Viscosity Calculation for Binary Mixtures of Organic ... - ACS Publications

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Viscosity Calculation for Binary Mixtures of Organic Solvents Based on Modified Eyring-Modified Two-Suffix-Margules (MTSM) Model Xiaojie Wang, Xiaopo Wang,* and Jiajia Yin Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, Xi’an Jiaotong University, Xi’an 710049, China S Supporting Information *

ABSTRACT: Recently, the Eyring-modified two-suffix-margules (MTSM) model for calculating viscosities of ionic liquid mixtures was proposed by Atashrouz et al. (Atashrouz, S.; Zarghampour, M.; Abdolrahimi, S.; Pazuki, G.; Nasernejad, B. J. Chem. Eng. Data 2014, 59, 3691−3704). In this work, a simple modification of the Eyring-MTSM model was presented and applied to the viscosity calculation of binary mixtures of organic solvents. The accuracy of the modified model was assessed by comparing experimental viscosities at atmospheric pressure for 182 binary mixtures, and the overall average relative deviation (ARD) between the calculated results and literatures is 0.61%. In addition, the relationship between the parameters in the model and the boiling point of mixtures was established. Experimental viscosities containing 478 binary mixtures were used to evaluate the reliability of the relations, and good agreement was obtained between experimental and calculated values with ARD of 1.62%. Furthermore, the modified Eyring-MTSM model was extended to the viscosities of high pressures. The ARD is 1.61% for the high-pressure viscosities of 63 binary mixtures. The predictive ability of the model was also tested, and the model is suitable for the prediction of the homologous mixtures.

1. INTRODUCTION Organic solvents, including alkanes, aromatics, alcohols, esters, and ethers, are widely used in industrial production and daily life. Viscosity is a fundamental property used to describe the flow resistance of substances. The viscosity of binary mixtures of organic solvents is particularly important for the study of flow systems such as crude oil extraction and transportation, nanofluid science, food engineering, and alternative fuel development. On the other hand, the viscosity of the mixtures is beneficial for understanding the intermolecular interactions. As it is impractical and extremely time-consuming to provide the viscosity data of any composition at given conditions via the experimental method, many schemes have been proposed to establish reliable viscosity model for the binary mixtures. Generally, the schemes for viscosity calculation of liquid mixtures can be summarized as empirical or semitheoretical models. Excellent reviews are available in the literature, as described by Mehrotra et al.1 and Poling et al.2 Empirical methods include the equation of Grunberg and Nissan,3 Herráez,4 Haghbakhsh,5 Saini,6 and Haj-Kacem et al.7 Semitheoretical models are usually based on certain theories, for example, hard sphere theory, square well theory, the principle © XXXX American Chemical Society

of corresponding state, free-volume theory, and Eyring’s absolute rate theory. The Eyring’s absolute rate theory combined with equation of state (EOS) or excess Gibbs energy model, i.e. Eyring-EOS,8 Eyring-Wilson,9 Eyring-UNIQUAC,10 Eyring-COSMOSPACE,11 and Eyring-NRTL,12,13 is considered as one of the most widely accepted models to calculate the viscosities of liquid mixtures. In recent years, Atashrouz et al.14 proposed a viscosity model (the Eyring-MTSM model) based on the Eyring’s theory couple with modified Gibbs energy model (two-suffix-margules model). Calculated results from the model were compared to the experimental values of 122 ionic liquid mixtures, and good agreement was observed for 5512 experimental data points. Moreover, a comparison with the Eyring-NRTL model indicated that the accuracy of the Eyring-MTSM model is more competitive. In this work, a modified Eyring-MTSM model was established, and the new model was applied to calculate the Received: November 14, 2017 Accepted: April 3, 2018

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DOI: 10.1021/acs.jced.7b00994 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

pressure. Figure 1 shows that the interaction parameter τ12 and τ21 are approximately equal in absolute value with the opposite

viscosities of 660 binary mixtures at atmospheric pressure and 63 binary mixtures at elevated pressures, and the results are discussed.

2. THEORY BACKGROUND Based on Eyring’s absolute reaction rate theory,15 the calculative equation for the viscosities of binary liquid mixtures can be expressed as follows16 ln(ηmix ) = x1ln(η1) + x 2 ln(η2) +

GE RT

(1)

The third term of the above equation is a nonideal term, which represents the interaction between the components as well as the effects of molecular size and shape. Among different excess Gibbs free energy models, the two-suffix-margules model is the simplest. For the binary system, the two-suffix-margules model can be expressed as2 G E = AX12X 21

Figure 1. Interaction parameters for different binary systems. Methylcyclohexane/2,2,4-trimethylpentane:18 ■ (τ12), □ (τ21); nundecane/1-pentanol:19 ● (τ12), ○ (τ21); diethyl carbonate/1hexane:20 ▲ (τ12), △ (τ21); anisole/1-decane:21 ▼ (τ12), ▽ (τ21); ethanol/1-heptanol:22 ⧫ (τ12), ◇ (τ21); butyl acetate/2-propanol:23 ◀ (τ12), ◁ (τ21); phenetole/1-pentanol:24 ▶ (τ12), ▷ (τ21); ethyl benzoate/butyl vinyl ether:25 ★ (τ12), ☆ (τ21).

(2)

where A is the coefficient which is related to temperature. Atashrouz et al.14 proposed a viscosity model combined with Eyring’s theory and two-suffix-margules model.17 The model is called Eyring-MTSM: ln(ηmix ) = x1ln(η1) + x 2 ln(η2) + α12 x1x 2G12G21 (x1G12 + x 2)(x 2G21 + x1)

(3)

⎛ τ ⎞ G12 = exp⎜ − 12 ⎟ , τ12 = g12 − g22 ⎝ RT ⎠

(4)

⎛ τ ⎞ G21 = exp⎜ − 21 ⎟ , τ21 = g21 − g11 ⎝ RT ⎠

sign for the same system at isothermal condition. Therefore, we assume: τ12 = ΔU , τ21 = −ΔU

When applying eqs 6 and 7 to eqs 3−5, a modified EyringMTSM model can be obtained as follows, α ⎞ ⎛ ln(ηmix ) = x1ln(η1) + x 2 ln(η2) + ⎜α0 + 1 ⎟ ⎝ T⎠ x1x 2

(5)

where gij represents the potential energy parameter among different components. α12, τ12, and τ21 are the interaction parameters of the binary system. In Eyring-MTSM model, α12 was considered as temperature-dependent: α12 = α0 + α1/T

(7)

(x exp(− ) + x )(x exp( ) + x ) ΔU RT

1

2

2

ΔU RT

1

(8)

The above model has only three parameters (α0, α1, and ΔU), and ΔU is independent of the temperature. To evaluate its performance, the model was applied to the viscosities of 182 binary mixtures. The correlating results are summarized in Table S3 in the Supporting Information. For a certain of binary system, the parameters are obtained by fitting the literature experimental data, and the objective function is

(6)

α0 and α1 were determined from the viscosities of the mixtures.

3. MODIFIED EYRING-MTSM METHOD Based on the work of Atashrouz et al.,14 we established a modified Eyring-MTSM model to improve its predictive ability. We first collected the viscosities of a variety of binary mixtures from the published literatures. The database comprised of 660 binary mixtures (containing hydrocarbons + hydrocarbons, hydrocarbons + alcohols, hydrocarbons + esters, hydrocarbons + ethers, alcohols + alcohols, alcohols + esters, alcohols + ethers, and esters + ethers) with a total of 29 146 experimental data points at atmospheric pressure and 63 binary mixtures (including carbon dioxide + alkanes, carbon dioxide + alcohols, hydrocarbons + hydrocarbons, alkanes + alcohols, and alcohols + alcohols) with 9051 data points covering the temperature range from 215 to 473 K and pressures up to 510 MPa. Basic information for each binary system is available in Tables S1 and S2 in the Supporting Information. 3.1. Modified Model at Atmospheric Pressure. The interaction parameters τ12 and τ21 in eqs 4 and 5 were calculated from the viscosities of binary mixtures at atmospheric

obj =

1 N

N

∑

ηi lit − ηical

i=1

ηi lit

(9)

The deviations between the calculated results and literature data, including the average relative deviation (ARD) and the maximum relative deviation (MRD), are given in Table S3. In addition, for comparison purposes, Table S3 also lists the deviations of the original Eyring-MTSM model and EyringNRTL model with the literature. The expressions of ARD and MRD are ARD/% =

100 N

N

∑ i=1

ηi lit − ηical ηi lit

(10)

⎛ η lit − η cal MRD/% = max⎜⎜100 i lit i ηi ⎝ B

⎞ ⎟ ⎟ ⎠

(11)



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and 1-dodecane), n-undecane + 1-alcohols19,27 (1-propanol, 1butanol, 1-pentanol, 1-hexanol, 1-heptanol, 1-octanol, 1-nonanol, and 1-decanol), methyl benzoate + aromatic hydrocarbons28 (ethylbenzene, p-xylene, m-xylene, and o-xylene), anisole + n-alkanes21,29 (n-hexane, n-heptane, n-octane, nnonane, n-decane, n-dodecane, and n-tetradecane), ethanol + 1alkanols30 (1-butanol, 1-pentanol, 1-heptanol, 1-octanol, 1nonanol, and 1-decanol), dimethyl carbonate + 1-alkanols31,32 (ethanol, 1-propanol, 1-butanol, 1-pentanol, 1-hexanol, and 1octanol), and methyl tert-butyl ether + 1-alkanols33(ethanol, 1propanol, 1-butanol, 1-pentanol, and 1-hexanol). We found that, for each homologous mixtures, the parameters (α0, α1, and ΔU) in the modified model exhibit regular trends with the corresponding binary mixtures boiling point Tb‑eq. Figure 3 shows the variation of the parameters with Tb‑eq, and the relationship between them can be expressed as

The 182 binary mixtures studied herein can be divided into 8 kinds of systems, that is, hydrocarbon/hydrocarbon system (40), hydrocarbon/alcohol system (32), hydrocarbon/ester system (34), hydrocarbon/ether system (7), alcohol/alcohol system (18), alcohol/ester system (34), alcohol/ethers system (5), and ester/ether system (12). Figure 2 shows the ARD of

Y = Ci + Bi Tb ‐ eq

Y represents the parameters (α0, α1, and ΔU), and Ci and Bi are coefficients. Thus, when eq 14 and eq 8 are combined, the modified Eyring-MTSM model can be converted to be predictive. However, it should be pointed out that the applicability and scope of the predictive model is only suitable for the corresponding homologous mixtures. Consequently, 117 homologous mixtures (including 478 binary mixtures), as listed in Table S4 of the Supporting Information, were used to further assess the reliability of the predictive model. The mixtures can be divided into seven kinds of systems, which are hydrocarbon/hydrocarbon system (82), hydrocarbon/alcohol system (139), hydrocarbon/ester system (69), hydrocarbon/ether system (25), alcohol/alcohol system (53), alcohol/ester system (82), and alcohol/ether system (28). In the Supporting Information, Table S4 gives the obtained parameters for each of the homologous mixtures. In addition, the ARD and the MRD are also given. The ARD for each system is shown in Figure 4. It can be seen that the ARD is lower than 2.2% for all the systems, and the overall ARD is 1.62%. Statistical results, as plotted in Figure 5, show that the number of MRD values which is less than 10% for the studied mixtures is dominant. 3.3. Model at High Pressures. In this work, we further extend the modified model to high-pressure viscosities of 63 binary mixtures. The studied mixtures include carbon dioxide/ alkanes, carbon dioxide/alcohols, hydrocarbons/hydrocarbons, alkanes/alcohols, and alcohols/alcohols. Results show that, for binary systems of carbon dioxide (or methane, ethane) with hydrocarbons and carbon dioxide with alcohols, the form of the parameters (α12 and ΔU) can be expressed as

Figure 2. ARD between the calculated values from the modified model and the literature data. (A) hydrocarbon/hydrocarbon, (B) hydrocarbon/alcohol, (C) hydrocarbon/ester, (D) hydrocarbon/ether, (E) alcohol/alcohol, (F) alcohol/ester, (G) alcohol/ether, (H) ester/ether.

calculated values from eq 8 with literature data for each system. As depicted in Figure 2, the ARD is about 1.3% for hydrocarbon/alcohol system, and for other systems, the ARD is lower than 0.55%. From Table S3, we can see that, for most of the mixtures, the MRD is less than 5%. The overall ARD for the model of this work, the original Eyring-MTSM model, and Eyring-NRTL model is 0.61, 0.54, and 0.68%, respectively. It indicates that the modified model has a similar accuracy with other models; however, the new model has only three parameters. 3.2. Predictive Model at Atmospheric Pressure. The major drawback of the modified Eyring-MTSM model and the original Eyring-MTSM model is only correlative, and the parameters of the model are obtained from experimental data. Therefore, the development of a model to predict the viscosities of homologous mixtures would be an important improvement. In this work, the boiling point (Tb‑eq) of the binary mixtures was introduced and calculated by a volume fraction linear mixing rule of the boiling point of the pure components. That is, Tb − eq = ϕ1Tb1 + ϕ2Tb2

ϕi =

Vi V1 + V2

(14)

(12)

α12 = α0 + α1/T +α2/p

(15)

ΔU = d0 + d1p

(16)

where, α0, α1, α2, d0, and d1 are the coefficients, and p is the pressure in MPa. For the other binary systems studied in this work, eq 6 and 16 are used. All the parameters are obtained from the high pressure experimental viscosity data. The parameters of the model, the average relative deviation, and the maximum relative deviation are reported in Table S5 of the Supporting Information. The overall ARD between the literature data and calculated results is 1.61% for all the mixtures. Figure 6 shows the ARD values for each series of mixtures. It is obvious that the ARD is relatively high for the binary systems of carbon dioxide (methane or ethane) with

(13)

where Tb1 and Tb2 represent the boiling point of component 1 and component 2, respectively. V1 and V2 are the molar volume of the two substances at 298.15 K or neighboring temperature when the density data are not available at 298.15 K. ϕ1 and ϕ2 are the volume fraction. Consequently, we studied the viscosities of several homologous mixtures, including methylcyclohexane + 1alkanes26 (1-hexane, 1-heptane, 1-octane, 1-nonane, 1-decane, C



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Figure 4. ARD between the calculated values and the literature for different systems. (A) Hydrocarbon/hydrocarbon, (B) hydrocarbon/ alcohol, (C) hydrocarbon/ester, (D) hydrocarbon/ether, (E) alcohol/ alcohol, (F) alcohol/ester, and (G) alcohol/ether.

Figure 5. MRD number distribution for the studied mixtures.

Figure 3. Variation of parameters of eq 14 with boiling point: (a) α0 ∼ Tb‑eq, (b) α1 ∼ Tb‑eq, and (c) ΔU ∼ Tb‑eq. ■, Methylcyclohexane/1alkane; □, n-undecane/1-alcohol; ●, methyl benzoate/aromatic hydrocarbon; ○, anisole/1-alkane; ▲, ethanol/1-alkanol; △, dimethyl carbonate/1-alkanol; ★, methyl tert-butyl ether/1-alkanol. Solid line represents the fitting results.

alkanes or hydrocarbons. The reason is probably due to the asymmetrical behavior of those systems. Also, for each mixture, Figure 7 indicates that the number of MRD values that are lower than 10% exceed 80%, illustrating the reliability of the new model.

Figure 6. ARD between the calculated values and experimental viscosities under high pressure for different systems. (A) Carbon dioxide/alkanes, (B) carbon dioxide/alcohols, (C) methane/hydrocarbons, (D) ethane/alkanes, (E) hydrocarbons/hydrocarbons, (F) alkanes/alcohols, and (G) alcohols/alcohols.

4. VALIDATION AND COMPARISON OF THE MODIFIED MODEL In this section, the above modified model was used to predict the viscosities of other substances of homologous mixtures to validate its reliability. Therefore, we collected 259 viscosity data points of seven binary mixtures at atmospheric pressure from literature; the mixtures include methylcyclohexane (1) + pentane (2),34 cyclohexane (1) + propyl acetate (2),35 methyl tert-butyl ether (1) + 1-hexane (2),36 methyl tert-butyl ether (1)

+ 1-heptanol (2),37 anisole (1) + 1-butanol (2),38 anisole (1) + 1-pentanol (2),38 and ethanol (1) + 1-propanol (2) systems.39 It should be noted that the seven mixtures were not involved in the model fitting. Figures 8 and 9 manifest the viscosity predictive results. Good agreement can be observed between the calculated results and the literature data. Detailed D



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Figure 7. MRD number distribution for the studied mixtures at high pressures. Figure 10. Viscosity prediction results for methane/1-decane binary mixtures at different temperatures and 75 MPa. □, 303.15 K; ○, 323.15 K; △, 353.15 K; ▽, 393.15 K.

work. For the binary mixtures, the GN equation can be expressed as ln ηmix = x1ln η1 + x 2 ln η2 + x1x 2G12

(17)

where G12 is the binary interaction parameter. Here, the G12 is calculated using the method of group contribution presented by Isdale41 and Poling et al.2 At 298 K, G12(298) is calculated by G12(298) =

∑ Δ1 − ∑ Δ2 + W

(18)

where the terms ∑Δ1 and ∑Δ2 are the sum of the group contributions in the components 1 and 2, respectively. The term W can be calculated from

Figure 8. Viscosity predictive results for methylcyclohexane (1) + pentane (□)34 at 293.15 K, cyclohexane (1) + propyl acetate (○)35 at 308.15 K, methyl tert-butyl ether (1) + 1-hexane (△)36 at 303.15 K.

W=

(0.3161)(N1 − N2)2 − (0.1188)(N1 − N2) N1 + N2

(19)

Once G12(298) was calculated, the values of G12(T) at desired temperature can be given by G12(T ) = 1 − [1 − G12(298)]

573 − T 275

(20)

Table S6 also gives the deviations between the predictive results of GN model and literature data. For methylcyclohexane/1-pentane and cyclohexane/propyl acetate systems, it is obvious that there is a larger discrepancy; the ARD are 13.01 and 19.22%, respectively. Other systems such as methyl tertbutyl ether with 1-hexane or 1-heptanol or anisole with 1butanol or 1-pentanol have acceptable results. However, the predictive accuracy of GN model is still higher than that of the model of this work. It can be shown that the viscosity model obtained in this work is suitable for the viscosity prediction of homologous mixtures.

Figure 9. Viscosity predictive results for methyl tert-butyl ether (1) + 1-heptanol (□)37 at 308.15 K, anisole (1) + 1-butanol (○)38 at 303.15 K, anisole (1) + 1-pentanol (△)38 at 303.15 K, and ethanol (1) + 1propanol (◇) systems39 at 323.15 K.

comparison results are given in Table S6 of the Supporting Information, and the overall ARD is 2.65%. For the viscosities under high pressures, Table S7 of the Supporting Information gives the calculated results of three binary mixtures. The overall ARD is 2.35%. Figure 10 shows the comparison between the predicted results and the literature data40 at different temperatures and at 75 MPa for methane/1decane binary mixtures. It can be reasonably concluded that the new viscosity model can meet the expected accuracy requirements. In addition, the Grunberg−Nissan (GN) equation3 was chosen for comparison with the new predictive model of this

5. CONCLUSIONS A modified Eyring-MTSM viscosity model was given in this work to calculate the viscosities of binary mixtures. The parameters of the new model are associated with the boiling point of mixtures or pressure (for high-pressure viscosities), and the calculated viscosities for 660 binary mixtures at atmospheric pressure and 63 binary mixtures at elevated pressures were compared with the experimental data from the literature. The results indicate that the overall ARD is less than 2% for all the cases. Furthermore, the model predictive ability for the homologous mixtures was verified by the experimental E



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values that are not involved in the fitting, and the overall ARD between the predicted and literature values is within 2.65%.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b00994. Basic information on the binary mixtures and the calculated results from the improved Eyring-MTSM viscosity model (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; Fax: 86-29-82668789; Tel: 86-29-82668210. ORCID

Xiaopo Wang: 0000-0002-5550-2193 Funding

This research was supported by the National Natural Science Foundation of China (Grant 51476129). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge the anonymous reviewers for the valuable comments.

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REFERENCES

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