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Estimation of Vapor Pressures and Enthalpies of Vaporization of Biodiesel-Related Fatty Acid Alkyl Esters. Part 1. Evaluation of Group Contribution and Corresponding States Methods Nathan S. Evangelista,† Frederico R. do Carmo,*,†,‡ and Hosiberto B. de Sant’Ana† †

Grupo de Pesquisa em Termofluidodinâmica Aplicada, Departamento de Engenharia Química, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza, CE Brazil ‡ Departamento de Ciências Exatas, Tecnológicas e Humanas, Universidade Federal Rural do Semi-Á rido, Campus de Angicos, 59515-000 Angicos, RN Brazil S Supporting Information *

ABSTRACT: Several models based on the Group Contribution concept and on the Corresponding States Principle were reviewed for the estimation of vapor pressures and enthalpies of vaporization of fatty acid methyl and ethyl esters commonly occurring in biodiesel. The accuracy of each model was tested by comparing its output values to experimental data obtained from the literature. For vapor pressures, the number of occurrences where the methods failed completely have also been analyzed. Efforts were made to identify the best method for the estimation of each property and it was observed that the models’ reliability depend on the type of ester and on the operational conditions (pressure and temperature) considered.

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INTRODUCTION

The composition of biodiesel is mainly affected by the feedstock used in its production. Different vegetable oils may generate biodiesels with very scattering compositions, and thus, with diverging physical properties.7 Hence, the experimental determination of vapor pressures and enthalpies of vaporization for all the existing or conceiving biodiesel or biodiesel/biodiesel blends is impractical. The solution is to rely on thermodynamic models for the calculation of these properties within a desirable accuracy. Estimated values of mixtures of physical properties usually stem from the application of mixing rules, which consider the behavior of the pure constituents at the desired conditions. Therefore, accurate values of vapor pressures and enthalpies of vaporization of the various fatty acid alkyl esters (FAAES) comprising a biodiesel are required to obtain reliable values of these properties for the overall fuel. For this reason, the selection of an appropriate estimation model is extremely important for industrial applications. Most of the existing models capable of estimating vapor pressures and enthalpies of vaporization of FAAES can be divided into three main categories: a. correlations8 b. Group Contribution techniques9−23 c. methods based on the Corresponding States Principle24−38

Depleting of world’s oil resources and environmental issues have prompted the search of new sources of energy. For this reason, alternative fuels, such as biodiesel, have been of considerable interest to both the scientific and industrial communities. Biodiesel is defined as a mixture of long-chain fatty acid alkyl esters usually obtained from the transesterification reaction of a vegetable oil or animal fat with a short-chain alcohol.1 Its characteristics, such as low volatility, renewability, and biodegradability have encouraged the application of this fuel, either alone or blended with petrodiesel, in diesel engines.2 Moreover, as biodiesel feedstock can be derived from agriculture surplus, it can offer social inclusion and economic development, especially in poor rural areas of emerging countries.3 The knowledge of the physical properties of biodiesel is essential to the development of new processes and to the optimization of existing processes involving these fuels.4 The liquid vapor pressure has been pointed out as one of the most important quantities for many tasks in process simulation. Additionally, accurate values of this property are required for the determination of the number of theoretical stages in distillation columns and the calculation of temperature profiles. Furthermore, a carefully determined vapor pressure−temperature curve is essential to the assessment of other relevant properties of biodiesel, such as the cold weather properties and the heat of vaporization.5,6 This property influences the rates of vaporization and injection characteristics of a fuel and, therefore, is crucial to the modeling of combustion processes.6 © 2017 American Chemical Society

Received: Revised: Accepted: Published: 2298

December 9, 2016 January 25, 2017 February 6, 2017 February 6, 2017 DOI: 10.1021/acs.iecr.6b04772 Ind. Eng. Chem. Res. 2017, 56, 2298−2309

Article

Industrial & Engineering Chemistry Research Table 1. Selected Models for Vapor Pressure Estimation required compound informationa method 19

Pankow/Asher Ceriani et al.18 Wang et al.20 Wang et al.20 Nannoolal et al.21 M?ller et al.22 Tu23 Reid et al.44 Lee/Kesler29 Ambrose/Walton38 Edalat et al.37 Riedel30,40 Vetere31 Gomez/Thodos32,33 Frost et al.34,44 Riedel et al.35,36,44

b

acronym

conceptual basis

molecular structure

SIMPOL CGL WMJS1c WMJS2d NRR MRR T RPS LK AW EJM R V GT FKT RPM

GC GC GC GC GC GC GC CSP CSP CSP CSP CSP CSP CSP CSP CSP

X X X X X X X

MW

Tb

X X X

Tc

Pc

ω

X X X X

X X X

X

X X X X X

X X X X X X X X X

X X X X X X X X X

X X

Notation: MW, molecular weight; Tb, normal boiling temperature; Tc, critical temperature; Pc, critical pressure; ω, acentric factor. bGC, group contribution; CSP, Corresponding States Principle. cOnly experimental Tc values were taken as input. dOnly estimated Tc values were taken as input.

a

Despite the highest accuracy of the methods in the first category, their applicability is limited by the lack of correlation coefficients for all FAAES. Yuan et al.8 have reported a compilation of Antoine equation parameters for 14 fatty acid methyl esters (FAMES). These coefficients were regressed on the basis of experimental data and on estimated data from the Ceriani and Meirelles’39 model for some compounds. The proposed correlations were used to estimate the normal boiling point of six saturated FAMES. The reported results showed that the maximum relative deviation between calculated and experimental data was about 1%. The Group Contribution (GC) approach considers that the properties of molecules are established from the contributions of its functional groups. The conceptual basis is that the intermolecular forces, which strongly impacts on the behavior of substances, depend mostly on the nature of the atoms and on the types of chemical bonds within the molecules.40 The application of these methods is usually broad and simple, because only structural formula and, in some cases, basic properties of a compound are needed.4 Another versatile approach successfully applied over the years is the Corresponding States Principle (CSP). As determined from a small amount of experimental information (e.g., critical properties, acentric factors), methods based on the CSP can reproduce reliable estimates of equilibrium properties, such as vapor pressures and enthalpies of vaporization, of a wide variety of chemical substances.41 These methodologies are predictive; that is, the specific correlated parameters of the compounds are not required. As aforementioned, most of the available methods fall into these two categories and, indeed, some of them combine features of both techniques. An et al.42 estimated the vapor pressures and enthalpies of vaporization of methyl oleate using the corresponding states methods. The predicted results were compared to pseudoexperimental data generated by correlations taken from a data compilation.43 The authors reported that the methods of Lee and Kesler29 and Pitzer24,44 were more suitable than the methods of Ambrose and Walton38 and Fish and Lielmezs45 for vapor pressure and enthalpy of vaporization estimations, respectively. Although they suggest the application of the first

two methods for the modeling of biodiesel combustion, the methods were not sufficiently tested because the authors used a databank containing only one FAME. A more comprehensive study was published by Wallek et al.46 These authors applied seven different models for the estimation of FAMES vapor pressures: three of them were based on the GC concept18,21,22 and the others were based on the CSP.29,30,32,33,38 These methods were evaluated by comparing the estimated data with 478 experimental vapor pressure points of nine different FAMES. By dividing the databank into different vapor pressure intervals, the authors showed that the accuracy of the models varied drastically in these regions. It was observed that most of the methods had output better results at the highest pressure region (>100 kPa). In this interval, the model proposed by Moller et al.22 showed superior results when compared to the others. However, the model presented by Ceriani et al.18 was the most accurate at low pressures ( 10% Ndata

(4)

Figure 1. Percentage of estimated data by each model (Pvap).

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where %UE denotes the percentage of unreliable estimates, N%ARD>10% is the number of absolute relative deviations higher than 10% and Ndata is the quantity of data estimated by each model.

RESULTS AND DISCUSSION Vapor Pressure. Figure 1 illustrates an analysis of the overall data that could be predicted by each method. As it can be seen, most of the models could not be applied to the whole

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DOI: 10.1021/acs.iecr.6b04772 Ind. Eng. Chem. Res. 2017, 56, 2298−2309

Article

Industrial & Engineering Chemistry Research Table 4. %AARD for Vapor Pressure Estimation Using the GC Models methoda

a

compound

SIMPOL (%)

CGL (%)

WMJS1 (%)

WMJS2 (%)

NRR (%)

MRR (%)

ME-C6:0 ME-C7:0 ME-C8:0 ME-C9:0 ME-C10:0 ME-C11:0 ME-C12:0 ME-C13:0 ME-C14:0 ME-C15:0 ME-C16:0 ME-C17:0 ME-C18:0 ME-C18:1 ME-C18:1,OH ME-C18:2 ME-C18:3 ME-C19:0 ME-C20:0 ME-C22:0 ME-C22:1 ME-C24:0 EE-C6:0 EE-C8:0 EE-C10:0 EE-C12:0 EE-C14:0 EE-C16:0 EE-C18:0 EE-C18:1 EE-C18:2 FAMES FAEES Overall

19.45 23.45 43.46 77.61 53.69 87.02 73.99 141.95 85.09 175.24 125.65 232.42 122.61 84.84 99.09 85.95 83.16 152.16 117.66 99.84 99.91 59.87 19.54 21.29 46.58 46.87 59.44 92.92 148.35 99.22 99.48 82.99 58.93 78.02

3.20 2.84 3.29 2.65 5.27 11.81 9.88 22.99 12.61 32.03 20.27 42.24 21.97 15.22 41.07 8.42 20.83 44.84 38.63 19.28 8.77 22.99 12.14 13.41 17.20 13.96 18.89 15.66 15.37 10.19 11.68 13.70 14.84 13.93

27.34 15.20 7.13 8.95 12.62 3.67 4.45

3.21 7.24 4.10 3.49 4.16 3.74 4.46 2.92 3.79 2.54 6.92 10.31 6.55 7.10

2.07 5.83 3.78 4.01 23.01 4.08 15.80

2.03 4.83 3.11 4.58 21.66 15.66 12.58

6.74

3.91

36.26

45.07

14.13 6.49

8.88 5.49

10.39 13.67 9.99 11.70 20.39 9.66 51.58 7.37 5.40 5.73 4.35 3.74 5.88 26.81 10.14 12.79 5.80 7.62 6.18

9.22 13.05

7.22 6.79

42.99 12.28 10.68

19.54 7.24 6.11

5.58 7.57 17.57 14.70 5.28

4.44 3.83 7.71 12.06 7.24

14.10 9.11 13.25

13.26 6.73 12.14

21.52 7.29 9.66 26.67 39.54 74.15

5.90 20.64 15.35 29.92 15.14 27.23 16.50

14.68 21.31 16.01

T (%) 45.46 52.78 33.15 18.12 10.26 47.35 49.40 81.10 69.33 92.32 82.54 97.03 85.37 82.75 81.67 88.60 93.78 84.42 23.15 2.91 2441.46 16.94 9.84 48.30 66.96 81.10 84.16 81.82 73.92 65.72 80.20 54.76 74.86

Empty fields indicate that the model could not be applied.

pattern was very similar to the FAMES’. Except for the CGL, WMJS1, and WMJS2, all the models performed better for FAEES than for FAMES. This result is probably related to the database used in the development of these methods. Although Ceriani et al.18 did not present detailed information about the quantity of FAMES and FAEES vapor pressure data used to regress their model’s parameters, Wang et al.20 employed a total of 1199 experimental points (935 for FAMES and 264 for FAEES). Surprisingly, the AW model was the most accurate among the corresponding states methods. Although the set of equations of this model was fit to vapor pressure data of nalkanes, its results for esters were of comparable and, for several compounds, of higher accuracy than the results generated by the GC methods. Like the GC models, most of the CSP methods performed worse for FAMES than for FAEES. For the former esters, the ranking of accuracy was: AW (15.49%) > V (27.65%) > EJM (29.32%) > RPM (33.44%) > LK (46.53%) > FKT (52.43%) > R (54.79%) > GT (109.36%) > RPS (391.16%). For the latter: AW (6.96%) > V (16.85%) > RPM (16.92%) > LK (21.64%) > FKT (24.85%) > R (26.12%) > EJM (38.26%) > GT (64.57%) > RPS (186.30%). The overall ranking was similar to that of the FAMES.

database because of the lack of experimental values of some needed properties of the esters (Tb, Tc and Pc). We did not apply the models of Tu23 and Wang et al.20 to estimate the data of methyl ricinoleate, because the former is not recommended for calculations of multifunctional compounds and the latter cannot represent the structure of hydroxyls with the available functional groups. Although the WMJS1 and WMJS2 models refer to the same equation, Figure 1 shows that the latter covered more data than the former. The reason is related to the values of Tc applied in these models: in the former, experimental data have been considered; in the latter, only estimated values were accepted, as suggested in the original work.20 The performances of the studied models are presented in Tables 4 and 5. With a few exceptions, the GC methods showed higher accuracy in comparison to the CSP ones. Considering the FAMES’ database, the following decreasing order of accuracy was obtained by the GC models: WMJS2 (5.80%) > MRR (13.26%) > CGL (13.70%) > NRR (14.10%) > WMJS1 (14.68%) > T (80.20%) > SIMPOL (82.99%). For FAEES, a similar order was obtained: MRR (6.73%) > WMJS2 (7.62%) > NRR (9.11%) > CGL (14.84%) > WMJS1 (21.31%) > T (54.76%) > SIMPOL (58.93%). The overall accuracy 2302


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Industrial & Engineering Chemistry Research Table 5. %AARD for Vapor Pressure Estimation Using the CSP Models methoda

a

compound

RPS (%)

LK (%)

AW (%)

EJM (%)

R (%)

V (%)

GT (%)

FKT (%)

RPM (%)

ME-C6:0 ME-C7:0 ME-C8:0 ME-C9:0 ME-C10:0 ME-C11:0 ME-C12:0 ME-C13:0 ME-C14:0 ME-C15:0 ME-C16:0 ME-C17:0 ME-C18:0 ME-C18:1 ME-C18:1,OH ME-C18:2 ME-C18:3 ME-C19:0 ME-C20:0 ME-C22:0 ME-C22:1 ME-C24:0 EE-C6:0 EE-C8:0 EE-C10:0 EE-C12:0 EE-C14:0 EE-C16:0 EE-C18:0 EE-C18:1 EE-C18:2 FAMES FAEES Overall

83.75 139.40 71.74 173.91 350.92 804.14 515.26

18.21 5.99 7.38 16.28 51.23 10.56 94.05

25.77 15.53 8.48 14.36 7.56 8.18 10.33

29.32 27.21 22.24 115.13 44.18 53.95 11.96

21.20 11.71 10.57 23.63 60.69 28.05 108.05

14.40 5.53 3.85 1.74 28.04 30.63 53.26

34.60 42.87 28.37 63.21 114.27 147.14 180.84

22.31 11.40 10.97 22.81 58.38 18.26 105.44

21.94 6.36 7.13 10.28 40.33 21.99 69.62

354.88

42.42

6.90

16.59

53.25

10.01

119.19

48.78

12.82

360.04

43.99

9.01

21.55

37.09

62.16

38.93

42.90

66.40

1389.37 138.93

113.06 15.60

8.15 39.36

18.82 43.64

142.39 23.58

28.87 15.44

332.58 66.57

125.86 18.54

22.68 19.83

174.93 157.16

19.79 24.72

60.30 164.96

67.66 184.62

28.23 15.82

14.77 50.82

77.89 41.53

22.51 24.65

19.03 59.00

84.79

31.64

21.93

22.21

36.71

10.04

58.17

34.96

3.78

147.08 130.03 315.32 227.98 155.85

6.71 4.80 28.40 74.47 12.76

6.95 5.32 5.35 15.58 3.80

17.91 185.31 18.59 11.61 6.41

3.64 9.43 38.70 82.38 20.20

14.93 8.05 4.47 47.36 13.91

31.04 38.54 100.12 120.46 62.27

3.99 8.49 35.22 83.17 17.05

7.32 3.56 10.77 59.39 14.40

391.16 186.30 355.49

46.53 21.64 42.19

15.49 6.96 14.00

29.32 38.26 30.87

54.79 26.12 49.80

27.65 16.85 25.77

109.36 64.57 101.56

52.43 24.85 47.63

33.44 16.92 30.56

Empty fields denote that the model could not be applied.

Table 6. Performances of All Models in Different Vapor Pressure Regions pressure intervals [kPa] I −6

II −4

−4

III −2

IV

overall

method

[1.76 × 10 ; 1.00 × 10 ] (%)

[1.00 × 10 ; 1.00 × 10 ] (%)

[1.00 × 10 ; 1.00] (%)

[1.00; 175] (%)

[1.76 × 10−6; 175] (%)

SIMPOL CGL WMJS1 WMJS2 NRR MRR T RPS LK AW EJM R V GT FKT RPM

183.40 36.81 11.78 14.89 33.65 37.23 90.05 1117.25 109.60 13.07 21.83 126.53 62.66 260.20 118.21 61.58

154.56 27.23 16.33 8.70 24.96 22.79 143.90 1289.88 84.11 8.12 39.55 101.47 42.12 255.67 93.57 45.38

56.74 12.58 17.48 5.25 11.57 9.97 77.06 311.80 40.07 18.57 35.53 48.02 25.00 97.98 45.77 30.91

61.35 8.75 14.97 5.35 10.64 9.93 48.93 139.71 30.82 11.59 25.57 35.43 20.71 61.16 34.97 25.24

78.02 13.93 16.01 6.18 13.25 12.14 74.86 355.49 42.19 14.00 30.87 49.80 25.77 101.56 47.63 30.56

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Industrial & Engineering Chemistry Research

Figure 2. Unreliable estimates per model (Pvap).

Figure 5. Percentage of estimated data by each model (ΔHv).

pressure increased. In the low-pressure regions (I and II), the WMJS2 and AW methods showed significantly better results in comparison to the others. In the intermediate regions (III and IV), the best results were obtained by the WMJS2 and MRR models. Overall, the most accurate models can be synthesized as follows: WMJS2 (6.18%) > MRR (12.14%) > NRR (13.25%) > CGL (13.93%) > AW (14.00%). Figure 2 illustrates the percentages of estimation failures for each method. Excepting the CGL, WMJS1, and WMJS2, the number of flawed results for FAMES was higher than those for FAEES for all the methods. Most of the models generated a high quantity of unreliable estimates for FAMES’ vapor pressures. The WMJS2 was unquestionably the most reliable in this case (%UE = 16%). The MRR, CGL, and NRR models, which were among the most accurate methods for FAMES, generated numerous flawed results: 44%, 45%, 47% of the estimated data, respectively. For FAEES, the WMJS2, MRR and AW models were the most reliable: they failed in 19%, 23%, and 23% of the estimates, respectively. In Figure 3 we plotted a simple distribution of relative deviations considering the overall database. This chart illustrates the frequencies of data points that have been under or overestimated by each method. The distribution is strongly shifted to the positive deviations for the CGL, T, and AW models, which suggests that an application of these methods would probably underestimate the value of a FAAE’s vapor pressure. The distributions of the EJM, SIMPOL, NRR, WMJS2, and MRR methods are balanced between positive and negative values. Finally, one should probably expect overestimated values of vapor pressures from all the other methods. We did not observe any pattern in these distributions regarding to the vapor pressure intervals presented in Table 6. To illustrate the best and worst level of obtained predictions, experimental and calculated vapor pressure−temperature curves are presented in Figure 4 for methyl oleate, a common FAME that occurs in different biodiesel. Given the previous results, we concluded that, before estimating the vapor pressure of a compound, one should first check the pressure region of interest, since the uncertainty of the models varied significantly in these intervals. Considering the overall results, we recommend the method of Wang et al.20 (WMJS2), which was the most accurate and reliable among the ones analyzed. A comparison between the WMJS1 and WMJS2 models showed that the latter generated clearly superior results. Therefore, regardless of the availability of an esters’ critical temperature, one should always use estimated values of this

Figure 3. Distribution of relative deviations per model (Pvap).

Figure 4. Experimental and calculated Pvap vs T curve for methyl oleate.

Following the methodology of Wallek et al.,46 we divided the databank into four pressure regions. At this point, we decided not to distinguish the FAMES and FAEES database due to the similarity of the methods’ accuracy ranking for both types of esters. The %AARD being output by the models in different vapor pressure regions are presented in Table 6. From these results, it could be observed that the accuracy of the models varied drastically in the chosen intervals. As it was expected, most of the models showed significantly better results as the 2304


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Industrial & Engineering Chemistry Research Table 7. %AARD for Enthalpy of Vaporization Estimation Using the GC Models methodsa

a

compound

CG (%)

MG (%)

KRG (%)

BS (%)

TL (%)

BRSAF (%)

DSG (%)

UNIVAP (%)

CGL (%)

WMJS2 (%)

ME-C6:0 ME-C7:0 ME-C8:0 ME-C9:0 ME-C10:0 ME-C11:0 ME-C12:0 ME-C13:0 ME-C14:0 ME-C15:0 ME-C16:0 ME-C17:0 ME-C18:0 ME-C18:1 ME-C18:2 ME-C18:3 ME-C19:0 ME-C20:0 ME-C21:0 ME-C22:0 ME-C22:1 ME-C24:0 ME-C25:0 ME-C26:0 EE-C6:0 EE-C7:0 EE-C8:0 EE-C9:0 EE-C10:0 EE-C11:0 EE-C12:0 EE-C14:0 EE-C16:0 EE-C18:0 EE-C18:1 EE-C18:2 FAMES (T = 298.15 K) FAEES (T = 298.15 K) FAMES (all data) FAEES (all data) overall (298.15 K) overall (all data)

3.01 2.13 1.40 1.83 1.69 1.51 2.43 2.32 2.68 2.86 3.13 1.89 3.30 2.89 4.16 6.32 2.67 3.61 3.62 3.91 4.00 4.49 4.85 4.98 1.88 0.76 2.06 2.79 2.91 4.37 0.03 1.64

0.76 1.15 1.68 1.42 1.72 2.04 1.22 1.62 1.15 1.74 0.84 3.16 0.80 3.14 3.76 3.44 1.51 0.58 0.62 0.37 1.86 0.16 0.50 0.61 2.95 3.90 3.30 6.35 4.62 8.26 3.87 2.32

1.54 2.19 2.72 2.08 2.05 2.09 1.02 1.41 0.65 1.38 0.51 2.07 0.39 1.22 1.12 0.10 0.58 0.83 0.89 1.23 0.34 1.91 2.31 2.49 3.73 4.97 3.97 6.70 4.67 8.05 3.44 1.53

1.32 0.46 1.20 0.25 0.82 1.06 2.54

6.55 6.00 7.64 7.09 6.77 7.11 6.12

0.31 1.44 0.70 1.72 2.86 3.14 4.66

2.73 2.34 3.95 3.36 2.96 3.14 2.01

2.45 2.46 0.82 1.25 1.07 0.81 1.62

4.16

5.61

6.42

1.13

5.29

5.50

7.51

0.57

6.21 4.54 4.09 4.55

5.58 8.82 10.67 11.55

8.42 12.80 17.97 23.56

0.19

7.81 7.56 5.84 6.11 5.82 5.49 6.20 5.95 6.14 6.18 6.30 4.08 6.21

2.75 2.08 2.75 2.08 2.64 2.64

1.37 4.03 1.37 4.03 1.83 1.83

1.40 4.38 1.40 4.38 1.92 1.92

2.81 2.33 3.15 1.86 1.21 0.65 1.07 1.50 2.44 3.16 3.92 2.24 4.95 5.62 14.92 23.46 4.71 5.98 6.31 6.88 2.34 7.95 8.52 8.85 2.00 0.57 1.54 0.40 2.47 0.37 3.56 7.12 15.95 22.07 11.32 17.98 3.82 2.51 3.28 4.59 3.59 3.57

1.58 1.68 1.44 0.16 0.59 1.92

5.49 6.29 6.21 6.39 7.36

7.51

15.80

0.51

2.80 2.67 1.19 3.00 1.53 3.78 1.74 6.46 18.37

7.36 9.29 8.53 10.98 9.11 13.07 12.60 7.00 4.71

2.75 1.97 1.78 2.35 2.23 3.38 1.29 6.77 17.90

2.77 3.42 3.09 5.23 3.63 7.10 6.75 4.95 10.38

2.70 2.72 1.80 3.25 2.71 2.14

6.79 8.80 6.50 8.77 7.27 7.04

5.08 2.49 2.84 3.36 4.46 2.96

2.34 3.33 2.40 4.28 2.60 2.87

6.78 7.05 7.11 2.46 0.93 2.58 2.57 2.97 3.97 3.71 8.83 25.90 44.65

6.19 2.07 6.89 5.85 5.44 6.67

1.93 1.35 2.45 2.71 5.23 6.01 4.23 9.81

1.40 2.09 2.00 3.36 1.56 2.31


⎡ 6DT7 ⎤ ⎥ ΔH v = R ⎢BTc + CT + Tc 6 ⎦ ⎣

property as suggested in the original paper.20 If the WMJS2 method cannot be applied, we suggest the application of the MRR model for FAMES and FAEES. If none of these can be used, one should rely on a model whose application only require the structure of a compound. Among the ones considered in this work (CGL, SIMPOL and T), the model proposed by Ceriani et al.18 is outstandingly more reliable than the two others are, which encourages its application. Enthalpy of Vaporization. As previously mentioned, the WMJS2 model was unquestionably the best method among the ones analyzed for vapor pressure estimations. Starting with this method’s original equation, we applied the ideal gas approximation in the Clausius−Clapeyron equation in order to derive the WMJS2 enthalpy of vaporization model presented as follows:

(5)

where ΔHv denote enthalpy of vaporization. R is the ideal gas constant. B, C, and D are parameters determined for each compound by group contributions. T and Tc denote experimental and critical temperature, respectively. Contrarily, the CGL equation was taken from the original work published by Ceriani et al.18 In this case, the authors approximated the change in molar volume using another methodology. Similar to what was observed for vapor pressure estimates, most of the enthalpy of vaporization models could not be applied to the entire database. Figure 5 illustrates the percentages of the overall data predicted by each method. 2305


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Industrial & Engineering Chemistry Research Table 8. %AARD for Enthalpy of Vaporization Estimation Using the CSP Models Methodsa compound ME-C6:0 ME-C7:0 ME-C8:0 ME-C9:0 ME-C10:0 ME-C11:0 ME-C12:0 ME-C13:0 ME-C14:0 ME-C15:0 ME-C16:0 ME-C17:0 ME-C18:0 ME-C18:1 ME-C18:2 ME-C18:3 ME-C19:0 ME-C20:0 ME-C21:0 ME-C22:0 ME-C22:1 ME-C24:0 ME-C25:0 ME-C26:0 EE-C6:0 EE-C7:0 EE-C8:0 EE-C9:0 EE-C10:0 EE-C11:0 EE-C12:0 EE-C14:0 EE-C16:0 EE-C18:0 EE-C18:1 EE-C18:2 FAMES (T = 298.15 K) FAEES (T = 298.15 K) FAMES (all data) FAEES (all data) Overall (298.15 K) Overall (all data) a

CK (%)

SMK (%)

MK (%)

M (%)

CM1 (%)

CM2 (%)

0.67 3.37 2.94 4.39 6.70 6.93 8.80

1.39 1.57 0.76 1.95 4.16 4.15 5.86

4.52 2.60 4.12 3.73 2.11 3.10 2.11

2.28 0.83 0.63 0.99 3.22 3.28 5.09

2.32 0.90 0.57 1.26 3.55 3.68 5.53

2.37 0.79 0.65 0.91 3.12 3.13 4.87

11.90

8.70

0.60

8.10

8.60

7.77

13.28

9.69

1.03

9.37

9.91

8.85

15.14 16.13 17.38 19.42

11.26 12.48 13.91 16.17

0.67 1.44 3.53 6.54

11.20 12.31 13.70 15.92

11.75 12.84 14.21 16.41

10.50 11.71 13.15 15.43

21.17

17.16

4.54

17.40

17.94

16.55

3.72 3.14 4.15 4.54 5.66 3.88 3.81 11.98

2.45 0.95 2.21 2.01 3.45 0.85 2.55 9.94

2.96 3.58 1.83 4.12 2.06 7.31 3.91 4.92

1.89 0.08 1.34 0.98 2.10 0.02 3.87 7.46

1.95 0.09 1.29 1.31 2.45 0.44 3.08 8.63

1.99 0.11 1.34 0.93 2.42 0.21 2.75 8.38

8.31

5.90

2.68

5.55

5.86

5.32

5.24

2.89

3.51

2.36

2.50

2.33

4.95 5.20 7.57

3.65 3.38 5.18

3.21 3.19 2.88

3.62 2.57 4.78

3.80 2.72 5.05

3.52 2.64 4.60

5.01

3.59

3.21

3.37

3.55

3.32

Figure 6. Distribution of relative deviations per model (ΔHv).

“ME-C18:1” and “ME-C22:1” generated math errors in the calculations at the temperature of interest (298.15 K). As an attempt to overcome this problem, the calculation routines, the functional groups used to draw these compounds as well as the groups’ parameters were carefully checked and no problem was found. Other reasons for the nonapplication of the models include the unavailability of experimental values of some needed esters’ properties (Tb, Tc, and Pc). Tables 7 and 8 contain the %AARD values that were output by the group contribution and the corresponding states models, respectively. Given the well-known importance of obtaining accurate values of enthalpy of vaporization at 298.15 K,13 the accuracies of all models at this temperature have been first analyzed and the following decreasing ranking of accuracy was obtained for FAMES: MG (1.37%) > CGL = KRG (1.40%) > DSG (2.34%) > MK (2.68%) > BS (2.70%) > CG (2.75%) > WMJS2 (3.82%)> BRSAF (5.08%) > CM2 (5.32%) > M (5.55%) > CM1 (5.86%) > SMK (5.90%) > UNIVAP (6.19%) > TL (6.79%) > CK (8.31%). Particularly, the MK model, which was originally developed focusing on long-chain hydrocarbons, produced more accurate results than most of the methods, including the group contribution ones. For FAEES, a very different order was obtained: UNIVAP (2.07%) > CG (2.08%) > CGL (2.09%) > CM2 (2.33%) > M (2.36%) > BRSAF (2.49%) > CM1 (2.50%) > WMJS2 (2.51%) > BS (2.72%) > SMK (2.89%) > DSG (3.33%) > MK (3.51%) > MG (4.03%) > KRG (4.38%) > CK (5.24%) > TL (8.80%). Considering the data at all temperatures and excluding the models applicable only at 298.15 K (CG, MG, and KRG), the following order of accuracy was obtained for FAMES: BS (1.80%) > CGL (2.00%) > DSG (2.40%) > BRSAF (2.84%) > MK (3.21%) > WMJS2 (3.28%) > CM2 (3.52%) > M (3.62%) > SMK (3.65%) > CM1 (3.80%) > CK (4.95%) > TL (6.50%) > UNIVAP (6.89%). These results show that the BS method was very superior at other temperatures, because its performance at 298.15 K was not among the best. For FAEES, the accuracy ranking was M (2.57%) > CM2 (2.64%) > CM1 (2.72%) > MK (3.19%) > BS (3.25%) > BRSAF = CGL (3.36%) > SMK (3.38%) > DSG (4.28%) > WMJS2 (4.59%) > CK (5.20%) > UNIVAP (5.85%) > TL (8.77%). Notably, the corresponding states methods outperformed the group contribution ones at this situation. From the above analysis, we concluded that none of the methods showed superior results in all the analyzed circumstances (FAMES at 298.15 K, FAEES at 298.15 K, FAMES at all temperatures, and FAEES at all temperatures). Therefore, one should always check the type of ester and the temperature


Despite the simplicity in the application of the CG, MG, and KRG models, they were developed for estimates at 298.15 K only, which explains why these were the least applied. The UNIVAP model could not be applied to unsaturated esters, because the interaction parameters between the functional groups “CH” and “-CH2(COO)CH3” groups are not available. The DSG method was not used to estimate the data of “ME-C18:2” and “ME-C18:3”, because none of the available functional groups can represent the methylene (“CH2”) occurring between the double bonds in the fatty acid chain. We noticed that the application of this model to 2306


Industrial & Engineering Chemistry Research

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ACKNOWLEDGMENTS The authors acknowledge the financial support provided by ́ CAPES (Coordenaçaõ de Aperfeiçoamento de Pessoal de Nivel Superior, Brazil).

at which the enthalpy of vaporization value is desired. Despite that, it is worth analyzing the results obtained considering two more scenarios: (1) data of FAMES and FAEES at 298.15 K; (2) data of FAMES and FAEES at all temperatures. For the former, the best results were generated by the CGL (1.56%), MG (1.83%), and KRG (1.92%) models. For the latter, by the BS (2.14%), CGL (2.31%), and DSG (2.87%) models. Given these overall results, we concluded that the group contribution methods outperformed the corresponding states ones. The distribution of relative deviations obtained by the application of all models to their estimated database is presented in Figure 6. The distribution is strongly shifted to the positive deviations for the models in the left side: CK, UNIVAP, CG, CGL, BRSAF, SMK, CM1, CM2, and M. It suggests that when these models are applied, it is more probable to obtain underestimated values of FAAE’s enthalpies of vaporization. Very even distributions were obtained for the BS and WMJS2 models, which indicates a randomness in the relative deviations. Finally, the application of the models in the right side will very likely generate overestimated values of enthalpy of vaporization.

■

CSP = Corresponding States Principle FAAE = fatty acid alkyl ester FAME = fatty acid methyl ester FAEE = fatty acid ethyl ester GC = group contribution MW = molecular weight Ndata = number of estimated data Pc = critical pressure Pvap = vapor pressure Tb = normal boiling temperature Tc = critical temperature Greek Letters

ΔHv = enthalpy of vaporization ω = Pitzer acentric factor

CONCLUSIONS In this work, several models have been evaluated for the estimation of the vapor pressures and enthalpies of vaporization for FAMES and FAEES. Data estimated by various methods were compared to experimental points collected in an extensive literature survey. Given the difficulty in estimating vapor pressures, we have also analyzed the number of flawed percentages produced by each model. For this property, the WMJS2, MRR, and NRR methods output the best overall results. Care should be taken when applying the CGL model for FAEES, because of the high quantity of unreliable estimates that it produced. Considering the enthalpy of vaporization data at 298.15 K of the FAMES and FAEES, the best results were generated by the CGL, MG, and KRG methods. For the whole database, the BS, CGL, and KRG models were the most accurate. In spite of these overall conclusions, one should always check the type of ester and the operational conditions (pressure region and temperature) at which values of vapor pressures and enthalpies of vaporization are desired.

Superscripts

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calc = calculated exp = experimental

REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b04772. Detailed information about the developed databank, including literature references, the number of data points and data sets for each compound (PDF)

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ABBREVIATIONS

Roman Letters

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +55-85999998649. Fax: + 55-84-33178503. ORCID

Frederico R. do Carmo: 0000-0003-0666-2956 Notes

The authors declare no competing financial interest. All calculations were performed by using the OCTOPUS v1.0 computational tool. It is available for free at https://github. com/thegibbsproject/octopus. 2307


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