Mathematical modeling for Thermodynamics

7/25/2019 Mathematical modeling for Thermodynamics

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Universit Politecnica delle MarcheScuola di Dottorato di Ricerca in Scienze dellIngegneria

Curriculum in Energetica----------------------------------------------------------------------------------------

Mathematical modeling for

Thermodynamics:

Thermophysical Properties and Equation

of State

Ph.D. Dissertation of:

Mariano Pierantozzi

Advisor:

Prof. Giovanni Di Nicola

Curriculum supervisor:

Prof. Renato Ricci

XII edition


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Universit Politecnica delle MarcheDipartimento di Ingegneria Industriale e Scienze Matematiche

Via Brecce Bianche 60131 - Ancona, Italy


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A Stefano, Cecilia, e.

Da giovane studiai per ostentazione.Poi, un poco, per istruirmi.

Ora per divertirmi. Mai, per, per guadagno.Montaigne


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Contents

1.1 Introduction

1.2 Terminology and units

1.3 Methods of measurement

1.4 An overview of existing equations

1.5 Results

1.5.1 A Scaled Equation for the Surface Tension of Pure Fluids 1.5.2 A new scaled equation for the calculation of surface tension ofketones

1.5.3 A new Scaled Equation and an Artificial Neural Network forcalculation of surface tension of Alcohols 1.5.4 Surface tension prediction for refrigerant binary systems

1.6 Bibliografia

2.1 Introduction



2.3.1 Equation of Weber 2.3.2 First equation of Smith 2.3.3 Equation of Palmer

2.3.4 Equation of Vargaftik 2.3.5 Equation of Robbins and Kingrea 2.3.6 Equation of Bridgmann 2.3.7 Equation of Osida 2.3.8 Equation of Hirschfelder, Curtiss and Bird 2.3.9 Equation of Viswanath 2.3.10 Equation of Kardos 2.3.11 Equation of Sakiadis and Coates 2.3.12 First equation of Badea 2.3.13 Equation of Narasimhan, Swamy and Narayama 2.3.14 Second equation of Smith 2.3.15 Equation of Denbigh

2.3.16 Equation of Sheffy and Johnson 2.3.17 Equation of Missenard 2.3.18 Second equation of Badea 2.3.19 First equation of Pachaiyappan, Ibrahim, and Kuloor 2.3.20 Second equation of Pachaiyappan, Ibrahim, and Kuloor

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2.3.21 First equation of Pachaiyappan

2.3.22 Second equation of Pachaiyappan 2.3.23 Equation of Pachaiyappan, and Vaidyanathan 2.3.24 Equation of Vaidyanathan, and Velayutham 2.3.25 Equation of Reid and Sherwood 2.3.26 First Equation of Reid, Prausnitz, and Poling 2.3.27 Second Equation of Reid, Prausnitz, and Poling 2.3.28 Equation of Sastri 2.3.29 Equation of Gharagheizi, Ilani-Kashkouli, Sattari, Mohammadi,Ramjugernath, and Richon 2.3.30 Equation of Latini

2.4 Results

.

2.5 A Critical Survey of Thermal Conductivity Literature Data for Organic

Compounds at Atmospheric Pressure and a modification of Latini Equation for

Aromatic Compounds

2.6 Bibliografia

3.1 Introduction


3.3 Results

3.4 References

4.1 Introduction


4.3 Methods of measurement

4.4 An overview of existing equations 4.5 Results Viscosity model based on Patel-Teja equation of state for pure

liquid refrigerants 4.6 References

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Table 2.11. Statistical overview for each family.................................................................147

Table 2.12. Summary of the experimental data ranges for the aromatic compounds andcoefficients for Eq. (2.42)..........................................................................................150Table 2.13. Summary of the coefficients. ...........................................................................151Table 2.14. AAD% comparison between Equation(2.42), Eq. (2.42) Attempt 1 and Eq.

(2.42) Attempt 2 ........................................................................................................151Table 2.15. AAD% comparison between different equations.............................................153Table 3.1. Summary of experimental data for each family.................................................163Table 3.2. Coefficients ai, bi, ci, di, and eifor each group...................................................165Table 3.3. Summary of deviations for Equations (3.5) (3.8), (3.11), (3.16), (3.18) and

(3.20). ........................................................................................................................171Table 4.1. Summary of collected data, constant coefficients of the model, and AAD%

deviations, and ,e0, e1coefficients..........................................................................180Table 5.1 Most important modification of attractive term..................................................192

Table 5.2 List of modification of repulsive term ................................................................195Table 5.3. Properties of compounds chosen for comparison ..............................................196Table 5.4. AAD% butane comparison between equations PR, CSD, PRSVI.....................200Table 5.5. AAD% pentane comparison between equations PR, CSD, PRSVI...................201Table 5.6. AAD% propane comparison between equations PR, CSD, PRSVI ..................202Table 5.7. propene comparison between equations PR, CSD, PRSVI ...............................202Table 5.8. R14 comparison between equations PR, CSD, PRSVI......................................203Table 5.9. R32 comparison between equations PR, CSD, PRSVI......................................204Table 5.10. Comparison between new method and classical method for n-butane ............212Table 5.11. Comparison between new method and classical method for n-pentane ..........213Table 5.12. Comparison between new method and classical method for propane .............215Table 5.13. Comparison between new method and classical method for propene .............216Table 5.14. Comparison between new method and classical method for R14 ...................217

Table 5.15. Comparison between new method and classical method for R32 ...................218

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Figure 1.1 Capillary Rise Method

This phenomenon provides a simple and accurate technique for the measurement of thesurface tension of a liquid as shown inThis phenomenon is exploited both in differential and singular capillary rise method(DCRM and SCRM, respectively). The experimental cell generally contains multiplecircular tubes of small section in the case of DCRM and a single pipe in the case of SCRM.The accuracy of the method is of the order of 0.05-0.1 mN m-1in most cases.

The well known Young-Laplace equation, states that the capillarity rise hof a liquid in acapillary tube is given by

(1.1)

where , and are the liquid and vapor density respectively, is the liquid height into thetube, is the acceleration due the gravity, is the contact angle between the liquid and thewall of the tube, and is the radius of the capillary tube. The method is only accurate for = 0. If the radius of tube is appreciable, then r is replaced by b where b refers to the radiusof curvature of the liquid, and a relation between b and r must be found. The tube must beof uniform bore and the height h must be measured from the flat surface in the reservoir.A recently adopted method of measurement is the Surface Light Scattering (SLS),consisting of the Dynamic Light Scattering (DLS) application to fluid surfaces. The DLSmakes it possible to determine, under conditions of balance, various thermodynamicproperties of fluids, including surface tension, without any direct contact. The method isbased on the analysis of light diffusion caused by microscopic fluctuations on the fluid

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Tbr is the reduced normal boiling temperature, Tb/Tc, and Tb is the normal boilingtemperature (K).In some cases, the surface tension can be represented in terms of an additional parameter.One example is the so-called acentric factor, , initially introduced as an empiricalparameter by Pitzer to explain the deviation from the corresponding states principle, asdefined for noble gases, when applied to larger molecules. Pitzers [17] relation in terms ofTc(K),Pc(bar), and leads to the following corresponding state relationship for :

=Pc

2

3Tc

1

31.86 +1.18

19.05

3.75 + 0.91

0.291 0.08

2

3

1 Tr( )

11

9 (1.8)

For compounds that exhibit hydrogen bonds, Sastri and Rao [18] proposed a modifiedexpression for the Brock and Bird correlation:

= KPc

xT

b

yT

c

z 1 Tr1 T

br

m

(1.9)

wherePcis in bar. Coefficients were regressed for different families separately. The valuesfor the constant of the equation (1.9) in reported in Table 1.1

Table 1.1. Coefficients for Sastri Rao Equation

K x y z mAcids 0.125 0.50 -1.5 1.85 11

9

Alcohols 2.28 0.25 0.175 0 0.8Allothers

0.158 0.50 -1.5 1.85 11

9

To underline the work done in this research field, other recent equations have beenproposed. Miqueu et al. [19] proposed the following:

( ) ( )

2

3

1.26 0.5

4.35 4.14 1 0.19 0.487a

c

c

N

kT t t t V = + +

(1.10)

where , , , , , are the Boltzmann constant, Avogadro number, the

critical volume, and the acentric factor, respectively, and it represents the surface tension

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)

Radius ofryration (A)

Criticaldensity

(molcm-3)Tb (K) m q

@T

b

373.15

1-heptene Alkenes 19 14.35-36.735154.12-353.15

537.40 4.08 0.0025 366.79 -0.11 53.50 12.31

1-hexene Alkenes 12 14.4-38.182133.39-333.15

504.00 3.66 0.0029 336.63 -0.12 53.25 13.49

1-octene Alkenes 14 14.1-36.024

171.45-

373.15 566.90 4.46 0.0022 394.41 -0.11 54.09 11.25

1-pentadecene Alkenes 11 20.37-28.17283.15-373.15

708.00 6.64 0.0011 541.61 -0.09 52.71 5.77

1-tetradecene Alkenes 11 19.77-28.56273.15-373.15

691.00 6.40 0.0012 524.25 -0.09 52.56 6.50

Propene Alkenes 12 12.73-21.41193.15-253.1

364.85 2.23 0.0054 225.45 -0.14 49.42 16.78

1,2-dimethylbenze

ne

Aromatics

33 21.3-32.28273.1-373.15

630.30 3.84 0.0027 417.58 -0.11 61.89 16.62

1,3-dimethylbenze

ne

Aromatics

22 19.8-30.88273.1-373.15

617.00 3.94 0.0027 412.27 -0.11 61.00 15.71

1,4-dimethylbenze

ne

Aromatics

19 19.6-28.98288.15-373.15

616.20 3.83 0.0026 411.51 -0.11 60.13 15.63

Antracene Aromatic 11 4.5689-26.792 488.93- 873.00 4.98 0.0018 615.18 -0.07 62.86 16.79

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

1,3-dioxane Epoxides 11 3.9716-44.216228.15-

531.590.00 2.97 0.0042 378.15 -0.13 72.45 22.46

1,3-propyleneOxyde

Epoxides 11 4.305-27.85 298.-468. 520.00 2.23 0.0053 321.00 -0.14 68.75 24.08

1,4-dioxane Epoxides 10 22.32-34.45288.15-373.15

587.00 3.02 0.0042 374.47 -0.14 75.36 22.06

2,3-dihydrofuran

Epoxides 11 4.2252-42.174 176.-471.6

524.00 2.60 0.0049 327.65 -0.13 63.70 21.76

2,5-dihydrofuran

Epoxides 11 4.2779-46.232171.-487.8

542.00 2.58 0.0049 339.00 -0.13 67.52 22.80

2-methyl-1,3-dioxolane

Epoxides 164.11141-39.773

212.1-494.28

549.20 3.03 0.0040 355.15 -0.13 65.50 20.71

2-methylbenzofu

ranEpoxides 11 4.436-35.55 290.-620. 697.00 4.05 0.0025 470.15 -0.09 61.79 17.59

3,5-dimethylbenzo

furanEpoxides 16

3.81159-39.128

257.7-642.33

713.70 4.49 0.0022 491.60 -0.09 61.16 16.24

Dibenzofuran Epoxides 9 8.0113-33.543355.31-664.34

824.00 4.71 0.0020 558.31 -0.08 61.92 15.95

EthyleneOxyde

Epoxides 26 24.3-36.4221.15-293.15

469.15 1.94 0.0071 283.60 -0.16 72.00 25.93

- 193.15- -

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

IsobutylMercaptan

Mercaptans

14 17.87-24.52288.15-353.15

559.00 3.46 0.0033 361.64 -0.10 54.05 17.00

IsopropylMercaptan

Mercaptans

10 18.39-22.5288.15-323.15

517.00 2.99 0.0039 325.71 -0.12 56.81 18.10

MethylMercaptan

Mercaptans

7 20.73-26.44282.95-316.65

469.95 1.61 0.0069 279.11 -0.17 74.44 27.08

N-butylMercaptan

Mercaptans

14 18.93-26.36 288.15-353.15

570.10 3.40 0.0033 371.61 -0.12 60.34 16.73

N-decylMercaptan

Mercaptans

11 4.6697-33.332247.56-617.56

696.00 5.71 0.0016 512.35 -0.08 51.57 12.15

N-dodecylMercaptan

Mercaptans

11 3.4938-32.715265.15-651.6

724.00 6.37 0.0014 547.75 -0.08 51.19 10.04

N-heptylMercaptan

Mercaptans

16 3.6026-34.483229.92-579.92

645.00 4.64 0.0022 450.09 -0.09 53.18 13.91

N-hexylMercaptan

Mercaptans

25 4.2611-38.273192.62-552.62

623.00 4.25 0.0024 425.81 -0.09 54.97 15.04

N-nonylMercaptan

Mercaptans

11 4.123-33.08253.05-603.05

681.00 5.34 0.0018 492.95 -0.08 52.70 12.16

N-octylMercaptan

Mercaptans

24 4.0005-35.399223.95-593.95

667.30 5.01 0.0019 472.19 -0.08 52.32 13.13

N-pentylMercaptan

Mercaptans

16 19.51-27.26284.05-363.15

598.00 3.80 0.0028 399.79 -0.10 55.50 15.92

N-propyl Mercapta - 288.15- -

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

Hydroperoide s 613.15DicumylPeroxide

Peroxydes

11 12.09-38.32311.15-661.15

884.00 6.54 0.0011 669.00 -0.07 60.74 10.60

Di-t-butylPeroxide

Peroxydes

11 3.319-22.45233.15-483.15

547.00 4.53 0.0020 384.15 -0.08 39.84 10.54

Ethylbenzene

Hydroperoxide

Peroxyde

s 10 4.3664-37.832

290.-

663.3 737.00 4.49 0.0024 536.00 -0.09 62.42 14.71M-

diisopropylbenzene

Hydroperoxide

Peroxydes

11 3.277-29.99 264.-724. 812.00 5.49 0.0015 620.00 -0.06 43.93 8.24

N-butylbenzenehydroperoxyde

Peroxydes

15 4.39-42.266.7-686.25

762.50 5.53 0.0019 574.90 -0.09 64.05 12.88

N-butylhydroper

oxide

Peroxydes

11 3.9114-36.706213.-508.5

565.00 3.52 0.0034 405.00 -0.11 58.99 14.48

N-propylbenzeneHydroperoxyd

e

Peroxydes

15 3.95-37.1259.4-672.12

746.80 5.09 0.0021 555.30 -0.08 56.32 12.11

P-diisolpropylbe

Peroxydes

8 6.3371-31.489315.67-677.33

810.00 5.38 0.0015 616.00 -0.07 52.41 9.79

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

nzeneHydroperoxide

P-menthaneHydroperoxide

Peroxydes

11 4.87-36.17 250.-630. 711.00 4.60 0.0018 532.00 -0.08 55.54 12.14

T-butylHydroperoxyd

e

Peroxyde

s

11 3.4978-23.336277.45-

496.8

552.00 3.30 0.0034 388.15 -0.09 47.97 13.00

2,3,5-trimethylthiop

hene

Sulphides

15 3.7363-39.158215.-

591.12656.80 3.92 0.0025 437.65 -0.09 57.96 16.88

2,3-dimethylbenzo

thiophene

Sulphides

15 4.128-42.12282.15-707.4

786.00 4.49 0.0022 538.00 -0.09 65.68 17.85

2,5-dimethylthiop

hene

Sulphides

4 23.45-30.57293.15-358.15

629.90 3.50 0.0030 409.95 -0.11 63.11 17.28

2,7-dimethylbenzo

thiophene

Sulphides

164.83083-40.463

341.5-695.16

772.40 4.52 0.0021 528.70 -0.10 73.69 20.54

2-butylbenzothio

phene

Sulphides

15 3.337-21.6425.65-736.83

818.70 5.36 0.0016 581.80 -0.06 46.20 12.04

2- Sulphide - 282.35- -

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

ethylbenzothiophene

s 696.6

2-ethyl-tetrahydrothio

phene

Sulphides

18 3.965-44.556201.2-592.02

657.80 3.74 0.0026 429.15 -0.10 62.37 19.01

2-

ethylthiophene

Sulphide

s 7 23.45-31.1

293.15-

358.15 626.60 3.59 0.0030 407.15 -0.12 65.42 17.212-menthyl

Benzothiophene

Sulphides

16 3.955-36.485324.65-691.83

768.70 4.19 0.0022 514.70 -0.09 63.96 18.41

2-methylthiacycl

opentane

Sulphides

3 28.61-30.63293.15-313.15

628.40 3.35 0.0030 405.60 -0.10 60.23 19.27

2-methylThiaindan

Sulphides

15 4.0597-42.239273.3-709.74

788.60 4.24 0.0022 521.80 -0.09 64.54 19.00

2-methylthiophe

ne

Sulphides

7 22.67-31.7209.79-327.07

609.00 3.30 0.0036 385.71 -0.07 48.33 20.04

2-n-propylthiophe

ne

Sulphides

4 23.12-29.67293.15-358.15

649.20 4.06 0.0026 431.65 -0.10 59.22 15.70

2-pentanehiolSulphide

s15 3.68521-43.16

160.8-525.9

584.30 3.87 0.0026 385.15 -0.11 58.55 16.98

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

2-propylbenzothi

ophene

Sulphides

15 2.9792-21.044387.2-726.75

807.50 4.90 0.0017 563.60 -0.05 41.26 11.29

3,4-dimethylthiop

hene

Sulphides

16 3.68913-39.67216.-

578.52642.80 3.64 0.0029 418.15 -0.10 59.73 18.37

3,5-dimethylbenzo

thiophene

Sulphides

164.20012-43.415

265.19-706.32

784.80 4.66 0.0021 537.21 -0.09 65.20 17.70

3-ethyl-2,5-dimethylthiop

hene

Sulphides

15 4.0682-45.198195.-602.1

669.00 4.30 0.0024 456.15 -0.10 62.67 16.73

3-ethylbenzothio

phene

Sulphides

164.92354-52.651

252.2-698.85

776.50 4.68 0.0021 531.50 -0.11 77.46 21.02

3-methylthiophe

ne

Sulphides

10 24.51-32.62293.15-358.15

613.00 3.33 0.0036 388.55 -0.12 66.31 20.98

4-methyldibenzo

thiophene

Sulphides

15 4.5879-45.194339.15-813.69

904.10 5.06 0.0018 622.80 -0.09 72.47 19.43

Benzothiophene

Sulphides

163.40542-47.584

304.5-687.6

764.00 3.87 0.0026 494.05 -0.11 79.40 22.69

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

CarbonDisulfide

Sulphides

16 26.45-42.47230.75-333.15

552.00 1.57 0.0063 319.38 -0.16 78.01 28.36

Dibenzothiophene

Sulphides

11 4.0321-35.664371.82-807.3

897.00 4.94 0.0020 604.61 -0.07 61.68 17.89

DicyclohexylSulfide

Sulphides

15 3.1638-36.267283.05-721.8

802.00 5.41 0.0014 559.00 -0.08 56.02 14.02

DiethylDisulfide

Sulphides

30 4.299-45.39 171.63-571.63

642.00 3.62 0.0028 427.13 -0.10 60.94 17.39

DiethylSulfide

Sulphides

14 17.3-26.5283.15-363.15

557.15 3.21 0.0031 365.25 -0.11 58.36 17.05

Diisopentylsulfide

Sulphides

9 17.64-26.6288.35-394.95

664.40 5.78 0.0015 488.45 -0.09 52.06 9.32

DiisopropylSulfide

Sulphides

16 3.09-33.78195.07-526.68

585.20 3.98 0.0024 393.16 -0.09 50.56 14.34

DimethylDisulfide

Sulphides

16 3.9269-49.19188.44-553.5

615.00 2.94 0.0040 382.90 -0.12 69.96 23.04

DimethylSulfide

Sulphides

11 19.87-26.5284.25-330.85

503.04 2.37 0.0050 310.48 -0.14 64.40 22.48

Di-n-butylSulfide

Sulphides

12 20.94-27.4291.45-360.95

652.00 4.89 0.0019 461.95 -0.09 53.95 11.73

Di-n-octylSulfide

Sulphides

15 7.2851-31.004285.-604.2

760.00 7.39 0.0011 601.00 -0.08 51.66 6.41

Di-n-propyl Sulphide - 187.66- -

27


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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

Disulfide s 607.5Di-n-propyl

SulfideSulphide

s15 6.7657-40.131

170.44-505.34

608.00 4.09 0.0024 415.98 -0.10 55.36 14.38

Di-tert-butylDisulfide

Sulphides

4 21.22-27.33290.65-361.05

689.70 4.69 0.0019 473.60 -0.09 52.60 11.36

Di-tert-butyl

Sulfide

Sulphide

s 15 4.34-32.

264.15-

560.88 623.20 4.32 0.0019 426.15 -0.09 56.25 16.13

Dyallil SulfideSulphide

s15 3.6941-41.96

190.15-588.18

653.53 3.94 0.0026 412.15 -0.10 58.80 19.16

Ethyl MethylDisulfide

Sulphides

164.17121-46.6678

183.22-566.01

628.90 3.41 0.0032 407.15 -0.11 65.35 20.41

Ethyl N-octylSulfide

Sulphides

163.01057-33.942

241.3-630.

700.00 5.53 0.0014 505.54 -0.08 51.52 11.54

Ethyl N-pentylSulfide

Sulphides

16 3.517-37.39210.-

569.88633.20 4.41 0.0022 442.70 -0.09 55.71 14.23

Ethyl PropylDisulfide

Sulphides

164.02508-47.492

165.-594. 660.00 4.30 0.0024 446.85 -0.10 62.29 17.39

Ethyl PropylSulfide

Sulphides

163.44585-41.135

156.12-525.51

583.90 3.59 0.0027 391.65 -0.10 55.31 15.60

Ethyl T-butylSulfide

Sulphides

12 16.85-23.54293.15-363.15

588.00 3.92 0.0024 393.56 -0.10 51.54 13.95

Ethyl Tert-butyl Disulfide

Sulphides

16 3.72-40.391206.44-593.91

659.90 4.26 0.0021 448.81 -0.09 58.45 16.21

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

Methyl EthylSulfides

Sulphides

13 3.7437-42.848167.23-479.7

533.00 2.85 0.0039 339.80 -0.12 61.87 19.65

MethylIsopropylSulfide

Sulphides

3 23.1-24.2293.15-303.15

553.10 3.23 0.0030 357.91 -0.11 56.43 17.06

Methyl N-

butyl Sulfide

Sulphide

s 7 18.84-26.4

293.15-

364.45 593.00 3.82 0.0028 396.58 -0.11 57.52 15.36Methyl N-

propyl SulfideSulphide

s13 3.3757-43.383

160.17-508.5

565.00 3.37 0.0033 368.69 -0.11 59.63 17.62

MethylPenthylSulfide

Sulphides

173.38629-39.161

179.15-550.71

611.90 4.15 0.0023 418.15 -0.10 54.65 14.73

Methyl PhenylSulfide

Sulphides

8 32.14-40.31293.15-359.55

712.80 3.90 0.0026 467.46 -0.12 75.64 19.09

Methyl T-butyl Sulfide

Sulphides

13 3.1947-35.039190.84-

513.570.00 3.51 0.0028 372.05 -0.10 52.57 16.04

Methyl T-pentyl Sulfide

Sulphides

10 3.1615-36.794195.-568.8

632.00 3.99 0.0023 423.00 -0.09 52.92 15.01

Tetrahydrothiophene

Sulphides

23 25.-38.4273.15-373.15

631.95 2.90 0.0040 394.27 -0.13 75.05 22.23

Thiacyclopropane

Sulphides

11 4.5977-55.155164.15-492.3

547.00 2.06 0.0060 328.08 -0.15 78.36 27.95

Sulphide - 269.92- -

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name familyN

Pointsrange

(mNm-1)T range

(K)Tc(K)


Criticaldensity

(molcm-3)Tb (K) m q

@T

b

s 683.82

ThiopheneSulphide

s32 21.-34.

273.15-373.15

579.35 2.78 0.0046 357.31 -0.14 74.28 22.80

TrimethyleneSulfide

Sulphides

14 5.2691-49.487199.91-541.8

602.00 2.54 0.0046 368.12 -0.13 74.24 26.88

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Figure 1.5-Figure 1.7show the relationship between the selected physical properties and

the surface tension at boiling point. From Figure 1.6 to Figure 1.7, a general trend isclearly evident for surface tension at the boiling point when plotted against the selectedparameters. In particular, surface tension at boiling point was found to be decreasing withradius of gyration with an evident common trend for all the fluids. Surface tension atboiling point was found to be decreasing with critical temperature and increasing withcritical density, but in these cases points were found to be rather scattered. From the generalanalysis, three alkanes, namely n-heneicosane, n-docosane and n-tricosane, were found notto exactly follow the common trends of other alkanes. In addition, ethyl mercaptan wasfound to be always out of common trends: this was probably due to its very high boilingpoint temperature, that was found to be out of the range of the experimental temperatures.The following equation was finally proposed:

Figure 1.5 Surface tension at the boiling point vs. critical density

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Figure 1.6 Surface tension at the boiling point vs. radius of gyration

Figure 1.7 Surface tension at the boiling point vs. critical temperature

(1.17)

where 0 is a scaled factor = (kTc)/Gr2, Tr is the reduced temperature, k is Boltzmannsconstant, is an adimensional term = NacGr31012, Gris the radius of gyration (m), Na is

Avogadros number (mol

-1

), cis the critical density (molm

-3

) and 10

12

is a scaling factor.In order to deeper analyze the cross-correlations between the selected parameters, in Figure1.8-Figure 1.10the critical density versus radius of gyration, the critical temperature versusradius of gyration and the critical density versus critical temperature are plotted,respectively.

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Figure 1.8 Cross-correlation between radius of gyration and critical density

Figure 1.9 Cross-correlation between radius of gyration and critical temperature

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Figure 1.10 Cross-correlation between critical density and critical temperature

Figure 1.8a noticeable cross-correlation between radius of gyration and critical density forall the families, suggesting that one of the parameters could be substituted by a correlation,giving the possibility to remove a physical parameter from Equation (1.17). The followingvery simple correlation between critical density and radius of gyration was considered:

4.2808-Gr41520e = .c (1.18)

To optimize the coefficients, the Levenberg-Marquardt curve-fittingmethod was adopted

[28]. This method is actually a combination of two minimization methods: the gradientdescent method and the Gauss-Newton method.In order to compare the proposed equation with the literature ones, deviations werecalculated as follows:

(1.19)

The respective AAD% values were reported in Table 1.4. In spite of its simplicity (reducedtemperature and radius of gyration are the only two physical parameters needed), theproposed equation is generally giving the lowest deviations. To keep the proposed equationas general and simple as possible, we decided not to regress the coefficients for each family

separately.The residuals produced by Equation (1.17) + (1.18) are reported in Figure 1.9. Theprediction of surface tension of organics is generally good, clearly increasing with thereduced temperature. Excluding few fluids that generally show very low experimentalsurface tension data (namely 1,2-ethanedithiol, m-diisopropylbenzene hydroperoxide, n-propylbenzene hydroperoxide, dibenzothiophene, tert-dodecyl mercaptan, 2-

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propylbenzothiophene and 3-ethylbenzothiophene), deviations are generally well below 5

mNm-1.

Figure 1.11 Deviations of Equation (1.17)+ (1.18)as function of reduced temperature

Inorganic Compounds

In order to understand if the equation proposed for organics was extendable for inorganiccompounds, the surface tension data for inorganics were collected, again from the onesaccepted by DIPPR database [2]. A total of 2032 surface tension data were finally analyzedfor the following families named as in the DIPPR database: elements, silanes, halides,

gases, acids, bases and other inorganics.In Figure 1.12the surface tension data for inorganics divided by families are shown. Sincethe collected data for the family of elements are clearly out of commons order ofmagnitudes for surface tension (up to 1450 mNm-1) and reduced temperatures (up to 5764K), they were not reported in Figure 1.12. The experimental ranges of collected data aresummarized in Table 1.6From Figure 1.12, it is clear that a common trend for inorganics,even neglecting the elements, is not easily evident as it was for organics. In particular, thereare many compounds which behavior is out of trend. For this reason, Equation (1.17)wasseparately regressed for the different families.

Table 1.6. Summary of the collected data for inorganic compounds.

Family

N. of

Fluids

N. of

Points

range

(mNm-1)

T range

(K)Acids (inorg.) 7 79 1.26-66.816 158.97-1276.2Bases 1 22 0.2-43.39 198.15-403.15Elements 47 516 0.-1450 1.13-5764Gases 8 93 0.06-36.32 81.15-472.01

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FamilyN. of

Fluids

N. of

Points

range

(mNm-1

)

T range

(K)Halides 18 297 1.5-97.01 121.85-1280.Other Inorganics 4 193 0.07-79.77 272.74-652.74Silanes 55 832 1.6253-75. 88.5-810.27

Figure 1.12. Scatter plot of the collected surface tension data of inorganics vs. (1-Tr).

A good prediction was obtained for the following families: acids, bases and gases. In

particular for inorganic acids (hydrogen bromide, hydrogen chloride, hydrogen iodide,sulfaminic acid and sulfuric acid) an AAD% of 10.71% was obtained with the followingcoefficients:A = 0.00038,B = 1.34635, C = 0.06488 andD = 23.05590. Poorer results wereobtained only for hydrogen fluoride and (123.37%) nitric acid (52.37%). For inorganicgases (carbon dioxide, nitric oxide, nitrogen dioxide, nitrogen tetroxide, nitrous oxide,sulfur dioxide, sulfur trioxide) an AAD% of 6.97% was obtained with the followingcoefficients:A = 20.09860,B = 1.20986, C = 0.27195 andD = 9.81645. All gases showeddeviations below 10%. To avoid any division by zero, gases and elements with a null radiusof gyration (namely Helium-3, Helium-4, Argon, Neon, Krypton, Xenon, Sulfur,phosphorus white, germanium, bismuth, iron, lithium) were not considered during thecalculations. Very good results were obtained for the unique inorganic base available(ammonia), with an AAD% of 1.90%.Other inorganics (deuterium oxide, hydrogen peroxide, tin(iv) chloride and water) were

also well predicted (4.64%) with the following coefficients: A = 7.48842,B = 1.08465, C =0.27053 andD = 11.87120. Deviations for water were found to be 10.23%.Bad results were obtained, as expected, with elements (17.78%), halides (32.78%) andsilanes (24.78%). However, for these families, the same kind of deviations were obtainedwith the equations available from the literature [15-18].

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As a final step, in order to give reliable coefficients for a selection of relevant compounds,

9 inorganics were selected and regressed separately. Results are given in Table 1.7, wherethe coefficients for Equation (1.17), not considering equation (1.18), and the AAD% foreach fluid are also given.

Table 1.7.Deviations for inorganic compounds and coefficients for Equation (1.17)

Name NPoints AAD% a b c d

Sulfuric Acid 5 0.99 0.35032 -2.16443 0.91502 0.06566Ammonia 22 0.75 1.21393 -1.09690 0.89715 1.89821

Nitrous Oxide 11 0.66 1.18465 -1.30609 0.79176 2.45910Carbon Dioxide 25 0.83 1.28280 -1.33738 0.84276 4.11487Sulfur Dioxide 16 0.56 1.18333 -1.91077 0.91009 0.62518Sulfur Trioxide 13 0.29 1.21787 -3.04139 0.85797 0.80687Nitric Oxide 5 0.22 1.55173 -1.24242 0.82278 0.06844

Water 48 0.53 1.15617 -0.90070 0.91476 7.52403Deuterium Oxide 113 0.63 0.90714 -1.05708 0.90652 0.61183

An Artificial Neural Network

For the modeling of surface tension of organics and the selected inorganics, the WolframMathematica artificial neural network (ANN) toolbox was employed. The general area ofANN has its roots in our understanding of the human brain. In this regard, initial conceptswere based on attempts to mimic the brains way of processing information. Efforts thatfollowed gave rise to various models of biological neural network structures and learningalgorithms. A neural network is a structure involving weighted interconnections among

neurons, or units, which are most often nonlinear scalar transformations, but they can alsobe linear. A neuron is structured to process multiple inputs, including the unity bias, in anonlinear way, producing a single output. Specifically, all inputs to a neuron are firstaugmented by multiplicative weights. These weighted inputs are summed and thentransformed via a nonlinear activation function, s.The goal of training is to find values of the parameters so that, for any input x, the networkoutput is a good approximation of the desired output y. Training is carried out via suitablealgorithms that tune the parameters so that input training data map well to correspondingdesired outputs. These algorithms are iterative in nature, starting at some initial value forthe parameter vector and incrementally updating it to improve the performance of thenetwork.There are many models that can be used by ANN such as: linear models, perceptrons,feedforward neural networks, radial basis function networks, etc. The ANNs used to solve

the approximation problem concerning surface tension are the feedforward neural networks,known by many different names, including multi-layer perceptrons (MLP).The network is generally divided into layers. The input layer consists of just the inputs tothe network. Then follows a hidden layer, which consists of any number of neurons, orhidden units, placed in parallel. The network output is formed by another weighted

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with the training points, but the structure of this network is too rigid and too adapting to

those values of database provided.Often it is more convenient to use the RMSE when evaluating the quality of a model duringand after training, because it can be compared with the output signal directly. Differentneural networks were compared adopting the Levenberg-Marquardt algorithm [28] usingtheir root mean square errors (RMSE), defined as follows:

( )( )=

=N

i

ii x,gyN

RMSE1

21 (1.23)

where N is the number of data points.When the network is trained with new data there is the risk of committing unacceptablylarge errors. These problems occurring during neural network training are called overfitting.

To avoid and overcome this overfitting problem it is possible to divide the data points intothe training and validation or test data.Therefore, before the trained network is accepted, it should be validated. Roughly, thismeans running a number of tests to determine whether the network model meets certainrequirements. In this case, it was decided to split the database into three parts: one thatcomprises 70% of the data, which were used for the training, a second which employs20% of the data, which were used to perform the validation test, and a third whichemploys 10% of the data which were used for the test carried out to investigate theprediction capability and validity of the model obtained. In all cases, data were randomlychosen within the database.Figure 1.13illustrates a diagram of a one-hidden-layer feedforward network adopting thefollowing inputs: experimental temperature, T, critical temperature, Tc, critical density, c,and the radius of gyration, Gr. The output is surface tension, . Each arrow in the figuresymbolises a parameter in the network.

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Input Layer

Hidden Layer

Figure 1.13 Schematic diagram of the ANN model. Inputs: Tc(K), critical temperature; Gr(), radiusof gyration; T(K), temperature; c(molcm-3), critical density. Output: (mN/m), surface tension

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Figure 1.14. Absolute average percent deviation between data collected and ANN results vs. numberof neurons in the hidden layer

Table 1.8.The parameters of the hidden layer and of the output layer of the ANN

hidden layer output layer

output neuronsNeurons bias Tc Pc Gr c bias -19.7141

1 -13.0 6.3 11.4 -1.2 35.0 5.22 3.7 10.4 -18.7 0.1 -17.0 3.73 -4.7 7.6 6.5 -0.6 -10.7 7.34 22.7 -23.2 3.5 -0.5 -25.2 -16.15 -1.4 -15.7 20.1 0.0 -4.3 22.4

6 30.1 -9.4 -12.7 -33.8 -9.4 0.07 -3.7 12.1 -14.1 1.8 -17.3 2.78 -3.9 5.2 7.8 -0.6 -8.7 -12.69 -1.6 21.3 -23.4 -0.2 14.1 6.510 14.1 26.4 -74.3 1.4 -24.7 0.411 -5.7 0.3 2.2 -0.4 7.6 24.412 -28.9 33.4 12.9 -2.9 1.1 7.813 -22.7 24.7 9.9 -4.7 16.2 -0.514 19.0 -11.3 -16.7 2.0 -43.8 2.315 8.8 -2.6 -22.5 3.3 12.2 -0.316 -15.1 7.5 -0.3 0.3 58.4 -0.317 2.5 4.7 -20.0 27.9 -13.4 0.0

18 24.6 -25.6 5.1 -0.6 -27.7 15.019 -10.9 18.1 -16.5 0.0 10.8 4.720 -4.6 -14.1 21.5 -0.1 6.1 -15.821 -13.8 1.7 6.6 9.9 8.6 0.522 6.2 -9.6 6.0 -0.3 -4.1 10.6

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23 -17.1 3.7 7.5 11.4 6.9 -0.3

24 25.4 -43.7 13.6 0.5 28.5 -9.925 -30.8 34.7 14.5 -3.0 3.4 -6.826 -24.4 35.4 5.8 -1.2 -41.3 -0.527 48.9 -55.2 -7.8 -0.8 0.2 -0.328 -26.8 23.6 1.7 -1.4 5.1 7.529 25.7 1.0 -6.3 -4.3 -26.1 3.830 26.8 -43.9 13.0 0.3 22.5 10.331 13.6 -2.3 -23.5 -14.2 -11.6 0.132 -10.0 6.8 -8.2 -0.5 14.1 -12.033 6.0 -1.6 -17.3 1.0 0.6 -5.634 -9.5 4.2 7.3 0.2 -21.1 -1.6

Table 1.9.Summary of the deviations of the ANN

AAD%RMSE(mNm-1)

Sum of Residuals(mNm-1)

Entire dataset 4.06 0.0093 0.000012Training Set 3.43 0.0092 -0.000043Validation Set 5.11 0.0092 -0.000149Test Set 6.39 0.0106 0.000713

Because of the discussed cross-correlation between reduced density and radius of gyration,a configuration with a one-hidden-layer feedforward network without critical density as aninput parameter was also tried, but results in terms of AAD% for the entire dataset were a

little poorer (AAD%=4.6%) and this configuration was finally neglected.Figure 1.15 shows the correlation between the predictions of the validated, trained andtested MLP and the corresponding experimental surface tension data. From the figure alinear fit is evident, witnessing a good correlation between predictions and data.

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As summarised in Table 1.4, the equation proposed gave the best results in terms of

deviations when compared to the equations existing in literature.Different ANN approaches were adopted for surface tension prediction in literature [31-35].The use of artificial neural networks was coupled with the group contribution method todetermine surface tension of pure compounds [31]. The compounds investigated belong to78 chemical families containing 151 functional groups. A good compromise was reachedby considering one hidden layer and 10 neurons. The method gave noticeable results, withan overall AAD%=1.7%. In other papers, the parachor was determined [32-33]. A larger setof data for surface tension was considered adopting a quantitative structurepropertyrelationship strategy [34]. In this case, the required parameters of the model are temperatureand the numbers of 19 molecular descriptors in each molecule investigated. A three layerfeedforward artificial neural network was finally optimised. In another paper [35], a widevariety of materials such as alkanes, alkenes, aromatics, and sulfur, chlorine, fluorine, andnitrogen containing compounds were used in an ANN with one hidden layer of 20 neurons.

The AAD% obtained for 1048 data points related to 82 compounds is 1.57%. It is worthnoting that here [35], the ANN adopted about half of our data and an additional inputparameter (a total of five): critical pressure, acentric factor, reduced temperature, reducednormal boiling temperature and specific gravity at normal boiling point. In addition, about70% of data was employed as training data, while the remaining 30% was applied fornetwork verification.At the end of this part, a new scaled equation and a multi-layer perceptron neural networkare proposed to predict surface tension of organics and 9 inorganics. The equation is verysimple, containing only reduced temperature and radius of gyration as parameters. Themultilayer perceptron proposed adopts as input parameters the same parameters proposedfor the scaled equation and has one hidden layer with 34 neurons, determined according tothe constructive approach. The model was validated, trained and tested for a wide set ofdata.

1.5.2 A new scaled equation for the calculation of surface tension ofketones

In this part is presented a new formula for the surface tension calculation of ketones.As a first step, an analysis of the available data of the experimental surface tension data forketones was performed. The experimental data were collected for the following pure fluids:acetone, 2-butanone, 2-pentanone, 3-pentanone, 2-hexanone, 3-heptanone, 4-heptanone, 2-octanone, 6-undecanone. Then, the experimental data were regressed with the most reliablesemi-empirical correlating methods based on the corresponding states theory existing in theliterature.The finally proposed equation is very simple and gives a noticeable improvement with

respect to the existing equationsKetones are organic compounds with a carbonyl group (C=O) bonded to two other carbonatoms. Many ketones are known and they are of great importance in industry (acetone) andin biology.

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To provide a reliable database of surface tensions experimental data for ketones, as a first, a

literature review was performed. The list of analyzed ketones is summarized in Table 1.10,together with the physical properties adopted for the calculation.From Table 1.10, it is possible to observe that the measurements were performed over afairly large temperature (from 252.85 to 499.99 K) and surface tension (from 2.99 to 30.84mN/m) ranges. After the data selection, a total of 154 reliable experimental points wereconsidered in the regressions.The main goal of this study was to find a new equation that minimizes the deviationsbetween the predicted data and the experimental data for ketones. For this purpose, webased the new equation on the well known corresponding states principle, according to theVan der Waals theory.

Table 1.10: Studied ketones and their adopted physical properties

CompoundT Range

[K]Range[mN/m]

N. ofdata

Tc[K] ZcRadius of

Gyration [m]

CriticalDensity[mol/cm

3]

Acetone273.15-353.15

16.2-26.21 20 508.2 0.233 2.7510-10 0.00478

2-butanone298.14-323.15

21.16-23.96 6 535.5 0.249 3.1410-10 0.00374

2-pentanone288.15-363.15

17.66-25.3 31 561.08 0.238 3.6210-10 0.00332

3-pentanone273.15-499.99

4.22-26.9 33 560.95 0.269 3.5810-10 0.00297

2-hexanone288.14-363.15

18.4-26.54 19 587.61 0.254 4.0910-10 0.00264

3-heptanone 289.25-360.35

19.55-26.55 11 606.6 0.251 4.5410-10 0.0023

4-heptanone297.95-333.15

21.93-25.47 11 602 0.253 4.5710-10 0.0023

2-octanone252.85-569.43

2.99-30.84 16 632.7 0.249 4.8910-10 0.00201

6-Undecanone292.55-360.85

21.43-27.4 7 678.5 0.252 5.7210-10 0.00144

As a preliminary step, a statistical overview for surface tension experimental data ofketones was performed. The choice of finding a new equation specifically oriented toketones was due to the generally quite high deviations showed by the existing equationsduring the representation of the ketones experimental data. In order to understand if some

data had to be rejected, the distribution of the entire dataset was statistically analyzed. InFigure 1.17, the descriptive statistics that refers to properties of distribution such aslocation, dispersion and shape of the data are reported.In Figure 1.18, the scatter plot of the surface tension data versus reduced temperature foreach fluid was reported, showing a rather negative correlation of all fluids, that underlinesthe importance of the reduced temperature for the surface tension calculation also for the

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family of ketones. Then, to analyze the linear correlation between all the variables and the

surface tension, a correlation matrix was created, as reported Table 1.11. From the table,the strong inverse correlation between the reduced temperature and the surface tension wasconfirmed. All others variables showed a rather low linear correlation.

5 10 15 20 25 300

10

20

30

40

50

/ -

/

-

0

5

10

15

20

25

30

/

-

0

5

10

15

20

25

30

-

Figure 1.17: Descriptive statistics of ketones set of experimental data

(

-

)

0.5 0.6 0.7 0.8 0.9

5

10

15

20

25

30 r2 =0.925351

2-butanone

2-hexanone

2-octanone

2-pentanone

3-heptanone

3-pentanone

4-heptanone

6- undecanone

acetone

Figure 1.18. Scatter Plot of experimental surface tension data versus reduced temperature. Confidencebands at 1standard deviation, 2standard deviation and 3standard deviation

Table 1.11.Correlation matrix between all the variables. : surface tension, Tc: critical temperature,Pc: critical pressure, :acentric factor, Tb: temperature at boiling point, c:critical density, M:molecular weight, Zc: compressibility factor, Ttp: temperature at triple point, Ptp: pressure at triplepoint, Gr: radius of gyration

46

Tc Pc Tr Tb c M Zc Ttp Ptp Gr

1.00 0.02 -0.03 0.04 -0.96 0.04 -0.03 0.04 0.03 0.01 -0.04 0.03

Tc 0.02 1.00 -0.99 0.95 -0.27 0.99 -0.95 0.99 0.28 0.88 -0.16 0.99

Pc -0.03 -0.99 1.00 -0.91 0.29 -0.97 0.97 -0.97 -0.30 -0.86 0.15 -0.99

0.04 0.95 -0.91 1.00 -0.28 0.98 -0.84 0.97 0.17 0.845 -0.22 0.95


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Finally, the proposed equation is the following:

(1.24)

Where 0is a scaled factor = (KbTc)/Gr, Tris the reduced temperature, =NacGr3106,Zcis the compressibility factor, Gris Radius of Gyration, Kbis the Boltzmann Constant, Nais Avogadro number and c is the critical density. For the present equation, after theregression the following coefficients were found: A=39.6326, B= 1.2416, C=0.8422,D=7.8612 and E=40.2106.As summarized in Table 1.12, the proposed equation gave the best results in terms ofdeviations when compared to equations existing in the literature. It is worth noting that theSastri Rao equation shows lower deviations if compared with other equations, because thecoefficients were specifically given for different families, as reported in the original paper[18]. Deviations for all the equations are reported in Figure 1.19. Calculation of surface

tension is clearly improved, especially at the lower reduced temperatures, where literatureequations generally show the higher deviations. In addition, deviations for each ketone arereported in Table 1.13. It is clear that the deviation from experimental data are very low foralmost all the compounds belonging to the ketones family. Only 2-octanone showedslightly higher deviations. Finally, Figure 1.20, deviations for the different ketonesconsidered in this paper are presented.Experimental surface tension data for ketones were collected. The ketones family wasselected because the equations existing in the literature are not able to represent theexperimental data of this family with sufficient precision. According to the extendedcorresponding states principle, a new correlation for the surface tension calculation ofketones was proposed. This equation well represents the experimental data and deviationswere found to be much lower than similar equations based on the corresponding statesprinciple present in the literature.

47

Tr -0.96 -0.27 0.29 -0.28 1.000 -0.29 0.27 -0.29 -0.07 -0.21 0.10 -0.29

Tb 0.043 0.992 -0.975 0.982 -0.29 1.000 -0.91 0.999 0.233 0.87 -0.20 0.99

c -0.03 -0.95 0.973 -0.84 0.279 -0.91 1.000 -0.927

-0.50 -0.90 -0.03 -0.94

M 0.045 0.992 -0.97 0.979 -0.29 0.999 -0.92 1.000 0.25 0.88 -0.17 0.99

Zc 0.030 0.287 -0.30 0.172 -0.07 0.233 -0.50 0.254 1.00 0.63 0.82 0.25

Ttp 0.010 0.880 -0.86 0.845 -0.21 0.872 -0.90 0.881 0.63 1.00 0.29 0.87

Ptp -0.04 -0.16 0.158 -0.22 0.108 -0.20 -0.03 -0.175

0.82 0.29 1.00 -0.19

Gr 0.039 0.996 -0.99 0.952 -0.29 0.992 -0.94 0.992 0.25 0.87 -0.19 1.00


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Figure 1.19. Comparison of residuals between different equations

Figure 1.20. Deviations for the different ketones

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1.5.3 A new Scaled Equation and an Artificial Neural Network for

calculation of surface tension of Alcohols

This part presents a new formula to calculate the surface tension of alcohols.As a first step, an analysis of the data available on the surface tension of alcohols wasmade. A total of 1643 data were collected for n-alcohols, aromatic alcohols, cycloaliphaticalcohols, 2-alkanols and methyl alkanols. The data were then regressed with the mostreliable semi-empirical correlation methods in the literature based on the correspondingstates theory. The scaled equation proposed is very simple and gives noticeableimprovement with respect to existing equations.The same physical parameters considered in the scaled equation were also adopted as inputparameters in a multi-layer perceptron neural network, to predict the surface tension ofalcohols. The multilayer perceptron proposed has one hidden layer with 29 neurons,determined according to the constructive approach. The model developed was trained,

validated and tested for the set of data collected, showing that the accuracy of the neuralnetwork model is higher than that of the methods proposed in the literature.An alcohol is an organic compound in which the hydroxyl functional group (-OH) is boundto a carbon atom. Due to the presence of the hydroxyl functional group, alcohols canhydrogen bond. In addition, because of the difference in electronegativity between oxygenand carbon, they are polar, and therefore hydrophilic, showing the ability of dissolving bothpolar and non-polar substances. For this reason, between the several uses of alcohols, ofparticular interest are the fields of solvents and reagents. Other important uses are in fuels,antifreezes, preservatives and beverages. In organic synthesis, alcohols serve as versatileintermediates.Surface tension is an important fluid property for the study of industrial applications. Inparticular, it plays a fundamental role in the study of phase transitions and technicalprocesses such as boiling and condensation, absorption and distillation, microscale channel

flow processes and detergents. Liquidvapour interfaces are also important for particularnaturally occurring phenomena, such as the performance of biological membranes.The raw surface tension data (experimental, smoothed and predicted) available from DIPPRdatabase [2] were analysed for 94 alcohols. During the data collection, a fluid by fluidanalysis was performed and data showing deviations higher than three times the standarddeviation were rejected. All the fluids containing the alcohol functional group wereconsidered and divided into the following subfamilies: n-alcohols, aromatic alcohols(phenols), cycloaliphatic alcohols and other aliphatic alcohols (2-alkanols and methylalkanols). Although phenols, whose hydroxyl group is attached to an unsaturated aromaticring system, are more reactive than alcohols and act more like acids, they were stillconsidered in the data treatment.Despite there being no surface tension calculation/estimation methods specifically orientedto alcohols, several theories that have been proposed to date can be applied to describe the

surface tension of alcohols. It is necessary to emphasise that, because of association, theequations for the surface tension calculation of polar molecules, such as alcohols or acids,often pose serious problems in terms of predictability.

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Table 1.14. Summary of the collected data and calculated surface tension at the boiling point,@Tb=mT+q.

range T range Tc Gr c m q @Tb TbAAD%

Alcohol FamilyNPoints

(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

1-methyl-3-hydroxy-5-isopropyl Benzene Aromatic 15

3.61897-25.183

321.65-654.12 726.8 4.872 0.00202 514.15 -0.065 45.485 12.18 2.43

1-methyl-3-hydroxy-6-isopropyl Benzene

Aromatic15

3.5624-19.85

384.15-650.25 722.5 4.702 0.00202 511.15 -0.061 43.132 11.79 1.56

1-phenyl-1-propanolAromatic

1826.62-34.93

283.15-373.15 677 4.434 0.00227 492.15 -0.092 61.050 15.58 0.13


114.5992-36.769

285.-610.2 678 4.457 0.00227 493.15 -0.098 63.743 15.27 3.69

2,3-xylenolAromatic

114.479-28.41

345.71-645.71 722.9 4.069 0.00278 490.07 -0.079 55.597 16.67 0.84

2,4-xylenolAromatic

1317.2-31.23

313.15-473.15 707.6 4.143 0.00256 484.13 -0.087 58.495 16.18 0.19

2,5-xylenolAromatic

719.72-29.92

353.15-473.15 707.0 4.128 0.00286 484.33 -0.085 59.938 18.77 0.00

2,6-di-tert-butyl-p-cresolAromatic

115.335-28.152 344.-644. 724 5.682 0.00132 541.15 -0.076 53.980 13.08 0.35

2,6-xylenolAromatic

114.005-28.59

318.76-628.76 701.0 4.068 0.00256 474.22 -0.079 53.397 15.92 1.46


114.4326-42.363

257.-616.5 685 4.483 0.00227 497.65 -0.105 67.726 15.62 5.39

2-phenyl-2-propanol*Aromatic

119.834-61.97

309.15-589.15 660 4.29 0.00227 475.15 -0.185 118.28 30.32 1.48

2-phenylethanolAromatic

246.4706-46.94

246.15-615.6 684 4.279 0.00258 492.05 -0.106 71.869

19.656 1.35

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

3,4-xylenolAromatic

1617.55-29.02

348.15-473.15

729.95 4.156 0.00286 500.15 -0.091 60.663 15.08 0.06

3,5-xylenolAromatic

1617.96-28.42

347.15-473.15

715.65 4.318 0.00208 494.89 -0.082 56.553 16.09 0.27


125.0642-47.852

255.15-624.6 694 4.737 0.00227 508.15 -0.115 75.634 17.05 5.11

4-hydroxystyreneAromatic

113.4884-25.644

346.65-662.4 736 4.103 0.00267 502.00 -0.070 49.479 14.35 2.01

Alpha-methylbenzylAlcohol

Aromatic24

4.61-41.61

293.15-629.1 699 4.079 0.00256 477.15 -0.110 72.605 20.24 3.39

Benzyl AlcoholAromatic

504.61-43.33

257.85-648.14

720.15 3.8 0.00262 478.60 -0.100 68.246 20.59 2.78

Dinonylphenol*Aromatic

1015.663-87.566 350.-746. 902 11.82 0.00081 735.00 -0.181

146.876 14.04 5.82

M-cresolAromatic

2521.37-38.01

288.15-453.15

705.85 3.87 0.00321 475.43 -0.096 65.229 19.49 1.85

M-ethylphenolAromatic

114.174-40.499

269.15-644.8

716.45 4.278 0.00258 491.57 -0.096 65.083 17.80 4.86

M-tolualcoholAromatic

113.5007-28.454

252.-612.9 681 4.23 0.00258 490.15 -0.068 45.055 11.49 2.64

NonylphenolAromatic

115.2288-35.489

279.15-679.15 770 6.929 0.00132 590.76 -0.075 55.484 11.14 3.17

O-cresolAromatic

25 13.-36.7313.15-533.15

697.55 3.787 0.00355 464.15 -0.102 67.777 20.24 1.33

O-ethylphenolAromatic

113.9638-42.256

269.84-632.7 703 4.043 0.00285 477.67 -0.105 68.648 18.55 7.60

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

O-tolualcoholAromatic

114.9864-40.067

309.15-621.9 691 4.107 0.00258 497.15 -0.112 73.290 17.83 4.31

P-cresolAromatic

53 14.-37.44303.15-533.15

704.65 3.762 0.00361 475.13 -0.094 64.637 19.74 0.71

P-cumylphenolAromatic

114.4478-40.961

346.3-750.6 834 5.739 0.00142 608.15 -0.090 70.702 16.08 4.90

P-ethylphenolAromatic

114.827-34.74

318.23-638.23

716.45 4.13 0.00267 491.14 -0.093 63.656 17.94 2.41

PhenolAromatic

16 23.6-38.2323.15-453.15

694.25 3.415 0.00437 454.99 -0.108 72.845 23.66 0.77

P-isopropenyl Phenol*Aromatic

1510.17-80.25

356.65-668.43 742.7 4.381 0.00271 513.70 -0.225

157.631 42.27 4.57

P-tert-amylphenol*Aromatic

1110.782-66.361 366.-666. 752 4.913 0.00183 535.65 -0.185

132.607 33.69 2.35

P-tert-butylphenolAromatic

11 4.6-27.41371.56-651.56 734 4.614 0.00203 512.88 -0.081 57.174 15.45 2.00

P-tert-octylphenolAromatic

115.12-34.12

358.55-678.55 765 5.703 0.00142 563.60 -0.090 65.554 14.69 3.37

P-tolualcoholAromatic

1126.32-34.09

343.15-433.15 681 4.148 0.00258 490.15 -0.088 64.345 21.12 0.36

ThymolAromatic

23 17.9-34.2273.15-484.15

698.25 4.679 0.00229 505.65 -0.078 55.147 15.71 1.50

1,4-cyclohexanedimethanol

Cycloaliphatic 11

5.0512-35.795

320.65-651.6 724 4.639 0.00225 556.15 -0.092 64.296 13.07 3.48

1-methylcyclohexanolCycloaliphatic 11

3.7215-32.893

299.15-617.4 686 3.67 0.00267 441.15 -0.092 59.609 19.10 3.84

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

Beta-cholesterol*Cycloaliphatic 11

5.203-47.189

421.65-863.1 959 3.852 0.00074 771.00 -0.095 84.817 11.91 7.47

Cis-2-methylcyclohezanol

Cycloaliphatic 14

5.0802-33.835

280.15-552.6 614 3.821 0.00267 438.15 -0.104 62.403 16.80 1.45

Cis-3-methylcyclohexanol

Cycloaliphatic 12

5.9555-34.26

267.65-547.65 625 3.931 0.00267 446.15 -0.100 60.218 15.76 1.57


Cycloaliphatic 16

3.9934-31.625

263.95-559.8 622 3.869 0.00267 444.15 -0.094 55.869 14.25 3.28

CyclohexanolCycloaliphatic 12

25.67-34.23

289.35-373.15 650.1 3.601 0.00311 434.00 -0.104 64.243 19.22 0.80

L-mentholCycloaliphatic 11

4.754-24.66

315.65-585.65 658 4.754 0.00188 489.55 -0.073 47.639 11.80 0.33

Sitosterol*Cycloaliphatic 11

6.1359-58.764

413.15-857.7 953 8.288 0.00068 778.00 -0.118

104.283 12.73 8.67

Stigmasterol*Cycloaliphatic 15

8.4585-54.122

443.15-830.57 953.6 8.23 0.00069 778.00 -0.117

103.666 12.46 5.28

Trans-2-methylcyclohexanol

Cycloaliphatic 14

5.3333-34.637

269.15-555.3 617 3.824 0.00267 440.15 -0.101 61.370 16.99 0.81


Cycloaliphatic 12

5.0479-31.718

272.65-564.3 627 3.937 0.00267 447.15 -0.091 56.290 15.74 0.42


Cycloaliphatic 16

4.4357-31.773

265.-559.8 622 3.854 0.00267 444.15 -0.092 55.488 14.82 1.74

1-butanol n-Alcohol 3814.2-26.28

273.15-413.15 563.1 3.225 0.00366 391.90 -0.085 49.568 16.37 1.21

1-decanoln-Alcohol

2122.4-29.61

283.15-373.15 688 5.499 0.00155 503.00 -0.076 50.910 12.90 0.76

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

1-dodecanoln-Alcohol

1823.8-29.75

293.15-373.15 718.7 6.119 0.00127 537.10 -0.076 52.166 11.19 0.21

1-eicosanoln-Alcohol

86.9271-28.767

338.55-641.53 807.7 8.501 0.00072 645.50 -0.072 51.982 5.681 4.34

1-heptadecanol n-Alcohol 102.8761-29.588

327.05-702. 779.2 7.59 0.00086 611.30 -0.071 50.984 7.818 10.13

1-heptanoln-Alcohol

17 20.2-28.6273.15-373.15 632.3 4.38 0.00225 448.60 -0.087 52.648 13.63 0.70

1-hexadecanoln-Alcohol

103.168-29.331

322.35-693. 768.6 7.31 0.00093 598.00 -0.070 50.563 8.668 7.47

1-hexanoln-Alcohol

2518.23-28.08

273.15-388.15 611.3 4.144 0.00262 429.90 -0.083 50.798 14.90 0.39

1-nonadecanol n-Alcohol 102.8791-29.071

334.85-719.1 798.8 8.208 0.00076 635.10 -0.068 50.018 7.116 10.04

1-nonanol n-Alcohol 22 21.7-29.8273.15-373.15 670.9 5.108 0.00174 485.20 -0.078 51.218 13.13 0.60

1-octadecanoln-Alcohol

115.0965-29.3

331.05-668.78 789.3 7.93 0.00081 623.60 -0.072 52.220 7.045 6.20

1-octanoln-Alcohol

39 21.1-29.1273.15-373.15 652.3 4.787 0.00196 467.10 -0.080 50.956 13.58 0.22

1-pentadecanoln-Alcohol

103.256-30.033

317.05-683.1 757.3 7.086 0.00099 583.90 -0.073 51.723 9.309 7.22

1-pentanol n-Alcohol 1218.8-26.67

283.15-373.15 588.1 3.679 0.00307 410.90 -0.086 51.067 15.55 0.21

1-propanoln-Alcohol

1718.27-24.48

283.15-353.15 536.8

2.7359 0.00457 370.35 -0.083 48.191 17.38 0.70

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

1-tetradecanoln-Alcohol

87.8911-29.94

310.65-591.93 745.3 6.73 0.00107 569.00 -0.078 53.366 8.955 2.98

1-tridecanoln-Alcohol

113.4629-30.779

293.15-660.6 732.4 6.417 0.00116 553.40 -0.076 52.473 10.62 5.27

1-undecanol n-Alcohol 11 22.9-29.2293.15-373.15 703.9 5.808 0.00140 520.30 -0.079 52.265 11.28 0.10

2-butanoln-Alcohol

822.14-23.89

288.15-303.15 535.9 3.183 0.00370 372.90 -0.088 49.151 16.22 1.38

2-heptanol n-Alcohol 12 19.-27.2273.15-373.15 608.3 4.51 0.00224 432.90 -0.082 49.642 14.12 0.15

2-hexanoln-Alcohol

2714.15-26.5

273.15-413.15 585.3 4.09 0.00260 412.40 -0.087 50.357 14.45 0.79

2-methyl-2-butanoln-Alcohol

718.51-23.22

288.15-338.15 543.7 3.42 0.00309 375.20 -0.095 50.570 15.00 0.14

2-pentanoln-Alcohol

522.96-24.42

288.15-303.15 561 3.655 0.00307 392.20 -0.083 48.477 15.77 0.81

2-propanol n-Alcohol 1220.46-21.79

288.15-303.15 508.3 2.76 0.00450 355.30 -0.085 46.378 16.09 0.87

3-butanoln-Alcohol

616.36-19.96

298.15-338.15 506.2 3.067 0.00364 355.57 -0.086 45.363 14.89 0.56

3-heptanoln-Alcohol

203.68291-37.4885

203.15-544.86 605.4 4.381 0.00230 429.15 -0.098 55.999 13.83 5.00

Ethanoln-Alcohol

48 1.1-24.4273.097-503.052 513.9 2.259 0.00599 351.39 -0.100 51.933 16.96 4.56

Methanol n-Alcohol 44 0.34-24.5273.1-508.15 512.5 1.552 0.00860 337.63 -0.100 52.208 18.42 12.83

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

2,2-dimethyl-1-propanolOtherAliphatic 15

2.885-14.94

328.15-497.43 552.7 3.388 0.00306 386.25 -0.071 38.298 10.75 0.32

2,6-dimethyl-4-heptanolOtherAliphatic 11

4.186-35.3 208.-538. 603 4.915 0.00186 451.00 -0.093 53.615 11.46 4.22

2-butanolOtherAliphatic 8

22.14-23.89

288.15-303.15 535.9 3.183 0.00370 372.90 -0.088 49.151 16.22 1.38

2-buthyl-1-nonanolOtherAliphatic 15

4.147-33.476

271.2-613.35 681.5 6.866 0.00129 538.00 -0.085 55.278 9.461 4.91

2-buthyl-1-octanolOtherAliphatic 16

3.8233-38.935

193.15-600.57 667.3 6.412 0.00139 520.00 -0.085 53.488 9.109 6.67

2-butyl-1-decanolOtherAliphatic 15

4.0195-33.062

277.7-636.93 707.6 7.159 0.00117 563.00 -0.080 54.031 8.818 5.21

2-heptanolOtherAliphatic 12 19.-27.2

273.15-373.15 608.3 4.51 0.00224 432.90 -0.082 49.642 14.12 0.15

2-hethyl-1-butanolOtherAliphatic 15

15.29-25.9

278.15-418.15 591.2 3.681 0.00273 420.15 -0.073 46.007 15.49 0.75

2-hethyl-1-hexanolOtherAliphatic 10

9.253-36.01

203.15-499.15 640.6 4.809 0.00206 457.75 -0.090 53.339 12.28 2.17

2-hexanolOtherAliphatic 27

14.15-26.5

273.15-413.15 585.3 4.09 0.00260 412.40 -0.087 50.357 14.45 0.79

2-methyl-1-butanolOtherAliphatic 17

4.7515-35.41

195.-517.86 575.4 3.437 0.00304 401.85 -0.094 53.314 15.52 0.73

2-methyl-1-dodecanolOtherAliphatic 15

4.0807-34.43

271.-614.25 682.5 6.928 0.00129 538.80 -0.088 56.874 9.656 4.74

2-methyl-1-hexanolOtherAliphatic 11

4.4343-35.623

220.-535.5 595 4.796 0.00231 437.15 -0.098 56.403 13.52 2.56

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

2-methyl-1-pentanolOtherAliphatic 40 15.79-27.

273.15-408.15 604.4 3.967 0.00263 421.15 -0.082 49.325 14.86 0.15

2-methyl-1-propanolOtherAliphatic 15

16.58-23.73

283.15-373.15 547.8 3.332 0.00365 380.81 -0.080 46.433 15.94 0.11

2-methyl-1-tridecanolOtherAliphatic 30

3.8665-33.062

277.7-636.93 707.7 6.96 0.00117 563.10 -0.081 54.126 8.764 5.04

2-methyl-1-undecanolOtherAliphatic 16

4.0173-32.115

264.3-599.31 665.9 6.312 0.00140 521.00 -0.084 53.433 9.521 4.91


18.51-23.22

288.15-338.15 543.7 3.42 0.00309 375.20 -0.095 50.570 15.00 0.14

2-methyl-2-propanolOtherAliphatic 6

16.36-19.96

298.15-338.15 506.2 3.067 0.00364 355.57 -0.086 45.363 14.89 0.56

2-nonanolOtherAliphatic 24 20.5-28.6

273.15-373.15 649.5 5.281 0.00180 471.70 -0.081 50.830 12.49 0.08

2-octanolOtherAliphatic 29

19.8-27.14

273.15-373.15 629.8 4.908 0.00195 452.90 -0.079 49.180 13.62 0.47

2-pentanolOtherAliphatic 5

22.96-24.42

288.15-303.15 561 3.655 0.00307 392.20 -0.083 48.477 15.77 0.81

3-heptanolOtherAliphatic 20

3.68291-37.4885

203.15-544.86 605.4 4.381 0.00230 429.15 -0.098 55.999 13.83 5.00

3-hethyl-1-heptanolOtherAliphatic 14

1.97007-36.4491

240.1-608.37 636.7 5.28 0.00186 480.15 -0.093 57.255 12.44 9.74

3-hexanolOtherAliphatic 15

14.14-25.74

278.15-409.15 582.4 3.973 0.00261 406.15 -0.088 50.164 14.54 0.43

3-methyl-1-butanolOtherAliphatic 53 15.1-25.3

273.15-403.15 577.2 3.636 0.00304 404.15 -0.076 46.106 15.24 1.56

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

3-methyl-1-pentanolOtherAliphatic 52

15.05-26.28

278.15-423.15 588 4.18 0.00263 425.55 -0.077 47.725 15.06 0.52


0.4308-35.254

188.-546.1 556.1 3.427 0.00306 383.88 -0.097 52.358 15.03 24.80


14.87-25.4

273.15-388.15 575.6 3.703 0.00263 394.06 -0.089 49.748 14.73 0.46

3-pentanolOtherAliphatic 17

7.8805-36.706

204.15-447.68 559.6 3.562 0.00308 388.45 -0.117 59.272 13.92 2.32

4-methyl-1-octanolOtherAliphatic 40

4.0576-34.763

240.1-568.53 631.7 5.31 0.00172 477.90 -0.093 55.764 11.49 4.08


13.65-24.67

273.15-405.15 574.4 3.843 0.00263 404.85 -0.082 47.185 13.83 0.13

4-methyl-cyclohexane-methanol

OtherAliphatic 10

2.62854-28.89

303.15-582.84 647.6 4.263 0.00239 463.09 -0.096 57.777 13.18 13.16

5-methyl-1-hexanolOtherAliphatic 22

3.109-26.12

293.15-543.15 605 4.563 0.00231 445.15 -0.092 51.876 11.13 6.02

6-methyl-1-octanolOtherAliphatic 40

4.1213-36.382

240.1-570.06 633.4 5.28 0.00186 479.15 -0.097 58.270 11.84 4.48

Allyl AlcoholOtherAliphatic 16 19.2-27.6

273.15-368.15 545.1 2.577 0.00481 370.23 -0.089 51.972 18.85 0.27

Alpha-terpineolOtherAliphatic 16

2.8892-20.975

293.15-607.5 675 4.653 0.00197 492.95 -0.057 37.254 8.93 3.26

Beta-terpineolOtherAliphatic 16

3.0126-21.18

291.15-598.68 665.2 4.615 0.00199 483.15 -0.059 37.898 9.38 2.74

IsopropanolOtherAliphatic 43

16.59-22.9

273.15-353.15 508.3 2.76 0.00450 355.30 -0.079 44.479 16.45 0.43

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(mNm-1) (K) (K) () (molcm-3)(mNK-1m-1)

(mNm-1)

(mNm-1)

(K)

Propargyl AlcoholOtherAliphatic 5

30.97-36.05

293.15-333.15 588.8 2.524 0.00526 386.65 -0.130 74.404 24.00 0.27

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Starting from the consideration that experimental data can generally be well described by

adopting a linear dependence on a function of (1-Tr), in Figure 1.22 a scatter plot ofsurface tension data of alcohols divided by family versus (1-Tr) is presented; it shows arather positive correlation for all fluids, which underlines the importance of the reducedtemperature in calculating the surface tension, as confirmed by Equazione (1.16)

Figure 1.22. Data on the surface tension of alcohols vs. reduced temperature. The marked/selectedsymbols are the fluids not considered during the data regression, as in Table 1.14

From Figure 1.22, it is also evident that three cycloaliphatics (namely sitosterol,stigmasterol and beta-cholesterol) and four aromatics (namely 2-phenyl-2-propanol,dinonylphenol, P-isopropenylphenol and P-tert-amylphenol) are not following the commontrend. In particular the three cycloaliphatics were found to have a much higher criticaltemperature value than the other fluids. The same thing was noticed for dinonylphenol,while the other aromatic alcohols (2-phenyl-2-propanol, P-isopropenylphenol and P-tert-amylphenol) were found to have higher surface tension values than the other fluids, as itcan be clearly noticed also from Table 1.14.Besides the experimental temperature and the critical temperature, we searched for somespecific additional fluid properties that could be involved in the model to improve the

representation of data for all the alcohols, which represent various chemical structures.According to equation (1.5), and to the classical parachor approach, the critical density wasselected as a parameter. Furthermore, according to equation (1.24) the radius of gyrationwas also considered.

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From the general definition, the radius of gyration can be calculated as the root mean

square distance of the objects' parts from either a given axis or its center of gravity. Inparticular, it can be referred to as the radial distance from a given axis at which the mass ofa body could be concentrated without altering the rotational inertia of the body about thataxis. For a planar distribution of mass rotating about some axis in the plane of the mass, theradius of gyration can be considered as the equivalent distance of the mass from the axis ofrotation. Radius of gyration plays an important role in polymers chemistry: it is well knownthat for a polymer in a poor solvent, the radius of gyration is smaller than in a good solventsuch as alcohols are. In fact, the radius of gyration is usually a better estimate of the chaindimensions than the root-mean-squared end-to-end distance, as it accounts for thehydrodynamic interactions between the polymer chains and the solvent. The end-to-enddistance is difficult to measure, while the radius of gyration can be measured by a lightscattering technique.It is worthwhile to investigate this matter in order to provide a more thorough explanation

as to how surface tension for alcohols is related to the selected parameters, namely criticaltemperature, critical density and radius of gyration. Since the boiling point was generallyfound to be within the range of the experimental temperatures, as witnessed in Table 1.14,a linear regression of surface tension against the reduced temperature of each fluid wasmade; then the surface tension at the boiling point was calculated for each fluid; lastly, thesurface tension at the calculated boiling point was plotted for the three parametersconsidered. The regression coefficients m and q are reported in Table 1.14along with therange of data collected for surface tension and temperatures, the number of points, and thecalculated surface tension at the boiling point for each alcohol.From Figure 1.23 to Figure 1.25 show the relationship between the selected physicalproperties and the surface tension at the boiling point. From Figure 1.23to Figure 1.25, ageneral trend is clearly evident for surface tension at the boiling point when plotted againstthe selected parameters. In particular, surface tension at the boiling point was found to

decrease with critical temperature and radius of gyration and increase with critical density:a common trend was evident for all the alcohols, excluding the already pointed out sevenfluids: sitosterol, stigmasterol, beta-cholesterol, 2-phenyl-2-propanol, dinonylphenol, P-isopropenylphenol and P-tert-amylphenol. Such fluids were marked in Figures 8-10. Sincethese seven fluids presented a very limited number of data (a total of 84 points), we decidednot to consider these data in the regression. Aromatic alcohols show slightly differentbehaviour when surface tension at boiling point is plotted versus critical temperature, radiusof gyration and critical density (From Figure 1.23to Figure 1.25)

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Figure 1.23. Surface tension at boiling point vs. critical temperature

Figure 1.24. Surface tension at boiling point vs. radius of gyration

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Figure 1.25. Surface tension at boiling point vs. critical density

Starting from equation (1.24) and not considering the term containing the criticalcompressibility factor, the following equation for alcohols was finally proposed:

(1.25)

where 0 is a scaled factor = (kTc)/Gr2, Tr is the reduced temperature, k is Boltzmannsconstant, is an nondimensional term =NacGr31012, Gris the radius of gyration (m), Nais Avogadros number (mol-1), c is the critical density (molm-3) and 1012 is a scalingfactor. To optimize the coefficients, the Levenberg-Marquardt curve-fitting method wasadopted [28]. This method is actually a combination of two minimization methods: thegradient descent method and the Gauss-Newton method. The following coefficients werefound after regression: A=0.13823, B=1.16570, C=0. 144755, D=16.5905.

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Figure 1.26. Cross-correlation between radius of gyration and critical temperature

Figure 1.27. Cross-correlation between critical density and critical temperature

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Figure 1.28. Cross-correlation between critical density and radius of gyration

In order to further analyze the cross-correlations between the selected parameters, fromFigure 1.26 to Figure 1.28 the radius of gyration versus critical temperature, the criticaldensity versus critical temperature and the critical density versus radius of gyration areplotted, respectively. Again, aromatic alcohols exhibit slightly different behaviour inFigure 1.26and Figure 1.27.

Figure 1.28 shows a noticeable cross-correlation between radius of gyration and criticaldensity for all the families including aromatics, suggesting that one of the parameters couldbe substituted by a correlation, giving the possibility to remove a physical parameter fromequation (1.25). The following correlation between critical density and radius of gyrationwas considered:

(1.26)

Substituting equation (1.26)into equation (1.25), again adopting the Levenberg-Marquardtcurve-fitting method [28], the following new values for the coefficients were found:A=0.151271, B=1.17126, C=0.148516, D= 16.5579.In order to compare the equations, deviations were calculated as follows:

The respective AAD values are reported in Table 1.15, while deviations for each fluid arereported in Table 1.16. Deviations are generally rather similar for n-alcohols,cycloaliphatic and other aliphatics, while aromatic alcohol prediction clearly improves withthe correlation found in equation (1.26). Comparing equation (1.25) and equation (1.25)with equation (1.26) it has to be considered that the equation obtained has one physical

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parameter less, but four additional coefficients. In both cases, to keep the proposed

equations as general and simple as possible, we decided not to regress the coefficients foreach sub-family separately.

Table 1.15.Summary of deviations for equations

FamilyN. ofPoints

AAD%Eq

(1.9)

AAD%Eq

(1.6)

AAD%Eq

(1.8)

AAD%Eq

(1.10)

AAD%Eq.

(1.25)

AAD%Eq

(1.25)+Eq(1.26)

AAD%ANN

Cycloaliphatic 129 17.2 26.1 35.1 19.9 9.4 8.0 3.5Other

Aliphatic 487 11.4 27.4 35.1 29.2 5.2 5.8 1.5

n-Alcohol 528 20.3 26.9 32.6 23.2 15.0 9.7 2.6

Aromatic 893 17.4 28.8 38.3 24.9 7.2 8.4 2.1Mean - 16.7 27.8 35.9 25.1 8.9 8.1 2.1

The residuals produced by equation (1.25)are reported in Figure 1.29. The prediction ofsurface tension of alcohols is good, especially at high reduced temperatures. The behaviourof n-alcohols, cycloaliphatic and other aliphatic is generally well described by the equation(within 3 mNm-1), while aromatic alcohols are, as expected, poorly predicted at lowreduced temperatures (up to 10 mNm-1). However, prediction of aromatic alcoholsdrastically improves with the increase of the reduced temperature.

Figure 1.29. Deviations of equation Eq (1.25)+Eq (1.26)as function of reduced temperature

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Alcohol FamilyN

Points

AAD%Eq

(1.9)

AAD%Eq

(1.6)

AAD%Eq

(1.8)

AAD%Eq

(1.10)

AAD%Eq.

(1.25)

AAD%Eq

(1.25)+Eq(1.26)

AAD%

ANN

Deviationsexp-calc(mNm1)

Eq(1.25)+Eq

(1.26)

Deviationsexp-calc

(mNm1)ANN

l

1-methylcyclohexanolCycloaliphatic

alcohols11 21.00 1.30 4.66 2.79 3.92 13.98 2.77 -2.02 0.03

Cis-2-methylcyclohezanol

Cycloaliphaticalcohols 14 13.77 17.17 24.42 9.93 20.20 14.01 4.93 2.70 -0.05


Cycloaliphaticalcohols

12 13.18 25.67 34.75 18.63 11.17 8.28 2.75 1.53 -0.01



16 16.77 39.88 49.03 30.95 3.40 3.02 5.16 -0.41 -0.51

CyclohexanolCycloaliphatic

alcohols12 23.34 9.65 11.81 10.59 4.52 2.98 1.82 0.90 -0.08

L-mentholCycloaliphatic

alcohols11 28.61 27.08 36.71 20.39 6.72 5.58 1.47 -0.67 -0.07



14 12.72 16.59 23.69 11.16 20.40 14.31 1.93 2.69 0.13



12 11.01 25.33 33.42 18.61 10.57 7.88 3.16 1.01 0.20



16 12.77 35.12 44.65 26.95 7.12 3.11 5.86 -0.01 0.24

1-butanol n-Alcohols 38 2.27 42.61 47.86 39.22 3.79 3.99 1.03 -0.21 0.03

1-decanol n-Alcohols 21 3.36 6.42 14.25 6.10 4.27 1.41 0.89 -0.24 -0.05

1-dodecanol n-Alcohols 18 2.49 1.09 8.88 0.85 7.71 0.87 1.52 -0.23 -0.13

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Alcohol FamilyN

Points

AAD%Eq

(1.9)

AAD%Eq

(1.6)

AAD%Eq

(1.8)

AAD%Eq

(1.10)

AAD%Eq.

(1.25)

AAD%Eq

(1.25)+Eq(1.26)

AAD%

ANN


Eq(1.25)+Eq

(1.26)

Deviationsexp-calc

(mNm1)ANN

1-eicosanol n-Alcohols 8 43.02 6.56 7.68 4.38 6.10 27.03 2.99 -3.93 0.26

1-heptadecanol n-Alcohols 10 54.70 4.78 11.24 4.74 7.05 18.90 2.25 -1.96 0.17

1-heptanol n-Alcohols 17 3.50 20.26 27.21 19.34 1.19 3.18 0.99 -0.80 -0.07

1-hexadecanol n-Alcohols 10 45.05 2.21 7.50 2.23 4.96 12.13 5.66 -1.40 -0.361-hexanol n-Alcohols 25 2.79 25.30 31.85 24.38 3.58 1.93 1.40 -0.24 -0.04

1-nonadecanol n-Alcohols 10 59.53 4.47 9.39 4.08 6.86 25.50 0.54 -2.87 0.00

1-nonanol n-Alcohols 22 3.58 10.38 17.85 9.82 3.39 2.24 0.77 -0.58 0.04

1-octadecanol n-Alcohols 11 37.22 4.63 7.04 3.85 6.46 16.93 2.06 -2.42 0.00

1-octanol n-Alcohols 39 3.61 15.89 23.12 15.01 1.03 3.02 0.39 -0.80 0.04

1-pentadecanol n-Alcohols 10 39.91 2.80 6.39 2.75 5.44 8.06 7.11 -0.83 -0.09

1-pentanol n-Alcohols 12 2.51 36.34 42.38 31.92 2.22 2.08 1.49 -0.34 0.18

1-propanol n-Alcohols 17 9.79 28.44 63.31 54.29 2.85 3.02 1.12 -0.13 0.06

1-tetradecanol n-Alcohols 8 24.06 1.76 8.17 1.84 5.70 6.57 0.71 -0.88 0.07

1-tridecanol n-Alcohols 11 28.75 2.44 5.87 2.81 7.56 3.44 1.57 -0.24 -0.13

1-undecanol n-Alcohols 11 2.33 4.15 11.51 3.55 5.37 1.57 1.24 -0.42 -0.07

2-butanol n-Alcohols 8 3.18 36.89 43.52 36.70 2.99 1.61 1.38 0.35 0.08

2-heptanol n-Alcohols 12 2.40 17.86 24.89 15.73 1.59 1.71 0.73 0.26 0.09

2-hexanol n-Alcohols 27 3.34 20.09 26.53 18.62 2.58 2.56 1.23 0.36 -0.02

2-methyl-2-butanol n-Alcohols 7 3.56 22.78 27.38 19.95 8.41 1.06 0.74 -0.17 -0.16

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Alcohol FamilyN

Points

AAD%Eq

(1.9)

AAD%Eq

(1.6)

AAD%Eq

(1.8)

AAD%Eq

(1.10)

AAD%Eq.

(1.25)

AAD%Eq

(1.25)+Eq(1.26)

AAD%

ANN


Eq(1.25)+Eq

(1.26)

Deviationsexp-calc

(mNm1)ANN

2-hethyl-1-hexanolOther Aliphatic

Alcohols10 12.40 16.35 23.02 16.93 5.49 3.06 1.92 -0.59 -0.22

2-hexanolOther Aliphatic

Alcohols27 3.34 20.09 26.53 18.62 2.58 2.56 1.23 0.36 -0.02

2-methyl-1-butanol Other AliphaticAlcohols

17 11.08 26.93 34.55 22.12 11.50 5.07 2.13 -0.12 -0.22

2-methyl-1-dodecanolOther Aliphatic

Alcohols15 29.93 8.96 27.21 7.65 4.45 15.24 1.46 2.82 0.00

2-methyl-1-hexanolOther Aliphatic

Alcohols11 17.03 19.27 29.89 14.39 3.66 16.26 1.42 2.93 0.09

2-methyl-1-pentanolOther Aliphatic

Alcohols40 2.66 23.11 29.01 22.31 2.05 4.32 1.11 -1.02 -0.03

2-methyl-1-propanolOther Aliphatic

Alcohols15 1.60 42.55 49.43 40.48 4.80 3.37 3.93 -0.14 -0.03

2-methyl-1-tridecanolOther Aliphatic

Alcohols30 38.29 13.58 34.66 12.47 5.15 7.30 1.25 1.35 0.01

2-methyl-1-undecanolOthe

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