Artificial Neural Network Modeling of Surface Tension for Pure Organic

Dec 9, 2011 - dx.doi.org/10.1021/ie2017459 |Ind. Eng. Chem. Res. 2012, 51, 561-566. ARTICLE pubs.acs.org/IECR. Artificial Neural Network Modeling of S...
0 downloads 9 Views 313KB Size
ARTICLE pubs.acs.org/IECR

Artificial Neural Network Modeling of Surface Tension for Pure Organic Compounds Aliakbar Roosta, Payam Setoodeh, and Abdolhossein Jahanmiri* School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran ABSTRACT: Surface tension as an important characteristic in much scientific and technological research is a function of liquid materials’ physical properties. Thus, it is desirable to have an accurate correlation between effective parameters and surface tension. This study investigates the applicability of artificial neural networks as an efficient tool for the prediction of pure organic compounds’ surface tensions for a wide range of temperatures. The experimental data gathered for training and verification of the network are related to a wide variety of materials such as alkanes, alkenes, aromatics, and sulfur, chlorine, fluorine, and nitrogen containing compounds. The most accurate network among several constructed configurations has one hidden layer with 20 neurons. The average absolute deviation percentage obtained for 1048 data points related to 82 compounds is 1.57%. The results demonstrate that the multilayer perceptron network could be an appropriate lookup table for the determination of surface tension as a function of physical properties.

’ INTRODUCTION Surface tension is a phenomenon in which the surface of a liquid, where the liquid is in contact with a vapor, i.e., its own vapor, or a gas, acts like a thin elastic sheet. If the surface is between two liquids or between a liquid and a solid, the phenomenon is called “interfacial tension”. Various intermolecular forces, such as van der Waals forces, draw the liquid particles together. Along the surface, the particles are pulled toward the rest of the liquid. There is always a tendency for the surface layer to minimize its area according to the total mass, constraints posed by the container, and external forces. The surface tension, generally employed as a quantitative index of this tension, is defined as the force exerted in the plane of the surface per unit length. Scientific and technological research in many areas needs data on surface tension of the used materials, e.g., thin layer technologies, microelectronics, electronic functional units, solgel technologies for material production, development of compound materials, phase separation techniques, matrix systems for chemical reactions, drug carriers, treatment of raw materials, chemical synthesis catalyzed by micelles, washing processes, and tertiary oil recovery. Many handbooks, e.g., the CRC Handbook of Chemistry and Physics1 and Surface Tension of Pure Liquids and Binary Liquid Mixtures,2 contain data or, at least, constants of fitting functions for the surface tension of liquids. Furthermore, several attempts have been made to predict surface tension based on thermodynamic properties. One of the simplest methods to estimate the surface tension of pure liquids and liquid mixtures was proposed by Macleod.3 It expresses the surface tension of a liquid in equilibrium with its own vapor as a function of the liquid and vapor phase densities according to eq 1. σ ¼ KðFl  Fv Þ4

Figure 1. Schematic diagram of the ANN model. Pc (bar), critical pressure; ω, acentric factor; Tr, reduced temperature; Tnbr, reduced normal boiling temperature; s, specific gravity at normal boiling point; σ (mN/m), surface tension.

Sugden4 modified this expression as follows: σ ¼ ½PðFl  Fv Þ4

Sugden called the temperature-independent parameter P the parachor and described a way to estimate it based on molecular structure. According to Escobedo and Mansoori’s study,5 there are various shortcomings associated with the use of eq 2. To overcome the shortcomings of the above equations, Escobedo and Mansoori5 proposed an expression for surface tension based on statistical thermodynamics. Their parameters are molar densities of liquid and vapor, reduced temperature, critical temperature, critical

ð1Þ

Received: August 6, 2011 Accepted: December 9, 2011 Revised: November 21, 2011 Published: December 09, 2011

where K is a constant which is independent of temperature and is a characteristic of the liquid and Fl and Fv are the molar densities of liquid and vapor, respectively. r 2011 American Chemical Society

ð2Þ

4

561

dx.doi.org/10.1021/ie2017459 | Ind. Eng. Chem. Res. 2012, 51, 561–566

Industrial & Engineering Chemistry Research

ARTICLE

pressure, and normal boiling temperature. The relation is similar to Sugden’s,4 where P is given by the following equations:   0:30066 9 þ 0:86442Τr P ¼ P0 ð1  Τr Þð0:37ÞTr exp Tr

deviation (AAD/%) was 1.05 for 94 organic compounds, mostly comprised of saturated and unsaturated paraffinic and aromatic compounds. Kavun et al.6 developed a quantitative structure activity relationship model for calculation of surface tension ranging from 9.49 to 67 mN/m. this model had eight parameters and was fitted to data for 72 organic compounds. The predictions for 22 test compounds using this model were 30% of the measured surface tension. Kumar et al.7 used artificial neural networks (ANNs) as an approach to predict surface tensions of 166 organic compounds at 20 °C, mostly comprised of saturated and unsaturated paraffinic and aromatic ones. Their model parameters were the liquid density, the refractive index at 20 °C, and the parachor. AAD/% of 38 test compounds was 3.24. Lin et al.8 used the gradient theory for prediction of the surface tensions of pure fluids and binary mixtures. In their study, the volume-translated Peng Robinson (VTPR) and SoaveRedlichKwong (VTSRK) equations of state were applied to determine the Helmholtz free energy density and the bulk properties. AAD values of the surface tensions for 83 pure components were less than 0.24 (mN/m). Fu et al.9 proposed a method for prediction of surface tensions of nonpolar fluids based on the density functional theory. They used the BarkerHenderson perturbation theory and statistical associating fluid theory to establish an equation of state. The

ð3Þ    13=12 R Tc P0 ¼ 39:6431 0:22217  0:00291042 Tbr 2 Pc 5=6 ð4Þ where P0 is a temperature-independent parameter and Tbr is the reduced normal boiling point. R* is equal to Rm/Rm,ref, where Rm is the molar refraction and Rm,ref is the molar refraction of the reference fluid (methane). The overall average absolute percent

Table 2. AAD/% Values, MSE Values, and R2 Values for Training Data and Test Data training data

Figure 2. Absolute average percent deviation between experimental data and ANN results vs number of neurons in hidden layer.

test data

AAD/%

MSE

R2

AAD/%

MSE

R2

1.41

0.09

0.999

1.95

0.11

0.998

Table 1. Parameters of Hidden and Output Layers hidden layer weight matrix

output layer bias vector

0.548 0.365

76.163 7.311

7.629 14.297

406.356 1.230

transposed weight vector

0.195 12.529

265.503 33.733

1.383 9.784 224.382

0.212

33.137

0.242

163.930

5.066

111.260

2.505

17.123

88.321

15.393

39.863

3.964

3.071

0.656

30.748

22.138

1.235

6.071

24.195

17.831

0.022

13.219

29.971

46.770

15.046

33.661

66.361

0.002

4.659

0.407

21.264

0.042

12.354

292.126

0.743

4.004

0.372

15.750

14.425

20.971

26.916

0.987 0.247

33.927 39.903

2.983 0.055

135.835 215.847

49.319 4.905

23.020 140.694

2.036 132.437

0.149

20.287

10.083

151.032

9.785

101.916

0.866

0.534

101.300

6.605

64.561

2.743

8.173

1.354

22.570

465.544

86.398

183.407

746.415

157.602

0.525

3.960

842.332

8.515

1.873

789.895

172.631

2.615 294.382

0.002

3.728

0.605

16.251

0.021

9.373

0.319

52.871

1.214

150.093

0.904

100.652

8.906

8.147 0.214

270.342 33.650

1.891 0.179

211.231 166.542

77.755 5.105

82.378 113.021

1.724 223.785

0.073

14.951

0.123

81.994

0.895

51.610

22.836

0.614

37.916

8.162

30.288

22.470

51.449

34.284

562

bias 0.685

dx.doi.org/10.1021/ie2017459 |Ind. Eng. Chem. Res. 2012, 51, 561–566

Industrial & Engineering Chemistry Research

ARTICLE

Table 3. Comparison between Correlated and Experimental Data of Surface Tension no.

compound name

formula

Tmin (K)

Tmax (K)

AAD/%

1

bromochlorodifluoromethane

CBrClF2

200

400

1.13

2

dichlorodifluoromethane

CCl2F2

130

370

2.72

3

bromotrifluoromethane

CBrF3

110

340

4.45

4

trifluoromethane

CHF3

210

270

2.02

5

methane

CH4

6

methanol

CH4O

7

methylamine

CH5N

204.15

308.15

0.45

8 9

ethene ethyne

C2H4 C2H2

116 183.15

180.65 223.15

0.52 1.73

10

ethane

C2H6

89.87

274.49

1.24

11

ethanol

C2H6O

158.5

513.92

1.37

12

acetic acid

C2H4O2

293.15

523.15

1.26

13

1,2-dichloropropane

C3H6Cl2

287.25

335.45

4.62

14

propylamine

C3H9N

292.35

314.75

0.98

15

propene

C3H6

193.15

250.25

2.15

16 17

2-propanone propanoic acid

C3H6O C3H6O2

200.15 238.05

353.15 413.45

1.76 2.68

18

propane

C3H8

273.15

360.63

5.40

19

1-propanol

C3H8O

243.16

513.15

1.55

20

2-methylpropene

C4H8

187.15

257.15

0.77

21

butanoic acid

C4H8O2

281.75

404.45

1.31

22

2-methylpropane

C4H10

113.74

370

2.79

23

1-butene

C4H8

218.15

293.15

0.73

24 25

2-methylpropanoic acid acetic acid ethyl ester

C4H8O2 C4 H8O2

273.15 293.15

405.35 513.15

0.46 4.90

26

2-methyl-2-propanol

C4H10O

298.15

338.15

1.04

27

diethyl sulfide

C4H10S

286.65

350.55

1.57

28

1-pentene

C5H10

243.15

298.15

1.45

29

2-pentanone

C5H10O

198.95

372.75

1.93

30

pentane

C5H12

144.18

407.25

0.96

31

pyridine

C5H5N

273.15

381.35

0.83

32 33

2-methyl-2-butene 1-pentanol

C5H10 C5H12O

203.15 243.52

343.15 553.15

0.84 1.50 5.83

96.84

183.68

1.69

175.59

512.64

1.22

34

tetradecafluorohexane

C6F14

293.15

432.25

35

chlorobenzene

C6H5Cl

227.95

580

1.90

36

benzene

C6H6

293.15

553.15

1.87

37

phenol

C6H6O

273.15

423.15

1.13

38

aniline

C6H7N

273.15

453.15

1.46

39

methylcyclopentane

C6H12

283.15

333.15

1.15

40 41

1-hexene 2-hexanone

C6H12 C6H12O

283.15 298.15

333.15 323.15

1.79 0.24

42

hexane

C6H14

175.12

447.13

1.38

43

2-methylpentane

C6H14

273.15

333.15

0.39

44

1-hexanol

C6H14O

242.95

593.15

2.08

45

hexadecafluoroheptane

C7F16

303.15

434.35

2.69

46

toluene

C7H8

178.15

270

0.79

47

2-methylphenol

C7H8O

285.65

457.35

0.97

48 49

2,4-dimethylpyridine methylcyclohexane

C7H9N C7H14

293.15 146.58

358.15 540

1.30 2.86

50

1-heptene

C7H14

273.15

348.15

0.62

51

2-methylpropanoic acid propyl ester

C7H14O2

288.45

360.05

0.91

52

heptane

C7H16

183.21

507.94

4.08

53

3-methylhexane

C7H16

253.15

363.15

0.79

54

1-heptanol

C7H16O

243.67

613.15

1.83

563

dx.doi.org/10.1021/ie2017459 |Ind. Eng. Chem. Res. 2012, 51, 561–566

Industrial & Engineering Chemistry Research

ARTICLE

Table 3. Continued no.

compound name

formula

Tmin (K)

Tmax (K)

AAD/%

55 56

1,2-dimethylbenzene ethylbenzene

C8H10 C8H10

273.15 178.16

373.15 620

0.51 1.45

57

1-octene

C8H16

283.15

373.15

1.46

58

octane

C8H18

218.51

522.48

2.27

59

2,2,3-trimethylpentane

C8H18

283.15

373.15

1.30

60

1-octanol

C8H18O

257.97

613.15

2.83

61

quinoline

C9H7N

303.15

673.15

2.57

62

propylbenzene

C9H12

283.15

373.15

0.83

63 64

1,2,4-trimethylbenzene 1-nonene

C9H12 C9H18

283.15 283.15

373.15 373.15

1.48 0.68

65

nonane

C9H20

283.15

393.15

0.85

66

2,2,3,4-tetramethylpentane

C9H20

283.15

333.15

0.71

67

1-nonanol

C9H20O

273.15

633.15

1.62

68

naphthalene

C10H8

363.15

673.15

3.03

69

butylbenzene

C10H14

283.15

373.15

0.96

70

decane

C10H22

283.15

393.15

0.82

71 72

1-decanol 1-methylnaphthalene

C10H22O C11H10

278.15 242.27

653.15 710

1.64 1.35

73

undecane

C11H24

283.15

393.15

0.60

74

1,1-biphenyl

C12H10

343.15

673.15

2.13

75

dodecane

C12H26

283.15

393.15

0.19

76

diphenylmethane

C13H12

293.15

373.15

1.18

77

tridecane

C13H28

283.15

393.15

0.33

78

tetradecane

C14H30

283.15

393.15

0.14

79 80

pentadecane hexadecane

C15 H32 C16H34

293.15 293.15

393.15 393.15

0.16 0.50

81

heptadecane

C17H36

273.15

393.15

0.95

82

1-ethoxyhexadecane

C18H38O

273.15

393.15

overall AAD/%

surface tensions for 18 pure nonpolar fluids were predicted with an AAD/% of 3.3. Romero-Martínez et al.10 developed a method to estimate and predict the surface tensions for isomers of pure hydrocarbons by using the surface tension value of the linear or normal member of a given hydrocarbon homologous series. This method was used to calculate surface tension values for 60 isomers with an AAD/% of 1.5. Artificial neural networks (ANNs) with different structures have been proven to be universal function approximators. This ability has been exploited for the approximation of chemical and physical properties.11 The major advantage of the ANN model is construction without detailed information of the underlying process. ANNs as black box modeling tools have already been used for many applications in industry, business, and science.12 Since in white box modeling approaches the model of development is based on the information of relevant equations and detailed knowledge for a specific system is usually not directly available, most efforts in the white box modeling approach are devoted to revealing all relevant mechanisms and quantifying these mechanisms correctly. This usually requires an extensive research program (including experiments, which can also be very time- and money-consuming). Here a compromise must be made in order to save time and money. Therefore, white box models often have limited accuracy because in developing the models minor mechanisms are neglected and only the major mechanisms are taken into account. The major advantage of the artificial neural networks is that they can be constructed

0.69 1.57

without the need for detailed knowledge of the underlying system. One of the applications of neural network models is to map an input space to an output space and function as a lookup table. Thus, in recent years, artificial neural networks have been applied to formulation of chemical and physical properties.1319

’ METHODS: ARTIFICIAL NEURAL NETWORK APPROACH In this study, applicability of a multilayer perceptron (MLP) neural network is investigated to develop an appropriate model for the prediction of surface tensions of organic compounds. The needed experimental data on surface tension were gathered from Surface Tension of Pure Liquids and Binary Liquid Mixtures,2 and the physical properties were gathered from the CRC Handbook of Chemistry and Physics.1 The data set prepared consists of the surface tension values ranging from 0 to 45 mN/m at various temperatures for 82 organic compounds. The total data set contains 1048 data points. About 70% of data is employed as training data, and the remaining 30% is applied for network verification. The MATLAB artificial neural network toolbox is employed in this study for model development and MLP training. A schematic diagram of the ANN model is illustrated in Figure 1. Several physical properties were considered and checked as input parameters, and after many attempts the most effective ones were selected as the inputs of the proposed network, which are critical 564

dx.doi.org/10.1021/ie2017459 |Ind. Eng. Chem. Res. 2012, 51, 561–566

Industrial & Engineering Chemistry Research

ARTICLE

(AAD) and mean square error (MSE) values: AAD=% ¼

MSE ¼

1 N jσexp  σ cal j  100 N i¼1 σ exp



1 N ðσ exp  σcal Þ2 N i¼1



ð5Þ

ð6Þ

where N is the number of data points and σexp and σcal are the available experimental data within the literature and calculated values for surface tension, respectively.

Figure 3. Surface tension as a function of temperature for different compounds. Solid lines represent model results, and points are experimental data.

Figure 4. Absolute average percent deviation as function of reduced temperature.

pressure (Pc, bar), acentric factor (ω), reduced temperature (Tr), reduced normal boiling temperature (Tnbr = Tnbp/Tc), and specific gravity at normal boiling point (s), and also the output is surface tension (σ, mN/m). The required number of training data points and hidden layer neurons are the two challenges that must be tackled appropriately. According to Cybenko,20 a network that has only one hidden layer is able to approximate almost any type of nonlinear mapping. However, the determination of the approximate number of neurons for the hidden layers is difficult and is often done by trial and error. The LevenbergMarquardt algorithm was used in the training procedure.2023 Different neural network topologies were compared using their average absolute deviation

’ RESULTS After many attempts, the best ANN obtained is a one-hiddenlayer MLP with 20 neurons in the hidden layer, as illustrated in Figure 2. The transfer function of the first layers is hyperbolic tangent sigmoid, and that of the second layer is a positive linear function. The parameters of the ANN structure are shown in Table 1. In addition, the AAD/%, MSE, and R2 values for training and test data are listed in Table 2. The results of the ANN model are reported in Table 3. It is noticed from this table that the overall AAD/% of 1.57 in surface tension estimation is obtained for all 82 compounds regarding 1048 data points. In order to show the predicting capabilities of the ANN model, the results relevant to eight different compounds are plotted in Figure 3 in order to demonstrate the model's accuracy. Figure 4 illustrates the effect of reduced temperature as an effective parameter on the network accuracy. As seen, when Tr approaches 1 (near critical temperature), the AAD/% values mostly increase. This is due to the fact that near critical temperature the exact experimental surface tension values which appear in the denominator for calculation of average absolute deviation are too low. ’ DISCUSSION One of the most significant advantages of the proposed approach is fewer input variables and more available parameters employed in comparison to the previous studies. For instance, specific gravity at the normal boiling point (a temperatureindependent term) is used as an input variable instead of (Fl  Fv), which is a function of temperature. Moreover, the current model is more general than the previous ones due to the wider range of temperature. To be noted, the average difference between the upper and the lower temperature bounds regarded in this study is 170 K, while that of Escobedo and Mansoori, as the most global and accurate study, is 97 K. In addition, this work considers greater variety for the components than the other studies. Besides the alkanes, alkenes, and aromatics, the model is trained by data sets for sulfur compounds as well as fluorine, nitrogen, and chlorine compounds. At first, it seems that the model proposed by Escobedo and Mansoori is more accurate than this one, regarding the values for AAD/% reported, 1.05 versus 1.57. However, this fact stems from the difference between the corresponding temperature ranges considered. In order to make crystal clear the ability of the current model, Table 4 comprises the AAD/% values related to the models for a number of compounds with similar temperature ranges regarded in both studies. 565

dx.doi.org/10.1021/ie2017459 |Ind. Eng. Chem. Res. 2012, 51, 561–566

Industrial & Engineering Chemistry Research

ARTICLE

Table 4. Comparison between Present Study and Escobedo and Mansoori’s work AAD/% formula

name

temperature range (K)

present study

Escobedo and Mansoori5

CH4

methane

90170

1.69

3.39

C2H6

ethane

90275

1.24

3.18

C3H8

propane

273360

5.40

4.38

C5H12

pentane

156440

0.96

2.30

C6H12

methylcyclopentane

283333

1.15

0.31

C6H14

hexane

175450

1.38

1.55

C7H16

heptane

183508

4.08

3.87

C8H10 C9H12

1,2-dimethylbenzene propylbenzene

273343 273373

0.51 0.83

0.62 1.04

C9H12

1,2,4-trimethylbenzene

283373

1.48

0.58

C9H18

1-octene

283373

1.46

0.25

C10H14

butylbenzene

273373

0.96

1.74

C10H22

decane

273393

0.82

1.28

C11H24

undecane

273393

0.60

1.31

C13H28

tridecane

283393

0.33

1.32

1.53

1.81

average

’ CONCLUSION An MLP network was constructed based on experimental data available for organic compounds to correlate the critical pressure, acentric factor, reduced temperature, reduced normal boiling temperature, and specific gravity at the normal boiling point with surface tension. The results demonstrate that the proposed model is appropriate to correlate the surface tensions of a wide variety of organic compounds such as alkanes, alkenes, aromatics, and sulfur compounds as well as fluorine, chlorine, and nitrogen compounds in a wide range of temperatures.

(11) Bernazzani, L.; Duce, C.; Micheli, A.; Mollica, V.; Sperduti, A.; Starita, A.; Tine, M. R. Predicting Physical-Chemical Properties of Compounds from Molecular Structures by Recursive Neural Networks. J. Chem. Inf. Model. 2006, 46, 2030. (12) Widrow, B; Rumelhart, D. E.; Lehr, M. A. Neural Networks: Applications in Industry, Business and Science. Commun. ACM 1994, 37, 93. (13) Alabi, S. B.; Williamson, C. J. Centrifugal Pump-Based Predictive Models for Kraft Black Liquor Viscosity: An Artificial Neural Network Approach. Ind. Eng. Chem. Res. 2011, 50, 10320. (14) Ghanadzadeh, H.; Fallahi, S.; Ganji, M. LiquidLiquid Equilibrium Calculation for Ternary Aqueous Mixtures of Ethanol and Acetic Acid with 2-Ethyl-1-hexanol Using the GMDH-Type Neural Network. Ind. Eng. Chem. Res. 2011, 50, 10158. (15) Gharagheizi, F.; Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Representation and Prediction of Molecular Diffusivity of Nonelectrolyte Organic Compounds in Water at Infinite Dilution Using the Artificial Neural Network-Group Contribution Method. J. Chem. Eng. Data 2011, 56, 1741. (16) Gharagheizi, F.; Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Determination of Critical Properties and Acentric Factors of Pure Compounds Using the Artificial Neural Network Group Contribution Algorithm. J. Chem. Eng. Data 2011, 56, 2460. (17) Sedighi, M.; Keyvanloo, K.; Towfighi, J. Modeling of Thermal Cracking of Heavy Liquid Hydrocarbon: Application of Kinetic Modeling, Artificial Neural Network, and Neuro-Fuzzy Models. Ind. Eng. Chem. Res. 2011, 50, 1536. (18) Gharagheizi, F.; Abbasi, R. A New Neural Network Group Contribution Method for Estimation of Upper Flash Point of Pure Chemicals. Ind. Eng. Chem. Res. 2010, 49, 12685. (19) Kumar, K. V. Neural Network Prediction of Interfacial Tension at Crystal/Solution Interface. Ind. Eng. Chem. Res. 2009, 48, 4160. (20) Cybenko, G. V. Approximation by superpositions of a sigmoidal function. Math. Control, Signals, Syst. (MCSS) 1989, 2, 303. (21) Marquardt, D. An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 1963, 11, 431. (22) Levenberg, K. A method for the solution of certain problems in least squares. SIAM J. Numer. Anal. 1944, 16, 588. (23) Hagan, M.; Menhaj, M. Training feedforward networks with the Marquardt algorithm. IEEE Trans. Neural Networks 1994, 5, 989.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: +98-711-2303071. Fax: +98-711-6474619. E-mail: jahanmir@ shirazu.ac.ir.

’ REFERENCES (1) Lide, D. R. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 2004. (2) Wohlfarth, C.; Wohlfarth, B. Surface Tension of Pure Liquids and Binary Liquid Mixtures; Springer: New York, 1997. (3) Macleod, D. B. Relation between surface tension and density. Trans. Faraday Soc. 1923, 19, 38. (4) Sugden, S. The Variation of surface tension with temperature and some related functions. J. Chem. Soc. 1924, 125, 32. (5) Escobedo, J.; Mansoori, G. A. Surface tension prediction for pure liquids. AIChE J. 1996, 42, 1425. (6) Kavun, S. M.; Chalykh, A. E.; Palyulin, V. A. Prediction of surface tension of organic liquids. Colloid J. 1995, 57, 767. (7) Kumar, D.; Gupta, S.; Basu, S. Prediction of surface tension of organic liquids using artificial neural networks. Indian Chem. Eng. 2005, 47, 219. (8) Lin, H.; Duan, Y. Y.; Min, Q. Gradient theory modeling of surface tension for pure fluids and binary mixtures. Fluid Phase Equilib. 2007, 254, 75. (9) Fu, D.; Lu, J. F.; Liu, J. C.; Li, Y. G. Prediction of surface tension for pure non-polar fluids based on density functional theory. Chem. Eng. Sci. 2001, 56, 6989. (10) Romero-Martínez, A.; Trejo, A.; Murrieta-Guevara, F. Surface tension of isomers of pure hydrocarbons: a method for estimation and prediction. Fluid Phase Equilib. 2000, 171, 1. 566

dx.doi.org/10.1021/ie2017459 |Ind. Eng. Chem. Res. 2012, 51, 561–566