Modeling the Dynamic Viscosity of Ionic Solutions - ACS Publications

Correlation pubs.acs.org/IECR

Modeling the Dynamic Viscosity of Ionic Solutions Aimee Ruiz-Llamas and Ricardo Macías-Salinas* ESIQIE, Departamento de Ingeniería Química, Instituto Politécnico Nacional, Zacatenco, México D.F. 07738, Mexico S Supporting Information *

ABSTRACT: In the present work, an Eyring-theory model based on concepts of excess Gibbs energy of activation of the viscous flow has been developed for the accurate correlation or prediction of the dynamic viscosity of ionic solutions: inorganic salt (electrolyte) + solvent and organic salt (ionic liquid) + solvent. For the excess Gibbs energy of activation (GEX,≠), both thermal and mechanical contributions to the viscous flow were considered. Accordingly, a thermal GEX,≠ term was described by mixing rules of the Redlich−Kister-type, whereas the mechanical GEX,≠ term was computed from a simple cubic equation of state in an attempt to overall represent the main molecular interactions (between the ionic species and the solvent) affecting viscosity. The resulting model was successfully validated during the representation of experimental dynamic viscosities of various nonaqueous and aqueous ionic solutions within wide ranges of temperature and composition (or salt molality).

■

INTRODUCTION In general, ionic solutions comprise those liquid mixtures containing either inorganic or organic ionic species plus a solvent that could be water or an organic compound. The viscosity of ionic solutions plays a very important role in a number of geophysical and engineering applications; its precise knowledge is therefore paramount. Some examples of these applications include the transport of geothermal fluids through wells and ducts, the modeling of geothermal reservoirs, the design of both mass- and heat-transfer equipment, and the simulation of petroleum reservoirs, among others. A significant amount of experimental data have been published in the literature dealing with the dynamic viscosity of binary ionic solutions containing either strong or weak electrolytes, mostly at atmospheric pressure and aqueous conditions.1,2 On the other hand, during the last 12 years, a number of experimental works have increasingly appeared regarding viscosity measurements of aqueous and nonaqueous ionic solutions containing ionic liquids (ILs).3−10 Unlike inorganic salts, ILs belong to a particular class of organic salts that behave as liquids below 100 °C. They usually comprise a large organic cation in combination with an organic or inorganic anion of smaller size. The various combinations of cations and anions conveniently allow tailoring IL properties for specific engineering applications. As far as viscosity models for ionic solutions are concerned, many modeling efforts have been reported in the literature regarding the viscosity of electrolyte solutions. Most of these models are essentially empirical or semiempirical in nature and apply to aqueous and mixed-solvent electrolyte systems.11−18 As a matter of fact, Laliberté16 published a comprehensive review on the available approaches for modeling the viscosity of electrolyte solutions. In general, these modeling efforts range from simple viscosity correlations that are only applicable to single solutes in dilute solutions (e.g., the Jones−Dole equation19) to more comprehensive models that are not only valid for concentrated solutions, but they are also able to handle mixed-solute or mixed-solvent electrolyte solutions.12,14,15,18 In fact, a survey of modern viscosity models for electrolyte solutions shows that the © 2015 American Chemical Society

Eyring’s absolute rate theory has been up to now the most utilized modeling framework. In regards to available viscosity models for liquid mixtures containing an IL, to the best of our knowledge, very few modeling efforts have appeared so far in the literature. Wang et al.20 seem to be the first investigators to propose a viscosity model of this type based on the Eyring’s absolute rate theory in combination with a well-known excess Gibbs free energy model (universal quasichemical (UNIQUAC) and nonrandom two-liquid (NRTL)). The application of their model yielded satisfactory results in the correlation of liquid viscosities of various nonaqueous mixtures containing diverse ILs. Later, Wang and Anderko,21 using their own previously developed thermodynamic model coupled with transport property models, presented the viscosity modeling of a single aqueous solution containing an IL, which in turn was treated as a dissociable species. Their viscosity correlation results were reasonable by using the chemical speciation of the dissolved IL estimated from their own thermodynamic model. Very recently, Bajić et al.22 compared the performance of various modeling approaches in both correlating and predicting the viscosity of 10 binary mixtures containing an imidazolium-based IL and an organic solvent. In doing so, the authors considered the Seddon equation,23 the Grunberg−Nissan model,24 the three- and four-body McAllister models, and four viscosity models based on the Eyring’s theory incorporating activity coefficient methods25 such as analytical solution of groups (ASOG), universal functional activity coefficient (UNIFAC), NRTL, and UNIQUAC. According to the authors, the three-parameter models (four-body McAllister and Eyring-NRTL models) yielded the best correlating performances, whereas the UNIFAC-VISCO and ASOG-VISCO group contribution methods gave comparable predictive results using a new set of group interaction parameters regressed by the same authors. Received: Revised: Accepted: Published: 7169

May 4, 2015 June 15, 2015 June 22, 2015 June 22, 2015 DOI: 10.1021/acs.iecr.5b01664 Ind. Eng. Chem. Res. 2015, 54, 7169−7179

Correlation

Industrial & Engineering Chemistry Research

potentials) that are responsible for the GEX,≠ of the viscous flow. In this context, the term (GEX/RT), which is highly temperature dependent, represents the thermal contribution to GEX,≠ of the viscous flow, whereas the term (PVEX/RT), being mostly pressure dependent, stands for the mechanical contribution to the activated state of the viscous flow. Accordingly, eq 6 can be also expressed as

Although the application of the aforementioned approaches usually yields satisfactory results, they still present some drawbacks. Practically, the majority of the available viscosity models for ionic solutions are only valid at atmospheric pressure; further, they are not capable of representing (at least qualitatively) complex viscosity-composition behaviors that some binary ionic solutions may exhibit (e.g., the occurrences of both viscosity maximums and minimums over the whole composition range). On the basis of the capabilities and limitations of the viscosity models for ionic solutions described earlier, in this paper, we developed an unified modeling approach for the viscosity of ionic solutions (containing either an electrolyte or an IL) under the basis of the Eyring’s theory26 coupled with a suitable expression for the excess Gibbs energy of activation to properly correlate and predict the dynamic viscosity of aqueous and nonaqueous ionic solutions over wide ranges of temperature, pressure, and composition. Further, the predictive capabilities of the present approach were partly confirmed during the representation of experimental viscosities of selected ionic solutions at pressures much higher than 1 bar.

ηm =

G≠,EX,thermal GEX ≈− RT RT G≠,EX,mechanical PV EX P ⎛ ⎜Vm − ≈ = RT RT RT ⎜⎝

(ηV )ID

(1)

+

(2)

n

am =

⎛ GEX (ηV )ID PV EX ⎞ exp⎜ − + ⎟ Vm RT ⎠ ⎝ RT

P ⎛ ⎜Vm − RT ⎜⎝

(9)

n

⎞⎞

i=1

⎠⎠

∑ xiVi 0⎟⎟⎟⎟

(10)

n

∑ ∑ xixj i=1 j=1

(4)

aiaj (1 − ki , j) (12)

n

bm =

∑ xibi i=1

(5)

(13)

As a matter of fact, the energy interaction parameter (ki,j = kj,i) was set equal to zero in eq 12 in all viscosity calculations performed here. Models for the Thermal Contribution to GEX,≠. As mentioned earlier, the second term of the exponential argument of eq 10, namely GEX/RT, is the one that introduces the primary effect of temperature on GEX,≠. This term is also thought to be mainly responsible in capturing the nonideal viscosity behavior; accordingly, a suitable compositional dependence of this term should be devised to properly represent the maximums and

The last two expressions are incorporated into the present viscosity model: ηm =

⎠

where δ1 = 1, δ2 = 0 for SRK, and δ1 = 1 + √2, δ2 = 1 − √2 for PR. Mixture parameters (am and bm) appearing in the above equation were computed by using classical mixing rules of the van der Waals type:

(3)

From well-known thermodynamic relation, AEX can be expressed as follows: AEX = GEX − PV EX

i=1

Prior to application of the above equation, viscosity values of the pure components (ηi0) should be available at the temperature and pressure of interest; this makes the present viscosity model quite interpolative. On the other hand, the volumetric properties needed in eq 10, that is, Vi0 and Vm, were conveniently estimated via the use of two simple cubic equations of state: Soave28 (SRK) and Peng−Robinson29 (PR), knowing T, P, and the liquid composition of the mixture constituents xi. The general form of these equations for mixtures is given by am RT − P= Vm − bm (Vm + δ1b)(Vm + δ2bm) (11)

where σ is a proportionality factor. In an attempt to introduce pressure effects on the value of GEX,≠, a more versatile thermodynamic potential such as the excess Helmholtz free energy (AEX) can be used in place of GEX (assuming σ = 1): GEX, ≠ = −AEX

⎞

⎛ n 1 GEX ηm = exp⎜⎜∑ xi ln(ηi 0Vi 0) − Vm RT ⎝ i=1

In the two equations above, ηm is the solution viscosity, Vm is the molar volume of the mixture, and xi is the liquid composition of component i. On the other hand, ηi0 and Vi0 stand for the viscosity and molar volume of the pure component, respectively. The excess activation free energy of flow (GEX,≠) in eq 1 depends on temperature, pressure, and composition. Wei and Rowley27 related this quantity to the excess Gibbs free energy of the mixture (GEX) through GEX, ≠ = −σGEX

n

∑ xiVi 0⎟⎟

Upon substitution of eqs 2, 8, and 9 into eq 7, the final form of the present viscosity model is obtained as follows:

where (ηV)ID is the kinematic viscosity of an ideal solution given by ⎡ n ⎤ = exp⎢∑ xi ln(ηi 0Vi 0)⎥ ⎢⎣ i = 1 ⎥⎦

(8)

and

MODELING APPROACH The present model is based on the following Eyring’s viscosity expression for liquid mixtures: ⎛ GEX, ≠ ⎞ (ηV )ID exp⎜ ⎟ Vm ⎝ RT ⎠

(7)

where

■

ηm =

⎛ G≠,EX,thermal (ηV )ID G≠,EX,mechanical ⎞ exp⎜ + ⎟ Vm RT RT ⎝ ⎠

(6)

The form of the exponential argument appearing in the above equation should be regarded as an attempt to roughly identify the main intermolecular forces (in terms of thermodynamic 7170

DOI: 10.1021/acs.iecr.5b01664 Ind. Eng. Chem. Res. 2015, 54, 7169−7179

Correlation


mixtures comprising a salt + solvent, an acid + solvent, and an IL + solvent. Accordingly, Table 1 lists the ionic solutions

minimums of the viscosity-composition function usually exhibited by highly nonideal binary systems. In contrast, the mechanical effects on GEX,≠, introduced by the third term of the exponential argument of eq 10, may be significant particularly at elevated pressures; however, they do not determine the shape of the viscosity-composition function over the whole composition range (as long as the value of VEX is computed from the equations of state used in this work). A literature search for the most suitable mixing rule for GEX/RT reveals that the Gering’s expression15 is able to describe both maximums and minimums in viscosity at any temperature. For a binary mixture, his GEX expression is given by ⎛ T ⎞5 GEX = x1x 2A(1 + 2x1B)(1 + 2x 2C)⎜ ref ⎟ ⎝T ⎠ RT

Table 1. Experimental Dynamic Viscosities of Selected Ionic Solutions ionic solution Salts NaCl + H2O KBr + H2O LiI + H2O CaCl2 + H2O Na2SO4 + H2O Acids acetic acid + H2O HNO3 + H2O H2SO4 + H2O formic acid + acetone Ionic Liquids [hmim][BF4] + H2O [hmim][Br] + H2O [emim][BF4] + H2O [bmim] [NO3] + 1-propanol

(14)

The three mixing parameters (A, B, and C) are necessary to properly represent the complex viscosity behavior that some binary mixtures may exhibit; Tref is a reference temperature (equal to 298.15 K). As stated by Gering,15 the fifth power applied to the ratio Tref/T is consistent with previous expressions developed by Planck to obtain spectral energy distributions for a blackbody. Gering15 successfully applied eq 14 within the Eyring’s theory framework to estimate first the mixed-solvent (salt-free) viscosity and then used this value in another model to predict the viscosity of aqueous and nonaqueous electrolyte systems. In this work, we chose eq 14 as a suitable GEX/RT expression; however, we found out that to obtain accurate viscosity estimations over much wider temperature ranges, the fifth power appearing in eq 14 should be treated as another adjustable parameter, that is, ⎛ T ⎞D GEX = x1x 2A(1 + 2x1B)(1 + 2x 2C)⎜ ref ⎟ ⎝T ⎠ RT

maximum m [mol/kg]

source Lobo1 Isono31 Abdulagatov and Azizov32 Isono31 Abdulagatov et al.33

15−55 15−55 20−90

1 1 1−400

6.00 4.00 3.10

15−55 25−75

1 1

6.00 2.25

15−55 10−40 0−75 25−45

1 1 1 1

Lobo1 Lobo1 Lobo1 Irving34

15−45 20−60 15−45 10−60

1 1 1 1

Rilo et al.8 Li et al.9 Rilo et al.8 Mokhtarani et al.7

considered in this study along with their corresponding temperature and pressure ranges, the maximum molality at which the viscosity of the salt solutions was measured, and the source of the experimental viscosity data. As a matter of fact, the majority of the ionic solutions listed in Table 1 are of the aqueous type; their experimental viscosity data cover a moderate temperature range (0−90 °C) at atmospheric conditions (1 bar) with the only exception being the LiI solution, for which its highpressure viscosity data (up to 400 bar) served to verify the predictive capabilities of the present model. As mentioned earlier, prior to the application of the viscosity model developed here (eq 10), some pure-component quantities should become available. For example, critical properties (Tc and Pc) and acentric factors needed in the two cubic equations of state (CEoS) for each ionic species and the three solvents are given in Table 2. Regarding the critical properties and acentric factors of the four ILs, these were obtained from the most recent work of Valderrama et al.,36 who modified their own previously extended group contribution method37 for more consistent estimations of the aforementioned properties. Eq 10 also requires the viscosities of the mixture constituents (ηi0) at the temperature and pressure of interest. For most solutes (acids and ILs) and solvents considered in this work, their viscosities were either estimated using the NIST REFPROP version 7 or taken directly from the data sources given in Table 1. For the case of the five inorganic salts, however, their “pure component” viscosities cannot be determined directly; infinite dilution viscosities may be used instead, but from an experimental point of view, this is difficult to establish. In this work, it was rather reasonable to approximate the viscosity of the pure salt as being equal to 100-times the viscosity of the solvent at the given temperature and pressure. The present viscosity model (eq 10) was first applied to the correlation of experimental liquid viscosities of all ionic solutions at 1 bar and over their corresponding temperature ranges given in Table 1. In doing so, the mixing parameters (A, B, C, and D) of the thermal contribution (eq 15 or 17) were regressed by

(15)

Another mixing rule commonly used to represent highly nonideal liquid behavior can be derived from a Redlich−Kister expansion.30 As in the Gering’s equation, the corresponding three-parameter Redlich−Kister expansion of a binary mixture yields the following equation: GEX = x1x 2(A + B(x1 − x 2) + C(x1 − x 2)2 ) RT

T [°C] P [bar]

(16)

Temperature dependence in the above equation was introduced in a similar manner as in the Gering’s equation: ⎛ T ⎞D GEX = x1x 2(A + B(x1 − x 2) + C(x1 − x 2)2 )⎜ ref ⎟ ⎝T ⎠ RT (17)

Eqs 10, 11, and either eq 15 or 17 contain the complete description of the viscosity model proposed in this work. A quick inspection of these equations indicates that the present modeling approach contains only four adjustable parameters, namely the mixing parameters (A, B, C, and D) of the GEX/RT expression given by either the Gering equation (eq 15) or the Redlich− Kister equation (eq 17). These parameters were adjusted here to fit experimental viscosities of selected binary ionic solutions (containing either an electrolyte or an IL) within wide ranges of temperature and composition (or salt molality).

■

RESULTS AND DISCUSSION To assess the performance of the present modeling approach, three types of ionic solutions were considered, namely, binary 7171


Correlation


As a matter of fact, model calculations were carried out using two different thermal contributions (Gering or Redlich−Kister) in combination with two different mechanical contributions (SRK or PR), thus yielding four versions of the present modeling approach: Gering−SRK, Gering−PR, Redlich−Kister−SRK, and Redlich−Kister−PR. The correlating results obtained at 1 bar are summarized in Table 3 for the ionic solutions considered here. Table 3 shows, for each solute−solvent pair, the resulting percents of both absolute average deviations (%AAD) and statistical biases (%Bias) between calculated and experimental mixture viscosities for the four versions of the present model. Overall, as revealed by Table 3, the correlative abilities of eq 10 using the Redlich− Kister expression were better than using the Gering equation with an overall AAD value of 2.83% (as compared to Gering’s AAD value of 4.26%) based on a total of 752 data points. Furthermore, the two CEoS (SRK and PR) yielded nearly identical AAD and Bias values for the 13 ionic solutions when used in combination with either the Gering or the Redlich− Kister equations; presumably, the capability differences between the two CEoS vanished during the regression process of the viscosity data while being finally absorbed by the regressed parameters (A, B, C, and D) appearing in the thermal contribution. As a matter of fact, the Supporting Information gives the regressed values of the model parameters obtained at 1 bar using the Gering−SRK/PR and the Redlich−Kister−SRK/ PR approaches, respectively. Solutions Containing an Inorganic Salt. Unlike previously developed Eyring-based viscosity models for single electrolyte solutions,11,13 in which the solvent is treated as a continuum and the ions as charged hard spheres (the McMillan− Mayer framework), the present model does not take into account ion−ion interactions or electrostatic forces that are thought to be the main interactions responsible for the anomalous viscosity behavior of certain strong electrolyte solutions over the dilute region. Moreover, in a series of systematic studies, Good38−42 as

Table 2. Pure-Component Properties Needed as CEoS Inputs component Solvents watera acetonea 1-propanola Salts sodium chlorideb potassium bromideb lithium iodideb calcium chloridec sodium sulfatec Acids acetic acida nitric acidd sulfuric acida formic acida Ionic Liquids [hmim][BF4]e [hmim][Br]e [emim][BF4]e [bmim][NO3]e

Tc [K]

Pc [bar]

ω

647.096 508.0655 536.765

220.64 47.03 51.3

0.3443 0.308 0.615

3400.0 2697.31 2352.04 3709.4 2568.0

355.0 222.42 193.95 429.51 322.0

0.134 0.596 0.556 0.323 0.706

593.0 520.0 578.0 588.0

57.86 68.9 68.92 89.38

0.457 0.714 0.909 0.506

690.0 873.5 596.2 954.8

17.94 25.01 23.59 27.33

0.9625 0.5715 0.8087 0.6436

a

NIST Standard Reference Database 103b, TDE version 8 (2013). Yaws.35 cAspen One, Pure-Component Databank, version 2006.1. d DIPPR Diadem Professional Database, version 8.1.0 (2014). e Valderrama et al.36 b

performing a least-squares fit based on the Levenberg− Marquardt method. The minimization of the following objective function served for this purpose: N

min f =

2

∑ ⎡⎣1 − ηical /ηiexpt⎤⎦

(18)

i=1

ηexpt i

where N is the number of experimental points, whereas and ηcal i stand for the observed and calculated mixture viscosities. Table 3. Regression Results for All Ionic Solutions at 1 Bara Gering ionic solution Salts NaCl + H2O KBr + H2O LiI + H2O CaCl2 + H2O Na2SO4 + H2O Acids acetic acid + H2O HNO3 + H2O H2SO4 + H2O formic acid + acetone Ionic Liquids [hmim][BF4] + H2O [hmim][Br] + H2O [emim][BF4] + H2O [bmim][NO3] +1-propanol overall a

Redlich−Kister

N

%AAD SRK

%AAD PR

56 49 42 56 25 336

0.479 0.082 0.316 1.729 0.644 0.467

0.479 0.084 0.316 1.731 0.644 0.468

0.027 0.030 0.022 0.476 0.042 0.094

157 26 73 27 317

2.707 3.519 5.946 0.725 3.060

2.706 3.519 5.946 0.724 3.060

32 75 36 98 279 752

6.420 18.700 3.851 3.342 7.434 4.257

6.419 18.699 3.848 3.339 7.432 4.256

%Bias SRK

%Bias PR

%AAD SRK

%AAD PR

%Bias SRK

%Bias PR

0.027 0.031 0.023 0.477 0.042 0.095

0.479 0.082 0.316 1.729 0.644 0.467

0.479 0.084 0.316 1.731 0.644 0.468

0.027 0.030 0.022 0.476 0.042 0.094

0.027 0.031 0.023 0.477 0.042 0.095

1.082 0.044 0.967 0.283 0.786

1.081 0.044 0.967 0.282 0.786

0.821 3.519 5.946 0.725 2.126

0.821 3.519 5.946 0.724 2.126

0.209 0.044 0.967 0.283 0.354

0.208 0.044 0.967 0.282 0.353

1.755 10.696 1.019 −0.330 3.092 1.521

1.755 10.694 1.017 −0.328 3.092 1.521

3.952 9.446 3.758 3.342 4.651 2.830

3.951 9.443 3.755 3.339 4.649 2.830

0.048 2.790 0.392 −0.330 0.690 0.447

0.048 2.789 0.392 −0.328 0.691 0.448

expt N cal expt %AAD = (100/N) × Σi N= 1|1 −(ηcal i /ηi )|; %Bias = (100/N) × Σi = 1(1 −(ηi /ηi )).

7172


Correlation


Figure 1. Correlated and experimental viscosities for the NaCl+water system at 1 bar and at various temperatures. The regressed parameters in the Gering−SRK model are A = −0.751785, B = 15.2703, C = 0, and D = −1.58079.

well as Good and Ingham43 employed the Eyring’s theory to identify the main molecular interactions affecting fluidity (the reciprocal of viscosity) of a wide variety of binary electrolyte solutions. The authors found that ion−solvent interactions (ion solvation and other short-range forces) in comparison to ion− ion interactions were the most dominant effects on the fluidity of electrolyte solutions. Therefore, it turned out to be reasonable not to include a contribution term accounting for electrostatic forces in the present modeling approach. It can be seen from Table 3 that the ability of the present model in representing the experimental viscosities of the five salt solutions at 1 bar was remarkably good with an overall AAD value of 0.47% using either the Gering−SRK/PR approach or the Redlich−Kister−SRK/PR approach. To achieve these results, only parameters A, B, and D appearing in the thermal contribution (eq 15 or 17) were regressed; parameter C was set equal to zero. The resulting parameter values are given in the Supporting Information for the two modeling approaches. On the other hand, Figure 1 graphically depicts the comparison between experimental and calculated mixture viscosities (using the Gering−SRK approach) for the NaCl solution at seven temperatures (15, 20, 25, 30, 35, 45, and 55 °C). As shown by this figure, there is an excellent agreement between the calculated and observed viscosity values at all temperatures within a molality range of 0−6 mol/kg. The present approach also gives an accurate representation of the experimental viscosities in the dilute region (0 < m < 0.5 mol/kg), as demonstrated by Figure 1, panel b, which uses a logarithmic x-axis for better visualization over this region. Similarly, Figure 2 demonstrates the excellent job of the Redlich−Kister−SRK approach in correlating the “anomalous” experimental viscosity behavior of the KBr solution at seven temperatures (15, 20, 25, 30, 35, 45, and 55 °C). The viscosity behavior of the KBr solution is considered “anomalous” in that its viscosity-concentration function exhibits a minimum; in contrast, for other salt solutions such as NaCl+water, their viscosities

Figure 2. Representation of viscosities for the KBr+water system at 1 bar and at various temperatures. The regressed parameters in the RK−SRK model are A = −7.98254, B = −14.7266, C = 0, and D = 5.37041.

increase monotonically with concentration at any temperature (see Figure 1). Another challenging electrolyte solution is the aqueous CaCl2; although it exhibits monotonic behavior, its viscosity increases slightly up to a molality of about 1.0 and then starts increasing sharply. Accordingly, Figure 3 graphically shows the aforementioned viscosity behavior of the CaCl2 solution at seven temperatures (15, 20, 25, 30, 35, 45, and 55 °C). As revealed by this figure, the present model (Gering−PR approach) 7173


Correlation


H2SO4+water) and two more solutions comprising a weak acid (acetic acid+water and formic acid+acetone) served to test the capabilities of the present model in correlating their viscosities at 1 bar. For these particular acid solutions, both modeling approaches (Gering−SRK/PR and Redlich−Kister−SRK/PR) yielded identical AAD values except for the acetic acid+water solution, for which the Redlich−Kister equation was clearly superior to the Gering equation with an AAD value of 0.82% (almost three-times smaller than Gering’s AAD value of 2.71%). To obtain accurate representations of experimental viscosities for highly nonideal acid solutions such as the ones considered here, in particular the aqueous solutions containing acetic acid, HNO3, and H2SO4, it was necessary to adjust the four parameters (A, B, C, and D) appearing in the thermal contribution. The Supporting Information lists the values of these parameters for the Gering−SRK/PR and Redlich−Kister−SRK/PR approaches, respectively; note that for the formic acid+acetone solution, only parameters A, B, and D were regressed since this particular mixture exhibits moderate deviations from ideal viscosity behavior. Figure 5 depicts the typical nonideal viscosity behavior exhibited by the acetic acid+water mixture over the whole Figure 3. Correlated and experimental viscosities for the CaCl2+water system at 1 bar and at various temperatures. The regressed parameters in the Gering−PR model are A = −10.0359, B = 4.59419, C = 0, and D = 0.124082.

correctly captures the complex viscosity variation with molality and temperature for the CaCl2 solution. Lastly, Figure 4 depicts the percent of relative deviations of correlated viscosities (using Redlich−Kister−PR approach) to

Figure 5. Representation of viscosities for the acetic acid+water system at 1 bar and at various temperatures. The regressed parameters in the RK−SRK model are A = −4.22791, B = 0.638184, C = −1.79195, and D = 1.55316. Figure 4. Relative errors between correlated and experimental viscosities for all the salt solutions considered in this work.

composition range in which a viscosity maximum occurs at about equimolar conditions and at all temperatures (15, 20, 25, 30, 35, 45, and 55 °C); this peculiar but rather anomalous viscosity behavior is mainly attributed to hydrogen-bond formation. As shown by Figure 5, the present model (Redlich−Kister−SRK approach) provides a very good fit to experimental data over the entire range of composition, although it tends to underestimate the viscosity maximum as temperature decreases. The aqueous HNO3 solution is another binary mixture exhibiting not only a maximum, but also a strongly asymmetric behavior in its viscosity with respect to composition. Figure 6 shows the variation of viscosity with composition of the aforementioned solution at 10,

experiments versus molality for the five salt solutions based on a total of 336 data points. According to this figure, the majority of the model deviations fall within ±2% except for the CaCl2 solution, for which some deviations are as large as 5.3%. Figure 4 also shows that all model deviations are very well yet randomly distributed around zero mean, as evidenced by the low Bias value (0.095%) previously reported in Table 3. Solutions Containing an Acid. As shown in Table 3, two solutions containing a strong acid (HNO 3 +water and 7174


Correlation


Figure 6. Correlated and experimental viscosities for the HNO3+water system at 1 bar and at various temperatures. The regressed parameters in the Gering−PR model are A = −0.05713, B = 0.826845, C = 32.2945, and D = 0.90645.

20, and 40 °C where all the experimental maximums take place at an acid mole fraction of about 0.3. Figure 6 also reveals that the present modeling approach (Gering−PR) represents reasonably well the experimental data; however, it estimates the maximum at a higher composition (about 0.4 mole fraction) for the three isotherms. Lastly, the aqueous H2SO4 solution served to ultimately test the capabilities of the present model in representing unusual viscosity extrema. As depicted by Figure 7, this particular binary mixture experimentally exhibits both a maximum and a minimum in viscosity at four temperatures (0, 25, 50, and 75 °C). As seen in Figure 7, although the present model (Redlich−Kister−PR approach) qualitatively yields the occurrence of both maximum and minimums at all temperatures, it does not accurately represent such viscosity extrema particularly at low temperatures (0 and 25 °C). The highly complex viscosity behavior that this acid solution displays is indeed one of the most difficult to model; the results obtained from the present model in regards to this solution should be therefore considered quite acceptable. To support this conclusion, a brief performance comparison of the present model was made against other similar modeling efforts. The comprehensive model of Wang et al.14 is one of the very few modeling approaches so far reported in the literature dealing with the viscosity of fully miscible aqueous acids such as the H2SO4 solution considered here. For this particular aqueous system, the authors were able to accurately represent its complex viscosity behavior (both maximums and minimums were very well represented) but at the cost of adjusting 14 interaction parameters, ten more parameters than the present approach. Furthermore, their correlating results yielded an AAD value of 6.08%, quite comparable with the one obtained in this work (5.95%, as shown in Table 3), both of them using the same temperature range (0−75 °C). Solutions Containing an Ionic Liquid. Experimental evidence indicates that viscosities of both aqueous and

Figure 7. Representation of viscosities for the H2SO4+water system at 1 bar and at various temperatures. The regressed parameters in the RK−PR model are A = −5.7482, B = 3.47205, C = 1.99751, and D = 0.069808.

nonaqueous solutions containing an IL have been extensively measured and that the imidazolium-based ILs are the most preferred by many investigators. Accordingly, Table 3 gives the three aqueous solutions ([hmim][BF4]+water, [hmim][Br]+water, and [emim][BF4]+water) and one nonaqueous solution ([bmim][NO3]+1-propanol) chosen here for modeling purposes; note that all cations of the ILs are of the imidazolium type. It can be seen from Table 3 that both modeling approaches (Gering−SRK/PR and Redlich−Kister−SRK/PR) yielded comparable results for the [emim][BF4]+water and 7175


Correlation

Industrial & Engineering Chemistry Research [bmim][NO3]+1-propanol solutions with similar AAD values; however, for the other two solutions ([hmim][BF4]+water and [hmim][Br]+water), the Redlich−Kister−SRK/PR approach definitely provided much better viscosity correlations than the Gering−SRK/PR approach with AAD values nearly 50% less than the ones from Gering. ILs are in general highly structured molecules with large molar masses; therefore, solutions containing an IL, aside from possibly displaying association effects, may also exhibit strong molecular structure effects on their viscosity behavior. Furthermore, binary mixtures comprising an IL are usually considered highly asymmetric systems since the viscosity of the IL in its pure state is significantly high; for example, from two up to nearly four orders of magnitude larger than the solvent’s viscosity based on the solutions studied here. In this context, Figure 8 shows the experimental viscosity behavior of the ionic

Figure 9. Representation of viscosities for the [hmim][Br]+water system at 1 bar and at various temperatures. The regressed parameters in the RK−PR model are A = −5.67355, B = 6.39386, C = −8.80274, and D = 0.366947.

viscosity under these conditions is more challenging. As evidenced by Figure 9, the present model (Redlich−Kister−PR approach) captures reasonably well the complex viscosity behavior exhibited by the [hmim][Br]+water solution at all temperatures yielding a relatively high AAD value of 9.45%. However, for such a nonideal ionic system, the general fit should be considered quite well. Lastly, unlike the two aqueous systems described earlier, the nonaqueous solution considered here ([bmim][NO3]+1-propanol) exhibits only a moderate departure from ideal viscosity behavior. Hence, sufficiently accurate results were obtained by only regressing the model parameters A, B, and D (C was set equal to zero) thus yielding identical AAD values of 3.34% for both modeling approaches (Gering−SRK/PR and Redlich− Kister−SRK/PR). Figure 10 gives the graphical comparison between observed and correlated viscosity data for the [bmim][NO3]+1-propanol system at 10, 20, 30, 40, 50, and 60 °C. As seen in this figure, the present three-parameter model (Gering−SRK approach) provides a satisfactory fit to experimental data at all temperatures and over the entire composition range. Prediction of Viscosity Data at Elevated Pressures. Finally, what remains is to verify the capability of the present approach in predicting the experimental viscosities of a selected ionic system at high pressures. For this purpose, the Redlich− Kister−PR three-parameter model was applied to predict the viscosity behavior of the aqueous LiI solution at higher pressures (100 and 400 bar) using the same model parameters previously obtained here at 1 bar. The corresponding model parameters are A = 21.6879, B = −29.2896, C = 0, and D = −2.69916; they are also valid over a temperature range of 20−90 °C. Figure 11 shows the percent of relative deviations between experimental and predicted viscosities versus pressure for the aqueous LiI solution at a fixed molality (0.0159 m) and at six temperatures (20, 25, 40, 50, 70, and 90 °C). As revealed by this figure, model predictions

Figure 8. Correlated and experimental viscosities for the [hmim][BF4]+water system at 1 bar and at various temperatures. The regressed parameters in the RK−SRK model are A = −5.42174, B = 5.85865, C = −3.63492, and D = −0.169088.

system [hmim][BF4]+water at 1 bar and at 15, 25, 35, and 45 °C where the logarithm of viscosity varies almost linearly with IL composition from a molar fraction of about 0.4 up to pure IL. It is precisely within this composition region where the whole departure from ideal viscosity behavior takes place. As also depicted by this figure, the present model (Redlich− Kister−SRK approach) does a good job in correlating all the experimental viscosity isotherms with an AAD value of 3.95% (see Table 3). On the other hand, Figure 9 gives the variation of viscosity versus composition for the highest asymmetric mixture considered in this work: the [hmim][Br]+water solution at 20, 30, 40, 50, and 60 °C. Unlike the previous case, the shape of each viscosity isotherm of this particular mixture is even more complex over the entire composition range, likely due to the formation of hydrogen bonds between water and IL molecules as well as self-association and dissociation of the IL. Thus, modeling 7176


Correlation


Figure 12. Variation of thermal and mechanical GEX contributions with pressure for the LiI+water system at various temperatures.

Figure 10. Correlated and experimental viscosities for the [bmim][NO3]+1-propanol system at 1 bar and at various temperatures. The regressed parameters in the Gering−SRK model are A = −1.1087, B = −0.241267, C = 0, and D = −6.94024.

hand, all mechanical GEX contributions are very close to zero at atmospheric pressure and then start decreasing almost linearly with pressure; in fact, the largest mechanical GEX effects occur at the highest temperature and pressure; nevertheless, they are not even comparable to the thermal GEX effects, which represent the most dominant contribution for this particular mixture (note that all thermal GEX values plotted in Figure 12 have been previously divided by 100). Even though the mechanical term barely contributes to the total GEX,≠, it actually serves to safely predict the viscosity at elevated pressures as in the present case.

■

CONCLUSIONS A unified yet versatile modeling approach to represent the dynamic viscosities of ionic solutions is presented here. It makes use of the Eyring’s theory as a modeling framework coupled with a liquid GEX model (Gering or Redlich−Kister) along with a cubic equation of state (Soave or Peng−Robinson). The following conclusions can be drawn from this work: (1) The present approach unifies the calculation of the mixture viscosities for a broad variety of ionic solutions: inorganic salt+solvent, acid+solvent, and IL+solvent over wide temperature and composition ranges. (2) The two proposed thermal GEX equations (Gering or Redlich−Kister), in either their three- or four-parameter versions, correctly captured the highly complex viscosity behavior of practically all of the ionic solutions considered here, thus confirming the suitability of the mathematical expressions chosen for such a purpose. In this context, the Redlich−Kister equation yielded better results than the Gering approach. (3) Model predictions at pressures higher than 1 bar were also satisfactory thanks to the use of a mechanical GEX term (eq 9), which in turn was calculated from a well-known cubic equation of state (Soave or Peng−Robinson). This conclusion, however, is based solely on the present analysis of a single ionic system; namely, the aqueous LiI

Figure 11. Relative errors between predicted and experimental viscosities for the aqueous LiI solution at high pressures and at various temperatures.

mostly yielded low deviations that fall within ±1% and are very well distributed around zero mean with a Bias value of 0.086%. For a given temperature and pressure, each GEX term appearing in eq 7 distinctively contributes to the total value of the excess activation free energy of the viscous flow (GEX,≠). For example, Figure 12 gives the variation of both thermal and mechanical GEX contributions with pressure for the aqueous LiI solution at 0.0159 m, and at 25, 50, 70, and 90 °C. It can be seen from this figure that all thermal GEX contributions remain always positive and constant with pressure; they also increase linearly with temperature, thus indicating that the thermal GEX effects become more important as temperature increases. On the other 7177


Correlation

Industrial & Engineering Chemistry Research Superscripts

solution. Consequently, to confirm the extrapolation abilities of the present viscosity model at elevated pressures, several more ionic systems should be further considered. (4) Finally, the use of either the Soave or Peng−Robinson CEoS within the present Eyring’s formulation yielded almost identical results; their beauties and limitations during the calculation of molar volumes (of pure species and the mixture) were canceled out by being finally absorbed by the regression parameters appearing in the thermal GEX contribution.

■

■

REFERENCES

(1) Lobo, V. M. M. Handbook of Electrolyte Solutions; Elsevier Science: New York, 1989. (2) Zaytsev, I. D.; Aseyev, G. G. Properties of Aqueous Solutions of Electrolytes; CRC Press: Boca Raton, FL, 1992. (3) Heintz, A.; Klasen, D.; Lehmann, J. K. Excess Molar Volumes and Viscosities of Binary Mixtures of Methanol and the Ionic Liquid 4Methyl-N-butylpyridinium Tetrafluoroborate at 25, 40, and 50 °C. J. Solution Chem. 2002, 31, 467. (4) Wang, J. J.; Zhu, A. L.; Zhao, Y.; Zhuo, K. L. Excess Molar Volumes and Excess Logarithm Viscosities for Binary Mixtures of the Ionic Liquid 1-Butyl-3-methylimidazolium Hexafluorophosphate with Some Organic Compounds. J. Solution Chem. 2005, 34, 585. (5) Arce, A.; Rodil, E.; Soto, A. Physical and Excess Properties of Binary Mixtures of 1-Methyl-3-octylimidazolium Tetrafluoroborate, [omim][BF4], Ionic Liquid with Different Alcohols. J. Solution Chem. 2006, 35, 63. (6) Gonzalez, E. J.; Gonzalez, B.; Calvar, N.; Dominguez, A. Physical Properties of Binary Mixtures of the Ionic Liquid 1-Ethyl-3methylimidazolium Ethyl Sulfate with Several Alcohols at T = 298.15, 313.15, and 328.15 K and Atmospheric Pressure. J. Chem. Eng. Data 2007, 52, 1641. (7) Mokhtarani, B.; Sharifi, A.; Mortaheb, H.; Mirzaei, M.; Mafi, M.; Sadeghian, F. Density and Viscosity of 1-Butyl-3-methylimidazolium Nitrate with Ethanol, 1-Propanol, or 1-Butanol at Several Temperatures. J. Chem. Thermodyn. 2009, 41, 1432. (8) Rilo, E.; Pico, J.; García, G.; Segade, L.; Varela, L. M.; Cabeza, O. Electrical Conductivity and Viscosity of Aqueous Binary Mixtures of 1Alkyl-3-methylimidazolium Tetrafluoroborate at 4 Temperatures. J. Chem. Eng. Data 2010, 55, 639. (9) Li, J.-G.; Hu, Y.-F.; Sun, S.-F.; Liu, Y.-S.; Liu, Z.-C. Densities and Dynamic Viscosities of the Binary System (Water + 1-Hehyl-3methylimidazolium Bromide) at Different Temperatures. J. Chem. Thermodyn. 2010, 42, 901. (10) Tomida, D.; Kenmochi, S.; Qiao, K.; Bao, Q.; Yokoyama, C. Viscosity of Ionic Liquid Mixtures of 1-Alkyl-3-methylimidazolium Hexafluorophosphate + CO2. Fluid Phase Equilib. 2011, 307, 185. (11) Esteves, M. J. C.; Cardoso, M. J. E.; De, M.; Barcia, O. E. A Debye−Huckel Model for Calculating the Viscosity of Binary Strong Electrolyte Solutions. Ind. Eng. Chem. Res. 2001, 40, 5021. (12) Hu, Y.-F.; Lee, H. Prediction of Viscosity of Mixed Electrolyte Solutions Based on the Eyring’s Absolute Rate Theory and the Semiideal Hydration Model. Electrochim. Acta 2003, 48, 1789. (13) Jiang, J.; Sandler, S. I. A New Model for the Viscosity of Electrolyte Solutions. Ind. Eng. Chem. Res. 2003, 42, 6267. (14) Wang, P.; Anderko, A.; Young, R. D. Modeling Viscosity of Concentrated and Mixed-Solvent Electrolyte Systems. Fluid Phase Equilib. 2004, 226, 71. (15) Gering, K. L. Prediction of Electrolyte Viscosity for Aqueous and Nonaqueous Systems: Results from a Molecular Model Based on Ion Solvation and a Chemical Physics Framework. Electrochim. Acta 2006, 51, 3125. (16) Laliberté, M. Model for Calculating the Viscosity of Aqueous Solutions. J. Chem. Eng. Data 2007, 52, 321. (17) Sadeghi, R. Extension of the Electrolyte NRTL and Wilson Models for Correlation of Viscosity of Strong Electrolyte Solutions at Different Temperatures. Fluid Phase Equilib. 2007, 259, 157. (18) Zafarani-Moattar, M. T.; Majdan-Cegincara, R. Viscosity Modeling and Prediction of Aqueous Mixed Electrolyte Solutions. Ind. Eng. Chem. Res. 2009, 48, 5844.

ASSOCIATED CONTENT

S Supporting Information *

Regressed parameters for the Gering and Redlich−Kister models using two different mechanical contributions (SRK or PR). The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b01664.

■

cal = calculated value expt = experimental value EX = excess property 0 = pure species ≠ = activation state

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge the Instituto Politécnico Nacional and CONACyT for providing financial support for this work.

■

LIST OF SYMBOLS a = attractive parameter in the CEoS A = Helmholtz free energy, regression constant in thermal GEX term b = covolume parameter in the CEoS B = regression constant in thermal GEX term C = regression constant in thermal GEX term D = regression constant in thermal GEX term f = objective function G = Gibbs free energy k = interaction parameter in the CEoS m = molality n = number of components N = number of data points P = pressure R = universal gas constant T = temperature V = molar volume x = liquid mole fraction

Greek Letters

δ1, δ2 = CEoS-specific constants η = viscosity σ = proportionality factor in eq 3 ω = acentric factor Subscripts

c = critical property i = ith specie i,j = i−j pair interaction ID = ideal solution m = mixture property 7178


Correlation

Industrial & Engineering Chemistry Research (19) Jones, G.; Dole, M. The Viscosity of Aqueous Solutions of Strong Electrolytes with Special Reference to Barium Chloride. J. Am. Chem. Soc. 1929, 51, 2950. (20) Wang, Y.-G.; Chen, D.-X.; OuYang, X.-K. Viscosity Calculations for Ionic Liquid−Cosolvent Mixtures Based on Eyring’s Absolute Rate Theory and Activity Coefficient Models. J. Chem. Eng. Data 2010, 55, 4878. (21) Wang, P.; Anderko, A. Modeling Chemical Equilibria, Phase Behavior, and Transport Properties in Ionic Liquid Systems. Fluid Phase Equilib. 2011, 302, 74. (22) Bajik, D. M.; Serbanovic, S. P.; Zivkovic, E. M.; Jovanovic, J.; Kijevcanin, M. L. Prediction and Correlation of Viscosity of Binary Mixtures of Ionic Liquids. J. Mol. Liq. 2014, 197, 1. (23) Seddon, K. R.; Stark, A.; Torres, M. J. Influence of Chloride, Water, and Organic Solvents on the Physical Properties of Ionic Liquids. Pure Appl. Chem. 2000, 72, 2275. (24) Grunberg, L.; Nissan, A. H. Mixture Law for Viscosity. Nature 1949, 164, 799. (25) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (26) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (27) Wei, I. C.; Rowley, R. L. A Local Composition Model for Multicomponent Liquid Mixture Shear Viscosity. Chem. Eng. Sci. 1985, 40, 401. (28) Soave, G. Equilibrium Constants from a Modified Redlich− Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (29) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (30) Redlich, O.; Kister, A. T. Algebraic Representation of Thermodynamic Properties and the Classification of Solutions. Ind. Eng. Chem. 1948, 40, 345. (31) Isono, T. Density, Viscosity, and Electrolytic Conductivity of Concentrated Aqueous Electrolyte Solutions at Several Temperatures. Alkaline-Earth Chlorides, LaCl3, Na2SO4, NaNO3, NaBr, KNO3, KBr, and Cd(NO3)2. J. Chem. Eng. Data 1984, 29, 45. (32) Abdulagatov, I. M.; Azizov, N. D. Viscosities of Aqueous LiI Solutions at 293−525 K and 0.1−40 MPa. Thermochim. Acta 2005, 439, 8. (33) Abdulagatov, I. M.; Zeinalova, A.; Azizov, N. D. Viscosity of Aqueous Na2SO4 Solutions at Temperatures from 298 to 573 K and at Pressures up to 40 MPa. Fluid Phase Equilib. 2005, 227, 57. (34) Irving, J. D. Viscosities of Binary Liquid Mixtures: The Effectiveness of Mixture Equations; Natl. Eng. Lab. Rept. 63: East Kilbride, Glasgow, Scotland, 1977. (35) Yaws, C. L. Chemical Properties Handbook: Physical, Thermodynamic, Environmental, Transport, Safety, and Health Related Properties for Organic and Inorganic Chemical; McGraw-Hill: New York, 1999. (36) Valderrama, J. O.; Forero, L. A.; Rojas, R. E. Critical Properties and Normal Boiling Temperature of Ionic Liquids. Update and a New Consistency Test. Ind. Eng. Chem. Res. 2012, 51, 7838. (37) Valderrama, J. O.; Rojas, R. E. Critical Properties of Ionic Liquids. Revisited. Ind. Eng. Chem. Res. 2009, 48, 6890. (38) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of ElectrolytesI. Alkali-Metal and Ammonium Halides. Electrochim. Acta 1964, 9, 203. (39) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of ElectrolytesII. Alkaline Earth Chlorides, Cupric Chloride, and Ferric Chloride. Electrochim. Acta 1965, 10, 1. (40) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of ElectrolytesIII. Nitrates and Chlorates. Electrochim. Acta 1966, 11, 760. (41) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of ElectrolytesIV. Sulphates and Carbonates. Electrochim. Acta 1966, 11, 767. (42) Good, W. The Effect of Solute Concentration on Fluidity and Structure in Aqueous Solutions of ElectrolytesV. Strong Acids and Alkali-Metal Hydroxides. Electrochim. Acta 1967, 12, 1031.

(43) Good, W.; Ingham, D. B. Enthalpy−Entropy Relationships in the Fluid Kinetics of Aqueous Electrolyte Solutions Containing Positively Hydrated Ions. Electrochim. Acta 1975, 20, 57.

■

NOTE ADDED AFTER ASAP PUBLICATION This paper was originally posted ASAP on July 8, 2105. Minor corrections were made to the Introduction and footnote of Table 3. The paper was reposted on July 22, 2015.

7179


Modeling the Dynamic Viscosity of Ionic Solutions - ACS Publications

Recommend Documents