Density and Viscosity in Biodiesel + Diesel Mixtures from Recycled Feedstocks

Abstract

The objective of this work was to study the volumetric and transport properties of mixtures made up of biodiesel and diesel, in order to meet the desirable properties of these formulations for their practical applications. The volumetric and transport properties were analyzed for two pseudobinary mixtures constituted of diesel + beef tallow butyl ester biodiesel and diesel + waste cooking oil methyl ester biodiesel in the whole range of composition at 0.078 MPa. The study of butyl ester biodiesel was motivated by the scarcity of these properties’ data for butyl esters and the improvement of some of its physicochemical properties. The biofuels were previously transesterified from waste materials and alcohols, beef tallow with 1-butanol and cooking oil with methanol. Density measurements were performed in a vibrating tube densimeter from 293.15 to 363.15 K; the kinematic viscosity experiments were carried out in Cannon-Fenske viscometers from 293.15 to 343.15 K. The derived thermophysical properties evaluated were the excess molar volume, the partial molar volume, the thermal expansion coefficient, the dynamic viscosity and the viscosity deviation. The excess molar volumes presented positive and negative values. The Redlich–Kister correlation and the theoretical ERAS (Extended Real Association Solution) model were applied for modelling the excess molar volume. Both approaches resulted in good agreement. For viscosity, the McAllister model was implemented and yielded lower deviations for the butyl ester biodiesel.

1. Introduction

Since the beginning of the 21st century, global greenhouse gas (GHG) emissions have been steadily increasing due to the development of various countries. It is estimated that 71.6% of total GHG emissions come from the burning of fossil fuels [1]. Due to the desire to reduce or stabilize GHG emissions, together with the decrease in oil reserves, renewable energy sources such as biofuels have received increasing attention.

An alternative renewable energy source is waste materials; fats and cooking oils are some of them. Through a transesterification process, these can be converted into biodiesel. This biofuel is classified as low-toxic, biodegradable and renewable. It has a higher cetane index, no sulfur content, good lubricity and about 10–12 wt% of oxygen [2,3,4,5]. On the opposite hand, biodiesel has poor cold flow properties, tends to oxidize with air at high temperatures, and its calorific value is lower than that of fossil diesel [6,7,8,9,10,11]. Based on the above, the current design of compression engines allows biodiesel or biodiesel + fossil diesel mixtures to be successfully applied without altering the durability and reliability of the engine if the material specifications are complied [5]. Some of these crucial specifications are volumetric and transport properties; the density of biodiesel must be in the range of 860–900 kg·m⁻³ and the viscosity between 3.50 and 5.00 mm²·s⁻¹ at 313.15 K. Fuel viscosity affects the pump specifications, the atomization process and the performance of injectors [12,13]; meanwhile, the density also affects the injector systems, the cetane number and the calorific value [14,15].

All of these make it necessary to account for information about volumetric and transport properties for biodiesel + diesel mixtures at atmospheric pressure [3,6,13,16,17,18,19,20,21,22,23,24,25,26]. The temperature interval ranges from 288.15 to 373.15 K, based on fatty acid methyl esters from vegetable feedstocks (canola, castor, coconut, corn, cotton, hazelnut, jatropha, mustard, neem, palm, rapeseed, safflower, soybean, sunflower) or waste materials (cooking oil, edible tallow) [19,23]. Nevertheless, a limited number of research studies are focused on the analysis of derived volumetric properties for biodiesel + diesel systems [24,25,26]. The (fish or sunflower) biodiesel + diesel systems were reported to have negative viscosity deviation and positive excess molar volumes [24], while the (coconut or soybean) biodiesel + diesel mixtures exhibited positive and negative excess molar volumes [25,26]. For mixtures of rich-unsaturated biodiesel and diesel, negative deviations for volume occurred [27]. Regarding viscosity, it has been found that predictive models can provide good results considering specific conditions and the biodiesel feedstock, which yields differences in the values of viscosity of biodiesel (from different raw materials) and diesel blends [28].

Besides, it was reported that the use of long-chain alcohols (i.e., butanol) improved some properties of biodiesel, like the cloud point, which decreased up to 8 °C (butyl esters) [29] from 20 °C (methyl esters) [30] for the same raw material, hence the necessity of studying the production of biodiesel using other long-chain alcohols. Apart from that, the development of models and correlations to describe properties of pure compounds and mixtures requires the knowledge of experimental data, which makes their study necessary.

Considering the availability of waste materials as sources of biodiesel production, the need to study their volumetric and transport properties when mixed with diesel and the testing and development of models that represent those properties, in the present work, the density and viscosity of biodiesel + diesel pseudobinary mixtures are reported at 0.078 MPa. The biodiesel samples were based on fatty acid butyl esters (FABE) from waste beef tallow and fatty acid methyl esters (FAME) from waste cooking oil. Some data of density and viscosity were contrasted with literature data. The molecular behavior of each mixture was analyzed in terms of the coefficient of thermal expansion (${\alpha }_p$), excess molar volume ($V^{\mathrm{E}}$), partial molar volume (${\overline{V}}_i$) and viscosity deviation ($\Delta \eta$). The modelling of $V^{\mathrm{E}}$ was performed with the Redlich–Kister correlation and the ERAS model, while the viscosity deviation was calculated with the McAllister four-body models, in order to test their performance for modelling the mixtures.

2. Materials and Methods

This section contains the characteristics of the materials used in this work, as well as the experimental methodology and the equations for the derived properties. The flowchart depicted in Figure 1 summarizes the experimental methodology followed in this work.

Commercial diesel was purchased from a local fuel service station. The fatty acid butyl ester biodiesel was synthetized at 633.15 K and 21.1 MPa using waste beef tallow and 1-butanol (Sigma-Aldrich, St. Louis, MO, USA) at a molar ratio of 1:45, respectively [29]. The waste cooking oil collected from local restaurants and the waste beef tallow collected from local markets were filtered to remove solid particles before transesterification with supercritical methanol (Sigma-Aldrich, St. Louis, MO, USA) at 623.15 K, 15 MPa, 45:1 molar ratio. The chemical composition of biodiesel samples is listed in

Table 1. Chemical composition of biodiesel and properties of diesel.

Cooking Oil Methyl Ester Biodiesel Molar Mass: 277.57 g·mol−1	Beef Tallow Butyl Ester Biodiesel [29] Molar Mass: 260.18 g·mol−1
Ester profile (mol%) 1	Ester profile (area%) 1
(C15H30O2, methyl tetradecanoate) 1.063	(C18H36O2, butyl tetradecanoate) 1.904
(C17H34O2, methyl hexadecanoate) 8.763	(C20H40O2, butyl hexadecanoate) 25.549
(C19H38O2, methyl octadecanoate) 4.116	(C22H44O2, butyl octadecanoate) 50.282
(C19H36O2, methyl 9-octadecenoate) 12.025	(C23H44O2, methyl tert-butyl 9-octadecenoate) 22.264
(C19H34O2, methyl 9,12-octadecadienoate) 74.033
Yield (%) 2
99.46	87.7
Alcohol (%) 3
0.15	0.20
Water content (wt%) 4
0.125	0.121
Iodine value (g·100 g−1) 5
80.57	76.01
Saponification number (mgKOH·gFAEE−1) 5
191.81	167.56

¹ Gas chromatography–mass spectrometry (GC-MS). ² Nuclear magnetic resonance spectroscopy (1H-NMR). ³ Gas chromatography with FID detector and elite 624 capillary column. ⁴ Coulometric Karl Fischer titrator. ⁵ Titration method.

, while the properties of diesel are mentioned in

Table 2. Properties of diesel.

Diesel Molar Mass: 184.33 g·mol−1
Distillation curve (K) 1
(Initial)	427.85
(10%)	456.95
(50%)	516.25
(90%)	592.85
(Final)	615.85
Flash (K) 2	325.88
Sulfur (ppm) 3	11.52
Cetane index 4	50

¹ ASTM D86 standard test method [31]. ² ASTM D93 standard test method [32]. ³ ASTM D7039 standard test method [33]. ⁴ ASTM D4737 standard test method [34].

. The physicochemical properties of beef tallow were reported in Ref. [30], the procedures are described in the same reference. For the waste cooking oil, physicochemical properties are indicated in

Table 3. Physicochemical properties of waste cooking oil.

Property	Value
ρ at 308.15 K (kg·m−3)	936.7
Molar mass (g·mol−1)	790.76
High heating value (J·g−1)	39,562.42
Water content (ppm)	265
Saponification index	211.8
%free fatty acid (hexadecanoic acid)	3.98
%free fatty acid (octadecanoic acid)	4.19
%free fatty acid (9-octadecenoic acid)	4.35

.

Analytical methods for biodiesel characterization have been already detailed elsewhere [29,30], and the properties of diesel are based on ASTM methods. Regarding the biodiesel from waste cooking oil, the NMR spectrum is shown in Figure 2, and according to an analysis of the spectrum by a previous work [30], the biodiesel contains fatty acid methyl esters.

The experiments were carried out at 0.078 MPa (DPI 145, Druck, Leicester, UK); A vibrating tube densimeter (VTD) performed measurements in the range of 293.15–363.15 K. The VTD (DMA 4500 M, Anton Paar, Graz, Austria) contains a U-shaped measurement cell built in borosilicate. Two integrated platinum thermometers 100-Ω together with Peltier elements provide temperature control of the sample. VTD has an accuracy of 0.01 kg·m⁻³ provided by the supplier. Uncertainty in temperature was ±0.01 K. VTD was previously calibrated using dry air and water as reference fluids, as described previously [35]; air densities were extracted elsewhere [36], while the water densities were obtained from the equation proposed by Wagner and Pruß [37]. The standard uncertainties for density and temperature were 0.2 kg·m⁻³ and 0.02 K; all the uncertainties reported in this work were determined according to the Guide to the Expression of Uncertainty in Measurement [38].

The kinematic viscosity ($\upsilon$) was experimentally determined in two calibrated capillary viscometers (Cannon-Fenske sizes 75 and 100) with a standard uncertainty of 0.14 mm²·s⁻¹ by relating the elapsed time ($t$) with the viscometer constant ($K$) for each temperature, $\upsilon =K·t$. The viscometers were immersed in a water container, which was subsequently thermally controlled by a circulating bath (PD07R, with a thermal stability of ±0.005 K, Polyscience, Niles, IL, USA); thermal stability of the water container was ±0.02 K. Temperature was measured by a calibrated platinum resistance probe 100-Ω, and the estimated standard uncertainty was 0.04 K.

The mole fraction compositions for the binary mixtures were prepared gravimetrically by successive loadings by using an analytical balance (EP 520A, Precisa Gravimetrics AG, Zurich, Switzerland). Each pseudocomponent contained in a flask was degassed by a vacuum pump accompanied by heating (but preventing evaporation) and mixing by a hot-stirrer plate. For each specific composition, the biodiesel was first poured into an amber bottle and weighed; the same was subsequently performed for diesel. The estimated standard uncertainty in mole fraction was 0.0004, considering as uncertainty sources the accuracy of the balance, the repeatability of the weightings and the uncertainty of the molar masses. Part of the homogeneous mixture was loaded to a Buchner flask to configure a round circuit with the VTD, a syringe, check valves and dry-air cartridge. The flask was connected on the upper part to a syringe (10 cm³) and subsequently coupled to one port of the VTD; the other port was connected to either the dry-air cartridge or vacuum entry and the side arm flask. The configuration guaranteed the atmospheric pressure with dry air, the feeding of the mixture from the flask to the VTD through the syringe and the recovery of the sample by returning it to the flask. Isothermal density was evaluated five times as minimum, followed by temperature upwards.

Regarding the viscosity experiments, these were attained by following the ASTM D445 standard test [39] in the Cannon-Fenske viscometers previously calibrated. Briefly, a specific amount of the mixture was loaded to the viscometer. It was immersed in a water container, which was subsequently regulated by the recirculating bath; the immersed platinum probe allowed monitoring of the stabilization of temperature. Afterwards, the elapsed time in the viscometer was registered with five measurements to reduce deviations and calculate the kinematic viscosity. For density and viscosity, five measurements were performed for each point, respectively, which is the average of the measurements, while the reproducibility was considered as the standard deviation, obtaining 0.05 kg·m⁻³ for density and 0.02 mm²·s⁻¹ for viscosity.

All the characteristics of the equipment used in this work are included in

Table 4. Characteristics of the equipment used in this work.

	Model	Accuracy	Standard Uncertainty
Synthesis of mixtures by successive loadings method
Balance	Precisa model EP 520A	±0.1 mg	0.0004 in mole fraction
Density measurement
Vibrating tube densimeter	Anton Paar DMA 4500M	0.01 kg·m−3	0.2 kg·m−3
Thermometer	100-Ω platinum thermometer	0.01 K	0.02 K
Viscosity measurement
Capillarity viscometer	Cannon-Fenske	0.01 mm2·s−1	0.14 mm2·s−1
Thermometer	Polyscience, model PD07R	0.02 K	0.04 K

.

The excess molar volume ($V^{\mathrm{E}}$) was calculated by the difference between the real and the ideal molar volumes in terms of Equation (1):

$$V^{\mathrm{E}}=\left({\displaystyle \frac{P{M}_{\mathrm{M}}}{{\rho }_{\mathrm{M}}}}\right)-\left({\displaystyle \frac{x_{1}P{M}_1}{{\rho }_1}}+{\displaystyle \frac{x_{2}P{M}_2}{{\rho }_2}}\right),$$

(1)

where $PM$ denotes the molecular weight, $x$ is designated to the mole fraction, while the subscripts 1, 2 and $\mathrm{M}$ respectively correspond to biodiesel, diesel and mixture. The $V^{\mathrm{E}}$ behavior can be fitted to the Redlich–Kister polynomial function expressed in Equation (2):

$$V^{\mathrm{E}}=x_1\left(1-x_1\right){\sum }_{j=0}^{j=k}{A}_j{\left(2x_1-1\right)}^j,$$

(2)

where $k$ symbolizes the degree for the polynomial expression, and $A_j$ corresponds to each parameter; Equation (2) was delimited to a third-degree function.

The Extended Real Association Solution (ERAS) model is useful for predicting the excess molar volume of systems, which considers a physical and a chemical contribution for interactions [40,41]. The application of this model involves the use of volumetric properties of pure fluids, namely thermal expansion coefficient ($\alpha$) and isothermal compressibility ($\kappa$). For the former, the values were determined from the data of this work; for the latter, it was necessary to find density data at different pressures from the literature. Diesel density was extracted from [42], density data of FAME were measured in our laboratory, and FABE density was taken from a previous work [7]. The ERAS model’s aim is to correlate the excess molar volume of a mixture fitting three parameters, namely the association constant (${\kappa }_{\mathrm{AB}}$), the cross-association volume ($\Delta v_{\mathrm{AB}}^{*}$) and the interaction parameter ($X_{\mathrm{AB}}$). The fitting was accomplished in a spreadsheet, with Gauss–Newton method. A brief description of the algorithm used for obtaining the parameters of ERAS model is as follows: reduction molar volume ($v^{*}$) is calculated with the temperature, molar volume and isobaric thermal expansion coefficient for each component. This reduction molar volume is used for calculating the stoichiometric hard-core volume fractions of the components (${\mathit{\Phi }}_{\mathrm{i}}$), which also depends (separately) on the hard-core volume fraction of a component in the mixture (${\varphi }_{\mathrm{i}}$). A system of two equations with two unknowns (the volume fractions) is solved for the chemical contribution of the excess molar volume. Besides, the stoichiometric volume fractions lead to obtaining the surface fraction of each component (${\mathit{\Theta }}_{\mathrm{i}}$), a part of the calculation of the reduction temperature and reduction pressure of the mixture, dependent on the reduction property of each component. For reduction pressure, the isothermal compressibilities are necessary. Reduction temperature also depends on the reduced volume of the mixture ($\tilde{v}$), defined as the quotient of the molar volume of the mixture and the reduction molar volume ($v/v^{*}$). The value of molar volume that equals the reduction temperature functions is part of the physical contribution of the excess molar volume. The detailed equations can be found elsewhere [43].

Partial molar volumes for biodiesel and diesel were calculated based on Equations (3) and (4):

$${\overline{V}}_1=V^{\mathrm{E}}+V_1^0+\left(1-x_1\right){\left({\displaystyle \frac{\mathit{\partial }{V}^{\mathrm{E}}}{\mathit{\partial }{x}_1}}\right)}_{P, T, x_2},$$

(3)

$${\overline{V}}_2=V^{\mathrm{E}}+V_2^0-x_1{\left({\displaystyle \frac{\mathit{\partial }{V}^{\mathrm{E}}}{\mathit{\partial }{x}_1}}\right)}_{P, T, x_2},$$

(4)

where the molar volumes for pure components biodiesel and diesel are denoted by $V_1^0$ and $V_2^0$, respectively.

The isobaric expansion coefficient was calculated according to Equation (5):

$$\alpha =-{\displaystyle \frac{1}{\rho }} \left({\displaystyle \frac{\mathit{\partial }\rho }{\mathit{\partial }T}}\right).$$

(5)

The experimental densities were fitted to a first order temperature function ($\rho ={\kappa }_0+{\kappa }_1 T$), hence,

$$\alpha ={\displaystyle \frac{{\kappa }_1}{{\kappa }_0+{\kappa }_1 T}}.$$

(6)

Respectively, ${\kappa }_0$ and ${\kappa }_1$ are the $y$-intercept and the slope.

Once the kinematic viscosity was determined, it was possible to calculate dynamic viscosity ($\eta =\nu ·\rho$), then to evaluate the deviation ($\Delta \eta$) as follows:

$$\Delta \eta =\eta -\left(x_1{\eta }_1+x_2{\eta }_2\right),$$

(7)

where the subscripts 1 and 2 denote the components biodiesel and diesel, in that order.

Regarding the McAllister four-body model [44], being an empirical model, its implementation was also performed in a spreadsheet with the Gauss–Newton method, which yielded the optimized parameters, the average deviation and the standard deviation of the model. The expression for this model is

$$\begin{array}{c}\ln \eta =x_1^4\mathit{ln}{\eta }_1+4x_1^3{x}_2\mathit{ln}{\eta }_{1112}+6x_1^2{x}_2\mathit{ln}{\eta }_{1122}+4x_1{x}_2^3\mathit{ln}{\eta }_{222}+x_2^4\mathit{ln}{\eta }_2-\\ \mathit{ln}\left(x_1+x_{2}P{M}_2/P{M}_1\right)+4x_1^3{x}_2\mathit{ln}\left[\left(3+P{M}_2/P{M}_1\right)/4\right]+\\ 6x_1^2{x}_2^2\mathit{ln}\left[\left(1+P{M}_2/P{M}_1\right)/2\right]+4x_1{x}_2^3\mathit{ln}\left[\left(1+3P{M}_2/P{M}_1\right)/4\right]+\\ x_2^4\mathit{ln}\left(P{M}_2/P{M}_1\right),\end{array}$$

(8)

where the optimized parameters are ${\eta }_{1112}$, ${\eta }_{1122}$ and ${\eta }_{222}$.

3. Results and Discussions

This section details the results of the density measurements of the biodiesel + diesel systems, which allowed the calculation of the derived properties, such as excess molar volume and partial molar volume, with the Redlich–Kister equations and the ERAS model. The kinematic viscosity was also discussed; their values were used to calculate dynamic viscosities and the excess viscosity by means of the McAllister four-body model.

3.1. Density

The experimental density values ($\rho$) for waste beef tallow butyl ester diesel (1) + diesel (2) and waste cooking oil methyl ester biodiesel (1) + diesel (2) pseudobinary systems are depicted isothermally in Figure 3 and summarized in Table S1.

Density was observed to increase as the biodiesel amount incremented in the pseudobinary solution at a stated temperature; in the case of temperature variations, density decreased as the temperature was rising. This last was ascribed to the constant increment of the space between molecules but without abrupt alterations that could evaporate the mixture. Contrasting the density between both systems at fixed composition and temperature, densities were lower for the system containing waste beef tallow butyl esters. This behavior was associated with the different alkyl radical; the biodiesel density for methyl ester group (FAME) is superior to longer alkyl carbon chains such as ethyl (FAEE), propyl (FAPE) or butyl (FABE) [7]. Moreover, the waste beef tallow butyl ester biodiesel contained a high proportion of saturated esters of about 77.735% of the relative area (constituted by butyl tetradecanoate, butyl hexadecanoate and butyl octadecanoate), as commonly occurs in fat materials. For the waste cooking oil methyl ester biodiesel, the saturated ester components comprised 13.942 mol% (methyl tetradecanoate, methyl hexadecanoate, methyl octadecanoate). The cooking treatment of the fresh vegetable oils in restaurants could mean that the double C=C bonds of the unsaturated esters were converted to simple C–C bonds. Then, the saturated esters possess lower density than unsaturated ones, as previously reported for pure methyl esters; for instance, contrasting the saturated component methyl palmitate (C₁₇H₃₄O₂) [45] and the unsaturated chemical methyl palmitoleate (C₁₇H₃₂O₂) [46], the respective densities are 854.5 and 857.6 kg·m⁻³ at 308.15 K or 817.3 and 819.8 kg·m⁻³ at 358.15 K. In summary, those effects caused the densities of FAME biodiesel and their mixtures with diesel to be even higher than the corresponding for FABE + diesel blends. Complementarily, Figure 4 summarizes the trends for density of waste cooking oil biodiesel (FAME) [23,42,47,48] and tallow biodiesel (FAME [23,49] or FABE) obtained in this work and those reported elsewhere as a function of temperature. This comparison yielded percent relative deviations from 0.24 to 3.71% for FAME and from 0.42 to 0.97% for FABE. In general, the density of biodiesel varies from 810 to 920 kg·m⁻³; these differences are attributed to the different chemical compositions of the ester profile, which alters the corresponding properties, such as cetane number and oxidation stability; a major presence of unsaturated bonds leads to lower density, cetane number and oxidation stability [50,51].

3.2. Volumetric Properties

The excess molar volume ($V^{\mathrm{E}}$) behavior for both mixtures is contained in Table S2 and also shown in Figure 5, along with the excess molar volumes fitted to the Redlich–Kister polynomial function. The experimental (symbols) excess molar volume and the corresponding calculated (lines) with the Redlich–Kister equation are depicted as a function of biodiesel composition for both systems.

The corresponding parameters are listed in

Table 5. Redlich–Kister parameters for the excess molar volume correlation.

Parameter	A0 1	A1 1	A2 1	A3 1	$\mathit{s}$ 1	AAD (%)
T (K)	waste cooking oil methyl ester biodiesel (1) + diesel (2)
293.15	0.1123	−1.4763	−0.6015	0.8471	0.0188	13.94
303.15	0.1618	−1.5012	−0.7173	0.6607	0.0222	12.69
313.15	0.1653	−1.6925	−0.7872	0.7788	0.0249	11.60
323.15	0.0841	−1.8618	−0.6734	0.8722	0.0420	94.60
333.15	0.0892	−1.9400	−0.5437	0.8304	0.0332	31.50
343.15	0.3930	−2.2957	−1.1090	1.2698	0.0372	14.37
353.15	0.3798	−2.4213	−1.0682	1.3524	0.0420	13.72
363.15	0.4071	−2.5143	−1.2786	1.1875	0.0420	11.75
T (K)	waste beef tallow butyl ester biodiesel (1) + diesel (2)
293.15	−0.3844	−1.7615	−0.1971	0.4791	0.0183	11.39
303.15	−0.3726	−1.7793	−0.1586	0.4493	0.0207	12.81
313.15	−0.3623	−1.8012	−0.0256	0.3727	0.0199	10.80
323.15	−0.3722	−1.8347	0.1367	0.5746	0.0163	8.98
333.15	−0.3235	−1.9433	0.2209	0.6131	0.0116	5.96
343.15	−0.2798	−1.9425	0.2941	0.4672	0.0016	0.68
353.15	−0.1983	−1.9498	0.2679	0.3144	0.0078	4.30
363.15	−0.1390	−2.1558	0.7994	−0.1146	0.0182	7.25

¹ Units are 10⁶ × m³·mol⁻¹.

, which gave residuals in terms of the average absolute deviation of 94.60% (FAME system) and 0.68% (FABE system) as maximum. This deviation was calculated as $AAD \left(\%\right)=\left(\frac{100}{N}\right){\sum }_{i=1}^N\left|\frac{V_{\exp }^{\mathrm{E}}-V_{\mathrm{cal}}^{\mathrm{E}}}{V_{\exp }^{\mathrm{E}}}\right|$, where $N$ refers to the number of data, and the superscripts exp and cal denote the experimental and calculated values. The standard deviation of the model was also calculated as $\sqrt{{\sum }_{i=1}^N{\left(V_{\exp }^{\mathrm{E}}-V_{\mathrm{cal}}^{\mathrm{E}}\right)}^2/\left(N-m\right)}$, with $m$ as the number of parameters.

As seen in Figure 5, the excess molar volumes exhibited both positive and negative values. The former consisted of an expansion region (biodiesel mole fraction from zero to approximately 0.4–0.5) associated with predominant dispersion forces between esters (polar molecules) and diesel components (mostly non-polar compounds such as linear and cyclic hydrocarbons, olefins and some benzene derivatives), which do not tend to create bonds with the esters; this effect was caused by the weak interaction between the components of the mixture causing predominant repulsion forces. The latter belonged to a compression region from about 0.4–0.5 to 1.0 of biodiesel composition, which could be ascribed to possible interactions between the -OH groups (traces of alcohol, water and glycerol) with low concentration and the functional groups from biodiesel causing less dominant effect of the steric hindrance of esters [52]. Nevertheless, the polarity of esters was hindered as the alkyl carbon chain was increasing; thus, there could be predominant attraction forces between the esters and the non-polar molecules of diesel. Regarding the temperature, $V^E$ increased as the temperature increases at any composition for the FABE-containing system; meanwhile, the excess molar volumes for the FAME-containing systems exhibited a crossover point along the composition as temperature changed. $V^E$ increased with temperature on the expansion region; thereafter, the excess molar volumes tended to be near ideality at low temperatures, going more negative at higher temperatures.

Positive excess molar volumes for biodiesel + diesel systems have been reported [24,26,53]. The binary systems based on pure chemicals agreed with the preceding information: alkane + methyl ester [54,55], alkyl ester + 1-alkanol [56,57,58], alkanol + cycloalkane [55]. Conversely, positive and negative excess molar volumes have been reported along the composition range for coconut biodiesel + diesel [25], 2-octanol + octane [59] and methyl ester biodiesel (from beef tallow) + alcohol [52,60] mixtures, where the negativity was found at low alcohol compositions for the last ones. There is no doubt that interaction molecular effects between chemical groups can alter the excess properties, for instance, non-polar (hydrocarbons from diesel), esters, alcohols, water and glycerol; all of this was confirmed in the study of $V^{\mathrm{E}}$ for ternary systems (biodiesel + diesel + alcohol or their corresponding base components) [52,53,55] where regions with negative $V^{\mathrm{E}}$ values were observed.

Regarding the ERAS model, the results are presented in

Table 6. ERAS model parameters and deviations by fitting the excess molar volumes.

	$\mathit{T}=$ 293.15 K	$\mathit{T}=$ 313.15 K	$\mathit{T}=$ 333.15 K	$\mathit{T}$ = 353.15 K
	waste cooking oil methyl ester biodiesel (1) + diesel (2)
${\kappa }_{AB}$ (–)	−0.4139	−0.4246	−0.4066	−0.4293
$\Delta v_{AB}^{*}\times {10}^6$ (m3·mol−1)	39.8175	44.9050	44.4452	50.5017
$X_{AB}$ (MPa)	219.5644	237.7053	200.1880	225.9977
$AAD$ (%)	13.81	10.03	27.19	13.72
$s\times {10}^6$ (m3·mol−1)	0.0333	0.0441	0.0573	0.0751
	waste beef tallow butyl ester biodiesel (1) + diesel (2)
${\kappa }_{AB}$ (–)	0.1500	0.2630	0.3182	0.3198
$\Delta v_{AB}^{*}\times {10}^6$ (m3·mol−1)	−103.4619	−61.3002	−47.3863	−46.4457
$X_{AB}$ (MPa)	112.2140	101.1186	84.3279	77.4472
$AAD$ (%)	10.67	10.19	7.35	4.27
$s\times {10}^6$ (m3·mol−1)	0.0293	0.0324	0.0242	0.0131

. Deviation was evaluated in terms of $AAD$ and standard deviation $s$ (m³·mol⁻¹), $\sqrt{{\sum }_{i=1}^N{\left(V_{\exp }^{\mathrm{E}}-V_{\mathrm{cal}}^{\mathrm{E}}\right)}^2/\left(N-m\right)}$, where $m$ refers to the number of parameters.

The cross-association constant (${\kappa }_{AB}$) had some dependence in temperature for FABE + diesel system. Its values span from −0.4293 to 0.3198, which are characteristic of non-associating compounds [61]. The cross-association volume ($\Delta v_{AB}^{*}$) for the FAME + diesel system was positive at any temperature, reflecting the weak intermolecular interactions. For the system FABE + diesel, the cross-association volume was negative, which indicated the interaction of diesel with FABE was stronger than its interaction with FAME. However, the values were much lower than those of associating compounds, even if they exhibit small association degree [61]. The interaction parameter ($X_{AB}$) reflects the difference of dispersive interactions and presents higher values than the ones used for associating compounds (<12 MPa) [62]. Regarding the physical and chemical contributions of the ERAS model, for all the temperatures and systems, the chemical interactional contributions dominated when more alkyl ester was present in the mixture, as the associating phenomenon (important to the chemical contribution) predominated the intermolecular interactions. The opposite occurred when the mole fraction of diesel was larger.

Partial molar volumes for biodiesel and diesel are presented in Figure 6 and Table S3. The behavior of partial molar volumes for each component was proportional to temperature changes, both ${\overline{V}}_1$ and ${\overline{V}}_2$ rose as temperature increased. There was noticed a slight increase in partial molar volume of biodiesel (${\overline{V}}_1$) and a small decrease in partial molar volume of diesel (${\overline{V}}_2$) at $x_1$ = 0.1 and 0.9, in that order. This performance was likely caused by the predominant effect of dispersion forces found for ${\overline{V}}_1$ and the strong attractive forces settled for ${\overline{V}}_2$, which agreed with the excess molar volume results.

For the isobaric expansion coefficient, the values are plotted as a function of composition (mole fraction of biodiesel) in Figure 7 and listed in Table S4.

It can be noticed that a linear relationship exists between the temperature and the $\alpha$ value with a positive slope, indicating the intermolecular forces tended to be weaker as temperature increased, preserving the aggregation state in the liquid phase. It confirmed the expansion of the solution; hence, the sample could occupy more volume. Besides, $\alpha$ values were higher for diesel than biodiesel, indicating that it is much easier to break intermolecular bonds, and, therefore, diesel expands more when heated. Referring to both systems, higher biodiesel content limits the expansion of the solution, occupying a smaller volume at high temperature. The isobaric expansion coefficient was lower for the FAME biodiesel + diesel than the FABE biodiesel + diesel system at the same temperature and composition. The former exhibited a smoothed curve behavior when diminishing as a function of composition, while the latter presented an almost linear relation. The effect of ester composition was also confirmed on the values for this property; the waste cooking oil biodiesel was constituted of more saturated esters than beef tallow biodiesel, which induced a lower $\alpha$, as the content of this kind of esters was superior. Although there is no normalized $\alpha$ value for this property, a high coefficient of expansion is translated into lower densities as the temperature increases, leading to a loss of engine power when heating the fuel [48,63].

3.3. Viscosity

The kinematic viscosity ($\upsilon$) obtained via the Cannon-Fenske viscometers for the two binary systems is presented in Table S5 and Figure 8.

The kinematic viscosity for biodiesel samples (FAME and FABE) was greater than the one for diesel at a fixed temperature; consequently, $\upsilon$ for binary mixtures decreased as the composition of diesel incremented. Besides, the samples tended to be less viscous with temperature increasing; this behavior was attributed to the influence of the ester carbon chain length, the number of double and triple bonds and the conformation of the fatty acids used as feedstock. Viscosities for waste cooking oil methyl ester biodiesel were compared with some selected data from the literature [23,64,65,66,67,68], ranging deviations of 2.28 and 0.55 mm·s⁻¹ at 303.15 and 343.15 K, respectively; even at 313.15 K, the deviations reach 1.83 mm·s⁻¹ as maximum. All the data sets correspond to waste cooking oil methyl ester biodiesel; the difference in viscosity values confirms that the ester profile for each sample is not the unique one that alters viscosity. Impurities such as glycerides and water content, the unreacted material indicated in the reaction yield and the presence of polyunsaturated alkyl esters can change the viscosity of biodiesel [69]. Moreover, Sabudak and Yildiz [68] reported different viscosity data for waste cooking oil biodiesel by distinct reaction and purification methods that varied from 4.63 to 5.82 mm·s⁻¹. The comparison is displayed in Figure 9.

The calculated dynamic viscosity ($\eta =\nu ·\rho$) is contained in

Table 7. Dynamic viscosity for biodiesel (1) + diesel (2) systems.

T (K)	293.15	303.15	313.15	323.15	333.15	343.15
x1	η × 103 (Pa·s) waste cooking oil methyl ester biodiesel (1) + diesel (2)
0.0000	2.74	2.24	1.79	1.51	1.31	1.12
0.1006	3.18	2.53	2.02	1.72	1.47	1.27
0.2003	3.56	2.81	2.23	1.86	1.60	1.38
0.2998	3.92	3.11	2.45	2.03	1.74	1.47
0.4996	4.72	3.75	2.97	2.42	2.04	1.71
0.7007	6.36	4.69	3.69	3.06	2.54	2.13
0.7998	7.27	5.20	4.10	3.35	2.77	2.32
0.8990	8.21	5.74	4.52	3.66	3.02	2.52
1.0000	9.25	6.29	4.93	3.98	3.27	2.74
x1	η × 103 (Pa·s) waste beef tallow butyl ester biodiesel (1) + diesel (2)
0.0000	2.74	2.24	1.79	1.51	1.31	1.12
0.1001	3.14	2.54	2.03	1.70	1.46	1.26
0.2003	3.67	2.91	2.30	1.93	1.65	1.41
0.2999	4.28	3.33	2.61	2.18	1.84	1.57
0.5001	5.57	4.25	3.28	2.72	2.26	1.90
0.6998	6.97	5.26	3.96	3.23	2.67	2.22
0.8001	7.66	5.75	4.30	3.48	2.87	2.36
0.9001	8.45	6.29	4.68	3.74	3.07	2.50
1.0000	9.43	7.02	5.19	4.16	3.42	2.80

.

For excess viscosity, the isothermal values are plotted in Figure 10 as a function of composition along with $\eta$ calculated by the McAllister four-body model, whose parameters are declared in

Table 8. Parameters for the McAllister model for the correlation of the viscosity deviation ¹.

Parameter	η1112 1	η1122 1	η222 1	$\mathit{s}$ 1	AAD (%)
T (K)	waste beef tallow butyl ester biodiesel (1) + diesel (2)
293.15	1.62	2.64	3.83	0.02	2.93
303.15	1.49	2.32	3.01	0.02	3.72
313.15	1.32	2.06	2.41	0.01	2.48
323.15	1.06	2.05	1.97	0.01	6.17
333.15	0.99	1.87	1.68	0.01	7.34
343.15	0.90	1.75	1.45	0.02	15.51
T (K)	waste cooking oil methyl ester biodiesel (1) + diesel (2)
293.15	4.58	1.40	4.34	0.08	7.61
303.15	2.56	1.61	3.16	0.02	4.33
313.15	2.35	1.46	2.50	0.01	4.25
323.15	2.40	1.29	2.12	0.03	21.12
333.15	2.17	1.21	1.82	0.03	16.82
343.15	2.29	1.04	1.62	0.02	25.92

¹ Units are mm²·s⁻¹.

.

Dynamic viscosity presented the same behavior as kinematic viscosity. The value of this property decreases with temperature rising; the viscosity increases as the value of $x_1$ increases, since diesel is less viscous than biodiesel. Therefore, a mixture with a higher biofuel content has a greater resistance to deformation, implying greater effort for its transport in the engine injection.

The molecular interactions can be analyzed by means of viscosity deviation where molecules with different shapes and size coexisted in both systems. This parameter presented negative values throughout the composition range; therefore, the repulsion forces between molecules were stronger than attraction ones, caused by the smaller molecules were inserted on the structure of the bigger molecules in the absence of interactions and the steric hindrances [57,70,71]. The viscosity deviation increased with respect to temperature rising and tended to zero due to a better interaction between the components because of the molecular motion generated by the solution heating [52]. The minimum value was reached at about equimolar composition in both systems, being the system that contained waste cooking oil biodiesel, the one with the lowest value (highest negative). The greater dispersion forces observed for waste cooking oil biodiesel + diesel in contrast with the waste beef tallow biodiesel + diesel system could be associated with (1) the greater number of saturated esters and (2) the lower alkyl carbon chain esters in the waste cooking oil biodiesel that increased its polarity. This behavior could be indirectly confirmed by the relation found in the binary mixtures hexadecane + fatty acid methyl ester [54], alkyl propanoate + alkane [72] and hexadecane + fatty acid ethyl ester [73]; the ester polarity decreased as its carbon chain was larger, allowing a better molecular arrangement.

4. Conclusions

This work was a first effort to study the volumetric and transport properties for diesel + waste material biodiesel systems at 0.078 MPa and different temperatures and compositions. The density of waste cooking oil biodiesel ranged from 837.9 to 888.3 kg·m⁻³; for waste beef tallow biodiesel, it ranged from 782.0 to 832.2 kg·m⁻³, values that were similar to those found in the literature. Thermodynamic and transport-derived properties were analyzed to understand the molecular interactions of waste beef tallow butyl ester biodiesel or waste cooking oil ester biodiesel with fossil diesel. The calculated properties were the excess molar volume, partial molar volume, isobaric expansion coefficient and viscosity deviation.

The excess molar volume performed a sigmoidal shape that tended to change from positive to negative values as the composition of biodiesel increased, considering that such behavior was interpreted as a contribution of opposing effects. The weak interactions induced by repulsion forces were the most relevant on the former region, and the stronger interactions predominated over the effect of steric hindrance of esters on the latter. The slight variations on the partial molar volumes confirmed the mentioned effects on volumetric properties. The negative values on the viscosity deviations were ascribed to minimum molecular interactions accompanied by the insertion of the smaller molecules into the structure of the bigger ones. Finally, the ERAS model was applied successfully to the excess molar volumes of this work, while the McAllister four-body model depicted the excess viscosity with deviation lower than 25.92%. The application of the ERAS model required some thermophysical properties, hence the necessity for experimental data of volumetric properties for the development and application of theoretical models. The measurement was limited at atmospheric pressure. In future work, it will be desirable to extend the pressure interval for a better understanding of the behavior of the mixtures and a wider availability of data for the use and testing of models.