Methyl Ester Composition and Other Factors
6.14 Prediction of Density
Density can be defined as the mass of an object divided by its volume. Fuel
density is a key property that affects engine performance.36 Because fuel-
injection pumps meter fuel by volume, not by mass, a greater or lesser mass of fuel is injected depending upon its density. Thus, the air-to-fuel ratio and energy content within the combustion chamber are influenced by fuel density. The densities of biodiesel fuels are slightly higher than those of petroleum diesel, and increasing the biodiesel level of biodiesel blends increases the blend’s density.
In the work by Alptekin and Canakci, two different commercially available diesel fuels were blended with the biodiesels produced from six different vegetable oils (sunflower, canola, soybean, cotton seed, corn oils and waste
palm oil).14The blends (B2, B5, B10, B20, B50 and B75) were prepared on a
volume basis. The key fuel properties such as density and viscosities of the blends were measured by following ASTM test methods. Eqn (58) fits the data sufficiently well that higher degree equations are not required:
D~AxzB ð58Þ
where D is density (g cm23), A and B are coefficients and x is the biodiesel
fraction. The calculated density values from eqn (58) were validated by using the measured density values for all the blends. There is strong agreement between the measured and estimated values. The maximum absolute error
between the measured and estimated values is 0.42% and the minimum R2is
0.9984 for biodiesel–N.D fuel blends. The maximum difference between the
measured values and estimated values and the minimum R2 are 0.20% and
0.9996 for biodiesel–S.E fuel blends, respectively.
The model developed in MATLAB was also used to predict the densities of
biodiesels.25
A traditional statistical technique of linear regression (PLS) was used to
estimate the density of diesel and biodiesel mixtures.23A set of seven neural
network architectures, three training algorithms along with 10 different sets of weights and biases were examined to choose the ANN to best predict the density. The performances of both of the traditional linear regression and ANN techniques were then compared to check their validity to predict the properties of various mixtures of diesel and biodiesel.
In another study, near-infrared spectroscopy was used to determine the
density at 15 uC.15
Principal component analysis was used to perform a qualitative analysis of the spectra, and PLS regression was used to develop the calibration models between analytical and spectral data. The results support the notion that near-infrared spectroscopy, in combination with multivariate calibration, is a promising technique to be applied to biodiesel quality control, in both laboratory- and industrial-scale samples. An accurate knowledge of biodiesel density permits the estimation of other properties such as the CN, whose direct measurement is complex and presents low repeatability and low
reproducibility.46In this study densities of methyl and ethyl esters published in the literature, were complied and equations were proposed to convert them to
15 uC and to predict the biodiesel density based on its chain length and
unsaturation degree. Calculations also proved that the introduction of high- biodiesel-content blends in the fuel market would force the refineries to reduce the density of their fossil fuels. Most of the density data obtained from the
literature were measured at 20, 25 and 40uC. Three options have been studied
for converting all data to 15uC (because this is the temperature specified in the
biodiesel standards). In order to consider the effect of temperature on the
density of alkyl esters, the following equation was proposed:46
D~DðTÞ 0:29056{0:08775vð Þ 1{T=Tð cÞ2=7{ 1{288:15=Tð cÞ2=7
h i
ð59Þ
where D is determined at 288.15 K. D(T) and Tc is temperature and Tc is
critical temperature inuC and v is acentric factor. However, this equation was
originally proposed for a saturated liquid over a wide temperature range (up to the critical temperature); therefore its application to a liquid at a lower temperature than its saturation temperature (at a given pressure) is not correct.
Moreover, in the range 0–50 uC the equation underestimates the decrease of
the density with temperature compared to that derived from experimental density values of different methyl esters. Second, a linear correlation was proposed specifically for biodiesel fuels in EN 14214 in 2009. This is given as:
D~DðTÞz0:723 T{15ð Þ ð60Þ
where D is determined at 15uC. However, this is being currently revised by the
European Committee for Standardization (Working Group 24) since it overestimates the decrease with temperature and does not account for the different nature of the esters like the type of alkyl ester, the number of double bonds and the chain length. Third, considering the shortcomings of the
aforementioned methods, a new equation [eqn (61)] has been developed.46
Similar to eqn (60) a linear dependence of the density and temperature is considered.
Therefore, the proposed equation is linear:
D~DðTÞza T{15ð Þ ð61Þ
where D is determined at 15uC. The coefficient a is dependent on the type of
ester. The final correlation proposed here is a five-coefficient function which depends on the number of carbon atoms in the original fatty acid (n), the number of carbon atoms in the original alcohol used for the transesterification process (m) and the number of double bonds in the acid molecule (db):
D~851:471z 250:718dbz280:899{921:180 m{1½ ð Þ=1:214zn ð62Þ
The fit of this equation to the collected data provides a correlation
coefficient R25 0.969.
In similar work, the aim was to present new density data for different biodiesels and use the reported data to evaluate the predictive capability of
models previously proposed to predict biodiesel or FAME densities.67Densities
were measured for 10 biodiesel samples, for which the detailed composition was reported, at atmospheric pressure and temperatures from 278.15 to 373.15 K. The density dependence with temperature was proposed for the biodiesels, and isobaric expansivities were presented. It was shown that Kay’s mixing rules and a revised form of the group-contribution volume (GCVOL) model are able to
predict biodiesel densities with average deviations of only 0.3%.67,68 It was
shown that it can predict the densities of biodiesel fuels with average deviations less than 0.4%. Kay’s mixing rules are the simplest form of mixing rules by which mixture properties are obtained by summing the products of the component properties using weighting factors, which are usually the concentrations of the components in a mixture. For example:
D~Xmicipi ð63Þ
where ci is the concentration of component i. The major drawback in the
application of linear mixing rules is that they require knowledge of the experimental densities of the pure components present in the mixture and assume that the mixture excess volumes are negligible. This may not be easy for many real fluids because they are either composed of a large number of compounds or have different natures, and subsequently, the excess volumes are non-negligible. However, biodiesels are simple mixtures composed, in general, of fewer than 10 fatty acid esters all from the same family, and thus, excess volumes are very small. The density data measured here were correlated using a linear temperature dependency and an optimization algorithm based on the least- squares method:
D~bTza ð64Þ
where T is measured in K, and the parameter values along with their confidence limits were estimated.
In another study, to predict the density, a mixing rule was evaluated as a
function of the volume fraction of biodiesel in the blend.69The effects of the
biodiesel fraction on each of these properties in addition to the effects of temperature on the density and viscosity were investigated. The blends (B2, B5, B10, B20, B50 and B75) were prepared on a volume basis. Generalized equations and the Arrhenius equation for predicting the density of the blends were used. The low values of the AADs and the maximum absolute deviations obtained confirmed the suitability of the mixing rule used. For all the blends, it was observed that the results from the measured and estimated values of density and viscosities were in good agreement. From the results, the density of
the blends decreased with increasing temperature while these properties increased with increasing fraction of biodiesel in the fuel blend. The measured density of the biodiesels and their blends with commercial grade No. 2 diesel was correlated as a function of biodiesel fraction and temperature, respectively using the linear square method. The linear regression equation formulated is as follows:
D~a T zbð Þ ð65Þ
where T is the temperature (uC), D is the density (kg m23
) and a and b are correlation coefficients.
Biodiesel–diesel fuel mixtures have been commonly used in recent years especially in the transport industry, in order to reduce environment pollution,
and dependency on imported fossil fuels.31 Some of the basic properties of
these mixtures, especially density and viscosity, strongly influence spray properties, atomization and combustion processes, engine deposits formation, engine behavior in cold weather conditions, and are used as input data for predictive engine combustion models. In this study, pseudo-binary biodiesel– diesel fuel and biodiesel–benzene mixtures were prepared and the densities, viscosities, and refractive indices of mixtures were measured at 298.15 K. The accuracy of the different mixing rules and empirical equations used to estimate these properties was evaluated. The density of the studied mixtures could be predicted with very good accuracy using Kay’s mixing rule or empirical equations obtained from regression analysis. Density measurements were carried out at atmospheric pressure and 298.15 K according to the ASTM D- 4052 test method and using an Anton Paar DM4500 density meter.
The Kay equation is:
Dm~v1D1zv2D2 ð66Þ
where Dmis the density of the mixture (g cm23), D1and D2are the densities of
components 1 and 2 of the mixture (g cm23), and v1 and v2 are the volume
fractions of components 1 and 2. An increased accuracy of density correlation can be obtained using empirical polynomial equations obtained from regression analysis of the measured values:
Dm~av1zb ð67Þ
Dm~av2zbv1zc ð68Þ
where a, b and c are the regression coefficients.
Using eqn (67), the densities for mixtures of seven types of biodiesels with diesel fuel were predicted, and errors were in the range 0–2%. The experimental data for pseudo-binary mixtures of rapeseed oil biodiesel–diesel fuel were
correlated as a function of biodiesel volume fraction in order to determine the values of regression coefficients for the first-order equation, eqn (67). Regression analysis was performed to determine the equation which best fits the experimental data in the case of rapeseed oil biodiesel–benzene mixtures. For this system, eqn (68) was used because it gave better results than eqn (67). The experimental data were correlated with the empirical eqns (67) and (68).
The fitting coefficients from eqn (67) were a 5 0.0388 and b 5 0.8376 (R25
0.9997), and the fitting coefficients from eqn (68) used to calculate the density
of biodiesel–benzene mixtures were a 5 0.0024, b 50.0008 and c 5 0.8736 (R2
5 0.9971). The correlation between density and composition given by the
empirical eqns (67) and (68) is very high. The calculated density of the two pseudo-binary mixtures from the densities of the mixture components, and the mixture compositions had a good accuracy: 0.012 and 0.014% for biodiesel– diesel fuel mixtures and 0.022 and 0.003% for biodiesel–benzene mixtures in terms of AAD. It may be noted that for the biodiesel–diesel fuel system, only two parameters were sufficient, while for the biodiesel–benzene system, three parameters were needed for density correlation.
Another empirical correlation was proposed to estimate the density:32
D~aV zbTzd ð69Þ
where T is the temperature in K, and b and d are adjustable parameters. The estimated values of the density were in good agreement with the experimental data and absolute average prediction errors of 0.02 and 2.10%
were obtained for the biodiesel(1)+ ULSD(2) system studied in this work.
PLS models using near- and mid-infrared spectrometry were developed to
predict the density of diesel–biodiesel blends.37 Practical aspects were
discussed, such as calibration set composition; model efficiency using different infrared regions and spectrometers; and the calibration transfer problem. The root-mean-square errors of prediction, employing both regions and equipment, were comparable with the reproducibility of the corresponding standard method for the properties investigated. Calibration transfer between the two instruments, using DS, yielded prediction errors comparable to those obtained with complete recalibration of the secondary instrument.
Density is considered to one of the important properties that affects the utilization of biodiesel fuels, because it is involved in the definition of fuel
quality and required as input data for predictive engine combustion models.35
In this work, the researchers presents the characterization of two biodiesel samples made from beef tallow and soybean oil through their FAME profile. Empirical equations were developed to estimate the density of the methyl esters; and an AAD of 0.11% density was obtained. The density decreases as the molecular weight increases and density increases as the degree of unsaturation increases. Two general mixing rules and five biodiesel samples were used to study the influence of FAMEs over the physical properties of biodiesel. The prediction of the density of biodiesels was very close to the
experimental values. The expression for the density of saturated and unsaturated FAMEs is:
Di~0:8463z4:9=Miz0:018N ð70Þ
where Diis the density at 20uC of the ith FAME in g cm23, Mirepresents the
molecular weight of the ith FAME, and N is the number of double bonds in a given FAME, therefore N is equal to zero, one, two and three for the methyl esters C18 : 0, C18 : 1, C18 : 2 and C18 : 3, respectively.
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