Article pubs.acs.org/EF
Estimation of Density of Biodiesel Suriya Phankosol,† Kaokanya Sudaprasert,† Supathra Lilitchan,‡ Kornkanok Aryusuk,§ and Kanit Krisnangkura*,§ †
Division of Energy Technology, School of Energy, Environment and Materials, and §Division of Biochemical Technology, School of Bioresources and Technology, King Mongkut’s University of Technology Thonburi, Bangkok, 10140, Thailand ‡ Department of Nutrition, Faculty of Public Health, Mahidol University, Rachathewi, Bangkok 10400, Thailand S Supporting Information *
ABSTRACT: Density of biodiesel is an important physical property of liquid fuel and biodiesel. A slight change in density can affect engine output power. In this work, density of a saturated, unsaturated fatty acid methyl ester or a biodiesel can be estimated from either (1) its number of carbon atoms (of fatty acid, z) and number of double bonds (nd): ln ρ = −0.435 − 0.0025z + 85.98/T + 0.792z/T + 4.0 nd/T or (2) its saponification number (SN) and iodine value (IV): ln ρ = −0.427 − 10/SN + 83.38/T + 3168.95/(T × SN) + 11 IV/(T × SN), where T is absolute temperature. The predicted densities at different temperatures from both equations agree well with the reported literature values. where ρ and t are density and temperature (Celsius); a and b are dependent empirical constants. Equation 1 does not meet the criteria of Halvoson et al.6 Rackett equation8 is a classical model for estimation of liquid density. The equation had been modified by Spenser and Danner,9 Yamada and Gunn,10 Soave11 and Meng et al.,12 as shown in eq 2.
1. INTRODUCTION Biodiesel is clean and renewable fuel for diesel engine. It has a higher cetane number and flash point than those of petrodiesel.1 Also, it is more environmentally friendly than petrodiesel.2 However, vegetable oils, which are the main feed stocks for biodiesel, vary from country to country. Thus, a national standard for commercial biodiesel is necessary for each country, and density is included in the specification. Density of biodiesel is an important property for liquid fuel. Its delivery from the fuel tank to the engine is generally measured by volumetric flow, while the energy released in the combustion chamber is calculated on the weight basis. These two parameters are interrelated via the density. In addition, density data are also required in the calculations related to storage facilities, fluid flow, distillation units, separation process, storage tanks, design of reactors, and process piping.3,4 Also, density of biodiesel has been correlated to its cetane number by Lapuerta et al.5 Although simple equipment can be used for measurement of density of biodiesel at a specific temperature, it takes time, especially when measurements are carried out at different temperatures. Thus, a mathematical model would be helpful in obtaining the density data at different temperatures. As it was pointed out that density is an important property of a liquid fuel, models for estimation of density of liquid and biodiesel are plentiful. Halvoson et al.6 pointed out that two criteria were important for estimating the density of fatty acids. First, all of the commonly encountered saturated and unsaturated fatty acids must be included in the density estimation scheme. Second, the estimation scheme must account for the temperature dependency of density. Fisher7 proposed a simple model for estimating the density of fatty acid and its ester at different temperatures. ρ = at + b
ρ=
(2)
where M, Tc, and Pc are molecular mass, critical temperature and pressure of fatty acid methyl esters; T is the measured temperature in Kelvin; ZRA is the Rackett compressibility factor, and R is the universal gas constant. Equation 2 contains molecular mass and temperature, while the degree of unsaturation is not conspicuous. It is possibly hidden in the Tc, Pc, or/and ZRA terms. However, the model is very complex. It requires the knowledge of critical temperature, critical pressure and compressibility factor, which in turn requires the acentric factor of the molecule. A very early correlation for predicting the density of vegetable oils was attributed to Lund given in ref 6. Lund correlated the specific gravity (Sg) of the vegetable oil at 15 °C to its saponification number (SN) and iodine value (IV) as Sg = 0.8475 + 0.00030SN + 0.00014IV
(3)
Thus, the densities of vegetable oils were related to their fatty acid compositions. Elbro et al.13 approached the problem differently by introducing group contribution (GCVOL) method for predicting saturated density of a liquid. Received: May 6, 2014 Revised: June 16, 2014 Published: June 16, 2014
(1) © 2014 American Chemical Society
M RTc [1 + (1 − T / Tc)]2/7 Z Pc RA
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1000M ∑ niΔvi
ΔG1...ΔGz are the free energies of the methylene and methyl groups, which are not very different. Thus, they are average to δG and eq 10 is shortened to eq 12.
(4)
where ni is the number of group i and Δvi is the molar group volume. Equation 4 had been used for estimation of density of different FAMEs and biodiesels by many researchers,14,15 but the Δvi values were slightly modified, and the double bonds group was added by Pratas et al.15 However, Meng et al.12 pointed out that the GCVOL method was good for predicting liquid density below 373 K. Recently, Ramirez-Verduzco16 proposed a very interesting model (eq 5) for predicting density of fatty acid methyl esters(FAMEs) and biodiesel. The model includes all the necessary terms specified by Halvoson et al.6 (temperature, molecular mass and number of double bonds (nd) of FAMEs). ρ = 1.069 +
3.575 + 0.0113nd − 7.41 × 10−4T M
ΔG = ΔGf + zδG
where ΔGf is the free energy of the functional group, X; δG is the change in free energy/carbon atom; z is the number of carbon atoms (for fatty acid ester, z is the number of carbon atoms of the fatty acid). The Martin’s rule of free energy additivity has been successfully extended to other physical phenomena, gas chromatographic peak width, vapor pressure of a pure liquid, viscosity, and surface tension of FAME and biodiesel.18−21 In this study, the Martin’s rule of free energy additivity is extended to cover the density of a liquid, where ΔG is defined as free energy of volumetric expansion of a liquid. Expanding the ΔG to its enthalpy and entropy forms and substitution eq 12 into eq 10, eq 13 is obtained.
(5)
The Ramirez-Verduzco empirical model is a good model for prediction of density of pure liquids or pure FAME. For biodiesel, which is a mixture of FAMEs, the Kay’s rule for mass fraction was used for the calculation. Predicting the physical properties of biodiesel from its chemical composition is nowadays an interesting issue. It would greatly help manipulate the proper composition for use as the diesel fuel. Thus, in this study, two empirical equations based on free energy additivity are derived and proposed for predicting density of biodiesel at different temperatures from its fatty acid composition.
ln ρ = ln ρ0 +
⎛ δG ⎞ ⎜ ⎟ ⎝ δP ⎠T
ln ρ = a + bz +
−
dG̃ 1 = − dρ ρ RT
a = ln ρ0 +
ρi ρ0
=−
ΔG̃ RT
(6)
ΔGunsat = ΔGsat + ndΔGdb
(7)
(16)
And ΔG = ΔGf + zδG + ndΔGdb
(8)
(17)
where nd is number of double bonds in FAME. Expanding eq 17 to the enthalpic and entropic forms, one gets,
(9)
fn c dz + + end + d T T T
(18)
where
(10)
where ρi and ρ0 are densities of component i and reference, respectively. 2.1. Density of Saturated Fatty Acid Methyl Ester. For a compound having the molecular structure of CH3-(CH2)z−1X, Martin17 divided the molecule into different groups; X, CH2, CH3. The free energy of transfer from solution to gas of the molecule (in gas chromatography) was derived from the sum of the free energies of all the contribution groups. ΔG = ΔGf + ΔG1 + ΔG2 ... + ΔGz
(15)
Thus,
ln ρ = a + bz + ρi = ρ0 e
(14)
ΔSf ΔHf δS δH , b= , c=− and d = − R R R R
ΔGunsat − ΔGsat = ndΔGdb
or −ΔG̃ / RT
c dz + T T
Hence, eq 14 is proposed for calculation of density of saturated fatty acid methyl ester at different temperatures. 2.2. Density of Unsaturated Fatty Acid Methyl Ester. For unsaturated FAME, ΔGdb is assigned for the free energy of volumetric expansion of the double bond (group), and it is the difference between the volumetric expansion of unsaturated FAME (ΔGunsat) and saturated FAME (ΔGsat).
where n = m/M and G̃ = G/n and the integral form of eq 8 is
ln
(13)
where
For an ideal solution, eq 6 can be transformed to eq 7. dG m1 dP = RT MP
ΔHf ΔSf zδH zδS + − + RT R RT R
Grouping,
2. THEORY From the definition of different form of energies, U (internal energy), H (enthalpy), and G (Gibbs free energy or just free energy), the relationship between ΔG and liquid volume can be written as V=
(12)
e=
−ΔHdb ΔSdb and f = R R
Equation 18 is used for estimation of density of unsaturated FAME. However, eq 18 will be reduced to eq 14, when there is no double bond in the molecule. 2.3. Density of Liquid Mixture/Biodiesel. Models for calculate of biodiesel are few; Pratas et al.15 and Ramirez− Verduzco et al.1,16 used the Kay’s mixing rule (eq 19) for the calculation of density of biodiesel.
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n
∑ wiρi
ρbiodiesel =
i=1
where xi, zi and nd(i) are the percent composition, carbon numbers, and number of double bonds of fatty acid ester i, respectively. 3.2. Statistical Analysis. Statistical analysis was performed on Microsoft Excel 2010. The average absolute deviation (AAD) was calculated from eq 26.
(19)
where wi is mole fraction or mass fraction of the component (FAME) i. Ramrez−Verduzco et al.1 showed that the difference of using mole or mass fraction for predicting density of biodiesel was marginal. For biodiesel, the free energy of volumetric expansion (ΔGbiodiesel) is the sum of free energies of volumetric expansion of individual FAME contributed to the mixture. Thus, eq 20 is obtained by summation of all the free energies contribution from different FAMEs in the mixture.
n
AAD =
i=1
∑ yi ΔGi i=1
ρlit
N (26)
4. RESULTS AND DISCUSSION Generally, the linear free energy relationship in eq 11 is limited to a narrow range of carbon numbers, and the lowest number is about 5,20 but there is no report on the relationship between free energy of volumetric expansion and carbon numbers of FAME. Hence, it is safer to separate FAMEs into two groups, C6−C12 (short chain) and C12−C24 (long chain). 4.1. Density of Saturated FAMEs. Using density data of Liew et al.22 (C6−C12) and Pratas et al.23 (C12−C24) the numeric values of a, b, c, and d for short chain saturated FAMEs are −0.546, −0.0078, 128.7, and −3.093, and long chain saturated FAMEs are −0.436, −0.0025, 85.98, and 0.792, respectively. Substitution these numeric values into eq 14, eq 27, and eq 28 are obtained, and they are the numeric form of eq 14 for estimation the density of short and long chain saturated FAMEs, respectively. 128.7 3.093z ln ρ = −0.546 − 0.0078z + − (27) T T
(20)
where yi is the mole or mass fraction of component (FAME) i, ΔGi is the free energy of volumetric expansion of component (FAME) i. Alternatively, density of a biodiesel can be estimated by eq 18 if the carbon numbers and number of double bonds of fatty acids in biodiesel are average and the mixture is arbitrarily treated as a single FAME.
3. METHODOLOGY The density values of pure FAMEs and biodiesels are obtained from Liew et al.22 and Pratas et al.15,23 Liew et al.22 determined density of FAMEs with a 10 mL flask-type pycnometer, while Pratas et al.23 measured with an automated SVM 3000 Anton Paar rotational Stabinger viscometer-densimeter. The absolute uncertainty of was 0.0005 g·cm−3. All the four numeric constants of eq 14 for saturated fatty acid esters were solved according to Krisnangkura et al.,24 and they are briefly described below. At constant T, eq 14 is reduced to eq 21. ln ρ = a′ + b′z
⎣
⎤ × 100%⎥ ⎥⎦
where ρlit and ρcal are the experimental and predicted density, respectively. N is the number of data points.
n
ΔG biodiesel =
⎡ |ρ − ρ | lit cal
∑ ⎢⎢
85.98 0.792z + (28) T T 22,23 Percent differences between literature and calculated densities of saturated FAMEs (by eqs 27 and 28), at different temperatures are shown in Figure 1. The calculated density ln ρ = −0.435 − 0.0025z +
(21)
where
a′ = a +
c T
(22)
d T
(23)
and
b′ = b +
Thus, a and c are the intercept and the slope of the a′−1/T plot. Similarly, b and d are the intercept and the slope of the b′−1/T plot. The four numeric values (a, b, c, and d) of for unsaturated FAME (eq 18) are the same as those of eq 14. Two additional constants (e and f) of eq 18 are solved by two simultaneous equations according to Phankosol et al.21 3.1. Determination of Average Carbon Numbers and Number of Double Bonds. Biodiesel is the mixture of FAMEs and its density can be directly estimated without a prior knowledge of individual ester when the average carbon number (zave) and the average number of double bonds (nd(ave)) are used in place of z and nd. The zave and nd(ave) are calculated according to eq 24 and eq 25, respectively.
Figure 1. Relative deviations (%Δ) between densities of saturated FAMEs predicted by eq 27 (short chain), eq 28 (long chain), and the experimental values reported by Liew et al.22 and Pratas et al.23
values for both short and long chain saturated FAMEs at 10− 100 °C are in good agreement with the literature values. The AAD was 0.11%. The highest and lowest absolute percent differences are 0.43 and 0.00, respectively. Methyl dodecanoate (C12:0) estimated by eqs 27 and 28 gives similar AAD of 0.13%. The plot between the calculated (cal) and literature (lit)) values is linear with the intercept, slope, R2 and standard
n
zave =
∑i = 1 xizi n
∑i − 1 xi
(24)
n
nd(ave) =
∑i = 1 xind(i) n
∑i − 1 xi
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error of 0.0081, 0.9907, 0.9965, and 0.0018, respectively (Figure 2).
ln ρ = −0.435 − 0.0025z + − 0.00001nd +
85.98 0.792z + T T
4.0nd T
(29)
Densities of unsaturated FAMEs are summarized in Table 1. The differences between the literature and calculated values are listed in the parentheses. The calculated densities for unsaturated FAMEs at 283.15−353.15 K are in good agreement with the literature values. The AAD is 0.68% and the highest absolute difference is 2.24% (methyl erucate at 5 °C). The plot between ρcal and ρlit for unsaturated FAMEs is linear with a slope, intercept, R2 and the standard error of 1.104, −0.085, 0.947, and 0.0058, respectively. Methyl palmitoleate showed high differences from literature values23 at all temperatures. 4.3. Density of Biodiesel. Biodiesels are mixture of FAMEs prepared from different fats and oils. Thus, densities of biodiesels vary according to the sources or fatty acid compositions. Fatty acid compositions of different biodiesels are summarized in Table 2. 4.3.1. Estimation of Density of Biodiesel from zave and nd(ave). As it was pointed out in the above section that eq 29 could be used for estimation of density of saturated FAME, unsaturated FAME and biodiesel. In the case of biodiesel, there are two alternatives. The first method is the density of individual FAME is calculated by eq 29 and summed by the Kay’s mixing rule (eq 19). The second method is by treating biodiesel as a single FAME having the average carbon numbers and number of double bonds of zave and nd(ave), respectively. Subsequently, density of the biodiesel is estimated by eq 29. It was found that densities of biodiesels estimated by both methods are statistically insignificant. In addition, the numeric value for coefficient e is very small (−0.00001) and the nd(ave) of most biodiesels are not greater than 2. Therefore, the product of e × nd(ave) can be neglected without affecting the accuracy of
Figure 2. Correlation of the estimated densities (ρcal) to the literature values (ρlit) of saturated FAMEs (C6:0−C24:0) at 10−100 °C.
4.2. Density of Unsaturated FAMEs. Short chain fatty acids are minor components found in naturally occurring fats and oils, and they are not suitable for production of biodiesel. Long chain fatty acids are much more abundant in nature, and they are feed stock for biodiesel. In addition, long chain fatty acids occurred in nature are both saturated and unsaturated with different number of double bonds. Thus, eq 18 was derived for unsaturated FAMEs, but it can be used for saturated FAMEs as well. The numeric constants for eq 18 are listed in eq 29. The first four numeric values (a, b, c, and d) are the same as those of eq 28. Therefore, eq 29 is reduced to eq 28 if nd is zero.
Table 1. Estimated Densities (g/cm3) of Unsaturated FAMEs at 5−100 °Ca T (K)
C16:1b
C18:1
C18:2
C18:3
C20:1
C22:1
278.15 283.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 328.15 333.15 338.15 343.15 348.15 353.15 358.15 363.15 368.15 373.15 AAD
0.899 (−2.17) 0.894 (−1.94) 0.888 (−1.74) 0.883 (−1.57) 0.877 (−1.42) 0.873 (−1.29) 0.868 (−1.18) 0.863 (−1.09) 0.859 (−1.02) 0.854 (−0.96) 0.850 (−0.92) 0.846 (−0.91) 0.842 (−0.91) 0.839 (−0.92) 0.835 (−0.95) 0.832 (−0.99) 0.828 (−1.04) 0.825 (−1.11) 0.822 0.819 (−1.23)
0.900 0.894 (−1.45) 0.888 (−1.23) 0.883 (−1.02) 0.878 (−0.84) 0.873 (−0.68) 0.868 (−0.55) 0.863 (−0.43) 0.859 (−0.32) 0.854 (−0.24) 0.850 (−0.17) 0.846 (−0.12) 0.842 (−0.08) 0.838 (−0.06) 0.835 (−0.06) 0.831 (−0.06) 0.828 0.825 0.821 0.818 (−0.49)
0.913 (−1.78) 0.907 (−1.50) 0.901 (−1.23) 0.895 (−1.01) 0.890 (−0.81) 0.884 (−0.63) 0.879 (−0.46) 0.874 (−0.32) 0.870 (−0.19) 0.865 (−0.08) 0.861 (0.02) 0.856 (0.08) 0.852 (0.14) 0.848 (0.18) 0.844 (0.20) 0.841 (0.21) 0.837 (0.19) 0.834 (0.17) 0.830 0.827 (−0.38)
0.926 (−1.44) 0.920 (−1.13) 0.913 (−0.86) 0.907 (−0.61) 0.902 (−0.38) 0.896 (−0.17) 0.891 (0.00) 0.886 (0.17) 0.881 (0.31) 0.876 (0.44) 0.871 (0.56) 0.867 (0.65) 0.862 (0.72) 0.858 (0.78) 0.854 (0.82) 0.850 (0.85) 0.847 (0.86) 0.843 (0.87) 0.839 0.836 (0.13)
0.901 (−1.80) 0.895 (−1.54) 0.889) (−1.30) 0.883 (−1.09) 0.878) (−0.90) 0.873 (−0.73) 0.868) (−0.58) 0.863 (−0.43) 0.859) (−0.32) 0.854 (−0.23) 0.850 (−0.14) 0.846 (−0.08) 0.842 (−0.03) 0.838 (0.01) 0.834 (0.02) 0.831 (0.03) 0.827 (0.03) 0.824 (0.01) 0.821 (0.00) 0.818 (−0.03) (−0.45)
0.901 (−2.24) 0.895 (−1.97) 0.889 (−1.72) 0.884 (−1.49) 0.878 (−1.29) 0.873 (−1.10) 0.868 (−0.94) 0.863 (−0.79) 0.859 (−0.66) 0.854 (−0.56) 0.850 (−0.47) 0.846 (−0.39) 0.842 (−0.32) 0.838 (−0.27) 0.834 (−0.24) 0.830 (−0.22) 0.827 (−0.21) 0.823 (−0.21) 0.820 0.817 (−0.84)
a
Numbers in parentheses are percent differences from literature values. Density data were reported by Pratas et al.15 bCz:nd, fatty acid (methyl ester) of z carbon atoms of fatty acid with nd double bonds. 4636
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Table 2. Fatty Acid Composition of Different Fats and Oils fats and oils (methyl esters) 2
palm sunflower25 coconut26 colza26 soybean26 coconut + colza (0.4974 + 0.5026)a,26 coconut + colza (0.6952 + 0.3048)26 coconut + colza (0.9017 + 0.0983)26 cotton27 babassu27 cotton + babassu (0.299 + 0.701)27 cotton + babassu (0.684 + 0.316)27 cotton + babassu (0.900 + 0.100)27 soybean + babassu (0.404 + 0.596)27 soybean + babassu (0.605 + 0.395)27 soybean + babassu (0.901 + 0.099)27 rapeseed28 soybean + rapeseed29 palm + rapeseed29 soybean + palm29 soybean + rapeseed + palm29 soybean + sunflower29 a
C8:0b C10:0
C12:0
C14:0
C16:0
C16:1
C18:0
C18:1
C18:2
C18:3
C20:0
C20:1
C22:0
C22:1
C24:0
0.00 0.00 4.08 0.00 0.00 2.03
0.00 0.00 3.65 0.00 0.00 1.82
0.00 0.00 35.35 0.00 0.00 17.58
0.00 0.00 19.84 0.00 0.00 9.87
41.50 7.10 13.83 3.99 11.32 8.88
0.00 0.00 0.00 0.00 0.00 0.00
4.90 4.80 3.94 3.91 0.00 3.92
40.10 22.60 14.30 56.67 25.68 35.60
13.50 65.50 4.73 23.61 54.94 14.22
0.00 0.00 0.00 9.88 8.07 4.97
0.00 0.00 0.00 1.94 0.00 0.98
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00
2.84
2.54
24.58
13.79
10.83
0.00
3.93
27.21
10.48
3.01
0.59
0.00
0.00
0.00
0.00
3.68
3.29
31.88
17.89
12.86
0.00
3.94
18.46
6.59
0.97
0.19
0.00
0.00
0.00
0.00
0.00 0.00 0.00
0.00 5.10 3.58
0.00 28.11 19.71
0.62 25.56 18.10
24.09 15.41 18.01
0.00 0.00 0.00
2.56 5.04 4.30
15.74 20.79 19.28
56.99 0.00 17.04
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00
1.61
8.88
8.50
21.35
0.00
3.34
17.34
38.98
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.51
2.81
3.11
23.22
0.00
2.81
16.25
51.29
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3.04
16.75
15.23
13.75
0.00
4.60
20.46
23.59
2.37
0.00
0.00
0.21
0.00
0.00
0.00
2.01
11.10
10.10
12.92
0.00
4.39
20.30
35.33
3.55
0.00
0.00
0.31
0.00
0.00
0.00
0.01
0.03
0.03
1.13
0.00
0.40
2.00
5.78
0.58
0.00
0.00
0.05
0.00
0.00
0.00 0.00 0.00 0.00 0.00
0.02 0.00 0.02 0.01 0.01
0.03 0.03 0.20 0.18 0.14
0.83 0.09 0.54 0.01 0.38
3.82 8.90 23.09 25.56 18.97
0.12 0.15 0.17 0.11 0.14
1.77 2.76 3.02 4.04 3.28
58.33 41.82 52.92 33.13 42.51
21.27 37.51 15.47 31.72 27.93
11.22 7.02 3.08 3.58 4.66
0.55 0.46 0.49 0.39 0.45
0.67 0.68 0.67 0.20 0.52
1.15 0.46 0.24 0.32 0.33
0.22 0.12 0.09 0.12 0.14
0.00 0.00 0.00 0.63 0.53
0.00
0.00
0.02
0.07
6.40
0.09
4.22
23.90
64.16
0.12
0.03
0.15
0.76
0.08
0.00
b
Numbers in parentheses are mass fractions. Cz:nd, fatty acid (methyl ester) of z carbon atoms of fatty acid with nd double bonds.
tension of mixed biodiesels.21 The model cannot differentiate a mixed biodiesel from pure biodiesels. Once they are mixed, the model would treat it as a single and pure biodiesel with a different fatty acid composition or a new biodiesel. Thus, impurity in one biodiesel would be diluted or percent deviation was average. This would be considered as an advantage for the model that density of a FAME, a pure or a mixed biodiesel at different temperatures can be estimated by the same equation. In addition, for a biodiesel, it does not require a prior knowledge of the density of individual fatty acid methyl esters. The zave and nd(ave) estimated according to eq 30 for each biodiesel are also included in Table 3. Among the oils collected in this work, the fatty acids of biodiesels in this work are mainly 16 and 18 carbon atoms with some minor components of shorter and longer carbon chain length, the range and the average of zave and nd(ave) are 14.10−17.96 and 0.21−1.60, respectively. However, beside temperature, the density of a biodiesel varies with fatty acid chain length, Pratas et al.23 pointed out that densities of FAMEs decreased with increasing chain length and increased with the degree of unsaturation. 4.3.2. Estimation of Density of Biodiesel from SN and IV. Although eq 30 is a good and simple model for estimation of biodiesel at different temperature, determination of the z(ave) and average nd(ave) requires the knowledge of fatty acid composition. The analysis must be done with a GC or a HPLC. Saponification number (SN) and Iodine value (IV) have long been used for characterization of fat and oil and they require no special instrumentation. However, both SN and IV
the calculation and eq 30 is good for estimation of density of biodiesel. ln ρ = −0.435 − 0.0025zave + +
0.792z(ave) 85.98 + T T
4.0nd(ave) T
(30)
Hence, only the densities determined by using the zave and of nd(ave) are summarized in Table 3. Experimental values (from literatures) and those estimated by using Ramirez-Verduzco16 model (eq 5) are included for comparison. Figure 3 is the correlation between the reported and estimated values (104 data points). The correlation is linear with the slope, intercept, R2 and standard error of 1.079, −0.069, 0.982, and 0.0023, respectively. The maximum absolute deviation was 1.22% (for soybean + babassu mass ratio 0.901:0.099). The AAD was 0.39%. The %AAD was lower than that estimated by RamirezVerduzco’s eq (0.45%). When more data points were analyzed (380 data points), similar results were obtained. The correlation is linear with the slope, intercept, R2 and standard error of 1.101, −0.086, 0.971, and 0.001, respectively. The maximum absolute deviation was 1.61% (for Jatropha). The AAD was 0.36%. The %AAD was lower than that estimated by Ramires-Verduzco’s eq (0.41%). Interestingly, the %AAD for mixed biodiesel (0.38) is lower than those of pure (0.41%) and total biodiesels. This may not be irregular. The phenomena was also observed for the surface 4637
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Table 3. Estimated Densities of Biodiesel from zave and nd(ave) (eq 30), SN and IV (eq 34), and (eqs 5 + 19) at Different Temperatures T (K)
eq 30
eq 34
eq 5 + 19
exp
%Δ1b
%Δ2c
%Δ3d
palm zave = 17.17 nd(ave) = 0.67 SN = 196.53 IV = 57.60
293.15 313.15 333.15 353.15
0.8789 0.8595 0.8428 0.8283
0.8803 0.8608 0.8441 0.8295
0.8719 0.8571 0.8423 0.8274
0.8726 0.8579 0.8434 0.8288 %AAD
−0.72 −0.19 0.06 0.06 0.26
−0.88 −0.35 −0.08 −0.08 0.35
0.08 0.09 0.13 0.16 0.12
sunflower25 zave = 17.86 nd(ave) = 1.54 SN = 190.95 IV = 132.25
293.15 313.15 333.15 353.15 373.15
0.8895 0.8691 0.8516 0.8363 0.8229
0.8937 0.8730 0.8552 0.8397 0.8261
0.8813 0.8665 0.8517 0.8369 0.8220
0.8830 0.8683 0.8537 0.8389 0.8247 %AAD
−0.73 −0.09 0.25 0.31 0.21 0.32
−1.22 −0.54 −0.18 −0.10 −0.17 0.44
0.19 0.21 0.24 0.24 0.32 0.24
coconut26 zave = 14.10 nd(ave) = 0.24 SN = 236.29 IV = 20.45
293.15 313.15 333.15 353.15 373.15
0.8732 0.8547 0.8388 0.8249 0.8127
0.8729 0.8545 0.8387 0.8249 0.8127
0.8696 0.8547 0.8399 0.8251 0.8103
0.8709 0.8555 0.8402 0.8246 0.8093 %AAD
−0.26 0.09 0.17 −0.04 −0.42 0.20
−0.23 0.11 0.18 −0.03 −0.42 0.20
0.15 0.09 0.03 −0.06 −0.12 0.09
Colza26 zave = 17.96 nd(ave) = 1.34 SN = 189.78 IV = 114.93
293.15 313.15 333.15 353.15 373.15
0.8870 0.8669 0.8495 0.8344 0.8211
0.8908 0.8704 0.8527 0.8374 0.8240
0.8790 0.8642 0.8493 0.8345 0.8197
0.8846 0.8701 0.8556 0.8412 0.8268 %AAD
−0.28 0.37 0.71 0.81 0.68 0.57
−0.71 −0.03 0.33 0.45 0.34 0.37
0.64 0.68 0.73 0.79 0.86 0.74
soybean26 zave = 17.77 nd(ave) = 1.60 SN = 191.85 IV = 137.68
293.15 313.15 333.15 353.15 373.15
0.8902 0.8698 0.8522 0.8369 0.8235
0.8946 0.8738 0.8559 0.8404 0.8267
0.8821 0.8673 0.8524 0.8376 0.8228
0.8853 0.8707 0.8562 0.8416 0.8272 %AAD
−0.55 0.11 0.47 0.55 0.44 0.42
−1.05 −0.36 0.03 0.14 0.05 0.33
0.36 0.40 0.44 0.47 0.53 0.44
coconut + Colza (0.4974 + 0.5026)a,26 zave = 16.04 nd(ave) = 0.79 SN = 212.91 IV = 67.94
293.15 313.15 333.15 353.15 373.15
0.8801 0.8608 0.8442 0.8297 0.8170
0.8809 0.8616 0.8449 0.8305 0.8177
0.8743 0.8595 0.8447 0.8298 0.8150
0.8779 0.8630 0.8481 0.8332 0.8184 %AAD
−0.25 0.25 0.46 0.42 0.18 0.31
−0.34 0.16 0.37 0.33 0.08 0.26
0.41 0.41 0.41 0.40 0.41 0.41
coconut + Colza (0.6952 + 0.3048)26 zave = 15.27 nd(ave) = 0.57 SN = 222.12 IV = 49.25
293.15 313.15 333.15 353.15 373.15
0.8774 0.8584 0.8420 0.8278 0.8153
0.8776 0.8586 0.8423 0.8281 0.8156
0.8724 0.8576 0.8428 0.8280 0.8131
0.8756 0.8605 0.8452 0.8300 0.8149 %AAD
−0.20 0.24 0.37 0.26 −0.05 0.23
−0.22 0.22 0.34 0.23 −0.09 0.22
0.36 0.34 0.29 0.24 0.22 0.29
coconut + colza (0.9017 + 0.0983)26 zave = 14.48 nd(ave) = 0.35 SN = 231.72 IV = 29.74
293.15 313.15 333.15 353.15 373.15
0.8745 0.8559 0.8398 0.8259 0.8136
0.8744 0.8558 0.8398 0.8259 0.8136
0.8705 0.8557 0.8408 0.8260 0.8112
0.8726 0.8572 0.8419 0.8265 0.8112 %AAD
−0.22 0.15 0.24 0.08 −0.29 0.20
−0.20 0.16 0.25 0.08 −0.30 0.20
0.24 0.18 0.13 0.06 0.00 0.12
cotton27 zave = 17.49 nd(ave) = 1.30 SN = 194.22 IV = 111.71
293.15 313.15 333.15 353.15 373.15
0.8865 0.8664 0.8492 0.8341 0.8210
0.8897 0.8694 0.8519 0.8367 0.8234
0.879 0.864 0.849 0.834 0.820
0.8816 0.8672 0.8528 0.8382 0.8234
−0.56 0.09 0.43 0.48 0.30
−0.92 −0.26 0.10 0.18 0.00
0.31 0.37 0.42 0.46 0.47
biodiesel 2
4638
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Table 3. continued biodiesel
T (K)
eq 30
eq 34
eq 5 + 19
%Δ1b
%Δ2c
%Δ3d
%AAD
0.37
0.29
0.41
exp
Babassu27 zave = 14.58 nd(ave) = 0.21 SN = 228.43 IV = 17.79
293.15 313.15 333.15 353.15 373.15
0.8729 0.8544 0.8384 0.8245 0.8123
0.8728 0.8543 0.8384 0.8245 0.8123
0.8687 0.8539 0.8391 0.8242 0.8094
0.8762 0.8608 0.8454 0.8303 0.8146 %AAD
0.38 0.74 0.82 0.69 0.28 0.58
0.39 0.76 0.83 0.70 0.28 0.59
0.86 0.80 0.75 0.73 0.63 0.75
cotton + babassu (0.299 + 0.701)27 zave = 15.45 nd(ave) = 0.53 SN = 218.20 IV = 45.88
293.15 313.15 333.15 353.15 373.15
0.8769 0.8580 0.8416 0.8274 0.8149
0.8772 0.8583 0.8420 0.8277 0.8152
0.8717 0.8569 0.8421 0.8273 0.8125
0.8780 0.8629 0.8478 0.8327 0.8172 %AAD
0.12 0.57 0.73 0.64 0.28 0.47
0.09 0.53 0.69 0.60 0.24 0.43
0.71 0.69 0.67 0.65 0.58 0.66
cotton + babassu (0.684 + 0.316)27 zave = 16.57 nd(ave) = 0.95 SN = 205.03 IV = 82.03
293.15 313.15 333.15 353.15 373.15
0.8822 0.8626 0.8458 0.8311 0.8182
0.8837 0.8641 0.8471 0.8324 0.8195
0.8756 0.8608 0.8460 0.8312 0.8164
0.8799 0.8653 0.8507 0.8357 0.8203 %AAD
−0.26 0.31 0.58 0.55 0.25 0.39
−0.43 0.14 0.42 0.39 0.10 0.30
0.48 0.52 0.55 0.54 0.48 0.52
cotton + babassu (0.900 + 0.100)27 zave = 17.20 nd(ave) = 1.19 SN = 197.64 IV = 102.32
293.15 313.15 333.15 353.15 373.15
0.8851 0.8652 0.8481 0.8332 0.8201
0.8878 0.8677 0.8504 0.8353 0.8221
0.8778 0.8630 0.8482 0.8334 0.8185
0.8815 0.8668 0.8521 0.8373 0.8227 %AAD
−0.41 0.18 0.47 0.49 0.32 0.37
−0.71 −0.10 0.20 0.24 0.08 0.27
0.42 0.44 0.46 0.47 0.51 0.46
soybean + Babassu (0.404 + 0.596)27 zave = 15.88 nd(ave) = 0.75 SN = 213.56 IV = 64.34
293.15 313.15 333.15 353.15 373.15
0.8796 0.8603 0.8438 0.8293 0.8166
0.8803 0.8611 0.8445 0.8300 0.8173
0.8739 0.8590 0.8442 0.8294 0.8146
soybean + babassu (0.605 + 0.395)27 zave = 16.52 nd(ave) = 1.02 SN = 206.16 IV = 87.50
293.15 313.15 333.15 353.15 373.15
0.8829 0.8633 0.8464 0.8317 0.8188
0.8845 0.8648 0.8478 0.8331 0.8201
0.8764 0.8616 0.8468 0.8320 0.8171
soybean + babassu (0.901 + 0.099)27 zave = 17.48 nd(ave) = 1.41 SN = 195.27 IV = 121.60
293.15 313.15 333.15 353.15 373.15
0.8879 0.8677 0.8503 0.8352 0.8220
0.8913 0.8709 0.8533 0.8380 0.8246
0.8802 0.8654 0.8505 0.8357 0.8209
rapeseed28 zave = 17.96 nd(ave) = 1.36 SN = 189.91 IV = 116.60
293.15 313.15 333.15 353.15 373.15
0.8873 0.8671 0.8497 0.8346 0.8213
0.8911 0.8706 0.8530 0.8376 0.8242
0.8793 0.8644 0.8496 0.8348 0.8200
soybean + rapeseed29 zave = 17.86 nd(ave) = 1.39 SN = 190.85 IV = 119.49
293.15 313.15 333.15 353.15 363.15
0.8877 0.8675 0.8501 0.8349 0.8281
0.8915 0.8710 0.8533 0.8380 0.8310
0.8796 0.8648 0.8500 0.8352 0.8278
palm + rapeseed29
293.15
0.8822
0.8845
0.8748
0.8802 0.8653 0.8503 0.8350 0.8201 %AAD 0.8824 0.8676 0.8530 0.8381 0.8226 %AAD 0.8853 0.8709 0.8563 0.8418 0.8321 %AAD 0.8814 0.8669 0.8524 0.8378 0.8234 %AAD 0.8820 0.8673 0.8528 0.8384 0.8313 %AAD 0.8784
0.07 0.57 0.77 0.68 0.42 0.50 −0.06 0.49 0.77 0.76 0.46 0.51 −0.29 0.37 0.70 0.78 1.22 0.67 −0.66 −0.03 0.31 0.39 0.25 0.33 −0.64 −0.02 0.32 0.41 0.39 0.36 −0.43
−0.01 0.49 0.69 0.60 0.34 0.43 −0.24 0.32 0.61 0.60 0.31 0.42 −0.68 0.00 0.35 0.45 0.91 0.48 −1.10 −0.43 −0.07 0.02 −0.09 0.34 −1.08 −0.43 −0.06 0.05 0.03 0.33 −0.70
0.72 0.72 0.72 0.67 0.67 0.70 0.68 0.69 0.73 0.73 0.66 0.70 0.58 0.64 0.67 0.72 1.35 0.79 0.25 0.28 0.32 0.36 0.41 0.32 0.27 0.28 0.33 0.38 0.42 0.34 0.42
4639
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Table 3. continued T (K)
eq 30
eq 34
eq 5 + 19
exp
%Δ1b
%Δ2c
%Δ3d
zave = 17.54 nd(ave) = 0.94 SN = 193.45 IV = 80.73
313.15 333.15 353.15 363.15
0.8625 0.8455 0.8308 0.8241
0.8646 0.8475 0.8326 0.8259
0.8599 0.8451 0.8303 0.8229
soybean + palm29 zave = 17.54 nd(ave) = 1.08 SN = 193.64 IV = 92.70
293.15 313.15 333.15 353.15 363.15
0.8839 0.8640 0.8469 0.8321 0.8253
0.8865 0.8665 0.8492 0.8342 0.8274
0.8763 0.8615 0.8467 0.8319 0.8244
soybean + rapeseed + palm29 zave = 17.66 nd(ave) = 1.13 SN = 192.53 IV = 97.31
293.15 313.15 333.15 353.15 363.15
0.8845 0.8646 0.8475 0.8326 0.8258
0.8875 0.8673 0.8500 0.8349 0.8280
0.8769 0.8620 0.8472 0.8324 0.8250
soybean + sunflower29 zave = 17.90 nd(ave) = 1.53 SN = 190.59 IV = 131.63
293.15 313.15 333.15 353.15 363.15
0.8894 0.8690 0.8515 0.8363 0.8294
0.8937 0.8730 0.8551 0.8396 0.8326
0.8812 0.8664 0.8516 0.8368 0.8294
0.8637 0.8492 0.8346 0.8273 %AAD 0.8782 0.8635 0.8490 0.8345 0.8274 %AAD 0.8793 0.8646 0.8500 0.8356 0.8284 %AAD 0.8835 0.8689 0.8544 0.8399 0.8328 %AAD overall
0.14 0.43 0.46 0.39 0.37 −0.64 −0.06 0.24 0.29 0.25 0.30 −0.59 0.00 0.30 0.37 0.31 0.31 −0.67 −0.01 0.34 0.43 0.41 0.37 0.39
−0.11 0.20 0.24 0.18 0.28 −0.95 −0.35 −0.03 0.04 0.00 0.27 −0.93 −0.31 0.00 0.09 0.04 0.28 −1.15 −0.47 −0.09 0.03 0.02 0.35 0.34
0.44 0.48 0.52 0.53 0.48 0.21 0.23 0.27 0.32 0.36 0.28 0.28 0.30 0.33 0.38 0.41 0.34 0.26 0.29 0.33 0.37 0.41 0.33 0.44
biodiesel
a
Numbers in parentheses are mass fractions. b= 100 × (eq 30 − exp)/exp. c= 100 × (eq 34 − exp)/exp. d= 100 × (eqs 5 + 19 − exp)/exp.
ln ρ = −0.427 −
10 83.38 3168.95 11 × IV + + + SN T T × SN T × SN (34)
The densities of pure, mixed, and total biodiesels estimated by eq 34 (at different temperatures) were very close to those calculated using the zave and nd(ave) (eq 34). The maximum and average percent deviations were 1.22 (for sunflower) and 0.34%, respectively. The slight differences in the calculated values between eq 30 and eq 34 are due to the conversion equation in eq 31 and eq 32, which are approximate. The correlation between the reported density and estimated values using eq 34 (104 data points) is linear with the slope, intercept, R2 and standard error of 1.099, −0.085, 0.980, and 0.002, respectively.
Figure 3. Correlation of the estimated densities (ρcal) to the literature values (ρitl) of biodiesels at 20−100 °C.
5. CONCLUSIONS This work provides an empirical correlation of density of a liquid to its chemical structure or chemical composition at different temperatures (10−100 °C). Hence, density of a biodiesel can be predicted either from the zave and nd(ave) (eq 29) of fatty acids or from the SN and IV (eq 34) with approximately the same accuracy. Besides the accuracy, both models provide two additional advantages: (1) density of biodiesel can be estimated without a prior knowledge of the densities of individual FAMEs, and (2) all the coefficients of the equations are well-defined. This would allow further refinement of the models.
can also be calculated from fatty acid composition.30 In addition these two values have been correlated to cetane index31 and heat of combustion32 of biodiesel. Thus, it would be convenient if the density of biodiesel can be estimated from these two values. The nd and z can be converted to IV and SN according to eqs 31−33. nd =
IV × M 25400
z=
M + 2nd − 46 14
M=
56000 SN
(31)
■
(32)
ASSOCIATED CONTENT
* Supporting Information S
(33)
Estimated densities of biodiesel from zave and nd(ave) eq 30, SN, and IV (eq 34) and eqs 5 + 19 at different temperatures. This material is available free of charge via the Internet at http:// pubs.acs.org/.
Equation 34 is obtained by combining eqs 30, 31, 32, eq 33, and it can be used for estimation of density of biodiesel at different temperatures from the SN and IV value. 4640
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was financially supported by National Research University Project of Thailand’s Office of the Higher Education Commission and Energy Planning and Policy Office.
■
REFERENCES
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