New Excess Gibbs Energy Equation for Modeling the Thermodynamic

ARTICLE pubs.acs.org/IECR

New Excess Gibbs Energy Equation for Modeling the Thermodynamic and Transport Properties of Polymer Solutions and Nanofluids at Different Temperatures Mohammed Taghi Zafarani-Moattar* and Roghayeh Majdan-Cegincara Physical Chemistry Department, University of Tabriz, Tabriz 51664, Iran ABSTRACT: A local composition model is developed for the representation of the excess Gibbs energy of polymer solutions. The model consists of two contributions due to the configurational entropy of mixing, represented by the Freed FloryHuggins relation, and to the enthalpic contribution, represented by local compositions through nonrandom factor. The model is applied to correlate the solvent activity of binary polymer solutions. The new excess Gibbs energy equation was used along with the absolute rate theory of Eyring for modeling the dynamic viscosity of binary polymer solutions in the entire concentration range at different temperatures considering different molar mass of polymers. The fitting quality of new model has favorably been compared with polymer-NRTL, segment-based-liquid-NRTL, polymer-Wilson, polymer-NRF and polymer-NRF-Wilson models. The validity of the proposed model is especially demonstrated for the whole range of polymer concentrations at different temperatures using different molar masses of polymers. The segment-based approach provides a more physically realistic model for large molecules when diffusion and flow are viewed to occur by a sequence of individual segment jumps into vacancies rather than jumps of the entire large molecule. Therefore, the correlation of viscosity values for nanofluids was also tested with the proposed Eyring-modified NRF model developed with respect to the segment-based approach. The performance of this model in the fitting of viscosity values of nanofluids are compared with the previously used liquid-NRTL model. Results show that this segment-based model is most valid in the fitting of viscosity values of nanofluids in the entire concentration range at different temperatures.

1. INTRODUCTION An understanding of the thermodynamics of the polymer solutions is important in practical applications such as polymerization, devolatilization, and the incorporation of plasticizers and other additives. Proper design and engineering of many polymer processes depend greatly upon accurate modeling of thermodynamic parameters. Knowledge of the thermodynamic and transport properties of polymer solutions is important for practical and theoretical purposes. These quantities provide invaluable data in polymer research, development, and engineering. Furthermore, the simultaneous investigation of viscosity and volume effects on mixing can be a powerful tool for the characterization of the intermolecular interactions present in these mixtures. Also, knowledge of the dependence of thermodynamic and transport properties of polymer solutions on composition is of great interest from a theoretical standpoint because it may lead to better understanding of the fundamental behavior of polymer solutions. There are two categories of models available for description of thermodynamic properties of polymer solutions: the excess Gibbs energy (Gex) models and the equation of state (EOS) models. Principal Gex models for phase equilibrium calculations of polymer solution are those of Flory1 and Huggins,2 who developed an expression based on lattice theory to describe the nonidealities of polymer solutions and of Edmond and Ogston,3 who modeled nonidealities with a truncated osmotic virial expansion based on McMillan Mayer theory.4 Local composition models such as UNIQUAC (universal quasi chemical),5 UNIFAC-FV (UNIQUAC functional group activity coefficient-free volume),6 NRTL (nonrandom twoliquid),7 NRF (nonrandom factor),8 Wilson,9 and NRF-Wilson10 r 2011 American Chemical Society

have also been used to describe the thermodynamics of polymer solutions. Recently viscosity values of binary polymer solutions have been correlated with segment-based liquid-NRTL,11 Wilson,12 NRF,13 and NRF-Wilson10 models based on the absolute rate theory of Eyring14 at only one temperature. Using excess Gibbs energy of these models in the correlation of viscosity values of binary solutions has been done with consideration of the close analogy between the thermodynamic excess Gibbs energy and excess Gibbs free energy of activation for flow similar to that suggested by Eyring et al.14 Close examination indicated that although the performance of Eyring-segment-based-liquid-NRTL,11 Eyring-polymer-Wilson,12 Eyring-polymer-NRF,13 and Eyringpolymer-NRF-Wilson10 models in the correlation of viscosity values of polymer solutions is good, the fitting procedures have been made at only one temperature and correlation of polymers with larger molar mass has only been tested with Eyringsegment-based-liquid-NRTL.11 For some systems the deviations between experimental and calculated viscosity values obtained by this model11 are larger than 20%. Nanofluids are composites considering solid nanoparticles with sizes varying generally from 1 to 100 nm dispersed in heat transfer liquids. These materials are routinely used in chemical production, manufacturing, power generation, transportation, and many other aspects of modern life. Viscosity is important in designing nanoflids for flow and heat transfer applications Received: August 18, 2010 Accepted: May 25, 2011 Revised: May 25, 2011 Published: May 25, 2011 8245

dx.doi.org/10.1021/ie200003c | Ind. Eng. Chem. Res. 2011, 50, 8245–8262

Industrial & Engineering Chemistry Research because the pressure drop and the resulting pumping power depend on the viscosity. As a nanofluid is a two-phase fluid, one may expect that it would have common features with solid liquid mixtures. Various models are widely used for prediction and correlation of the viscosity values of nanofluids without considering temperature dependency. Einstein’s viscosity model15 can be used for predicting the viscosity values of nanofluids with relatively low volume fraction of nanoparticle. Brinkman16 extended Einstein's formula for use at moderate particle concentration. KriegerDougherty17 proposed a model considering aggregation of nanoparticles in the high concentration ranges. Frenkel and Acrivos18 have also proposed a new model for predicting the viscosity of nanofluids for low concentrations of nanofluids without considering temperature dependency. An equation in the form of a Taylor series in terms of the volume fraction of nanoparticle was used by Lundgren19 in the correlation of viscosity values of nanofluids. Batchelor’s equation20 considered the effect of Brownian motion of particles on the bulk stress of an isotropic suspension of spherical particles. A simple expression was proposed by Kitano et al.21 to predict the viscosity of nanofluids. Graham22 proposed a generalization form of the Frenkel and Acrivos equation18 that agreed well with Einstein’s equation for a low value of volume fraction of nanoparticle. Chen et al.23 modified the KriegerDougherty equation17 by considering the changing of packing density with radial position. Masoumi et al.24 presented a new analytical model to predict nanofluid viscosity considering the Brownian motion of nanoparticles. A new dimensionless group model has also been used for determining the viscosity values of nanofluids at one temperature.25 Kole et al.26 explained the various correlations of nanofluid viscosities with temperature dependency proposed by some authors. Hosseini et al.27 used the Eyring-liquid-NRTL model28 for correlating the viscosity values of nanofluids considering the temperature and particle volume fraction impacts on viscosity values. It was found that the performance of the Eyring-NRTL model in the correlation of nanofluid viscosity at different temperatures is better than other equations proposed by Einstein,15 Brinkman,16 and Lundgren.19 Close examination of the performance of local composition models in the modeling of thermodynamic and transport properties of polymer solutions indicates that for some systems satisfactory results are not obtained using these models. Therefore, it is necessary to develop a new model for this purpose. Here, we proposed a new local composition model (modifiedNRF) for the excess Gibbs energy of polymer solutions using local cell theory and considering different reference state assumptions and correction terms. The model consists of the configurational entropy of mixing, represented by the Freed Flory Huggins29 relation, and the enthalpic contribution, represented by local compositions. The performance of the developed model has been tested using experimental vaporliquid equilibrium (VLE) data for a variety of polymer solutions in the entire concentration range with different molar masses of polymer considering temperature dependency. The fitting quality of the proposed model has favorably been compared with NRTL,7 NRF,8 Wilson,9 and NRF-Wilson10 models. The model presented in this work produced good results in the fitting of VLE data. Then, the new excess Gibbs energy relation has been used along with the Eyring absolute rate theory in the correlation of viscosity values of binary polymer solutions. The performance of the proposed model has been compared with that for Eyringpolymer-NRTL,7 Eyring-segment-based-liquid-NRTL,11 Eyringpolymer-Wilson,12 Eyring-polymer-NRF,13 and Eyring-polymer-

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NRF-Wilson10 models in the correlation of the viscosity data of binary polymer solutions. The segment-based approach provides a more physically realistic model for large molecules when diffusion and flow are viewed to occur by a sequence of individual segment jumps into vacancies rather than jumps of the entire large molecule. Therefore, we assumed that flowing of nanofluids occurs similar to flowing of segments in vacancies and this permitted us to use a new model for the correlation of viscosity data of nanofluids. The fitting quality of the new model was compared with the previously used Eyring-NRTL model.27

2. THEORETICAL FRAMEWORK Following Chen’s approach7 used in the development of the NRTL model for expression of the local physical interaction between the solvent and a segment of polymer chains, we assume the existence of two types of cells, depending on the central species. One of the cells has a solvent central molecule with segment and solvent molecules in the surrounding and the other cell has a polymer segment as the central species with other polymer segments and solvent molecules in the surrounding. Thus, g ex, mNRF g ex g ex ¼ xw w þ rp xp s RT RT RT

ð1Þ

where xw and xp are the mole fraction of solvent and polymer. The number of polymer segments, rp, approximates the ratio of the molar volume of the polymer and that of the solvent molecules. T is temperature and R is the universal constant of ex gases. gex s and gw represent respectively the contributions of the cells with central polymer segments s and central solvent molecules w to the excess Gibbs energy arising from short-range interactions that are defined as gsex ¼ gs gs0

ð2aÞ

gwex ¼ gw gw0

ð2bÞ

where gs and gw are residual Gibbs energies per mole of a cell with a central segment of polymer and solvent, respectively. g0s and g0w are reference state Gibbs energies for a cell with a central segment and solvent, respectively. The residual Gibbs energies are defined as gs ¼ Xws gws þ Xss gss

ð2cÞ

gw ¼ Xww gww þ Xsw gsw

ð2dÞ

where gji and gii are energies of interaction between ji and ii pairs of species, respectively. Xji and Xii are the effective local mole fractions of species j and i. Subscripts w and s represent the solvent and segment of polymer chain, respectively. For the two type of cells, and in our case, for the reference cells in the random case, g0s and g0w were defined as

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gs0 ¼

Xw gws þ Xs gss Xs þ Xw

ð2eÞ

gw0 ¼

Xw gww þ Xs gsw Xs þ Xw

ð2f Þ

dx.doi.org/10.1021/ie200003c |Ind. Eng. Chem. Res. 2011, 50, 8245–8262

Industrial & Engineering Chemistry Research

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where Xs ¼

xp xw þ rp xp

ð2gÞ

xw xw þ rp xp

ð2hÞ

Xw ¼

1 Xw þ Xs βsw, ww λw ¼ exp þ ωw Z

Γww ¼ βsw, ww

The nonrandomness of the mixture is represented by means of nonrandomness factor (NRF) formally similar to those defined by Haghtalab and Vera.30 Thus in general, for ij interactions Xij ¼ Xi Γij

ð3aÞ

Xlj ¼ Xl Γlj

Because in the systems containing mixed polymers and solvents, one can assume the existence of different types of local cells, we extended eq 5 for multicomponent systems and derived the following equation: X X X Xw λsw, w0w ð Xw Γw0w Xs Γw0w þ 1Þ X g ex, mNRF xw00 ¼ Xs þ Xw RT w00 s w0 w þ

XX w00

ð3bÞ

rp, w00 xp

p

ð3cÞ

Xij Xi ¼ βij, lj Xlj Xl

ð3dÞ

Γis ¼ P w0

g ex, mNRF Xw λw ð Xw Γww Xs Γww þ 1Þ ¼ xw Xs þ Xw RT Xs λs ðXs Γss Xw Γss 1Þ þ rp xp ð5aÞ Xs þ Xw where gss gws RT kss kws ¼ RT

ð5cÞ

βss, ws

ð6cÞ

λs ¼ exp þ ωs Z

ð5dÞ

s

ð6dÞ

ωij, kl ¼ ωij, lk ¼ ωji, kl ¼ ωji, lk ¼ ωkl, ij ¼ ωlk, ij ¼ ωkl, ji ¼ ωlk, ji ð6eÞ βij, lk

λij, lk þ ωij, lk ¼ exp Z

! ð6f Þ

where subscripts s and s0 represent the segments of polymer chain; w, w0 , and w00 show the solvent molecules. rp,w00 approximates the ratio of the molar volume of the polymer and corresponding solvent molecule, w00 . The activity coefficient of component i in a polymer solution, γi, can be considered as the sum of two contributions: conf ig

ln γi ¼ ln γi

þ ln γmNRF i

ð7Þ

The expression for the activity coefficient of the solvent due to , is represented by following Freed configurational entropy, γconfig w FloryHuggins relation:29 ig ln γconf w

ð5bÞ

gsw gww λw ¼ λsw, ww ¼ RT ksw kww ωw ¼ ωsw, ww ¼ RT βss, ws Γss ¼ Xw þ Xs βss, ws

1 P Xw0 βiw0, iw þ Xs βsi, iw

s0

λij, kl ¼ λij, lk ¼ λji, kl ¼ λji, lk ¼ λkl, ij ¼ λlk, ij ¼ λkl, ji ¼ λlk, ji

λs ¼ λss, ws ¼ ωs ¼ ωss, ws

ð6bÞ

Γiw ¼ P

ð4bÞ

Here, Z is the nonrandom factor which was set to 8 in this work. Considering the above assumptions and following the same procedure as our previous work8 used in obtaining the excess Gibbs energy expression for polymer solutions, we obtained the mNRF equation for polymer solutions as

Xs þ Xw

s

βis, ws P Xw0 βw0i, ws þ Xs0 βis0, ws

w0

kji kli RT

s0

ð6aÞ

The local composition factor, βji,li, is modified compared to the original NRF model8 by use of an empirical energy correction term ωji,li as gji gli þ ωji, li βji, li ¼ exp ð4aÞ ZRT ωji, li ¼

X X X Xs λs0s, ws ðXs Γs0s Xw Γs0s 1Þ w

from which Xij Xi Γij ¼ Xlj Xl Γlj

ð5eÞ

! !2 Xw 1 1 ¼ ln X s 2 rp 2 X s rp þ R 1 þ 1 rp rp xw ð8Þ

where R is the nanrandomness factor. Different values for R varied in the range 0.10.4 have been used in the NRTL,7 NRF,8 Wilson,9 and NRF-Wilson10 models. We also tested the reliability of the mNRF model with different values for R in the range 0.10.4 for some systems and found that a better quality of fitting with the mNRF model is obtained with R = 0.2. Therefore, the value of R was set to 0.2 in this work. The expression for the 8247



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activity coefficient of the solvent due to enthalpic contribution can be derived from the eq 5. The derived equation for the activity coefficient of the solvent is as follows: ln γmNRF w ¼ 2Xw λw Xw 2 λw

Xw 2 λw ð1 Xs Xw Þ ðXs þ Xw Þ

!

Xw Γww Xs Γww þ 1 ðXs þ Xw Þ

þ ðXw ðXw þ Xs 1ÞΓww

Xw λw þ Xw ðXs þ Xw ÞΓww 2 ð1 Xw Xs βsw, ww ÞÞ ðXs þ Xw Þ ! 2 X r λ ð1 X X Þ X s p s s w s Γss Xw Γss 1 þ Xs 2 rp λs ðXs þ Xw Þ ðXs þ Xw Þ Xs Γss ð1 Xw Xs βss, ws Þ ðXw þ Xs βss, ws Þ ðXs þ Xw Þ

ðXs ðXw Xs 1ÞÞΓss þ ðXw Xs Þ þ Xs rp λs

ð9Þ The proposed model on the basis of Eyring’s absolute rate theory can be utilized in the correlation of viscosity values of polymer solutions with the following equation: ! 2 X g ex Xi lnðηi Vi Þ þ ð10Þ lnðηV Þ ¼ RT i¼1 where η and V are the viscosity and molar volume of mixture, respectively. Subscript i represent the pure component i. gex* is molar excess Gibbs energy of activation for flow.

3. RESULTS AND DISCUSSION 3.1. Correlation of Solvent Activity. The solvent activity data of polymer solutions in the entire concentration range with different molar mass of polymer have been correlated by the proposed model with eqs 79. The model parameters were estimated by minimizing the following objective function: X exp 2 ðlnðaw, i Þ lnðacal ð11Þ OF ¼ w, i ÞÞ i

where aw is solvent activity in binary polymer solution; superscript exp and cal denote the experimental and calculated values, respectively. The obtained adjustable parameters of this model along with the absolute average relative deviations, AARD, have been collected in Table 1. From the obtained AARD values, we conclude that in the correlation of solvent activity, the performance of the mNRF model is good for the different polymer solutions in the composition and temperatures ranges given in Table 1. The reliability of mNRF model with two parameters has also been examined by removing the correction term from the local composition factor. Then the solvent activity data were fitted by the mNRF model with two parameters. It was found that the correction term is generally necessary for correlating the VLE data of polymer solutions investigated in this work. This indicates that the interaction between polymerpolymer and polymer solvent molecules in the local composition cells is higher than presented by gji. The fitting quality of the new model was compared with polymer-NRTL,7 polymer-NRF,8 polymerWilson,9 and polymer-NRF-Wilson10 models with the results collected in Table 2. As can be seen from this table, the performance of the proposed model in the fitting of activity values of binary polymer solutions is better than the other local composition models. To see the performance of the mNRF

model in a better manner, experimental and calculated solvent activity, aw, data have been plotted in Figure 1 for systems of polydecene213900 þ toluene at 303.15 K, PIB1200000 þ cyclohexane at 298.15 K, 1,4-cis-polyisoprene100000 þ CCl4 at 296.65 K, PS-b-PEO (diblock copolymer)45000 þ toluene at 342.65 K, poly(N-vinylcarbazole)94000 þ benzene at 318.15 K, and PVP13750 þ methanol at 298.15 K as examples. The difference between experimental and calculated values of the solvent activity obtained from polymer-NRTL, polymer-NRF, polymer-Wilson, and polymer-NRF-Wilson models has also been shown in Figure 2 for the PEG200 þ H2O system at 298.15 K as an example. From Figures 1 and 2 we concluded that the reliability of mNRF model in correlating of aw values of systems considered is better than the other local composition models. To see the reliability of the new model in light of temperature dependency, the following Wu31 type equations were used: 2 T T a0w þ a1w T0 T0 λw ¼ RT 2 0 T 1 T as þ as T0 T0 λs ¼ ð12aÞ RT 2 T T 1 þ bw T0 T0 ωw ¼ RT 2 T T b0s þ b1s T0 T0 ð12bÞ ωs ¼ RT where a0w, a1w, a0s , a1s , b0w, b1w, b0s , and b1s are adjustable parameters of the mNRF model. The value of T0 is fixed at 273.15 K in this work. The temperature range for the systems investigated in this work is not large; therefore, we considered a1w = a1s and b1w = b1s as stated by Wu et al.31 The obtained parameters for mNRF model along with absolute average relative deviations are given in Table 3. The performance of new model using eq 12 in the correlation of solvent activity values of polymer solutions was compared with that obtained from the polymer-NRTL,7 polymerNRF,8 polymer-Wilson,9 and polymer-NRF-Wilson10 models using the temperature dependency of parameters similar to eq 12; the results are reported in Table 4. From this table one can conclude that the performance of new model in light of temperature dependency of the correlation of solvent activity values of polymer solutions is generally better than other local composition models. We also tested the fitting quality of new model along with other local composition models with considering temperature dependency of parameters as eq 13, but the satisfactory results were not obtained. aw as λs ¼ ð13aÞ λw ¼ RT RT b0w

ωw ¼

bw RT

ωs ¼

bs RT

ð13bÞ

3.2. Correlation of Viscosity. By incorporating gex* from

different models in eq 10, one can obtain the necessary equation for correlating the viscosity values with the corresponding model. When we replace gex* with our mNRF model, the Eyring-mNRF 8248



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Table 1. Parameters of mNRF Model along with Absolute Average Relative Deviation, 100 3 AARD,a Obtained from Correlating the Solvent Activity of Binary Polymer Solutions at Different Temperatures range of polymer systems cellulose acetate þ acetone

34

cellulose acetate þ pyridine34 cellulose triacetate þ dichloromethane34 dextran þ H2O34 ethylenevinyl acetate (39.5%) þ benzene34 ethylenevinyl acetate (41.8%) þ o-xylene34

Mn

T

(g 3 mol1)

(K)

weight NP

fraction

λw

λs

ωw

ωs

100 3 AARD

104000

303.15

9

0.10000.5000

0.004501 136.600

11.170

1.614

0.05

104000

308.15

9

0.10000.5000

0.004154 138.800

11.350

1.614

0.05

104000

303.15

7

0.15000.4500

0.01546

0.1243

0.01535

6.816

0.08

104000

308.15

7

0.15000.4500

0.000

0.1875

0.000

6.797

0.09

157000

293.15

9

0.10000.5000

0.2505

0.2846

0.2518

7.007

0.05

157000

298.15

9

0.10000.5000

1.002

0.2901

1.007

6.991

0.05

101000 10100

313.15 303.15

15 13

0.08060.3055 0.24700.7790

1.002 361.500

0.5698 1.101

1.008 45.470

8.027 8.387

0.001 0.95

0.1046

8.275

5.126

1.38

19.560

12.200

0.96

14550

363.15

8

0.38000.8000

99.480

130000

293.15

18

0.32370.7407

119.000

polybutadiene þ CCl434

65200

296.65

6

0.58100.7506

1.243

5.254

5.552

16.260

0.05

poly(n-butyl acrylate) þ benzene34

33000

296.65

7

0.63570.8238

0.2528

0.5598

15.230

6.319

0.22

poly(ε-caprolactone) þ CCl434

10700

338.15

9

0.38350.9127

0.1733

0.0106

5.562

35.780

0.09

213900

303.15

17

0.13830.9774

0.009926

0.2252

250.000

7.473

2.13

26000 6000

303.15 298.15

12 14

0.97970.2321 150.600 0.23060.5417 1676.000

0.2009 0.873

18.960 9.880

5.524 3.613

0.54 0.04

PEG þ H2O37

200

298.15

11

0.05020.8981

6.043

0.0944

1.133

3.257

0.09

PEG þ H2O37

600

298.15

11

0.04980.8972

0.3599

5.656

PEG þ H2O37

1000

298.15

7

0.04930.4929 2620.000

PEG þ H2O37

1450

298.15

9

0.04960.6941

2.787

0.1232

PEG þ H2O37

3350

298.15

7

0.04970.4964

0.000

0.1935

PEG þ H2O37

8000

298.15

7

0.04990.4984

0.000

PEG þ H2O37 PEG þ H2O37

10000 20000

298.15 298.15

7 7

0.04970.4967 0.04970.4971

0.003923 1.002

nitrocellulose þ ethyl formate34

polydecene þ toluene34 PDMS þ n-hexane35 PEG þ acetonitrile36

PEG þ 2-propanol38 PEO þ benzene34

0.01319

110.000

0.9062

0.25

3.287

0.09

2.341

5.220

0.07

0.000

5.195

0.05

0.1988

0.000

6.191

0.07

0.1964 0.2054

0.003953 1.008

6.408 7.228

0.13 0.03

22.610 167.400

296

298.15

20

0.06580.4529

5.338

2.815

1.615

600000

343.15

8

0.27580.9033

0.1467

0.5769

8.983

16.280

0.03

0.08673

0.09

1800

303.15

9

0.09910.8981

6.195

0.4546

3.579

2.303

0.26

poly(ethylene glycol) methacrylate þ ethanol39

361

298.15

27

0.08880.6529

0.8437

0.08991

1.505

1.395

0.08

poly(ethylene glycol) methacrylate þ methanol39

361

298.15

27

0.12390.8505

1.577

0.2976

1.516

0.272

0.15

poly(ethylene glycol) dimethyl ether þ CCl434

360

303.15

9

0.09830.9000

0.4193

2.078

0.27

38600 224100

296.65 303.15

8 20

4.261 2062.00 0.01739 7.348

0.43 1.73

1200000

298.15

11

0.97320.2466

100000

296.65

7

0.56310.7547

1.189

poly(R -methylstyrene) þ R-methylstyrene34

17000

338.15

9

0.27950.7718

265.200

poly(methyl acrylate) þ chloroform34

63200

296.65

8

0.44680.7352

1.693

PPG þ acetonitrile36

425

298.15

16

0.07170.5691

4.964

PPG þ 1-butanol40

976

298.15

16

0.13580.5884

9.026

PPG þ ethanol40 PPG þ H2O41

976 404

298.15 298.15

24 8

0.08570.7265 0.18280.4261

37.830 0.2218

0.2335 1.362

4.856 2.947

PEO-b-PPO-b-PEO (triblock copolymer) þ CCl438

poly(ethyl acrylate) þ chloroform34 polyheptene þ toluene34 PIB þ cyclohexane35 1,4-cis-polyisoprene þ CCl434

0.42740.6862 5.354 0.16490.9985 11.070 0.2082

0.06168 0.3202 36.330 0.02796 0.1019

9.308

2.735

0.43

4.466

5.844

16.260

0.08

0.4395 48.450

4.184

0.47

5.840

1.131

0.67

0.2666

2.713

1.911

0.05

0.1315

5.056

4.993

0.27

2.610 0.01348

0.08 0.04

12.460

2000

303.15

7

0.92390.9932

1.201

0.02407

3.137

2.980

0.26

290000

343.14

9

0.47600.9370

0.3639

1.968

7.636

5.995

0.36

45000

342.65

13

0.56280.8155

1.750

750

318.15

9

0.31100.8980

3.431

poly(N-vinylcarbazole) þ benzene34

94000

318.15

8

0.30000.8480

42.370

10.150


324000

318.15

7

0.28600.8670

28.620

5.971

poly(vinyl acetate) þ vinyl acetate34 poly(vinyl alcohol) þ H2O35

150000 88000

303.15 303.15

9 8

poly(vinyl chloride) þ toluene34

34000

316.35

poly(vinyl methyl ether) þ ethylbenzene34

14600

398.15

PPO þ methanol35 PS b þ 1-propyl acetate34 PS-b-PEO (diblock copolymer) þ toluene34 poly(N-vinylcarbazole) þ benzene34

4.295

6.476

0.26

0.3931 120.500

1.767

0.13

9.893

2.058

0.49

9.749

4.787

1.58

0.43400.9260 1543.000 0.66590.9806 0.6483

0.3048 481.900 0.1253 8.456

7.858 16.280

0.86 4.06

8

0.62210.9434

295.500

2.091

37.480

8.232

0.71

21

0.52180.7986

22.590

0.2691

3.423

5.056

0.15

8249

0.2658



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Table 1. Continued range of polymer systems

T (K)

NP

fraction

weight λw

λs

PVP þ acetonitrile

10000

298.15

20

0.19530.7370

766.400

PVP þ 1-butanol42

13750

298.15

16

0.13580.5884

0.02956 45.680

PVP þ methanol42

13750

298.15

16

0.12250.7753 88.560

36

a

Mn (g 3 mol1)

ωw

0.7487 19.860 34.120

ωs

100 3 AARD

3.979

0.17

8.098

17.570

0.90

8.560

16.280

0.24

cal exp exp b p AARD = (1/Np)ΣN i=1(|ai ai |/ai ). PS is polystyrene.

Table 2. Absolute Average Relative Deviation, 100 3 AARD, of Different Models Obtained from Correlating the Solvent Activity of Binary Polymer Solutions at Different Temperatures polymerrange of polymer systems

Mn (g 3 mol1) T (K) NP

weight fraction

polymer-

polymer-

polymer-

NRF-

NRTL (2)a Wilson (2) NRF (2) Wilson (2) mNRF (4)

cellulose acetate þ acetone34

104000

303.15

9

0.10000.5000

0.11

0.32

0.16

0.55

0.05

cellulose acetate þ pyridine34

104000 104000

308.15 303.15

9 7

0.10000.5000 0.15000.4500

0.11 0.08

0.31 0.14

0.16 0.08

0.50 0.08

0.05 0.08

104000

308.15

7

0.15000.4500

0.08

0.16

0.09

0.09

0.09

157000

293.15

9

0.10000.5000

0.07

0.06

0.08

0.15

0.05

157000

298.15

9

0.10000.5000

0.07

0.06

0.23

0.16

0.05

101000

313.15 15

0.08060.3055

0.02

0.002

0.004

0.16

0.001

10100

303.15 13

0.24700.7790

1.43

1.81

1.79

2.11

0.95

14550

363.15

cellulose triacetate þ dichloromethane34 dextran þ H2O34 ethylenevinyl acetate (39.5%) þ benzene34 ethylenevinyl acetate (41.8%) þ o-xylene34 nitrocellulose þ ethyl formate34

130000

8

0.38000.8000

1.61

1.37

1.36

4.77

1.38

293.15 18

0.32370.7407

2.13

4.38

4.77

3.75

0.96

polybutadiene þ CCl434

65200

296.65

6

0.58100.7506

0.11

0.34

0.28

0.31

0.05

poly(n-butyl acrylate) þ benzene34

33000

296.65

7

0.63570.8238

0.49

1.51

1.21

0.56

0.22

poly(ε-caprolacton) þ CCl434

10700

338.15

9

0.38350.9127

3.11

2.68

1.44

0.96

0.09

303.15 17

0.13830.9774

10.38

3.15

2.03

2.50

2.13

polydecene þ toluene34

213900

PDMS þ n-hexane35

26000

303.15 12

0.97970.2321

1.13

6.40

0.38

0.74

0.54

PEG þ acetonitrile36 PEG þ H2O37

6000 200

298.15 14 298.15 11

0.23060.5417 0.05020.8981

0.08 0.17

0.13 0.20

0.21 0.20

0.28 0.95

0.04 0.09

PEG þ H2O37

600

298.15 11

0.04980.8972

0.50

0.47

0.82

2.57

0.25

PEG þ H2O37

1000

298.15

7

0.04930.4929

0.10

0.15

0.10

0.19

0.09

PEG þ H2O37

1450

298.15

9

0.04960.6941

0.14

0.19

0.16

0.34

0.07

PEG þ H2O37

3350

298.15

7

0.04970.4964

0.05

0.09

0.04

0.14

0.05

PEG þ H2O37

8000

298.15

7

0.04990.4984

0.06

0.13

0.09

0.15

0.07

PEG þ H2O37

10000

298.15

7

0.04970.4967

0.13

0.16

0.14

0.18

0.13

PEG þ H2O37 PEG þ 2-propanol38

20000 296

298.15 7 298.15 20

0.04970.4971 0.06580.4529

0.04 0.10

0.15 0.10

0.09 0.10

0.17 0.51

0.03 0.09

PEO þ benzene34 PEO-b-PPO-b-PEO

600000

343.15

8

0.27580.9033

2.21

0.04

0.03

0.27

0.03

1800

303.15

9

0.09910.8981

0.65

0.42

0.32

1.24

0.26

(triblock copolymer) þ CCl438 poly(ethylene glycol) methacrylate þ ethanol39

361

298.15 27

0.08880.6529

0.08

0.08

0.09

2.94

0.08

poly(ethylene glycol) methacrylate þ

361

298.15 27

0.12390.8505

0.17

0.21

0.23

3.35

0.15

303.15 296.65

0.09830.9000 0.42740.6862

0.50 1.10

0.58 1.24

0.65 1.40

0.12 0.52

0.27 0.43 1.73

methanol39 poly(ethylene glycol) dimethyl ether þ CCl434 poly(ethyl acrylate) þ chloroform34 polyheptene þ toluene34

360 38600

9 8

224100

303.15 20

0.16490.9985

23.45

7.12

2.85

4.47

1200000

298.15 11

0.97320.2466

5.75

1.37

0.44

2.00

0.43

100000

296.65

7

0.56310.7547

0.10

0.19

0.73

0.11

0.08

poly(R -methylstyrene) þ R-methylstyrene34

17000

338.15

9

0.27950.7718

1.07

0.81

0.85

1.11

0.47

poly(methyl acrylate) þ chloroform34

63200

296.65

8

0.44680.7352

0.65

1.01

0.68

0.99

0.67

PIB þ cyclohexane35 1,4-cis-polyisoprene þ CCl434

8250



ARTICLE

Table 2. Continued polymerrange of polymer systems

weight fraction

polymer-

polymer-

NRF-

NRTL (2)a Wilson (2) NRF (2) Wilson (2) mNRF (4)

PPG þ acetonitrile36

425

298.15 16

0.07170.5691

0.05

0.05

0.05

1.04

0.05

PPG þ 1-butanol40

976

298.15 16

0.13580.5884

0.39

1.31

0.31

0.34

0.27

PPG þ ethanol40

976

298.15 24

0.08570.7265

0.08

0.08

0.12

1.03

0.08

PPG þ H2O41

404

298.15

8

0.18280.4261

0.11

0.04

0.04

0.27

0.04

2000

303.15

7

0.92390.9932

3.46

0.62

3.60

48.57

0.26

290000

343.14

9

0.47600.9370

2.86

3.02

0.59

2.39

0.36

45000 750

342.65 13 318.15 9

0.56280.8155 0.31100.8980

0.26 0.41

0.41 0.62

0.32 0.44

5.74 11.62

0.26 0.13


94000

318.15

8

0.30000.8480

5.02

10.14

10.88

10.72

0.49


324000

318.15

7

0.28600.8670

5.69

14.27

16.31

12.37

1.58

poly(vinyl acetate) þ vinyl acetate34

0.86

PPO þ methanol35 PS þ 1-propyl acetate34 PS-b-PEO (diblock copolymer) þ toluene34 poly(N-vinylcarbazole) þ benzene34

a

Mn (g 3 mol1) T (K) NP

polymer-

150000

303.15

9

0.43400.9260

3.32

3.14

1.09

5.46

poly(vinyl alcohol) þ H2O35

88000

303.15

8

0.66590.9806

7.43

4.41

4.26

23.65

4.06

poly(vinyl chloride) þ toluene34

34000

316.35

8

0.62210.9434

1.41

0.97

3.07

4.31

0.71

poly(vinyl methyl ether) þ ethylbenzene34

14600

398.15 21

0.52180.7986

0.28

0.16

0.16

0.37

0.15

PVP þ acetonitrile36 PVP þ 1-butanol42

10000 13750

298.15 20 298.15 16

0.19530.7370 0.13580.5884

0.26 1.21

0.21 1.47

0.30 1.47

2.66 1.20

0.17 0.90

PVP þ methanol42

13750

298.15 16

0.12250.7753

0.27

0.25

0.62

0.77

0.24

Number of model parameters are presented in the parentheses.

Figure 1. Experimental and calculated solvent activity data, aw, with mNRF model for different systems: (0) polydecene213900 þ toluene at 303.15 K; (]) PIB1200000 þ cyclohexane at 298.15 K; (Δ) 1,4-cispolyisoprene100000 þ CCl4 at 296.65 K; (/) PS-b-PEO (diblock copolymer)45000 þ toluene at 342.65 K; () poly(N-vinylcarbazole)94000 þ benzene at 318.15 K; (O) PVP13750 þ methanol at 298.15; (—) mNRF model.

equation is obtained for viscosity. In the Eyring-mNRF equation we considered gex*/RT = (N/Ns)(gex,mNRF/RT) similar to that considered by Novak et al.11 In this equation N is the total number of polymer and solvent and Ns is the total number of polymer segment and solvent moles in the mixture. Compilations of the viscosities of the binary polymer solutions can be

Figure 2. Difference between experimental and calculated solvent activity data, aw, for PEG200 þ H2O system at 298.15 K with different models: (Δ) polymer-NRTL; (O) polymer-Wilson; () polymer-NRF; (/) polymer-NRF-Wilson; (2) mNRF.

found in the literature, and reliability of the new local composition model has been tested with these data. The molar volumes of pure components at different temperatures have been determined from the density data of pure solvents and polymers in the corresponding references. For PEG þ H2O, PVP þ H2O, and polystyrene þ styrene systems, the pure polymer viscosity was not available. In these cases, the polymer viscosity was treated as an adjustable parameter. In the case of PEG þ H2O systems at different polymer molar masses, the following MarkHouwink type relation was used to account for the polymer molar mass dependence of the pure polymer viscosity, ηP, as proposed by 8251



ARTICLE

Table 3. Parameters of mNRF Model Using Temperature Dependency as Eq 12 along with Absolute Average Relative Deviation, 100 3 AARD, Obtained from Fitting the Solvent Activity Values of Binary Polymer Solutions Mn system dextran þ H2O dextran þ H2O34 34

polyethylene þ chlorobenzene34

(g 3 mol1) 46300 64800

T (K)

NP

293.15333.15 23 293.15333.15 47

a0w

a1w

73.710 73.650

0.4859 0.4859

a0s

b0w

1158.000 1199.000

18330.000 19210.000

16720.000 17490.000

95.780 918.200

393.15413.15 20

565.800 2514.000

3874.000

400

298.15328.15 62

4299.000 3425.000

4106.000

3966.000

PEG þ H2O37,44

400

298.15338.15 35

923.800

972.700

398.700

8429.000

PEG þ H2O37,44

6000

298.15328.15 21

3119.000

63.510 7302.000

8566.000

400

298.15328.15 62

1784.000 1466.000

PEG þ methanol45 poly(4-hydroxystyrene) þ acetone34 PPG þ ethanol46 PPG þ 2-propanol46

100 3 AARD 0.006 0.006

8853.000

0.71

189.200 1074.000

0.28

104.000

195.300

0.06

2630.000

652.000

0.04

1844.000 22770.000 26740.000 32270.000

0.63

1500

293.15318.15 95

1351.000 537.000 1004.000

400 400

298.15328.15 97 298.15328.15 78

4497.000 1894.000 5327.000 1418.000

1784.000 928.000

298.15353.20 137 642900.000 3342.000

4839.000

5010.000 477.500

895.300

925.600

5008.000

10220.000

1.94

3873.000 5811.000

10710.000

4.179

127.300

0.26

PIB þ benzene34

50000

PS þ benzene34

218000

PVP þ H2O47

b0s

332.400 333.000

6220

PEG þ ethanol43

b1w

4088

313.20353.20 70

298.15328.15 66 2826.000

976.900

6176.000 4398.000

0.46

2380.000 3668.000

407.900 1345.000 155.200 2017.000

0.24 0.09

83910.000 2729.000

17610.000

2.83

Table 4. Absolute Average Relative Deviation, 100 3 AARD, of Different Models Using Temperature Dependency as Eq 12 for Correlating the Solvent Activity of Binary Polymer Solutions Mn

polymer-

polymer-

polymer-

polymer-

(g 3 mol1)

T (K)

NP

NRTL (3)a

Wilson (3)

NRF (3)

NRF- Wilson (3)

mNRF (6)

34

46300

293.15333.15

23

0.009

0.03

0.006

0.08

0.006

dextran þ H2O34

64800

293.15333.15

47

0.01

0.009

0.01

0.15

0.006

6220

393.15413.15

20

2.89

0.76

0.95

0.78

0.71

400

298.15328.15

62

0.87

0.34

0.39

1.13

0.28 0.06

system dextran þ H2O

polyethylene þ chlorobenzene34 PEG þ ethanol43 PEG þ H2O37,44

400

298.15338.15

35

0.52

0.14

0.17

0.97

PEG þ H2O37,44

6000

298.15328.15

21

0.23

0.06

0.05

0.04

0.04

PEG þ methanol45 poly(4-hydroxystyrene) þ acetone34

400 1500

298.15328.15 293.15318.15

62 95

1.76 1.74

0.66 1.93

0.70 0.63

0.54 1.08

0.63 0.46

PPG þ ethanol46

400

298.15328.15

97

0.77

0.24

0.28

1.03

0.24

PPG þ 2-propanol46

400

298.15328.15

78

0.11

0.14

0.13

0.15

0.09

PIB þ benzene34

50000

298.15353.20

137

17.17

2.79

4.43

15.51

2.83

PS þ benzene34

218000

313.20353.20

70

8.48

2.56

2.69

2.63

1.94

4088

298.15328.15

66

1.07

0.29

0.39

1.62

0.26

PVP þ H2O47 a

Number of model parameters are presented in the parentheses.

Sadeghi10 1

ηp ðmPa 3 sÞ ¼ bðMn ðg 3 mol ÞÞ

c

ð14Þ

where Mn is the number-average polymer molar mass. The parameters b and c were calculated to be 6.5727 103 and 3.1542, respectively.10 Also, in the case of PVP þ H2O systems at different temperatures, the following relation was used to describe the temperature dependency of the pure polymer viscosity10 c1 ηp ðmPa 3 sÞ ¼ b1 exp ð15Þ T ðKÞ The parameters b1 and c1 were calculated to be 1.5252 105 and 7.1446 103, respectively.10 Density and viscosity values of pure styrene and density values of polystyrene at different temperatures have been taken from literature.32,33 Novak et al.11 obtained the value of 106 Pa 3 s for viscosity of pure polystyrene

from extrapolating of mixture viscosity; we also used this value for viscosity of pure polystyrene. The viscosity values of binary polymer solutions have been correlated with new model by incorporating of eq 5 in eq 10. The model parameters were estimated by minimizing the following objective function: X ðlnðηi Vi Þexp lnðηi Vi Þcal Þ2 ð16Þ OF ¼ i

The evaluated parameters of Eyring-mNRF model along with the corresponding absolute average relative deviation, AARD, for the systems studied are listed in Table 5. The results obtained from fitting of viscosity values with the new model without considering the correction terms have also been reported in parentheses in Table 5. On the basis of the AARD given in Table 5, we concluded that the proposed model represents the experimental viscosity data of the polymer solutions, with good accuracy, and the performance of the proposed model with two 8252


8253

298.15 308.15 318.15

400 400 400 400 400 400 400 400 400

4088

4088

4088

4088

PPG þ ethanol46

PPG þ ethanol46

PPG þ ethanol46

PPG þ ethanol46

PPG þ 2-propanol46 PPG þ 2-propanol46

PPG þ 2-propanol46

PPG þ 2-propanol46

PVP þ H2O47

PVP þ H2O47

PVP þ H2O47

PVP þ H2O47 328.15

318.15

308.15

298.15

328.15

328.15

318.15

308.15

298.15

308.15

303.15

303.15 298.15

303.15

303.15

303.15

303.15

400

303.15

303.15

PEG þ methyl acetate51

408

PEG þ1,3-dioxolane50

298.15 303.15


408 192

PEG þ dimethoxymethane49 PEG þ1,3-dioxolane50

298.15

408 400

192

PEG þ dimethoxymethane49

298.15

298.15

192

408

PEG þ 1,2-dimethoxyethane49

PEG þ oxane50 PEG þ methyl acetate51

192

PEG þ 1,2-dimethoxyethane49

298.15

PEG þ oxane50

6000

PEG þ H2O48

298.15

298.15

408

4000

PEG þ H2O48

PEG þ oxolane50

3000

PEG þ H2O48

298.15 298.15

192

1500 2000


298.15

408

1000

PEG þ H2O48

298.15

298.15

PEG þ oxolane50

900

PEG þ H2O48

PEG þ 1,4-dioxane50

600

PEG þ H2O48

298.15

298.15

192

400

T (K)


300

PEG þ H2O48

Mn (g 3 mol1)

PEG þ H2O48

systems

27

27

27

27

27

27

28 27

22

22

22

22

11

11

14 11

14

14

14

14

14

14

14 14

14

14

14

10

10

10

10 10

10

10

10

10

10

NP

0.00270.4529

0.00270.4529

0.00270.4529

0.00270.4529

0.04950.9920

0.04950.9920

0.04950.9920 0.04950.9920

0.04910.9885

0.04910.9885

0.04910.9885

0.04910.9885

0.36580.9813

0.36580.9813

0.16660.9852 0.36580.9813

0.11840.9831

0.16750.9895

0.11650.9818

0.04470.9864

0.00780.9797

0.09500.9939

0.16200.9841 0.03570.9770

0.10200.9810

0.26650.9860

0.11710.9832

0.00380.0599

0.00610.0959

0.00760.1201

0.01010.1586 0.00870.1357

0.01200.1872

0.01380.2136

0.01950.2984

0.01690.2583

0.01880.2885

range of polymer weight fraction

4.644 18.600 9.661 18.620 18.670 0.0741

0.3008 4.812 1.232 4.812 4.812 0.00007345

10.010

14.820 4.008 18.550 2.375 4.812 0.4528

0.6221 (0.2043) 0.3488 (0.1841) 0.226 (0.1703) 0.1488 (0.102) 57.730 (10.420)

80.470 (17.620)

0.006568 (18.170)

0.2323 (18.000)

180.500 (18.870)

2.652 (3.987)

3.181 (4.327)

6.311 (5.805) 4.425 (5.045)

0.424 (29.580)

0.8417 (8.550)

0.003791

250.000

1.468

0.1579 (0.03357)

0.8381 (12.310)

18.590

15.040

0.6451 (0.09943) 7.205 (6.036) 1.095 (5.123)

1.333

8.182

500.000

0.7832

1.116

1.014 0.8711

125.000

1.167

0.7893

3.139 15.070

0.5968 (0.176) 7.691 (6.394)

11.230 (7.216)

0.75 (0.84)

1.58 (1.68)

1.23 (1.34)

1.31 (1.31)

0.71 (0.85)

1.81 (1.75)

1.00 (1.00) 0.64 (0.73)

1.38 (1.80)

1.15 (1.23)

0.60 (0.61)

1.41 (1.47)

0.97 (1.32)

0.96 (1.01)

0.6164 8.234

7.268

0.09262 (0.1527)

0.4167

0.21 (0.85) 1.07 (1.08)

0.22 (0.93)

0.55 (0.57)

1.05 (0.47)

0.41 (0.72)

0.19 (0.46)

0.48 (0.78)

0.78 (0.80) 0.26 (0.31)

0.31 (0.42)

0.61 (0.77)

0.28 (0.35)

0.09 (0.09)

0.19 (0.19)

0.13 (0.12)

0.15 (0.16) 0.12 (0.13)

0.25 (0.28)

0.08 (0.08)

0.10 (0.10)

0.18 (0.31)

0.07 (0.14)

EyringmNRFb

125.000 125.000

125.000

125.000

0.08311 (0.267)

2.610 1.298

1.494

0.7251

12.700

2.209

125.000

125.000

125.000 125.000

125.000

31.250

0.9456 (0.2091)

3.993 (3.958)

58.530 (4.249)

0.001913 (61.570) 0.3818 (0.1929)

0.1777 (0.1719)

0.5566 (2.619) 0.2543 (228.300) 0.6412 (4.475)

0.2889 (0.1457)

1.804 (4.855)

9.583

4.074 0.2403 (0.1807)

76.850 (3.989)

0.9376 (1.319)

1.088

0.296 (0.08366)

0.3137 (0.1403)

1.786 (5.689)

1.356 1.028 0.663

0.4696 (0.2265) 0.674 (0.3394)

0.9137 (5.082) 1.167 (4.851)

1.686

8.190

0.1825 (0.1325)

0.6858 (0.3344)

0.3124 (3.835)

1.143 (2.442)

0.4273 (0.2833)

0.213 (0.339)

10760.000 (10850.000)

3750.000 (3944.000)

1579.000 (1847.000)

184.900 (419.300) 533.600 (795.500)

59.6500 (230.000)

0.462

0.4106

0.00475

7.579

18.590

4.812

11.730 (10.860)

1.368 (0.6671)

500.000

250.000

ωs

250.000

62.500

ωw

8.935 (0.8989)

8.454 (0.5503)

λsb

65.720 (4.550)

1.597 (3.309)

831.500 (27.530)

1040.000 (28.890)

1072.000 (29.080)

685.100 (26.370) 849.600 (27.660)

458.200 (24.020)

428.300 (428.200)

277.800 (277.800)

0.004477 (175.300)

0.002664 (124.300)

λwb

Table 5. Parameters of Eyring-mNRF Model along with Absolute Average Relative Deviation, 100 3 AARD,a Obtained from Correlating the Viscosity of Binary Polymer Solutions at Different Temperatures

Industrial & Engineering Chemistry Research ARTICLE


0.89 1.383 1.456 0.1213 0.4622 0.0810.942 10 303.15 112000 PDMS þ PDMS53e

c cal exp exp b p AARD = (1/Np)ΣN i=1(|ηi ηi |/ηi ). Parameters and 100 3 AARD values of the Eyring-mNRF model considering two parameters have been given in parentheses. Molar mass of polydimethylsiloxane, d 1 e 1 PDMSO, used as solvent is 1900. Molar mass of PDMSO used as solvent is 44000 (g 3 mol ). Molar mass of PDMSO used as solvent is 76000 (g 3 mol ).

8.429 0.080- 0.911 10 112000 PDMS þ PDMS53d

303.15

6.728

4.214 0.094100.4457

0.0810.953 10

12

112000 PDMS þ PDMS53c

303.15

166000 PS þ styrene52

333.15

3.392 0.094100.5075 13 166000 PS þ styrene52

333.15

3.584

3.493 0.004740.5075

0.004740.4457 18

19


323.15


323.15

3.469 3.295

3.039 0.004740.4457

0.004740.5075 0.004740.4457 21 20

20

166000 166000 PS þ styrene52 PS þ styrene52

313.15 313.15


303.15

2.722

3.467 0.004740.5075

0.061500.3193 8

21 303.15 166000 PS þ styrene52

293.15 166000 PSþ styrene52

weight fraction

range of polymer

NP T

(K)

Mn

(g 3 mol1) systems

Table 5. Continued

a

4.31 1.493 0.1474

10.47

6.53 24.990

5.793

8.238

10.260 0.9532

0.5659

5.80

10.79 10.630

24.990

8.07

0.3146

24.980 0.2917

41.900 10.550

10.480

344.300

5.73

8.55 5.63 10.610 10.610

24.930 10.720 2.172

329.500 1.192

39.210 24.990

8.99

4.31 10.240

10.650

44.770

11.030 3.594

376.600

Eyring-

λwb

λsb

ωw

ωs

mNRFb


ARTICLE

parameters in the correlation of viscosity values of investigated polymer solutions is comparable with that considering four parameters. The exceptions are polystyrene þ styrene and PDMS þ PDMS systems containing a large molar mass of polymer. This indicated that, in the fitting of viscosity values with Eyring-mNRF model for polymer solutions with larger molar mass, the correction term is necessary. The viscosity values of polystyrene þ styrene system at high concentration is very large; therefore, we repeated the fitting procedure for this system by removing the one datum point corresponding to the highest concentration and a rather lower AARD value is obtained, which is reported in Table 5 in parentheses for each temperature. We also examined the reliability of the Eyring-polymer-NRTL model7 in the correlation of viscosity values of binary polymer solutions for first time. In Table 6 the obtained absolute average relative deviations, AARD, of the aforementioned model in the correlation of the viscosity values are given along with the AARD values determined from other local composition models (Eyringsegment-based-liquid-NRTL,11 Eyring-polymer-Wilson,12 Eyring-polymer-NRF,13 and Eyring-polymer-NRF-Wilson10 models), which have been previously used for the correlation of the viscosity values of some polymer solutions. The fitting quality of Eyring-segment-based-liquid-NRTL11 and Eyring-polymerNRTL7 models is sensitive to the value of the nonrandomness factor, R. With these models the best quality fitting is obtained by fixing R at 0.25, as discussed by Sadeghi.10 The value of 0.25 was used for the value of nonrandomness factor in the Eyringsegment-based-liquid-NRTL11 and Eyring-polymer-NRTL7 models and produced good results, but for polystyrene þ styrene and PDMS þ PDMS systems, with a large molar mass of polymer, we found that the best fitting quality is obtained with R = 0.05. However, the fitting quality of the new model is not sensitive to value of Z. To show the reliability of the proposed model, comparison between experimental and correlated viscosity data are shown in Figure 3 for PEG192 þ 1,2-dimethoxyethane and PEG408 þ 1,2-dimethoxyethane systems at 298.15 K as examples. To see the performance of the aforementioned models in a better manner, the difference between calculated and experimental viscosity values versus the mole fraction of polymer has been plotted in Figure 4 for PPG þ 2-propanol system at 298.15 K. From Table 6 and Figures 3 and 4 one can conclude that the reliability of Eyring-mNRF model in the correlation of viscosity values of binary polymer solutions is better than other local composition models investigated in this study, especially at the higher temperatures and concentrations and for the solutions containing polymer with large molar mass. For some polymer solutions the viscosity data are available at different temperatures. The performance of the Eyring-mNRF model along with other local composition models is tested by considering the temperature dependency of parameters as in eq 13 for these systems; it was found that the performance of all local composition models, especially the Eyring-mNRF model, in the correlation of viscosity values of polymer solutions at different temperatures and low molar mass of polymer is good. However, the fitting quality for the polystyrene þ styrene system is not satisfactory, especially with the Eyring-polymer-NRF and Eyring-polymer-NRF-Wilson models. Therefore, we decided to use the temperature dependency in eq 12 using a1w = a1s and b1w = b1s . The obtained parameters for the Eyring-mNRF model along with absolute average relative deviations are given in Table 7. The performance of the new model using eq 12 in the 8254


49

2000

3000

4000 6000

192

PEG þ H2O48

PEG þ H2O48


PEG þ

49

8255

400

400

400

400 400

PPG þ ethanol46

PPG þ ethanol46

PPG þ 2-propanol46

PPG þ 2-propanol46 PPG þ 2-propanol46

4088

400

PPG þ ethanol46

PVP þ H2O47

400

PPG þ ethanol46

400

400


4088

400 400

PEG þ methyl acetate51 PEG þ methyl acetate51

PVP þ H2O47

408

PEG þ oxane50

PPG þ 2-propanol46

192

408

PEG þ oxolane50

PEG þ oxane50

192

PEG þ oxolane50

50

408

192


PEG þ 1,4-dioxane

192 408

408

192

PEG þ1,3-dioxolane50 PEG þ1,3-dioxolane50

PEG þ dimethoxymethane

49

PEG þ dimethoxymethane49

1,2-dimethoxyethane

PEG þ

408

1500

PEG þ H2O48

1,2-dimethoxyethane

1000

900

PEG þ H2O48

PEG þ H2O

600

PEG þ H2O48

48

300 400

308.15

298.15

328.15

308.15 318.15

298.15

328.15

318.15

308.15

298.15

308.15

298.15 303.15

303.15

303.15

303.15

303.15

303.15

303.15

303.15 303.15

298.15

298.15

298.15

298.15

298.15 298.15

298.15

298.15

298.15

298.15

298.15

298.15

298.15 298.15

T (K)

Mn

(g 3 mol1)


systems

27

27

27

27 27

28

22

22

22

22

11

11 11

14

14

14

14

14

14

14 14

14

14

14

14

10 10

10

10

10

10

10

10

10 10

NP

0.00270.4529

0.00270.4529

0.04950.9920

0.04950.9920 0.04950.9920

0.04950.9920

0.04910.9885

0.04910.9885

0.04910.9885

0.04910.9885

0.36580.9813

0.36580.9813 0.36580.9813

0.16660.9852

0.11840.9831

0.16750.9895

0.11650.9818

0.04470.9864

0.00780.9797

0.03570.9770 0.09500.9939

0.16200.9841

0.10200.9810

0.26650.9860

0.11710.9832

0.00610.0959 0.00380.0599

0.00760.1201

0.00870.1357

0.01010.1586

0.01200.1872

0.01380.2136

0.01950.2984

0.01880.2885 0.01690.2583

weight fraction

range of polymer

16.31c

16.97c

11.13

11.12 12.35

9.76

6.74

6.95

7.50

6.11

2.17c

0.99c 1.33c

4.28c

4.13c

3.31c

3.05c

1.97

c

1.99c

1.16c 1.43c

5.06

c

4.34c

6.39c

5.50c

8.76c 10.01c

4.26c

1.66c

3.06c

3.20c

5.81

c

5.21c

7.43c 2.53c

NRTLa (2)

Eyring-polymer-

1.77

1.35

1.04

0.84 2.25

0.99

1.38

1.17

0.88

1.41

2.70

0.92 0.88

1.72

3.51

0.57

0.23

0.52

2.10

0.26 0.46

0.71

0.60

0.59

0.28

1.70 1.21

1.57

1.03

0.69

0.49

0.12

0.10

0.12 0.29

based-liquid-NRTLa (2)

Eyring-segment-

1.58c

3.17c

1.46

1.66 2.81

1.28

1.38

1.42

1.80

1.63

6.26c

4.63c 3.82c

5.25c

3.70c

5.47c

3.35c

3.55

c

2.32c

2.27c 3.35c

8.90

c

6.41c

13.51c

8.38c

8.90c 10.09c

4.46c

1.75c

2.83c

2.96c

5.30

c

5.43c

8.22c 3.33c

Wilson (2)

Eyring- polymer-

7.11c

6.91c

1.17

0.90 1.84

1.75

2.53

2.22

1.19

2.75

2.36c

0.85c 1.36c

4.21c

4.24c

3.20c

3.18c

1.96

c

2.05c

1.22c 1.40c

4.89

c

4.40c

6.02c

5.35c

9.42c 10.43c

5.13c

1.08c

2.05c

2.09c

4.30

c

1.64c

8.36c 3.45c

NRF (2)

Eyring- polymer-

1.22c

1.41c

0.92

1.23 1.86

1.49

1.61

1.60

1.54

2.75

1.89c

0.96c 1.24c

3.26c

3.44c

1.23c

1.31c

1.66c

1.37c

0.94c 1.08c

1.48c

1.62c

1.97c

2.05c

7.95c 9.31c

3.44c

1.31c

3.85c

3.95c

6.38c

4.43c

5.65c 1.10c

NRF- Wilson (2)

Eyring- polymer-

1.23 (1.34)

1.31 (1.31)

0.71 (0.85)

0.64 (0.73) 1.81 (1.75)

1.00 (1.00)

1.38 (1.80)

1.15 (1.23)

0.60 (0.61)

1.41 (1.47)

0.97 (1.32)

1.07 (1.08) 0.96 (1.01)

0.21 (0.85)

0.22 (0.93)

0.55 (0.57)

1.05 (0.47)

0.41 (0.72)

0.19 (0.46)

0.26 (0.31) 0.48 (0.78)

0.78 (0.80)

0.31 (0.42)

0.61 (0.77)

0.28 (0.35)

0.19 (0.19) 0.09 (0.09)

0.13 (0.12)

0.12 (0.13)

0.15 (0.16)

0.25 (0.28)

0.08 (0.08)

0.10 (0.10)

0.07 (0.14) 0.18 (0.31)

mNRFb (4)

Eyring-

Table 6. Absolute Average Relative Deviation, 100 3 AARD, of Different Models Obtained from Correlating the Viscosity of Binary Polymer Solutions at Different Temperatures



T (K) 318.15 328.15 293.15 303.15 303.15 313.15 313.15 323.15 323.15 333.15 333.15 303.15 303.15 303.15

Mn

4088

4088

166000

166000 166000

166000

166000

166000

166000

166000

166000

112000 112000

112000

PSþ styrene52

PS þ styrene52 PS þ styrene52

PS þ styrene52

PS þ styrene52

PS þ styrene52

PS þ styrene52

PS þ styrene52

PS þ styrene52

PDMS þ PDMS53 PDMS þ PDMS53

PDMS þ PDMS53 10

10 10

12

13

18

19

20

21

21 20

8

27

27

NP 15.22c

0.00270.4529 14.71 80.79 (3.76) 70.88 (9.92) 69.95 (6.13) 68.61 (8.92) 67.89 (6.56) 64.95 (8.83) 64.46 (7.61) 71.54 (11.09) 72.05 (9.53) 68.10(18.32) 18.43(5.39) 5.22(6.24)

0.00270.4529 0.061500.3193 0.004740.5075 0.004740.4457 0.004740.5075 0.004740.4457 0.004740.5075 0.004740.4457 0.094100.5075 0.094100.4457 0.0810.953 0.0800.911 0.0810.942

0.86(0.82)

54.05(4.90) 2.45(2.44)

72.04 (9.53)

71.53 (11.09)

69.77 (8.02)

64.94 (9.69)

72.42 (6.55)

68.61 (8.92)

70.88 (9.92) 69.95 (6.13)

80.79 (3.76)

1.59

1.91

NRTLa (2)

weight fraction c

Eyring-segmentbased-liquid-NRTLa (2)

Eyring-polymer-

range of polymer

6.48

5.24 2.44

26.45

24.83

19.78

19.85

17.83

19.57

18.26 14.90

12.34

2.94

c

2.48c

Wilson (2)

Eyring- polymer-

10.79 6.53

0.95c 90.08 80.28 79.48 79.30 78.55 77.26 76.49 88.39

c

90.08 80.28 79.48 79.30 78.55 77.26 76.49 88.39

88.39 76.09 24.46 7.63

88.39 75.31 20.93 0.88

6.10

1.55c

6.64c

Eyring-

NRF (2)

0.89

4.31 2.05

5.80

8.07

5.63

8.55

8.99 5.73

4.31

0.75 (0.84)

1.58 (1.68)

mNRFb (4)

Eyring- polymerNRF- Wilson (2)

Eyring- polymer-

a

100 3 AARD values reported in paranthesis have been obtained with R = 0.05. b 100 3 AARD values of Eyring-mNRF model with considering two parameters have been given in parentheses. c 100 3 AARD values have been taken from literature.10,13

PVP þ H2O

47

PVP þ H2O47

systems

(g 3 mol1)

Table 6. Continued


Figure 3. Experimental and calculated viscosity data, η, with EyringmNRF model for PEG192 þ 1,2-dimethoxyethane and PEG408 þ 1,2-dimethoxyethane systems at 298.15 K: (O) PEG192 þ 1,2-dimethoxyethane; (Δ) PEG408 þ 1,2-dimethoxyethane; (—) EyringmNRF model.

Figure 4. Difference between experimental and calculated viscosity data, η, for PPG þ 2-propanol system at 298.15 K with different models: (0) Eyring-segment-based-liquid-NRTL; (Δ) Eyring-polymer-NRTL; (O) Eyring-polymer-Wilson; () Eyring-polymer-NRF; (/) Eyringpolymer-NRF-Wilson; (2) Eyring-mNRF.

correlation of viscosity values was compared with that obtained from the other investigated local composition models using the temperature dependency of parameters similar to that in eq 12; the results are collected in Table 8. As can be seen from this table, it is obvious that local composition models considering temperature dependency as in eq 12 produce good results for viscosity. However, the molar mass of polymer in PS þ styrene system is large and the fitting quality of different local composition models considering temperature dependency as in eq 12 using a1w = a1s and b1w = b1s is not satisfactory. Therefore, we fitted the viscosity values of this system with conditions a1w 6¼ a1s and b1w 6¼ b1s ; the obtained results are reported in Tables 7 and 8. These conditions

8256



ARTICLE

Table 7. Parameters of Eyring-mNRF Model Using Temperature Dependency as Eq 12 along with Absolute Average Relative Deviation, 100 3 AARD, Obtained from Fitting of Viscosity Values of the Polymer þ Solvent Systems Mn

Eyring-

(g 3 mol1)

system

T (K)

NP

103a0w

103a1w

103a0s

2.967 (11.23)

1.934 (1.451)

1.245 (1.147) 8.925


400

298.15 308.15

33

PPG þ

400

298.15

88

ethanol46

328.15

PPG þ

400

2-propanol46

298.15

4088

0.2123

4.240 (6.742)

298.15 328.15

(11.110)

(22.290)

109

328.15

PVP þ H2O47

7.268

1.841 (33.910)

0.4369 (330.900) 18.260

PS þ styrene52

166000

293.15 333.15

82

PS þ styrene52

166000

293.15

78

103b0w

103b1w

103b0s

61.320

52.010

98.750

14.300

10.970

41.040

103b1s

1.53 (4.48)

5.450

19.170

15.200

22.000

1.21

(7.728)

(3.11) 78.010

1.901

(25.320)

1.377

114.100

1.26

(34.660)

8.874

mNRFa 1.38 (1.49)

(12.250)

0.4376

108

103a1s

(1.37)

27.780

10.180

2.018

23.700

26.100

15.130

10.300

7.065

0.938

24.450

0.3624

25.560

1.391

314.800

202.500

17.600

5.309

74.920

62.340

8.137

11.00 2.875

2.573

8.89

0.005571

8.80

333.15 a

0.451

17.13

0.2362

6.33

Parameters and 100 3 AARD values of Eyring-mNRF model with considering two parameters have been given in parentheses.

Table 8. Absolute Average Relative Deviation, 100 3 AARD, of Different Viscosity Models Using Temperature Dependency as Eq 12 for the Polymer þ Solvent Systems EyringMn (g 3 mol1)

system

T (K)

NP

Eyring-

segment-

Eyring-

Eyring-

polymer-

based-liquid-

polymer-

polymer-

Eyring- polymer-

Eyring-

NRTL (3)

NRTL (3)

Wilson (3)

NRF (3)

NRF- Wilson (3)

mNRFa (6)


400

298.15308.15

33

12.77

1.77

2.15

1.54

1.45

1.38 (1.49)a

PPG þ ethanol PPG þ2-propanol46

400 400

298.15328.15 298.15328.15

88 109

6.97 1.86

1.59 1.42

2.06 1.37

1.93 1.30

2.23 1.45

1.53 (4.48)a 1.21 (3.11)a

4088

298.15328.15

108

1.70

1.87

1.36

1.36

1.26 (1.37)a

19.50

81.49

81.49

11.00 (8.89)c

18.12

80.96

80.96

8.80 (6.33)c

46

PVP þ H2O47 PS þ styrene

52

166000

PS þ styrene

52

166000

293.15333.15 293.15333.15

1.68

81

70.05 (11.23)

77

b

b

70.05 (11.23)

b

b

69.77 (9.12)

69.77 (9.15)

a

b

100 3 AARD values of Eyring-mNRF model with considering two parameters have been given in parentheses. 100 3 AARD values reported in paranthesis have been obtained with R = 0.05. c 100 3 AARD values reported in paranthesis have been obtained with considering a1w 6¼ a1s and b1w 6¼ a1s in eq 12.

Table 9. Parameters of Eyring-mNRF Model Using Temperature Dependency as Eq 13 along with Absolute Average Relative Deviation, 100 3 AARD, of Eyring-mNRF and Eyring-NRTL Models Obtained from Viscosity Fitting of Nanofluids system

NP

range of j2

aw

CuOH2O

33

0.010.09

Al2O3H2O

88

0.010.094

as

bw

bs

Eyring-mNRF-

Eyring-NRTL

164700.000

42690.000

25550.000

7077.000

3.32

5.22

1682000.000

23930.000

211800.000

6618.000

9.02

9.80

improved the obtained results of the local composition models, especially for mNRF. However, in this case the obtained results with the Eyring-polymer-NRF and Eyring-polymer-NRF-Wilson models are not satisfactory. For examination of the performance of the Eyring-mNRF model for systems of nanofluids, we also used this model in the correlation of viscosity values of nanofluids CuOH2O and Al2O3H2O. The viscosity data of only two systems of nanofluids are available in the literature27 as Supporting Information; for other systems of nanofluids the viscosity data are only schematically shown in the literature. Viscosity data of nanofluids

are measured in different temperatures; therefore, the temperatures dependency given by eq 13 was used for fitting viscosity values with the Eyring-mNRF model. The obtained results are collected in Table 9, and these are compared with those obtained from the Eyring-NRTL model.27 We also considered eq 12 and other equations suggested by Kole et al.26 for temperature dependency in the fitting of viscosity values of nanofluids with our model; however, rather high AARD values obtained indicated that these equations are not suitable for this purpose. To see the performance Eyring-mNRF and Eyring-NRTL models in a better manner, the difference between calculated and 8257



ARTICLE

Table 10. Parameters of Eyring-mNRF Model Using Temperature Dependency as Eq 13 along with Absolute Average Relative Deviation, 100 3 AARD, of Eyring-mNRF and EyringNRTL Models Obtained from Viscosity Fitting of Nanofluids at Different Particle Volume Fractions, u2 Eyring- Eyring103aw

j2

103as

103bw

103bs mNRF NRTL

CuOH2O 0.01 17940.000

161.200 2245.000

22.910 2.59

0.045 2759.000

90.250 347.600

12.700 2.78

2.471 3.487

0.07 8466.000

916.300 62.440

21.530 2.11

2.301

0.09 792.300

35.670 12.920

8.485 3.05

3.370

Al2O3H2O

Figure 5. Difference between experimental and calculated viscosity data, η, for nanofluids of CuOH2O at different temperatures and concentration ranges with different models: (0) Eyring-NRTL; (Δ) Eyring-mNRF.

experimental viscosity values versus the particle volume fraction, j2, has been plotted in Figure 5 for the CuH2O system at different temperatures as an example. As can be seen from Table 9 and Figure 5, the fitting quality is improved by utilizing the Eyring-mNRF model, especially for nanofluids CuOH2O. The fitting procedure has also been made with the Eyring-mNRF model at each particle volume fraction with the results reported in Table 10. From this table one can concluded that in each particle volume fraction the performance of our model is better than that of the Eyring-NRTL model recently used27 for such nanofluid systems. 3.3. Interpolation and Extrapolation Ability of mNRF Model. As a further check for the reliability of our proposed model, we also examined the behavior of eqs 9 and 10 in the interpolation and extrapolation outside of the available experimental solvent activity and viscosity data of binary polymer solutions for different conditions regarding composition and temperature. The results obtained from the interpolation and extrapolation of solvent activity and viscosity values are respectively given in Tables 11 and 12 for some systems. As can be seen from these tables, the performance of the proposed equation is satisfactory in the interpolation and extrapolation outside of the available experimental solvent activity and viscosity data of binary polymer solutions considered in these tables. Similar results are obtained for other polymer solutions. We also tested the behavior of mNRF model in the interpolation and extrapolation outside of the available experimental viscosity data of investigated nanofluids for different composition and temperature ranges. The results are also collected in Table 12 for CuOH2O nanofluid system as an example. From the obtained AARD values for this system it can be concluded that the mNRF model has a good performance in the interpolation of viscosity values outside particle volume fraction and temperature ranges. Similar results are obtained for the Al2O3H2O nanofluid system. In the extrapolation of the viscosity values, however, a rather high AARD value is obtained for these nanofluid systems.

0.01 53230.000

2.903 6720.000

25.550 2.29

2.570

0.04 38260.000

702.300 4821.000

88.920 1.08

1.808

0.07 3184.000

66.360 400.200

10.350 2.21

2.324

0.094 6.250 1010

3.615 6.250 1010 7.002 2.55

2.652

Table 11. Absolute Average Relative Deviation, AARD, of mNRF Model (Eq 9)a in the Interpolation and Extrapolation of Solvent Activity of Binary Polymer Solutions at Different Concentration and Temperature Ranges range

all data used in fitting

interpolation

extrapolation

dextran64800 þ H2O data in range of wp
0.2

range of wp < 0.25

or t < 30 °C and t > 40 °C

or t < 50 °C used in

used in fitting

fitting

wpb < 0.1c

0.002

0.002

0.002

0.1 e wp < 0.2

0.003

0.004

0.005

0.2 e wp
30 °C

0.003 0.009

0.003 0.009

0.003 0.01

Poly(4-hydroxystyrene)1500 þ Acetone data in range

data in range of

of wp < 0.35 and

wp < 0.46 or

wp > 0.46 or t < 35 °C

t e 35 °C

and t > 40 °C

used in

used in fitting

fitting

wp < 0.2 0.2 e wp < 0.3

1.00 0.23

1.52 0.83

1.08 0.11

0.3 e wp < 0.4

0.28

0.38

0.18

0.4 e wp < 0.5

0.22

0.70

0.20

wp g 0.5

0.47

0.67

0.80

t e 25 °C

0.32

0.65

0.29

25 < te 35

0.24

0.99

0.20

0.82

1.64

1.01

°C t g 40 °C 8258



ARTICLE

Table 11. Continued

Table 12. Continued

all data used in range

all data used in

fitting

interpolation

extrapolation

range

PPG400 þ 2-Propanol data in range of

data in range

wp < 0.3 and wp > 0.4

of wp < 0.46 or

or t e 35 °C and

t < 55 °C

t g 55 °C

used in fitting

fitting

interpolation

wp g 0.9

1.86

1.73

1.98

t < 35 °C

1.06

1.11

1.06

t g 30 °C

1.54

1.87

2.80

PPG400 þ Ethanol

used in fitting wp < 0.2 0.2 e wp < 0.3

0.06 0.08

0.06 0.09

0.04 0.07

0.3 e wp < 0.4

0.08

0.10

0.08

0.4 e wp < 0.5

0.22

0.21

0.21

wp g 0.5

0.19

0.21

0.69

t e 35 °C

0.07

0.06

0.06

35 < te 45

0.07

0.08

0.08

°C t > 45 °C

0.16

0.17

0.18

PS218000 þ Benzene data in range of

data in range of

wp < 0.8 and wp > 0.86

wp < 0.9 or

or t < 60 °C and t g 80

t < 80 °C used

extrapolation

data in

data in

range of

range of

wp < 0.4

wp < 0.85

and wp > 0.65

or t < 50 °C

or t < 40 °C

used

and t > 50 °C

in

used in fitting

fitting

wp < 0.2 0.2 e wp < 0.4

1.32 2.32

1.33 2.23

1.33 2.29

0.4 e wp < 0.6

1.81

1.97

1.84

0.6 e wp < 0.8

1.46

1.48

1.46

wp g 0.8

1.04

1.08

1.66

t < 30 °C

1.62

1.65

1.65

30 < t e 50 °C

1.39

1.36

1.35

t > 50 °C

1.71

1.74

1.82

PVP4088 þ H2O

°C used in fitting

in fitting

wp < 0.7

2.15

1.44

1.41

data in range

0.7 e wp < 0.8

1.32

1.61

1.29

of wp < 0.1 and

range

0.8 e wp < 0.9

1.56

2.22

1.02

wp > 0.3 or

of wp < 0.25

wp g 0.9 t < 60 °C

3.08 1.92

2.95 0.77

5.11 0.75

t < 40 °C and t > 50 °C

or t < 50 °C used

t g 60 °C

1.98

2.53

5.42

used in

in

fitting

fitting 0.03

a

Equation 12 was used for the temperature dependency of model parameters. b wp is weight fraction of polymer. c Interpolation and extrapolation have been made using specified values of weight fraction in the fitting. d Interpolation and extrapolation have been made using specified values of temperature in the fitting.

Table 12. Absolute Average Relative Deviation, AARD, of Eyring-mNRF Model (Eqs 5 and 10)a in the Interpolation and Extrapolation of Viscosity of Binary Polymer Solutions and the CuOH2O Nanofluid System at Different Concentration and Temperature Ranges

wp < 0.01

0.09

0.10

0.01 e wp < 0.1

0.71

0.65

0.58

0.1 e wp < 0.2

1.14

1.44

1.11

0.2 e wp < 0.3

1.73

2.02

1.63

wp g 0.3 t e 30 °C

2.39 1.49

2.28 1.49

2.77 1.49

30 < te 50 °C

1.43

1.42

1.42

t > 50

0.71

0.73

0.71

b

CuOH2O

all data used in range

fitting

interpolation

PEG400 þ Methyl Acetate data in

data in range

wp < 0.6 and

of wp < 0.88

wp > 0.88 or

or t < 35 °C

t < 30 °C and

used

t g 35 °C

in

wp < 0.6

1.28

used in fitting 0.78

fitting 1.17

0.6 e wp < 0.8

1.22

2.66

1.25

0.8 e wp < 0.9

0.94

1.01

1.20

data in range of

data in range of

j2c < 0.04 and

j2 < 0.08 or

j2 > 0.06 or t < 26.85 °C t < 56.85 °C

extrapolation

range of

data in

and t > 36.85 °C used in fitting

used in fitting 3.69

j2 < 0.04

4.09

3.89

0.04 e j2 < 0.08

3.01

3.96

2.37

j2 g 0.08

2.86

2.66

11.27

t e 26.85 °C

4.78

4.94

4.80

26.85 < t e 46.85

2.72

2.68

2.17

4.03

4.02

4.30

°C t g 46.85 °C a

Equation 12 was used for the temperature dependency of model parameters. b Equation 13 was used for the temperature dependency of model parameters. c j2 is volume fraction of CuO nanoparticle. 8259



4. CONCLUSION New excess Gibbs energy model, mNRF, has been developed on the basis of the local cells theory. The performance of the proposed model in the correlation of thermodynamic and transport properties of the binary polymer solutions has been tested. Fitting quality of mNRF model in the correlation of solvent activity and viscosity values have also been compared with the NRTL, Wilson, NRF, and NRF-Wilson models. It was found that the performance of mNRF model in the correlation of thermodynamic and transport properties of the binary polymer solutions is better than other local composition models. The proposed model has also been used in the correlation of viscosity values of nanofluids; and the obtained results were compared with the previously used Eyring-NRTL model. The obtained results indicated that Eyring-mNRF model in the fitting of viscosity of nanofluids accomplished better than Eyring-NRTL model. ’ AUTHOR INFORMATION Corresponding Author

*Fax: þ98 411 3340191. E. mail: [email protected].

’ ACKNOWLEDGMENT We are grateful to University of Tabriz Research Council for the financial support of this research. ’ NOMENCLATURE aw = solvent activity as = adjustable parameter a0s = adjustable parameter a1s = adjustable parameter aw = adjustable parameter a0w = adjustable parameter a1w = adjustable parameter b = adjustable parameter b1 = adjustable parameter bs = adjustable parameter bw = adjustable parameter b0s = adjustable parameter b1s = adjustable parameter b0w = adjustable parameter b1w = adjustable parameter c = adjustable parameter g = interaction energy gs = residual Gibbs energies per mole of cell with central segment of polymer gss = energies of interaction between segmentsegment gsw = energies of interaction between segment-solvent g0s = reference state Gibbs energy for cell with central segment gw = residual Gibbs energies per mole of cell with central solvent gws = energies of interaction between solvent-segment gww = energies of interaction between solventsolvent g0w = reference state Gibbs energy for cell with central solvent gex = molar excess Gibbs energy gex,mNF = molar excess Gibbs energy of modified-NRF equation gex* = molar excess Gibbs energy of activation for flow Mn = number-average polymer molar mass N = total number of polymer and solvent Ns = total number of polymer segment and solvent moles Np = total number of data points

ARTICLE

R = gas universal constant rp = number of polymer segments T = absolute temperature T0 = 273.15 K V = molar volume of solution Vi = molar volume of pure i Xs = effective mole fractions of segment Xss = effective local mole fractions of segment and segment Xsw = effective local mole fractions of segment and solvent Xw = effective mole fractions of solvent Xww = effective local mole fractions of solvent and solvent Xws = effective local mole fractions of solvent and segment xp = mole fraction of polymer xw = mole fraction of solvent Z = nonrandom factor

’ GREEK LETTERS R = nonrandomness factor η = solution viscosity ηi = viscosity of pure i ηp = polymer viscosity k = correction energy λw = adjustable parameter of Eyring-mNRF model λs = adjustable parameter of Eyring-mNRF model γ = activity coefficient j2 = particle volume fraction ωw = adjustable parameter of Eyring-mNRF model ωs = adjustable parameter of Eyring-mNRF model ’ SUBSCRIPTS 0 = reference state i = solute species j = solute species n = number p = polymer s = segment of polymer chain s0 = segment of polymer chain w = solvent molecule w0 = solvent molecule w00 = solvent molecule ’ SUPERSCRIPTS cal = calculated cofig = configurational ex = excess exp = experimental mNRF = modified NRF model * = activation state ’ REFERENCES (1) Flory, P. J. Thermodynamics of high polymer solutions. J. Chem. Phys. 1941, 9, 660–661. (2) Huggins, M. L. Solutions of long chain compounds. J. Chem. Phys. 1941, 9, 440–441. (3) Edmond, E.; Ogston, A. G. An approach to the study of phase separation in ternary aqueous systems. Biochem. J. 1968, 109, 569–576. (4) McMillan, W. G.; Mayer, J. E. The statistical thermodynamics of multicomponent systems. J. Chem. Phys. 1945, 13, 276–306. (5) Kang, C. H.; Sandler, S. I. Thermodynamic model for two-phase aqueous polymer systems. Biotechnol. Bioeng. 1988, 32, 1158–1164. 8260


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New Excess Gibbs Energy Equation for Modeling the Thermodynamic

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