Estimation of Biodiesel Physical Properties Using Local

Article pubs.acs.org/IECR

Estimation of Biodiesel Physical Properties Using Local Composition Based Models Hamed Abedini Najafabadi,† Gholamreza Pazuki,‡ and Manouchehr Vossoughi*,†,§ †

Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran Department of Chemical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran § Institute for Nanoscience and Nanotechnology, Sharif University of Technology, Tehran, Iran ‡

ABSTRACT: In this study, the local composition based models such as the Wilson, the nonrandom two-liquid (NRTL), and the Wilson-NRF have been applied in correlation and estimation of density, viscosity, and surface tension of biodiesels. The thermodynamic models have been used in correlating the thermophysical properties for 215 experimental data points. These models have the interaction energy between each pair that is considered as adjustable parameters. To decrease the number of these adjustable parameters, it is assumed that the biodiesels are composed of two hypothetical components. The average absolute deviation (AADs) of the correlated density of biodiesels for the Wilson, the NRTL, and the Wilson-NRF models are 0.0141, 0.0136, and 0.0148, respectively. The AADs of the correlated viscosity of biodiesels for the Wilson, the NRTL, and the Wilson-NRF models are 0.638, 0.547, and 0.621, respectively. Also, the AAD of the correlated surface tension of biodiesels for the Wilson, the NRTL, and the Wilson-NRF models are 0.402, 0.392, and 0.479, respectively. Comparisons between the results of the models previously proposed in the literature with those obtained in the present study confirm the effectiveness of the local composition based models in estimating the physical properties of biodiesels. Among these models, the NRTL model can estimate physical properties of biodiesels the most accurately. not provide sufficient lubrication for the precision fit of fuel injection pumps, resulting in leakage or increased wear.9 The surface tension is an important parameter in the formation of oil droplets.10 High surface tension causes difficulties in formation of droplets from the liquid fuel. Also, surface tension has a major impact on proper mixing and complete combustion in an injection engine.11 So far, many researchers have studied methods for predicting physical properties of biodiesels. Pratas et al.12 used the cubicplus-association equation of state (CPA EoS) to predict the density of seven biodiesels in high-pressure conditions. Lapuerta et al.13 used several experimental data and developed a new correlation to predict the density of pure fatty acid alkyl esters. Freitas et al.14 investigated the prediction capability of several empirical models previously proposed in the literature for a description of the viscosities of several biodiesels and introduced the best one. Yuan et al.15 developed a method for predicting temperature-dependent viscosities of biodiesel based on fatty acid ester composition. Freitas et al.16 reported experimental surface tensions for 10 biodiesel fuels in a wide range of temperatures and evaluated the ability of two models to predict the experimental data. Allena et al.17 presented a method to predict the surface tension of biodiesel fuels based on their fatty acid ester composition. In this paper, the excess Gibbs free energy models such as the Wilson, the nonrandom two-liquid (NRTL), and the WilsonNRF are used for estimation of density, viscosity, and surface

1. INTRODUCTION In recent decades, biodiesel has been focused on by many researchers because of its rising use as a new energy source to replace petroleum-based fuels. The main advantages of biodiesel with respect to fuel oil are: (1) fewer greenhouse gases released by its use,1 (2) more favorable combustion profile,2 (3) mixing in all proportions with regular diesel with no need for motor changes,3 and (4) easier storage and transportation.4 Although renewable fuels cannot replace fossil fuels yet, they can only contribute to reduce their consumption.5 Biodiesel consists of a blend of fatty acid alkyl esters that are industrially produced through the transesterification reaction. In this process, vegetable oil or animal fat reacts with an alcohol at elevated temperature to produce biodiesel and glycerol. The rate and yield of this reaction can be increased by using a typical catalyst such as sodium hydroxide. Knowledge and prediction of biofuel physical properties are very important in the design and optimization of engine injection systems as well as in the process of production and purification of biodiesel. Density is an important fuel property, which directly affects the engine performance characteristics such as cetane number and heating value.6 Also, density has a major role in delivering a precise amount of fuel to the engine. So changes in the fuel density will influence engine output power due to a different mass of fuel injected.7 Viscosity strongly affects the operation of fuel injection equipment. High viscosity fuels tend to form larger droplets upon injection, leading to poorer atomization during the spray and cause operation problems, such as increased carbon deposition.8 On the other hand, a fuel with low viscosity may © 2012 American Chemical Society

Received: Revised: Accepted: Published: 13518

June 5, 2012 September 5, 2012 September 6, 2012 September 6, 2012 dx.doi.org/10.1021/ie301464g | Ind. Eng. Chem. Res. 2012, 51, 13518−13526

Industrial & Engineering Chemistry Research

Article

tension of biodiesels. The adjustable parameters of these models are obtained for each property by minimization of an objective function. Also, the results of these models are compared with the experimental data and the results of other models previously proposed in the literature such as the group contribution method (GCVOL), the modified Yuan model, and the MacLeod−Sugden model.

v ex =

i

g (T , P ) =

∑

P) + RT ln xi) + g

i

⎛ ∑ xj ΛijΔλijv ⎞ j ⎟ ∑ xΛ ⎟ ⎝ k k ki ⎠

i

ex

(8)

Δgvji

Δgji and = ∂Δgji/∂P are adjustable parameter. αji represents nonrandomness in the mixture and varies from about 0.2 to 0.47. When experimental data are scarce, the value of αji can set to 0.3. Finally, excess volume of a solution based on the WilsonNRF model can be expressed by the following equation

(1)

v ex =

⎛ ∑ x H Δhv ji j j ji − x ∑ i⎜⎜ ∑ x H k k ki ⎝ i

⎞ v⎟ Δ x h ∑ j ji⎟ ⎠ j

(9)

In the above equation ⎛ −Δhji ⎞ Hji = exp⎜ ⎟ ⎝ CRT ⎠

(10)

Δhji and Δhvji = ∂Δhji/∂P are adjustable parameters for the proposed model which should be regressed by minimizing the error between experimental density and the results of the proposed model. C is coordination number that is set to be 10. 2.2.2. Viscosity. According to the Eyring theory, the viscosity of solution can be obtained as23

(2)

ln(ηv) =

∑ xi ln(ηivi) + i

g ex, * RT

(11)

where η and v are the viscosity and molar volume of a mixture, and ηi and vi are viscosity and molar volume of component i. The molar volume of the solution is given as v=

∑ xivi i

(12)

gex,*is activation excess Gibbs free energy which can be expressed by the Wilson,18 the NRTL,19 and the Wilson-NRF20 models, respectively. 2.2.3. Surface Tension. Chunxi et al.24 represent a new model for prediction of surface tension of several binary solutions. They related surface tension to Gibbs free energy using eq 13

(3)

(4)

where in eq 4 Λij can be calculated by the following equation ⎛ Δλij ⎞ Λij = exp⎜ − ⎟ vi ⎝ RT ⎠

(7)

RT

Gji = exp( −αjiτji)

On the basis of the Wilson model, excess volume of solution is given by

∑ xi⎜⎜

k

Δgji

τji =

ex

v ex =

j

where:

where in the above equation ρ is the density of solution and Mwi and vi are molecular mass and molar volume of component i, respectively. The excess volume of solution can be obtained using the excess Gibbs free energy at constant temperature as follows ⎛ ∂g ex ⎞ v =⎜ ⎟ ⎝ ∂P ⎠T , n

∑ xjGjiαjiτji]}/[(∑ xkGki)2 ]) (6)

∑i xi Mwi ∑i xivi + v ex

k

xkGkiΔgkiv][

k

where xi, g0i (T,P), and gex are the mole fraction, molar Gibbs free energy of component i, and excess Gibbs free energy, respectively, at T and P. In this research, three types of local composition based models including the Wilson,18 the NRTL (nonrandom twoliquid),19 and a modified version of the Wilson model proposed by Pazuki et al.,20−22 so-called the Wilson-NRF, are applied for expressing excess Gibbs free energy. These models consider the interaction energy between the i−j pair which is assumed to be adjustable parameters. 2.2. Physical Properties. According to the fundamental relations of thermodynamics, the physical properties of mixture can be considered as a function of Gibbs free energy. In the next paragraphs, an expression for density, viscosity, and surface tension will be defined using the excess Gibbs free energy model. 2.2.1. Density. The relation between density and excess volume of solution can be written as21

ρ=

j

+ [∑

2. MATHEMATICAL MODELING 2.1. Thermodynamic Model. For a multicomponent liquid mixture at system temperature T and pressure P, the molar Gibbs free energy of the system can be expressed as xi(gi0(T ,

∑ xi({[∑ xjGjiΔg jiv(1 − αjiτji)][∑ xkGki]

σ=

∑ xiσi + σ ex i

vj

(13)

where in the above equation σ is surface tension of solution; xi and σi are mole fraction and surface tension of each component, respectively. σex is obtained by the following equation

(5)

where Δλij is the interaction energy between the i−j pair which is assumed to be independent of temperature and Δλvij = ∂Δλij/ ∂P. Here, Δλij and Δλvij are considered as adjustable parameters. According to the NRTL model, the excess volume of the mixture can be expressed by eq 6:

⎛ ∂g ex ⎞ σ ex = ⎜ ⎟ ⎝ ∂A ⎠T , P , n 13519

(14)

dx.doi.org/10.1021/ie301464g | Ind. Eng. Chem. Res. 2012, 51, 13518−13526


Article

where in eq 14 A is surface area. On the basis of this theory, σex can be expressed by eqs 15−17 according to the Wilson, the NRTL, and the Wilson-NRF models, respectively σ

ex

⎛ ∑ xj ΛijΔλijσ ⎞ j ⎟ = ∑ xi⎜⎜ ⎟ ∑ Λ x k ki ⎝ ⎠ k i

σ ex =

Table 1. Name, Temperature Range, Number of Experimental Data and References of Biodiesels Studied in This Paper

soybean 1 sunflower rapeseed palm GPa soybean 2 soybean 1 sunflower rapeseed palm GPa soybean 2 soybean 1 sunflower rapeseed palm GPa soybean 2

(15)

∑ xi({[∑ xjGjiΔg jiσ (1 − αjiτji)][∑ xkGki] i

j

+ [∑

k

xkGkiΔgkiσ ][

k

∑ xjGjiαjiτji]}/[(∑ xkGki)2 ]) j

k

(16)

σ

ex

⎛ ∑ x H Δhσ ji j j ji − = ∑ xi⎜⎜ ∑ xH k k ki ⎝ i

⎞

∑ j

Δλij, Δλσij

xjΔhjiσ ⎟⎟ ⎠

(17)

Δgji, Δgσij

In the above equations = ∂Δλij/∂A, = ∂Δgij/ ∂A, Δhji, and Δhσij = ∂Δhij/∂A are adjustable parameters of surface tension models. 2.3. Parameter Estimation. The adjustable parameters of density, viscosity, and surface tension models can be regressed by minimizing one of the following objective functions OF1 =

∑ (Miexpt − Micalc)2 i

⎛ M expt − M calc ⎞2 i ⎟⎟ OF2 = ∑ ⎜⎜ i expt M ⎝ ⎠ i i

a

Micalc − Miexpt × 100 Miexpt

(19)

(20)

∑i AD n

viscosity

surface tension

temperature range (K)

18 18 18 16 18 6 18 17 18 16 18 6 6 5 5 6 6 5

278.15−363.15 278.15−363.15 278.15−363.15 288.15−363.15 278.15−363.15 293.15−373.15 278.15−363.15 283.15−363.15 278.15−363.15 288.15−363.15 278.15−363.15 293.15−373.15 303.15−353.15 313.15−353.15 303.15−343.15 303.15−353.15 303.15−353.15 293.15−343.15

reference Pratas et al.25

Feitosa et al.26 Freitas et al.14

Feitosa et al.26 Freitas et al.16

Blangino et al.27

GP = blending of soy and rapeseed.

lot of adjustable parameters should be assigned to the thermodynamic models. Ordinarily, by raising the number of adjustable parameters, the accuracy of the models would be increased, although in the present study we try to decrease the number of these parameters in an acceptable manner. So the results of the proposed models would be accurate and also less sensitive to the adjustable parameters. To reduce the number of adjustable parameters, it is assumed that the biodiesels are composed from two hypothetical components. C10 to C16:1 are considered as the first component, while C18 to C22:1 are considered as the second one. By considering this assumption, the number of adjustable parameters will be decreased which makes mathematical calculation easier. The schematic diagram of this assumption is shown in Figure 1. Physical properties of the hypothetical components can be calculated using the simple mixing rule as below

Also, the average absolute deviation (AAD) was calculated by averaging the ADs over n experimental data points AAD =

density

n

(18)

where Mi is a physical property of biodiesel at each data point and subscripts ″expt″ and ″calc″ stand for experimental data and calculated results, respectively. To study the performance of the investigated models, the absolute deviation (AD) for the predicted physical property of each biodiesel was expressed according to eq 20 AD =

physical property

biodiesel

(21)

MHyCo =

∑ xjMFA, j j

3. EXPERIMENTAL DATA BANK The experimental data for density, viscosity, and surface tension of six biodiesels are obtained from the literature.14,16,25−27 These biodiesels are synthesized by transesterification of vegetable oils with methanol using sodium hydroxide as the catalyst. The information of the experimental data is presented in Table 1. Also, compositions in terms of methyl ester of these biodiesels are reported in Table 2. It is noted that soybean 1 is used for correlating the physical properties of biodiesel, while soybean 2 is used for prediction to show the validity of the proposed models.

(22)

where MHyCo is physical property (density, viscosity, and surface tension) of the hypothetical components. MFA,j and xj are the physical property and mole fraction of each fatty acid alkyl ester, respectively. 4.1. Prediction of Density. According to Lapuerta et al.,13 the density of each fatty acid alkyl ester as a function of temperature can be estimated using the following equation 0 ρFA (kg/L) = ρFA − a(T /K − 288.15)

(23)

where

4. RESULTS AND DISCUSSION The local composition based models have been used to correlate and predict the physical properties of biodiesels. As these biodiesels are composed of several fatty acid alkyl esters, a

0 ρFA = 851.471

+ 13520

250.718db + 280.889 − 92.180(nA − 1) 1.214 + nF

(24)



Article

Table 2. Compositions of the Biodiesels Gathered from the Literature, In Mass Fraction methyl esters

soybean 1

sunflower

rapeseed

palm

GP

soybean 2

C10 C12 C14 C16 C16:1 C18 C18:1 C18:2 C18:3 C20 C20:1 C22 C22:1

0 0 0.0007 0.1078 0.0007 0.0395 0.2302 0.5366 0.0703 0.0038 0.0023 0.008 0

0 0.0002 0.0007 0.0641 0.0009 0.0423 0.2393 0.6425 0.0012 0 0.0003 0.0077 0.0008

0.0001 0.0004 0.0007 0.0526 0.002 0.0163 0.6249 0.2094 0.0699 0.006 0.0123 0.0135 0.0019

0.0003 0.0025 0.0057 0.4252 0.0013 0.0403 0.4199 0.0981 0.0009 0.0036 0.0015 0.0009 0

0 0.0002 0.0013 0.1057 0.0013 0.0266 0.4105 0.3667 0.071 0.0044 0.0067 0.0045 0.0012

0 0 0 0.1132 0 0 0.2568 0.5494 0.0807 0 0 0 0

Figure 1. Schematic diagram of two hypothetical components.

Table 3. Results of Density Prediction According to the Local Composition Based Models and the Group Contribution Method Wilson

a

NRTL

Wilson-NRF

GCVOL

biodiesel

Δλ12

Δλv12

AAD

Δg12

Δgv12

AAD

Δh12

Δhv12

AAD

ARDa

soybean 1 sunflower rapeseed palm GP overall

−5626 −5713 −6314 −1820 2016

−0.0214 −0.6934 −0.6974 −7.1128 −3.4478

0.0256 0.0185 0.0132 0.0034 0.0096 0.0141

−13499 −13710 −15594 −3385 3144

−0.0177 −0.5194 −0.4829 −6.9767 −3.3729

0.0256 0.0171 0.0122 0.0035 0.0097 0.0136

−60659 −61576 −67838 −27357 53956

−0.0132 −0.4897 −0.5162 −3.7111 −1.7868

0.0256 0.0203 0.0145 0.0034 0.0100 0.0148

0.2390 0.0390 0.0430 0.1730 0.0680 0.1124

expt ARD = 100/n Σi(Mcalc − Mexpt i i )/Mi .

a=

7.536 − 0.446 ln(nF) + 3.584

investigated by Pratas et al.25 are presented in Table 3. As can be seen, the local composition based models are far superior to the GCVOL in estimating the density of biodiesels. Also, the results show that the performance of the local composition based models is nearly the same. In between, the results of the NRTL model with overall AAD of 0.0136 are closer to the experimental data. Figure 2 shows the AD values for density of soybean 1 biodiesel based on the studied models at various temperatures. To verify the accuracy of the studied models, they were used to predict the density of soybean 2 biodiesel which has composition similar to soybean 1 biodiesel. It should be emphasized that parameters of the models are assigned from data of Table 3, and no experimental data of soybean 2 were used to correlate these parameters. The AAD for the Wilson, the NRTL, and the Wilson-NRF models are 0.0697, 0.0706, and 0.0687, respectively. The results indicate that the proposed models can predict the density of biodiesels with good accuracy. 4.2. Prediction of Viscosity. Yuan et al.15 proposed a model for the description of the viscosity−temperature

(25)

In the above equations, nF is the number of carbon atoms in the original fatty acid; nA is the number of carbon atoms in the original alcohol used for the transesterification process; db is the number of double bonds in the fatty acid molecule; and subscript FA stands for fatty acid alkyl ester. After predicting the density of each fatty acid methyl ester, the density of the two hypothetical components can be calculated using eq 22. Equation 2 is used for predicting the density of the biodiesels. The adjustable parameters for the density model according to the Wilson model are Δλ12, Δλ21, Δλv12, and Δλv21. For decreasing the number of adjustable parameters, it is assumed that Δλ12 = Δλ21 and Δλv12 = Δλv21 (symmetric condition of adjustable parameters). The same assumption is made for the density model according to the NRTL and the Wilson-NRF models. These parameters can be obtained by minimizing eq 18 as the objective function. Adjustable parameters and AADs for density correlation according to the local composition based models and also the results of the group contribution method (GCVOL) 13521



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Figure 3. AD values for correlating the viscosity of soybean 1 biodiesel at various temperatures.

Figure 2. AD values for correlating the density of soybean 1 biodiesel at various temperatures.

relationship of pure fatty acid methyl esters commonly present in biodiesel fuels A2 ln ηFA (mPa s) = A1 + T /K − T0 (26) In eq 26, A1, A2, and T0 are parameters of the model for which their values are reported by Yuan et al.15 Using the model, the viscosity of each fatty acid methyl ester can be estimated and based on them; the viscosity of the two hypothetical components can be calculated. Viscosities of biodiesels are predicted according to eq 11. Δλ12 and Δλ21 are adjustable parameters for the viscosity model according to the Wilson model; Δg12 and Δg21 correspond to the NRTL model; and Δh12 and Δh21 correspond to the Wilson-NRF model. These parameters are obtained by minimizing eq 18. Table 4 represents the results of viscosity estimation according to the Wilson, the NRTL, the Wilson-NRF models and the modified Yuan model investigated by Freitas et al.14 According to the results reported in Table 4, the NRTL model has the best performance in correlation of biodiesel viscosity. Also, the results of the Wilson and the Wilson-NRF models are better than the modified Yuan model. The AD values for viscosity correlation of soybean 1 biodiesel versus temperature are plotted in Figure 3. Figure 4 shows the experimental and correlated viscosity of biodiesels in a wide range of temperatures (273−373 K). The R2 value for the viscosity of the biodiesels is 0.9943. As can be inferred from the figure, a reasonable agreement between the results of the proposed models and the experimental data was attained.

Figure 4. Experimental vs correlated viscosity for five biodiesels.

Also, the performance of the presented models for predicting the viscosity of soybean 2 biodiesel is shown in Figure 5. As can be seen from this figure, there is good agreement between the experimental data and those obtained from the local composition based models. 4.3. Prediction of Surface Tension. Allen et al.17 represented a model for estimating the surface tension of fatty acid alkyl esters. This model can be written as below

Table 4. Results of Viscosity Prediction According to the Local Composition Based Models and the Modified Yuan Model Wilson

NRTL

Wilson-NRF

revised Yaun

biodiesel

Δλ12

Δλ21

AAD

Δg12

Δg21

AAD

Δh12

Δh21

AAD

ARD


681.32 566.52 2289.84 2886.32 369.20

114.80 92.56 390.51 519.06 319.08

0.493 0.325 1.036 0.813 0.523 0.638

−3243.4 −2352.4 −5132.2 −4910.9 −2529.5

4083.6 2473.3 7060.2 5693.0 2670.5

0.616 0.415 0.603 0.484 0.618 0.547

46.55 38.76 95.51 102.71 152.70

344.59 286.04 1183.93 1504.40 180.12

0.488 0.325 1.011 0.773 0.508 0.621

2.77 2.48 5.64 6.34 5.59 4.56

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Figure 6. AD values for correlating the surface tension of soybean 1 biodiesel at various temperatures.

Figure 5. Experimental and predicted viscosity for soybean 2 biodiesel versus temperature.

⎛ PchρFA ⎞4 σFA (N m−1) = ⎜ ⎟ ⎝ Mw ⎠

(27)

In the above equation, ρFA is the density of each fatty acid alkyl ester which can be calculated using eq 23; Mw is the molecular mass; and Pch is the parachor constant, for which the value for each fatty acid methyl ester is reported by Allen et al.17 Equation 14 is used for correlating the surface tension of the biodiesels. Adjustable parameters for the surface tension model according to the Wilson model are Δλ12, Δλ21, Δλσ12, and Δλσ21. For decreasing the number of adjustable parameters, it is assumed that Δλ12 = Δλ21 and Δλσ12 = Δλσ21. The same assumption is made for the surface tension model according to the NRTL and the Wilson-NRF models. These parameters are obtained by minimizing the eq 18. The results of estimated surface tension according to the local composition based models and the MacLeod−Sugden model16 are reported in Table 5. As can be seen, the performance of the local composition based models is nearly the same and far better than the MacLeod−Sugden model. Of course, the NRTL model is more accurate than the other models. The variations of AD vs temperature of soybean 1 biodiesel are plotted in Figure 6. Also, the local composition based models are used for predicting the surface tension of soybean 2 biodiesel, and the results are shown in Figure 7. The AAD values according to the Wilson, the NRTL, and the Wilson-NRF models are 2.452, 2.455, and 2.386, respectively. So the ability of these models to predict the surface tension is nearly the same. To show the sensitivity of the various objective functions on the adjustable parameters, eq 19 is used as the objective

Figure 7. Experimental and predicted surface tension for soybean 2 biodiesel versus temperature.

function, and density, viscosity, and surface tension of the biodiesels are estimated again. In Figure 8 the overall AAD values for correlating these physical properties using eqs 18 and 19 as the objective function are compared to each other. The results show that for density and surface tension eq 18 has better performance, while for viscosity eq 19 gets better results. However, overall AAD values are close to each other, and using eq 18 or 19 would not change the results very much. The results of density, viscosity, and surface tension estimation confirm the effectiveness of the local composition based models in predicting the physical properties of several biodiesels with different composition in a broad range of temperatures (273−373 K).

Table 5. Results of Surface Tension Prediction According to the Local Composition Based Models and the Macleod−Sugden Model Wilson

NRTL

Wilson-NRF

MacLeod−Sugden

biodiesel

Δλ12

Δλσ12

AAD

Δg12

Δgσ12

AAD

Δh12

Δhσ12

AAD

ARD


−5828.7 3712.8 16561.7 274.6 −3491.3

−5.6762 −9.3369 −471.572 −52.8951 −2.9996

0.078 0.411 0.659 0.469 0.392 0.402

−10868.4 5496.6 16373.9 445.0 −7389.5

−5.4079 −9.0848 −91.8366 −52.8475 −2.7101

0.074 0.411 0.621 0.469 0.387 0.392

−63311.1 65560.8 88964.5 6268.1 −37830.0

−3.9873 −3.7419 −7.3668 −26.5678 −1.8066

0.161 0.430 0.911 0.469 0.423 0.479

0.47 0.67 1.50 1.10 2.70 1.29

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Figure 8. Comparison between eqs 18 and 19 as the objective function for estimation of (a) density, (b) viscosity, and (c) surface tension of biodiesels.

5. CONCLUSION In this research, the excess Gibbs free energy models such as the Wilson, the NRTL, and the Wilson-NRF are applied to estimate the physical properties (density, viscosity, and surface tension) of biodiesels. It is assumed that the biodiesel is divided into two hypothetical components. Thus, there are interactions between two hypothetical components. The adjustable parameters of the excess Gibbs energy models can be obtained

from nonlinear regression between the experimental data, and the results obtained form the thermodynamic models. The results emphasized that the NRTL model can more accurately predict physical properties of biodiesels.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +98-021-66164104. E-mail: [email protected]. 13524



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Notes

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The authors declare no competing financial interest.

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NOMENCLATURE A = surface area (m2) a = parameter in eq 23 AAD = average absolute deviation AD = absolute deviation A1, A2, and T0 = parameters of Yaun’s model C = coordination number db = number of double bonds in fatty acid g = Gibbs free energy g0i = molar Gibbs free energy of component i Gji = parameter of the NRTL model Hji = parameter of the Wilson-NRF model M = physical property Mw = molecular weight n = number of experimental data nA = number of carbon atoms in alcohol nF = number of carbon atoms in fatty acid alkyl ester OF = objective function P = pressure (bar) Pch = parachor constant R = universal gas constant T = temperature (K) v = molar volume (L/mol) x = mole fraction

Greek Letters

ρ = density (kg/L) η = viscosity (mPa s) σ = surface tension (N m−1) ρ0 = parameter of eq 23 Λij = parameter of the Wilson model Δλij = interaction energy between the i−j pair in the Wilson model Δgji = interaction energy between the i−j pair in the NRTL model Δhji = interaction energy between the i−j pair in the WilsonNRF model τji = parameter of the NRTL model αji = nonrandomness parameter in the NRTL model Subscripts

i, j, and k = component FA = fatty acid methyl ester HyCo = hypothetical component Superscripts

calc = calculated ex = excess expt = experimental v = derivative with respect to molar volume σ = derivative respect to surface tension * = activation

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REFERENCES

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(26) Feitosa, F. X.; Rodrigues, M. D. L.; Veloso, C. B.; Cavalcante, C. L.; Albuquerque, M. C. G., Jr.; Sant’Ana, H. B. D. Viscosities and densities of binary mixtures of coconut + colza and coconut + soybean biodiesel at various temperatures. J. Chem. Eng. Data 2010, 55, 3909− 3914. (27) Blangino, E.; Riverós, A. F.; Romano, S. D. Numerical expressions for viscosity, surface tension and density of biodiesel: analysis and experimental validation. Phys. Chem. Liq.: .Int. J. 2008, 46, 527−547.

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Estimation of Biodiesel Physical Properties Using Local

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