Volumetric and Transport Properties of Commercial Diesel + FAME from Residual Chicken Fat in the Interval of 293.15 to 353.15 K

Abstract

This study presents the experimental characterization of the volumetric and transport properties of pseudo-binary mixtures of commercial diesel and residual chicken fat methyl ester biodiesel over the temperature range of 293.15–353.15 K at 0.078 MPa. Density measurements were performed using a U-shaped vibrating-tube densimeter; kinematic viscosities were obtained using Cannon–Fenske capillary viscometers. The results show that density decreased with increasing temperature and diesel content. The excess molar volume (V^E) was negative for all mixtures; the strongest volumetric contraction took place at around x₁ ≈ 0.4–0.6. The Redlich–Kister equation and the Prigogine–Flory–Patterson (PFP) model were applied to represent excess molar volumes, with an absolute average deviation (AAD) lower than 14.92%. The thermal expansion coefficient (${\alpha }_P$) and its excess property (${\alpha }_P^E$) further confirmed the existence of non-ideal mixing driven by polar–apolar interactions. The kinematic viscosity ($\nu$) was confirmed to be temperature-dependent and increased with the amount of FAMEs; this effect can be associated with the greater polarity and structural rigidity of esters. The McAllister model also adequately reproduced the dynamic viscosity ($\eta$) with an AAD < 4.2%. Furthermore, an increase in the activation enthalpy (${\Delta H}^{\ne }$) was observed at higher FAME fractions, indicating a high energy demand is required to overcome the internal energy barrier for the initial displacement of the molecules.

1. Introduction

Biofuels produced from biomass represent a renewable energy source whose utilization has been consolidated as a strategic alternative to fossil fuels [1]. Among biofuels, bioethanol and biodiesel are the most relevant; bioethanol is obtained through the fermentation of sugars or starches while biodiesel is produced from transesterification of oils or fats. Both are compatible with existing internal combustion engine technologies and the fuel distribution network [2].

The operation of an engine with vegetable oil was demonstrated at the end of the nineteenth century, while the first vehicles were designed based on ethanol as fuels. However, the development and low cost of petroleum-derived fuels throughout the twentieth century relegated biofuels to a secondary role until following the energy crisis of the 1970s and the rise in environmental awareness, research and application in this field gained renewed momentum [3]. Nowadays, their use is regulated in various regions of the world through mandatory blending policies in variable proportions. For instance, blends such as E10 and E85 (ethanol in gasoline) or B5, B10, and B20 (biodiesel in conventional diesel) have been widely adopted in countries such as Brazil, the United States, and members of the European Union, bringing benefits in terms of reduced greenhouse gas emissions and diversification of the energy matrix [4].

The use of liquid fuels derived from crude oil, especially diesel, contributes to a decrease in air quality and the production of greenhouse gases, as well as the generation of suspended particles, nitrogen oxides (NOx), carbon monoxide (CO), and unburned organic compounds. The amount of these pollutants emitted into the environment depends on the combustion efficiency, fuel atomization, and the quality of the air–fuel mixture within the piston [5]. Some of the hydrocarbon properties that have the greatest influence on these processes are viscosity and volumetric properties, as these determine the thermodynamic state of the fuel [6]. The kinematic viscosity plays a decisive role in the formation of the injection spray, the mean droplet diameter, jet penetration, and its dispersion with air, thereby influencing combustion efficiency, ignition delay, and the formation of carbonaceous residues [7]. Regarding volumetric properties, isobaric thermal expansivity is essential for understanding fuel behavior [8].

Therefore, the fatty acid methyl esters (FAMEs) produced from waste oils or fats have become increasingly established as a renewable alternative for the partial substitution of petroleum-derived diesel. Based on the oxygen content of the FAME molecules, their combustion reduces unburned material and the CO amount, and their lubricity is also enhanced [9]. In contrast, FAMEs have limitations, such as high viscosity, that affect atomization during engine operation, especially in cold climates, and they tend to oxidize during long periods of storage, generating peroxides and sediments [10]. Morita and Sugiyama [11] determined that density and viscosity alter the average droplet diameter and the concentration of pollutants in exhaust gases. An increase in biodiesel in blends with diesel increases the density and viscosity of the mixture, which promotes high NOx emissions but reduces CO emissions under certain conditions [12]. For this reason, the systematic measurement of kinematic viscosity and density for commercial diesel and fatty acid methyl ester (FAME) pseudo-binary systems, together with their derived volumetric and transport properties, constitutes a relevant research subject for understanding the molecular interactions in the temperature range of 293.15 to 353.15 K at atmospheric pressure.

2. Materials and Methods

2.1. Fuel Characterization

The diesel was collected from a local service fuel station in Mexico City.

Table 1. Properties of commercial diesel.

Test	Method	Results	Expanded Uncertainty
Cetane index	ASTM D4737-10 [13]	41.2	1.00
Flash point	ASTM D93-20 [14]	325.95 K	1.14 K
Total sulfur	ASTM D7039-15a [15]	6.15 mg·kg−1	0.80 mg·kg−1
Density @ 15 °C	ASTM D1298-12b [16]	0.8565 g·cm−3	0.0010 g·cm−3
Distillation temperature/K
Initial boiling	ASTM D86-18 [17]	440.15	1.64
10% recovered		466.20	0.97
50% recovered		532.05	0.95
90% recovered		615.75	1.10
Final boiling		654.55	1.06
Distillation residue/%v/%v		3.5	0.05

shows the tests, methods [13,14,15,16,17], and results of the experimental characterization of the diesel sample carried out in our laboratory, where distillation tests were performed at the atmospheric pressure of Mexico City (0.078 MPa).

The chicken fat sample consisted of the residues left after cooking and was collected from several Mexican rotisserie establishments located in the Tehuantepec region of Oaxaca, Mexico. This fat was subject to pretreatment that consisted of washing the fat with distilled water, followed by centrifugation, filtering and drying at 373.15 K. The latter stages were done to remove the suspended solids, proteins, inorganic salts, water-soluble compounds and moisture, aiming to attain a high yield of transesterification; in contrast, water can promote the hydrolysis of triglycerides that tend to produce free fatty acids (FFAs) and subsequent secondary saponification [9,18]. Afterwards, the saponification index (SI) for the residual chicken fat was 195.71 mg KOH·g⁻¹ oil. Literature data indicate an SI content between 168 and 196 mg KOH·g⁻¹ fat for pork lard and chicken fat [19,20].

Pretreated chicken fat was esterified with methanol at 333.15 K, aiming to diminish the free fatty acids; esterification was carried out using sulfuric acid as a catalyst at 0.7 wt% for 2 h and a molar ratio of methanol:fat of 6:1 based on a literature survey [21,22]. Afterwards, the remaining fluid was stabilized for 24 h to allow for phase separation, yielding a pretreated fat (heavy) phase and an FFA-rich phase (light). The fat fraction was washed with distilled water to remove the excess alcohol and traces of catalyst, followed by drying at 373.15 K.

The esterified fat was mixed with methanol at a molar ratio of 6:1 to carry out the transesterification using potassium hydroxide (1 wt%) as a catalyst for 1 h at 338.15 K. Afterwards, the fluid was kept stable for 24 h for further phase separation [21]. The upper phase rich in FAMEs was extracted to be washed with distilled water to remove catalyst and alcohol at 323.15 K. Water was removed from biodiesel by decantation, and traces of moisture together with volatile compounds were removed by evaporation at 373.15 K [9,23].

Fourier Transform Infrared Spectroscopy (FTIR) analysis was carried out with the purpose of identifying the characteristic functional groups present in the FAME. The FTIR spectrum of the biodiesel produced from residual chicken fat is depicted in Figure 1 and exhibits the distinct absorption bands associated with fatty acid methyl esters (FAMEs). A strong band at 1737 cm⁻¹ corresponds to the carbonyl stretching vibration ν(C=O), which is typical of ester functional groups [24]. The transesterification that produced the fatty acid methyl esters was also confirmed by the C–O–C stretching vibrations ν(C–O–C) identified by the bands at 1249, 1169, 1112, and 1013 cm⁻¹ [25,26], as well as the CH₃ groups adjacent to the carbonyl group within the ester molecule identified at 1190 cm⁻¹ [27]. The presence of long hydrocarbon chains derived from fatty acids was confirmed by the bending vibration $\rho$(CH₂) at 722 cm⁻¹; the aliphatic stretching vibrations ν(C–H) were associated with the peaks at 2923 and 2853 cm⁻¹. Cis-type unsaturation (oleate/linoleate) seems to be confirmed by the weak signal at 3005 cm⁻¹ that was produced by unsaturated fatty (oleic and linoleic) acids [26]. Low concentrations of impurities can be indirectly confirmed in the biodiesel sample since no significant signals were observed at 3600–3200 cm⁻¹, associated with water or alcohols, at 1705 cm⁻¹ for free fatty acids or at 1560 and 1410 cm⁻¹ for carboxylate salts, as well as any signals associated with monoglycerides or diglycerides [28,29].

The transesterification reaction was verified by ¹H-NMR analysis (Bruker spectrometer at 750 MHz, using CDCl₃ as a solvent); the spectrum for the FAME is depicted in Figure 2. The multiplet (D) between 5.30 and 5.40 ppm was ascribed to the olefinic protons (–CH=CH–) of unsaturated fatty acids; it indicates that the lipids have double bonds [30]. The highest peak was the singlet (M) at 3.65–3.70 ppm that corresponds to the methoxy group (–O–CH₃) of the methyl ester [30,31]. The weak triplet signal (A) observed between 2.75 and 2.85 ppm represents an allylic group with protons adjacent to double bonds that corroborates unsaturated chains [31,32]; the other triplet (A) between 2.2 and 2.4 ppm was attributed to the two protons of the methylene group α next to the carbonyl (–CH₂–COO–). A triplet (F) between 1.90 and 2.10 ppm revealed bis-allylic protons, which confirmed polyunsaturated fatty acids. The multiplet (B) between 1.55 and 1.70 ppm indicates β-methylene protons related to the carbonyl group. The signal (C) between 1.20 and 1.46 ppm was ascribed to the aliphatic –(CH₂)ₙ– chain of the long-chain fatty acid esters. The triplet (E) at 0.80–0.95 ppm indicates a methyl group [33]. Finally, the yield was determined by characteristic signals identified in the ¹H-NMR spectrum based on Equation (1):

$$Y(\%)=100\times {\displaystyle \frac{2I_M}{3I_A}}$$

(1)

where $I_M$ is the integrated area of the methoxy singlet (M), and $I_A$ is the integrated area of the triplet A (2.2–2.4 ppm). The calculated yield was 94.5%.

2.2. Density Measurements

The molecular weight for diesel and biodiesel was estimated by cryoscopy (WR 5009, Precision Systems, Natick, MA, USA); the reported values were 182.37 g·mol⁻¹, $U_c\left(PM\right)$ = 10.6 g·mol⁻¹ (k = 2) for diesel and 296.11 g·mol⁻¹, $U_c\left(PM\right)$ = 12.4 g·mol⁻¹ (k = 2) for the FAME. Since both materials are multicomponent mixtures, the reported values correspond to the average molecular weights of the samples analyzed. Binary mixtures of diesel ($x_1$) and FAME ($x_2$) were prepared gravimetrically by successive weighing on a high-precision analytical balance (sensitivity ±0.1 mg, EP 520A, Precisa Gravimetrics AG, Zurich, Switzerland). Each pseudo-component, stored in separate flasks, was degassed under vacuum with controlled heating to avoid evaporation and stirred magnetically to ensure homogeneity. For each composition, biodiesel was first transferred to an amber flask and weighed, followed by diesel, ensuring controlled and reproducible mixture preparation.

The density measurements of the commercial diesel + FAME mixtures were carried out with four repetitions for each experimental point by transferring a portion of the homogeneous mixture to a Büchner flask, which was configured as part of a closed-loop circuit, based on Figure 3, and connected to a (Anton Paar DMA 4500M, Graz, Austria) vibrating-tube densimeter (VTD). The VTD has an internal temperature sensor; previous calibrations indicated uncertainties of $2.6\times {10}^{-5}$ g·cm⁻³ and 0.02 K. The top of the Büchner flask was connected to a 10 cm³ syringe (S), which was attached to one inlet of the VTD; the other inlet was alternately coupled to the dry air cartridge (F), vacuum pump (VP), or lateral flask. This configuration ensured dry-air operation under atmospheric pressure, a careful mixture feed from the flask to the VTD and reproducibility of measurements by minimizing evaporation of light components.

2.3. Calculations of Volumetric Properties

The excess molar volume ($V^E$) for the mixture in cm³·mol⁻¹ was determined as the difference between the corresponding experimental molar volume and the ideal molar volume, which is the sum of the molar fraction multiplied by the molar volume of each species i as a function of the number of species, according to Equation (2):

$$V^E=\left({\displaystyle \frac{PM}{\rho }}\right)-\left({\displaystyle \frac{x_1{PM}_1}{{\rho }_1}}+{\displaystyle \frac{x_2{PM}_2}{{\rho }_2}}\right)$$

(2)

where PM is the molecular weight of the mixture, $\rho$ is the density of the mixture, and $x_{\mathrm{i}}$, ${PM}_{\mathrm{i}}$, and ${\rho }_{\mathrm{i}}$, correspond to the mole fraction, molecular weight, and density, in that order, of commercial diesel (superscript 1) and biodiesel (superscript 2). Furthermore, the behavior of $V^E$ was correlated using the Redlich–Kister polynomial function as expressed in Equation (3):

$$V^E=x_1\left(1-x_1\right){\sum }_{j=0}^{j=k}{A}_{\mathrm{j}}{\left({2x}_1-1\right)}^j,$$

(3)

where $k$ denotes the order of the polynomial expression limited to a fifth-order function and $A_j$ represents each fitting parameter.

In addition, the Prigogine–Flory–Patterson (PFP) model is a very useful tool for the analysis of the behavior of non-ideal mixtures, since it relates ideal volumetric deviations to the physical mechanisms that dominate intermolecular interactions. This model is based on Flory’s solution theory [34,35,36,37], later expanded by Prigogine [38,39] and Patterson [40], incorporating into the model the effects of molecular size, interaction energy, and the internal cohesion of the components.

According to the PFP model, the excess molar volume ($V^E$) is composed of three independent physical contributions. These contributions represent different molecular mechanisms that simultaneously intervene during the mixing process, which is expressed by the following equation:

$$V^E=V_{int}^E+V_{free}^E+V_{P^{*}}^E$$

(4)

where $V_{int}^E$ (the first contribution) is the term representing the energetic interaction between different molecules and indicates the cohesive energies, $V_{free}^E$ (the second contribution) is the term related to the free volume, and $V_{P^{*}}^E$ (the third contribution) is the term linked to the internal pressures, related to the difference in internal cohesive forces of each component. The developed model is expressed as [41]:

$$\begin{gathered} {\displaystyle \frac{V^E}{x_i{V}_i^{*}+x_j{V}_j^{*}}}={\left[{\displaystyle \frac{\left({\overline{V}}^{\frac{1}{3}}-1\right){\overline{V}}^{\frac{2}{3}}{\Psi }_i{\theta }_i\left(\raisebox{1ex}{${\chi }_{12}$}\!\left/ \!\raisebox{-1ex}{$p_i^{*}$}\right.\right)}{\left(\left(\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right){\overline{V}}^{-\frac{1}{3}}-1\right)}}\right]}_i+{\left[-{\displaystyle \frac{{\left({\overline{V}}_1-{\overline{V}}_2\right)}^2\left(\left(\raisebox{1ex}{$14$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\right){\overline{V}}^{-\frac{1}{3}}-1\right){\Psi }_1{\Psi }_2}{\left(\left(\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\right){\overline{V}}^{\frac{1}{3}}-1\right)\overline{V}}}\right]}_{ii} \\ +{\left[{\displaystyle \frac{\left({\overline{V}}_1-{\overline{V}}_2\right)\left(p_1^{*}+p_2^{*}\right){\Psi }_1{\Psi }_2}{p_2^{*}{\Psi }_1+p_1^{*}{\Psi }_2}}\right]}_{iii} \end{gathered}$$

(5)

where $V^E$, cm³ mol⁻¹, is the excess molar volume; $x$ is the mole fraction; $V^{*}$, cm³ mol⁻¹, is the characteristic volume; $p^{*}$, MPa, is the characteristic pressure; $\Psi$ is the molecular contact energy fraction; $\theta$ is the molecular surface fraction; and ${\chi }_{12}$, J·cm⁻³, is the interaction parameter of the PFP model, determined by nonlinear least-squares regression minimizing the difference between experimental and calculated excess molar volumes.

Meanwhile, the reduced volume is represented by Equation (6):

$${\overline{V}}_i={\left({\displaystyle \frac{1+\left(\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right){\alpha }_{i}T}{1+{\alpha }_{i}T}}\right)}^3 , i=1,2$$

(6)

where ${\alpha }_i$, in K⁻¹, is the isobaric thermal expansivity of the pure component. On the other hand, the characteristic pressure is expressed as follows

$$p_i^{*}={\displaystyle \frac{T{{\overline{V}}_i^2\alpha }_i}{{\kappa }_{T_i}}}$$

(7)

where ${\kappa }_{T_i}$, in MPa⁻¹, is the isothermal compressibility of the pure components; these values were taken from the literature and are shown in

Table 2. Values of the isothermal compressibility of the pure components reported in the literature.

$\mathit{T}/\mathbf{K}$	${\mathit{\kappa }}_{{\mathit{T}}_{\mathit{i}}}\times {10}^{-4}/{\mathbf{M}\mathbf{P}\mathbf{a}}^{-1}$
	Diesel [42]	Biodiesel (FAME) [43]
293.15	7.58	6.66
303.15	8.04	7.04
313.15	8.53	7.44
323.15	9.07	7.87
333.15	9.65	8.34
343.15	10.30	8.84
353.15	11.00	9.37

.

Likewise, the characteristic volume is represented by Equation (8), where $V_i={PM}_i/{\rho }_i$:

$$V_i^{*}={\displaystyle \frac{V_i}{{\overline{V}}_i}}$$

(8)

Meanwhile, the fraction of molecular contact energy is given by the following equation:

$${\Psi }_1=1-{\Psi }_2={\displaystyle \frac{{\varphi }_1{p}_1^{*}}{{\varphi }_1{p}_1^{*}+{\varphi }_2{p}_2^{*}}}$$

(9)

The reduced volume of the binary mixture is defined as [44]:

$$\overline{V}={\Psi }_1{\overline{V}}_1+{\Psi }_2{\overline{V}}_2$$

(10)

$\varphi$ is a hard-core volume fraction and is expressed by the following equation:

$${\varphi }_1=1-{\varphi }_2={\displaystyle \frac{x_1{V}_1^{*}}{x_1{V}_1^{*}+x_2{V}_2^{*}}}$$

(11)

The molecular surface fraction, ${\theta }_{\mathrm{i}}$, is given by Equation (12):

$${\theta }_1=1-{\theta }_2={\displaystyle \frac{x_1{V}_1^{*}\left(\frac{S_1}{S_2}\right)}{x_1{V}_1^{*}\left(\frac{S_1}{S_2}\right)+x_2{V}_2^{*}}}$$

(12)

In addition, $S_{\mathrm{i}}$ is the surface-to-volume ratio of the molecular fraction of the components determined by the Bondi method [45]:

$${\displaystyle \frac{S_1}{S_2}}={\left({\displaystyle \frac{V_1^{*}}{V_2^{*}}}\right)}^{-1/3}$$

(13)

In addition, the partial molar volumes of diesel and biodiesel ($\overline{V_i}$) were calculated according to Equations (14) and (15):

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

(14)

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

(15)

where $V_1^0$ is the molar volume of the FAME and $V_2^0$ is the molar volume of diesel.

The isobaric thermal expansion coefficient for each mixture was calculated according to the function ${\alpha }_p=(-1/\rho ){\left(\partial \rho /\partial T\right)}_p$ and the first-order function to calculate density ($\rho =k_0+k_{1}T$); therefore,

$${\alpha }_p={\displaystyle \frac{k_1}{k_0+k_{1}T}}$$

(16)

where $k_0$ and $k_1$ correspond to the interception and the slope, respectively.

Meanwhile, the excess isobaric thermal expansion coefficient for the mixture, ${\alpha }_p^E$, was determined by

$${\alpha }_p^E={\alpha }_p-\sum _i{x}_i{\alpha }_{p,i}^0.$$

(17)

This expression depends on the mole fraction ($x_i$) and the thermal expansion coefficient (${\alpha }_{p,i}^0$) of the component $i$. This expression allows quantification of deviations from ideal behavior, which are attributed to intermolecular interactions between the components [46].

2.4. Transport Properties

The kinematic viscosity ($\nu$) was determined experimentally using two calibrated capillary viscometers with sizes of 75 and 100 (Cannon Instrument Company, State College, PA, USA). The viscosity was calculated by relating the efflux time ($t$) to the corresponding viscometer constant ($K$) at each temperature, according to the expression $\nu =K·t$, repeating the measurement four times for each temperature and composition point. The viscometers (1) were immersed in a water bath, as depicted in Figure 4, and the temperature was controlled by a recirculating bath (4) (Polyscience, PD07R, Niles, IL, USA) with a thermal stability of ±0.005 K. The temperature stability of the water bath container was ±0.02 K, ensuring uniform measurement conditions. A PT-100 (2) thermometer (Thermo-Est, Maizières-les-Metz, France) measured the system temperature and was connected to a temperature indicator (3) (ASL, F200, Redhill, UK).

Once the kinematic viscosity was measured, the dynamic viscosity ($\eta =v\rho$) was calculated and subsequently the dynamic viscosity deviation ($∆\eta$) was determined using the following expression:

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

(18)

where $\eta$ is the dynamic viscosity of the mixture and the variables $x_i$ and ${\eta }_i$ are the mole fraction and the dynamic viscosity of diesel (component 1) and biodiesel (component 2), respectively.

The four-body McAllister model [47] was employed to represent the experimental dynamic viscosity data, which is expressed by:

$$\begin{gathered} \ln \left(\eta \right)=x_1^4\ln {\eta }_1+4x_1^3{x}_2\ln {\eta }_{1112}+6x_1^2{x}_2\ln {\eta }_{1122}+4x_1{x}_2^3\ln {\eta }_{1222}+x_2^4\ln {\eta }_2-\ln \left(x_1+{\displaystyle \frac{x_2{PM}_2}{{PM}_1}}\right)+ \\ 4x_1^3{x}_2\ln \left({\displaystyle \frac{3+\frac{{PM}_2}{{PM}_1}}{4}}\right)+6x_1^2{x}_2^2\ln \left({\displaystyle \frac{1+\frac{{PM}_2}{{PM}_1}}{2}}\right)+4x_1{x}_2^3\ln \left({\displaystyle \frac{1+\frac{{3PM}_2}{{PM}_1}}{4}}\right)+x_2^4\ln \left({\displaystyle \frac{{PM}_2}{{PM}_1}}\right) \end{gathered}$$

(19)

where ${\eta }_{1112}$, ${\eta }_{1122}$, and ${\eta }_{1222}$ denote the optimized parameters.

The thermodynamic activation parameters associated with viscous flow were determined using Eyring’s equation which is derived from the transition-state theory for liquids [48]:

$$\eta ={\displaystyle \frac{h N_A}{V_m}} {\mathrm{e}}^{\mathsf{\Delta }{G}^{\ne }/\left(RT\right)}$$

(20)

where $\eta$ is the dynamic viscosity (Pa·s), $h$ denotes Planck’s constant $(6.62607015\times {10}^{-34} \mathrm{J}·\mathrm{s})$, $N_A$ is Avogadro’s number $(6.02214076\times {10}^{23} {\mathrm{mol}}^{-1})$, $V_m$ is the molar volume $({\mathrm{m}}^3·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1})$, $\mathsf{\Delta }{G}^{\ne }$ symbolizes the Gibbs free energy of activation for viscous flow $(\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1})$, $R$ is the universal gas constant (8.3144 $\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}·{\mathrm{K}}^{-1}$) and $T$ expresses the temperature ($\mathrm{K}$).

Moreover, the Gibbs free energy of activation is related to the enthalpy and entropy of activation according to $\mathsf{\Delta }{G}^{\ne }={\mathsf{\Delta }H}^{\ne }-T{∆S}^{\ne }$; hence, by substituting the last equation into Equation (20) and linearizing:

$$\ln \left({\displaystyle \frac{\eta V}{h{N}_A}}\right)={\displaystyle \frac{{\mathsf{\Delta }H}^{\ne }}{R}}{\displaystyle \frac{1}{T}}-{\displaystyle \frac{{∆S}^{\ne }}{R}}$$

(21)

Thus, the graphical representation of $1/T$ versus $\ln (\eta V/h{N}_A)$ enables the determination of the activation parameters; the slope of the linear fit corresponds to ${\mathsf{\Delta }H}^{\ne }/R$, whereas the y-intercept is equal to $-{\mathsf{\Delta }S}^{\ne }/R$. From these values, ${\mathsf{\Delta }H}^{\ne }$ and ${\mathsf{\Delta }S}^{\ne }$ can be obtained for each composition, providing fundamental information on the energetic barriers and the degree of molecular organization associated with the viscous-flow process.

3. Results

3.1. Density Determination

The experimental density ($\rho$) for the mixture of commercial diesel (1) and residual chicken fat methyl ester biodiesel (2) is presented isothermally in Figure 5 and summarized in

Table 3. Densities of the mixture of commercial diesel ($x_1$) + FAME ($x_2$) derived from residual chicken fat.

x1	$\mathit{\rho }/\mathbf{g}·{\mathbf{c}\mathbf{m}}^{-3}$
	293.15 K	303.15 K	313.15 K	323.15 K	333.15 K	343.15 K	353.15 K
1.0000	0.83265	0.82674	0.82036	0.81345	0.80606	0.79718	0.78911
0.9001	0.84082	0.83494	0.82853	0.82159	0.81418	0.80527	0.79719
0.8001	0.84803	0.84203	0.83560	0.82866	0.82120	0.81225	0.80408
0.7001	0.85482	0.84886	0.84239	0.83541	0.82795	0.81898	0.81082
0.5998	0.86088	0.85492	0.84845	0.84147	0.83400	0.82500	0.81682
0.5001	0.86726	0.86135	0.85483	0.84783	0.84032	0.83131	0.82310
0.3998	0.87252	0.86656	0.86004	0.85303	0.84550	0.83648	0.82827
0.3005	0.87640	0.87028	0.86383	0.85679	0.84924	0.84017	0.83193
0.2000	0.88070	0.87470	0.86817	0.86114	0.85361	0.84453	0.83631
0.1008	0.88446	0.87844	0.87192	0.86488	0.85733	0.84826	0.84001
0.0000	0.88786	0.88176	0.87515	0.86807	0.86049	0.85134	0.84306

The expanded uncertainty of density measurements was $U_c\left(\rho \right)= 2.6\times {10}^{-5} \mathrm{g}·{\mathrm{c}\mathrm{m}}^{-3}$, with a 0.95 confidence level and coverage factor $k = 2$.

. In the temperature range of 293.15 to 353.15 K and 0.078 MPa, the density decreased with increasing temperature and with high amounts of commercial diesel ($x_1$). This phenomenon has been described for diesel and biodiesel mixtures, where biodiesel typically exhibited densities between 0.82 and 0.90 g·cm⁻³, compared to 0.74 to 0.86 g·cm⁻³ for diesel [42,49,50,51]. The synergistic contributions of dispersive and polar forces contributed to the described behavior, typical of non-ideal mixtures between nonpolar compounds (hydrocarbons) and oxygenated molecules [52,53].

The decrease in density with increasing amounts of commercial diesel represented a systematic transition from a system dominated by polar and rigid molecules, such as those in FAMEs, to a predominantly nonpolar system with flexible molecules, characteristic of commercial diesel [54]. In systems with higher amounts of FAME, carbonyl (C=O) and ester (–COO–) functional groups were present, which produced dipole–dipole interactions [55]. These interactions favored a more compact and oriented organization, as the dipoles partially aligned and minimized the system’s energy, increasing the cohesion of the system in the liquid phase. Therefore, greater molecular packing was achieved, resulting in higher density values compared to commercial diesel [55,56].

It can be noted that increasing the amount of commercial diesel fuel added predominantly nonpolar molecules that were incapable of strong dipolar interactions. Therefore, induced dipole–dipole interactions or Debye forces became relevant, since the permanent dipoles of FAME could induce temporary polarization in diesel molecules. These interactions were weaker and less ordered than permanent dipole interactions, leading to a progressive loss of order in the structure and less fluid packing. Additionally, increasing the temperature weakened both dipolar and induced dipole interactions, increased molecular motion and favored fewer compact configurations [54].

Furthermore, Figure 6 depicts the density trends of the FAMEs derived from residual chicken fat obtained in this work and those reported for fatty acid methyl esters in the literature [57,58,59,60] as a function of temperature, synthesized from various sources. The residual chicken fat biodiesel had one of the highest densities in contrast with the other biodiesel sources. Waste cooking oil methyl ester biodiesel [57] exhibited the second highest density, highlighting the nature of the recycled feedstock in that it is likely to contain multiple impurities. The density sets corresponding to the FAMEs derived from vegetable oils (soybean, canola, and sunflower) [58], as well as the FAMEs obtained from soybean, rapeseed and palm oil blends [60], have the lowest values. This behavior was consistent with biodiesel fuels derived from vegetable sources that generally contained a high composition of unsaturated esters instead of the saturated esters commonly contained in residuals or fats, as well as traces of glycerine and moisture that significantly increased the density values. Although the density of pure unsaturated esters was higher than saturated esters with the same carbon chain length, the molecular arrangement among different unsaturated components did not allow for complete packing due to the presence of cis double bonds, which introduce kinks and bending in the chains [61].

3.2. Volumetric Properties

The behavior of the excess molar volume ($V^E$) of the commercial diesel ($x_1$) + chicken fat biodiesel ($x_2$) system is depicted in Figure 7, with symbols corresponding to the experimental values and the solid lines representing the correlation using the Redlich–Kister polynomial function. The $V^E$ values were negative across the entire composition range with a pronounced minimum occurring at mole fractions between 0.4 and 0.6. This negative sign indicated volumetric contraction upon mixing diesel and FAMEs, which was interpreted as occurring due to the predominance of attractive intermolecular interactions and more efficient packing between molecules of dissimilar nature (polar esters versus nonpolar hydrocarbons). Such behavior has previously been reported in studies of diesel–biodiesel mixtures and was associated with the complementary packing of molecules differing in size and polarity [57,62]. The increasing temperature, interpreted as thermal energy augmentation, meant that excess molar volumes tended to high negative values. This was ascribed to the weakening of specific interactions and orientations between ester and hydrocarbon groups, disrupting partially ordered liquid structures [63,64]. The expanded uncertainty of $V^E$ was estimated as $U_c\left(V^E\right)=0.0054 {\mathrm{cm}}^3·{\mathrm{mol}}^{-1}$ (k = 2).

The Redlich–Kister model accurately reproduced the overall trend of the experimental data, including the position of the minimum values. The excellent agreement between the experimental and calculated values, represented by the solid lines, was confirmed by the low absolute average deviation (AAD), as shown in

Table 4. Parameters of Equation (3) for the excess molar volume correlation (the units are expressed in cm³ mol⁻¹).

T/K	A0	A1	A2	A3	A4	A5	STD	AAD/%
293.15	−18.425	2.0495	3.3946	7.0249	−5.7785	−6.1312	0.07	0.9
303.15	−18.425	2.0495	3.3946	7.0249	−5.7785	−6.1312	0.11	1.6
313.15	−18.985	2.1834	4.2763	5.6773	−7.5757	−4.3839	0.07	0.9
323.15	−19.27	2.2436	4.3292	5.7199	−7.7098	−4.5875	0.07	0.9
333.15	−19.27	2.2436	4.3292	5.7199	−7.7098	−4.5875	0.11	1.6
343.15	−19.964	2.4048	4.8038	5.0925	−8.7429	−3.8885	0.08	0.9
353.15	−20.312	2.5566	5.1633	3.8562	−9.5436	−2.0866	0.08	1.0

, together with the parameters. The results from the Redlich–Kister polynomial for describing excess molar volumes in fuel mixtures agreed with the findings reported elsewhere [42,65]. The negative A₀ values (from −18.425 to −20.312) confirmed the predominance of negative excess volumes, evidencing attractive interactions between FAME and diesel molecules. The progressive increase in A₀ and A₁ parameters with temperature confirmed the non-ideal behavior, where the molecules tended to lose order and attractive interactions were weaker. The A₃–A₅ coefficients, fitted with minor variations, were useful to improve the corrective function during fine-tuning [66,67].

The standard deviation (STD) was lower than 0.11 cm³·mol⁻¹ and the absolute average deviation was below 1.6%; these were estimated by

$$STD=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{\left(V_{exp}^E-V_{cal}^E\right)}^2}{N-m}}}$$

(22)

$$AAD=\left({\displaystyle \frac{100}{N}}\right){\sum }_{i=1}^N\left|{\displaystyle \frac{V_{exp}^E-V_{cal}^E}{V_{exp}^E}}\right|.$$

(23)

where N denotes the number of experimental data points, m is the number of fitted parameters, and subscripts $exp$ and $cal$ represent the experimental and calculated values.

In addition, Figure 8 shows, by means of the points in the graph, the excess molar volumes ($V^E$) of the commercial diesel ($x_1$) + FAME mixtures obtained from residual chicken fat, and by means of the solid lines, the excess molar volumes calculated using the PFP model ($V_{cal}^E$) in the temperature range of 293.15 to 353.15 K. Likewise,

Table 5. Contributions to the excess molar volume, $V_{\mathrm{c}\mathrm{a}\mathrm{l}}^E$, and interaction parameter, ${\chi }_{12}$, obtained using the Prigogine–Flory–Patterson (PFP) model.

${\mathit{x}}_1$	$$\begin{gathered} {\mathit{V}}_{\mathit{c}\mathit{a}\mathit{l}}^{\mathit{E}} \\ /{\mathbf{c}\mathbf{m}}^3·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1} \end{gathered}$$	$$\begin{gathered} {\mathit{V}}_{\mathit{i}\mathit{n}\mathit{t}}^{\mathit{E}} \\ /{\mathbf{c}\mathbf{m}}^3·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1} \end{gathered}$$	$$\begin{gathered} {\mathit{V}}_{\mathit{f}\mathit{r}\mathit{e}\mathit{e}}^{\mathit{E}} \\ /{\mathbf{c}\mathbf{m}}^3·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1} \end{gathered}$$	$$\begin{gathered} {\mathit{V}}_{{\mathit{P}}^{*}}^{\mathit{E}} \\ /{\mathbf{c}\mathbf{m}}^3·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1} \end{gathered}$$	$$\begin{gathered} {\mathit{\chi }}_{12} \\ /\mathbf{J}·{\mathbf{c}\mathbf{m}}^{-3} \end{gathered}$$	$$\begin{gathered} \mathit{A}\mathit{A}\mathit{D} \\ /\% \end{gathered}$$	$$\begin{gathered} \mathit{S}\mathit{T}\mathit{D} \\ /{\mathbf{c}\mathbf{m}}^3·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1} \end{gathered}$$
293.15 K
0.9001	−1.4760	−1.4633	0.0045	−0.0081	−129.65	14.54	0.4475
0.8001	−2.6888	−2.6653	0.0083	−0.0150
0.7001	−3.6162	−3.5840	0.0114	−0.0206
0.5998	−4.2368	−4.1984	0.0137	−0.0247
0.5001	−4.5224	−4.4804	0.0149	−0.0270
0.3998	−4.4498	−4.4075	0.0150	−0.0272
0.3005	−3.9949	−3.9559	0.0138	−0.0251
0.2000	−3.1169	−3.0855	0.0110	−0.0202
0.1008	−1.8086	−1.7898	0.0066	−0.0121
303.15 K
0.9001	−1.4992	−1.4854	0.0048	−0.0089	−124.94	14.41	0.4519
0.8001	−2.7307	−2.7051	0.0089	−0.0166
0.7001	−3.6716	−3.6366	0.0122	−0.0227
0.5998	−4.3008	−4.2589	0.0146	−0.0272
0.5001	−4.5894	−4.5436	0.0159	−0.0297
0.3998	−4.5144	−4.4683	0.0160	−0.0300
0.3005	−4.0516	−4.0091	0.0147	−0.0277
0.2000	−3.1600	−3.1258	0.0118	−0.0223
0.1008	−1.8329	−1.8124	0.0070	−0.0133
313.15 K
0.9001	−1.5224	−1.5073	0.0051	−0.0099	−120.28	14.00	0.4468
0.8001	−2.7722	−2.7442	0.0096	−0.0184
0.7001	−3.7265	−3.6881	0.0131	−0.0253
0.5998	−4.3639	−4.3179	0.0156	−0.0302
0.5001	−4.6553	−4.6051	0.0170	−0.0330
0.3998	−4.5776	−4.5272	0.0171	−0.0333
0.3005	−4.1068	−4.0603	0.0157	−0.0307
0.2000	−3.2017	−3.1643	0.0125	−0.0247
0.1008	−1.8562	−1.8338	0.0075	−0.0148
323.15 K
0.9001	−1.5481	−1.5312	0.0055	−0.0113	−115.60	13.81	0.4487
0.8001	−2.8181	−2.7868	0.0102	−0.0210
0.7001	−3.7869	−3.7441	0.0139	−0.0288
0.5998	−4.4328	−4.3817	0.0166	−0.0344
0.5001	−4.7269	−4.6711	0.0181	−0.0376
0.3998	−4.6460	−4.5898	0.0182	−0.0379
0.3005	−4.1660	−4.1143	0.0167	−0.0350
0.2000	−3.2460	−3.2045	0.0133	−0.0281
0.1008	−1.8807	−1.8559	0.0079	−0.0168
333.15 K
0.9001	−1.5502	−1.5319	0.0060	−0.0122	−110.22	13.73	0.4463
0.8001	−2.8211	−2.7874	0.0111	−0.0226
0.7001	−3.7900	−3.7438	0.0151	−0.0310
0.5998	−4.4351	−4.3800	0.0180	−0.0370
0.5001	−4.7278	−4.6677	0.0196	−0.0404
0.3998	−4.6452	−4.5847	0.0197	−0.0408
0.3005	−4.1637	−4.1080	0.0180	−0.0376
0.2000	−3.2428	−3.1981	0.0144	−0.0302
0.1008	−1.8780	−1.8512	0.0086	−0.0181
343.15 K
0.9001	−1.6086	−1.5877	0.0066	−0.0142	−106.45	13.37	0.4535
0.8001	−2.9262	−2.8877	0.0122	−0.0263
0.7001	−3.9294	−3.8766	0.0167	−0.0360
0.5998	−4.5961	−4.5331	0.0199	−0.0430
0.5001	−4.8968	−4.8281	0.0216	−0.0470
0.3998	−4.8084	−4.7393	0.0216	−0.0473
0.3005	−4.3071	−4.2436	0.0198	−0.0436
0.2000	−3.3521	−3.3011	0.0158	−0.0350
0.1008	−1.9397	−1.9092	0.0094	−0.0210
353.15 K
0.9001	−1.5908	−1.5907	0.0063	0.0062	−108.64	14.92	0.5118
0.8001	−2.9085	−2.9083	0.0118	0.0116
0.7001	−3.9271	−3.9268	0.0164	0.0161
0.5998	−4.6213	−4.6209	0.0197	0.0193
0.5001	−4.9564	−4.9559	0.0217	0.0212
0.3998	−4.9030	−4.9024	0.0221	0.0216
0.3005	−4.4277	−4.4271	0.0206	0.0200
0.2000	−3.4778	−3.4773	0.0168	0.0162
0.1008	−2.0332	−2.0328	0.0102	0.0098

presents these $V_{cal}^E$ values and the excess volume contributions $V_{int}^E$, $V_{free}^E$, and $V_{P^{*}}^E$. It can be observed that the excess molar volumes of the mixture were negative over the entire composition and temperature range studied, which indicated that the heteromolecular interactions of the diesel + FAME mixture exhibited greater packing; this can be ascribed to the addition of polar chains with ester groups from FAMEs into the diesel matrix. This favored greater compactness, reducing the total volume of the system.

In all isotherms, the minimum value of $V^E$ was located at $x_1\approx 0.47$, which is usually observed in non-ideal mixtures where there are significant differences in molecular size, shape, chain rigidity and moderate energetic interactions [68,69]. The $V_{cal}^E$ values adequately represented the experimental excess molar volumes both in shape and magnitude over the entire composition and temperature range. Therefore, it can be confirmed that the PFP model was capable of adequately representing the physical properties of the commercial diesel + FAME mixture.

Meanwhile, Table 5 presents the individual contributions to the excess molar volume calculated by means of the PFP model, as well as the values of the interaction parameter, ${\chi }_{12}$, for the different temperatures. The interactional contribution, $V_{int}^E$, had negative values over the entire composition and temperature range, which confirmed the predominant energetic interactions between the diesel and FAME molecules. Likewise, the values of ${\chi }_{12}$ were negative and progressively decreased with increasing temperature, which indicated that the heteromolecular forces were dominated by dispersive and dipole-induced forces, without the presence of strong specific chemical associations, and that the increase in thermal energy diminished the net effect of energetic interactions [70].

In addition, the free volume contribution, $V_{free}^E$, was positive over the entire composition and temperature range, and showed a systematic increase with increasing temperature. This indicated a thermal expansion effect and greater conformational freedom of the molecular chains. The contribution associated with the internal pressure, $V_{P^{*}}^E$, represented a corrective energetic term that arises from differences in the internal cohesive forces of the pure components; its aim is to explain how each component present in the mixture responded differently to thermal and mechanical perturbations after the mixing process. According to Table 5, the $V_{P^{*}}^E$ contribution was negative over practically the entire temperature range, except for at the temperature of 353.15 K; the negative value of this contribution indicated that the differences in the internal pressure of the pure components favored an additional contraction of the molar volume of the mixture. The presence of FAMEs may contribute to higher internal cohesive forces, associated with the presence of polar ester groups and long hydrocarbon chains, compared to commercial diesel, whose composition was predominantly paraffinic and aromatic. In addition, it can be observed that the value of $V_{P^{*}}^E$ increased with increasing temperature, which can be attributed to the fact that the internal pressure of the mixture was more sensitive to thermal expansion.

Isothermal trends for the partial molar volumes are depicted in Figure 9 as a function of the diesel mole fraction. The corresponding variable for biodiesel ($\overline{V_2}$) increased with both temperature and diesel content, reaching values of around 330 cm³·mol⁻¹ at 293.15 K and up to 380 cm³·mol⁻¹ at 353.15 K, indicating a less polar mixture due to the addition of nonpolar hydrocarbons with reduced affinity to the carbonyl (C=O) groups. This confirmed the thermal expansion reduced intermolecular cohesion and weakened the dipole–dipole interactions between the ester groups of the FAME and the hydrocarbon chains of diesel; this produced a liquid structure with low density [55,71,72]. Regarding the behavior of partial molar volumes for diesel ($\overline{V_1}$), the values were higher under diluted diesel and decreased as the diesel fraction increased, approaching a minimum at $x_1=1$. This behavior described the partial expansion of diesel when dissolved in a biodiesel medium; the differences in polarity and molecular size reduced the molecular packing and led to more free space volume, with positive deviations from ideal behavior [72]. The combined analysis of partial molar volume for each component confirmed that biodiesel tended to suffer a relative expansion when mixed with diesel, while it seemed to experience a compaction with high quantities of diesel [73].

The trends in the isobaric thermal expansivity (${\alpha }_p$) as a function of the diesel molar composition ($x_1$) are illustrated in Figure 10. A moderate increase in ${\alpha }_p$ values occurred within $x_1$ = 0–0.4, suggesting dominant specific (dipole–dipole and dispersive) interactions that reduced molecular mobility and compacted liquid structures [58,73]. As the temperature increased, ${\alpha }_p$ rose progressively, reaching its maximum values at 353.15 K. The increase in ${\alpha }_p$ with temperature and diesel content demonstrated the general volumetric expansion linked to the structural differences between the components. The thermal increase was associated with the breakdown of some intermolecular interactions, which were replaced by weaker dispersive forces. The increase in ${\alpha }_p$, linked to the liquid density decrease and structural disorder, has been studied for mixtures containing fatty acid methyl esters with alcohols or hydrocarbons [42,73,74,75]. The increasing presence of diesel favored thermal expansion due to the high content of aliphatic hydrocarbons that possessed great rotational and vibrational freedom, as well as its structural rigidity and low polarity [51]. The expanded uncertainty of isobaric thermal expansivity was $U_c\left({\alpha }_P\right)=2.6\times {10}^{-5}{\mathrm{K}}^{-1}$, with a 0.95 confidence level and coverage factor $k=2$.

The excess thermal expansivity (${\alpha }_p^E$) is illustrated in Figure 11 for the studied temperature and composition range. The negative trend for ${\alpha }_p^E$ with a distinguishable well-defined minimum of $x_1\approx 0.4$ indicates the specific and non-additive molecular interactions that allowed for relative molecular packing and a reduction in the configurational freedom of the fluid when the compositions are near $x_1=0$ or $1$. Noticeable differences between ${\alpha }_p^E$ isotherms were not observed; hence, the interactions responsible for this non-ideality were not fully weakened by the increase in thermal energy. This can be interpreted as evidence that ester–ester or ester–hydrocarbon interactions were sufficiently strong to persist even at elevated temperatures. Such mechanisms have previously been discussed in theoretical models for diesel–biodiesel mixtures, where ester polarity and chain length contributed to the formation of more ordered microstructures [57,76]. The expanded uncertainty of ${\alpha }_p^E$ was estimated as $U_c\left({\alpha }_p^E\right)=1.37\times {10}^{-4} {\mathrm{K}}^{-1}$ (k = 2).

3.3. Kinematic Viscosity

The kinematic viscosity ($\nu$) at each composition and temperature is presented in Figure 12 and

Table 6. Kinematic viscosities (ν) for a diesel (1) + biodiesel (2) mixture.

T/K	$\mathit{\nu }/{\mathbf{m}\mathbf{m}}^2·{\mathbf{s}}^{-1}$
	x1
	0	0.1008	0.2000	0.3005	0.3998	0.5001	0.5998	0.7001	0.8001	0.9001	1
293.15	9.806	9.600	8.950	8.350	7.651	6.750	5.718	3.705	2.636	2.079	1.814
303.15	7.527	7.369	6.920	6.500	6.012	5.250	4.496	3.161	2.102	1.696	1.494
313.15	6.000	5.834	5.500	5.200	4.850	4.252	3.718	2.605	1.751	1.392	1.232
323.15	4.856	4.732	4.450	4.200	3.950	3.480	3.063	2.215	1.442	1.173	1.043
333.15	4.035	3.937	3.700	3.450	3.310	2.920	2.609	2.029	1.347	1.067	0.908
343.15	3.434	3.350	3.100	2.900	2.800	2.450	2.242	1.820	1.174	0.991	0.788
353.15	2.942	2.840	2.620	2.450	2.400	2.100	1.947	1.650	1.080	0.890	0.702

The expanded uncertainty of the kinematic viscosity was $U_c\left(\nu \right)=0.14 {\mathrm{m}\mathrm{m}}^2{\mathrm{s}}^{-1}$ (k = 2).

. The general behavior indicated a monotonical viscosity decrease with increasing temperature, commonly reported in organic liquids; the thermal energy barrier associated with the molecular arrangement was lowered to facilitate the relative mobility of hydrocarbon chains and ester structures [77,78]. For each isotherm, a clear and systematic increase in $\nu$ was observed as the biodiesel mole fraction increased. The methyl esters possessed high molar masses, polar carbonyl and ester groups, and long aliphatic chains; these molecular characteristics promoted strong molecular interactions and a great structural rigidity that together induced flow resistance. The kinematic viscosity was not a linear function of the composition and a sharp increase in $\nu$ was observed in the interval of $x_1\approx 0.6 \mathrm{t}\mathrm{o} 0.8$. This effect seemed to be caused by heteromolecular interactions between the ester groups from biodiesel and the aliphatic chains from diesel, which led to ordered configurations and an increase in the intermolecular friction.

The isothermal ν can be observed in Figure 13, which covered each fatty acid methyl ester biodiesel sample obtained from different vegetable oil or animal fat feedstocks. Biodiesel samples produced from residual sources [78] were expected to have the highest kinematic viscosities, reflecting a strong dependence on the high content of long-chain saturated and monounsaturated fatty acids as well as material affected by oxidation, thermal cycling and the formation of polar degradation products. On the contrary, the viscosity magnitude diminished for biodiesel synthetized from vegetable sources including soybean oil, canola oil, sunflower oil and palm oil; the high unsaturation degree tended to reduce the kinematic viscosity in biodiesel from vegetables due to the presence of multiple double bonds that increased the molecular flexibility and weakened the molecular interactions [79,80].

The variation in the dynamic viscosity ($\eta$) is shown in Figure 14. This variable presented a pronounced and systematic decrease with an increase in temperature; the specific interactions weakened and the compositional variation in viscosity became smoother [71,82,83]. Nevertheless, the dependence of dynamic viscosity on composition is not strictly linear, particularly at lower temperatures, where the differences between adjacent compositions were more pronounced. Heteromolecular interactions increased the viscosity between the ester groups of FAMEs and the aliphatic chains of diesel, allowing for ordered configurations and an increase in intermolecular friction. The dashed lines in Figure 14 correspond to the values calculated by the McAllister model, commonly employed to describe the viscosity of non-ideal liquid mixtures. The excellent agreement between the experimental data and the model predictions yielded an AAD < 4.2%, as listed in

Table 7. Parameters for the McAllister model.

	${\mathit{\eta }}_{1112}$	${\mathit{\eta }}_{1122}$	${\mathit{\eta }}_{1222}$	AAD/%
T/K	Diesel (x1) + FAME (x2)
293.15	$1.65\times {10}^{-3}$	$1.55\times {10}^{-2}$	$6.19\times {10}^{-3}$	3.3
303.15	$1.34\times {10}^{-3}$	$1.21\times {10}^{-2}$	$4.70\times {10}^{-3}$	3.4
313.15	$1.10\times {10}^{-3}$	$1.00\times {10}^{-2}$	$3.60\times {10}^{-3}$	3.6
323.15	$0.92\times {10}^{-3}$	$0.81\times {10}^{-2}$	$2.85\times {10}^{-3}$	4.1
333.15	$1.01\times {10}^{-3}$	$0.55\times {10}^{-2}$	$2.48\times {10}^{-3}$	4.0
343.15	$1.02\times {10}^{-3}$	$0.39\times {10}^{-2}$	$2.16\times {10}^{-3}$	4.2
353.15	$0.98\times {10}^{-3}$	$0.31\times {10}^{-2}$	$1.79\times {10}^{-3}$	4.0

, along with the corresponding parameters. The deviation seemed to increase as temperature increased; this was a consequence of the gradual reduction in intermolecular interactions. The expanded uncertainty of dynamic viscosity was estimated using standard error propagation from the uncertainties of density and kinematic viscosity. The expanded uncertainty (k = 2) was $U_c\left(\eta \right)=0.23 \mathrm{m}\mathrm{P}\mathrm{a}·\mathrm{s}$, corresponding to a confidence level of 95%.

Figure 15 illustrates the deviation in dynamic viscosity (Δη), where the symbols correspond to the experimental values, whereas the dotted lines represent the values calculated by the McAllister model. The viscosity deviations exhibit two general regions: the first is characterized by positive deviations in the biodiesel-rich region from a diesel mole fraction of zero to 0.6, and the second region exhibits negative values that correspond to the diesel-rich mixture with diesel mole fractions of 0.6 to 1. The positive deviations indicated that the experimental viscosity of the mixtures was higher than expected from ideal behavior; this suggested strong heteromolecular interactions between the polar ester groups of biodiesels and the hydrocarbon chains of diesel, causing more ordered configurations and an increase in the effective intermolecular friction and leading to enhanced resistance to flow [55,71,73]. The thermal exposure reduced intermolecular cohesion, which diminished specific ester–hydrocarbon associations and facilitated molecular motion [71,84]. On the contrary, the negative deviations observed for diesel-rich mixtures could be explained by a plasticizing effect; the FAME molecules mainly contained in biodiesel partially disrupted the hydrocarbon–hydrocarbon interactions of diesel, inducing a local reduction in viscosity [58]. Therefore, the asymmetric Δη behavior occurred due to a combination of structural and energetic effects [85]. The McAllister model accurately reproduced both the shape and the location of maximum and minimum viscosity deviation since the model incorporates interaction parameters related to molecular size, molar mass, and intermolecular forces.

3.4. Thermodynamic Activation Parameters

The plot of $\mathrm{l}\mathrm{n}(\eta V/h{N}_A)$ as a function of $1/T$ is depicted in Figure 16, where a well-defined linear relationship is observed over the investigated temperature range. This confirmed the suitability of Eyring’s transition-state theory to properly describe the viscous flow and the control of the activated process over the momentum transport, where molecules must overcome an energy barrier to reorganize and enable macroscopic liquid flow. The slopes of the straight lines increased as the biodiesel content in the mixture increased; this implied values of high activation enthalpy $\left(\mathsf{\Delta }{H}^{\ne }\right)$, that is, the energetic barrier that must overcome to ensure molecular motion, as listed in

Table 8. Gibbs free energy of activation ($\mathsf{\Delta }{G}^{\ne }$), activation enthalpy (${\mathsf{\Delta }H}^{\ne }$), and activation entropy (${∆S}^{\ne }$) for the viscous-flow process of diesel (1) + biodiesel (2) mixtures.

${\mathit{x}}_1$	${∆\mathit{G}}^{\ne }/\mathbf{J}·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1}$							$$\begin{gathered} {∆\mathit{H}}^{\ne } \\ /\mathbf{J}·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1} \end{gathered}$$	$$\begin{gathered} {∆\mathit{S}}^{\ne } \\ /\mathbf{J}·{\mathbf{m}\mathbf{o}\mathbf{l}}^{-1}·{\mathbf{K}}^{-1} \end{gathered}$$
	293.15 K	303.15 K	313.15 K	323.15 K	333.15 K	343.15 K	353.15 K
0.0000	295.529	297.112	298.696	300.279	301.863	303.446	305.030	249.108	−0.158
0.1008	296.940	298.503	300.067	301.630	303.194	304.758	306.321	251.105	−0.156
0.2000	296.790	298.258	299.726	301.194	302.663	304.131	305.599	253.751	−0.147
0.3005	296.488	297.926	299.365	300.804	302.242	303.681	305.119	254.316	−0.144
0.3998	295.045	296.910	298.774	300.638	302.503	304.367	306.232	240.388	−0.186
0.5001	292.209	293.952	295.695	297.439	299.182	300.925	302.668	241.110	−0.174
0.5998	287.860	290.099	292.337	294.576	296.815	299.054	301.293	222.227	−0.224
0.7001	274.159	277.740	281.321	284.903	288.484	292.065	295.646	169.175	−0.358
0.8001	262.641	265.326	268.010	270.694	273.378	276.062	278.746	183.957	−0.268
0.9001	255.742	258.518	261.294	264.070	266.845	269.621	272.397	174.371	−0.278
1.0000	253.536	255.431	257.326	259.221	261.117	263.012	264.907	197.980	−0.190

The expanded uncertainty was calculated using a coverage factor of k = 2, at $U_c\left({∆G}^{\ne }\right)= 0.3 \mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$, $U_c\left({∆H}^{\ne }\right)= 0.6 \mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ and $U_c\left({∆S}^{\ne }\right)= 2 \mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}·{\mathrm{K}}^{-1}.$

. This increase was associated with the high molar mass of methyl esters and the content of polar functional groups (carbonyl and ester) and long aliphatic chains; all these factors promoted stronger intermolecular interactions and greater structural cohesion of the fluid [86].

The y-axis intercepts of the linear fits in Figure 16 were related to the activation entropy $\left(\mathsf{\Delta }{S}^{\ne }\right)$, and the values are summarized in Table 8. The negative magnitude of $\mathsf{\Delta }{S}^{\ne }$ indicated a more ordered transition state, associated with viscous flow, in comparison to the initial state, a characteristic feature of liquids exhibiting directional interactions or transient molecular associations [87]. Mixtures with high methyl ester biodiesel content attained more negative $\mathsf{\Delta }{S}^{\ne }$ values, suggesting increased configurational restriction during the flow process. This behavior was consistent with the formation of more organized structures placed locally, induced by dipole–dipole interactions between FAME ester groups and heteromolecular interactions with diesel hydrocarbon chains at intermediate compositions for the latter [88].

Figure 17 complements this analysis by illustrating the variation in the Gibbs free energy of activation for viscous flow $\left(\mathsf{\Delta }{G}^{\ne }\right)$ as a function of composition at different temperatures. For all isotherms, $\mathsf{\Delta }{G}^{\ne }$ increased systematically with increasing biodiesel mole fractions, reaching the highest values in the ester-rich region; this confirmed that viscous flow became progressively less energetically favorable as the FAME content increased, in full agreement with the observed increase in both the dynamic and kinematic viscosities of the system. Furthermore, $\mathsf{\Delta }{G}^{\ne }$ decreased with increasing temperature for all compositions, reflecting the dominant role of thermal energy in facilitating molecular rearrangement. The increase in thermal energy weakened intermolecular interactions and enhanced the molecular mobility required to reach the transition state, thereby reducing the free-energy barrier for flow [89]. Nevertheless, the separation between the curves corresponding to different compositions remained nearly constant with temperature, indicating that the structural differences between diesel and FAMEs continued to exert a significant influence on the flow mechanism even at elevated temperatures.

4. Conclusions

• Based on physicochemical and thermophysical characterization, the fatty acid methyl ester biodiesel synthetized from residual chicken fat exhibited a composition dominated by medium- and long-chain alkyl esters, in contrast with biodiesels produced from vegetable sources.
• Density decreased with increases in temperature and diesel content because of thermal exposure and the proper diesel structural compaction.
• The negative excess molar volumes indicated significant attractive interactions between the carbonyl groups of methyl esters and the hydrocarbon chains of diesel. The local minimum observed near x₁ ≈ 0.4–0.45 confirmed the efficient packing and the enhancement of heteromolecular interactions at intermediate compositions.
• The partial molar volume of biodiesel increased with temperature, consistent with its higher polarity and structural rigidity, mainly ascribed to esters.
• Mixtures with a higher biodiesel content exhibited low thermal expansivity, indicative of more compact molecular arrangements, while the excess thermal expansion coefficient (${\alpha }_P^E$) was negative for all composition ranges, confirming the non-ideal behavior controlled by ester–hydrocarbon interactions.
• The high activation enthalpies (${\mathsf{\Delta }H}^{\ne }$) and increasingly negative activation entropies (${\mathsf{\Delta }S}^{\ne }$) for mixtures rich in FAMEs indicated more ordered transition states and higher energetic barriers to viscous flow. Properties such as viscosity have an important impact on atomization behavior and the quality of fuel injection in diesel engines, since they affect droplet size, spray formation, and the overall combustion process.