Estimation of Vapor Pressures and Enthalpies of Vaporization of

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Estimation of Vapor Pressures and Enthalpies of Vaporization of Biodiesel-Related Fatty Acid Alkyl Esters. Part 2. New Parameters for Classic Vapor Pressure Correlations Nathan Sombra Evangelista, Frederico Ribeiro do Carmo, and Hosiberto Batista de Sant Ana Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01539 • Publication Date (Web): 29 Jun 2017 Downloaded from http://pubs.acs.org on July 1, 2017

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Industrial & Engineering Chemistry Research

Estimation of Vapor Pressures and Enthalpies of Vaporization of Biodiesel-Related Fatty Acid Alkyl Esters. Part 2. New Parameters for Classic Vapor Pressure Correlations Nathan S. Evangelista1, Frederico R. do Carmo1,2,*, Hosiberto B. de Sant’Ana1 1

Grupo de Pesquisa em Termofluidodinâmica Aplicada, Departamento de Engenharia Química, Universidade Federal do Ceará, Campus do Pici. 60455-760 Fortaleza – CE, Brazil. 2

Departamento de Ciências Exatas, Tecnológicas e Humanas, Universidade Federal Rural do Semi-Árido, Campus de Angicos. 59515-000 Angicos – RN, Brazil.

*Corresponding author: Frederico R. do Carmo, Departamento de Ciências Exatas, Tecnológicas e Humanas, Universidade Federal Rural do Semi-Árido, Rua Gamaliel Martins Bezerra, 587, Alto da Alegria. 59515-000 Angicos – RN, Brazil; e-mail: [email protected]; Tel: +55-85-999998649; Fax: + 55-84-33178503

ACS Paragon Plus Environment

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ABSTRACT

In Part I of this series of articles, predictive models capable of estimating vapor pressures and enthalpies of vaporization of biodiesel-related esters have been reviewed. In this study, we propose component-specific parameters for classic vapor pressure correlations. First, the previously collected database was evaluated and used in the regression of Antoine coefficients for all compounds. Then, a constrained regression methodology was employed to generate extrapolatable Wagner parameters for the esters whose experimental critical properties were available. Finally, expressions derived from the Clausius-Clapeyron equation have been investigated for the estimation of enthalpies of vaporization. A comparative analysis of the results produced by our parameters and by others available in the literature encourages the use of our values in future applications.


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INTRODUCTION In a previous paper,1 we have examined how group contribution and corresponding states models perform in computing vapor pressures and enthalpies of vaporization of fatty acid alkyl esters (FAAES). The accuracy of each model was verified by comparing estimated values to experimental data obtained in an extensive literature survey. Despite the good results obtained for enthalpy of vaporization, one should not expect very high accuracies from the analyzed vapor pressure methods. For this property, even the best models may generate estimates outside the desirable accuracy for engineering calculations. Therefore, the application of correlations becomes essential. Given its simplicity, the Antoine equation2 has been extensively used over the years. Antoine parameters for fatty acid methyl esters (FAMES) and fatty acid ethyl esters (FAEES) have been reported in the literature.3–13 As long as they are used for the temperature range at which the coefficients were regressed, these correlations should generate reliable estimates. Extrapolation beyond these limits may lead to absurd results and, therefore, should be avoided.14 If the vapor pressure behavior must be known in a wider range of temperature, a more robust correlation has to be applied.15 The Wagner equation16 has proven to be successful in correlating vapor pressure curves of many substances over the entire liquid range.14 Unlike Antoine’s, Wagner’s equation can be applied outside the stated temperature limits depending on the methodology used to generate their coefficients. McGarry17 proposed a constrained regression procedure to obtain extrapolatable parameters. According to him, three constraints, that reproduce features of vapor pressure curves supposedly valid for all substances, must be applied in the correlation of experimental data. Although

reliable

Wagner

constants

are

readily

available

for

common

industrial


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components,14,15,17–27 there is a lack of values for FAAES. To our knowledge, there are very few works regarding this issue,12,28,29 which may be related to the lack of accurate critical properties for these substances. In this work, a vapor pressure database1 was critically checked and used in the fitting of Antoine coefficients for all the compounds studied (18 FAMES and 6 FAEES). Following the methodology of Forero and Velásquez,27 the Antoine correlations were applied to generate pseudo-experimental data later used in the regression of Wagner constants for the esters whose critical properties were available (12 FAMES and 6 FAEES). These parameters were determined by a procedure similar to that proposed by McGarry,17 which allows extrapolations of the correlations in a wide-ranging temperature domain.

METHODOLOGY The equations involved The following form of Antoine’s equation has been adopted:

Pvap

log (

P0

B

(1)

) = A − T+C

where T and Pvap denote temperature [K] and vapor pressure [bar], respectively. P0 is a reference pressure (1 bar). A, B and C are component-specific constants. There are certain restrictions involving these parameters:15 1) The slope of the vapor pressure-temperature curve must be positive (dPvap /dT > 0), which leads to B > 0.


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2) The temperature derivative of the enthalpy of vaporization (∆Hv ) is always negative (d∆Hv /dT < 0) for non-associating substances. Considering the ideal gas approximation, the application of equation (1) to the Clausius-Clapeyron equation leads to C < 0. 3) At T = −C, equation (1) has a pole. Therefore, the lower application limit of the Antoine equation should be well above this temperature. The original form of Wagner’s equation has been considered:

ln (

Pvap Pc

)=

Aτ+Bτ1.5 +Cτ3 +Dτ6

(2)

Tr

where τ = 1 − Tr and Tr = T⁄Tc . Pc and Tc denote the critical pressure [bar] and the critical temperature [K], respectively. Accurate values of these properties are necessary to obtain reliable results from equation (2). Table 1 shows the critical properties adopted in this work, which were taken from recent publications.28–31 It should be noted that these properties are not available for some compounds, thereby their Wagner constants (A, B, C and D) have not been regressed. McGarry17 suggested empirical constraints involving these coefficients, which are summarized as follows: 1) The coefficients should be restricted to those sets yielding a solution for the expression “0.75B(1 − Tr )−0.5 + 6C(1 − Tr ) + 30D(1 − Tr )4 = 0” that lies within a specified range of reduced temperature (Tr ). The author obtained different ranges depending on the normal boiling points (Tb ) of the substances. For those whose Tb > 273 K, which is the case of all FAAES in this work, the interval should be 0.82 ≤ Tr ≤ 0.88. This restriction is based on the observations of Ambrose et al.32 and Waring.33


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2) The term ln(Pvap ⁄P′), where Pvap is calculated by equation (2) at Tr = 0.95 and P′ is P′

obtained from the expression “ln (P ) = −0.28278(1 + ω)” should be within the c

following interval for substances whose Tb > 273 K: -0.020 ≤ ln(Pvap ⁄P′) ≤ -0.001. −RTc [A 0.375 r)

3) The expression “g = (1−T

+ Bτ0.5 (1.5 − 0.5τ) + Cτ2 (3 − 2τ) + Dτ5 (6 − 5τ)]”

must be calculated at Tr = 0.5 and Tr = 0.6. If a substance’s Tb > 273 K, the ratio g(Tr = 0.5)⁄g(Tr = 0.6) should fall into a narrow range: [0.99; 1.03]. This restriction ensures that the low-temperature behavior of the vapor pressure curve matches the temperature dependence of the enthalpy of vaporization predicted by Watson’s correlation.14,34 In his original work,17 McGarry applied this methodology to regress Wagner constants for compounds containing limited data (usually in the range 0.55 ≤ Tr ≤ 0.65). After comparing calculated vapor pressure values to highly accurate experimental data, the author verified that his parameters yielded reliable results in a much broader interval (0.5 ≤ Tr ≤ 1).

Unconstrained fit of Antoine parameters The vapor pressure database1 was carefully checked before starting the regression of Antoine parameters. Initially, we excluded the low-temperature data (Tr < 0.5) for the substances whose Wagner parameters would be further regressed. Like McGarry,17 we focused our attention on a specific temperature domain (0.5 ≤ Tr ≤ 1) for these compounds. After that, the initial number of data points was decreased from 1095 to 839. Experimental values reported by different authors may have large discrepancies, which may be related to the purity of the samples or the method carried out during experimental measurements.


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For this reason, the remaining data were taken into an evaluation procedure based on that suggested by Gmehling et al.:15 I.

The internal consistency of each dataset was verified by separately fitting them to equation (1). Throughout this step, the following criteria were adopted: •

To guarantee a reasonable ratio between the number of data points and the number of adjustable parameters, each dataset had to contain at least 6 experimental data points at different temperatures. Smaller datasets have been rejected.

•

If the absolute relative deviation (%ARD – equation (6)) between an experimental and a calculated value was above 1%, that data point would be removed and a new fitting would be performed. If less than six points remained after removing all outliers, that dataset would be completely excluded.

After this procedure, 442 experimental values remained in the database. At total, 397 data points were excluded: 138 due to the first criterion and 259 due to the second criterion. II.

The remaining data were combined into global datasets for each compound, which were simultaneously fitted to equation (1). Like in the previous step, the calculations were performed several times for each compound. In each attempt, outlier experimental values (%ARD>1%) were removed and a new regression was performed until the mean absolute relative deviation (%AARD – equation (7)) would be below 1% for each substance. After all, 348 data remained in the databank. Up to this point, regressions were based on the minimization of the following objective function:

N

data OF = ∑i=1 (

exp

Pvap −Pcalc vap exp Pvap

2

(3)

)


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exp

calc where Pvap and Pvap denote the experimental and the calculated vapor pressures, respectively.

The summation in this equation went over all the available experimental data (Ndata ) for each reference (in step I) and for each compound (in step II). These calculations were initially carried out by a computer code based on the E04CBF routine from the NAG Library, 35 which uses the Simplex method36 to solve non-linear optimization problems. Then, the validated data of each compound were again fitted to equation (1) by using the previous Antoine constants as an initial guess to a modified Gauss-Newton method (from the E04GZF routine35). Additional information regarding determination of the Antoine parameters, including each step of the databank evaluation, can be found in the Supporting Information.

Constrained fit of Wagner parameters After evaluation of the entire database, a limited amount of experimental data remained in the datasets of some compounds. To ensure a reasonable ratio between the number of data points and the number of adjustable parameters, pseudo-experimental data generated by the Antoine correlations were used in the regression of Wagner parameters. This procedure was originally proposed by Forero and Velásquez.27 Therefore, a slightly different objective function was minimized in this case:

OF =

2 pexp Npdata Pvap −Pcalc vap ∑i=1 ( ) pexp P

(4)

vap

pexp

where Pvap

calc and Pvap denote the pseudo-experimental and the calculated vapor pressures,

respectively. The pseudo-experimental values were determined between the limiting temperatures


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of the Antoine correlations spaced by a 1 K step for all compounds. The constraints used in these regressions were very similar to those suggested by McGarry.17 However, two modifications were assumed and ought to be highlighted: 1) In the first constraint, we observed that the value of Tr that minimized the function was in the range 0.86 < Tr < 0.90 instead of that previously mentioned. This modification was strictly empirical. 2) Values of acentric factors (ω) were not readily available for FAAES. Unlike in our previous work,1 estimated values of these parameters were not used in the second constraint. Instead, we applied Pitzer’s definition37 to calculate P′. Therefore, the original expression was T =0.7

P′

r Pvap

c

Pc

replaced by “ln (P ) = 0.28278 [log (

T =0.7

r )]”, where Pvap

was calculated by equation

(2). To carry out the regressions, we applied the E04USF routine from the NAG Library35, which uses a sequential quadratic programming method to solve constrained optimization problems. More details about the regression of Wagner parameters are presented in the Supporting Information.

Comparison with other correlations The accuracy of the vapor pressure correlations was determined by comparing experimental and estimated data based on the following statistical parameters:

Xexp −Xcalc

%RD = 100 (

Xexp

Xexp −Xcalc

%ARD = 100 |

100

%AARD = N

data

(5)

)

Xexp data ∑N i=1 |

(6)

|

Xexp −Xcalc Xexp

|

(7)


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where X exp and X calc denote the experimental and the calculated properties, respectively. The summation in equation (7) went over all the validated data (Ndata ). To test the quality of the proposed parameters, we compared our results to those obtained from other correlations presented in Table 2. The Wagner constants reported by Nikitin and Popov28,29 have not been analyzed, because we were not certain about their range of applicability.

RESULTS AND DISCUSSION Antoine correlations Table 3 presents the regressed Antoine constants. The performances of all correlations are summarized in Table 4. Vapor pressure correlations should be extremely accurate to be used in process simulation. Ideally, the average deviations should be below 0.5%.15 Although exceptions can be made for compounds whose vapor pressures are extremely low (i.e., FAAES), the reported %AARD indicate the satisfactoriness of the proposed parameters. Except for two compounds (“ME-C12:0” and “ME-C16:0”), the parameters proposed in this work showed the highest accuracy in comparison to the others. Even for these specific esters, our results were almost as accurate as the best: 0.42% (TW) vs 0.16% (RS) for “ME-C12:0” and 0.40% (TW) vs 0.27% (RS) for “ME-C16:0”. Considering the complete database, the following decreasing order of accuracy was obtained: DU (0.11%) > TW (0.30%) > RS (0.41%) > SMM (0.71%) > RV (0.72%) > BEN (0.80%) > TDHX (0.84%) > RAJS (0.89%) > YHZ (1.66%) > SAH (2.35%) > FOLC (2.41%) > SFMK (7.77%). Although the DU correlation generated the best overall results, it was only applied to methyl ricinoleate. For this substance, our parameters showed slightly better results (%AARD = 0.10%).


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It is important to emphasize that the accuracy of a correlation is deeply related to the temperature interval at which it is applicable. It is very likely that the accuracy will increase as the application range becomes more restricted. Therefore, another useful parameter for qualifying a correlation is the breadth of its validity interval. For simplicity purposes, this interval should be as wide as possible. To contrast the correlations in terms of this criterion, we defined the temperature span as the difference between the maximum and minimum temperature at which they are applicable. Figures 1 and 2 depicts a comparative analysis between our correlations and the best ones in terms of this parameter. As it can be seen, the proposed correlations cover the widest temperature interval for half of the FAMES. Yuan et al.’s13 correlations showed the highest broadness for most of the long-chain FAMES, especially those containing more than 18 carbon atoms in the fatty acid chain. However, these authors used vapor pressure data determined by Ceriani and Meirelles’ model38 to obtain the parameters of these longest compounds, which may have deteriorated their accuracy. For FAEES, we can see that, except for “ME-C16:0”, the parameters proposed by Benziane et al.3 are the most complete in terms of this criterion. Although we did not plot it in the chart, their correlations are the most comprehensive for the “EE-C18:0”, “EE-C18:1” and “EE-C18:2” esters as well. The detailed values of %RD and %ARD output by all correlations, along with their parameters and temperature limits are available in the Supporting Information. To estimate the vapor pressure of a substance, one should always pick the most accurate correlation covering the specified operational conditions. Although the Antoine parameters proposed in this work showed great results, they have a significant drawback: their restricted validity range for some compounds. To overcome this sort of limitation, a common practice consists in combining two correlations for a substance. However, Gmehling et al.15 warns us to avoid this practice once it is not acceptable for process simulation, because these equations usually


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do not have the same value and the same slope at the point where the change-over between them takes place. Due to this fact, we propose Wagner constants in the next section.

Wagner correlations The regressed constants, as well as their performance in correlating experimental and pseudoexperimental data are reported in Table 5. The average deviations confirm the accuracy of the proposed parameters, which are within the ideal engineering specification for most substances.15 The lowest and highest %AARD were obtained for “ME-C7:0” and “ME-C16:0”: 0.12% and 1.14%, respectively. These results are probably related to the temperature range at which the Antoine correlations of these compounds are applicable. To illustrate the best and worst level of estimation, calculated vapor pressure-temperature curves are presented in Figures 3 and 4. In Figure 5, we plotted the %AARD distribution output by the Antoine and Wagner equations proposed in this work. As expected, the Antoine correlations output more accurate results for most compounds. However, the Wagner correlations generated better results for “ME-C9:0”, “MEC10:0”, “ME-C18:1”, “EE-C6:0”, “EE-C8:0”, “EE-C10:0”. Surprisingly, these results indicate that, for these compounds, Wagner’s equation should replace Antoine’s equation even if the operational temperature is within the boundaries reported in Table 3. Even though Duan et al.12 have proposed Wagner parameters for methyl ricinoleate, their application is limited to the temperature range presented in Table 3. Additionally, these authors’ Wagner correlation (%AARD = 0.22%) generated less accurate results in comparison to their Antoine correlation (%AARD = 0.11%). Differently, we attempted to generate accurate and, particularly, extrapolatable Wagner constants. If the vapor pressure behavior of FAAES obeys the considered constraints, our parameters will possibly be valid in the interval 0.5 ≤ Tr ≤ 1. To


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demonstrate this, reliable experimental data in the referred interval would be necessary. However, to our knowledge, these values are not available in the literature. As an alternative in analyzing the extrapolability of our parameters, we adopted a methodology divided in two parts. In the first part, we applied the Antoine correlations in the regression of new Wagner constants for each substance. At this step, pseudo-experimental data were generated in very restricted temperature ranges, as reported in Table 6. Then, these Wagner correlations were applied to estimate vapor pressures in broader intervals, at which validated data was available. The deviations presented in Table 6, specifically those relative to experimental data, are a first indication of the extrapolability. In the second part, the Wagner correlations were employed in the estimation of the normal boiling temperatures of the compounds. The calculated values were compared to experimental data, whose sources have been previously discussed.1 Table 7 presents a summary of the results. The difference between the reduced normal boiling temperature (Tb,r) and the maximum Tr used in each regression reveals the level of extrapolation provided by this test. For most esters, the extrapolation was even more severe than in the first part. Except for “EE-C16:0”, whose experimental Tb has not been found,1 small deviations were obtained for all substances. This confirms that the methodology adopted in this work produces extrapolatable parameters. Despite these results, we cannot affirm that our correlations will yield accurate results in the range 0.5 ≤ Tr ≤ 1. Nevertheless, in light of the applied constraints, we can ensure the consistency of the generated vapor pressure curves in the referred interval. To avoid misunderstandings, the latest Wagner parameters, obtained solely to validate the constrained regression methodology, are presented only in the supplementary file, which also contains a fluxogram presenting detailed information regarding the extrapolation test. In


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conclusion, due to their higher accuracy, we recommend the use of the parameters reported in Table 5 and the constants in Table 1 for future applications.

Enthalpy of vaporization As previously observed, estimation models can yield impressive results in calculating enthalpies of vaporization.1 Although predicting this property with vapor pressure correlations was not the main goal of this paper, we examined the results generated by the proposed parameters. Starting with equations (1) and (2), we applied the ideal-gas approximation in the Clausius-Clapeyron equation to derive the following expressions, respectively:

∆Hv =

RT2

B

(8)

log(e) (T+C)2

∆Hv = RT 2 [−

[A+1.5Bτ0.5 +3Cτ2 +6Dτ5 ] T

− Tc (

Aτ+Bτ1.5 +Cτ3 +Dτ6 T2

)]

(9)

where ∆Hv denotes the enthalpy of vaporization. R is the ideal gas constant and (e) denotes Euler’s number. These expressions are valid only for the forms of the vapor pressure equations adopted in this work. The forms adopted by other authors are presented in the Supporting Information. Figure 6 illustrates the quantity of the overall data that could be estimated by each correlation. As it can be seen, none of them could be applied to the entire database, because of their validity intervals or the lack of parameters for some compounds. These limitations also explain why the “SMM”, “RV”, “FOLC”, “DU” and “YHZ” correlations have not been analyzed at this step. In Table 8, we report the overall results. The most accurate correlations followed the order: SFMK (0.01%) > RAJS (0.28%) > SAH (0.44%) > RS (1.81%) > BEN (2.33%) > TW_WAG (2.87%) > TW_ANT (3.06%) > TDHX (11.39%). For this analysis, we considered our Wagner parameters


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to be valid in the interval 0.5 ≤ Tr ≤ 1. Readers can consult the supplementary file to check the detailed statistical parameters. Despite these results, it should be mentioned that some of the enthalpy of vaporization data considered in this analysis were not strictly experimental, that is, the reported values were not obtained with an experimental apparatus. Instead, the authors applied the Clausius-Clapeyron equation to calculate the data points from vapor pressure-temperature curves. As these values were in agreement with data obtained from calorimetry, they were included in the database.1 This fact explains the exactness of the results output by the “SMFK” correlations, which could only be compared to the data points reported by their authors. Thus, in this case, the results are not realistic. For the other correlations, the test was not effective either, given the low quantity of applications illustrated in Figure 6. For these reasons, we strongly recommend a careful reading of our previous paper before the estimation of enthalpy of vaporization of FAAES. Since the models studied in the previous work1 could predict a significantly larger quantity of the available experimental data, the conclusions regarding their accuracy are more trustworthy.

CONCLUSIONS In this work, we have continued investigating the estimation of vapor pressure and enthalpy of vaporization of fatty acid alkyl esters. Here we proposed component-specific parameters for wellknown vapor pressure correlations. An experimental vapor pressure database was critically evaluated to guarantee reliable Antoine correlations for the compounds studied. Further, pseudoexperimental data produced by these correlations were input into a methodology that has proven to generate extrapolatable Wagner constants for the esters whose critical properties were available.


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Finally, expressions were derived from the application of Antoine’s and Wagner’s equation in the Clausius-Clapeyron thermodynamic relationship. We contrasted our parameters with others available in the literature in terms of accuracy and breadth of their validity ranges. In light of the presented results, we suggest the use of our parameters for future vapor pressure estimations. Finally, we concluded that the analyzed correlations have not been really tested for enthalpy of vaporization calculations. Therefore, the application of the predictive models recommended in our previous work is more adequate.

Supporting Information. A MS-Excel spreadsheet is provided as a supplementary file. It contains complete information about the developed databank, including literature references and experimental data. Details of the database evaluation, of the Antoine/Wagner parameters determination, and of the extrapolation test are presented as well. This file also contains the results output by the analyzed correlations. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *Tel: +55-85-999998649. Fax: + 55-85-33178503. E-mail: [email protected] ACKNOWLEDGMENT Funding for this research was provided by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil).


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ABBREVIATIONS Roman Letters FAAE = Fatty acid alkyl ester. FAME = Fatty acid methyl ester. FAEE = Fatty acid ethyl ester. Ndata = Number of experimental data. Npdata = Number of pseudo-experimental data. Pc = Critical pressure. Pvap = Vapor pressure. T = Temperature. Tb = Normal boiling temperature. Tb,r = Reduced normal boiling temperature. Tc = Critical temperature. Tr = Reduced temperature.

Greek Letters ΔHv = Enthalpy of vaporization. ω = Acentric factor. Superscripts calc = calculated. exp = experimental. pexp = pseudo-experimental.


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Benziane, M.; Khimeche, K.; Mokbel, I.; Sawaya, T.; Dahmani, A.; Jose, J. Experimental Vapor Pressures of Five Saturated Fatty Acid Ethyl Ester (FAEE) Components of Biodiesel. J. Chem. Eng. Data 2011, 56, 4736.

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of Ethyl Myristate+ethyl Palmitate at 0.5, 1.0 and 1.5 kPa. Fluid Phase Equilib. 2013, 347, 8. (9)

Rose, A.; Schrodt, V. N. Vapor-Liquid Equilibria for the Methyl Oleate and Methyl Stearate Binary System. J. Chem. Eng. Data 1964, 9 (1), 12.

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Sahraoui, L.; Khimeche, K.; Dahmani, A.; Mokbel, I.; Jose, J. Experimental Vapor Pressures (from 1 Pa to 100 kPa) of Six Saturated Fatty Acid Methyl Esters (FAMEs): Methyl Hexanoate, Methyl Octanoate, Methyl Decanoate, Methyl Dodecanoate, Methyl Tetradecanoate and Methyl Hexadecanoate. J. Chem. Thermodyn. 2016, 102, 270.

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Duan, Y.; Nie, Y.; Gong, R.; Yu, S.; Deng, D.; Lu, M.; Chen, P.; Ji, J. Measurements and Correlations of Density, Viscosity, and Vapor Pressure for Methyl Ricinoleate. J. Chem. Eng. Data 2016, 61, 766.

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Yuan, W.; Hansen, A. C.; Zhang, Q. Vapor Pressure and Normal Boiling Point Predictions for Pure Methyl Esters and Biodiesel Fuels. Fuel 2005, 84, 943.

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Poling, B. E.; Prausnitz, J. M.; O’Connel, J. P. The Properties of Gases And Liquids, 5th ed.; McGraw-Hill, 2001.

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Gmehling, J.; Kolbe, B.; Kleiber, M.; Rarey, J. Chemical Thermodynamics for Process Simulation; Wiley-VCH: Weinheim, 2012.

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Wagner, W. New Vapour Pressure Measurements for Argon and Nitrogen and a New


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Method for Establishing Rational Vapour Pressure Equations. Cryogenics. 1973, 13, 470. (17)

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Ambrose, D.; Ghiassee, N. B.; Tuckerman, R. Vapour Pressure and Critical Temperature and Critical Pressure of 4-Methylpentan-2-One. J. Chem. Thermodyn. 1988, 20, 767.

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Ambrose, D.; Ewing, M. B.; Ghiassee, N. B.; Sanchez Ochoa, J. C. The Ebulliometric Method

of

Vapour-Pressure

Measurement:

Vapour

Pressures

of

Benzene,

Hexafluorobenzene, and Naphthalene. J. Chem. Thermodyn. 1990, 22, 589. (20)

Ambrose, D.; Ghiassee, N. B. Vapour Pressures and Critical Temperatures and Critical Pressures of Some Alkanoic Acids: C1 to C10. J. Chem. Thermodyn. 1987, 19, 505.

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Ambrose, D.; Ghiassee, N. B. Vapour Pressures and Critical Temperatures and Critical Pressures of C5 and C6 Cyclic Alcohols and Ketones. J. Chem. Thermodyn. 1987, 19, 903.

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Ambrose, D.; Ghiasse, N. B. Vapour Pressure, Critical Temperature, and Critical Pressure of Acetic Anhydride. J. Chem. Thermodyn. 1987, 19, 911.

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Ambrose, D.; Ghiasse, N. B. Vapour Pressures, Critical Temperatures, and Critical Pressures of 2-Chloro-1,1,2-Trifluoroethyl Difluoromethyl Ether (Enflurane) and of 1Chloro-2,2,2-Trifluoroethyl Difluoromethyl Ether (Isoflurane). J. Chem. Thermodyn. 1988, 20, 765.

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Ambrose, D.; Ghiassee, N. B. Vapor Pressure and Critical Temperature and Critical Pressure of 2,2,4,4,6,8,8-Heptamethylnonane. J. Chem. Thermodyn. 1988, 20, 1231.


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Ambrose, D.; Ghiasse, N. B. Vapour Pressures, Critical Temperatures, and Critical Pressures of Benzyl Alcohol, Octan-2-Ol, and 2-Ethylhexan-1-Ol. J. Chem. Thermodyn. 1990, 22, 307.

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Ambrose, D.; Walton, J. Vapour Pressures up to Their Critical Temperatures of Normal Alkanes and 1-Alkanols. Pure Appl. Chem. 1989, 61 (8), 1395.

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Nikitin, E. D.; Popov, A. P. Vapour-Liquid Critical Properties of Components of Biodiesel. 3. Methyl Esters of Linoleic, Linolenic, and Erucic Acids. Fuel 2016, 176, 130.

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Page 22 of 37

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22

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TABLES Table 1. Selected properties of FAMES and FAEESa

compound

acronym

methyl caproate

propertyc

b

Tc [K]

Pc [bar]

ME-C6:0

612.00

28.80

methyl enanthate

ME-C7:0

626.00

25.30

methyl caprylate

ME-C8:0

646.00

23.40

methyl pelargonate

ME-C9:0

665.00

20.60

methyl caprate

ME-C10:0

675.00

19.30

methyl laurate

ME-C12:0

709.00

15.20

methyl tridecanoate

ME-C13:0

-

-

methyl myristate

ME-C14:0

730.00

13.20

methyl pentadecanoate

ME-C15:0

-

-

methyl palmitate

ME-C16:0

760.00

11.70

methyl margarate

ME-C17:0

-

-

methyl stearate

ME-C18:0

785.00

10.80

methyl oleate

ME-C18:1

777.00

12.10

methyl ricinoleate

ME-C18:1,OH

-

-

methyl linoleate

ME-C18:2

778.00

12.40

methyl linolenate

ME-C18:3

779.00

14.40

methyl nonadecanoate

ME-C19:0

-

-

methyl arachidate

ME-C20:0

-

-

ethyl caproate

EE-C6:0

615.20

25.30

ethyl caprylate

EE-C8:0

655.00

21.60

ethyl caprate

EE-C10:0

687.00

17.40


23


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

compound

acronym

ethyl laurate

b

propertyc Tc [K]

Pc [bar]

EE-C12:0

718.00

13.70

ethyl myristate

EE-C14:0

740.00

12.70

ethyl palmitate

EE-C16:0

767.00

11.50

a

Page 24 of 37

Empty fields correspond to properties whose experimental values have not been found.

b

ME, methyl ester; EE, ethyl ester; In Cx:y, x and y denote the number of carbons and double bonds in the fatty acid chain, respectively. c

Tc, critical temperature; Pc, critical pressure.


24

Page 25 of 37

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Table 2. Selected correlations for vapor pressure estimation type

correlation

acronym

Benziane et al.3

BEN

X

Scott et al.4

SMM

X

Rose/Supina5

RS

X

Rose et al.6

RAJS

X

Silva et al.7

SFMK

X

Tang et al.8

TDHX

X

Rose/Schrodt9

RV

X

Freitas et al.10

FOLC

X

Sahraoui et al.11

SAH

X

Duan et al.12

DU

X

Yuan et al.13

YHZ

X

Antoine Wagner

X


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Page 26 of 37

Table 3. Antoine constants obtained in this work

compound

validity rangea

parameter A

B

C

Tmin

Tmax

ME-C6:0

4.2265

1482.7395

-72.6362

312.93

407.55

ME-C7:0

5.5460

2309.4487

-21.7563

316.75

340.35

ME-C8:0

4.6632

1887.3845

-60.2850

323.02

414.15

ME-C9:0

4.7467

1976.8461

-66.6868

333.21

357.68

ME-C10:0

4.5031

1907.0836

-80.6882

342.98

461.35

ME-C12:0

4.3401

1904.1529

-101.1651

361.64

531.15

ME-C13:0

4.1080

1952.2585

-98.3743

291.53

355.31

ME-C14:0

4.5445

2123.3729

-102.6919

444.25

555.11

ME-C15:0

3.6796

1859.7028

-118.0920

310.17

353.22

ME-C16:0

4.4729

2167.1463

-115.4810

385.78

600.90

ME-C17:0

3.4557

1786.4210

-139.1989

324.03

350.43

ME-C18:0

4.0741

1975.5982

-146.8913

412.55

503.85

ME-C18:1

4.6946

2345.7836

-118.2532

442.71

485.22

ME-C18:1,OH

3.3613

1427.0152

-213.6059

451.58

476.80

ME-C18:2

4.3349

2181.0885

-124.6626

399.84

440.17

ME-C18:3

4.3908

2191.0673

-126.5716

398.14

458.85

ME-C19:0

4.5105

2392.7545

-116.2024

350.51

367.74

ME-C20:0

5.0130

2661.2721

-110.0999

358.49

375.84

EE-C6:0

4.1424

1493.3126

-79.6426

313.15

462.43

EE-C8:0

4.5457

1885.1822

-65.7967

330.55

452.30

EE-C10:0

4.4724

1927.4988

-85.9831

353.16

462.37

EE-C12:0

4.3442

1943.0886

-104.0754

372.12

452.31

EE-C14:0

4.1166

1873.5106

-128.5317

398.52

462.30


26

Page 27 of 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


compound EE-C16:0 a

validity rangea

parameter A

B

C

Tmin

Tmax

3.2869

1521.0368

-170.4538

419.36

468.07

Tmin/Tmax, minimum/maximum temperature for the correlation to be used.


27


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Page 28 of 37

Table 4. %AARD for vapor pressure estimation using different Antoine correlationsa correlation compound

TWb

BEN

SMM

RS

RAJS

SFMK

TDHX

RV

FOLC

SAH

DU

YHZ

ME-C6:0

0.34%

-

-

0.86%

0.68%

-

-

-

-

2.77%

-

-

ME-C7:0

0.04%

-

-

-

-

-

-

-

-

-

-

-

ME-C8:0

0.27%

-

1.24%

0.31%

1.67%

-

-

-

-

0.57%

-

0.33%

ME-C9:0

0.24%

-

-

-

-

-

-

-

-

-

-

-

ME-C10:0

0.29%

-

0.31%

0.32%

-

-

-

-

-

2.58%

-

1.38%

ME-C12:0

0.42%

-

0.79%

0.16%

-

-

-

-

2.07%

0.95%

-

0.44%

ME-C13:0

0.45%

-

-

-

-

-

-

-

-

-

-

-

ME-C14:0

0.21%

-

-

0.25%

-

-

-

-

2.87%

4.83%

-

0.44%

ME-C15:0

0.45%

-

-

-

-

-

-

-

-

-

-

-

ME-C16:0

0.40%

-

0.28%

0.27%

-

-

-

-

1.92%

9.90%

-

0.30%

ME-C17:0

0.36%

-

-

-

-

-

-

-

-

-

-

-

ME-C18:0

0.45%

-

1.28%

1.00%

-

-

-

1.24%

-

-

-

0.77%

ME-C18:1

0.28%

-

1.23%

-

-

-

-

0.34%

-

-

-

3.16%

ME-C18:1,OH

0.10%

-

-

-

-

-

-

-

-

-

0.11%

-

ME-C18:2

0.20%

-

0.29%

-

-

-

-

-

-

-

-

6.17%

ME-C18:3

0.19%

-

0.23%

-

-

-

-

-

-

-

-

8.43%

ME-C19:0

0.31%

-

-

-

-

-

-

-

-

-

-

-

ME-C20:0

0.26%

-

-

-

-

-

-

-

-

-

-

14.76%

EE-C6:0

0.17%

0.47%

-

-

-

-

-

-

-

-

-

-

EE-C8:0

0.29%

0.99%

-

-

-

-

-

-

-

-

-

-

EE-C10:0

0.22%

0.70%

-

-

-

-

-

-

-

-

-

-

EE-C12:0

0.26%

1.58%

-

-

-

2.14%

-

-

-

-

-

-

EE-C14:0

0.31%

0.69%

-

-

-

11.30%

0.53%

-

-

-

-

-

EE-C16:0

0.22%

-

-

-

-

17.59%

1.23%

-

-

-

-

-

FAMES

0.31%

-

0.71%

0.41%

0.89%

-

-

0.72%

2.41%

2.35%

0.11%

1.66%

FAEES

0.24%

0.80%

-

-

-

7.77%

0.84%

-

-

-

-

-


28

Page 29 of 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


correlation compound Overall a

TWb

BEN

SMM

RS

RAJS

SFMK

TDHX

RV

FOLC

SAH

DU

YHZ

0.30%

0.80%

0.71%

0.41%

0.89%

7.77%

0.84%

0.72%

2.41%

2.35%

0.11%

1.66%

Empty fields indicate that the correlation could not be applied.

b

This work.


29


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Page 30 of 37

Table 5. Wagner constants obtained in this work

compound

parameter

%AARD

A

B

C

D

pseudo-expa

expb

ME-C6:0

-7.9246

1.6869

-6.0334

-0.4114

0.08%

0.35%

ME-C7:0

-9.1692

3.6529

-9.3901

6.7507

0.07%

0.12%

ME-C8:0

-8.6588

1.8820

-6.8408

2.5885

0.13%

0.28%

ME-C9:0

-8.8243

2.1058

-7.6156

0.6239

0.00%

0.24%

ME-C10:0

-9.2928

1.8516

-6.6162

-1.4891

0.11%

0.23%

ME-C12:0

-9.5474

2.8484

-10.4746

4.4904

0.44%

0.48%

ME-C14:0

-10.3120

3.5559

-13.3248

12.7104

0.50%

0.58%

ME-C16:0

-10.4846

3.8796

-14.5147

12.3759

1.02%

1.14%

ME-C18:0

-11.5152

5.4662

-16.1545

8.3469

0.73%

0.91%

ME-C18:1

-10.9924

3.4053

-12.4853

2.5246

0.01%

0.27%

ME-C18:2

-10.9103

2.6411

-9.4945

-3.2474

0.18%

0.28%

ME-C18:3

-11.1159

2.3978

-8.5344

-5.3765

0.35%

0.51%

EE-C6:0

-8.7504

1.8910

-4.5501

-7.1515

0.03%

0.14%

EE-C8:0

-8.8981

1.5328

-5.4694

-0.2417

0.11%

0.22%

EE-C10:0

-9.3625

2.1645

-7.8193

0.1060

0.11%

0.20%

EE-C12:0

-10.3450

4.3808

-11.9833

2.9768

0.09%

0.29%

EE-C14:0

-11.0074

4.6790

-13.1112

3.7784

0.27%

0.47%

EE-C16:0

-11.2914

4.1370

-10.9820

-0.6198

0.51%

0.76%

a

Deviations were calculated considering the pseudo-experimental data.

b

Deviations were calculated considering the validated experimental data.


30

Page 31 of 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 6. Analysis of the constrained regression methodology in terms of extrapolation (Part 1)

compound

regression

validation

Tr range

no. of dataa

%AARDb

Tr range

no. of datac

%AARDd

ME-C6:0

0.55-0.60

62

0.00%

0.51-0.67

32

0.39%

ME-C7:0

0.52-0.53

46

0.00%

0.51-0.54

6

0.15%

ME-C8:0

0.58-0.62

54

0.02%

0.50-0.64

31

0.34%

ME-C9:0

0.52-0.53

51

0.00%

0.50-0.54

6

0.24%

ME-C10:0

0.59-0.64

65

0.00%

0.51-0.68

30

0.57%

ME-C12:0

0.55-0.65

71

0.03%

0.51-0.75

26

0.68%

ME-C14:0

0.65-0.70

70

0.05%

0.61-0.76

26

0.53%

ME-C16:0

0.55-0.68

99

0.09%

0.51-0.79

25

1.56%

ME-C18:0

0.56-0.61

86

0.13%

0.53-0.64

17

1.12%

ME-C18:1

0.58-0.60

53

0.00%

0.57-0.62

14

0.29%

ME-C18:2

0.53-0.54

71

0.01%

0.51-0.57

8

0.40%

ME-C18:3

0.53-0.55

53

0.03%

0.51-0.59

7

0.72%

EE-C6:0

0.57-0.63

75

0.00%

0.51-0.75

16

0.21%

EE-C8:0

0.58-0.63

66

0.01%

0.50-0.69

10

0.35%

EE-C10:0

0.55-0.59

71

0.00%

0.51-0.67

8

0.41%

EE-C12:0

0.54-0.57

74

0.02%

0.52-0.63

7

0.32%

EE-C14:0

0.57-0.60

71

0.03%

0.54-0.62

11

0.42%

EE-C16:0

0.56-0.59

76

0.11%

0.55-0.61

7

0.82%

a

Pseudo-experimental.

b c

Deviations were calculated considering the pseudo-experimental data.

Experimental.

d

Deviations were calculated considering the validated experimental data.


31


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Page 32 of 37

Table 7. Analysis of the constrained regression methodology in terms of extrapolation (Part 2)

acronym

normal boiling pointa

reduced temperatureb

exp.

calc.

%ARD

Tr,max(reg.)

Tb,r

ME-C6:0

422.89

424.32

0.34%

0.60

0.69

ME-C7:0

445.29

445.77

0.11%

0.53

0.71

ME-C8:0

465.91

466.88

0.21%

0.62

0.72

ME-C9:0

486.75

487.49

0.15%

0.53

0.73

ME-C10:0

498.94

505.30

1.28%

0.64

0.74

ME-C12:0

535.03

541.95

1.29%

0.65

0.75

ME-C14:0

568.15

572.05

0.69%

0.70

0.78

ME-C16:0

611.15

603.28

1.29%

0.68

0.80

ME-C18:0

625.15

631.59

1.03%

0.61

0.80

ME-C18:1

620.65

622.91

0.36%

0.60

0.80

ME-C18:2

619.15

623.43

0.69%

0.54

0.80

ME-C18:3

620.15

619.95

0.03%

0.55

0.80

EE-C6:0

440.07

440.56

0.11%

0.63

0.72

EE-C8:0

479.64

481.64

0.42%

0.63

0.73

EE-C10:0

515.95

519.04

0.60%

0.59

0.75

EE-C12:0

547.48

555.09

1.39%

0.57

0.76

EE-C14:0

581.95

583.78

0.32%

0.60

0.79

EE-C16:0

-

615.36

-

0.59

-

a

exp., experimental; calc., calculated.

b

Tr,max(reg.), maximum value used in the regression (see Table 6); Tb,r, reduced normal boiling temperature, defined as Tb,r = Tb (exp. )⁄Tc .


32

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 8. %AARD for enthalpy of vaporization estimation using vapor pressure correlationsa correlation compound

TW_ANTb

TW_WAGc

BEN

RS

RAJS

SFMK

TDHX

SAH

ME-C6:0

1.70%

1.33%

-

1.81%

0.28%

-

-

0.35%

ME-C7:0

-

0.74%

-

-

-

-

-

-

ME-C8:0

-

-

-

-

-

-

-

0.79%

ME-C10:0

-

-

-

-

-

-

-

0.92%

ME-C13:0

2.51%

-

-

-

-

-

-

-

EE-C6:0

2.55%

2.50%

3.52%

-

-

-

-

-

EE-C8:0

2.26%

2.12%

1.63%

-

-

-

-

-

EE-C10:0

0.80%

0.73%

1.48%

-

-

-

-

-

EE-C12:0

2.21%

2.73%

0.46%

-

-

0.01%

-

-

EE-C14:0

-

6.82%

1.17%

-

-

0.01%

5.56%

-

EE-C16:0

20.06%

17.89%

-

-

-

-

17.23%

-

EE-C18:0

-

-

-

-

-

0.00%

-

-

EE-C18:1

-

-

-

-

-

0.01%

-

-

EE-C18:2

-

-

-

-

-

0.00%

-

-

FAMES

2.06%

1.27%

-

1.81%

0.28%

-

-

0.44%

FAEES

4.18%

4.33%

2.33%

-

-

0.01%

11.39%

-

Overall

3.06%

2.87%

2.33%

1.81%

0.28%

0.01%

11.39%

0.44%

a

Empty fields indicate that the correlation could not be applied.

b c

Antoine correlations proposed in this work.

Wagner correlations proposed in this work.


33


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Page 34 of 37

GRAPHICAL ABSTRACT


34

Page 35 of 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


FIGURES

Figure 1. Temperature spans of Antoine correlations (FAMES)

Figure 2. Temperature spans of Antoine correlations (FAEES)



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Figure 3. Experimental and calculated Pvap vs T curve for methyl enanthate.

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


Page 36 of 37

Page 37 of 37

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Figure 5. %AARD distribution of the proposed correlations.

Figure 6. Percentage of estimated data by each correlation (ΔHv)


Estimation of Vapor Pressures and Enthalpies of Vaporization of

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