Predicting the kinematic viscosity of FAMEs and biodiesel: Empirical models

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Contents lists available at ScienceDirect

Fuel journal homepage: www.elsevier.com/locate/fuel 5 6

Predicting the kinematic viscosity of FAMEs and biodiesel: Empirical models

3 4 7 8

Q1

Juan C. Chavarria-Hernandez ⇑, Daniella E. Pacheco-Catalán Unidad de Energía Renovable, Centro de Investigación Científica de Yucatán, Calle 43 No. 130, Colonia Chuburná deHidalgo, Mérida, Yucatán CP 97200, Mexico

9 10 1 2 13 14

h i g h l i g h t s  Three complementary models were developed to predict the kinematic viscosity of FAMEs.

15

 The models apply for wide ranges of temperature and hydrocarbon chain length.

16 17

 Average and maximum deviations for saturated FAMEs were 1.33–4.01% respectively.

18

 AADs for pure and biodiesel blends were the lowest when compared to 6 previous models.

 Average and maximum deviations for unsaturated FAMEs were 2.88–9.34% respectively.

19

a r t i c l e 2 4 1 3 22 23 24 25 26 27 28 29 30 31 32 33

i n f o

Article history: Received 12 November 2013 Received in revised form 31 January 2014 Accepted 31 January 2014 Available online xxxx Keywords: Kinematic viscosity Temperature Biodiesel Fatty acid methyl ester Correlation

a b s t r a c t Three correlations are proposed to predict the kinematic viscosity of fatty acid methyl esters (FAMEs) in a wide temperature range. One correlation is derived for saturated species from C6:0 to C24:0 while two complementary correlations are used for unsaturated species, from C14:1 to C22:1, including C18:2– C18:3. The correlation for saturated FAMEs was derived from 247 viscosity experimental points taken from the literature, giving an average absolute deviation (AAD) of 1.33%. In the case of unsaturated species, 154 experimental data points reported in the literature were considered, obtaining an AAD of 2.88%. The predictive capacity of the proposed correlations was tested by calculating the viscosity of 31 pure biodiesels (193 data points) and four binary biodiesel blends (150 data points) whose compositions and viscosities at different temperatures were gathered from the literature. A global AAD of 5.33% was obtained for pure biodiesels while the global AAD for biodiesel blends was 7.58%. These results were compared to the AADs obtained for other different six models previously reported in the literature and applied to the same data bank. The comparison showed that the lowest AADs for both data sets (pure biodiesel and biodiesel blends) were obtained with the correlations of this work. Ó 2014 Published by Elsevier Ltd.

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

51 52

1. Introduction

53

Biodiesel is a mixture of monoalkyl esters of long chain fatty acids derived from vegetable oils or animal fats. A small percentage of impurities such as glycerides, sterols or polar lipids is commonly present. The identity of the fatty esters in a given biodiesel has a great effect on its viscosity, which can widely vary from one biodiesel to another, depending on the source oil or fat [1]. Viscosity is certainly one of the most significant properties of biodiesel because of its major effect on the engine performance. Although the higher viscosity of biodiesel compared to that of fossil diesel may eventually promote fuel spray penetration into the combustion chamber, there are more adverse effects including

54 55 56 57 58 59 60 61 62 63

⇑ Corresponding author. Tel.: +52 999 942 8330x327; fax: +52 999 981 3900.

excessive fuel injection pressures that leads to a poorer spray, which in turn can cause an incomplete combustion process among other problems [2]. Engine manufacturers have expressed concern about biodiesel’s higher viscosity and in particular about the viscosity–temperature behavior [3], which is highly influenced by the fatty ester composition. For this reason kinematic viscosity is a regulated property by biodiesel standards. Kinematic viscosity (m) is related to dynamic viscosity (l) and density (q) through Eq. (1). The European standard (EN 14,214) establishes lower and upper limits of 3.5–5.0 mm2/s respectively for kinematic viscosity at 40 °C, while the US standard (ASTM D6751) sets 1.9–6.0 mm2/ s as the corresponding lower and upper limits.

m¼

l q

ð1Þ

E-mail address: [email protected] (J.C. Chavarria-Hernandez). http://dx.doi.org/10.1016/j.fuel.2014.01.105 0016-2361/Ó 2014 Published by Elsevier Ltd.

Please cite this article in press as: Chavarria-Hernandez JC, Pacheco-Catalán DE. Predicting the kinematic viscosity of FAMEs and biodiesel: Empirical models. Fuel (2014), http://dx.doi.org/10.1016/j.fuel.2014.01.105

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J.C. Chavarria-Hernandez, D.E. Pacheco-Catalán / Fuel xxx (2014) xxx–xxx

Reliable biodiesel viscosity data as well as accurate viscosity models that incorporate the dependence on temperature are very important in the development of combustion models. Apart of that, viscosity models are also very useful for design purposes, as well as for simulation and optimization of process equipment such as heat exchangers, reactors, mixing vessels and process piping [4]. Several expressions reported in the literature have been successfully applied to model the temperature dependence of viscosity for individual fatty acid methyl or ethyl esters (FAMEs or FAEEs). These expressions include three-parameter derivations of the well-known Andrade equation [5]. The Vogel equation [6] given by the following expression is an example of an accurate three-parameter model:

92 94 95 96 97 98 99 100

ln l ¼ A þ

B CþT

ð2Þ

In Eq. (2) A, B and C are adjustable parameters while T is the temperature in Kelvin. Once the viscosity of individual fatty esters constituting a biodiesel is known, a mixing rule is required to estimate the viscosity of the mixture. A commonly used mixing rule that assumes ideal mixture behavior is the logarithmic equation given by Eq. (3) [1]

More recently, Ramirez-Verduzco [2] proposed an empirical model with four adjustable parameters to predict the dynamic viscosity of FAMEs. His model calculates viscosity as a function of molecular weight, number of double bonds, and temperature, having the advantage of being a single equation for saturated and unsaturated species. However, his model exhibited some significant deviations between experimental and calculated viscosities. The author reported minimum, maximum and average absolute deviations of 0.09, 29.63 and 6.04%, for 259 data points corresponding to samples of fifteen methyl ester at different temperatures. Moreover, using the mixing rule given by Eq. (3), that author predicted the viscosity of different pure biodiesel samples and several binary biodiesel blends, totalizing 296 data points, obtaining a global AAD of 6.39%. The aim of the present work is to improve the accuracy in the prediction of FAMEs kinematic viscosity for wide ranges of temperature and hydrocarbon chain length, by using a small set of equations and a limited number of parameters. In this way, it will be possible to predict with more accuracy the viscosity of pure biodiesels and biodiesels blends.

144

2. Methods

164

Kinematic viscosity, dynamic viscosity and density data of individual FAMEs at different temperatures were gathered from the literature. Dynamic viscosity and density data reported at the same temperature were used to calculate kinematic viscosity through Eq. (1). In this way, a total of 401 kinematic viscosity data points were considered, 247 for saturated FAMEs, whose values are shown in Table 1, and 154 data points for unsaturated FAMEs shown in Table 2. The development of the proposed correlations was performed as follows. In the first step Eq. (2) was used to fit the experimental viscosities of individual FAMEs considered in Tables 1 and 2, obtaining a set of three parameters (A, B and C) for each FAME. Subsequently, different two-parameter functions (linear, logarithmic, etc.) were proposed to describe the behavior of the A values, as a function of either NC or the number of double bonds in it (ND), or both. The same procedure was then followed for parameters B and C. In this way, several six-parameter models were obtained and then evaluated, founding that a single equation was not good enough to accurately describe the complete data set. Thus, it was decided to evaluate the models for separate groups of data, for example, considering only saturated or only unsaturated FAMEs. The models that best fitted the data points were then selected for a next evaluation stage. The described procedure was also performed starting from other two- or three-parameter models different from that of Eq. (2). At the end, more than 30 correlations were evaluated and from them, the ones that best fitted the data in Tables 1 and 2, were selected and are presented in Section 3.1 In order to evaluate the accuracy of the developed equations, deviations between calculated (mcalc) and experimental (mexp) kinematic viscosities (both given in mm2/s) of individual FAMEs were calculated through Eq. (4), while AADs were calculated by using Eq. (5) in which Np is the number of data points considered for a defined data set. Additionally, the predictive capacity of the proposed correlations was tested by calculating the viscosity of 31 pure biodiesels and four binary biodiesel blends whose composition and viscosity at different temperatures were taken from the literature.

165

145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163

101

103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143

ln lB ¼

n X xi  ln li

ð3Þ

i¼1

in which xi is the mass fraction of the ith alkyl ester, n is the number of esters present in the mixture, while li and lb are the viscosities of the ith alkyl ester and of the biodiesel, respectively. Molar fractions instead of mass fractions can also be used in Eq. (3) and this does not have a significant effect on the prediction of biodiesel viscosity, according to Ramirez-Verduzco [2]. Allen et al. [1] applied Eq. (3) to calculate the dynamic viscosity of binary, ternary and quaternary mixtures of standard FAEEs, as well as to predict the viscosity of several methyl ester biodiesel types. The error in their calculations for the standard FAEEs mixtures was within ±3.7%, while an average prediction error of ±3% was obtained for the biodiesel samples. Those results indicate that Eq. (3) is a useful mixing rule for predicting biodiesel viscosity with a relatively small deviation. In order to improve the viscosity prediction for fatty ester mixtures, Yuan et al. [7] applied the Grunberg–Nissan equation [8] which has the form of Eq. (3) but with an additional term accounting for the interactions among the esters. Those authors first calculated the dynamic viscosity of individual esters with Eq. (2), using a set of three parameters (A, B and C) for each FAME. Viscosities of nine biodiesel fuels in the range of 20–100 °C were then predicted using the Grunberg–Nissan equation, in which the interactions term was calculated through a modified group contribution method proposed by them. This methodology resulted in a maximum error of 7.2% for a temperature of 20 °C as reported by the authors [7]. On the other hand, Krisnangkura et al. [9] developed a fourparameter model to calculate the kinematic viscosity of saturated FAMEs as a function of temperature and the number of Carbons in the hydrocarbon chain (NC) of the homologous series. These authors estimated a total of eight parameters to predict the kinematic viscosity of saturated FAMEs from C6:0 to C18:0: Four parameters were estimated for short chain esters (C6:0–C12:0), and four more parameters for the long chain ones (C12:0–C18:0). A maximum deviation of 5.94% and an AAD of 2.21% were obtained for the calculated values of C12:0–C18:0, when compared to 18 experimental data reported in the literature at 20, 40 and 70 °C [9]. Additionally, those authors calculated the viscosity of eight different biodiesels using the mixing rule given by Eq. (3), reporting deviations for their calculated viscosities from 1.08 up to 9.20%.

mexp  mcalc Deviation ð%Þ ¼ 100  mexp  Np  100  ðmi;exp  mi;calc Þ 1 X   AAD ð%Þ ¼    Np i¼1 mi;exp

ð4Þ

166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

201 203 204

ð5Þ

Please cite this article in press as: Chavarria-Hernandez JC, Pacheco-Catalán DE. Predicting the kinematic viscosity of FAMEs and biodiesel: Empirical models. Fuel (2014), http://dx.doi.org/10.1016/j.fuel.2014.01.105

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J.C. Chavarria-Hernandez, D.E. Pacheco-Catalán / Fuel xxx (2014) xxx–xxx Table 1 Kinematic viscositya (mm2/s) of saturated FAMEs at different temperatures. Data gathered from the literature or calculated from dynamic viscosity and density data reports. T (K)

C6:0

C7:0

C8:0

263.15 268.15 273.15 278.15

2.421 2.437 2.449 2.45 2.196 2.227 2.23 2.004 2.037 2.05 1.832 1.871 1.87

3.627 3.54 3.641 3.63 3.225 3.261 3.29 2.892 2.942 2.95 2.618 2.668 2.69

1.207

1.765

2.487

1.170 1.16 1.215

2.384

1.099 1.138 1.028 1.069

1.686 1.69 1.726 1.71 1.566 1.598 1.452 1.485

2.433 2.41 2.193 2.229 2.014 2.050

0.9663 1.006

1.353 1.384

1.859 1.894

1.499

1.967 1.931

288.15

1.084

1.379

1.772 1.769

293.15

1.01 1.026 1.011

1.273

1.610 1.59 1.628

303.15

308.15

310.95 313.15

0.8828

0.8302

1.181

1.101

1.028

1.471 1.504 1.368 1.396 1.262 1.300

C14:0

C16:0

C20:0

C22:0

C24:0

5.201

4.611

4.120

3.698

4.929

3.456

4.688

0.81 0.785 0.7852

C18:0

5.881 5.61 0.9610

318.15

0.7422

0.9084

323.15

0.7014

0.8440

3.23 3.338

4.32 4.414

5.61 5.867

3.030

3.977

5.241

2.764

3.602

4.706 5.737

328.15

0.6668

0.7974

2.533

3.280

4.254 5.154

333.15

338.15

343.15

348.15

353.15

0.632 0.6332

0.6043

0.577 0.5778

0.5561

0.5327

3.666 0.7635

0.7285

0.9089 0.911 0.9500

1.263 1.276 1.294

1.724 1.732 1.756

0.8592 0.8989

1.184 1.213

1.605 1.633

2.323 2.330

2.152

2.998 3.001

2.758

3.861

0.6557

0.6254

0.8144 0.819 0.8525

1.114 1.13 1.140

1.502 1.5 1.524

0.7718 0.8103

1.052 1.074

1.403 1.426

0.7349 0.7717

0.9927 1.014

1.318 1.338

1.95 1.996

1.856

1.732

2.5 2.544

4.230

5.143

6.188

3.861

4.672

5.595

2.355

3.540

4.263

5.086

3.258

3.908

4.645

3.011

3.597

4.260

2.793 2.599

3.329 3.088

3.921 3.620

2.428

2.871

3.355

3.11 3.227

2.969

2.188

2.741

0.9600

2.038

2.540

363.15

0.9111

1.905

2.362

0.457

2.004 0.618

373.15

5.691

3.522

358.15

368.15 372.05

4.658

3.11 0.6892

0.826

Purity (%) b

5.45 4.654 4.635 4.79 4.093 4.094 4.07

1.179

0.9412

C12:0

5.4 4.68 4.04 3.378 3.49 3.010 3.014 3.1 2.689 2.708 2.71

283.15

298.15

C10:0

1.072

1.36

1.703

>99 >99b >99b 99c >99b >99 99c,d >99b >99 99c,d >99b >99.7e – >99 – 99c,d >99b >99 99c,d >99b >99 99c,d >99b >99 99c,d >99b – – >99.7e >99 – 99c,d >99b >99 99c,d >99 99c,d 99.5c >99 99c,d 99.5c – >99 – 99c,d P99c >99 99c,d P99c >99.7e >99 – 99c,d P99c >99 99c,d P99c >99 99c,d P99c 99c P99c 99c P99c P99c – – P99c

Accuracy

Ref.

SD < 0.025 SD < 0.025 SD < 0.025 A1 SD < 0.025 – A1 SD < 0.025 – A1 SD < 0.025 SD < 0.01 – – – A1 SD < 0.025 – A1 SD < 0.025 – A1 SD < 0.025 A1 SD < 0.025 – – SD < 0.01 – – A1 SD < 0.025 – A1 – A1 A2 – A1 A2 – – – A1 A2 – A1 A2 SD < 0.01 – – A1 A2 – A1 A2 A1 A2 A1 A2 A1 A2 A2 – – A2

[10] [10] [10] [11]f [10] [12]g [11]f [10] [12]g [11]f [10] [13] [14] [12]g [7] [11]f [10] [12]g [11]f [10] [12]g [11]f [10] [12]g [11]f [10] [14] [7] [13] [12]g [7] [11]f [10] [12]g [11]f [12]g [11]f [15]h [12]g [11]f [15]h [14] [12]g [7] [11]f [15]h [12]g [11]f [15]h [13] [12]g [7] [11]f [15]h [12]g [11]f [15]h [12]g [11]f [15]h [11]f [15]h [11]f [15]h [15]h [14] [7] [15]h

SD – standard deviation. A1 – absolute uncertainty of density: 0.0005 g cm3; relative uncertainty of dynamic viscosity <1.5%; max. standard deviation for viscosity: 0.15%. A2 – absolute uncertainty of density: 0.0005 g cm3; relative uncertainty of dynamic viscosity <0.5%; max. standard deviation for viscosity: 0.15%. a Kinematic viscosity data gathered from the literature are given with the number of significant figures of the original reports. Calculated kinematic viscosities are given to four significant figures, since this is the least number of significant figures for every density–dynamic viscosity data pair considered for their calculation [11,12,15]. b Confirmed by NMR and/or GC–MS. c Confirmed by GC-FID.

Please cite this article in press as: Chavarria-Hernandez JC, Pacheco-Catalán DE. Predicting the kinematic viscosity of FAMEs and biodiesel: Empirical models. Fuel (2014), http://dx.doi.org/10.1016/j.fuel.2014.01.105

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J.C. Chavarria-Hernandez, D.E. Pacheco-Catalán / Fuel xxx (2014) xxx–xxx d e f g h

With the exception of C12:0 and C14:0, whose purity of is 98%. Reported in [16]. Calculated from dynamic viscosity and density data reported in Ref. [11]. Calculated from dynamic viscosity and density data reported in Ref. [12]. Calculated from dynamic viscosity and density data reported in Ref. [15].

Table 2 Kinematic viscositya (mm2/s) of unsaturated FAMEs at different temperatures. Data gathered from the literature or calculated from dynamic viscosity and density data reports. T (K)

C14:1

C16:1

C18:1

C18:2

C18:3

263.15 268.15 273.15 278.15

9.92 8.37 7.01 6.13

14.77 12.19 10.15 8.55

21.33 17.22 14.03 11.66

14.1 11.8 9.84 8.47 8.322

10.19 8.81 7.33 6.59

283.15

5.35

7.33

9.91 9.869

7.3 7.236

288.15

4.73

6.38

8.51 8.490

6.43 6.355

293.15

4.13

5.56

7.33 7.379

5.61 5.622

7.23 6.44 6.472

5.58 5.03 5.017

298.15

3.71

4.94

303.15

3.37

4.42

5.72 5.724

4.53 4.508

308.15

3.04

3.96

5.08 5.099

4.08 4.075

313.15

2.73

3.67

4.51 4.573

3.65 3.703

318.15

4.45 4.125

3.64 3.383

323.15

3.742

3.103

328.15

3.410

2.859

333.15

3.121

2.644

338.15

2.871

2.453

343.15

2.651

2.283

348.15

2.6 2.457

2.25 2.132

353.15

2.284

1.996

358.15 363.15 368.15 373.15

C20:1

C22:1

6.966 5.53

16.21

20.52

6.177 5.14

13.59

17.02

5.524 4.57

11.55

14.36

4.972 4.84 4.07

9.918

12.24 12.5

4.501 3.88

8.605

10.54

4.099 3.32

7.533

9.156

3.750 3.09

6.649

8.043

3.298 3.27

5.911

6.956 7.21

3.028

5.291

6.333

2.811

4.766

5.604

2.629

4.318

5.120

2.434

3.932

4.642

2.263

3.597

4.229

2.100 2.09

3.306

3.871 3.92

1.960

3.051

3.559

1.845 1.742 1.649

2.826 2.626 2.449 2.292 2.150

3.267 3.042 2.826

Purity (%)

Accuracy

Ref.

>99b >99b >99b >99b 99c P2 >99b 99c P2 >99b 99c P2 >99b 99c P2 P1 >99b 99c P2 >99b 99c P2 >99b 99c P2 >99b 99c P2 P1 99c P2 99c P2 99c P2 99c P2 99c P2 99c P2 P1 99c P2 99c P2 P2 P2 P2 P2

A1 A1 A1 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 SD < 0.01 SD < 0.025 A2 A3 SD < 0.025 A2 A3 SD < 0.025 A2 A3 SD < 0.025 A2 A3 SD < 0.01 A2 A3 A2 A3 A2 A3 A2 A3 A2 A3 A2 A3 SD < 0.01 A2 A3 A2 A3 A3 A3 A3 A3

[10] [10] [10] [10] [11]d [15]e [10] [11]d [15]e [10] [11]d [15]e [10] [11]d [15]e [13] [10] [11]d [15]e [10] [11]d [15]e [10] [11]d [15]e [10] [11]d [15]e [13] [11]d [15]e [11]d [15]e [11]d [15]e [11]d [15]e [11]d [15]e [11]d [15]e [13] [11]d [15]e [11]d [15]e [15]e [15]e [15]e [15]e

SD – standard deviation. A1 – SD < 0.025 for viscosity values less than about 6–8 mm2 s1 and increasing proportionally for higher viscosity values; viscosity data are means of triplicate determinations. A2 – absolute uncertainty of density: 0.0005 g cm3; relative uncertainty of dynamic viscosity <1.5%; max. standard deviation for viscosity: 0.15%. A3 – absolute uncertainty of density: 0.0005 g cm3; relative uncertainty of dynamic viscosity < 0.5%; max. standard deviation for viscosity: 0.15%. P1 – C18:1 > 99.7%; C18:2 = 99%; C18:3 = 98.5%; C22:1 = 98%, reported in [16]. P2 – C18:3 = 99%; C20:1 = 98; C22:1 = 99. a Kinematic viscosity data gathered from the literature are given with the number of significant figures of the original reports. Calculated kinematic viscosities are given to four significant figures, since density and dynamic viscosity data used to calculate them [11,15] are reported with four and five significant figures, respectively. b Confirmed by NMR and/or GC–MS. c Confirmed by GC-FID. d Calculated from dynamic viscosity and density data reported in Ref. [11]. e Calculated from dynamic viscosity and density data reported in Ref. [15].

Please cite this article in press as: Chavarria-Hernandez JC, Pacheco-Catalán DE. Predicting the kinematic viscosity of FAMEs and biodiesel: Empirical models. Fuel (2014), http://dx.doi.org/10.1016/j.fuel.2014.01.105

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J.C. Chavarria-Hernandez, D.E. Pacheco-Catalán / Fuel xxx (2014) xxx–xxx Table 3 Parameter values of the proposed models to calculate the kinematic viscosity of FAMEs. Model for saturated FAMEs, Eq. (6) Parameter a Value 3.02918 Models for unsaturated FAMEs, Eqs. (7) and (8) Parameter g Value 0.452351

b 0.138813

c 186.962

d 0.400877

e

h

i

j 0.849646

k 156.712

0.452419

42.9765

207

3. Results and discussion

208

3.1. Prediction of FAMEs kinematic viscosity

209

By following the procedure described in Section 2, Eqs. (6)–(8) were selected as the ones that best describe the behavior of kinematic viscosity of FAMEs as a function of temperature, NC and ND, for the available experimental data (Tables 1 and 2). The three models were derived from the Vogel equation, Eq. (2), in which parameters A, B and C are expressed as a function of either NC or ND or both. Eq. (6) calculates the kinematic viscosity of saturated FAMEs from C6:0 to C24:0 in a wide temperature range. In it, msat is the kinematic viscosity in mm2/s, NC is the number of Carbons in the hydrocarbon chain, for example, NC = 18 for methyl stearate (C18:0), T is the temperature in Kelvin, while a to f are the parameters of the model whose values are given in Table 3.

210 211 212 213 214 215 216 217 218 219 220 221

222 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244

ln msat ¼ a  NCb þ

c  NCd e  ln NC þ f þ T

22.9221

f 88.9471 l 1.14044

Fig. 1. Kinematic viscosity of saturated FAMEs as a function of temperature. Experimental data (symbols) gathered from the literature (Table 1) and calculated values (lines) using Eq. (6).

ð6Þ

A total of 247 experimental data points (Table 1) were considered for the fitting of parameters in Eq. (6), obtaining an AAD between calculated and experimental values of 1.33%. The quality of the fitting was quite good, as appreciated in Fig. 1, in which the experimental data points and the curves calculated with the model are shown. On the other hand, the percent deviation for each data point was calculated with Eq. (4) and the obtained values are shown in Fig. 2. The highest absolute deviation was 4.01% determined for C8:0 at 372.05 K, followed by 3.66% for C6:0 at 293.15 K. The rest of the calculated viscosities showed absolute deviations below 3.5%. On the other hand, two complementary equations were selected for unsaturated FAMEs. Eq. (7) applies for mono-unsaturated esters, while Eq. (8) applies for the di- and tri-unsaturated ones. Eq. (7) include 5 adjustable parameters, named g–k, while Eq. (8) uses the same five parameters in Eq. (7) and an additional parameter l, giving a total of six adjustable parameters that are necessary to accurately describe the viscosity–temperature behavior of unsaturated species. The value of these parameters is given in Table 3.

Fig. 2. Deviations between experimental and calculated kinematic viscosity of saturated FAMEs as a function of temperature.

245 247

ln mmonounsat ¼ g  NCh þ

i  NCj kþT

ð7Þ

248 h

250 251 252 253 254 255 256 257 258

ln mpolyunsat ¼ g  ðNC  NDl Þ þ

i  ðNC  NDl Þ kþT

j

ð8Þ

In Eqs. (7) and (8), mmono-unsat and mploy-unsat are the kinematic viscosities in mm2/s for mono- and for poly-unsaturated FAMEs respectively, NC and ND are the number of Carbons and the number of double bonds in the hydrocarbon chain respectively, while T is the temperature in Kelvin. In Eq. (8) ND can take the value of 2 or 3. The six parameters in Eqs. (7) and (8) were fitted simultaneously using the 154 experimental data points given in Table 2. They encompass unsaturated fatty esters from C14:1 to C22:1, including

Fig. 3. Kinematic viscosity of unsaturated FAMEs as a function of temperature. Experimental data (symbols) gathered from the literature (Table 2) and calculated values (lines) using Eqs. (7) and (8).

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C18:2–C18:3. By comparing the calculated and experimental viscosities an AAD of 2.29% was obtained. Fig. 3 shows experimental data points for unsaturated FAMEs and the curves given by Eqs. (7) and (8) with parameters g–l from Table 3. Fig. 4 shows viscosity deviations between experimental and calculated values for unsaturated FAMEs at the considered temperature range. The highest absolute deviation was 9.34%, obtained for C14:1 at 313.15 K, while 98.1% of the calculated kinematic viscosities presented absolute deviations lower than 8%. AADs were also calculated for individual FAMEs. The results are shown in Table 4 in which the number of experimental data points gathered from the literature for each FAME is also indicated. The figures indicate that C18:0 presented the highest AAD among the saturated FAMEs having a value of 2.399%, while for the unsaturated FAMEs, C14:1 showed the maximum AAD with a value of 4.414%. Higher deviations mentioned in the last two paragraphs for saturated and unsaturated species may partially attributed to limitations of the proposed models to get a better fitting of the experimental information, but it is also very important to note that the experimental data for individual fatty esters was gathered from up to four different reports taken from the literature. This data bank was thus obtained from different laboratories using different analytical equipment, and some values for the same compound at the same temperature present significant differences. Just to give an example, three different data reports were considered for C8:0 at 313.15 K as can be seen in Table 1. The highest viscosity value is 1.215 mm2/s, which is 4.7% higher than the lowest value at the same temperature (1.16 mm2/s). This simple example indicates that some experimental information used in this report may present some significant inaccuracies. For certain compounds only one report was found without the possibility to contrast the information against other reported data. The variability in the experimental data certainly affects the deviations between experimental and calculated kinematic viscosities. Notwithstanding the foregoing, the percent deviations and AADs obtained by applying Eqs. (6)–(8) indicate that the proposed correlations are more accurate than comparable models such as Krisnangkura’s model [9] and Ramírez-Verduzco’s model [2] referred to in Section 1. Other methods for predicting viscosity of individual fatty esters have also shown higher deviations compared to the models of this work. Pratas et al. [11,15] measured the viscosity of fifteen FAMEs over a wide temperature range (data

Fig. 4. Deviations between experimental and calculated kinematic viscosity of Unsaturated FAMEs as a function of temperature.

Table 4 Average absolute deviation (AAD) between experimental viscosities of individual FAMEs given in Tables 1 and 2, and the calculated values using the models of this work.

a

FAME

Np

Saturated FAMEs C6:0 C7:0 C8:0 C10:0 C12:0 C14:0 C16:0 C18:0 C20:0 C22:0 C24:0

a

AAD (%)

T range (K)

22 15 36 50 43 18 17 18 11 9 8

2.057 0.572 1.665 1.468 0.983 1.166 1.199 2.399 0.290 0.474 0.915

[283.15, 372.05] [283.15, 353.15] [283.15, 372.05] [263.15, 372.05] [278.15, 372.05] [293.15, 372.05] [308.15, 372.05] [310.95, 372.05] [323.15, 373.15] [333.15, 373.15] [338.15, 373.15]

Unsaturated FAMEs C14:1 11 C16:1 11 C18:1 29 C18:2 30 C18:3 32 C20:1 20 C22:1 21

4.414 1.585 1.787 1.478 3.952 1.993 1.129

[263.15, 313.15] [263.15, 313.15] [263.15, 353.15] [263.15, 353.15] [263.15, 363.15] [278.15, 373.15] [278.15, 363.15]

Number of experimental data points considered. For references see Tables 1 and 2.

Table 5 Dynamic viscosity prediction (l) of five biodiesels at 40 °C. Comparison between experimental data reported in Ref. [1] and calculated values of this work. FAME

ma (mm2/S) c

C24:0 11.794 C22:1 7.140 C22:0 9.366c C20:1 5.741 C20:0 7.360c C18:3 3.152 C18:2 3.629 C18:1 4.616 C18:0 5.714 C16:1 3.711 C16:0 4.371 C14:0 3.285 C12:0 2.415 C10:0 1.724 C8:0 1.183 Measured viscosity (mPa s) [1] Predicted viscosity (mPa s) Deviation (%) Purity (% FAMEs) a b c

q [11,15] (Kg/m3) c

851.0 856.5 848.6c 859.5 850.0c 887.0 871.5 859.5 849.8 853.8 850.8 852.2 853.9 856.0 859.6

lb (mPa s)

Coconut (xi)

Peanut (xi)

Soya (xi)

Palm (xi)

Canola (xi)

10.036c 6.116 7.948c 4.935 6.256c 2.795 3.163 3.967 4.855 3.169 3.719 2.800 2.062 1.476 1.017

0 0 0 0 0.002 0 0.014 0.055 0.019 0 0.073 0.171 0.533 0.06 0.075 2.32 2.24 3.32 95.01

0.035 0 0.024 0.014 0.013 0.01 0.301 0.466 0.027 0.004 0.105 0 0 0 0 3.77 3.90 -3.44 99.81

0 0 0 0.016 0.007 0.096 0.199 0.6 0.017 0.008 0.058 0 0 0 0 3.67 3.69 -0.51 99.76

0 0 0 0.001 0.003 0.002 0.08 0.373 0.04 0.003 0.481 0.013 0.004 0 0 3.87 3.78 2.30 98.21

0 0 0.001 0.021 0.012 0.112 0.213 0.574 0.02 0.004 0.042 0 0 0 0 3.7 3.67 0.79 100.00

Calculated with Eqs. (6)–(8). Calculated with Eq. (1). Extrapolated below the temperature of crystals formation.

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considered in this work), and then assessed the accuracy of the method of Ceriani et al. [17] and that of Marrero and Gani (MG) [18] to predict their experimental viscosities. For the case of FAMEs from C8:0 to C18:2, average and maximum deviations of 4.53– 9.50% were obtained respectively by applying Ceriani’s method, while the corresponding deviations for the MG method were 12.0–25.5% [11]. For FAMEs from C16:1 to C24:0, the same authors reported average and maximum deviations in the viscosity prediction of 10.24–25.9% respectively when Ceriani’s method was applied, while the corresponding values for the less accurate MG method were 24.73–44.3% [15].

313

3.2. Prediction of biodiesel viscosity

314

The kinematic viscosity correlations of this work were applied to predict the viscosity of 31 pure biodiesels whose FAME

302 303 304 305 306 307 308 309 310 311

315

compositions at different temperatures were taken from the literature. The ideal mixture behavior given by Eq. (3) was assumed for all the calculations. Prediction of dynamic viscosity at 40 °C for five pure biodiesels is given as an example in Table 5. In this case density data reported by Pratas et al. [11,15] were used for the application of Eq. (1). For these five biodiesels, minimum and maximum absolute deviations between experimental and predicted values of 0.51–3.44% were obtained respectively. The % content of FAMEs in the biodiesels is also indicated, ranging from 95.01 to 100%. These results indicate that viscosity prediction of a high purity biodiesel (high % FAMEs) can be performed with good accuracy from the correlations proposed in this work even assuming ideal mixture behavior. The results obtained for the complete data sets of pure biodiesels are summarized in Table 6, in which AADs are compared with the corresponding values obtained for previously reported models.

Table 6 Comparison of AADs obtained by using equations of this work as well as other six different models applied to the same data bank of pure biodiesels and biodiesel blends. Ref.

Biodiesel

Pure biodiesels [1] Coconut Peanut Soya Palm Canola [21] Peanut Rapeseed Canola Coconut Palm Soybean [7] Soybean Yellow grease Coconut Palm Canola [22] Coconut Colza Soybean [19] Babassu Soybean Cotton seed [23] Fish oil Sunflower [24] Soy A Soy B B1 (methyl oleate 70%) Sunflower Rapeseed Palm GP (soy + rapeseed) Biodiesel blends [22] Coconut + Soybean Coconut + Colza [19] Soybean + Babassu Cotton seed + Babassu Total for pure biodiesels Total for biodiesel blends Total

Ester content (%)

Npa

T range (K)

AAD (%)b for different models using the same databank for pure biodiesels and biodiesel blends This work

do Carmo (two fluids) [20]

Revised Yuanc [11]

RamirezVerduzcod [2]

do Carmo (one fluid) [20]

Yuanc [7]

Cerianic [17]

95.01 99.81 99.76 98.21 100.00 – – – – – – –– – – – – 98.2 97.8 99.1 >98.1 >98.2 >98.3 – – – 99.4 –

1 1 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 5 15 18 15

[313.15] [313.15] [313.15] [313.15] [313.15] [313.15] [313.15] [313.15] [313.15] [313.15] [313.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [283.15, 353.15] [278.15, 363.15] [283.15, 353.15]

3.32 3.44 0.51 2.30 0.79 6.00 0.51 6.03 6.92 4.72 4.83 2.55 5.45 4.22 5.97 5.32 7.56 14.41 11.79 5.45 12.45 12.87 3.44 5.14 5.71 2.59 4.71

0.00 0.09 3.13 9.89 2.73 6.49 15.75 10.03 9.30 2.72 16.49 5.43 12.96 1.82 11.47 2.80 5.39 14.23 4.43 1.48 0.40 6.28 5.37 1.80 0.18 5.38 10.53

7.49 2.48 0.21 1.95 0.22 6.59 2.63 6.64 1.65 4.91 6.59 1.57 6.79 7.14 5.97 3.67 1.9 16.37 12.57 0.37 4.44 13.52 9.77 5.87 4.59 2.37 7.3

2.50 4.54 2.57 3.00 1.42 8.22 0.20 8.46 7.66 4.21 9.76 2.86 8.57 7.42 7.91 3.66 4.83 17.04 12.60 3.01 13.43 4.86 6.77 5.52 4.71 2.45 6.91

6.99 2.14 4.59 11.63 4.46 7.58 5.61 11.71 1.64 4.56 17.35 5.67 13.19 6.22 12.5 2.83 2.73 13.6 4.36 4.07 0.65 6.21 6.69 2.17 0.39 5.78 9.71

9.65 0.92 1.4 2.84 1.94 4.84 2.03 4.75 0.68 3.8 5.24 2.21 7.92 9.1 6.26 4.68 0.56 17.22 12.96 1.43 4.69 14.01 8.25 6.54 5.24 2.98 8.08

9.21 3.67 2.38 1.01 3.23 6.18 5.51 3.25 0.01 7.69 0.66 11.08 8.12 10.62 7.7 12.17 3.3 22.71 21.97 3.56 11.78 22.36 10.97 14.59 7.78 8.74 5.27

98.5 98.8 96.5 –

17 18 16 18

[283.15, 363.15] [278.15, 363.15] [288.15, 363.15] [278.15, 363.15]

5.29 3.69 4.89 2.50

2.52 9.72 11.89 1.56

5.41 5.92 5.48 2.63

4.03 5.50 7.03 2.72

3.28 10.77 13.12 2.06

6.19 6.03 5.94 3.38

11.29 6.65 4.79 7.27

35 45 35 35

[293.15, 373.15] [293.15, 373.15] [293.15, 373.15] [293.15, 373.15]

17.13 5.68 5.16 2.88

19.53 6.69 4.06 0.48

17.27 8.67 7.74 2.59

18.36 7.87 6.98 3.03

19.78 8.05 6.18 2.76

17.49 9.95 8.72 3.48

16.89 13.38 13.39 7.40

193

5.33

5.90

5.42

5.95

6.40

5.91

8.91

150

7.58

7.62

9.04

8.98

9.12

9.91

12.81

343

6.44

6.66

7.00

7.44

7.59

7.66

10.61

a

Number of experimental data points. All AAD values were calculated with Eq. (5), even though some authors use different nomenclature in their respective reports. Reported by do Carmo in Ref. [20]. d Most of the these values were reported by Ramirez-Verduzco in Ref. [2], the rest of them were calculated in this work using the equations reported by that author in the same reference. b

c

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The predictive capacity of the proposed equations was also tested by calculating the viscosity of four binary biodiesel blends. Figs. 5 and 6 graphically illustrate the comparative results between experimental and predicted kinematic viscosities for two biodiesel blends (babassu/cotton seed and babassu/soybean) reported by Nogueira et al. [19]. The calculation of viscosity for a biodiesel blend is not really different from that for a pure biodiesel, since only the percent content of individual FAMEs present in the final mixture has to be taken into account in the application of Eq. (3). Table 6 summarizes the results obtained for the biodiesel blends considered in this work. It is also given a comparison of the AADs obtained by applying the proposed models with the corresponding results obtained for other six different viscosity models when applied to the same experimental databank. It was found that the proposed equations of this work, i.e. Eqs. (6)–(8) and the utilization of the logarithmic ideal mixing rule resulted in the lowest total deviations between experimental and calculated viscosities for both data sets (pure biodiesel and biodiesel blends). Even though three equations and twelve parameters may seem to be too many equations and parameters for the viscosity prediction of FAMEs and biodiesels, the obtained results justify the use of the proposed correlations of this work, which can be more accurate and even simpler than some of the comparable models. As a final outlook on the subject of the improvement of viscosity models, and considering that in general, models for the prediction of individual fatty esters are more accurate than models for their mixtures, it can be identified at least two aspects that should be further studied in future works. One of them is the improvement in the estimation of the parameter that takes into account the interactions among biodiesel constituents, when applying mixing rules that consider this term. The second one is the inclusion of

Fig. 5. Kinematic viscosity of cotton seed + babassu biodiesel blends as a function of mass fraction of cotton seed biodiesel at different temperatures. Experimental data (symbols) taken from Ref. [19] and predicted values (lines) calculated with correlations of this work.

Fig. 6. Kinematic viscosity of soybean + babassu biodiesel blends as a function of mass fraction of soybean biodiesel at different temperatures. Experimental data (symbols) taken from Ref. [19] and predicted values (lines) calculated with correlations of this work.

impurities in the mixture. In order to consider impurities it could be useful to choose a single compound representative of them. The choice of such model compound should be based on experimental data. mono- and/or di-glycerides could be some of the species considered. Additionally, the development of more accurate viscosity models also demands more experimental viscosity reports for some individual fatty acid esters and mainly for biodiesels and their blends.

363

4. Conclusions

371

Viscosity is one of the most significant properties of biodiesel because of its major effect on the engine performance. For this reason reliable mathematical models that accurately describe the kinematic viscosity of biodiesel as a function of temperature are of great interest for the development of combustion models as well as for the design of process equipment. In this work three complementary correlations were proposed to accurately predict the kinematic viscosity of FAMEs for a wide temperature range (263.15–373.15 K), and for a wide range of hydrocarbon chain length (C6:0–C24:0, including unsaturated FAMEs). The correlations were derived from the Vogel equation by expressing the three parameters of this equation as a function of either the number of Carbons in the hydrocarbon chain or the number of double bonds in it, or both. Even though 12 parameters are required for the application of the developed equations (6 for saturated FAMEs and 6 more for unsaturated FAMEs), the proposed modes have proved to be more accurate than other comparable models (some of them more complex) not only for the prediction of individual FAMEs, but also for the prediction of viscosity of pure biodiesel and biodiesel blends when applied to a very extensive data bank. The 12 adjustable parameters of the developed equations were fitted using experimental information gathered from the literature. A total of 247–154 experimental data points were considered for saturated and unsaturated species, respectively. Average and maximum absolute deviations between calculated and experimental viscosities for the saturated FAMEs were 1.33–4.01% respectively, while the corresponding values for the unsaturated species were 2.88–9.34%. These deviations are lower than those obtained for models recently published in the literature such as Krisnangkura’s model [9] and Ramírez-Verduzco’s model [2]. The developed models were subsequently applied for the prediction of viscosity of 31 pure biodiesels (193 data points) and four biodiesel blends (150 data points) whose composition and viscosity at several temperatures were gathered from the literature. Average absolute deviations (AADs) between experimental and predicted viscosities were calculated for the models of this work in combination with the logarithmic ideal mixing rule. ADD for all pure biodiesels was 5.33%, while the corresponding AAD for all biodiesel blends was 7.58%. These results were compared to the AADs obtained for do Carmo’s models [20], Yuan’s models [7,11], Ramirez-Verduzco´s model [2], and Ceriani´s model [17] when applied to the same data bank. The comparison showed that the lowest AADs for both data sets (pure biodiesel and biodiesel blends) were obtained with the correlations of this work. In future works, two aspects should be further studied in order to improve the viscosity prediction of fatty ester mixtures: the improvement in the estimation of the parameter that takes into account the interactions among biodiesel constituents for mixing rules that consider that term, and the inclusion of impurities in the calculation for the mixture. For the latter, it could be useful to choose one or more compounds representative of impurities. Mono- and/or di-glycerides could be some of the species considered. Additionally, more experimental data for individual fatty esters and mainly for biodiesels and their blends are also necessary for the improvement of viscosity models.

372

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Acknowledgements

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The author acknowledges the support granted by Consejo Nacional de Ciencia y Tecnología (CONACyT), for Project No. 166640 (Ciencia básica). The author also acknowledges the support of Emily Barrett for providing language help.

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References

433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457

[1] Allen CAW, Watts KC, Ackman RG, Pegg MJ. Predicting the viscosity of biodiesel fuels from their fatty acid ester composition. Fuel 1999;78:1319–26. [2] Ramírez Verduzco LF. Density and viscosity of biodiesel as a function of temperature: empirical models. Renew Sustain Energy Rev 2013;19:652–65. [3] Tat ME, Van Gerpen JH. The kinematic viscosity of biodiesel and its blends with diesel fuel. J Am Oil Chem Soc 1999;76(12):1511–3. [4] Holloway MD, Nwaoha C, Onyewuenyi OA. Process plant equipment: operation, control, and reliability. John Wiley & Sons; 2012. [5] Noureddini H, Teoh BC, Clements LD. Viscosities vegetable oils and fatty acids. J Am Oil Chem Soc 1992;69(12):1189–91. [6] Vogel H. The law of the relation between the viscosity of liquids and the temperature. Phys Z 1921;22:645–6. [7] Yuan W, Hansen AC, Zhang Q. Predicting the temperature dependent viscosity of biodiesel fuels. Fuel 2009;88:1120–6. [8] Grunberg L, Nissan AH. Mixture law for viscosity. Nature 1949;164:799–800. [9] Krisnangkura K, Yimsuwan T, Pairintra R. An empirical approach in predicting biodiesel viscosity at various temperatures. Fuel 2006;85(1):107–13. [10] Knothe G, Steidley KR. Kinematic viscosity of biodiesel components (fatty acid alkyl esters) and related compounds at low temperatures. Fuel 2007;86(16):2560–7. [11] Pratas MJ, Freitas S, Oliveira MB, Monteiro SC, Lima AS, Coutinho JAP. Densities and viscosities of fatty acid methyl and ethyl esters. J Chem Eng Data 2010;55:3983–90. [12] Liew KY, Seng CE, Oh LL. Viscosities and densities of the methyl esters of some n-alkanoic acids. J Am Oil Chem Soc 1992;69(2):155–8.

429 430

9

[13] Gouw TH, Vlugter JC, Roelands CJA. Physical properties of fatty acid methyl esters. VI. Viscosity. J Am Oil Chem Soc 1966;43(7):433–4. [14] Bonhorst CW, Althouse PM, Triebold HO. Esters of naturally occurring fatty acids. Physical properties of methyl, propyl, and isopropyl esters of C6–C18 saturated fatty acids. Ind Eng Chem 1948;40(12):2379–84. [15] Pratas MJ, Freitas S, Oliveira MB, Monteiro SC, Lima AS, Coutinho JAP. Densities and viscosities of minority fatty acid methyl and ethyl esters present in biodiesel. J Chem Eng Data 2011;56(5):2175–80. [16] Gouw TH, Vlugter JC. Physical properties of fatty acid methyl esters. I. density and molar volume. J Am Oil Chem Soc 1964;41(2):142–5. [17] Ceriani R, Goncalves CB, Rabelo J, Caruso M, Cunha ACC, Cavaleri FW, et al. Group contribution model for predicting viscosity of fatty compounds. J. Chem. Eng. Data 2007;52(3):965–72. [18] Marrero J, Gani R. Group-contribution based estimation of pure component properties. Fluid Phase Equilib 2001;183:183–208. [19] Nogueira CA, Feitosa FX, Fernandes FAN, Santiago RS, de Sant’Ana HB. Densities and viscosities of binary mixtures of babassu biodiesel + cotton seed or soybean biodiesel at different temperatures. J Chem Eng Data 2010;55(11):5305–10. [20] do Carmo FR, Sousa PM, Santiago-Aguiar RS, de Sant’Ana HB. Development of a new model for biodiesel viscosity prediction based on the principle of corresponding state. Fuel 2012;92(1):250–7. [21] Shu Q, Yang B, Yang J, Qing S. Predicting the viscosity of biodiesel fuels based on the mixture topological index method. Fuel 2007;86:1849–54. [22] Feitosa FX, Rodrigues ML, Veloso CB, Cavalcante CL, Albuquerque MCG, de Sant’Ana HB. Viscosities and densities of binary mixtures of coconut + colza and coconut + soybean biodiesel at various temperatures. J Chem Eng Data 2010;55:3909–14. [23] Parente RC, Nogueira Jr CA, do Carmo FR, Lima LP, Fernandes FAN, Aguiar RSS, et al. Excess volumes and deviations of viscosities of binary mixtures of sunflower biodiesel + diesel and fish oil biodiesel + diesel at various temperatures. J Chem Eng Data 2011;56(7):3061–7. [24] Freitas SVD, Pratas MJ, Ceriani R, Lima AS, Coutinho JAP. Evaluation of predictive models for the viscosity of biodiesel. Energy Fuels 2011;25(1): 352–8.

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