Experimental Data and Modeling of Solution Density and Heat

Article pubs.acs.org/jced

Experimental Data and Modeling of Solution Density and Heat Capacity in the Na−K−Ca−Mg−Cl−H2O System up to 353.15 K and 5 mol·kg−1 Ionic Strength Adeline Lach,*,†,‡ Karine Ballerat-Busserolles,§,∥ Laurent André,† Mickael̈ Simond,⊥ Arnault Lassin,† Pierre Cézac,‡ Jean-Claude Neyt,⊥ and Jean-Paul Serin‡ †

BRGM − 3 avenue C. Guillemin − 45000 Orléans, France Univ Pau & Pays Adour, Laboratoire de Thermique, Energetique et Procedes-IPRA, EA1932, 64000, Pau, France § Université Clermont Auvergne, Université Blaise Pascal, Institut de Chimie de Clermont-Ferrand, BP 10448, F-63000 Clermont-Ferrand, France ∥ CNRS, UMR 6296, ICCF, F-63178 Aubière, France ⊥ Calnesis − 24 avenue Blaise Pascal − 63170 Aubière, France ‡

ABSTRACT: This work is on in the volumetric and thermal properties of brines in the quinary Na−K−Ca−Mg−Cl−H2O chemical system. Its objective is twofold. First, by acquiring original data for temperatures ranging from 278.15 to 353.15 K and ionic strengths ranging from 1.3 to 5.1 mol·kg−1 it aimed to add to the experimental data set, usually acquired only at high ionic strengths or at 298.15 K. Experimental solution density was measured using a vibrating-tube densitometer with relative uncertainty, Δρ/ρ, better than 6 × 10−6. This property, combined with volumetric heat capacity measurements, provided the isobaric heat capacity of solution determined with a mean relative deviation better than 0.3%. Second, we used PhreeSCALE software to compute the density and heat capacity of the chemical system of interest, simultaneously applying the Pitzer and the Helgeson−Kirkham−Flowers (HKF) equations. We propose a new set of specific interaction parameters so that published and newly measured experimental data can be described accurately. We show that only binary interaction parameters are necessary and that ternary interaction parameters could be set to zero. to design desalination systems.14 Density is also a key property for computing the capacity of an aquifer to sequester CO2.15 These two properties can be obtained experimentally,16 or calculated from the temperature or the pressure derivative of the excess Gibbs energy.17 In this work, we studied the Na−K−Ca−Mg−Cl−H2O system because natural brines are mostly chloride solutions that systematically include some or all of the four major cations1 Na+, K+, Ca2+ and Mg2+. Our first objective was to propose a model18 able to compute the excess properties of this complex system using as few adjustable ion-specific interaction parameters as possible, and standard properties of solutes computed from the HKF model,19,20 which is widely used among geochemists. The specific interaction parameters were computed with the PhreeSCALE software17 from available experimental literature data. We first considered the binary systems (Na−Cl, K−Cl, Ca−Cl and Mg−Cl) to produce binary interaction parameters. Then we applied these to calculate the density and the heat capacity of the ternary systems of interest, namely Na−K−Cl, Na−Ca−Cl, Na−Mg−Cl, K−Ca−Cl, K−Mg−Cl,

1. INTRODUCTION Brines present in natural systems are mostly chloride solutions.1 Their chemical compositions vary according to their hydrogeological environment and origin (surface water or groundwater). Seawater and brackish surface waters are increasingly used to produce drinking water via desalination processes,2,3 and saline groundwaters can be targeted for energy storage4 or for geothermal energy.5 These natural brines also contain some exploitable and valuable substances (e.g., magnesium, potassium).6 The exploitation of such systems produces waste that must be managed to limit their environmental impact.7 For all of these applications, it is necessary to anticipate the chemical behavior and reactivity of the brines involved in the various processes to be able to optimize them. This can be achieved if a reliable thermodynamic model can predict the physical, chemical and thermodynamic properties of such brines at various temperature conditions and ranges of chemical composition. Many studies on brine systems are dedicated to the measurement and characterization of properties such as water activity, activity and osmotic coefficients.8−12 This study focused on two other specific properties of brines: heat capacity and density. Knowledge of the heat capacity and density of electrolyte solutions are of interest in many fields. For instance, these properties are used to estimate the heat content of geothermal fluids13 or © 2017 American Chemical Society

Received: June 16, 2017 Accepted: August 11, 2017 Published: September 7, 2017 3561

DOI: 10.1021/acs.jced.7b00553 J. Chem. Eng. Data 2017, 62, 3561−3576

Journal of Chemical & Engineering Data

Article

and for the reference fluid (water in our case). The differences between the densities of solution, ρ, and water, ρref, were calculated by comparing the periods of vibration of the tube containing the selected solution, τ, and pure water, τref. Equation 1 gives the relationship between the period of vibration and the density of the solution measured.

and Ca−Mg−Cl. The model was then applied to the quinary system Na−K−Ca−Mg−Cl. We compared the results with experimental literature data and to new data that we acquired during this study. Our second objective was to measure new experimental data for the Na−K−Ca−Mg−Cl−H2O system, between 278.15 and 353.15 K and for ionic strength varying between 1.3 and 5.1 mol·kg−1. These new data supplement the values measured by Krumgalz et al.21 for higher ionic strengths (8.3 and 9.6 mol·kg−1). This work confirms the great interest of using calorimetric data, in addition to isopiestic data, to constrain thermodynamic models of saline systems. Because they correspond to the second derivative of the excess energy, they can be combined with fewer isopiestic data using adequate numerical tools, like PhreeSCALE.

ρ − ρref = K (T , p) × (τ 2 − τref2 )

(1)

The calibration constant, K(T,p), was determined at each experimental temperature and pressure by measuring a 1 mol·kg−1 NaCl solution. The density of water was calculated from the equation of Hill,22 and the density of the NaCl solution was obtained from the representative correlation by Archer.23 Relative uncertainties for density measurements Δρ/ρ are better than 6 × 10−6. Isobaric heat capacities were determined using a SETARAM differential scanning microcalorimeter (microSC) equipped with specific and noncommercial cells adapted to work under pressure. The detection was based on the Calvet principle. The experimental procedure was similar to the one described by Origlia-Luster et al.16 First a blank experiment was performed by filling both sample and reference cells with nitrogen (N2). Then the sample cell was filled with the test solution while the reference cell was filled with N2. An experimental run consisted of a 20 min isothermal step at 273.15 K followed by temperature scanning (0.4 K·min−1) up to 358 K. Experiments were carried out at constant pressure of 1.6 × 105 Pa in the sample, with the reference maintained at atmospheric pressure. Pressure was controlled using a nitrogen buffer volume (40 mL) connected to the outlet of the measuring cell. The solutions’ volumetric heat capacities (ρScp,S) were then obtained from the difference between the thermal flux (HF) of the test solution and the blank as indicated in eq 2.

2. EXPERIMENTAL PROTOCOL: MATERIALS AND METHODS 2.1. Sample preparation. Solutions were prepared by mass using water distilled and degassed prior to use and reagent-grade salts. The balance used was a high precision Mettler Toledo AE163, operating up to 160 g, with a readability of 0.1 mg and a reproducibility of 0.2 mg on the full scale. NaCl and KCl were obtained from Acros Organics (>99% purity). MgCl2·6H2O and CaCl2·2H2O were from Sigma-Aldrich (>99% purity). The water content of each salt was determined by coulometric tiration using the Karl Fisher technique (Mettler Toledo model DL32). Water content of NaCl and KCl was below 0.15% weight by weight. For hydrated salts (MgCl2·6H2O and CaCl2· 2H2O) the water quantity determined by Karl Fisher matched with supplier information and the final compositions of the salts were MgCl2·6.11H2O and CaCl2·2.00H2O. All the salts were used without further purification. Solutions were prepared 24 h before use to ensure good sample homogeneity and were stored under nitrogen. Each solution was used within a couple of days after preparation. Uncertainties on molalities were found to be better than 0.001 mol·kg−1. In Table 1, we reported the characteristics of the chemicals used for the sample preparation. 2.2. Apparatuses. Densities were obtained using an Anton Paar DMA vibrating-tube densitometer equipped with an HPM cell. The temperature of the densitometer was controlled by a circulating bath (JULABO F12). Measurements were performed at constant temperature from 278.15 K up to 353.15 K with a precision of 0.01 K and stability better than 0.02 K. At temperatures above 333.15 K, measurements were carried out under pressure to avoid the presence of any vapor phase in the vibrating tube. Pressure was maintained constant using a buffer volume filled with nitrogen and measured with a Swagelok pressure transducer with a precision of 3 × 103 Pa connected to the outlet tube of the densitometer. Measurements were performed at constant liquid flow rate of 0.4 mL·min−1, using an isochratic pump (P 4.1S Azura from Knauer). The tube’s period of vibration τ was measured directly using a TTi TF930 frequency meter. Vibration period values were continuously registered for about 20 min for each solution,

ρS cp , S(T ) = K ′(T ) × (HFsample(T ) − HFblank(T ))

(2)

The constant K′(T) of the apparatus was determined from water using data recommended by Hill.22 The calibration measurements were repeated three times to ensure the reproducibility of the constant value. Then the accuracy of K′(T) was tested by measuring the molar heat capacity of a 1 mol·kg−1 NaCl solution.23 The mean percent relative deviation (% RD) is better than 0.3%.

3. NUMERICAL AND THEORETICAL APPROACH 3.1. The Gibbs excess energy − Pitzer model. The Pitzer model18 aims to describe the excess properties of concentrated aqueous electrolyte solutions by considering specific interactions between solutes. It expresses the excess Gibbs energy, which depends on the specific interaction parameters presented below, and allows the calculation of thermodynamic properties of aqueous electrolyte solutions such as osmotic coefficient,24−26 heat capacity,17,27,28 and density.17,29,30 It is a semiempirical model that was developed to improve the work of Debye−Hückel31 to extend its applications to highly concentrated solutions. Initially, the model assumes the total

Table 1. Chemicals Used for Sample Preparation

a

Chemical reagent

Supplier

Purity

Purification method

Final composition

Analysis methoda

NaCl KCl MgCl2·6H2O CaCl2·2H2O

Acros Organics Acros Organics Sigma-Aldrich Sigma-Aldrich

>99% >99% >99% >99%

None None None None

NaCl KCl MgCl2·6.11H2O CaCl2·2.00H2O

Karl Fisher

To determine water quantity in the hydrated salt. 3562



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and Cϕ).18,32 An additional parameter β(2) was added by Pitzer and Mayorga33 to represent the behavior of the 2−2 type electrolytes above 0.1 mol·kg−1. Later, Pitzer and Kim34 extended the original equations to electrolyte mixtures (i.e., containing two or more electrolytes). To improve the calculation of electrolyte solution properties, some authors have included virial coefficients corresponding to interactions of orders higher than Cϕ.35−37 Others consider the partial dissociation of electrolytes, adding specific interactions between neutral and neutral species or neutral and ionic species.25,26,38 For a system containing an undefined number of both neutral and ionic solute species, the excess Gibbs energy is described by the expression given by Clegg and Whitfield39 and Felmy and Weare.40 In this study, all the electrolytes were considered totally dissociated and, because the only anion was

electrolyte dissociation and contains only three binary specific interaction parameters between anions and cations (β(0), β(1), Table 2. Keywords Used in the PhreeSCALE Software,17 and Their Significance Keyword

Related property

ENTHALAPP Apparent relative molal enthalpy, Lϕ CPSTAN(“i”) Standard partial molal heat capacity of the aqueous species i at infinite dilution, C0p,i CPAPP Apparent molal heat capacity of the solution, Cp,ϕ, CPSOL Heat capacity of aqueous solution, Cp VSTAN(“i”) Standard partial molal volume of the species i at infinite dilution, V0i VAPP Apparent molal volume, Vϕ DENSOL Density of the aqueous solution, ρ

Table 3. Composition of Experimental Solutionsa S1

S2

S3

S4

S5

S6

S7

a

S1-1 S1-2 S1-3 S2-1 S2-2 S2-3 S3-1 S3-2 S3-3 S4-1 S4-2 S4-3 S5-1 S5-2 S5-3 S6-1 S6-2 S6-3 S7-1 S7-2 S7-3

NaCl/mol·kg−1

KCl/mol·kg−1

MgCl2/mol·kg‑1

CaCl2/mol·kg‑1

I/mol·kg−1

0.5002 0.500 0.4999 3.2991 3.300 3.3000 0.3002 0.300 0.2999 1.0999 1.102 1.1001 3.0993 3.102 3.1004 4.4000 4.401 4.3998 4.7998 4.800 4.7999

0.5003 0.500 0.4997 0.1000 0.100 0.1001 0.0999 0.100 0.0999 0.1999 0.200 0.2001 0.4999 0.500 0.4999 0.0999 0.100 0.0999 0.0000 0.000 0.0000

0.5009 0.500 0.4998 0.1000 0.100 0.1000 0.1000 0.100 0.0999 0.9022 0.900 0.9000 0.1002 0.100 0.1001 0.1000 0.100 0.1001 0.1000 0.100 0.0999

0.5007 0.500 0.4999 0.1009 0.100 0.1000 0.2015 0.200 0.2001 0.0000 0.000 0.0000 0.4004 0.400 0.4000 0.1011 0.101 0.1000 0.0000 0.000 0.0000

4.0054 4.0015 3.9988 4.0018 4.0001 3.9999 1.3046 1.3006 1.2999 4.0063 4.0022 4.0003 5.1010 5.1018 5.1007 5.1035 5.1031 5.0998 5.0998 5.1003 5.0997

Standard uncertainties u are u(m) = 0.001 mol·kg−1.

Figure 1. Experimental density as a function of temperature for different ionic strengths and solution compositions. Each symbol represents a solution (see Table 3 for details): shaded diamonds, S1; ○, S2; shaded squares, S3; shaded triangles, S4; ◇, S5; shaded circles, S6; □, S7 ; , model. 3563



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chloride (Cl−), the Gibbs excess energy (Gex) was computed using eq 3:

The complete equation for Φcc′, which takes into account the interaction between two ions of the same sign charges, is

⎧ 4IA ϕ Gex /(wwRT ) = ⎨ ln(1 + bI1/2) − ⎪ b ⎩

Φcc ′ = θcc ′ + Eθ cc ′(I )

⎪

θcc′ is an adjustable parameter, and Eθcc′ takes into account electrostatic asymmetric mixing effects and only depends on ion charges, total ionic strength, and solvent properties.43 The third virial coefficient ψcc′Cl− is a mixed electrolyte parameter for each cation−cation−anion or anion−anion−cation triplet in a mixed electrolyte solution. This term is an adjustable parameter. 3.2. Excess properties - heat capacity and density. The derivative of eq 3 with respect to a system parameter (molality, temperature, pressure, etc.) gives access to the set of excess properties (osmotic coefficient, heat capacity, density, etc.). Expressions of the heat capacity and density for a system containing an undefined number of both neutral and ionic solute species are described by Lach et al.17 In this study, the equations for the apparent heat capacity (Cp,ϕ/J·mol−1·K−1) and apparent volume (Vϕ/cm3·mol−1) can be reduced to the following expressions:

+ 2 ∑ mc mCl−(BcCl− + (∑ mc zc)CcCl−) c

c

⎫

+

⎪

∑ ∑ mcmc′(2Φcc′ + mCl ψcc ′Cl )⎬ −

c

−

c′

⎪

⎭

(3)

where ww is the mass of water, R is the ideal gas constant (J·mol−1·K−1), T is the absolute temperature (K), I is the ionic strength (mol·kg−1), b is the universal Pitzer parameter (equal to 1.2 kg1/2·mol−1/2), mi is the molality of species i (mol·kg−1), with i = c for cations, and Aϕ is the Debye−Hückel parameter computed according to eq 4. Aϕ =

1 (2πN0ρw )1/2 (e 2 /(εkT ))3/2 3

(4)

N0 is Avogadro’s number, ρw, the density of water (g·cm−3), is computed with the IAPWS formulation,41 e is the elementary charge, k is the Boltzmann constant, and ε, the dielectric constant, is computed using Bradley and Pitzer’s equation.42 The BcCl− parameters are obtained from the adjustable (1) parameters β(0) cCl− and βcCl−.

⎛ RT 2 ⎞⎧ AJ ln(1 + bI1/2) ⎟⎟⎨I 2 + ⎜⎜ b ⎝ ∑i mi ⎠⎩ RT ⎪

⎪

∑i mi

J J − + (∑ m z )C − 2 ∑ mc mCl−(BcCl c c cCl −) c

⎫

(5)

where α is a Pitzer parameter equal to 2 kg ·mol is a function expressed by 1/2

−1/2

−

and g(x)

⎪

∑ ∑ mcmc′(2ΦccJ′ + mCl ψccJ′Cl )⎬ −

c

(6)

Vϕ =

The CcCl− third virial coefficient is independent of ionic strength but is dependent on the adjustable parameter CϕcCl−: ϕ −/(2 CcCl− = CcCl |zczCl−| )

∑i miC p0, i

Cp , ϕ =

c

(0) (1) 1/2 BcCl− = βcCl ) − + β − g (α I cCl

g (x) = 2[1 − (1 + x) exp(−x)]/x 2

(8)

−

c′

∑i miV i0

⎪

⎭

(9)

⎛ 10RT ⎞⎧ A ln(1 + bI1/2) ⎟⎟⎨I V + ⎜⎜ b ⎝ ∑i mi ⎠⎩ 10RT ⎪

⎪

∑i mi

V V − + (∑ m z )C + 2 ∑ mc mCl−(BcCl c c cCl −)

(7)

c

c

⎫

where zi is the charge of species i. Only these three parameters (β(0), β(1), and Cϕ) have to be considered for a fully dissociated binary system.

+

⎪

∑ ∑ mcmc′(2ΦVcc′ + mCl ψccV′Cl )⎬ −

c

c′

−

⎪

⎭

(10)

Figure 2. Experimental heat capacity as a function of temperature for different ionic strengths and solution compositions. Each symbol represents a solution (see Table 3 for details): shaded diamonds, S1; ○, S2; shaded squares, S3; shaded triangles, S4; ◇, S5; shaded circles, S6; □, S7; , model. 3564



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The standard properties, C0p,i and V0i , are computed using the revised-HKF model.19,20 The equations and parameters for the ionic species (Na+, K+, Ca2+, Mg2+, and Cl−) can be found in ref 44. The parameters AJ and AV are computed using the Aϕ parameters according to eqs 11 and 12. AJ = 8RT

∂Aϕ

+ 4RT 2

∂T

AV = −40RT

Y = a1 + a 2T + a3T 2 + a4T 3 + a5/T + a6 ln T + a 7 /(T − 263) + a8/(680 − T ) + a 9 /(T − 227) (15)

The expression of the temperature dependence of interaction parameters is described in section 4.2. The heat capacity (cp) or the density (ρ) of solution (ρ) can be calculated from the apparent properties (Cp,ϕ or Vϕ) and the water properties (cp,w or ρw) according to the following equations:17

∂ 2Aϕ ∂T 2

(11)

∂Aϕ (12)

∂P

In the same way, every parameter with superscript J or V is obtained by YJ =

cp =

Cp , ϕ ∑i ≠ w mi + 1000cp , w 1000 + ∑i ≠ w miMi

2

2 ∂Y ∂Y + 2 T ∂T ∂T

ρ=

(13)

V

Y = (∂Y /∂P)

(14)

(16)

(1000 + ∑i ≠ w miMi) (Vϕ ∑i ≠ w mi + 106(ρw )−1)

(17) 17

where Y, a temperature-dependent interaction parameter, is defined by

Equations 3−17 are used in the PhreeSCALE software developed on the basis of the geochemical code PHREEQC.45 PhreeSCALE can calculate the heat capacity and the density of

Table 4. Experimental and Modelled Density ρ at Different Temperatures T and Pressures pa

Table 5. Experimental and Modelled Heat Capacity cp at Different Temperatures T and at Pressure p = 0.16 MPaa

Solution

T/K

p/MPa

S1-2 S1-2 S1-1 S1-2 S1-2 S2-2 S2-2 S2-2 S2-2 S2-2 S3-2 S3-2 S3-1 S3-2 S3-2 S4-2 S4-2 S4-1 S4-2 S4-2 S5-2 S5-2 S5-1 S5-2 S5-2 S6-2 S6-2 S6-1 S6-2 S6-2 S7-2 S7-2 S7-2 S7-2 S7-2

278.19 293.18 313.13 333.12 353.2 278.19 293.18 313.13 333.12 353.2 278.19 293.18 313.16 333.12 353.2 278.19 293.18 313.14 333.12 353.2 278.19 293.18 313.14 333.12 353.2 278.19 293.18 313.14 333.12 353.2 278.19 293.18 313.13 333.12 353.2

0.101 0.101 0.101 0.137 0.167 0.101 0.101 0.101 0.137 0.167 0.101 0.101 0.101 0.137 0.167 0.101 0.101 0.101 0.137 0.167 0.101 0.101 0.101 0.137 0.167 0.101 0.101 0.101 0.137 0.167 0.101 0.101 0.101 0.137 0.167

ρexpb /g·cm−3 1.120191 1.115334 1.106580 1.097503 1.086130 1.141503 1.134955 1.124886 1.114573 1.102096 1.043234 1.040112 1.032845 1.023860 1.012388 1.116863 1.111685 1.103718 1.094504 1.083192 1.170882 1.164063 1.153808 1.142902 1.130546 1.175360 1.168025 1.157586 1.146534 1.133777 1.176952 1.169667 1.158549 1.147914 1.135104

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

6 5 7 7 6 5 5 6 5 6 3 6 5 6 5 5 6 6 6 5 6 6 6 4 6 6 7 6 6 5 6 4 5 4 6

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6

ρmodc /g·cm−3 1.119718 1.114105 1.106418 1.096786 1.085215 1.142076 1.134264 1.124902 1.114631 1.102782 1.043130 1.039773 1.033114 1.023719 1.012135 1.115783 1.110464 1.102728 1.093513 1.082919 1.171153 1.162815 1.153334 1.142936 1.130565 1.175885 1.167013 1.157204 1.146699 1.134458 1.177458 1.168449 1.158524 1.147996 1.135820

Solution

T/K

cpexpb/J·g−1·K−1

cpmodc/J·g−1·K−1

S1-3 S1-3 S1-3 S1-3 S1-3 S2-3 S2-3 S2-3 S2-3 S2-3 S3-3 S3-3 S3-3 S3-3 S3-3 S4-3 S4-3 S4-3 S4-3 S4-3 S5-3 S5-3 S5-3 S5-3 S5-3 S6-3 S6-3 S6-3 S6-3 S6-3 S7-3 S7-3 S7-3 S7-3 S7-3

278.15 293.15 313.15 333.15 353.15 278.15 293.15 313.15 333.15 353.15 278.15 293.15 313.15 333.15 353.15 278.15 293.15 313.15 333.15 353.15 278.15 293.15 313.15 333.15 353.15 278.15 293.15 313.15 333.15 353.15 278.15 293.15 313.15 333.15 353.15

3.39 3.42 3.44 3.46 3.48 3.41 3.43 3.44 3.45 3.45 3.86 3.88 3.89 3.90 3.92 3.42 3.45 3.48 3.49 3.50 3.26 3.28 3.29 3.30 3.30 3.31 3.33 3.34 3.34 3.34 3.32 3.34 3.34 3.34 3.34

3.42 3.43 3.44 3.46 3.47 3.45 3.44 3.45 3.46 3.46 3.88 3.88 3.89 3.90 3.91 3.46 3.48 3.49 3.50 3.51 3.30 3.29 3.30 3.31 3.32 3.35 3.34 3.34 3.35 3.35 3.37 3.35 3.36 3.36 3.35

a

Standard uncertainties u are u(T) = 0.1 K u(p) = 0.001 MPa and u(cp) = 0.01 J·g−1·K−1. bExperimental values obtained in this study. c Modeling data obtained with the interaction parameters determined in this study.

a

Standard uncertainties u are u(T) = 0.02 K and u(p) = 0.001 MPa. b Experimental values obtained in this study. cModeling data obtained with the volumetric interaction parameters determined in this study. 3565



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brines in consistency with the Pitzer formalism. To develop PhreeSCALE, the sources of the PHREEQC code were also

modified to implement all the equations needed to calculate standard water and solute properties. Details about these

Figure 3. Heat capacity cp of NaCl aqueous solutions as a function of the concentration at (a) T = 278.15 K, (b) T = 298.15 K, (c) T = 338.15 K, and (d) T = 473.15 up to the solubility limit. Δ, ref 61; ◇, ref 60; ○, ref 48; □, ref 68; ●, ref 52; , model; - - -, solubility limit (NaCl).

Figure 4. Osmotic coefficient ϕ of NaCl aqueous solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 298.15 K, (c) T = 373.15 K, and (d) T = 473.15 K up to the solubility limit. ○, ref 70; ◇, ref 9; □, ref 64; Δ, ref 65; ●, ref 67; , model; - - -, solubility limit (NaCl). 3566



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temperatures (between 278.15 and 353.15 K) were studied. The details on the solution compositions are reported in Table 3. The experimental data are plotted as a function of temperature in Figures 1 and 2, for the experimental density and heat capacity, respectively. The experimental uncertainties are reported in Tables 4 and 5. As expected, for a given solution, its density decreases while its heat capacity increases when increasing the temperature. Despite the seven solutions having a different composition, they can be classified into three groups according to their ionic strength. The ionic strength of solutions S1, S2, and S4 is close to 4 mol·kg−1. That of solutions S5, S6, and S7 is 5.1 mol·kg−1. S3 is the only solution to have an ionic strength of 1.3 mol·kg−1. These groups can be clearly observed on

equations are given in ref 17. The syntax for running PhreeSCALE is inherited from PHREEQC. However, several new output keywords have been created corresponding to each property (Table 2). They can be called in the SELECTED_OUTPUT block (declared in the input file) to write the corresponding properties in a selected output file. The format of the PhreeSCALE thermodynamic database has also been slightly modified compared to the classical PHREEQC databases released with the code: it contains the HKF parameters for ionic species and the modified temperature dependence of interaction parameters (eq 15).

4. RESULTS AND DISCUSSION 4.1. Experimental data. For each thermodynamic property (density and heat capacity) seven solutions at five different

Figure 6. Osmotic coefficient ϕ of CaCl2 solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 298.15 K, and (c) T = 348.15 K up to the solubility limit. Δ, ref 50; ◇, ref 12; , model; - - -, solubility limit (CaCl2·6H2O at 273.15 and 298.15 K, and CaCl2· 2H2O at 348.15 K).

Figure 5. Heat capacity cp of CaCl2 solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 298.15 K, and (c) T = 348.15 K up to the solubility limit. ○, ref 47; ◇, ref 48; □, ref 35; Δ, ref 49; , model; - - -, solubility limit (CaCl2·6H2O at 273.15 and 298.15 K, and CaCl2·2H2O at 348.15 K). 3567



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Figures 1 and 2 even though we observe some discrepancies. For instance, the density values for solutions S5, S6, and S7 are close whereas the experimental values of S2 are not so close to those of S1 and S4. Similarly for an ionic strength of 5.1 mol·kg−1, the experimental values of the heat capacity of S5 are shifted from those of S6 and S7. So, even if it is right to state that solution densities (respectively heat capacity) increase as ionic strength increases (respectively decreases), the ionic strength value does not define a density or heat capacity value: the composition must be taken into account. For similar composition typology, densities increase as the solution’s ionic strength increases. Indeed, when the main aqueous species come from the same salt (NaCl as major salt for example), density is closely related to ionic strength but almost independent of the

detailed composition of the solution, as can be seen when comparing solutions S5, S6, and S7, as well as S1 and S4. The

Figure 7. Heat capacity cp of MgCl2 solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 298.15 K, and (c) T = 333.15 K up to the solubility limit. ▲, ref 52; ○, ref 53; □, ref 48; ◇, ref 49; ●, ref 54; , model; - - -, solubility limit (MgCl2·6H2O).

Figure 8. Osmotic coefficient ϕ of MgCl2 solutions as a function of the concentration at (a) T = 298.15 K, (b) T = 323.15 K, (c) T = 353.15 K, and (d) T = 373.15 K up to the solubility limit. ○, ref 10; ▲, ref 55; ◇, ref 56; □, ref 57; Δ, ref 8; , model; - - -, solubility limit (MgCl2·6H2O). 3568


+

1

1.432348 × 10 −4.856717 × 10−1 −1.837872 × 10−1 2.673756 × 101 −7.415596 −3.305313 3.685043 × 101 −2.046575 × 102 −1.217600 × 101 −4.715317 × 102 5.477530 × 103 9.370024 × 101

a1 (−) −3

5.58360 × 10 1.40962 × 10−3 −5.51034 × 10−5 1.00721 × 10−2 0 −1.29808 × 10−03 1.12000 × 10−2 1.03700 × 10−1 −5.27900 × 10−3 −2.80282 × 10−1 3.35092 5.51401 × 10−2

a2 (T)

b

−2.60685 × 10−6 0 5.00154 × 10−8 −3.75994 × 10−6 0 4.95714 × 10−7 −2.51700 × 10−6 −1.30000 × 10−4 2.04000 × 10−6 1.57456 × 10−4 −1.90689 × 10−3 −3.08203 × 10−5

a3 (T2) a5 (1/T) −4.20754 × 102 1.19990 × 102 1.07646 × 101 −7.58485 × 102 3.22893 × 102 9.12712 × 101 −1.04869 × 103 1.09337 × 104 3.15051 × 102 9.85237 × 103 −1.10337 × 105 −1.95686 × 103

a6 (ln T) −2.50237 0 2.75478 × 10−2 −4.70624 1.16439 5.86450 × 10−1 −6.33800 2.62400 × 101 2.19600 8.95516 × 101 −1.04569 × 103 −1.77600 × 101 −1 c

5.85167 × 10−4 2.90840 × 10−3 8.68478 × 10−4 0 0 0 1.14200 × 10−2 −3.49100 × 10−2 −7.07600 × 10−4 3.94963 × 10−3 −1.98547 × 10−2 −7.77655 × 10−4

a7 (1/(T − 263)) 4.42757 0 4.90448 × 10−2 0 0 0 −1.25300 × 101 7.07200 × 102 2.53800 −6.71209 × 102 8.60017 × 103 1.31648 × 102

a8 (1/(680 − T)) −1.71180 −4.27715 2.81591 × 10−1 0 −5.94578 0 −4.60700 × 10−1 −3.78100 × 101 −1.87300 × 10−1 −3.70522 0 6.93898 × 10−1

a9 (1/(T − 227))

d

d

d

d

d

d

11 11 11

d

d

d

ref

3569

a

+

−2

8.024378 × 10 1.0 × 10−1 −1.25783 × 10−3 2.9396106 × 10−1 1.62740 × 10−4 8.49685183 × 10−1 9.6626648 −1.6925646 × 102 −8.6439002 × 10−1 4.23942 × 10−3 1.11380 × 10−8 1.5198388 × 10−2

b0 (1/T) −4.418 × 10 −1.173 × 10−3 1.49988 × 10−7 −3.1049 × 10−3 1.313215 × 10−2 −1.087516 × 10−2 −1.1751875 × 10−1 2.0729146 1.0531622 × 10−2 3.0532 × 10−4 −1.3930 × 10−3 −1.9433 × 10−4

−4

b1 (−)

b2 (T) −7

6.14820 × 10 4.76690 × 10−6 1.04437 × 10−8 1.0924 × 10−5 −1.1370 × 10−4 5.2411 × 10−5 5.3417635 × 10−4 −9.4727630 × 10−3 −4.7948191 × 10−5 −1.9210 × 10−6 1.1470 × 10−5 7.7048 × 10−7

Parameters are valid up to the solubility limit except for the CaCl2 valid up to 6 mol·kg−1.

(0)V

(Na /Cl ) β β(1)V (Na+/Cl−) CϕV (Na+/Cl−) β(0)V (K+/Cl−) β(1)V (K+/Cl−) CϕV (K+/Cl−) β(0)v (Ca2+/Cl−) β(1)V (Ca2+/Cl−) CϕV (Ca2+/Cl−) β(0)V (Mg2+/Cl−) β(1)V (Mg2+/Cl−) CϕV (Mg2+/Cl−)

−

Table 7. Volumetric Interaction Parameters Determined in This Studya 0 −6.3730 × 10−9 0 −1.2690 × 10−8 3.2628 × 10−7 −1.1269 × 10−7 −1.0744785 × 10−6 1.9139561 × 10−5 9.6611244 × 10−8 2.9745 × 10−9 −2.2640 × 10−8 −9.7405 × 10−10

b3 (T2)

0 0 0 0 −3.0990 × 10−10 9.1144 × 10−11 8.0649972 × 10−10 −1.4425454 × 10−08 −7.2646937 × 10−11 0 0 0

b4 (T3)

273.15−373.15

273.15−398.15

273.15−398.15

273.15−413.15

T range validity/K

Parameter valid for T = (273.15 to 523.15) K and up to the solubility limit. Parameter valid for T = (273.15 to 373.15) K and up to 6 mol·kg . Parameter valid for T = (273.15 to 373.15) K and up to the solubility limit. dThis study

a

(0)

β (Na /Cl ) β(1) (Na+/Cl−)a Cϕ (Na+/Cl−)a β(0) (K+/Cl−)a β(1) (K+/Cl−)a Cϕ (K+/Cl−)a β(0) (Ca2+/Cl−)b β(1) (Ca2+/Cl−)b Cϕ (Ca2+/Cl−)b β(0) (Mg2+/Cl−)c β(1) (Mg2+/Cl−)c Cϕ (Mg2+/Cl−)c

− a

Table 6. Binary Interaction Parameters Used To Compute Osmotic Coefficient and Heat Capacity as a Function of Temperature

Journal of Chemical & Engineering Data Article



Article

same conclusions can be drawn with heat capacity measurements. 4.2. Binary interaction parameters. Two sets of interaction parameters are needed to calculate brine properties: (i) the Y-type interaction parameters, corresponding to any of the specific interaction parameters used to calculate the osmotic coefficient (i.e., β(0), β(1), Cϕ, ...), together with the YJ-type interaction parameters, which are related to temperature derivatives of the Y parameters and are used to calculate the apparent heat capacity (eq 9); (ii) the YV-type interaction parameters which are used to calculate solution density. The temperature dependence of the YJ-type interaction parameter is given by the following expression (applying eqs 13 and 15):

In this study, since the standard volumes were computed using HKF equations, new volumetric interaction parameters had to be optimized for the sake of consistency. To this end, the PhreeSCALE software was coupled with the parameter estimation software PEST,72 and YV parameters were fitted on the experimental density of the binary systems (compilation of Laliberté73 and Zaytsev and Aseyev49). The bi coefficients are reported in Table 7. Since all the experimental data were recorded under or close to atmospheric pressure, the influence of pressure on volumetric properties is not relevant. Consequently, b5 is taken equal to zero. In Table 8, the relative deviation (δY) and the standard deviations (σ) computed by eq 20 and 21 for the binary systems are reported. δY = 100 ×

Y J = (2/T )a 2 + 6a3 + 12a4T + a6 /T 2 + 526a 7 /T (T − 263)3 + 1360a8/T (680 − T )3 3

+ 454a 9 /T (T − 227)

σ=

(18)

Recently, Toner and Catling46 proposed a Pitzer parametrization for Na−K−Ca−Mg−Cl. But their parameters are valid from 298.15 to 298.15 K. The existing and determined ai coefficients of the four binary systems are reported in Table 6. The volumetric YV-type interaction parameters are temperature and pressure dependent and obey the following equation:71 Y V = b0 /T + b1 + b2T + b3T 2 + b4T 3 + b5Pbar

(Y exp − Y mod) Y exp

(20)

∑ (Y exp − Y mod)2 n

(21)

where n is the number of experimental data. The closer these values are to zero, the better the parametrization. In Figures 9 to 12, the density of the binary systems is plotted. The interaction parameters optimized satisfactorily represent the experimental data with a relative error lower than 0.6%. Finally, all the properties of the four binary systems (NaCl− H2O, KCl−H2O, CaCl2−H2O, and MgCl2−H2O), namely the osmotic coefficient, the heat capacity, and the density, are represented, so we can tackle the case of multiconstituent systems. 4.3. Ternary interaction parameters. Before studying the or V quinary system, the ternary interaction parameters ψJcc′Cl − and J or V θcc′ must be determined. To do this, the various ternary subsystems were studied using experimental data provided by the literature. Saluja et al.76 measured the heat capacity of the Na−Ca−Cl, Na−Mg−Cl, and Ca−Mg−Cl ternary systems between T = (298.15 and 373.15) K. For density, the compilation of Laliberté73 and the data of Kumar,77 Zezin et al.,78,79 Qiblawey et al.,80,81 Deng et al.,74 Yu et al.75,82 and Badarayani et al.83 were used. First, the calculations were performed with PhreeSCALE using only the binary interaction parameters. The relative error and standard deviations obtained are reported in Table 9 for the heat capacity and Table 10 for the density. The maximum deviation is obtained with the study of Yu et al.75 However, in the MgCl2−H2O binary system, Yu et al. measured a density value higher than those of other studies (Figure 12). The same observation can be made on the study of Deng et al. (Figure 10). In principle, it is possible to reduce the difference between experimental and modeling data by optimizing the ternary interaction parameters. However, Saluja et al.76 and Kumar and Atkinson84 showed that, if any, the ternary interaction

(19)

Table 8. Standard Deviation σ and Relative Error |δY| Obtained with Our Model on the Experimental Density Reported in the Literature for Binary Systems σ

Systems NaCl KCl CaCl2 MgCl2

2.91 1.89 1.84 1.05 3.67 2.65 7.19 1.82

× × × × × × × ×

10−4 10−3 10−4 10−3 10−3 10−3 10−2 10−3

|δY|/%

T /K

m / mol·kg−1

n

ref

0.26 0.56 0.127 0.247 0.3 0.495 0.28 0.56

273.15−413.15 273.15−363.15 278.15−398.15 273.15−363.15 288.15−399.15 273.15−363.15 273.15−371.82 273.15−363.15

0−6 0−6 0−6 0−4.71 0−6 0−2.85 0−6 0−2.31

869 244 655 202 328 228 383 171

73 49 73 49 73 49 73 49

3570



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Figure 10. Density ρ of KCl solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 308.15 K, (c) T = 333.15 K, and (d) T = 368.15 K. ○, ref 73; ▲, ref 49; ×, ref 74; , model.

Figure 9. Density ρ of NaCl solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 303.15 K, (c) T = 333.15 K, and (d) T = 368.15 K. ○, ref 73; ▲, ref 49; , model. 3571



Article

Figure 12. Density ρ of MgCl2 solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 303.15 K, (c) T = 348.15 K, and (d) T = 368.15 K. ○, ref 73; ▲, ref 49; ×, ref 75; , model.

Figure 11. Density ρ of CaCl2 solutions as a function of the concentration at (a) T = 273.15 K, (b) T = 303.15 K, (c) T = 333.15 K, and (d) T = 399.85 K. ○, ref 73; ▲, ref 49; , model. 3572



Article

systematically slightly overestimated. The solutions studied had ionic strength ranging between 1.3 and 5.1 mol·kg−1. To check the validity of our approach, heat capacity and density for NaCl-KCl-CaCl2−MgCl2−H2O solutions were also computed using literature data. For this system, in his compilation Laliberté73 reported two studies for density21,85 and one for heat capacity.21 Their measurements were at temperatures between 288.15 and 318.15 K and for a higher ionic strength (8.3 and 9.6 mol·kg−1). Badarayani et al.83 measured the density for the complex system at 298.15 K and for ionic strength of about 3 mol·kg−1. In each case, the relative deviation, δY (eq 20) and the standard deviation, σ (eq 21) were computed. They are reported in Table 9 and Table 10 for the heat capacity and the density, respectively. Badarayani et al.83 modeled these data using the Young’s mixing rule, and they reported a standard deviation equal to 0.6261 for the density, versus the 0.0124 we obtained in this study. Our maximum absolute relative deviation was 1.52%. Again for the complex system and using the Pitzer model too, Krumgalz et al.86 reported an average difference between experimental and calculated densities (ρexp − ρmod) of 0.000229 at the most, at 298.15 K, where the calculated values were always underestimated. In this study, we obtained a maximum difference of 0.00135 at T = 298.15 K. This maximum deviation corresponds to a relative error of 0.11% which remains very satisfactory. Part of the discrepancy can be related to the fact that Krumgalz et al.86 did not consider any volumetric ternary interaction parameters. This means that not only were the θ J and ψ J parameters ignored, but also the interaction parameters E θij(I), which are required in the Pitzer theory. Indeed, this

parameters are ionic strength dependent. Consequently, we did not introduce these supplementary parameters for the following reasons. First, our objective is to develop a unique set of interaction parameters to compute the heat capacity, the osmotic coefficient, and activity coefficients. If Y J is ionic strength dependent, then Y should be too, and this is incoherent with the Pitzer formalism. Second, as the ionic strength dependence is not an option of the classical Pitzer model, it is not used in PhreeSCALE. Finally, considering Y J = 0 implies that Y = a1 + a5/T, according to eqs 13 and 15. This is, for instance, in full agreement with the θ(T) and ψ(T) functions proposed by Greenberg and Møller11 for Na−K−Cl. 4.4. Modeling results and discussion on the quinary system. As seen in the previous section, the ternary interaction J or V J or V parameters ψcc′Cl are set to zero. Table 4 and Table 5 − and θcc′ report both the experimental and modeling data for density and heat capacity, respectively, for the seven solutions at the different temperatures. In Figure 1 and Figure 2, we also plotted the results obtained with the model. For each data point, we calculated the relative deviation δY using eq 20. The relative deviation for each density measurement from this study is reported in Figure 13a. The biggest difference was obtained for solution S1 at 293.15 K with a maximum value of 0.11%. Most of the other relative deviations are lower than 0.1%. The relative deviation for each heat capacity measurement is reported in Figure 13b. The biggest difference was obtained for the solution 6 at 278.15 K with a relative deviation of −1.35%, and the average (absolute) value was below 0.5%. It is noteworthy that the relative gap was systematically negative, implying that the calculated value was

Table 9. Standard Deviation σ and Relative Error |δY| Obtained with Our Model on the Experimental Heat Capacity Provided by the Literature σ

Systems NaCl-CaCl2 NaCl-MgCl2 CaCl2-MgCl2 KCl-NaCl-CaCl2-MgCl2 a

1.23 1.01 9.18 1.45 1.59

× × × × ×

−2

10 10−2 10−3 10−2 10−2

|δY|/%

T /K

I

n

ref

0.75 0.65 0.65 0.8 1.12

298.15−373.15 298.15−373.15 298.15−373.15 288.15−318.15 278.15−353.15

3−5 3−5 3−5 8.3−9.6 1.3−5.1

72 72 72 73 35

76 76 76 73 a

This study.

Table 10. Standard deviation σ and relative error |δY| obtained with our model on the the experimental density provided by the literature σ

Systems NaCl-KCl NaCl-CaCl2

NaCl-MgCl2

KCl-MgCl2

KCl-CaCl2 CaCl2-MgCl2 KCl-CaCl2-MgCl2 KCl-NaCl-CaCl2-MgCl2

a

5.41 1.28 9.00 9.43 2.65 9.85 4.02 9.81 5.67 1.49 2.48 9.99 8.52 1.12 1.24 6.64

× × × × × × × × × × × × × × × ×

10−4 10−3 10−4 10−4 10−4 10−4 10−3 10−4 10−3 10−1 10−2 10−4 10−3 10−3 10−2 10−4

|δY|/%

T/K

I

n

ref

0.11 0.66 0.18 0.41 0.062 0.14 1.14 0.11 1.36 13.36 4.20 0.21 0.96 0.13 1.52 0.11

298.15 278.15−371.82 298.15 293.15−323.15 296.02−371.82 298.15 298.15−318.15 298.15 298.15 348.15 288.15−308.15 296.02−371.82 298.15 288.15−318.15 298.15 278.15−353.15

0.1−5.8 1−11 0.4−10 2−14 2.8−4.9 1.3−10 0.4−9.4 0.5−4.5 4.88−18 6.4−20.5 4.4−34.4 2.7−4.8 2.7−3 8.3−9.6 2.7−3 1.3−5.1

21 314 19 216 72 7 250 26 16 14 29 72 13 148 13 35

79 73 78 80 73 78 81 77 82 75 74 73 83 73 83 a

This study. 3573



Article

parameters were set to zero. The adjustable parameters values were optimized by fitting, with the combined use of PhreeSCALE and PEST software programs, and the available literature data on binary systems. The application of the model to calculations on ternary systems showed that the calculated values were close to the experimental ones without adding adjustable ternary interaction parameters. When we made the calculation for quinary system experimental data, the conclusion was identical. The comparison between experimental heat capacity and density measured in this study with the modeled values showed that the properties (heat capacity and density) of complex systems can be computed with the Pitzer model using only binary interaction parameters. Indeed, the hypothesis that ternary interactions are null gave results with a relative deviation less than 1%. The next step would be to perform the same type of measurements on wider ranges of concentrations and temperatures to check whether the hypothesis that ternary interaction parameters are not required still holds. Pressure could also be investigated to see the influence of this parameter on the experimental data, and to see if the interaction parameters determined at atmospheric pressure are able to represent data obtained at higher pressure. Taking pressure into account is of great interest for underground applications.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], Tel: +33 2 38 64 31 57. ORCID Figure 13. Relative deviation between the experimental value (this study) and the predicted value using PhreeSCALE: (a) for the density; (b) for the heat capacity. Each symbol corresponds to a solution (shaded diamonds, S1; ○, S2; shaded boxes, S3; shaded triangles, S4; ◇, S5; shaded circles, S6; □, S7), and each solution is represented by five values, each corresponding to a temperature, namely 278.15, 293.15, 313.15, 333.15, and 353.15 K from left to right.

Adeline Lach: 0000-0001-7651-8242 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was funded by the Research Department of BRGM (the French Geological Survey). Our thanks go to Karen M. Tkaczyk, who helped to improve the English of the manuscript.

parameter is only dependent on ion charges, total ionic strength and solvent properties, so that it is computed automatically in PhreeSCALE for the sake of consistency with the Pitzer theory. To compare our standard deviation with the values reported by Krumgalz et al.,86 this term can be ignored, which results in a slight improvement, with a maximum difference (ρexp − ρmod) reduced to 0.000735. Another part of the discrepancy is related to the fact that we used additional experimental data for binary systems, over a larger range of temperatures than Krumgalz et al.,86 which introduces further variability. Laliberté,73,87 who proposed a mathematical model to compute heat capacity and density in complex systems, reported a standard deviation of 0.00433 for the heat capacity and 0.91 for the density versus 0.0145 and 0.001123 in this study.

■

REFERENCES

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5. CONCLUSION This work focused on chloride solutions because they are representative of most natural brines. First, we added to the set of experimental data by measuring with high accuracy the density and heat capacity of complex chloride solutions, from 278.15 to 353.15 K and up to 5 mol·kg−1 ionic strength. Then we studied the ability of the Pitzer equations to predict two physical properties of importance to various scientific and industrial applications: heat capacity and density. The numerical model was parametrized using the binary interaction parameters only as adjustable parameters. For that, the standard properties were calculated using the HKF equations and ternary interactions 3574



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