A New Method for Predicting the Liquid Dynamic Viscosity of Biodiesel-Related Esters Based on the Corresponding State Principle

Abstract

This work proposes a new model based on the three-parameter corresponding states principle (CSP) for estimating the dynamic viscosity of biodiesel-related esters in the liquid state. A Tait-like equation was employed to extend the model to high-pressure conditions. The model’s parameters were fitted to 249 experimental viscosity data points from 11 biodiesel-related esters, with 204 obtained under high-pressure conditions. The average absolute relative deviations were 7.94% for high-pressure conditions and 12.95% for atmospheric conditions. The proposed method was rigorously compared with 13 of the most consolidated models available in the literature for the same property (9 based on the group contribution concept, 3 based on the CSP, and 1 based on both methodologies). Considering all criteria of accuracy and physical consistency, the proposed model is recommended for future applications.

1. Introduction

The world still has significant dependence on fossil fuels and their derivatives. According to International Energy Agency data, until 2022, fossil fuels and their derivatives accounted for approximately 75% of all primary sources used for energy supply. For instance, 60% of the economy is related to the transportation sector. Diesel engines are the major contributors to this consumption and are responsible for high carbon dioxide emissions and other environmentally harmful gases.

Biodiesel is a mixture of long-chain fatty acid alkyl esters (FAAEs) that is industrially obtained by a transesterification reaction. It is an alternative to diesel and can be used in diesel engines without further modifications. Biodiesel also presents many advantages, such as low flammability, less particulate matter in its combustion products, free of sulfur and aromatics, and a high cetane number. −

The knowledge of biodiesel thermophysical properties such as viscosity is fundamental for the development and optimization process and a better understanding of variables that can affect the efficiency of the combustion phenomenon. Furthermore, biodiesel properties are also important in the development, optimization, and control of processes in biorefineries.

Experimental investigations into the effects variables such as viscosity have on biodiesel emissions have been exhaustively carried out. However, performing such analyses for biodiesel derived from different feedstocks becomes impractical due to the wide variety of raw materials. Therefore, computational fluid dynamics (CFD) has emerged as an effective approach for carrying out these analyses. In CFD studies, the use of accurate predictive models for viscosity is essential to ensure reliable simulation results. In the literature, researchers frequently evaluate predictive models for the liquid viscosity of fatty acid methyl esters (FAMEs) and fatty acid ethyl esters (FAEEs), and subsequently extend their applicability to biodiesel via appropriate mixing rules. Liquid viscosity, which quantifies a fluid’s resistance to flow, significantly influences the performance of diesel engines. Elevated viscosity levels can hinder fuel atomization, increase injection pump pressures, and ultimately compromise the combustion efficiency. These effects often lead to higher pollutant emissions due to incomplete combustion reactions. ,,

Even though they are essential for the liquid phase, experimental viscosity data in a wide range of temperatures and pressures are scarce for FAAEs, and their measurements are expensive and time-consuming. Because of that, predictive models are a simple alternative to overcome the issues related to experimental investigations. Moreover, predictive models can be applied in CFD simulations to produce reliable results. ,

The most straightforward and rapid way to estimate liquid viscosity is by using Group Contributions (GC) and the Corresponding State Principle (CSP) models. The GC concept establishes that the macroscopic properties of the substances can be calculated by summation of individual groups that constitute the molecule, also called functional groups. This class of models is a helpful tool because the primary information needed is the structure of the species of interest. Another simple methodology to estimate properties is by using models from CSP, which states that substances at the same reduced variables (e.g., reduced temperature, pressure, and volume) must present similar behavior.

In this work, we propose a new model based on CSP to predict the dynamic liquid viscosity (η) of FAMEs and FAEEs. Thus, it minimizes the number of fitted parameters, enhances simplicity, and offers satisfactory outcomes and accuracy. To develop the model, we created an extensive database containing experimental viscosities of FAMEs and FAEEs in a wide range of temperatures and pressures. Additionally, we compared its output results with the other 13 predictive models based on GC and CSP concepts. Finally, an extension to high-pressure conditions was proposed using a Tait-like equation with four fitted parameters.

2. Methodology

2.1. FAAEs Experimental Database

Accurate experimental data on physical properties is fundamental to the model’s development. Experimental data are used as comparison criteria to evaluate existing models and to develop new ones. , Because of that, an experimental database containing experimental values of viscosities of fatty acid methyl esters and fatty acid ethyl esters that may occur in biodiesel has been created. Table summarizes all collected data, and further details are available in the Supporting Information.

1. Summary of the Experimental Dynamic Viscosity Data of FAMEs and FAEEs.

compound	acronym	Tmin/K	Tmax/K	Pmin/MPa	Pmax/MPa	number of data (accepted)	number of data (rejected)	references
methyl butyrate	ME-C4:0	273.10	373.07	0.10	0.10	37	2	−
methyl valerate	ME-C5:0	283.15	343.12	0.10	0.10	14	0	−
methyl caproate	ME-C6:0	283.10	372.02	0.10	30.01	60	6	,−
methyl enanthate	ME-C7:0	283.10	361.47	0.10	30.03	63	16	− ,
methyl caprylate	ME-C8:0	263.05	372.02	0.10	30.06	77	19	,−
methyl pelargonate	ME-C9:0	293.14	360.96	0.10	30.03	42	18	,,
methyl caprate	ME-C10:0	263.05	372.02	0.10	200.00	106	58	,−
methyl undecanoate	ME-C11:0	293.14	343.12	0.10	0.10	3	0
methyl laurate	ME-C12:0	278.10	372.02	0.10	30.29	109	9	,,,
methyl tridecanoate	ME-C13:0	293.14	343.12	0.10	0.10	3	0
methyl myristate	ME-C14:0	293.14	372.02	0.10	100.00	80	12	,,,
methyl pentadecanoate	ME-C15:0	293.14	343.12	0.10	0.10	14	0	,
methyl palmitate	ME-C16:0	303.15	372.02	0.10	0.10	30	1	,,,
methyl palmitoleate	ME-C16:1	263.05	363.15	0.10	0.10	18	12	,,
methyl heptanoate	ME-C17:0	310.93	333.12	0.10	0.10	2	0
methyl stearate	ME-C18:0	310.93	573.18	0.10	0.10	25	1	,,,
methyl oleate	ME-C18:1	263.05	353.15	0.10	0.10	51	2	,,,
methyl linoleate	ME-C18:2	263.05	353.15	0.10	0.10	35	3	,,,
methyl linolenate	ME-C18:3	263.05	373.15	0.10	0.10	32	10	,,,
methyl nonadecanoate	ME-C19:0	313.13	343.12	0.10	0.10	2	0
methyl arachidate	ME-C20:0	323.15	373.15	0.10	0.10	11	0
methyl gadoleate	ME-C20:1	278.15	373.15	0.10	0.10	20	0
methyl behenate	ME-C22:0	333.15	373.15	0.10	0.10	9	0
methyl erucate	ME-C22:1	278.15	363.15	0.10	0.10	18	0
methyl lignocerate	ME-C24:0	338.15	373.15	0.10	0.10	8	0
ethyl butyrate	EE-C4:0	288.15	348.12	0.10	0.10	32	2	,,,
ethyl valerate	EE-C5:0	293.15	313.15	0.10	0.10	4	0	,
ethyl caproate	EE-C6:0	288.15	358.15	0.10	0.10	14	14	,,,
ethyl enanthate	EE-C7:0	312.72	353.04	0.10	15.17	30	0
ethyl caprylate	EE-C8:0	263.05	358.15	0.10	15.24	52	0	,−
ethyl caprate	EE-C10:0	263.05	353.15	0.10	200.00	84	50
ethyl laurate	EE-C12:0	273.10	353.65	0.10	15.20	71	3	,,,
ethyl myristate	EE-C14:0	283.15	353.15	0.10	100.00	74	8
ethyl palmitate	EE-C16:0	298.14	573.18	0.10	0.10	24	0	,,,
ethyl palmitoleate	EE-C16:1	293.15	310.14	0.10	0.10	3	0
ethyl stearate	EE-C18:0	310.14	573.18	0.10	0.10	20	1	,,,
ethyl oleate	EE-C18:1	263.05	363.15	0.10	0.10	51	0	,−
ethyl linoleate	EE-C18:2	263.05	363.15	0.10	0.10	32	1	,,,
ethyl linolenate	EE-C18:3	263.05	373.15	0.10	0.10	35	0	,,,
ethyl arachidate	EE-C20:0	318.15	373.15	0.10	0.10	12	0
overall						1655	248

^aME and EE denote methyl ester and ethyl ester, respectively; In Cx/y, x and y denote the number of carbons and double bonds in the fatty acid chain, respectively.

A total of 1655 experimental viscosity values were collected for 25 FAMEs and 15 FAEEs, distributed in 47 references published between 1897 and 2020. The minimum and maximum temperatures and pressures covered were 263.0–573.18 K and 0.10–200.00 MPa. Most of the data are under atmospheric pressure (0.10 MPa), totaling 998 data, and the remaining 658 data are under this condition. Viscosity data under high pressure for ME-C18:1 and ME-C18:2 were not included because these were obtained using an extrapolative methodology.

As it is broadly known and based on previous experiences , when dealing with different data sets, multiple values may be provided for FAEEs from various references. Because of this, it is necessary to verify the agreement within these references because some variables, such as experimental apparatus/methodology and sample purities, can affect the uncertainty associated with the final value of η. Therefore, it was adopted a refinement procedure to select the best data for each FAME and FAEE in the database, and mitigate possible problems that could be associated with higher uncertainties.

The current database contains η under diversified pressure conditions. Two different approaches were adopted to refine these data. At atmospheric conditions, the correlation proposed by Vogel expressed by eq , which only considers η as temperature-dependent, and the refinement must be carried out at constant pressure. Additionally, all data at pressures less than 0.13 MPa were considered under atmospheric conditions.

$$\ln \eta (T)=\mathrm{A}+\frac{\mathrm{B}}{T+\mathrm{C}}$$ 1

where A, B, and C are constants obtained by optimization, and T is the observed temperature in Kelvin.

For viscosity at high pressures, a Tait-like equation was used as presented in eqs and .

$$\ln \eta (T,P)={\eta }_0(T,P_0)\mathrm{e}\mathrm{xp}\left[D\ln \left(\frac{E(T)+P}{E(T)+P_0}\right)\right]$$ 2

$$E(T)=D_0+D_{1}T+D_2{T}^2$$ 3

where P ₀ is the reference pressure (in this work, 0.10 MPa was selected), η₀ is the viscosity calculated at P ₀, P denotes the observed pressure in MPa, and D ₀, D ₁, and D ₂ are also constants obtained by optimization. As seen in eq , when P = P ₀, the models return the viscosity at the reference pressure η₀.

To perform both refinements, choosing the tolerance that should be followed as the refinement criteria was necessary. Although η is a property that does not require greater exactness than other thermodynamic properties, there are no reference values for it, only a few reports about accuracy from well-established methods.

Therefore, considering the maximum uncertainty found in the experimental database, a value of 4.0% was chosen as a refinement criterion. Only refined FAEEs had, at minimum, the number of data higher than double the optimizable parameters (in eqs or ). The refinement results are presented in Table . Supporting Information provides further details, such as optimized parameters and the number of accepted and rejected data.

2.2. Predictive Models

As highlighted in Section , our goal was to compare the model proposed in this study with established models already described in the literature by other authors. These models are summarized in Table . The input parameters for the models in Table , including the functional group assignments for each GC model, are provided in the Supporting Information.

2. Selected Models to Predict the Liquid Dynamic Viscosity.

			required inputs
method	acronym	conceptual basis	molecular structure	ρ	M W	T m	T b	T c	P c	V c	ω
joback and reid	JR	GC	×		×
orrick and erbar	OR	GC	×	×	×
thomas	thomas	GC	×	×				×
morris	morris	GC	×					×
hsu, sheu, and tu	HST	GC	×						×
yinghua et al.	YPP	GC/CSP	×		×		×
ceriani et al.	CGC	GC	×		×
souders	souders	GC	×	×	×
nannoolal et al.	NRR	GC	×				×
sastri and rao	SR	GC	×				×	××
van velzen et al.	VCL	GC	×
przezdziecki and sridhar	PS	CSP		×		×		×	×	×	×
letsou and stiel	LS	CSP			×			×			×

^aGC: Group contribution; CSP: Corresponding States Principle.

^bρ: density of the liquid at specific temperature; M _W: molecular weight; T _m : melting temperature; T _c : critical temperature; P _c : critical pressure; V _c : critical volume; ω: acentric factor.

The required parameters T _b , T _c , P _c , and ω, used in this work, were taken from the database suggested by Evangelista et al. Melting temperatures used were taken from do Carmo et al. The Alvarez e Valderrama method was used for V _c estimations. Additionally, for those models that require ρ as input, the CSP model proposed by Yen and Woods was used due to its performance for systems composed of polar compounds.

2.3. The New CSP Model

The three-parameter Corresponding States Principle (CSP) states that if two fluids have identical acentric factors (ω), at the same reduced temperature (T _r ) and reduced pressure (P _r ), their equilibrium properties will also be identical. − Thus, any reduced property of a fluid can be expressed as a function of three parameters: T _r , P _r , and ω. Teja and Rice derived the following expression for viscosity:

$$\ln (\xi \eta )=\ln {(\xi \eta )}^{R_1}+\frac{\omega -{\omega }^{R_1}}{{\omega }^{R_2}-{\omega }^{R_1}}[\ln {(\xi \eta )}^{R_2}-\ln {(\xi \eta )}^{R_1}]$$ 4

where R ₁ and R ₂ denote the properties of reference fluids 1 and 2, respectively. Viscosities of reference fluids (η ^{R _j} ) are calculated by using eq .

$$\ln {(\xi \eta )}^{R_j}=\mathrm{A}+\frac{\mathrm{B}}{T_r},j=[1,2]$$ 5

where A and B are constants obtained from fitting experimental data and T_r is the reduced temperature defined as T/T _c . The term ξ is equivalent to the critical viscosity. Teja et al. suggested the following equation to calculate it

$$\xi =\frac{T_c^{1/6}}{M_{\mathrm{W}}^{1/2}{P}_c^{2/3}}$$ 6

A computational procedure was developed to determine the optimal values of parameters A and B in eq for each of the 40 individual FAAEs with available experimental data in our database. Subsequently, all possible pairs among these esters were evaluated as reference fluids using eq . Based on this evaluation, ME-C4:0 and ME-C24:0 were selected as reference fluids 1 and 2, respectively. The resulting eq expressions for these two reference esters are shown below

$$\ln {(\xi \eta )}^{R_1}=-22.49+\frac{1.99}{T_{\mathrm{r}}}$$ 7

$$\ln {(\xi \eta )}^{R_2}=-22.99+\frac{2.76}{T_{\mathrm{r}}}$$ 8

Eqs and () provide η values in mPa·s. We will refer to the proposal as “da Silva 2f” hereafter.

The parameters were obtained by minimizing the following objective function

$$\mathrm{OF}=\sum _i{\left(\frac{{\eta }_i^{\exp }-{\eta }_i^{\mathrm{calc}}}{{\eta }_i^{\exp }}\right)}^2$$ 9

where η _i and η _i represent the experimental and calculated dynamic viscosity, respectively. Eq summation includes all available experimental data points for each FAAE. Optimization was conducted using a modified Levenberg–Marquardt algorithm for nonlinear least-squares regression, implemented through a Python script utilizing the least_squares routine from the SciPy library. The optimization process was repeated with different initial estimates for parameters A and B to guarantee convergence to a global minimum.

In certain applications, accurate dynamic viscosity values of FAAEs under high-pressure conditions are necessary. To accommodate such cases, the proposed model was extended to elevated pressures by calculating the term η₀ in eq using eqs –(). The parameters D ₀, D ₁, D ₂, and C were treated as global constants obtained through fitting to experimental data from selected compounds.

3. Evaluation Procedure

3.1. Statistical Performance Indicators

To evaluate the proposed model and compare it with the other models presented in Table , the statistical indicator average absolute relative deviation (%AARD) was used

$$\%\mathrm{AARD}=\frac{100}{N_{\mathrm{data}}}\sum _i^{N_{\mathrm{data}}}\left|\frac{{\eta }_i^{\mathrm{calc}}-{\eta }_i^{\exp }}{{\eta }_i^{\exp }}\right|$$ 10

The superscripts calc and exp denote dynamic viscosity calculated by the models and experimental, respectively. N _data is the number of experimental data. The %AARD was chosen because it can provide accuracy and quantify the degree of data scatter of the models.

3.2. Physical Behavior Evaluation

As noted in our earlier studies, ,,, a predictive model must demonstrate accuracy and align with empirical observations, exhibiting physical consistency. Hence, this section outlines two methodologies designed to assess the physical consistency of all predictive models.

For an FAAE of the same class (FAMEs or FAEEs), with an equal amount of unsaturation in the chain, it is observed that η increases as the number of carbons increases. The homologous series test aims to evaluate the performance of all models submitted to eq through 34 homologous pairs of FAAEs analysis that may occur in biodiesel. Further information is provided in the Supporting Information.

$${\eta }_{(i+1)}>{\eta }_{(i)}\mathrm{at}\mathrm{the}\mathrm{same}\mathrm{temperature}$$ 11

The terms (i+1) and (i) denote adjacent members of a homologous series. This test was performed between the T _m of the member (i+1) and T _b of the member (i). This temperature interval was selected to fairly evaluate GC models, as most of these models have been originally proposed for application within this specific range.

As criteria, if the ratio (η_(i) – η_(i+1))/η_(i+1) calculated by each model was greater than the more significant uncertainty found in the experimental database, which was 4.0%, at least one temperature, the model was considered inconsistent and will not pass to the next test.

The most accurate model for FAMEs is not necessarily the best for FAEEs. For this reason, “n × n” packages were created, where n corresponds to the number of models passed in the homologous series test. Each package analyzes one possible combination between the models, wherein one package (i–j) represents one model for FAMEs (i) and another for FAEEs (j). The accuracy of all packages was calculated using eq

$$\%\mathrm{ARRD}(i-j)=\frac{100}{N_{\mathrm{data}}}\left(\sum _{k=1}^{N_{\mathrm{FAME}}}\left|\frac{{\eta }_k^{\mathrm{calc}}-{\eta }_k^{\exp }}{{\eta }_k^{\exp }}\right|+\sum _{k=1}^{N_{\mathrm{FAEE}}}\left|\frac{{\eta }_k^{\mathrm{calc}}-{\eta }_k^{\exp }}{{\eta }_k^{\exp }}\right|\right)$$ 12

The summation in eq for models i and j covers the amount of data for FAMEs (N _FAME) and FAEEs (N _FAEE).

Similar to the homologous series, experimental observations show that when comparing members with the same chain length and unsaturation number and from different classes (FAMEs/FAEEs), the η value for an EE member is always higher than that for an ME member. Hence, all packages were submitted to eq . The procedure was performed between T _m of EE member and T _b of ME members for 34 ME/EE pairs of FAAEs. In addition, the ratio (η_ME – η_EE)/η _EE calculated by each model was greater than 4.0%, at least one temperature, the package was considered inconsistent.

$${\eta }_{(\mathrm{EE})}>{\eta }_{(\mathrm{ME})}\mathrm{at}\mathrm{the}\mathrm{same}\mathrm{temperature}$$ 13

Initially, all predictive models were submitted to statistical and physical consistency analysis for only atmospheric conditions. After identifying the most accurate and physically consistent models, these were applied to predict the η values for FAAEs at elevated pressures.

4. Results and Discussion

4.1. Evaluating Optimum Models for FAAEs at Atmospheric Conditions

The proposed model (da Silva 2f) and the other 13 models mentioned in Section were tested at atmospheric conditions (P ≤ 0.13 MPa) for all FAAEs in the experimental database. The performance of the models studied is presented in Table . With some exceptions, the GC models showed better predictions than CSP. This result is not surprising, considering that this class of models is more suggested at the temperature range: T _m ≤T ≤ T _b . Further details can be found in the Supporting Information.

3. %AARD for Liquid Viscosity Obtained Using GC and CSP Models.

	models
FAAE	JR (%)	OR (%)	thomas (%)	Morris (%)	HST (%)	YPP (%)	CGC (%)	souders (%)	NRR (%)	SR (%)	VCL (%)	PS (%)	LS (%)	da Silva 2f (%)
ME-C4:0	5.93	4.27	3.67	3.00	1.93	5.39	6.31	74.16	9.08	1.94	1.08	114.04	23.37	1.04
ME-C5:0	11.98	1.95	1.69	12.57	4.88	3.55	1.17	47.12	14.89	5.66	1.85	109.16	34.40	4.06
ME-C6:0	10.57	2.82	7.87	25.66	6.41	3.56	1.37	96.66	12.18	4.31	5.22	108.52	35.84	8.50
ME-C7:0	11.35	3.38	5.53	25.57	6.49	3.80	1.49	148.47	12.29	10.09	6.63	106.79	45.98	0.92
ME-C8:0	7.79	3.16	5.55	27.92	10.03	4.93	2.75	133.35	8.41	14.55	10.78	105.85	50.24	4.44
ME-C9:0	9.18	2.02	12.08	37.18	14.23	5.31	1.76	193.81	10.66	27.32	10.34	105.24	53.61	6.22
ME-C10:0	8.61	3.28	9.22	36.93	16.86	8.07	1.51	171.55	4.11	22.53	11.56	103.75	63.51	13.51
ME-C11:0	9.82	3.24	18.16	50.76	20.02	5.83	0.59	175.46	8.90	54.57	10.77	103.22	66.62	12.65
ME-C12:0	9.45	3.19	23.33	59.78	29.32	7.01	1.87	114.86	4.10	44.13	11.46	102.68	72.04	18.14
ME-C13:0	10.84	4.22	32.34	71.81	23.75	6.24	1.02	139.71	5.67	78.93	10.73	102.27	73.29	13.88
ME-C14:0	10.40	4.37	32.91	71.30	34.15	5.19	2.11	70.89	3.89	123.32	11.34	102.11	75.98	21.93
ME-C15:0	21.54	13.67	51.48	96.07	28.62	8.35	1.42	76.35	8.81	182.44	3.41	101.70	77.87	16.70
ME-C16:0	15.86	8.87	56.19	102.66	27.58	7.09	1.83	95.03	7.53	138.55	7.82	101.83	77.21	12.65
ME-C16:1	13.72	7.98	64.88	109.33	36.71	6.29	2.60	62.51	4.66	154.20	9.23	101.97	73.58	15.08
ME-C17:0	24.15	12.14	4849.08	109.68	13183.29	30.71	13.74	88.50	24.48	243.93	15.86	101.53	78.49	18.50
ME-C18:0	32.27	24.49	105.46	160.75	25.40	22.17	12.72	80.92	19.10	251.29	5.98	102.78	72.12	8.92
ME-C18:1	15.74	13.15	85.00	126.41	40.04	13.48	2.60	45.81	5.64	252.95	14.23	101.08	82.55	11.15
ME-C18:2	24.87	7.75	6004.54	116.82	12518.62	18.56	4.41	65.79	11.53	314.66	11.92	101.19	80.24	7.09
ME-C18:3	33.68	3.54	283064.11	96.72	2255881.32	35.52	4.15	73.77	22.99	338.85	40.70	101.69	74.14	24.86
ME-C19:0	27.12	12.98	11501433.88	54.48	340107486.21	47.19	6.07	47.65	25.00	396.39	54.65	101.18	82.68	35.69
ME-C20:0	19.70	13.23	117.28	166.32	39.73	13.71	0.72	50.01	4.55	303.15	7.16	101.53	78.30	10.65
ME-C20:1	13.35	8.67	126.81	166.89	43.59	17.02	0.40	51.50	3.00	316.44	22276.55	101.20	81.57	7.52
ME-C22:0	21.61	9.11	7381.37	161.44	9348.87	31.61	1.88	44.89	12.70	459.28	26552.65	101.35	79.93	7.10
ME-C22:1	15.18	10.26	174.89	207.44	49.48	29.08	0.83	85.53	5.63	549.33	26837.00	100.89	84.45	3.93
ME-C24:0	37.12	21.19	10535.81	209.68	10454.05	42.91	3.94	27.51	15.30	743.73	34231.10	101.16	81.89	13.45
EE-C4:0	18.23	14.46	240.21	258.13	55.23	47.49	1.31	189.94	10.63	1226.09	32715.06	110.81	30.52	0.29
EE-C5:0	15.82	4.68	1.91	9.79	8.22	3.57	2.26	80.40	2.90	4.33	3.02	107.81	39.78	1.78
EE-C6:0	21.76	9.37	3.28	20.60	14.91	2.73	4.21	72.33	1.81	5.46	4.11	109.02	29.62	3.04
EE-C7:0	43.39	30.05	26.12	48.76	20.93	24.19	24.95	143.84	24.90	42.42	20.86	107.62	37.08	19.63
EE-C8:0	19.57	7.60	9.48	29.71	2.57	9.39	6.29	78.31	5.01	31.94	2.51	104.80	52.68	1.85
EE-C10:0	20.50	10.44	13.24	40.24	1.88	6.44	6.57	127.83	5.87	29.58	2.30	103.40	63.76	2.29
EE-C12:0	19.54	11.46	21.80	54.41	13.64	4.82	3.93	179.98	5.88	51.69	4.33	102.18	75.27	9.66
EE-C14:0	20.16	11.77	36.25	78.47	29.31	5.62	2.19	95.41	4.48	59.28	3.47	101.76	77.76	17.60
EE-C16:0	21.86	17.64	71.06	112.80	33.28	13.26	1.64	51.90	9.30	234.46	8.95	103.09	70.63	11.83
EE-C16:1	35.31	13.74	5858.13	124.72	11641.67	25.70	2.22	64.47	23.55	303.02	22.36	101.11	83.37	3.98
EE-C18:0	24.20	20.59	96.39	135.32	40.81	21.18	1.49	43.31	11.45	418.37	9.15	102.93	71.47	11.28
EE-C18:1	38.24	17.93	8322.63	160.52	10303.64	34.30	5.69	50.57	24.69	483.01	20.82	101.03	84.43	7.88
EE-C18:2	40.51	8.05	417820.35	130.66	1731664.19	55.08	4.59	50.32	36.01	533.51	45.19	101.24	81.35	26.80
EE-C18:3	42.32	3.78	23942321.02	96.20	296516627.72	79.00	4.98	51.57	46.28	588.80	72.44	101.56	76.73	53.63
EE-C20:0	23.52	17.15	155.82	194.68	44.54	34.98	1.34	72.44	14.20	774.22	26056.56	101.40	79.81	5.50
FAMEs	15.31	6.29	596376.10	75.55	17265600.26	14.56	3.08	96.83	10.11	175.33	2997.69	103.87	65.30	12.55
saturated	10.57	4.94	32.34	60.54	21.39	7.79	2.25	110.71	7.06	94.97	1631.78	104.84	59.87	10.58
unsaturated	27.87	9.87	2176344.34	115.31	63009461.52	32.51	5.29	60.04	18.18	388.25	6616.55	101.30	79.69	17.77
FAEEs	27.79	12.63	2340120.00	93.92	28665002.80	24.31	4.43	82.47	16.50	269.29	876.53	103.37	67.85	15.67
saturated	22.07	13.73	42.99	73.54	20.66	11.24	4.49	98.64	8.54	160.73	1561.20	104.89	58.16	9.04
unsaturated	34.85	11.29	5225736.43	119.05	64012618.81	40.43	4.36	62.54	26.32	403.14	32.25	101.48	79.79	23.85
overall	19.86	8.60	1231734.24	82.24	21419136.32	18.11	3.57	91.60	12.44	209.57	2224.81	103.69	66.23	13.69

Considering only GC models for the FAMEs class, it was observed the following accuracy order: CGC (3.08%) < OE (6.29%) < NRR (10.11%) < YPP (14.56%) < JR (15.31%) < Morris (75.55%) < Souders (96.83%) < SR (175.33%) < VCL (2997.69%) < Thomas (596376.10%) < HST (17265600.26%).

The Ceriani et al. (CGC) model presented the best performance of all models, presenting a %AARD lower than 4.0%, corresponding to the maximum uncertainty reported at P ₀. These results were expected since the CGC model was proposed for fatty compounds. The OE model presented the second-best performance, which could be related to the density data estimated by Yen and Woods model used as input. It is important to note that models that use experimental data as inputs could improve their accuracy. NRR also showed good results, which should be related to its good description of the interactions between –CH₃– and –CH₂– groups with electronegative atoms.

To CSP models, the crescent %AARD order for FAMEs is presented as follows: da Silva 2f (12.55%) < LS (65.30%) < PS (103.87%). Compared to GC models, da Silva 2f was as accurate as NRR. Such performance can be related to the chosen reference fluids, ME-C4:0 and ME-C24:0, which contain the experimental database’s smallest and largest carbon numbers.

Analyzing the %AARD profile varying the FAMEs’ carbon number, it was observed that by increasing the carbon number from ME-C4:0 until ME-C24:0, the %AARD behaved similarly to an interpolation profile. Therefore, minimum and maximum %AARD values were found to be around ME-C4:0 and ME-C24:0.

Comparing the models’ performances predicting η for FAEEs, it was observed that the results for FAAEs class were worse than for FAMEs. Besides, all models presented similar results for both classes. Models such as Thomas (5225736.43%), Morris (119.05%), HST (64012618.81%), Souders (82.47%), SR (269.29%), VCL (876.53%), PS (103.37%), and LS (67.85%) still presented large deviations.

Figure illustrates the distribution of %ARD and %AARD values as a function of carbon chain length. As shown in Figure a, most of the %ARD values obtained with the CGC model (27%) fall within the [0.0–1.0%] range, following a clear linear decreasing trend. Thus, as the range of %ARD values increases, the frequency of predictions within each interval declines. Upon examination of the predictions by carbon chain length (Figure b), it becomes evident that the CGC model does not exhibit a distinct accuracy pattern, maintaining consistent performance across all chain lengths studied. Nevertheless, FAEEs in the C₄–C₁₀ range observed the most significant deviations.

1.

(a) ARD distribution and (b) %AARD profile obtained by the most accurate models predicting η considering the carbon number of FAAEs.

The JR, OE, YPP, and NRR models predominantly exhibited deviations in the [5.0%–10.0%] range, with relatively few predictions falling within the [0.0–1.0%] interval. Most predictions by da Silva 2f, OE, and YPP models within the lowest deviation interval [0.0–1.0%] corresponded mainly to shorter-chain FAAEs (C₄–C₁₀). Conversely, the NRR model achieved its highest accuracy for medium-chain FAAEs (C₁₁–C₁₆). Furthermore, all models, except NRR, consistently showed their smallest and largest prediction errors for short- (C₄–C₁₀) and long-chain FAAEs (C₁₇–C₂₄), respectively.

Figure illustrates the influence of the temperature on the predictive performance of the models. Among the evaluated models, the CGC model provided the most reliable predictions across the temperature range investigated, exhibiting the lowest scattering of deviations. The OE model showed a similar deviation profile to CGC but with a slightly higher scatter. Both CGC and OE models presented their most significant outliers in the temperature interval between 263 and 301 K.

2.

ARD distribution by the models in the temperature interval. The colors navy, purple, cyan, green, yellow, and red are the models JR, OE, YPP, CGC, NRR, and da Silva 2f, respectively.

The da Silva 2f and NRR models presented similar scattering patterns. Specifically, the NRR model exhibited more deviations between 263 and 379 K, with a reduction observed at higher temperatures (>379 K). In contrast, the da Silva 2f model presented a uniform deviation profile, lacking regions with particularly dense scattering, and thus maintained a consistent accuracy across all temperatures analyzed. The JR model showed a deviation behavior similar to that observed in the da Silva 2f and NRR models, but with greater overall scatter. Finally, the YPP model displayed the highest scattering among all of the evaluated models, highlighting its limited reliability in predicting η at the lowest and highest temperature extremes.

All models exhibiting an overall %AARD below 20% (i.e., JR, OE, YPP, CGC, NRR, and da Silva 2f) underwent the consistency tests described in Section . All successfully passed the homologous series test. However, during this evaluation, the CGC model unexpectedly exhibited discontinuities in the predicted viscosity profiles for some homologous pairs. This behavior is illustrated in Figure , where the viscosity profiles for selected saturated and unsaturated FAMEs predicted by the CGC and da Silva 2f models are compared.

3.

Comparison between generated profiles by CGC (a, b) and da Silva 2f (c, d) models for selected saturated and unsaturated FAMEs.

Figure a,c illustrates the anomalous behavior of the CGC model near 273.50 K, where the predicted viscosity values tend toward infinity. This behavior can be explained by a part of the model expression that exhibits a hyperbolic function. This issue could significantly impact the estimation of cold flow properties, such as cloud point, pour point, and cold filter plugging point, which typically occur near this temperature region. On the other hand, Figure b,d shows that the da Silva 2f model maintains continuous viscosity predictions across the entire temperature range evaluated, highlighting its reliability for estimating biodiesel viscosity in this extreme condition.

In addition to the homologous series test, the six best-performing models (JR, OE, YPP, CGC, NRR, and da Silva 2f) were evaluated using the “FAME/FAEE pair test”. This test employs eq to confirm whether a FAEE consistently exhibits higher viscosity than its corresponding FAME (with an identical number of carbon atoms in the fatty chain) at the same temperature. A total of 36 model combinations (“packages”) were assessed, and only the following packages displayed consistent behavior: CGC/CGC (%AARD = 3.57%), OE/OE (%AARD = 8.60%), da Silva 2f/da Silva 2f (%AARD = 13.69%), YPP/YPP (%AARD = 18.11%), and JR/JR (%AARD = 19.86%). Additional information and complete results of all consistency tests are available in the Supporting Information.

4.2. Extension to High Pressures

Based on the %AARD analysis and the results of the consistency tests performed in previous sections, including the homologous series and ME/EE pair assessments, the proposed model (da Silva 2f) was selected as a reliable foundation for viscosity predictions under high-pressure conditions, as described by eqs and (). In this approach, the term η₀ in eq is calculated using the da Silva 2f model, while D ₀, D ₁, D ₂, and C are global constants derived from fitting experimental data.

Models validated against data sets distinct from those used for parameter fitting generally demonstrate enhanced reliability and improved predictive capabilities. ,, Therefore, the high-pressure viscosity data presented in Table were divided into two independent sets: a correlation data set and a prediction data set.

The complete high-pressure database consists of 408 experimental data points, randomly partitioned into the correlation and prediction data sets, each comprising 50% of the total data (204 points each). This random partition was performed using the train_test_split function from Python’s sklearn library. To estimate the global parameters, the least_squares function from Python’s SciPy library, which implements the Levenberg–Marquardt optimization algorithm, was employed. The objective function (f _obj) used for parameter fitting was defined by eq . The optimization results for the global constants (D ₀, D ₁, D ₂, and C) of eq are as follows: D ₀ = −1194.85, D ₁ = 7.62, D ₂ = −2.18 × 10^–2, and C = 1.40.

Figure presents the viscosity predictions generated by the high-pressure model for the correlation and prediction data sets and the corresponding relative deviations. Additionally, statistical analyses were conducted to thoroughly assess the accuracy and predictive reliability of the proposed model. It is important to highlight that the model constants (D ₀, D ₁, D ₂, and C) were optimized exclusively based on experimental data for selected FAMEs (C7:0-C10:0, C12:0, and C14:0) and FAEEs (C7:0-C14:0). As observed in Figure , the majority of relative deviations are within ± 10% and are randomly distributed around the baseline (0%), confirming the absence of systematic errors.

4.

Plots of experimental versus calculated P-η-T for FAAEs with high-pressure data using the high-pressure model: (a) correlation set; (b) relative deviations for correlation set; (c) prediction set; (d) relative deviations for prediction set. Dashed lines correspond to ± 10% relative deviation.

Furthermore, the predictive performance of the high-pressure model was quantitatively evaluated using the root-mean-square error (RMSE), the most widely adopted parameter for predictive model analysis. The RMSE is calculated as follows

$$\mathrm{RMSE}=\sqrt{\frac{\sum _i^N({\eta }_i^{\mathrm{calc}}-{\eta }_i^{\exp })}{N}}$$ 14

A lower RMSE value indicates a better predictive performance. In this study, the RMSE values obtained were 0.30 mPa·s for the correlation data set and 0.28 mPa·s for the prediction data set. The calculated %AARD values were 8.25 and 7.94% for the correlation and prediction data sets, respectively. These statistical indicators reinforce the robustness and reliability of the proposed high-pressure model. The complete set of predicted viscosity values used in this analysis is provided in the Supporting Information (SI).

5. Conclusions

In this work, a model based on the three-parameter Corresponding States Principle was proposed to estimate the dynamic viscosity of biodiesel-related esters (fatty acid methyl esters and fatty acid ethyl esters) over a wide range of temperatures and pressures. The proposed model exhibited strong accuracy, with average absolute relative deviations of 12.95% under atmospheric conditions and 7.94% under high-pressure conditions. A comprehensive analysis, including statistical indicators, deviation distribution, empirical trends, and temperature extrapolation, confirmed the model’s superiority, especially at low temperatures. Given its reliability, this model is highly recommended for future simulations, contributing to advancements in practical applications related to biodiesel.

Glossary

Abbreviations

Roman Letters

AARD

average absolute relative deviation

RMSE

root-mean-square error

FAAE

fatty acid alkyl esters

ME

methyl esters

EE

ethyl esters

GC

group contribution

CSP

corresponding state principle

Greek Letters

η

dynamic viscosity

Superscripts

calc

calculated

exp

experimental

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c03190.

• Comprehensive information on the developed database; including all registered fatty acid alkyl esters; corresponding experimental data points with literature references; details of functional group assignments for each compound; refinement procedures at both low and high pressures; results of all consistency tests performed, and the outputs from all applied models, and including the calculated deviations for each compound (XLSX)

The Article Processing Charge for the publication of this research was funded by the Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES), Brazil (ROR identifier: 00x0ma614).

The authors declare no competing financial interest.

Published as part of ACS Omega special issue “Chemistry in Brazil: Advancing through Open Science”.