High-Pressure Soybean Oil Biodiesel Density - ACS

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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High-Pressure Soybean Oil Biodiesel Density: Experimental Measurements, Correlation by Tait Equation, and Perturbed Chain SAFT (PC-SAFT) Modeling Rachid Aitbelale,*,†,‡ Younes Chhiti,‡ Fatima Ezzahrae M’hamdi Alaoui,‡ Abdelaziz Sahib Eddine,† Natalia Muñoz Rujas,§ and Fernando Aguilar§

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†

Laboratory of Catalysis and Corrosion of Materials (LCCM), Chemistry Department and ‡Science Engineer Laboratory for Energy (LabSIPE), National School of Applied Sciences, Chouaïb Doukkali University, 24000 El Jadida, Morocco § Department of Electromechanical Engineering, Superior Polytechnic School, Burgos University, E-09006 Burgos, Spain ABSTRACT: Biodiesel can easily become the crucial solution for environmental problems. The high production rate of soybean oil has been the subject of several research works to transform it into biodiesel. Knowledge of the thermodynamic properties of soybean oil biodiesel (SOB) such as densities and coefficients of expansivity and compressibility play an important role in the understanding of the intermolecular interactions between the different molecules, which in turn have an impact on fuel quality. The difficulty in measuring the thermodynamic properties of biodiesel is because they are complex structures and high-molecular-weight components. The experimental density (136 points) for SOB, as a pseudopure component, at several temperatures (298.15−393.15 K) and pressures up to 140 MPa is reported. An Anton Paar vibrating tube densimeter, calibrated with an uncertainty of ±0.7 kg m−3, was used to perform these measurements. To determine the chemical fatty acid methyl ester composition, SOB was analyzed by CHNS analysis, 1H NMR, 13C NMR, and gas chromatography−mass spectrometry and, then, the density experimental data were correlated by the Tait and perturbed chain-statistical associating fluid theory (PC-SAFT) equations of state (EoS). The experimental data were compared with correlated data, resulting in absolute average deviation (AAD = 0.01%), maximum deviation (MD = 0.03%), average deviation (Bias = −9.88 × 10−7%), and standard deviation (σ = 1.18 × 10−4 g cm−3) for the empirical Tait equation. Concerning PC-SAFT EoS, the density was reasonably correlated with AAD = 0.063%. On the other hand, isothermal compressibility, κT, and isobaric thermal expansivity, αp, were derived from the Tait equation. The same behavior is observed for κT and αp, consistent with the expected one. The isobaric thermal expansivity, αp, presents a crossing point at nearly 35 MPa, in agreement with what had been observed by other authors.

1. INTRODUCTION

The ability to predict biodiesel density and to determine several thermodynamic properties is important when designing equipment for synthesis processes and solving the engineering problems. This is of particular importance in the oil and gas sector where many flow measurement systems make use of volumetric flow measurement devices. The density at high pressures is essential to optimize and evaluate various chemical processes in the purification and production of biodiesel. The difficulty in predicting these properties is because biodiesels are complex structures and high-molecular-weight components. However, the development of robust equations to describe thermodynamic properties taking into account the effect of molecular interaction becomes paramount. The statistical associating fluid theory (SAFT) equations of state (EoS) has been successfully applied to a wide range of several

As an alternative to petrol and diesel, biodiesel can easily become the crucial solution for environmental problems.1 Biodiesel is the common name for a variety of ester-based oxygenated fuels from renewable biological sources. It can be obtained from vegetable oils and animal fats. To date, many vegetable oils have been used to produce biodiesel such as soybean, rapeseed, palm oils, etc. Soybean oil alone accounts for 61% of agricultural area in oilseeds, while those devoted to rapeseed, sunflower, and palm are 18, 14, and 7%, respectively.2 In principle, any source of fat can be used to prepare biodiesel. However, some sources are favored more than others according to the country.3 The literature reveals that several works are carried out on the synthesis and manufacturing of biodiesel from vegetable oils,4−10 but there is dearth of experimental data and/or prediction models for thermodynamic properties of vegetable oils.11 © XXXX American Chemical Society

Received: May 3, 2019 Accepted: August 22, 2019

A

DOI: 10.1021/acs.jced.9b00391 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Table 1. Fatty Acid Composition of SOB saturated fatty acid methyl esters (wt %) FAME SOB

C16:0 10.7

C18:0 3.9

unsaturatred fatty acid methyl esters (wt %) C18:1 22.8

C18:2 51

C18:3 6.8

others (wt %) aromatics and soybean oil residues 4.8

Table 2. Chemical Composition of SOB Obtained by GC/MS with Their Chemical Names, Molar Weights, Molecular Formulas, Percentage Weights, and Structures

fatty acid methyl esters. The main impurities noted are low soybean oil residues and the aromatics related to the hydrotreatment of polyunsaturated fatty acids of soybean oil. The molar weight (MW) of SOB is 264.9 g mol−1. 2.2. Materials and Methods. 2.2.1. CHNS Analysis. The method of CHNS analysis is based on the complete oxidation of the sample by instant combustion. The gases released from the combustion are transported by means of a carrier gas (He) through a reduction furnace. The technique consists of using a chromatographic column where the separation of different elements takes place. It then passes through a thermal conductivity detector that emits a signal directly proportional to the concentration of each component. The equipment used for this analysis is a Thermo Scientific FLASH 2000 elemental analyzer equipped with a SARTORIUS M2P ultramicrobalance (accuracy ± 0.001 mg) and a METTLER XP-6 precision microbalance (accuracy ±0.0001 mg). The combustion furnace can attain a temperature up to 1800 °C and is equipped with a thermal conductivity detector and a flame photometric detector for tracing sulfur analysis. The computer control, calculation, and data processing system was controlled with specialized software Eager Xperience. The CHNS analysis shows that SOB contains 76.39% wt carbon, 10.69% wt hydrogen, 0.134% wt nitrogen, 0.108% wt sulfur, and 12.078% wt oxygen. 2.2.2. Density Measurement. The densities of SOB were measured at pressures between 0.1 and 140 MPa and temperatures from 293.15 to 393.15 K using an Anton Paar DMA HPM high-pressure vibrating tube densimeter. Details of the equipment and its operation have been described previously.15−17 The densimeter was calibrated with water and vacuum as described in detail in a previous paper.18 The estimated value of uncertainty in measured temperature was ±0.03 K between 298.15 and 398.15 K using an Anton Paar MKT50 thermometer. The estimated value of uncertainty in the measured pressure was ±0.04 MPa using a Presens Precise Gold Plus pressure transmitter, and the estimated expanded uncertainty of density is around 0.07% (i.e., ±0.7 kg m−3, close

systems and provides the advantage of using parameters physically meaningful and independent of the thermodynamic conditions. The literature reveals several variants of the SAFT EoS depending on the type of reference fluid adopted. Oliveira et al.12 applied the soft version of the statistical associating fluid theory equation of state for the development, design, extension, and optimization of biodiesel production and purification processes. NguyenHuynh et al.13 applied the group contribution with perturbed chain-statistical associating fluid theory (GC-PC-SAFT) equation of state to predict atmospheric and high-pressure density data of several biodiesels and their mixtures. Corazza et al.14 applied the PC-SAFT equation (perturbed chain-statistical associating fluid theory) to predict the thermophysical properties, vapor−liquid equilibrium, and liquid−liquid equilibrium of binary and ternary systems related to biodiesel processing. This equation shows that the free energy of the system can be described as the effect due to the molecular form, the molecular force, and the molecular association. Consequently, this work will present a thermodynamic study concerning the soybean oil biodiesel (SOB) using the PC-SAFT model. In this work, the identification of soybean oil biodiesel is defined by CHNS analysis, NMR, and gas chromatography− mass spectrometry (GC/MS). The densities, isobaric thermal expansivity coefficient, and the isothermal compressibility coefficient data were reported at different pressures (0.1−140 MPa) and temperatures (298.15−393.15 K). However, Tait and PC-SAFT (perturbed chain-statistical associating fluid theory) EoS were fitted to the experimental density data.

2. EXPERIMENTAL SECTION 2.1. Materials. In this work, biodiesel derived from soybean oil was used. The sample was identified in the characterization section (Tables 1 and 2). After measuring the sample areas using the NMR spectra, we found that our sample is pure with 95.2% purity. The fatty acid methyl ester (FAME) compositions of soybean oil biodiesel (SOB) are presented in Table 1. It was found that SOB is composed of 14.6 wt % saturated fatty acid methyl esters and 80.6 wt % unsaturated B



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Figure 1. 1H NMR spectra of SOB.

Figure 2. 13C NMR spectra of SOB.

to water density with coverage factor k = 2). This uncertainty is similar to that reported in several studies.15−17,19−21 2.2.3. GC/MS Analysis. The identification of biodiesel compound is performed by gas chromatography−mass spectroscopy using an AGILENT 6890N GC/MS equipped with a

Micromass AutoSpec (Waters) mass analyzer. The injection volume was 2 μL at 240°C with a division ratio of 1:100. The temperature of the column has been programmed to raise to 240 °C at a rate of 15 °C min−1. The detector temperature was 280 °C. The mass spectra of the SOB biodiesel have been C



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Figure 3. Mass spectra of octadecadienoic acid methyl ester (a) and hexadecanoic acid methyl ester (b).

fragments. The triplet at around ∼2.3 ppm (B) is attributed to α-carbonyl hydrogen of all fatty acids (CH3−CO2−CH2− CH2−). Methylene groups in the carbon chain are represented by the peak at around 1.2 ppm, and β carbonyl methylene protons (−CH2−CH2−CO2−CH3) are attributed by peaks at 1.6 ppm (C). The protons in the CH2 groups between two carbon−carbon double bonds (CH−CH2−CH) are represented by peaks at 2.7 ppm (E), while the peak at 2.0 ppm (D) corresponds to allylic hydrogens of all fatty acids. The formation of fatty acid methyl ester (−C−O2−CH3) is confirmed by the appearance of a strong peak at 3.6 ppm (A). In addition, the vinylic (CH) protons (olefinic hydrogen) of the double bonds of fatty acids methyl esters are observed between 5.2 and 5.4 ppm. The 1H NMR spectrum of the SOB differs from that of the soybean oil cited in the literature22−26 by the presence of a peak at 3.6 ppm (Figure 1). This peak is attributed to methoxy protons of methyl ester. The low presence of peaks between 4.2 and 4.4 ppm signifies the formation of fatty acid methyl esters and also indicates the presence of soya oil residues. The 13 C NMR spectrum of biodiesel obtained is described in Figure

obtained using the electron ionization (EI) technique at 70 eV over the range 10−540 m/z in full scan mode. The constituents were identified by comparison with the mass spectra of fatty acid methyl esters of NIST mass spectral database. 2.2.4. NMR Analysis. The biodiesel was identified by 1H NMR and 13C NMR spectra using a VARIAN UNITY INOVA 400 MHz spectrometer at 399.94 and 100.58 MHz, respectively. The deuterated chloroform was used as a solvent to prepare experiments sample solution. The 1H NMR spectra were obtained with 32 scans, and 13C spectra were obtained with 100 000 scans.

3. CHARACTERIZATION The characterization of SOB was carried out by 1H NMR and 13 C NMR techniques. The 1H NMR spectra of SOB is shown in Figure 1; the characteristics peaks of the 1H NMR spectrum have been identified as follows: peaks at around 0.95 ppm (F) are associated with the terminal methyl protons (CHCH− CH2−CH3). The signals in the region 0.8−3 ppm are attributed to CH3, CH2, and allylic protons of the fatty acid D



i 1 yji ∂ρ zy κT = jjjj zzzzjjj zzz j z k ρ {k ∂p {T

2. Unsaturated fatty acids (olefinic carbons) are characterized by the presence of peaks between 127 and 132 ppm. The aliphatic resonances are between 14 and 50 ppm; the peaks between 20 and 35 ppm are due to CH2 groups. The CH3 group that terminates the fatty acid chains is represented by the signal at about 14 ppm. The carbonyl groups and carbon atoms of the methyl esters (−COO−) and (−OCH3) are characterized by peaks at 173 and 51 ppm, respectively. On the other hand, the measurement of the NMR areas shown that the SOB is a pseudopure component with 95.2% purity; the main impurities are soybean oil residues illustrated by peaks at 4.2 and 4.4 ppm, and the peaks of aromatics related to the hydrotreatment of polyunsaturated soybean oil are located at 6.8 and 7.2 ppm. The identification of the biodiesel compound is performed using gas chromatography−mass spectroscopy. The mass spectra of the octadecadienoic acid methyl ester and hexadecanoic acid methyl ester as an example have been obtained using EI mode at 70 eV and are shown in Figure 3. The five identified components are listed in Table 2 with their chemical names, molar weights, molecular formulas, percentage weights, and structures. Octadecadienoic acid methyl ester (51%) represents the main constituent, octadecenoic acid methyl ester (22.8%) was the second major constituent detected followed by hexadecanoic acid methyl ester (10.7%), and then come octadecatrienoic acid methyl ester and octadecanoic acid methyl ester with 6.8 and 3.9%, respectively.

=

(

B(T ) + p B(T ) + 0.1MPa

)

i ∂D(T , p) zy zz D′(T , p) = jjjj z k ∂T {p

ij 1 1 = CB′(T )jjj − j B(T ) + 0.1 + B ( T ) k 28

B(T ) = B0 + B1T + B2 T 2

(3)

(4)

(5)

yz zz p zz{

(6)

29

Cerdeiriña et al. and Troncoso et al. recommend deriving the isobaric thermal expansivity from the isobaric densities. Thus, in this work, we suppose that, at each pressure, ρP(T) = a0 + a1T + a2T2 and consequently (∂ρ/∂T)P = a1+2a2T. For each pressure, we get a set of (a0, a1, a2). By substituting the differentiated density and the calculated densities, ρP(T), into αp = −1/ρ(∂ρ/∂T)P, the isobaric thermal expansivity at different T and p conditions of each mixture has been derived αp = −

a1 + 2a 2T a0 + a1T + a 2T 2

(7)

As indicated previously in a similar high-pressure density study,15−17,21 the estimated uncertainty is 1% for the isothermal compressibility, κT, and 3% for the isobaric thermal expansivity, αp. 4.2. PC-SAFT Modeling. From the SAFT theory, many models have been developed by modifying terms of the equation of state. Using the perturbation theory developed by Barker and Henderson,30,31 Gross and Sadowski32 developed an equation of state to model the dispersive interactions generated by the long molecular chain, keeping the sphere term and chain term unchanged and using the new term to represent dispersive interactions, Gross and Sadowski33 published in 2001 the PC-SAFT (perturbed chain-statistical associating fluid theory) equation of state. It has been tested with the experimental data for numerous systems and has shown excellent results.34−37 The originality of PC-SAFT is to consider a chain of hard spheres as a reference term. The PC-SAFT equation expressed residual Helmholtz energy, ăres, as the sum of three terms of Helmholtz energy describing each molecular interaction in the fluid; the equation is written as

where ρ0(T, p = 0.1 MPa) is the dependency of the density on the temperature at the reference pressure (for this work, the atmospheric pressure was taken as the reference pressure) and B and C are two adjustable parameters with B(T) depending on the temperature and C being a parameter independent of the temperature and pressure, given as follows (2)

))(B(T) + p)

where D(T, p) is the denominator of Tait equation, eq 1, and D′(T, p) is its temperature derivative

(1)

ρ0 (T ) = A 0 + A1T + A 2 T 2 + A3T 3

(1 − C ln(

i 1 yi ∂ρ y αp = −jjjj zzzzjjj zzz k ρ {k ∂T { p 1 ij ρ′0 (T )D(T , p) − ρ0 (T )D′(T , p) yzz zz = − jjj z ρ jk D(T , p)2 {

ρ0 (T , 0.1MPa) 1 − C ln

C B(T ) + p B(T ) + 0.1 MPa

In addition, the coefficient of thermal expansivity, αp, corresponds to the derivation of the density with temperature at a constant pressure

4. CORRELATION AND MODELING OF EXPERIMENTAL DATA 4.1. Modified Tait Equation and the Derived Thermodynamic Properties. The empirical Tait equation was fitted to the experimental data to correlate correctly our values at our temperature and pressure ranges.27 This equation is frequently used to correlate the density of fluids at various temperatures and pressures, and it has the following form ρ0 (T , p) =

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It must be mentioned that the Ai, Bi, and C parameter values were determined by correlating simultaneously for all experimental density values (136 values). On the other hand, the Tait equation, eq 1, can be derived to obtain thermomechanical coefficients such as the isothermal compressibility, κT, and the isobaric thermal expansivity, αp. The knowledge of these two parameters provides useful information about the dependency of density on the temperature and pressure. The isothermal compressibility, κT, relates the derivation of the density with pressure at a constant temperature

a res − a ideal = a hc ̆ = a total ̆ ̆ ̆ + adisp ̆ + aassoc ̆

(8)

hc

where ă is the hard-sphere contribution (reference system), ădisp is the contribution of London forces that characterize interactions between segments called dispersion contribution, E



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Table 3. Experimental Densities, ρ (g cm−3), for SOB at various Temperatures, T, and Pressures, pa T (K)

p (MPa) 298.15 0.1 1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 80 90 100 110 120 130 140

313.15

0.8809 0.8814 0.8838 0.8867 0.8896 0.8923 0.8949 0.8975 0.9000 0.9024 0.9048 0.9071 0.9094 0.9116 0.9137 0.9159 0.9200 0.9240 0.9278 0.9315 0.9350 0.9385 0.9418

0.8699 0.8705 0.8731 0.8762 0.8792 0.8821 0.8847 0.8877 0.8903 0.8928 0.8953 0.8978 0.9002 0.9025 0.9048 0.9070 0.9113 0.9155 0.9194 0.9233 0.9270 0.9307 0.9342

333.15 ρ (g cm−3) 0.8552 0.8559 0.8587 0.8621 0.8655 0.8686 0.8716 0.8746 0.8774 0.8802 0.8828 0.8855 0.8880 0.8905 0.8929 0.8953 0.8999 0.9043 0.9085 0.9125 0.9164 0.9202 0.9239

353.15

373.15

0.8408 0.8414 0.8445 0.8483 0.8519 0.8554 0.8586 0.8618 0.8649 0.8678 0.8707 0.8736 0.8762 0.8789 0.8815 0.8840 0.8888 0.8935 0.8979 0.9022 0.9062 0.9102 0.9141

0.8269 0.8304 0.8345 0.8384 0.8422 0.8458 0.8492 0.8525 0.8557 0.8588 0.8618 0.8647 0.8675 0.8702 0.8728 0.8781 0.8829 0.8876 0.8920 0.8963 0.9005 0.9044

393.15

0.8123 0.8161 0.8206 0.8250 0.8290 0.8329 0.8366 0.8402 0.8437 0.8469 0.8501 0.8533 0.8562 0.8591 0.8620 0.8674 0.8726 0.8774 0.8822 0.8866 0.8909 0.8952

Estimated expanded uncertainties: temperature, U(T) = 0.03 K; pressure, U(p) = 0.04 MPa; density U(ρ) = 0.7 kg m−3.

a

and finally the contribution that illustrates the interaction between chains like hydrogen bonds is called association contribution, ăassoc.38 This association term has been the subject of a great debate for this work since all soybean oil biodiesels examined in the literature are nonassociating. However, in this work, the biodiesel studied can be considered as a mixture of fatty acid residues from soybean oil and fatty acid methyl esters. Therefore, the introduction of this term becomes necessary to describe the interactions between the residual fatty acids. hc a hc ̆ = ma ̅ ̆ −

the association energy, respectively; the association term, ăassoc, is defined by Ä ÑÉÑ ÅÅ i nc Å Ai y Ñ ÅÅ jj X 1 z assoc A zz + Mi ÑÑÑÑ = ∑ ÅÅÅ∑ jjln X i − ă z ÅÅ k 2 { 2 ÑÑÑÑ i=1 Å (11) ÇÅ A i ÖÑ The PC-SAFT parameters were estimated by fitting the PCSAFT equations to the experimental density presented in this work. The objective function (Obj.F) used in this work is as follows

nc

∑ (mi − 1)ln gijhs i=1

Obj. F =

i=1

The dispersion term, ădisp, accounts for the attraction between spherical segments; in a different way, it accounts for van der Waals forces. In this work, we use the dispersion expression defined by Gross and Sadowski, expressed as follows ̆ 1I2m2ε 2σ 3 adisp ̆ = −2πρ Ĭ 1m2εσ 3 − πρ mC

ij ρ exp − ρ calc yz zz i i zz z ρiexp k {

∑ jjjjj N

(9)

(12)

where N represents the number of data points and ρexp and i ρcalc represent the experimental and calculated densities, i respectively.

(10)

5. RESULTS AND DISCUSSION 5.1. Density. The densities were measured under highpressure conditions (up to 140 MPa) from 298.15 to 393.15 K (Table 3). The experimental data as a function of pressure and temperature (Figure 4) shows that the density of biodiesel behaved as expected, meaning that the density decreases as the temperature increases at constant pressure, and when the pressure decreases at a constant temperature (and vice versa). To compare the experimental data with those correlated by the Tait equation, we have used the absolute average deviation (AAD), the maximum deviation (MD), the average deviation (Bias), and the standard deviation (σ), which are defined as follows

where C1 depends on the compressibility expression and total number density of molecules, the integrals of the perturbation theory I1 and I2 are expressed by the reduced density and depend on the chain length, and σ and ε represent the segment diameter and the depth of pair potential for a pair of segments, respectively. The use of the PC-SAFT equation requires the determination of nonassociative parameters: m, segment number; ε/k, the segment energy parameter, and σ, the segment diameter by fitting the PC-SAFT equation to the experimental data. In case the sample is associating, it is necessary to add the association parameters kAiBi and εAiBi, signifying the association volume and F



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PC-SAFT model, the evaluation of the agreement between the measured and correlated results is performed by average absolute deviation (AAD) The biodiesel studied can be considered as a mixture of residual fatty acids (from soybean oil) and fatty acid methyl esters. To compare the necessity of using the association term, the densities of SOB were correlated by PC-SAFT EoS with (using the 2B scheme) and without using the association term; the AAD founded are shown in Table 5 using the parameters obtained by fitting; the average absolute deviation (AAD = 0.063%) found using the association term is more reliable than that used without this term (AAD = 0.1%). To explain the contribution of the association term, it is necessary to explain the concept of associating fluids; associating fluids can be described as fluids having an ability to form hydrogen bonds. This type of bonds can combine two molecules to form long chains. These intermolecular forces can be considered as intermediate between the weak electrostatic interactions (dispersion forces), and the forces are characteristic of chemical reactions forming the molecules. In another meaning, this type of phenomenon can significantly change the behavior of a fluid. The hydrogen bonding between the carboxy groups of a fatty acid in its pure liquid state is strong and intact as function temperatures.39 Concerning the fatty acid methyl esters, the resonance structures of the ester group result in weak interactions between the polarized hydrogen of the methyl carbon and the oxygen of the carbonyl on the opposite molecule; this molecular arrangement would maximize the polar interactions. Even though no strong hydrogen bonding exists in this ester, the polar interactions are sufficiently strong to limit the rotation of the head. To be more specific, the fatty acid methyl ester heads interact through weak polar interactions, while fatty acid heads interact through hydrogen bonding.40 Figure 5 presents the comparison between the experimental data and densities correlated by the Tait equation and PC-

Figure 4. Measured densities of SOB as a function of pressures and temperatures.

AAD =

100 N

ρiexp − ρicalc

N

∑

ρiexp

ij ρ exp − ρicalc j MD = maxjjj100 i jj ρiexp k Bias =

100 N

i=1

N

∑

(13)

yz zz zz zz {

(14)

ρiexp − ρicalc

i=1

ρiexp

(15)

N

σ=

∑i = 1 (ρiexp − ρicalc )2 N ‐m

(16)

where N represents the number of data points (N = 136) and m represents the number of parameters fitted by the Tait equation (m = 8). Table 4 presents the eight correlation parameters of the Tait equation, AAD, MD, Bias, and standard deviation, σ. For the Table 4. Obtained Parameters and Deviations for Density Correlation by the Tait Equation for SOB A0 (g cm−3) A1 (g cm−3 K−1) A2 (g cm−3 K−2) A3 (g cm−3 K−3) B0 (MPa) B1 (MPa K−1) B2 (MPa K−2) C σ (g cm−3) AAD (%) MD (%) Bias (%)

1.105 −8.073 × 10−4 2.808 × 10−7 −3.2066 × 10−10 436.5 −1.399 1.202 × 10−3 0.08682 1.18 × 10−4 0.01 0.03 −9.88 × 10−7

Figure 5. Comparison between the experimental densities of SOB and the density estimated by () PC-SAFT and (---) Tait equations as a , function of pressure at: □, 298.15 K; ×, 313.15; ○, 333.15 K; 353.15 K; ◇, 373.15 K; and Δ, 393.15 K

SAFT equation with the association term. A good correlation is provided by the Tait equation as a function of pressure and

Table 5. Characteristic Parameters of PC-SAFT Model for SOB compound

MW (g mol−1)

m (−)

σ (A)

ε/k (K)

εAB/k (K)

KAB (K)

association scheme

AAD % (ρLiq)

SOB (with the association term) SOB (without the association term)

265.1 265.1

8.24442 9.601

3.18116 3.566

203.552 281.298

1288.19

0.012656

2B

0.063 0.1

G



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temperature with a maximum standard deviation of σ = 9.6 × 10−8 g cm−3 at T = 313.15 K and p = 140 MPa. As the temperature interval considered here is sufficiently large, the variation of the density vs. temperature is nonlinear at low pressures, so, this justifies the use of eq 1. Moreover, the nonlinear shape of the isotherm of density vs. pressure is compatible with the logarithmic relationship used in the Tait equation to model the behavior of pressure on density. The compatibility of the estimated values with experimental data shows the effectiveness of this method to correlate the density of biodiesel. The values obtained are AAD (0.01%), MD (0.03%), Bias (−9.88 × 10−7%), and standard deviation, σ (1.18 × 10−4 g cm−3). Figure 6a illustrates the deviation

equation written as the sum of three terms of Helmholtz energy describes the microscopic molecular interactions of a fluid (see eq 8). Figure 5 shows the compatibility between the density correlated by the PC-SAFT model and the experimental data; it is clear that the correlated densities are similar to the experimental data, especially for pressures lower than 60 MPa. However, a small deviation is noted for pressures higher than 60 MPa. The SOB is a mixture of saturated and unsaturated fatty acid methyl esters (14.6 wt % of saturated fatty acid methyl esters and 80.6 wt % of unsaturated fatty acid methyl esters) in addition to the fatty acids of soybean oil residues. The dispersion forces are intermolecular forces of attraction acting between molecules and are generally due to the formation of instant dipoles resulting from the random movement of π−π electrons of unsaturated fatty acids. The reasonable attraction between unsaturated molecules is generated by the proximity of these instant dipoles. On the other hand, the bending due to cis-double bonds prevents chains from packing closely together. Therefore, unsaturated fatty acids can establish lower dispersion forces than saturated fatty acids. In other words, the molecules that are bent cannot pack close to each other along the length of the molecule and therefore, the surface charges are unable to approach close to each other to generate a reasonable force of attraction, which results in weaker attractive forces. In contrast, the saturated molecules are straight and packed together due to their linear chain configuration high flexibility, which allows molecules to minimize the steric hindrance of neighboring atoms and maximize the number of van der Waals interactions. This type of force has been translated by the dispersion term expressed by eq 10. This effect becomes weaker vs. temperature. On the other hand, this term still contributes to the residual Helmholtz energy in the PC-SAFT equation for the entire temperature range, which results in a deviation between the experimental and correlated data, especially at high temperatures. Additionally, considering the physical meaning of parameters used in PC-SAFT models, they are preferable to empirical ones, especially in the predictive description of mixtures.37 Figure 7 shows the deviation plots (using the Tait equation with our correlation parameters) between the measured and calculated densities determined in the present work compared with those proposed by some literature works.41−44 However, in this work, we presented the density values measured for SOB from 298.15 to 393.15 K as a function of pressure up to 140 MPa. As mentioned previously,

Figure 6. Deviation between the measured and calculated densities by the Tait equation (a) and PC-SAFT equation (b) as a function of pressure at: ◇, 298.15 K; □, 313.15; Δ, 333.15 K; ×, 373.15 K, and , 393.15 K.

between experimental and calculated densities by the Tait equation. This figure shows a deviation between −0.03 and 0.03%; the low deviations are observed in the ranges of 20−60 and 100−130 MPa. Figure 6b illustrates the deviation between the densities correlated by the PC-SAFT equation and the experimental data equation using the parameters with the association term. The comparison shows that this equation is powerful for density correlation, in particular, at the pressure range between 25 and 50 MPa; beyond this range, the deviation increases proportionally to the pressure. The maximum deviations are noted for low temperature (298.15 K) at high pressures with 0.60% and for high temperature (373.15 K) at low pressures with 0.49%. Generally, densities correlated by the PC-SAFT equation of state show a good agreement for the SOB compared with experimental data. However, the average absolute deviation (AAD) obtained was 0.063%. The SOB can be considered as fatty acid methyl esters of long chain with a complex structure and high molecular weight. However, the development of robust equations to describe the thermodynamic properties taking into account the effect of the molecular interaction becomes necessary. The PC-SAFT

Figure 7. Relative deviations between the experimental density data of SOB obtained in this work (×) and those obtained from the literature: ref 31, Δ; ref 32, ○; ref 33, ◇; and ref 34, + at atmospheric pressure as a function of temperatures. H



Article

Table 6. Isobaric Thermal Expansivity, αp, and the Isothermal Compressibility, κT, for SOB at Various Temperatures, T, and Pressures, p

the values obtained are AAD = 0.01%, MD = 0.03%, and Bias = 9.88 × 10−7%, according to number of data points, N = 136. This shows an excellent agreement with the experimental data compared to the values found by Nogueira et al.41 with AAD = 0.19%, MD = 0.23%, and Bias = 0.19% (N = 5); Feitosa et al.42 with AAD = 0.11%, MD = 0.13%, and Bias = 0.11% (N = 5); and Pratas et al.43,44 with AAD = 0.10%, MD = 0.12%, and Bias = −0.1% (N = 17) and AAD = 0.78%, MD = 0.85%, and Bias = −0.78% (N = 83). It can be seen that the maximum deviation is 0.78%, found by Pratas et al.44 and the best agreement is observed with our estimation with ADD = 0.01%. 5.2. Derived Thermodynamic Properties. From the Tait equation, it was also possible to obtain thermomechanical coefficients such as the isothermal compressibility, κT, and the isobaric thermal expansivity, αp. These coefficients are sensitive to changes in density and provide useful information about the dependency of density on temperature and pressure. These properties are expressed previously in eqs 4 and 7. Table 6 reports the isothermal compressibility, κT, and the isobaric thermal expansivity, αp, values as a function of temperature and pressure. Figure 8 shows the variation of αp as a function of temperature along the isobars, (αp, T)p (Figure 8a), and as a function of pressure along the isotherms, (αp, p)T, (Figure 8b), for the SOB. Note that, according to the expected, there is a decrease in isobaric thermal expansivity, αp, when the pressure increases by fixing the temperature and an increase when the temperature increases by fixing the pressure, mainly at low pressures (from 0.1 to 35 MPa). Beyond this range, a small decrease of αp with temperature was noticed. An intersection point close to 35 MPa is indicated by αp isotherms of value (7.21 ± 0.21) × 10−4 K−1. This intersection was observed for many liquids and was described for the first time by Bridgman.45 For the fatty acid methyl esters, Prieto et al.46 noted the same behavior for methyl linoleate biodiesel studied by Schedemann et al.47 and they found that the αp isotherms show an intersection point at p = 65 MPa with a value of (6.562 ± 0.011) × 10−4 K−1. This point means that αp was independent of the temperature at this pressure because it obeyed the condition ((∂αp/∂T)p = 0). Taravillo et al.48 also remarked this behavior and they indicated that it appears to be a general property of liquids and that these intersections frequently occur at pressures below 200 MPa. Concerning the isothermal compressibility κT, the variation as a function of temperature along the isobars, (κT, T)p, and the variation as a function of pressure along the isotherms, (κT, p)T, for the SOB are illustrated in Figure 9a,b, respectively. Parabolic curvatures are observed for κT when the temperature increases, especially at low pressures. A parabolic decrease in κT as the pressure increased to a fixed temperature was observed, especially at the highest pressures. The minimum and maximum κT values for SOB in the ranges 298.15−393.15 K and 0.1−140 MPa were (11.86± 0.12) × 10−4 MPa−1 and (3.49 ± 0.035) × 10−4 MPa−1, respectively.

p (MPa)

T (K) 298.15

0.1 1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 80 90 100 110 120 130 140

8.23 8.20 8.06 7.89 7.73 7.59 7.45 7.32 7.20 7.08 6.97 6.86 6.77 6.67 6.58 6.50 6.33 6.19 6.05 5.93 5.81 5.70 5.60

0.1 1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 80 90 100 110 120 130 140

6.87 6.83 6.64 6.41 6.21 6.01 5.83 5.66 5.50 5.35 5.21 5.07 4.95 4.82 4.71 4.60 4.40 4.21 4.04 3.89 3.74 3.61 3.49

313.15

333.15 353.15 αp × 104 (K−1) 8.34 8.49 8.66 8.30 8.45 8.61 8.14 8.27 8.40 7.96 8.06 8.16 7.79 7.86 7.94 7.63 7.68 7.73 7.47 7.51 7.54 7.33 7.35 7.37 7.20 7.21 7.21 7.08 7.07 7.06 6.96 6.94 6.92 6.85 6.82 6.78 6.74 6.70 6.66 6.64 6.59 6.54 6.54 6.49 6.43 6.45 6.39 6.33 6.28 6.21 6.13 6.13 6.04 5.96 5.99 5.89 5.80 5.85 5.75 5.65 5.73 5.63 5.52 5.62 5.51 5.40 5.51 5.40 5.28 κT × 104 (MPa−1) 7.46 8.35 9.39 7.41 8.29 9.31 7.18 8.01 8.96 6.92 7.69 8.56 6.68 7.39 8.19 6.46 7.12 7.86 6.25 6.87 7.55 6.06 6.63 7.27 5.87 6.41 7.01 5.70 6.21 6.77 5.54 6.02 6.55 5.39 5.84 6.34 5.25 5.68 6.14 5.11 5.52 5.96 4.98 5.37 5.78 4.86 5.23 5.62 4.63 4.97 5.33 4.43 4.74 5.06 4.24 4.52 4.82 4.07 4.33 4.60 3.92 4.16 4.41 3.77 3.99 4.23 3.64 3.85 4.06

373.15

393.15

8.79 8.55 8.27 8.02 7.79 7.58 7.39 7.21 7.05 6.89 6.75 6.62 6.49 6.37 6.26 6.06 5.88 5.71 5.56 5.42 5.29 5.18

8.98 8.70 8.39 8.11 7.85 7.62 7.41 7.22 7.04 6.88 6.72 6.58 6.45 6.33 6.21 6.00 5.81 5.64 5.48 5.34 5.21 5.09

10.48 10.04 9.54 9.09 8.68 8.31 7.97 7.66 7.38 7.11 6.87 6.64 6.42 6.23 6.04 5.70 5.40 5.13 4.88 4.67 4.47 4.28

11.86 11.29 10.66 10.11 9.61 9.16 8.75 8.38 8.04 7.72 7.44 7.17 6.92 6.69 6.48 6.09 5.75 5.45 5.18 4.93 4.71 4.51

a

The estimated uncertainty is 1% for the isothermal compressibility, κT, and 3% for the isobaric thermal expansivity, αp.

6. CONCLUSIONS In this paper, the thermophysical property data (density, isobaric thermal expansion, and isothermal compressibility) of soybean oil biodiesel were reported over the wide temperature range from 298.15 to 393.15 K and at pressure between 0.1 and 140 MPa. Density data were correlated by Tait and PCSAFT EoS, respectively, with good agreement. The Tait equation is used to correlate the density of the SOB, and it has provided an excellent correlation between the density values

over the experimental range of temperatures and pressures with ADD = 0.01%. The PC-SAFT model has a reasonable ability in predicting the properties relevant to biodiesel fuels with AAD = 0.063%. The deviations in densities have been calculated from the experimental data. The low deviations are observed in the ranges of 20−60 and 100−130 MPa, while for I



Article

The thermodynamic properties, i.e., isothermal compressibility, κT, and isobaric thermal expansivity, αp, were derived from the Tait equation. The same behavior is observed for κT and αp, consistent with the expected one. The isobaric thermal expansivity, αp, presents a crossing point at nearly 35 MPa, in agreement with what had been observed by other authors.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Rachid Aitbelale: 0000-0002-8813-8367 Notes

The authors declare no competing financial interest.

■

REFERENCES

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Figure 8. Thermal expansivity, αp, calculated from the Tait equation. (a) As a function of temperature along isobars: □, 0.1 MPa; ◇, 5 MPa; Δ, 15 MPa; ×, 25 MPa; , 35 MPa; +, 45 MPa; ○, 55 MPa; −, 65 MPa; -, 80 MPa; •, 100 MPa; ■, 120 MPa; ▲, 140 MPa. (b) As , 313.15; ×, a function of pressure along isotherms: +, 298.15 K; 333.15 K; Δ, 353.15 K; ◊, 373.15 K; □, 393.15 K.

Figure 9. Isothermal compressibility, κT, calculated from the Tait equation. (a) As a function of temperature along isobars: □, 0.1 MPa; ◇, 5 MPa; Δ, 15 MPa; -, 30 MPa; , 55 MPa; ×, 100 MPa; ○, 140 MPa. (b) As a function of pressure along isotherms: +, 298.15 K; , 313.15 K; ×, 333.15 K; Δ, 353.15 K; ◇, 373.15 K; □, 393.15 K.

PC-SAFT, the maximum deviation is recorded for low temperature (298.15 K) at high pressures with 0.60% and for high temperature (373.15 K) at low pressures with 0.49%. J



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