

5.6 Conclusion

**6 Conclusions and recommendations for future work **

6.1 Conclusion

6.2 Recommendations for future work

**Bibliography**