Determination of Cetane Number from Fatty Acid Compositions and Structures of Biodiesel

Abstract

Biodiesel, which possesses the dominant advantages of low emissions and environmental friendliness, is a competitive alternative fuel to petroleum-derived diesel. The cetane number, which indicates ignition delay characteristics, is considered the most significant fuel property of biodiesel. Determining the cetane number for biodiesel by general testing equipment is time-consuming and costly; hence, a simple and convenient predictive formula for the cetane number of biodiesel is a significant task to be carried out. A reliable and convenient predictive method for determining the cetane number is proposed in this study. The key parameters for the cetane number of biodiesel were first screened out. The analysis of multiple linear regressions using the available software SPSS for statistical analysis was carried out to obtain the regression coefficients of those key parameters and intercepts to establish the predictive model. Other available experimental data verified the validity of the proposed predictive equation. The determination coefficient of the formula reaches as high as 94.7%, and the standard error is 3.486. The key parameters, including the number of carbon atoms (NC), allylic position equivalent (APE), and double-bond equivalent (DBE), were more significant for influencing the cetane number of biodiesel. In addition, the increase of NC or the decrease of either APE or DBE results in the increase of the cetane number. Moreover, the present formula is found to obtain closer cetane numbers to those experimental data and features superior prediction capability.

1. Introduction

Biodiesel, also called fatty acid methyl esters (FAME) or fatty acid ethyl esters (FAEE), is produced from the transesterification reaction of vegetable oils, animal fats, or algae lipids, with short-chain alcohol catalyzed by a strong alkaline or acid catalyst or enzyme. While the fuel properties of biodiesel are similar to those of petro-derived diesel, no contents of polyaromatic hydrocarbons (PAHs) or nitrated-PAHs exist in biodiesel. Thus, the relatively low emissions of sulfur oxides (SOx), carbon dioxide (CO₂), unburnt hydrocarbons (UHC), particulate matters (PM), black smoke, etc., from burning biodiesel are primarily due to its oxygenated composition [1,2] and simple lipid structure. Biodiesel, which contains nearly no sulfur content, is a competitive alternative fuel to current heavy fuel oil to comply with the regulation of the IMO (International Maritime Organization) of a sulfur content lower than 0.5 wt.%, as enforced from January 2020 [3]. In addition, although replacing low-sulfur fuel oil as engine fuel would eliminate SOx emissions, the accompanied reduction of fuel lubricity with decreased fuel sulfur would cause abnormal wear and even breakdown of moving engine parts [4]. The addition of 1–2 wt.% biodiesel to heavy fuel oil could significantly enhance the lubricity of the moving parts or injection pumps of diesel engines, which is confirmed by the test results of the High-Frequency Reciprocating Rig (HFRR) [5]. Moreover, the high flash point of biodiesel would enhance the safety extent during the periods of fuel storage and transportation, while the superior biodegradability of biodiesel might reduce pollution disasters in soil and water sources after being leaked into the environment.

The cetane number (CN), which is an indicator for the time-lapse of compression-ignition engines between fuel injection and ignition, is regarded as the most significant characteristic for petro-derived diesel or biodiesel [6]. Liquid fuel with a higher cetane number implies its easy ignition by a high-pressure compressed air–fuel mixture and a shorter ignition delay period. The cetane number of biodiesel should be above 51 based on the EN 14214 biodiesel specifications [7]. Biodiesel is generally produced from the various dominant feedstocks available in different regions [8]; for example, soybean oil in the United States of America, rapeseed oil in most European regions, tallow and sheep lipids in New Zealand and Australia, and waste cooking oil in Japan and Taiwan are used mainly to manufacture biodiesel. Microalgae, which possess the predominant advantages of a high lipid production rate, fast growth rate, and non-competence with human food, have recently become a significant feedstock source for extracting lipids to produce biofuels [9]. The biodiesels made from various feedstocks have different chemical composition and structure profiles, which result in various fuel properties, particularly the cetane number and compression-ignition characteristics. The most used method to determine the cetane number is an engine test method based on the ASTM (American Society for Testing and Materials, West Conshohocken, PA, USA) D613 standard. A CFR F5 engine test rig feeding at least a 500 mL fuel sample should be used to acquire the cetane number of diesel fuel [10,11]; however, the shortcomings of this test method include insufficient reproducibility and repeatability due to the variations of feedstocks and compositions [12,13]. Bamgboye and Hansen [14] reported that the tested cetane number is obviously influenced by the chemical structures of raw materials, including animal fats and vegetable oils, which results in insufficient repeatability of the cetane number, as determined by the ASTM D613 standard method.

The alternative test method to the ASTM D613 standard method is the determination of the cetane index (CI) by the following formula according to the ASTM D4737 or ASTM D976 methods:

CI = 0.016G² − 420.34 + 0.192G (logT₅₀) + 65.01(logT₅₀)² − 0.0001809 T₅₀²(1)

in which G = 141.5/sg − 131.5 = American Petroleum Institute (API, Washington, DC, USA) gravity, sg is the specific gravity, and T₅₀ is the corresponding distillation temperature to 50 wt.% liquid fuel vaporized and condensed. Although the cetane index can represent the compression-ignition characteristics of diesel fuel to some extent [15], the fuel structure and composition between petro-derived diesel and biodiesel are still varied. In addition, as there is an obviously different distillation temperature range between those two liquid fuels [16], the cetane index is considered insufficient to indicate the ignition-delay characteristics of biodiesel [17].

Various types of fatty acids, such as C18:0 and C16:1, preserve the varied ignition-delay characteristics and cetane numbers; for example, the cetane numbers of C14:0 and C18:0 are 66.2 and 86.9, respectively [18]. Hence, a Gas Chromatography (GC) incorporated with a Flame Ionization Detector (FID) could be used to analyze the weight composition distribution of fatty acid methyl esters (FAME) or biodiesel. The combined cetane number is calculated by the summation of the weight percentage of each fatty acid multiplied by its corresponding cetane number. However, the reliability and repeatability of this method require further confirmation through more experimental data. Hence, more reliable and convenient approaches are still required to replace the tedious ASTM D613 engine test method. The cetane number of biodiesel was estimated based on the compositions of fatty acid methyl esters [19]. The biodiesel made from algae may have the highest ester content and cetane number [20]. Both artificial neural networks (ANN) and multiple linear regression are competitive approaches to analyze the cetane number [21] of biodiesel. The cetane number was proposed to be correlated with either the number of double bonds [22], the molecular weight, the ester saturation degree, etc. [23]. Since the fuel properties of biodiesel rely on the chemical compositions and structure of a variety of feedstocks, the specific attributes of biodiesel and feedstocks such as the number of double bonds, chain length, molecular weight, and fatty acid compositions [24,25] are considered to influence fuel characteristics, including the cetane number. There are too many feedstock sources available for biodiesel production. Non-edible feedstock oils such as Tamarindus indica seed oil were used to produce biodiesel to address the biofuel economy, and the Taguchi method was applied to the production parameters [26]. The blending of a few adequate feedstock oils might adjust some fuel properties of biodiesel. For example, sesame seed oil (SSO) contains natural antioxidants and preservatives. Blending sesame seed oil with palm oil before being transesterified would improve palm oil’s oxidation stability and cold flow properties [27]. The biodiesel mixed with diesel, fuel additives such as titanium dioxide (TiO₂) and dimethyl carbonate (DMC), and lubricants were found to significantly alter the engine emission, performance, lubricant tribology, and tribological characteristics in the aspects of coefficient of friction (COF), brake thermal efficiency (BTE), and emissions from the engine [28]. Hence, some compositions of fatty acid esters might influence typical fuel properties more significantly than others. In fact, the correlation formula proposed in the literature might be more adequate to predict the cetane number of biodiesel produced from some specific type rather than most types of feedstock oils. There is still a lack of a more comprehensive predictive formula for the cetane number of biodiesel.

The previous predictive formulae for cetane number in the literature might consider a single effect of either fatty acid compositions or ester structures, such as weight percentage of representative fatty acids or carbon chain length. Hence, those existing formulae might only apply to that biodiesel from specific feedstock oils. A more comprehensive understanding of biodiesel characteristics is essential in this study for the rapid and accurate determination of fuel properties, including the cetane number of biodiesel. It follows that the potential key influencing parameters could be much more widely studied to include all possible factors so that a more representative formula can be established to obtain the calculated cetane number so that the requirement of the engine test method can be reduced.

To screen out the key parameters for determining the cetane number, the structural features, compositions, and properties of biodiesel were first investigated, then the predictive formula for the cetane number was set up by multiple linear regression methods. Available experimental data were used to examine the validity of the formula. The novelty of this study includes a more comprehensive study for representative key influencing parameters which consider both compositions of fatty acids and ester structure so that the derived predictive formula can be applicable to biodiesel made from a variety of feedstock oils.

2. Materials and Methods

Giakoumis [29] found that the carbon chain length and unsaturation degree of fatty acid esters significantly influenced the cetane number. Biodiesels with a longer carbon chain are prone to have larger cetane numbers and superior compression-ignition characteristics; in contrast, a larger unsaturation extent of fatty acid esters tends to reduce the cetane number. In addition, biodiesel made from the transesterification of lipids with alcohol of a larger carbon number generally has a larger cetane number [30]. As the cis- or trans-structure of fatty acid esters does not significantly influence the cetane number, it is not considered in this study. Ardabili et al. [31] inferred that the carbon number of fatty acid esters is related to the cetane number. Hao et al. [32] considered that the cetane number of biodiesel is determined by the carbon number and the number of double bonds. The combined cetane number of fatty acid esters is determined either by summing up the weight percentages and the corresponding cetane numbers of various types of fatty acid esters [25,33] or compositions of fatty acid esters [21,34].

The organic chemical structure of the allylic position equivalent (APE) is considered a significant factor for compression-ignition characteristics [34]. The allylic position equivalent is located close to the alkene molecular groups of organic chemical compounds, which has a 10–15% weaker chemical bond strength than hydrocarbons’ general chemical bond strength. A chemical bond with a weaker bond strength is prone to be attacked by light, heat, and oxidants, resulting in structural damage of the fatty acid esters and deterioration of oxidation stability [35]. Biodiesel is composed of fatty acid esters with various lengths of carbon chains and a number of double bonds. The organic compounds with various allylic position equivalents (APE) tend to have varied oxidation stability of fatty acid esters, which implies that the number of double bonds is not the only factor for determining the characteristics of oxidation stability and other relevant fuel properties, such as the cetane number [36].

2.1. Screening out Key Parameters for Determining the Cetane Number

According to previous studies, the cetane number of biodiesel is primarily influenced by the compositions and structures of fatty acid esters. Hence, the key parameters of the cetane number of biodiesel are the carbon chain length and the saturation degree of fatty acid esters. The number of carbon atoms (NC) is considered the indicator of the length of the carbon chain of fatty acid esters. The degree of saturation of fatty acid esters is represented by the double-bond equivalent (DBE) and allylic position equivalent (APE). Thus, the function of the cetane number with independent parameters can be expressed quantitatively. Multiple linear regression analysis was carried out to derive the predictive formula of the cetane number of biodiesel. The cetane number tends to increase with the length of the carbon chains of fatty acid esters of biodiesel, which leads to superior combustion properties; for example, the cetane numbers of lauric acid (C12:0), palmitic acid (C16:0), and stearic acid (C18:0) increase from 61.4 through to 74.5 to 86.9 [25]. Hence, the number of carbon atoms (NC) is used in this study to indicate the length of the carbon chain of fatty acid esters, which is calculated by the following equation:

(2)$\mathrm{NC}={{\displaystyle \sum }}_{\mathrm{i}}\frac{{\mathrm{w}}_{\mathrm{i}}{\mathrm{C}}_{\mathrm{i}}}{100}$

where i is the corresponding type of fatty acid ester, w_i is the corresponding weight percentage of the fatty acid ester i, and C_i is the corresponding carbon number of fatty acid ester i.

In contrast, the increased degree of unsaturated fatty acid esters would decrease the cetane number of biodiesel. The extent of unsaturated fatty acid esters is indicated by the parameters of both the double-bond equivalent (DBE) and the allylic position equivalent (APE). Fatty acid esters consisting of compounds with more double bonds are prone to have a decreasing cetane number [25]; for example, the cetane numbers of oleic acid (C18:1), linoleic acid (C18:2), and methyl linolenate (C18:3) are 59.3, 38.2, and 23.0, respectively [25]. The DBE is defined by the following equation:

(3)$\mathrm{DBE}={ \mathrm{dp}}_1·{\mathrm{D}}_{\mathrm{C}1}+{\mathrm{dp}}_2·{\mathrm{D}}_{\mathrm{C}2}+{\mathrm{dp}}_3·{\mathrm{D}}_{\mathrm{C}3}+\dots +{\mathrm{dp}}_{\mathrm{i}}·{\mathrm{D}}_{\mathrm{Ci}}$

where ${\mathrm{dp}}_{\mathrm{i}}$ is the number of double bonds of the corresponding fatty acid ester, and ${\mathrm{D}}_{\mathrm{Ci}}$ is the weight percentage of the corresponding fatty acid ester.

The allylic position is located at the molecular group of a neighboring C=C olefin compound, and the hydrocarbon bond strength at the allylic position is weaker than the bond strength of a typical hydrocarbon chain. Hence, an allylic position with a weaker bond strength is prone to be attacked by the surrounding impact, which results in property deterioration or rancidity of fatty acid esters. The APE is calculated by the following equation:

(4)$\mathrm{APE}={\mathrm{ap}}_1·{\mathrm{A}}_{\mathrm{C}1}+{\mathrm{ap}}_2·{\mathrm{A}}_{\mathrm{C}2}+{\mathrm{ap}}_3·{\mathrm{A}}_{\mathrm{C}3}+\dots +{\mathrm{ap}}_{\mathrm{i}}·{\mathrm{A}}_{\mathrm{Ci}}$

where ${\mathrm{ap}}_{\mathrm{i}}$ is the number of the allylic position of the corresponding fatty acid ester, and ${\mathrm{A}}_{\mathrm{Ci}}$ is the weight percentage of the corresponding fatty acid ester.

2.2. Deriving Methods for a Predictive Formula of the Cetane Number

The multiple linear regression method was used to derive the predictive formula of biodiesel. The information on chemical compositions, carbon number, and the corresponding cetane number of biodiesel made from various feedstocks was collected and analyzed. The data of the relevant dependent variable (i.e., cetane number) and the independent variables (including NC, DBE, and APE) were calculated and grouped together to further derive the predictive formula of the cetane number of biodiesel. The chemical compositions and cetane numbers of 30 different biodiesels, which are primarily composed of 9 types of fatty acid esters [22,25,37,38], were collected to derive the formula.

In order to predict the development trend of a future event [39], multiple linear regression is generally applied to investigate the cause and effect relationship of an event due to the change in internal factors. Multiple linear regression analysis is frequently applied to establish predictive models to obtain more reliable expectation results. In this method, the linear formula of polynomial regression is expressed by a dependent variable (y) and two or more independent variables (i.e., x₁, x₂,…, x_n), which can be expressed as follows:

y = a₁x₁ + a₂x₂ + a₃x₃ +…+ a_nx_n + b(5)

where b is the intercept of the curve, a₁, a₂,…a_n are the regression coefficients, x₁, x₂,…, x_n are the independent variables, and y is the dependent variable.

An adequate statistical analysis software package was applied during multiple linear regression analysis. IBM Statistical Product and Service Solutions (SPSS, version 19.0, IBM Corp., Armonk, NY, USA) software was applied to analyze the cetane number with key parameters. Multiple linear regression is a powerful and widely used approach to resolve complicated interrelationships in engineering or applied science research when multiple predictor variables such as NC, DBE, and APE are introduced into the predictive model. Researchers frequently need to face challenges of multicollinearity once the results of multiple linear regression are interpreted [40]. The aid of adequate software is hence required to alleviate the laborious and time-consuming task of conducting regression analysis. Researchers may prefer to analyze multiple linear regression using traditional statistical software packages, particularly SPSS [41]. The predictive formula of the cetane number is rewritten from Equation (5) to be expressed as follows:

y = a₁ (NC) + a₂ (APE) + a₃ (DBE) + b(6)

The regression coefficients a₁, a₂, and a₃, and the intercept b, were calculated and inducted from multiple linear regression analysis using SPSS software by inputting the data of NC, APE, and DBE and their corresponding measured cetane numbers available from the literature. The predictive formula of the cetane number is thereafter derived.

The flow chart for deriving the predictive formula, as explained above, is shown in Figure 1. The key parameters which have been screened out for determining the cetane number include the number of carbon atoms (NC), the allylic position equivalent (APE), and the double-bond equivalent (DBE). Their values of 30 different biodiesels were calculated based on Equations (2)~(4) from their compositions of fatty acid esters available from the literature [22,25,37,42]. The corresponding measured cetane number of those 30 biodiesels was also collected from the literature [22,25,37,42].

2.3. Verifying the Regression Formula

To verify the effectiveness of the predictive formula for the cetane numbers of biodiesels, the fatty acid esters of seven types of biodiesels from the literature were inputted into the formula to obtain the calculated cetane numbers of the biodiesel. The compositions of fatty acid esters and their corresponding cetane numbers in [42,43,44] were not used at the previous stage to derive the predictive formula; instead, they were used to verify the formula. The values of key parameters, including NC, APE, and DBE, of those seven types of biodiesels were calculated based on their compositions of fatty acid esters in [42,43,44]. The predicted cetane numbers, based on the formula of three independent parameters of NC, APE, and DBE, and the data of measured cetane numbers of those biodiesels were compared to calculate the value of the determination coefficient (R²) of the predicted cetane number. A higher R² value indicates the superior prediction capability of the formula.

3. Results and Discussion

3.1. Effects of Key Independent Parameters on Fuel Characteristics

The three key independent parameters for the cetane number, which are NC, APE, and DBE, were observed to significantly influence the cetane number and other fuel characteristics of biodiesel. Biofuels with a larger carbon number frequently have a larger specific gravity, a higher melting point, and a higher energy density [45]. As the carbon numbers of gasoline and diesel fuel typically range between 6–10 and 11–20, the carbon number is one of the significant factors for determining the combustion application of biofuel composed of fatty acid esters. The number of carbon (NC) is used here to indicate the length of a carbon chain.

The stability characteristics of fatty acids generally include storage stability, oxidation stability, and thermal stability. A lipid composed of a larger content of unsaturated fatty acids is prone to oxidation under a surrounding environment of strong light and high heat, which results in a faster oxidation rate and earlier rancidity. The oxidation instability is primarily due to the occurrence of oxidation at the double bonds of C=C, which results in unstable lipid properties. Thermal stability deteriorates when the lipid is subjected to a high temperature, leading to the accelerated deterioration of lipid characteristics.

The APE, which is calculated by Equation (4), is located at the molecular group of a neighboring C=C olefin unit. As the oxidation reaction can occur on oleic fatty acid, the hydrogen attached to the carbon of the allylic position equivalent to the double bond of C=C can be activated by the oxygen compound in the surrounding air [46]. The activated hydrogen causes the oxygen to attach to the carbon structure to form organic peroxides, which are relatively unstable and contain a particularly weak bond strength at the O=O bond [47]. The bond strength between carbon and hydrogen at the allylic position is generally 10~15% weaker than the typical C-H bond [48], and this weaker C-H bond tends to cleave to produce free radicals of the oxygen compounds.

The oxidation rate of lipids is determined mainly by the number of double bonds in the fatty acid structure; for example, the oxidation rate of linoleic acid (C18:2) is 30 times faster than that of olefin acid (C18:1), while the oxidation rate of linolenic acid (C18:3) is 80 times faster than that of C18:1 [49]. In addition, the influences of the number of double bonds on the oxidation stability of fatty acids are frequently more important than the length of their carbon chain. Furthermore, the number of double bonds also influences the other fuel characteristics, such as pour point, cetane number, and kinematic viscosity. Hence, the DBE, as calculated by Equation (3), is also considered one of the significant factors for determining the cetane number of biodiesel.

3.2. Derivation of Predictive Formula for the Cetane Number of Biodiesel

The data of the compositions of fatty acid esters and the cetane numbers of 30 types of biodiesel were used to calculate the values regarding the length of the carbon chain (NC), the DBE, and the APE based on Equations (2)–(4). Then, the data of the three independent variables (NC, DBE, and APE) and their corresponding cetane numbers (CN) of the respective biodiesel types were used to carry out multiple linear regression analyses. The predictive formula derived from the above analysis is shown below:

CN = 5.212 NC − 0.075 APE − 0.139 DBE − 9.285

(R = 0.973, R² = 0.947, Sx= 3.486)(7)

where R is the correlation coefficient, and R² is the determination coefficient. R² is frequently used to indicate the adequacy degree of the linear regression model and the validity of the derived formula. There is no linear relationship between the dependent variable (y) and the independent variables (x_n) for R² = 0. The correlation degree between the dependent and independent variables is not sufficiently high when R² is less than 0.3. The coefficient of determination of the derived formula is 0.947, which implies a high degree of mutual correlation between the dependent variables (i.e., CN) and the independent variables in Equation (6). Sx is the standard error of estimates, which represents the standard deviation of the sampling distribution.

Figure 2 reveals the predictive formula for the cetane number, as derived from multiple linear regression based on the measured data of the cetane numbers of biodiesel from various sources [22,25,37,42]. The predicted formula for cetane numbers was derived accordingly, as shown in Equation (6). The isometric line was plotted based on the data of the measured cetane numbers in [22,25,37,42], which were used to justify the predictive formula for cetane numbers. As shown in Figure 2, a relatively high correlation exists between the measured and predicted data of cetane numbers, which implies the high prediction capability of the formula, as shown in Equation (6). The calculated and measured cetane numbers underwent linear fitting, where the determination coefficient (R²) was 0.947 and the standard error of estimates was 3.486, which indicates the superior regression result of the formula. In addition, the regression coefficients and standard error of estimates of the three key parameters in the presented predictive formula for cetane numbers are shown in Table 1. The regression coefficient of NC is positive, while those of the other two parameters, APE and DBE, are negative, which implies that the increased NC results in higher cetane numbers. However, an increased APE or DBE causes an increase in the unsaturated extent of the fatty acid esters, which leads to a reduced cetane number.

3.3. Verification of the Regression Formula

The regression formula for predicting cetane numbers is verified by the cetane numbers of seven types of biodiesel in the literature [42,43,44]. The data of the compositions of fatty acid esters in the above-mentioned seven biodiesels, and their corresponding cetane numbers, were not used to derive the predictive formula through multiple linear regression, but to verify the cetane numbers calculated by the formula. The compositions of the fatty acid methyl esters of the seven biodiesels were gathered to calculate the values of three independent parameters (NC, DBE, and APE) according to Equations (2)–(4). The calculated data of NC, DBE, and APE parameters were then input into Equation (6) to obtain the predicted cetane numbers. Then, the predictive formula for the cetane numbers was verified by the measured data of cetane numbers in the literature [42,43,44], as shown in Figure 3. The measured data were found to be highly agreeable with the predicted values of cetane numbers through Equation (6), which implies the significant validation of the predictive formula obtained in this study. In addition, the cetane number was observed to increase with the length of the carbon chain and the saturation extent of the biodiesel [20]. The curve trend of the predicted cetane numbers agrees well with previous research [21,24,50].

The cetane number of biodiesel was proposed to correlate with the chain length and the unsaturation degree [51]. The correlation of the cetane number with the number of double bonds and carbon atoms was studied previously [52,53]. The formula of the cetane number with the molecular weight and the number of double bonds was considered by Ramirez-Verduzco et al. [54]. The current predictive formula was compared with other formulae according to the virtue of their predicting capabilities of cetane numbers for biodiesel, as shown in Table 2. The determination coefficients and standard errors of the presented predictive formula were revealed to demonstrate the superior prediction function among those formulae in Table 2. The predictive capability of the presented formula of Equation (6) was compared with Table 2. The predictive capability of the presented formula of Equation (6) was compared to other relevant predictive formulae in the literature, as shown in Figure 4. An isometric line was plotted to compare the discrepancy extent among the various predictive formulae for cetane numbers, and the calculated cetane numbers from the presented predictive formula were found to be much closer to the isometric line than the calculated cetane numbers from other predictive formulae available in the literature [21,24,42]. It implies that the predicted cetane numbers of the formula obtained in this study can more accurately agree with practical cetane numbers. Moreover, the screened key independent parameters, including NC, APE, and DBE, are more influential than other parameters proposed in the literature. There is no literature to relate the key parameters of NC, APE, and DBE together with the cetane number of biodiesel and derive a representative formula [55,56]. In addition, the cetane number appeared to decrease with an increase in DBE, which implies that less saturated fatty acid esters exist in the biodiesel.

The data of the weight percentages of the oleic acid (C18:1), linoleic acid (C18:2), and α-linolenic acid (C18:3) of five various biodiesels, including soybean oil, rubber seed oil, jatropha oil, neem oil, and mahua oil, were collected from Gopinath et al. [43] to calculate the DBE of those biodiesels. The DBE values of those five biodiesels were 156.0, 143.5, 104.8, 75.3, and 69.2, respectively, where a higher cetane number indicates superior compression-ignition characteristics and combustion properties for diesel engines. A higher saturation extent of fatty acid esters causes a lower DBE value, which results in a higher cetane number and a shorter ignition delay in a diesel engine [45]. Hence, a negative correlation was observed between the cetane number and the DBE of biodiesel.

4. Conclusions

Biodiesel is composed of fatty acid methyl or ethyl esters, which differ from those of petro-derived diesel fuel; hence, the determination methods or predicted equations for the cetane numbers of petro-derived diesel are inadequate for biodiesel. A multiple linear regression equation predicts the cetane number of biodiesel with three independent variables in this study. The available experimental data of ester compositions and the measured cetane number were used to examine the validity of the predictive equation. The major results of this study are summarized as follows.

• The key parameters, which include characteristics of ester compositions and structures, were screened out to be the number of carbon atoms (NC), the allylic position equivalent (APE), and the double-bond equivalent (DBE), which influence cetane numbers of biodiesel more significantly and comprehensively than other parameters proposed in the literature.

• Either a higher APE or DBE of the biodiesel implies a higher unsaturation degree of the biodiesel. The cetane number of biodiesel was observed to increase with the NC while decreasing with the APE or the DBE.

• The predictive formula for the cetane number of biodiesel was successfully derived through quantitative analyses of various compositions of fatty acid methyl esters and their corresponding cetane numbers. Multiple linear regression analyses were conducted assisted by SPSS software to derive the regression coefficients and intercepts of the formula.

• The cetane number calculated by the predictive formula was found to be closer to the corresponding measured cetane numbers. The formula possesses superior prediction capability compared to other relevant formulae in the literature.

• The proposed predictive formula can more adequately predict the cetane number of biodiesel made from a variety of feedstock oils and significantly reduce the time and cost consumed for the engine test. The production procedures and operating conditions can be readily adjusted to meet biodiesel specifications.

• Those key parameters for cetane numbers would significantly influence other fuel properties, which require further extensive study. For example, the increase of the DBE might decrease the density and heating values of biodiesel.

• The C-C double bonds at the allylic position equivalent are prone to form unstable peroxides and thus have a weaker bond strength than other general hydrocarbon bonds.

Author Contributions

Conceptualization, C.-Y.L.; methodology, C.-Y.L.; validation, C.-Y.L. and X.-E.W.; formal analysis, C.-Y.L.; investigation, C.-Y.L. and X.-E.W.; resources, C.-Y.L.; data curation, C.-Y.L.; writing—original draft preparation, C.-Y.L.; writing—review and editing, C.-Y.L. and X.-E.W.; supervision, C.-Y.L.; project administration, C.-Y.L.; funding acquisition, C.-Y.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are contained within this article.

Acknowledgments

The authors gratefully acknowledge the assistance of Tin-Yu Yeh during the software operation for deriving the formula. The financial support from the Ministry of Science and Technology of Taiwan, ROC, under contract No. MOST 107-2221-E-019-056-MY2 and MOST 109-2221-E-019-024 is acknowledged.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Procedures for deriving the predictive formula for the cetane number of biodiesel.

Figure 2. Derivation of the predictive formula based on the fatty acid compositions and measured data of cetane numbers by multiple linear regression analysis using SPSS software [22,25,37,42].

Figure 3. Verification for the proposed predictive formula in this study by the measured cetane number data available from [42,43,44].

Figure 4. Comparison of the proposed formula’s predictive capability for cetane numbers with other competitive formulae in the literature [21,24,42].

Table 1

Regression coefficients and intercepts of the key parameters in this study.

Parameter	Coefficient	Standard Error
Intercept	−9.285	3.097
NC	5.212	0.249
APE	−0.075	0.027
DBE	−0.139	0.029

Table 2

Comparison of prediction capability of the present formula with competitive formulae in the literature.

Research Group	Prediction Equation	Comments	Ref.
Gopinath et al.	$\mathrm{CN}=62.2+0.017\mathrm{L}+0.074\mathrm{M}+0.115\mathrm{P}+0.177\mathrm{S}-0.103 \mathrm{O}-0.279\mathrm{LI}-0.366\mathrm{LL}$	R2 = 95.3%, Standard error = 2.27	[42]
Piloto-Rodríguez et al.	$\mathrm{CN}=56.16+0.07 \mathrm{La}+0.1 \mathrm{M}+0.15 \mathrm{P}-0.05 \mathrm{Pt}+0.23 \mathrm{S}-0.03 \mathrm{O}-0.19 \mathrm{Li}-0.31 \mathrm{Ln}+0.08 \mathrm{Ei}+0.18 \mathrm{Er}-$0.1Ot	R2 = 91.1%, Standard error = 4.6	[21]
Giakoumis andSarakatsanis	CN = 55.87 + 0.0747 × 1 + 0.098 × 2 + 0.164 × 3 + 0.176 × 4$-$0.050 × 5 + 0.001 × 6 $-$0.140 × 7 $-$0.273 × 8	R2 = 89.6%, Standard error = 3.04	[24]
Present study	$\mathrm{CN}=5.212\mathrm{NC}-0.075\mathrm{APE}-0.139\mathrm{DBE}-9.285$	R2 = 94.7%, Standard error = 3.486	-

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