Mathematical Correlations for Volumetric (Density and Specific Gravity) Properties of Diesel/Biodiesel Blends

Abstract

Biodiesel is a renewable and environmentally friendly alternative energy source to conventional diesel. The use of biodiesel blends with diesel to meet energy needs can significantly reduce greenhouse gas emissions, as biodiesel produces smaller amounts of carbon dioxide (CO₂) when burned. In addition, diesel/biodiesel blends can be used as fuel in existing diesel engines without the need to modify them, and their exploitation reduces dependence on oil imports and the impact of oil prices on the economy. Since increasing the percentage of biodiesel in diesel/biodiesel blends aims to increase the environmental and economic benefits, it is necessary to know the physicochemical properties of these blends, such as density, specific gravity, etc. The aim of the present work was to use appropriate mathematical equations that can predict the physicochemical properties of mixtures under different conditions of temperature and mixing ratios. Kay’s mathematical mixing expression, the Tammann–Tait equation, and empirical equations were used to describe the dependence of the density ($\rho$, kg/m³) of the mixtures on the volume percentage (v%) of biodiesel mixed with diesel and the temperature variance (T, K). In addition, mathematical equations were used to predict the specific gravity (Sg) of the mixtures. Mathematical estimations were based on experimental data obtained by blending diesel and animal or vegetable biodiesel volume percentages. These data showed the effect of different mixing volume percentages of biodiesel and diesel (from 0% to 100% biodiesel) on their physicochemical characteristics under different temperatures (278 to 298 K). The accuracy of the mathematical estimations was evaluated using factors such as the Nash and Sutcliffe coefficient (E) and relative root mean squared error (MSE%). The results showed that the selected mathematical equations were able to accurately estimate (E up to 0.9988 and MSE up to 0.4%) the increased density and specific gravity as the volume percentage of biodiesel increased and temperature decreased. The present study uses mathematical tools for choosing the right blending ratios and conditions, depending on the desired features of the finished product.

1. Introduction

The rising prices of fossil fuels, i.e., fuels derived from oil, such as diesel and petrol; the rapid depletion of known reserves due to their uncontrolled use; and the environmental problems caused by their use and the emission of harmful gases, such as CO₂, NO_x, and CO, make mandatory the development of other clean and renewable alternative fuels. This endeavour is becoming a new focus of research [1,2].

The most common fuels today are the products of the distillation of crude oil, i.e., petroleum, petrol, gasoline, kerosene, etc. Biofuels, produced from biomass, are considered renewable fuels, with lower emissions over their entire life cycle than conventional fossil fuels. In particular, they are solid, liquid, or gaseous fuels and are produced directly or indirectly from biomass. The most common biofuels today are biodiesel, bioethanol, and biogas. In particular, biodiesel, as an alternative fuel, has significant importance for diesel engines because it can be blended with diesel in various ratios, without the need for specific adjustments to the engine design [3].

Biodiesel can be produced (i) by feedstock that are usually derived from food crops such as maize (corn), sugarcane, and wheat (first-generation biofuels), (ii) by non-edible raw materials including biomass waste, wheat and corn stalks, wood, energy crops, and animal fats (second-generation biofuels), and (iii) by microalgae as a very good source of lipids (third-generation biofuels) [4]. Biodiesel is produced by the transesterification of lipids with a low-molecular-weight alcohol (methanol) and the use of strong basic homogeneous catalysts such as hydroxides (i.e., NaOH) [5]. The main feedstock is usually vegetable oils, while cooking oils (mainly frying oils) and animal fats are used to a lesser extent [6,7]. Biodiesel offers many advantages, including the following: (i) a 10 to 12% oxygen content by mass in its molecular structure, improving combustion efficiency and reducing emissions of CO, unburned hydrocarbons, and smoke; (ii) it reduces net CO₂ emissions by 78% over its life cycle compared to diesel fuel; (iii) it is biodegradable and non-toxic, protecting the environment; (iv) it has a higher ignition temperature and cetane number than diesel fuel; and (v) it improves lubricity, leading to the longer life of engine components [8,9]. Overall, biodiesel is a sustainable and environmentally friendly alternative for our energy needs [8,10]. Because animal fats include a large amount of saturated fatty acids, the biodiesels made from them have a greater melting point than those made from vegetable oil, making them unsuitable for diesel engine fuel. Nevertheless, biodiesels made from animal fats have a larger cetane number and calorific value than those made from vegetable oils, and they can be utilized in boilers to generate heat in their original form [11]. Another advantage concerns the fluidity of biodiesel at low temperatures [12,13]. While biodiesel applications offer many benefits, there are also significant challenges, such as competition with the food industry, lower energy content and volatility, and higher viscosity and NOx emissions compared to diesel fuel [14]. It should be noted that growing biofuel crops rather than food crops, as the demand for biodiesel rises, reduces the land availability for cultivating the basic food supply. Therefore, increased crop use for biodiesel may have an impact on international agricultural markets, which would further affect food access and affordability worldwide [15]. Also, the biodiesel produced from oils, no matter if it is neat vegetable oil or animal fat, is usually more expensive than diesel fuel. Therefore, the high cost of biodiesel is the major obstacle to its commercialization [16].

Fuel atomization and combustion in diesel engines, which take place at high pressures and temperatures as well as derived thermodynamic forces, are significantly influenced by the density, specific gravity, and viscosity of the fuel [17,18,19]. Density is the mass of a unit volume of fuel; it is a useful indicator of the composition and operational characteristics of diesel fuel, such as ignitability, power, economy, and low-temperature flow characteristics. Density depends on the type of hydrocarbon composing the fuel and the number of carbons in the tramp oil. A report from the National Renewable Energy Laboratory (NREL), published in 2024, investigated the properties of high-level biomass-derived biodiesel with renewable diesel (RD) and petroleum diesel. The study measured various characteristics, including flash point, cloud point, cetane number, surface tension, density, kinematic viscosity, distillation curve, lower heating value, water content, water solubility, lubricity, and oxidation stability across a broad temperature range. The results indicated that certain properties, such as density and viscosity, may limit the ability to blend high concentrations of biomass-derived biodiesel with diesel fuels [20]. Moreover, the higher biodiesel density than diesel fuel may increase blend density and cause a distinct spray pattern and fuel atomization. The increased density of biodiesel can influence combustion time and pressure within the cylinder. Biodiesel releases energy more slowly than diesel, potentially delaying combustion. This could impact engine performance [21]. In addition, the specific gravity of a fuel is proportional to its energy content. In general, biodiesel will produce less energy for the same volume as regular diesel since it has a lower energy density. As a result, when using biodiesel at higher quantities, power output is reduced. Therefore, density and specific gravity are important parameters of fuels, which, together with cetane number and calorific value, are related to fuel impurities and viscosity. Understanding the effects of blending biodiesel and diesel and predicting the density of the final blending fuel are key factors in selecting the appropriate blend for the final product. Additionally, the blending ratio has a significant role in determining the economic viability of the final fuel [16,22]. As a result, knowing fuel density as well as other parameters under different circumstances is crucial for forecasting and enhancing engine performance [23].

It should also be noted that modelling is a powerful tool that allows for the prediction, optimization, and management of complex systems and processes in a wide range of applications. Through the creation of models, we can analyze the individual parts of a system and understand how they interact with each other. Furthermore, models are used to predict the behaviour of systems under different conditions. This is useful for predicting the effects of changes and evaluating different scenarios. Moreover, modelling enables hypothesis testing and experimentation without the need for expensive or risky real-world trials. Ultimately, models contribute to process and system optimization, facilitating more informed and effective decision-making.

Based on the above, in recent years, research activity in the direction of the theoretical calculation and/or estimation of diesel/biodiesel blend properties has significantly developed. As an eco-friendlier option, the growing demand for biodiesel drives up the need for even higher fuel quality. Ivaniš et al. [24], in order to estimate combustion engine performance, determined some key fuel properties under various pressure and temperature conditions, with a particular focus on the density and viscosity of the mixtures. Their experimental data on density were obtained by blending diesel with 10% and 20% biodiesel (sunflower), with a temperature range of 293–413 K and a pressure of 0.1–60 MPa. In addition, Tesfa et al. [18] studied whether the physicochemical properties, such as density and temperature, of blending diesel with vegetable biodiesel (corn) affect the fuel properties. The blends (5% to 75% biodiesel) were prepared on a volume basis, and the densities were measured in the temperature range of 10–40 °C. The properties of vegetable biodiesel and its blends were evaluated by determining other important properties, such as flash point temperature and the highest temperature value.

The study performed by Brakora and Reitz [25] developed a comprehensive combustion model for biodiesel-fueled engine simulations. This model was designed for multidimensional simulations and incorporates realistic physical properties into an exhaust model, which involves multicomponent fuel sprays. In addition, an improved mechanism for biodiesel combustion chemistry is implemented. The study of Alahmer et al. [26] investigated the enhancement of compression ignition engine performance using various biodiesel blends. The research utilizes modelling and optimization techniques to assess key engine performance parameters, including brake torque, brake-specific fuel consumption, and brake thermal efficiency, while taking into account different biodiesel blend ratios. A recent study by Bousbaa et al. compared the physicochemical properties of two types of biodiesels (eucalyptus and a blend of palm oil with limonene) to conventional diesel, using mathematical models and experimental tests. The study developed a comprehensive analysis of the advantages and disadvantages of using biodiesel as an alternative to conventional fuels [27].

This paper examines how the increase in biodiesel fraction in diesel/biodiesel blends can affect the physicochemical characteristics of the fuel, maintaining the economic and environmental benefits. Therefore, the objective of the current study was to use a mathematical modelling tool that would be able to predict fuel properties, such as density and specific gravity, under various temperature conditions and biodiesel volumetric portions. This mathematical modelling study was based on experimental data that were obtained from specifically designed blending tests to investigate the effects of varying biodiesel fractions (0 to 100% vegetable or animal biodiesel) in the mixture under different temperature conditions (278 to 298 K) on the physicochemical properties. The mathematical expressions used in this study to predict the dependence of the density ρkg/m³) were Kay’s mathematical mixing expression and the Tammann–Tait equation [17,24]. The specific gravity was predicted using the API (American Petroleum Institute) values and the correlation between specific gravity and temperature, as per Tat and Gerpen [28]. The theoretical results were validated using part of the experimental data, while their accuracy was tested by evaluating the relative root mean squared error (MSE%) and the Nash and Sutcliffe efficiency index (E).

2. Experimental Study

The diesel and biodiesel fuels were not freely marketed; however, after a special request and customs clearance, they were given to the Environmental Technology Laboratory by Hellenic Petroleum together with their quality certificates. The physicochemical properties of diesel, vegetable (sunflower), and animal fat (sausage industry) biodiesel fuels are given in Table 1.

Mixtures, which were prepared volumetrically by blending portions of diesel and biodiesel (starting from 100 v% diesel and ending up to 100 v% biodiesel), were tested on their physicochemical properties by varying the temperature conditions from 278 to 298 K. Densities of the examined samples at atmospheric pressure were measured using an appropriate densitometer (Ludwig Schneider—Laboratory Hydrometers, DIN ISO 650 [29], ASTM D1298 [30] (Figure 1) following the procedure by the American Society for Testing and Materials (ASTM D6751) [31]. The density data were calculated experimentally at temperatures of 278, 283, 288, 293, and 298 K and at various blending ratios of diesel and biodiesel. Density and API values for both diesel/vegetable biodiesel and diesel/animal fat biodiesel are given in Supplementary Materials (Tables S1 and S2). The experimental data are reported with a 95% confidence interval.

Table 1. Physicochemical properties of diesel/biodiesel fuels based on quality certificates.

Parameter	Method—Reference	Value	Method	Value	Method	Value
	Diesel		Biodiesel Vegetable (Sunflower)		Biodiesel Animal Fat
Density 15°C (g/mL)	ISO 12185 [32]	0.821	ISO 12185 [32]	0.884	D1298	0.870
Ignition point (°C)	ISO 2719 [33]	56.0	ISO 3679 [34]	167	D93	172
Kinematic Viscosity (cSt)	ISO 3104 [35]	2.393	ISO 3104 [35]	6.0	D445	6.0
Cetane index	ISO 4264 [36]	55.1	ISO 5165 [37]	52.3	D613	52.5
Water Κ-F in products (mg/kg)	ISO 12937 [38]	105	ISO 14214 [39]	187	D6304	287
Conductivity (pS/cm)	ISO 6297 [40]	2	ISO 6297 [40]	4	D2624	122
Acidity (mgKOH/gr)	ISO 660 [41]	0.03	EN 14104 [42]	0.24	D1980	0.92

Table 1. Physicochemical properties of diesel/biodiesel fuels based on quality certificates.

Parameter	Method—Reference	Value	Method	Value	Method	Value
	Diesel		Biodiesel Vegetable (Sunflower)		Biodiesel Animal Fat
Density 15°C (g/mL)	ISO 12185 [32]	0.821	ISO 12185 [32]	0.884	D1298	0.870
Ignition point (°C)	ISO 2719 [33]	56.0	ISO 3679 [34]	167	D93	172
Kinematic Viscosity (cSt)	ISO 3104 [35]	2.393	ISO 3104 [35]	6.0	D445	6.0
Cetane index	ISO 4264 [36]	55.1	ISO 5165 [37]	52.3	D613	52.5
Water Κ-F in products (mg/kg)	ISO 12937 [38]	105	ISO 14214 [39]	187	D6304	287
Conductivity (pS/cm)	ISO 6297 [40]	2	ISO 6297 [40]	4	D2624	122
Acidity (mgKOH/gr)	ISO 660 [41]	0.03	EN 14104 [42]	0.24	D1980	0.92

3. Mathematical Modelling

3.1. Kay’s Mixing Rule for Density Behaviour in Biodiesel/Diesel Blends

According to Kay’s mixing rule [12,19], the density of a mixture increases linearly when the biodiesel volume percentage increases (Equation (1)). Diesel and biodiesel share comparable chemical characteristics, are entirely miscible, and are both non-polar liquids, indicating that they do not interact and that their volumes are essentially additive. This suggests that the volume percentage can be used in place of the molar fraction, thereby transforming Equation (1) into Equation (2).

$${\rho }_{blend}={\sum }_{i=1}^n{x_i\rho }_i$$

(1)

$${\rho }_{blend}={\sum }_{i=1}^n{v}_i{\rho }_i$$

(2)

where ${\rho }_{blend}$ is the density of the mixture, x_i and $vi$ are the molar and volume percentage, respectively, of the blend’s individual component, ${\rho }_i$ is its density, and n is the number of individual components of the mixture.

3.2. Tammann–Tait Equation for Density Estimation of Blends Versus Temperature

The density of liquid mixes, such as blends of diesel and biodiesel, is frequently predicted using the modified Tammann–Tait equation under specific conditions. The densities of diesel, biodiesel, and their blends at a specific pressure and temperature can be related by this equation. Therefore, the modified Tammann–Tait equation [24,43] was used to relate the densities of diesel, biodiesel, and their mixtures at a pressure of 0.1 MPa and in the temperature range 278–298 K.

$$\rho \left(T,p\right)={\displaystyle \frac{{\rho }^{ref}\left(T\right)}{1-C\ast \ln \left(\frac{B\left(T\right)+p}{B\left(T\right)+p^{ref}}\right)}}$$

(3)

where $C$ is a temperature-independent parameter and $B\left(T\right)$ is a temperature-dependent parameter, which can be calculated from the following second-order polynomial:

$$B\left(T\right)={\sum }_{i=0}^2{b}_i{T}^i$$

(4)

${\rho }^{ref}$ is the density of the exanimated sample at reference pressure ($p^{ref}=$1 MPa), which can be calculated using the second-order polynomial:

$${\rho }^{ref}\left(T\right)={\sum }_{i=0}^2{a}_i{T}^i$$

(5)

where $a_i$, $b_i$, and $C$ are parameters that can be adjusted.

3.3. Empirical Equation for Density Estimation of Blends Versus Temperature

For blends, Tesfa et al. [44] propose empirical equations to calculate the overall density based on the individual densities of diesel and biodiesel volume percentages. Each volume percentage (diesel or biodiesel) has its own density, which is affected by factors like temperature. Therefore, according to Tesfa et al. [41], empirical equations can be used for predicting the density of each volume percentage (diesel/biodiesel) separately, as well as equations for diesel/biodiesel blending:

$${\rho }_{biodiesel}=-0.69{\rm T}+1075$$

(6)

$${\rho }_{diesel}=-0.657{\rm T}+1051$$

(7)

$${\rho }_{mix}=\left(-0.033T+24.4\right)X-0.7950T+1052$$

(8)

where ${\rho }_{biodiesel}$ is the density of biodiesel, ${\rho }_{diesel}$ is the density of diesel, and ${\rho }_{mix}$ is the density of their mixture measured in g/cm³. $T$ is the temperature in K, and $X$ is the percentage volume (v%) of biodiesel in the mixture.

Preliminary modelling tests on the experimental data of this study using Equation (8) showed large discrepancies. Therefore, it was decided to set the variables a and b (Equation (9)) instead of the fixed values −0.033 and −0.7950, respectively. Parameters a and b were evaluated using the fitting process to the experimental data.

$${\rho }_{mix}=\left(aT+24.4\right)X-bT+1052$$

(9)

3.4. Specific Gravity (Sg) and API Behaviour Versus Temperature

The specific gravity of diesel oil and its blends is affected by temperature. API (American Petroleum Institute) is an empirical function of specific gravity, given by Equation (10):

$$APIgravity={\displaystyle \frac{141.5}{Sg}}-131.5$$

(10)

Specific gravity (Sg) values by solving Equation (10) for both diesel/vegetable biodiesel and diesel/animal fat biodiesel are given in the Supplementary Materials (Tables S3 and S4).

The association of temperature-dependent variations in the specific gravity of diesel/biodiesel blends is commonly represented by an empirical equation. Given that the density of the fuel has a direct influence on injection, combustion, and engine performance, these connections are crucial for comprehending how fuel behaves under various operating situations. Tat and Gerpen [28] present a correlation between specific gravity and temperature using the following equation (Equation (11)):

$$Sg=c+dT$$

(11)

Parameters c and d were evaluated using the fitting process to the experimental data.

4. Theoretical Estimations

4.1. Theoretical Estimations Tool

The parameters of the mathematical equations used in this study and the evaluation of the physicochemical properties were performed using the program Aquasim. The Aquasim program was developed by the Swiss Federal Institute of Environmental Science and Technology (EAWAG) (Eidgenössische Technische Hochschule, ETH, Zurich). Reichert et al. [45] developed this program for modelling chemical engineering systems, such as reactor operation, mixing, and adsorption processes. The aim of the software is to simplify the simulation of such systems, as in many cases, the models used to describe them are quite complex. The introduction and simulation of the various processes and/or systems is carried out using a system of ordinary and partial differential equations in time and space, combined with algebraic equations. The parameters of mathematical expressions were determined through a parameter estimation process using Aquasim. This was achieved by minimizing the sum of squared deviations between experimentally measured data and model-predicted values, following the Least Squares Method. The mathematical equations were then solved using the DASSL algorithm [46].

4.2. Experimental Data for Parameter Estimation

The parameters of the mathematical expression were estimated using the experimental data of different mixtures of biodiesel (vegetable or animal fat) and diesel at varying temperatures. It should be noted that for simplicity, from now on the biodiesel volume percentage will be referred as BD, while the diesel volume percentage will be referred as D. The theoretical calculations of Equations (3)–(5), (9), and (11) as well as the values of their parameters were determined by fitting through parameter estimation the experimental data of density, temperature, API, and Sg (Tables S1–S4).

In order to reduce the number of parameters to be fitted, the parameters b₀ (MPa), b₁ (MPa·K⁻¹), b₂ (MPa·K⁻²), and C in Equation (4) were adopted from the literature [24]. The parameters that were optimized in Equation (5) were a₀ (gr·cm⁻³), a₁ (gr·cm⁻³·K⁻¹), and a₂ (gr·cm⁻³·K⁻²). Note that Equations (4) and (5) are solved together with Equation (3). The parameter estimation was performed using data in six groups for vegetable biodiesel and six groups for animal fat biodiesel, while temperature was varied from 288 to 298 K:

Group 1: D100%-BD0%, D95%-BD5%, D90%-BD10%

Group 2: D85%-BD15%, D80%-BD20%, D75%-BD25%

Group 3: D70%-30%, D65%-BD35%, D60%-BD40%

Group 4: D55%-BD45%, D50%-BD50%, D45%-BD55%, D40%-BD60%

Group 5: D35%-BD65%, D30%-BD70%, D25%- BD75%, D20%-BD80%

Group 6: D15%-BD85%, D10%-BD90%, D5%-BD95%, D0%-BD100%

The parameters that were optimized in Equation (9) were a and b. The parameter estimation was performed using all the data at once for both vegetable and animal fat biodiesel. The parameters that were optimized in Equation (11) were c and d. The parameter estimation was performed using data for both vegetable and animal fat biodiesel separately.

4.3. Evaluation of the Model

The agreement of the model with the measured data was evaluated by determining the relative root mean squared error ($MSE\left(\%\right)$ (Equation (12)) [47] and the Nash and Sutcliffe efficiency index ($E$) (Equation (13)) [48]:

$$MSE\left(\%\right)={\displaystyle \frac{100}{\overline{Y}}}\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{\left(Y_i-S_i\right)}^2}{n}}}$$

(12)

$$E=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(Y_i-S_i\right)}^2}{{\sum }_{i=1}^n{\left(Y_i-\overline{Y}\right)}^2}}$$

(13)

where $n$ is the number of experimental data, $Y$ is the average of experimentally measured values, $S_i$ are the theoretical values produced by the model, and $Y_i$ are experimentally measured values.

5. Results and Discussion

5.1. Density Evaluation for Biodiesel/Diesel Blends at Constant Temperature

The correlation between the theoretical calculation of Kay’s mixing rule (Equation (2)) and the experimental density data obtained at a specific temperature (288 K) are shown for the vegetable BD blends in Figure 2a and for the animal fat BD blends in Figure 3a. Figure 2b and Figure 3b show the residuals of density versus volume percentage. It was observed that the decrease in density values with increasing D content in the diesel/biodiesel blends is adequately described by Equation (2). This was verified by the high value of E = 0.9902 with MSE = 0.4% for the blending of diesel with vegetable biodiesel and E = 0.9763 with MSE = 0.6% for blending diesel with animal fat biodiesel.

Figure 2a and Figure 3a show that the higher the volume percentage of diesel in the mixture, the lower the density of the mixture. This variation is expected as diesel has a lower density than biodiesel (Table 1), and increasing its percentage in the mixture decreases the density of the mixture. These results are in agreement with the findings reported by Ivaniš et al. [24].

5.2. Density Evaluation for Biodiesel/Diesel Blends at Varying Temperatures Using Tammann–Tait Equation

Figure 4 and Figure 5 present the experimental and the calculated dependence of density on temperature for different diesel/biodiesel blends, according to Equations (3)–(5). Figure 4a,b illustrate examples of the experimental data of diesel/vegetable biodiesel blends together with the estimated values using Equations (4) and (5) (see Section 4.2). In addition, Figure 5a,b present examples of the experimental and calculated values of diesel/animal fat biodiesel. The values of the parameters of the Tammann–Tait equation (b₀ (MPa), b₁ (MPa·K⁻¹), b₂ (MPa·K⁻²) and C) were adopted from Ivaniš et al. [19]. The parameters a₀ (gr·cm⁻³), a₁ (gr·cm⁻³·K⁻¹), and a₂ (gr·cm⁻³·K⁻²) were calculated using Aquasim and their values are shown in

Table 2. Results of the parameter estimation using Tammann–Tait equation (Equations (3)–(5)) for volume percentage vegetable biodiesel/diesel blends at varying temperatures.

	D100%-BD0%, D95%-BD5%, D90%-BD10% *	D85%-BD15%, D80%-BD20%, D75%- BD25%	D70%-BD30%, D65%-BD35%, D60%-BD40%	D55%-BD45%, D50%-BD50%, D40%-BD60% D45%-BD55%	D35%-BD65%, D30%-BD70%, D25%-BD75%, D20%-BD80%	D15%-BD85%, D10%-BD90%, D5%-BD95%, D0%-BD100%
Parameters	Group 1	Group 2	Group 3	Group 4	Group 5	Group 6
a0 (gr·cm−3)	−0.1543	2.4919	1.8164	2.3069	4.5449	5.2909
a1 (gr·cm−3·K−1)	0.0067	−0.0114	−0.0064	−0.0098	−0.0253	−0.0303
a2 (gr·cm−3·K−2)	−1.14 × 10−5	1.95 × 10−5	1.05 × 10−5	1.64 × 10−5	4.34 × 10−5	5.19 × 10−5
b0 (MPa)	400	400	400	400	400	400
b1 (MPa·K−1)	−1.31	−1.31	−1.31	−1.31	−1.31	−1.31
b2 (MPa·K−2)	0.001	0.001	0.001	0.001	0.001	0.001
C	0.0831	0.0831	0.0831	0.0831	0.0831	0.0831

* Data groups are given in order to relate the varying density of blends with the temperature given in Tables S1 and S2 in the Supplementary Materials.

(vegetable biodiesel) and

Table 3. Results of the parameter estimation using Tammann–Tait equation (Equations (3)–(5)) for volume percentage vegetable animal fat biodiesel/diesel blends at varying temperatures.

	D100%-BD0%, D95%-BD5%, D90%-BD10% *	D85%-BD15%, D80%-BD20%, D75%-BD25%	D70%-BD30%, D65%-BD35%, D60%-BD40%	D55%-BD45%, D50%-BD50%, D40%-BD60% D45%-BD55%	D35%-BD65%, D30%-BD70%, D25%-BD75%, D20%-BD80%	D15%-BD85%, D10%-BD90%, D5%-BD95%, D0%-BD100%
Parameters	Group 1	Group 2	Group 3	Group 4	Group 5	Group 6
a0 (gr·cm−3)	−0.5089	−0.3161	1.7982	0.7752	−2.8150	−2.9471
a1 (gr·cm−3·K−1)	0.0093	0.0084	−0.0059	0.0015	0.0264	0.0274
a2 (gr·cm−3·K−2)	−1.62 × 10−5	−1.52 × 10−5	9.07 × 10−6	−4.13 × 10−6	−4.73 × 10−5	−4.92 × 10−5
b0 (MPa)	400	400	400	400	400	400
b1 (MPa·K−1)	−1.31	−1.31	−1.31	−1.31	−1.31	−1.31
b2 (MPa·K−2)	0.001	0.001	0.001	0.001	0.001	0.001
C	0.0831	0.0831	0.0831	0.0831	0.0831	0.0831

* Data groups are given in order to relate the varying density of blends with the temperature given in Tables S1 and S2 in the Supplementary Materials.

(animal fat biodiesel) as determined by the parameter estimation process, respectively.

Equations (3)–(5) adequately describe the temperature dependence of density for the various mixing fractions of diesel and biodiesel, as shown in Figure 4 and Figure 5. The accuracy of the theoretical calculations was evaluated using Equations (12) and (13). It should be noted that the relative root mean squared error (MSE%) and the Nash and Sutcliffe efficiency index (E) for blending diesel with vegetable biodiesel were equal to MSE = 0.1–0.4% and E = 0.5363–0.8346, while for blends with animal fat biodiesel, these were equal to MSE = 0–0.5% and E = 0.5292–0.9605.

It was observed that the density of the mixture decreased as the volume percentage of diesel in the mixture increased. In addition, the density of the mixture decreased as the temperature in the mixture increased. The behaviour of mixtures is understandable because, as we know, the density ($\rho$) of a liquid is defined by the ratio of its mass (m) to its volume (V): $\rho ={\displaystyle \frac{m}{v}}$. When the mass of the liquid remains unchanged but the volume increases due to an increase in temperature, the density decreases. These results are in agreement with the findings reported by Ivanis et al. [24].

5.3. Density Evaluation for Biodiesel/Diesel Blends at Varying Temperatures Using Empirical Equation

The empirical Equation (9) was used to predict the effect of temperature and diesel volume percentage on blend densities. The empirical parameters of Equation (9) were determined using all the experimental data of blends for both vegetable and animal fat biodiesels in the fitting process of Aquasim. The result of parameter estimation for a and b values gave the values of a = 0.126 and b = 0.795. The experimental and theoretical results are presented in Figure 6a for diesel/vegetable biodiesel volume percentage blends and in Figure 6b for diesel/animal fat biodiesel volume percentage blends. It was observed that the theoretical values were close to the experimental values, achieving high values of E and low values of MSE. Specifically, for vegetable biodiesel mixtures, the values were E = 0.87 and MSE = 0.3–0.4%, while for animal fat biodiesel mixtures, the values were E = 0.8784 and MSE = 0.2–0.3%.

It is worth noting that empirical equations are widely used in engineering process design as a prior tool for decision-making.

5.4. Estimation of Specific Gravity for Biodiesel/Diesel Blends at Varying Temperatures Using Empirical Equation

Specific gravity of diesel and its blends with biodiesel is affected by temperature. In order to use a mathematical expression able to predict the specific gravity, API values were used as obtained from the experimental data (Tables S3 and S4), which were carried out with diesel and biodiesel blends. The values of specific gravity of each condition tested were calculated by using the API value (Equation (10)).

Tat and Gerpen [28] present a correlation between specific gravity and temperature (Equation (11)). The parameter estimation of constants c and d that are present in Equation (11) resulted in the values shown in

Table 4. Parameter values used in Equation (11) of diesel volume percentage mixed with vegetable biodiesel.

Par.	D100%-BD 0%	D95%-BD5%	D90%-BD10%	D85%-BD15%	D80%-BD20%	D75%-BD25%	D70%-BD30%	D65%-BD35%	D60%-BD40%	D55%-BD45%
c	0.8199	0.8244	0.8286	0.8335	0.8377	0.8427	0.8472	0.8515	0.8562	0.8586
d	0.0001	0.0001	0.0001	−0.0001	−0.0001	−0.0004	−0.0003	−0.0003	−0.0004	−0.0004
Par.	D50%-BD50%	D45%-BD55%	D40%-BD60%	D35%-BD65%	D30%-BD70%	D25%-BD75%	D20%-BD80%	D15%-BD85%	D10%-BD90%	D5%-BD95%
c	0.8613	0.8637	0.8679	0.8707	0.8726	0.8744	0.8772	0.8813	0.8850	0.8888
d	−0.0004	−0.0004	−0.0004	−0.0004	−0.0003	−0.0003	−0.0003	−0.0003	−0.0004	−0.0004

and

Table 5. Parameter values used in Equation (11) of diesel volume percentage mixed with animal fat biodiesel.

Par.	D100%-BD0%	D95%-BD5%	D90%-BD10%	D85%-BD15%	D80%-BD20%	D75%-BD25%	D70%-BD30%	D65%-BD35%	D60%-BD40%	D55%-BD45%
c	0.8199	0.8258	0.8292	0.8358	0.8404	0.8456	0.8526	0.8526	0.8562	0.8605
d	0.0001	−0.0001	−0.0001	−0.0003	−0.0004	−0.0006	−0.0007	−0.0007	−0.0008	−0.0009
Par.	D50%-BD50%	D45%-BD55%	D40%-BD60%	D35%-BD65%	D30%-BD70%	D25%-BD75%	D20%-BD80%	D15%-BD85%	D10%-BD90%	D5%-BD95%
c	0.8632	0.8668	0.8722	0.8756	0.8765	0.8789	0.8818	0.8845	0.8881	0.8901
d	−0.0009	−0.0009	−0.0010	−0.0010	−0.0009	−0.0009	−0.0009	−0.0009	−0.0009	−0.0008

. The predicted theoretical and experimental values of the specific gravity Sg using Equation (11) are presented in Figure 7. As shown in Figure 7, the theoretical calculations are in good agreement with the experimental values, which is confirmed by the results of the evaluation of the blends of diesel with vegetable biodiesel (E = 0.5375–0.9120, MSE = 0.1%) and with animal fat biodiesel (E = 0.5182–0.9988, MSE = 0.1%). It was observed that the specific gravity increased as the percentage of biodiesel fraction increased, in almost all the cases except for the D100%-BD0% and D90%-BD10%, for both vegetable and animal fat blends. Therefore, more in-depth study is suggested in order to understand these findings. The results of parameters c and d for each diesel/biodiesel blend, as presented in Table 4 and Table 5, are comparable to the values reported in the literature [28].

6. Conclusions

The goal of this study was to apply suitable mathematical equations to predict the physicochemical properties of mixtures under varying conditions of temperature and mixing ratios. To describe the relationship between the density (ρ, kg/m³) of the mixtures, the volume percentage (v%) of biodiesel mixed with diesel, and temperature (T, K), Kay’s mixing equation, the Tammann–Tait equation, and empirical formulas were employed. Mathematical models were also used to predict the specific gravity (Sg) of the mixtures. The information provided by the mathematical correlation of the physicochemical properties of diesel/biodiesel is a useful tool that can be used to select the appropriate blending rates of the different fuels based on the desired properties of the final product. Theoretical calculations were able to accurately predict (E up to 0.9988 and MSE up to 0.4%) the experimental findings. Therefore, based on Kay’s and Tammann–Tait equation estimations and the experimental findings, the density of the blends decreased as the percentage of diesel in the mixture increased and as the temperature in the mixture increased. Moreover, the density and specific gravity of the blends increased as the volume percentage of biodiesel increased and temperature decreased, as estimated by the Tat and Gerpen empirical equation. In any case, based on experimental data and theoretical calculations, blend densities (vegetable, 0.822–0.8823 gr/cm³; animal fat, 0.22–0.882 gr/cm³) were within the specifications given for biodiesel (0.86–0.9 gr/cm³) as its percentage in the blend increased and, respectively, for diesel (0.82–0.86 gr/cm³) as its percentage in the blend increased.