Kinematic viscosity prediction of jet fuels and alternative blending components via comprehensive two-dimensional gas chromatography, partial least squares, and Yeo-Johnson transformation

Abstract

This work presents an accurate yet simplified partial least squares model to predict the kinematic viscosity of conventional and alternative jet fuels at −20°C using comprehensive two-dimensional gas chromatography coupled to a flame ionization detector (GC × GC/FID). Three different normalization methods (mean-centering, logarithmic, and Yeo-Johnson) were evaluated to identify their impact in the prediction of middle distillates’ physical properties. Results using Yeo-Johnson transformation exhibited improved viscosity prediction capabilities over the validation set with a mean absolute percentage error of 5.3%, a root-mean-squared error of 0.23, and a coefficient of determination (R²) of 0.9404 using only 10 latent variables. Unlike previously reported correlations, this model allowed the identification of specific hydrocarbon groups and carbon numbers that drive jet fuel viscosity at low temperatures. The presence of even small amounts of large branched-alkanes (C₁₅–C₁₇), dicyclic-alkanes (C₁₀), and cycloaromatics (C₁₁) have the potential to strongly increase the kinematic viscosity of jet fuels. Contrastingly, light monocycloalkanes and branched-alkanes (≤ C₁₀) were associated with lower viscosity values. Novelly, this model suggests the implementation of Yeo-Johnson transformations to predict the physical properties of middle distillates to further improve the performance metrics of partial least squares models based on GC data.

Article Related Abbreviations

ATJ-SPK

alcohol-to-jet synthetic paraffinic kerosene

ASTM

American Society for Testing and Materials

CHJ

catalytic hydrothermolysis jet

FID

flame ionization detector

FTIR

Fourier-transform infrared spectroscopy

FT-SPK

Fischer-Tropsch hydroprocessed synthesized paraffinic kerosine

GC × GC

comprehensive two-dimensional gas chromatography

GC-MS

gas chromatography-mass spectrometry

H/C

hydrogen-to-carbon

HEFA

hydroprocessed esters and fatty acids

MAPE

mean absolute percentage error

MLR

multiple linear regression

MW

molecular weight

NIR

near-infrared spectroscopy

PCR

principal component regression

PLS

partial least squares

R2

coefficient of determination

RMSE

root-mean-squared error

RMSECV

root mean squared errors of cross-validation

SEP

standard error of prediction

SIP

synthesized iso-paraffins

1 INTRODUCTION

Viscosity is a key physical property that defines the lubricity, combustion, and emission characteristics of a fuel influencing atomization and ignition capabilities [1-3]. Specifically, pumpability and nozzle spray patterns are extensively dominated by fuel viscosity [4]. High viscosity values can cause filter plugging, pumpability issues, and spray flow deceleration conditions in fuel systems [4]. Fuels with adequate viscosity promote atomization forming thinner sprays with a smaller average droplet size promoting better combustion and emission performances [4, 5]. Contrastingly, extremely low viscosity values can prevent the forming of hydrodynamic film layers negatively affecting lubricity conditions boosting wear rates and scuffling of rubbing metal parts, or inducing power loss due to injector leakages [6].

Viscosity in liquid fuels inversely varies with temperature, exponentially increasing at low temperatures [7]. For that reason, distinctive technical requirements must be considered for different fuels depending on their service conditions. Specifically, ensuring adequate cold flow properties for jet fuels is critical given their extremely low operational temperature limits. Conventional jet fuels are usually composed of hydrocarbon molecules with 8–16 carbon atoms, encompassing n-alkanes, branched-alkanes, cycloalkanes, and aromatic compounds [8]. Other molecules including heteroatoms and olefins are limited to less than 1% vol./vol due to thermal and storage stability requirements [9]. Owing to this necessity, the kinematic viscosity of conventional Jet A/A-1, and alternative blending mixtures must be below 8.0 mm²/s at −20°C to ensure safe cold flow capabilities [10]. Although military and propellant jet fuels (JP-8, JP-5, and JP-10) have different chemical compositions compared to Jet A, having larger fraction of lower-carbon-number molecules in JP-8, heavier fractions to ensure higher flash points in JP-5 or target synthetic compositions such as the case of JP-10, their kinematic viscosity requirements are similar to conventional jet fuel to guarantee adequate technical performance, that is, 8.0 mm²/s at −20°C for JP-8, 8.5 mm²/s at −20°C for JP-5, and 10.0 mm²/s at −18°C for JP-10 [11-13].

The certification process of emerging alternative jet fuels is characterized by high costs (US$10 M—US$15 M) and extended times (3−5 years) requiring large amounts of sample volumes (up to 236 000 gallons) [8]. With that aim, American Society for Testing and Materials (ASTM) D4054 describes the testing program as divided into four tiers: basic specification properties; fit-for-purpose properties; engine/aircraft systems rig and component testing; and full-scale engine testing or aircraft flight testing, to be followed for the certification of new aviation turbine fuels and fuels additives to comply with the technical requirements established by ASTM D1655 [9, 14]. Once certified, the specification criteria for the alternative jet fuel or fuel additive is incorporated into the technical specifications of ASTM D7566 to be further authorized for commercial used under specific blending conditions [10]. Currently, Fischer-Tropsch hydroprocessed synthesized paraffinic kerosine (FT-SPK); synthesized paraffinic kerosine from hydroprocessed esters and fatty acids (HEFA); synthesized iso-paraffins from hydroprocessed fermented sugars (SIP); synthesized kerosine with aromatics derived by alkylation of light aromatics from non-petroleum sources (SPK/A); alcohol-to-jet synthetic paraffinic kerosene (ATJ-SPK); synthesized kerosine from hydrothermal conversion of fatty acid esters and fatty acids (CHJ), and synthesized paraffinic kerosine from hydroprocessed hydrocarbons, esters, and fatty acids (HC-HEFA) have been approved for use in gas turbine engines under specific blending ratios with conventional jet fuels [10].

Given the technical and economic importance of fuel viscosity, it is imperative to find technical alternatives for the rapid screening and prediction of the kinematic viscosity of jet fuels and alternative blending components [15]. The viscosity prediction based on functional groups and chemical composition analysis via near-infrared spectroscopy (NIR), Fourier-transform infrared spectroscopy (FTIR), and gas chromatography/mass spectrometry (GC/MS) has been reported as powerful options while utilizing minimum amounts of fuel [16, 17].

Based on spectroscopy data, Fodor and Kohl characterized and predicted the physical properties (including kinematic viscosity at 40°C) of over 300 fuel samples including diesel fuels and jet fuels (Jet A, Jet A-1, JP-5, and JP-8) via midband infrared spectroscopy (4000−650 cm⁻¹ wavenumber range) and partial least squares (PLS) [18]. This methodology effectively predicted the kinematic viscosity of middle distillate fuels based on spectral intensities obeying Beer's law with a coefficient of determination (R²) of 0.957 and a standard error of prediction (SEP) of 0.166 mm²/s [18]. Andrade et al., predicted, among other properties, the kinematic viscosity of 29 kerosene samples using multiple linear regression (MLR), principal component regression (PCR), and PLS using mean-centered FTIR data in the 1400−680 cm⁻¹ range [19]. Authors identified to PLS as the multivariate technique that offered the best precision figures for all cases with a SEP of 0.2 mm²/s [19]. Additionally, Fodor et al., predicted the kinematic viscosity of middle distillate fuels at 40°C via PLS based on FTIR through mean-centering and mean-centering of first- and second-difference spectra [20]. The calibration set consisted of 547 samples while 137 samples were used for validation achieving a R² of 0.9773 and a SEP of 0.07 mm²/s [20].

Complementarily, Xing et al., developed a quantitative near-infrared spectroscopy method (700−1100 nm) using PLS regression for the determination of the kinematic viscosity of 44 jet fuels [21]. The SEP between experimental values and those predicted for the validation group using NIR data was 0.027 mm²/s in a narrow viscosity range of 1.56−1.76 mm²/s [21]. The use of mid-infrared FTIR spectra (3300−3550 nm) via normalization and Lasso-regularized linear model yielded satisfactory estimations for the kinematic viscosity at −20°C for 64 hydrocarbon fuels, that is, distilled and synthetic jet fuels, pure hydrocarbons, and blending mixtures, with a cross-validation error of 0.697 or 14% using eight latent variables [22]. NIR spectroscopy data using a five step preprocessing methodology including mean centering was also used to predict the physical properties of jet and diesel fuels by Morris et al., [23]. In this case, coefficients of determination (R²) of 0.73 and 0.85 and root mean squared errors of cross validation (RMSECV) of 0.5202 and 0.195 were estimated for the kinematic viscosity of 50 jet (at −20°C) and 261 diesel (at 40°C) fuel samples using PLS, respectively [23]. This set of studies demonstrated the successful applicability of IR data to predict the viscosity of complex chemical mixtures. However, neither explicit predictive equations (except for Wang et al., [22]) nor direct relationships among viscosity and chemical structures are stated for any of the studies shown above using infrared spectroscopy limiting their direct external application.

Aiming to explore the influence of different hydrocarbon groups on the final viscosity of complex fuel mixtures, Wang et al. predicted the viscosity of 23 hydrocarbon fuels at −20°C (R² = 0.91) based on Grunberg-Nissan approach using an exponential regression model focused on hydrogen-to-carbon (H/C) molar ratio and molecular weight (MW) information [24, 25]. Thus, the authors stated that the kinematic viscosity of hydrocarbon fuels decrease with the increase of (H/C)/MW² [24]. Additionally, Cai et al. established that the viscosity of alkanes, alkenes, and aromatic compounds increases with the carbon number at one given temperature while cycloalkanes and aromatics are more viscous than other groups with the same carbon number [7]. Branched-alkanes generally have lower viscosity values than n-alkanes due to higher degrees of branching [7]. In terms of cycloalkanes, it has been shown that structural influences and position of substituents are key parameters influencing the viscosity of Jet-A/alkylcycloalkane mixtures [26]. In such cases, more sterically hindered structures in monocycloalkanes and cis-geometries in dicycloalkanes promoted higher kinematic viscosity values than less-hindered monocycloalkanes and trans-geometries in dicycloalkanes due to increased molecular interactions [26]. Aromatic compounds increase in general the viscosity of aviation fuels varying with molecular size and structures mainly due to hyperconjugation effects [27]. Additionally, ortho-positions of the substituents in aromatics increase the viscosity of the fuel due to more compact structures [25].

Expanding on the influence of different chemical structures, Fortin and Laesecke predicted the kinematic viscosity of nine conventional and alternative jet fuels in a temperature range from 20 to 100°C with a maximum absolute average deviation of 0.9% [28]. Despite stating general trends in terms of how molecular size and shape influence intermolecular interactions, via van der Waals forces in non-polar molecules (increasing for larger molecules), the authors did not identify specific trends linking kinematic viscosity values to the complex chemical composition of the fuels partially due to lack of carbon number and degree of branching information. Thus, they suggested the need for a more flexible and robust model able to capture the chemical composition influence on the final viscosity of complex fuel mixtures [28]. Striving to surmount this knowledge gap, Cai et al., developed the quantitative structure-property relationship model using the viscosity data collected at different temperatures of 261 model hydrocarbons including n-alkanes, branched-alkanes, alkenes, alkynes, monocyclic and polycyclic cycloalkanes, and aromatics to adjust Andrade's equation via an artificial neural network [7]. The authors proposed 36 fundamental chemical features based on MW, basic groups (hydrocarbon groups), and united (isomers) groups to correlate the relationship between molecular structures and Andrade's parameters [7]. For this case, the model was validated predicting the viscosity of gasoline and diesel fuels using a mixing rule based on the mole fraction and the viscosity of pure characteristic model compounds with an average absolute error of 0.21 mPa-s.

By using GC and MS data, the US Navy developed the fuel composition and screening tool based on hydrocarbon group distribution divided into saturates (normal-, branched-, and cyclic-compounds), alkenes (acyclic and cyclic), aromatics (alkylbenzenes, indans, naphthalenes, acenaphthenes, and tricycloaromatics), and heteroatomic compounds utilizing an uninformative variable elimination PLS model [29]. This approach helped to predict, among other properties, the kinematic viscosity of 76 diesel fuel samples with an RMSEP of 0.9902 [29]. However, no further available information is provided in the report regarding the latent variables used for such prediction. Complementarily, Benavides et al., developed a linear regression model (R² = 0.91) using 22 latent variables to predict the kinematic viscosity at −20°C of 70 Jet A/A-1 samples based on the semi-quantitation of alkanes, naphthenic, aromatic, naphthalenes, tetralin- and indane compounds via GC/MS [30]. Authors established that C₁₆ monobranched-alkanes and C₇ monobranched alkyl-benzenes strongly influenced the kinematic viscosity values of Jet A/A-1 [30]. More recently, Cain et al., showed the implementation of tile-based variance ranking to improve PLS predictive modeling via GC×GC time-of-flight MS (GC × GC/TOF-MS) [31]. Comprehensive two-dimensional gas chromatography overcome conventional one-dimensional GC offering enhanced separation of isomers, easier automated separation of saturated and aromatic fractions through structured chromatograms, higher peak capacity, and higher resolution [32, 33]. By such implementation, the authors predicted the viscosity of 58 fuel samples with a normalized root mean square error of cross validation (NRMSECV) and normalized root-mean-square error of prediction (NRMSEP) of 7.0% and 4.9%, respectively [31]. Latest advances on vacuum ultraviolet spectroscopy coupled to GC and GC × GC systems and specialized data processing algorithms profile this technology as a promising complement to MS to enhance isomer selectivity as a key factor towards the enhanced prediction of the physical properties of middle distillates in the future [34, 35].

Although different efforts have been explored to predict the kinematic viscosity of middle distillates using multiple analytical techniques and statistical methods, it is still fundamental to formulate complementary enhanced models aiming for better prediction capabilities while guaranteeing simplicity, low sample volumes, and short processing times. More importantly, new predictive models must drive the understanding of the linkage between kinematic viscosity and the chemical composition of complex fuel mixtures, as a key parameter for the design of new surrogates and sustainable alternative fuels while complying with ASTM requirements.

Aligned to these principles, this work proposes for the first time the use of GC × GC coupled to a flame ionization detector (GC × GC/FID) as a powerful option to fill this gap given the ability of this analytical technique to both separate at least an order of magnitude more compounds compared to conventional one-dimensional GC and to provide more detailed chemical information of complex mixtures compared to infrared spectroscopy [17, 36–38]. Additionally, GC × GC/FID provides remarkably better quantitation capabilities with response factors close to one for hydrocarbons unlike MS detectors while avoiding the challenge of different ionization efficiencies during electron and chemical ionization mechanisms and biases towards specific hydrocarbon groups especially when alternative blending components are analyzed via ASTM D2425 [39-43]. Thus, this work exhibits the applicability of GC × GC/FID and PLS to predict the kinematic viscosity of jet fuels and alternative blending components at −20°C. Additionally, this model shows the influence of different normalization methods (mean centering normalization, logarithmic, and Yeo-Johnson transformations) over the PLS performance metrics. Significantly, this work provides valuable information to identify specific hydrocarbon groups and carbon numbers that strongly shift the kinematic viscosity of complex fuel mixtures.

2 EXPERIMENTAL

2.1 Materials

A total of 18 fuel samples were used for analysis. The set of samples covered a broad spectrum of chemical compositions and kinematic viscosities including conventional Jet-A fuels, military petroleum-derived aviation jet fuels (JP-5, JP-8, and JP-10), and alternative jet fuel blending components (F-T, ATJ, SIP, and HEFA). Fifty-six percent of the samples were used for calibration purposes while the remaining fraction was used to validate the model as presented in Table 1. Samples were collected from the Zucrow Laboratory at Purdue University.

TABLE 1. List of samples, experimental kinematic viscosities, and quantitative chemical composition (training set^*).

		Chemical Composition [% wt./wt.]
Sample name	Kinematic Viscosity [mm^2/s]	n-Alkanes	Branched-alkanes	Monocycloalkanes	Dicycloalkanes	Tricycloalkanes	Alkyl-benzenes	Cyclo-aromatics	Alkyl-naphthalenes
Biojet (16POSF12816) ^*	1.796	20.2	47.9	0.2	0.0	0.0	31.8	0.0	0.0
F-T / IPK (11POSF7629)	3.438	0.3	94.7	4.4	0.1	0.0	0.2	0.2	0.1
F-T / ATJ (15POSF12344)	3.957	0.1	98.1	1.6	0.0	0.0	0.1	0.1	0.0
Jet A (13POSF10325) ^*	4.519	20.6	33.0	23.3	5.3	0.1	11.5	3.5	2.7
JP-5 (POSF 12553) ^*	5.002	21.0	30.0	24.7	6.4	0.1	9.8	6.3	1.7
50/50 Jet A / HRJ (POSF 11235)	4.348	14.0	60.9	10.9	2.2	0.0	7.0	3.6	1.5
Gevo–ATJ (13POSF11498)	5.136	0.0	99.7	0.3	0.0	0.0	0.0	0.0	0.0
ATJ ^*	5.142	0.0	99.5	0.3	0.0	0.0	0.2	0.0	0.0
JP-5 / Neat ATJ	5.159	0.1	99.6	0.3	0.0	0.0	0.0	0.0	0.0
HRJ-8 (Camelina 12POSF10301)	5.189	9.1	88.4	2.4	0.1	0.0	0.1	0.0	0.0
HEFA (Camelina) ^*	5.192	9.0	88.5	2.3	0.1	0.0	0.0	0.0	0.0
C-14 / TMB (POSF12223)	5.251	4.9	76.4	0.2	0.0	0.0	18.5	0.0	0.0
JP-8 (13POSF10264)	3.493	27.5	41.2	16.7	1.5	0.1	10.6	1.3	1.2
JP-10 (11POSF7478) ^*	8.842	0.0	0.0	0.0	100.0	0.0	0.0	0.0	0.0
JP-5 / CHCJ ^*	4.750	22.3	9.0	31.8	15.9	0.8	10.1	8.4	1.7
JP-5 / HRJ ^*	7.886	8.0	90.7	1.1	0.0	0.0	0.1	0.0	0.0
JP-5 / HDCJ ^*	8.294	0.2	0.2	11.8	37.1	7.1	5.9	28.0	9.7
JP-5 / Neat SIP ^*	13.587	0.0	99.4	0.5	0.0	0.0	0.1	0.0	0.0

• Notes:
• • Light fraction cut (< C₁₂) decreasing the kinematic viscosity of jet fuels and alternative blending components (in blue).
• • Heavy fraction cut (> C₁₂) increasing the kinematic viscosity of jet fuels and alternative blending components (in red).

2.2 Viscosity measurements

The viscosity of the samples was measured using an SVM 3001 Stabinger Viscometer (Anton Paar GmbH) [44]. The instrument was calibrated by using the APN2B Anton Paar standard at −20°C as recommended by the manufacturer. The measuring cell was cleaned using toluene before and after each injection. Triplicates were measured for each sample. A repeatability lower than 0.0856 mm²/s was guaranteed as specified by ASTM D7042 [44].

2.3 Chemical characterization via GC × GC/FID

The quantitative chemical characterization of the samples was carried out using an Agilent 7890B GC (GC × GC/FID) utilizing a liquid nitrogen-cooled thermal modulator. A reverse phase column configuration was selected, with a DB-17MS primary mid-polar column (30 m × 0.25 mm × 0.25 μm) and a DB-1MS non-polar secondary column (0.8 m × 0.25 mm × 0.25 μm) to optimize the separation of the saturated fraction. The front inlet flow was set at 1.5 mL/min. The inlet temperature was 280°C while the oven temperature was ramped from 40 to 250°C with a heating rate of 3°C/min. The modulation period was set to 2.5 s with a hot pulse duration of 0.42 s. Each sample was prepared using a 100-dilution fold in n-pentane. Data were processed in ChromaTOF software version 4.71.0.0 optimized for GC × GC/FID (LECO Corporation) with a signal-to-noise ratio of 50. Eight hydrocarbon groups: n-alkanes (NP: C₇-C₁₈), branched-alkanes (IP: C₇-C₂₁), monocycloalkanes (MCP: C₆-C₁₈), dicycloalkanes (DCP: C₈-C₁₆), tricycloalkanes (TCP: C₁₀-C₁₆), alkylbenzenes (AB: C₇-C₁₆), cycloaromatics (CA: C₉-C₁₅), and alkyl-naphthalenes (DA: C₁₀-C₁₅) were quantified (% wt./wt.) by peak area normalization [38].

2.4 Statistical analysis

The PLS model was used to describe the relationship between the chemical composition and the experimental viscosity of the samples. PLS establishes the relationship given by Equations (1) and (2) where $$X$$_nxm represents the matrix of latent variables, $$Y$$_nxp the matrix of responses, $$T$$_nxr and $$U$$_nxr the projections of $$X$$ and $$Y$$, respectively. Additionally, $$P$$_mxr and $$Q$$_pxr represent the orthogonal loading matrices, and $$E$$ and $$F$$ the random normal and independent error terms [45]. PLS decomposes $$X$$ and $$Y$$ maximizing the covariance between $$T$$ and $$U$$ in the way that the response variable $$Y$$ is expressed as the linear combination between the independent variable $$X$$ and the latent variables $$B$$, plus the intercept $$G$$ as shown in Equation (3).

$$$$\begin{equation}X\ = \ T\ {P}^T + E\end{equation}$$$$(1)

$$$$\begin{equation}Y\ = \ U\ {Q}^T + F\end{equation}$$$$(2)

$$$$\begin{equation}Y\ = \ X\ B + G\end{equation}$$$$(3)

PLS regression model was implemented via Jupyter Lab using Python Kernel through Anvil supercomputer at Purdue University [46]. The data imported to the model consisted of the response vector _(18x1) as the experimental kinematic viscosity values ($${\mu }_i$$) and the matrix of observations _(18x79) as the chemical composition measured via GC × GC/FID of the fuels ($${W}_{ij}$$). The response vector, with an intrinsic skewed and non-normal distribution nature, was transformed $${\tilde{\mu }}_i$$ using three different approaches: (i) mean centering by subtracting the mean of the observed data $$\bar{\mu }$$ of the actual viscosity $${\mu }_i$$ (Equation (4)), (ii) logarithmic transformation (Equation (5)) [47], and (iii) Yeo-Johnson transformation using $$\lambda $$ as transformation parameter (Equation (6)) [48]. Thus, this work evaluate the influence of each transformation in the PLS model output aiming for improving normality and reducing/stabilizing the variance.

$$$$\begin{equation}{\tilde{\mu }}_{MC\_i} = {\mu }_i\ - \bar{\mu }\end{equation}$$$$(4)

$$$$\begin{equation}{\tilde{\mu }}_{{\mathrm{log}}\_i} = \ {\mathrm{log}}\left( {1 + {\mu }_i} \right)\end{equation}$$$$(5)

$$$$\begin{equation}{\tilde{\mu }}_{YJ\_i} = \frac{{{{\left( {{\mu }_i + 1} \right)}}^\lambda - 1}}{\lambda }\ \ for\ {\mu }_i \ge 0\ and\ \lambda \ne 0\end{equation}$$$$(6)

The predicted transformed kinematic viscosity of each sample ($$\check{\mu}_i$$) corresponded to the linear combination of the respective coefficients of the latent variables ($${\beta }_i$$) times the chemical composition obtained via GC × GC/FID ($${W}_{ij}$$) plus the intercept ($${\beta }_o$$). Since the calculation is made over the transformed data, applying the inverse transformation was needed to obtain the actual predicted kinematic viscosities ($${\hat{\mu }}_i$$) as presented through Equations (7)–(9) for mean centering, logarithmic, and Yeo-Johnson transformations, respectively.

$$$$\begin{equation}{\hat{\mu }}_{MC\_i} = \left( {{\beta }_0 + \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^J {\beta }_i\ {W}_{ij}} \right)\ + \bar{\mu }\end{equation}$$$$(7)

$$$$\begin{equation}{\hat{\mu }}_{{\mathrm{log}}\_i} = \exp \left( {{\beta }_0 + \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^J {\beta }_i\ {W}_{ij}} \right)\ - 1\end{equation}$$$$(8)

$$$$\begin{equation}{\hat{\mu }}_{YJ\_i} = {\left( {\left( {{\beta }_0 + \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^J {\beta }_i\ {W}_{ij}} \right)\lambda + 1} \right)}^{\frac{1}{\lambda }}\ - 1\end{equation}$$$$(9)

The root-mean-square error (RMSE), mean absolute percentage error (MAPE), and coefficient of determination (R²) were used as performance metrics as presented by Equations (10)–(12) where $${\hat{\mu }}_i$$ refers to the predicted viscosity values and $$n$$ the number of samples under analysis.

$$$$\begin{equation}RMSE\ = \sqrt {\mathop \sum \limits_{i = 1}^n \frac{{{{\left( {{{\hat{\mu }}}_i - {\mu }_i} \right)}}^2}}{n}} \ \end{equation}$$$$(10)

$$$$\begin{equation}MAPE\ \left( \% \right) = \frac{1}{n}\ \mathop \sum \limits_{i\ = \ 1}^n \left| {\frac{{{{\hat{\mu }}}_i - {\mu }_i}}{{{\mu }_i}}} \right|*100\end{equation}$$$$(11)

$$$$\begin{equation}{R}^2 = \ 1 - \frac{{\mathop \sum \nolimits_{i\ = \ 1}^n \left( {{{\hat{\mu }}}_i - {\mu }_i} \right)}}{{\mathop \sum \nolimits_{i\ = \ 1}^n {{\left( {{{\hat{\mu }}}_i - \bar{\mu }} \right)}}^2}}\end{equation}$$$$(12)

3 RESULTS AND DISCUSSION

3.1 Kinematic viscosity and quantitative chemical characterization

The kinematic viscosity of the jet fuels and alternative blending components measured at −20°C together with the gravimetric distribution of the hydrocarbon groups are presented in Table 1. The broad spectrum of chemical compositions under study are visualized in the GC × GC/FID chromatograms of four representative samples, Jet A (13POSF10325), ATJ, HDCJ, and HEFA (Camelina) (Figure 1).

FIGURE 1

GC × GC/FID chromatograms of four representative samples exhibiting the broad range of chemical compositions of jet fuels and alternative blending components: (A) Jet A (13POSF10325), (B) ATJ, (C) HDCJ, and (D) HEFA (Camelina).

The average viscosity of all fuel samples was 5.61 mm²/s. The lowest viscosity was measured for biojet with 1.796 mm²/s, a fuel that presents the highest alkyl-benzene content of all samples (31.8% wt./wt). This result contrasts with the literature since higher content of aromatic compounds is in general associated with fuels with higher viscosities [27]. In this case, the presence of large fractions of C₁₀ branched-alkanes (46.3% wt./wt) and n-decane (19.7% wt./wt.) significantly decreased the viscosity of biojet. F-T/IPK and F-T/ATJ exhibited viscosities between 3.438 and 3.957 mm²/s. Their similar viscosity is ruled by their similar hydrocarbon group distributions. However, a higher monocycloalkanes content of 4.4% wt./wt. in F-T/IPK compared to 1.6% wt./wt. in F-T/ATJ decreased the viscosity of F-T/IPK especially due to the presence of a larger amount of light monocycloalkanes in the C₇–C₁₂ carbon number range of the former.

The viscosity of conventional Jet-A and JP-5 was 4.519 and 5.002 mm²/s, respectively. The higher viscosity of JP-5 compared to Jet-A agrees with the overall heavier nature of JP-5 and its higher cycloaromatic content, especially in the C₁₀–C₁₂ carbon number range [8]. Blending Jet-A with 50% HRJ decreased the viscosity to 4.348 mm²/s especially due to the increment in the content of branched-alkanes from 33.0% to 60.8% wt./wt. and the decrease of dicycloalkanes from 5.3 to 2.2% wt./wt.

ATJ samples (Gevo, ATJ, and JP-5 / Neat ATJ) exhibited a narrow viscosity range between 5.136 and 5.159 mm²/s clearly due to the practically identical hydrocarbon and carbon number distributions among these samples dominated by C₁₂ branched-alkanes (> 96.0% wt./wt.), specifically 2,2,4,6,6-pentamethylheptane [49]. The higher viscosity of ATJ samples compared to F-T/ATJ lies on the broader distribution of branched-alkanes between the C₈–C₁₆ carbon range but with a higher lighter fraction content (C₈ and C₁₁) for F-T/ATJ (29.8% wt./wt.) compared to ATJ (1.8% wt./wt.).

Camelina-derived fuels exhibited slightly higher viscosities (around 5.190 mm²/s) than ATJ fuels confirming that a higher content of normal-alkanes in HEFA and HRJ-8 fuels increases the viscosity given the reduction in the overall branching of the saturated alkane structures [27]. Differences among Camelina-based and ATJ fuels were also associated with a broader distribution of branched-alkanes between the C₈–C₁₇ carbon range for Camelina fuels, with a considerable decrease in the C₁₂ fraction from ∼80.0% wt./wt. (ATJ) to ∼8.0% wt./wt. (Camelina) but with a considerable increase in both, the lighter fraction C_7–C₁₁ from ∼1.8% wt./wt. (ATJ) to ∼32.8% wt./wt. (Camelina), and the heavier fraction C_13–C₁₈ from ∼16.3% wt./wt. (ATJ) to ∼47.5% wt./wt. (Camelina). Complementary, bimodal fuel C14/TMB exhibited a viscosity of 5.251 mm²/s, a value slightly higher than Camelina-based fuels, presumably due to the higher C₉ alkyl-benzene content (1,2,4-trimethylbenzene) and a significantly higher proportion of heavier branched-alkanes mainly dominated by the presence of iso-tetradecane (∼70.0% wt./wt.) in C14/TMB [50].

Military JP-8 exhibited a lower viscosity (3.493 mm²/s) than petroleum-based JP-5 (5.002 mm²/s) given its overall lighter hydrocarbon nature. In particular, JP-8 presented a lower viscosity due to considerably lower dicycloalkane (1.5% wt./wt. vs. 6.4% wt./wt.) and cycloaromatic contents (1.3% wt./wt. vs. 6.3% wt./wt.) compared to JP-5. Despite vast differences in the hydrocarbon group distributions, the viscosity of JP-8 was similar to F-T/IPK (3.438 mm²/s) confirming that the presence of target molecules even in only one hydrocarbon group could potentially emulate the viscosity of fuels with broader hydrocarbon distributions. Missile fuel JP-10 exhibited a viscosity of 8.842 mm²/s, that is, higher than all samples except JP-5/SIP. The high viscosity of JP-10 harmonized with the presence of dicycloalkane structures 97.5% wt./wt. of exo-tetrahydrodicylopentadiene and 2.5% wt./wt. endo-tetrahydrodicyclopentadiene [51].

Alternative blending components for kerosene-based fuels JP-5 of the type CHCJ, HRJ, HDCJ, and SIP completed the spectrum of viscosities ranging between 4.750 up to a maximum of 13.587 mm²/s. JP-5/CHCJ presented the lowest viscosity of this group with a distribution of n-alkanes (22.3% wt./wt.), monocycloalkanes (31.8% wt./wt.), and alkyl-benzenes (10.1% wt./wt.). JP-5/HRJ presented a considerable amount of heavy C₁₇ branched-alkanes (30.6% wt./wt.) which explained its higher viscosity (7.886 mm²/s). JP-5/HDCJ viscosity (8.294 mm²/s) was considerably higher than JP-5/CHCJ and still higher than JP-5/HRJ due to a larger content of dicycloalkanes, cycloaromatics, and alkyl-naphthalenes contents confirming that the presence of dicyclic saturated alkanes and aromatic compounds have the potential to increase fuel's viscosity [31]. Finally, JP-5/SIP exhibited the highest viscosity of all samples, due to the presence of farnesane (99.3% wt./wt.), a C₁₅ branched-alkane associated with higher viscosities when blended with jet fuels [49] but lower viscosity values when blended with diesel fuels [52] exhibiting the importance of the relative proportion among the light, bulk, and heavy fraction in middle distillates.

3.2 PLS predictive model

The results for the PLS model using 79 latent variables (Supporting Information material) encompassing the different combinations of all hydrocarbon groups and carbon numbers from GC × GC/FID together with ± 10% relative error deviation band is shown in Figure 2. Three different normalization methods were used for the experimental viscosity vector defined as the independent variable: mean centering (Figure 2A), logarithmic (log) transformation (Figure 2B), and Yeo-Johnson transformation (Figure 2C).

FIGURE 2

Partial least squares (PLS) model results for predicting kinematic viscosity of jet fuels and alternative blending components at −20°C using 79 latent variables and (A) mean centering, (B) logarithmic (log), and (C) Yeo-Johnson transformations including (D) performance metrics comparison.

Mean centering is commonly used to clarify regression coefficients while alleviating the interference of multicollinearity [53]. Figure 2A shows that PLS prediction using mean centering over the experimental viscosity values worked well for five out of eighteen samples, that is, JP-5/SIP, JP-5/HDCJ, JP-5/HRJ, JP-10, and F-T/IPK. However, relative errors between measured and predicted viscosities higher than ± 10% were estimated for the remaining samples with the largest relative deviation error for Biojet with 42.9%. In this case, no specific influence in terms of a specific type of hydrocarbon distribution was found to favor the prediction of viscosity by using mean centering.

Logarithmic (log) transformation is a statistical technique that rescales the observations equalizing the standard deviation of skewed data while normalizing the resulting theoretical distribution [47]. Figure 2b shows the PLS model for predicting viscosity by applying log transformation over the experimental viscosity measurements. Log transformation exhibited a remarkable improvement in prediction capability compared to mean centering exhibiting relative errors lower than ± 10% for 75% of the samples. For this case, C-14/TMB presented the largest deviation with a relative error of prediction of 16.3%.

Figure 2C presents the PLS model results using Yeo-Johnson transformation, as a technique used to potentially mitigate biases associated with non-normal distributions [54]. For this transformation, $$\lambda $$ was automatically estimated by the model to maximize the normality of the transformed variable adopting a value of −0.2227 (Equation (6)) [55]. In terms of relative error of predictions, the Yeo-Johnson transformation displayed the best prediction capabilities of the three normalization methods used with 85% of the samples exhibiting an error lower than ± 10%. However, the largest deviation was again estimated for bimodal C-14/TMB with a relative error of prediction of 12.2% indicating that the unique chemical composition of this sample driven by the presence of large quantities of 1,2,4-trimethylbenzene (C₉ alkyl-benzene) and iso-tetradecane (C₁₄ branched-alkane) represented a challenge in general for the proposed PLS model.

Performance metrics (RMSE and MAPE) among the different normalization methods for the training and validation sets (Figure 2D) confirmed the superior performance of Logarithmic and Yeo-Johnson transformations over mean centering. Yeo-Johnson transformation exhibited an RMSE of 0.48 and 0.25 and a MAPE of 6.56 and 7.03% over the training and validation sets, respectively. Overall, the Yeo-Johnson transformation performed slightly better than the log transformation when 79 latent variables were used. Thus, mean centering exhibited 1.6−2.0 and 1.7−2.4 times higher RMSE and MAPE compared to the Yeo-Johnson transformation.

Responding to the necessity of building simplified viscosity predictive models, a reduction of variables was performed selecting the ten key latent variables that minimized error performance metrics while avoiding underfitting and overfitting of the model. Table 2 shows the key hydrocarbon groups and carbon number combinations that optimized the performance of the PLS model together with the coefficients associated with each latent variable for each transformation. Thus, n-alkanes with 10 and 11 carbon atoms dominated decrements in viscosity, especially for petroleum-based fuels such as JP-5 and Jet-A. Light C₉ and C₁₀ branched alkanes were identified as key molecules that decreased the viscosity of JP-5/HRJ, JP-8, Camelina, and F-T-based fuels. Even small additions of heavy C₁₅ and C₁₇ branched-alkanes were the key group of molecules that increased the viscosity of ATJ, SIP, Camelina-based fuels, and blending Jet-A/HRJ mixtures [30]. Completing the saturated fraction, C₁₀ mono-, and dicyclo-alkanes induced contrasting effects with monocyclic molecules decreasing the viscosity of JP-5/HDCJ, JP-5/CHCJ, JP-8, JP-5, and Jet-A and dicyclic structures increasing considerably the viscosity of JP-5/HDCJ and JP-10. Cycloaromatics with 11 carbon atoms represented the group of molecules with the strongest influence in the aromatic fraction increasing the viscosity of JP-5/HDCJ, JP-5/CHCJ, and JP-5. The effect of alkyl-benzenes was secondary to the effects already discussed.

TABLE 2. List of coefficients for the 10 most significant latent variables for the partial least squares (PLS) models using mean centering, log transformation, and Yeo-Johnson transformation.

PLS model	Mean centering	Logarithmic transformation	Yeo-Johnson transformation (λ = −0.2227)
$${\beta }_0$$: Intercept	0.8909	1.9297	1.5555
$${\beta }_1$$: n-Alkane C10	−4.7441	−0.8065	−0.5528
$${\beta }_2$$: n-Alkane C11	−6.1074	−0.7429	−0.4621
$${\beta }_3$$: Branched-alkanes C9	−17.0051	−2.0472	−1.2650
$${\beta }_4$$: Branched-alkanes C10	−6.3728	−1.1839	−0.8329
$${\beta }_5$$: Branched-alkanes C15	7.9128	0.8211	0.5008
$${\beta }_6$$: Branched-alkanes C16	−1.1370	−0.1920	−0.1270
$${\beta }_7$$: Branched-alkanes C17	7.6801	1.1285	0.7384
$${\beta }_8$$: Monocycloalkanes C10	−5.6721	−0.7783	−0.4962
$${\beta }_9$$: Dicycloalkanes C10	3.1153	0.4223	0.2712
$${\beta }_{10}$$: Cycloaromatics C11	6.4373	0.9315	0.6041

The results of the simplified PLS models using 10 latent variables are shown in Figure 3. When the mean centering transformation was used (Figure 3A), only four (JP-5/SIP, JP-5/HRJ, JP-10, and F-T/IPK) out of eighteen samples exhibited relative errors of prediction lower than ± 10%. The remaining samples presented relative errors higher than ± 10%. Biojet presented the highest relative error of all samples with 45.1%. Log-transformation (Figure 3B) helped to improve the performance of the PLS model with 56% of the samples exhibiting a relative error of prediction lower than ± 10% and only three samples (JP-5/HDCJ, Biojet, and JP-8) exhibiting deviations higher than 15%. Similar to the case when a larger number of latent variables were used, Biojet presented the largest relative error of all samples (33.1%).

FIGURE 3

Partial least squares (PLS) model results for predicting kinematic viscosity of jet fuels and alternative blending components at −20°C using 10 latent variables and (A) mean centering, (B) logarithmic (log), and (C) Yeo-Johnson transformations including (D) performance metrics comparison.

Confirming the findings associated with the extended PLS model using 79 latent variables, Yeo-Johnson (Figure 3C) represented the best option for predicting the kinematic viscosity of jet fuels and alternative blending components at −20°C even when a smaller set of latent variables was used. For this transformation, 78% of the samples presented relative errors of prediction lower than ± 10%. Once again, Biojet presented the largest deviation with a relative prediction error of 30.9%. Yeo-Johnson transformation also offered the highest R² over the validation set with 0.9404 followed very closely by log-transformation with 0.9352 but remarkably higher than mean centering with 0.8951. These R² values for Yeo-Johnson and log-transformations are higher than others reported in the literature using GC/MS even after utilizing a considerably lower amount of latent variables.

The superior performance of Yeo-Johnson and logarithmic transformations over mean centering is evident when the general performance metrics of each transformation are compared using 10 latent variables (Figure 3D). Yeo-Johnson offered 1.3- and 1.9 times better RMSE (0.74 vs. 0.93) and MAPE (9.48% vs 18.26%) compared to mean centering for the training set. Analogously, Yeo-Johnson performed better over the validation set with a RMSE = 0.23 and a MAPE = 5.33%, i.e., values around 3.2 times lower than the values estimated using mean centering. Log-transformation closely followed the performance of Yeo-Johnson transformation, being around 1.1 times over the training set and 1.4 times higher over the validation set. Thus, Yeo-Johnson transformation is recommended to be used for improving the predictability of kinematic viscosity of jet fuels and alternative blending components using PLS models based on GC × GC/FID data. This observation could potentially be extended to other properties of interest during the certification process of aviation turbine fuels such as density, heat of combustion, distillation profile, freezing point, and flash point.

4 CONCLUSIONS

This work presented the implementation of PLS models using mean centering, logarithmic transformation, and Yeo-Johnson transformation to predict the kinematic viscosity of jet fuels and alternative blending components at −20°C via GC × GC/FID. The specific influence of different hydrocarbon groups on the overall kinematic viscosity of the samples was explained. It was identified that heavy branched-alkanes (C₁₅ and C₁₇), dicycloalkanes (C₁₀), and cycloaromatics (C₁₁) are key hydrocarbon molecules with the potential to strongly increase the kinematic viscosity of jet fuels and alternative blending components. Contrastingly, light branched-alkanes (C₉–C₁₀), linear-alkanes (C₁₀), and monocycloalkanes (C₁₀) are potential key molecules to decrease the kinematic viscosity of jet fuels and alternative blending components. Specifically, Yeo-Johnson transformation was identified as a suitable normalization method that offered superior performance metrics over mean-centering and logarithmic transformation to predict the kinematic viscosity of the samples via chemical composition analysis. A new simplified PLS model using only ten latent variables was proposed offering excellent kinematic viscosity prediction capabilities (MAPE = 5.3% and RMSE = 0.23) even for samples with highly divergent chemical compositions. These findings can be extended to other physical properties studied for the ASTM certification process of aviation turbine fuels. Further work in progress is focused on including isomers and degree of branching information to improve the prediction metrics based on GC × GC/FID data.

AUTHOR CONTRIBUTIONS

Louis Edwards Caceres-Martinez: Conceptualization; methodology; investigation; data curation; formal analysis; writing—original draft; writing—reviewing & editing. Gozdem Kilaz: Writing—reviewing & editing; validation; project administration; supervision; resources; funding acquisition.

ACKNOWLEDGMENTS

The authors thank Michelle Moody from Zucrow Labs at Purdue University for providing the set of samples here analyzed. We also thank Karen Zabala for her help with the viscosity measurements, GC × GC/FID sample preparation, and initial Python draft for the further development of the PLS model.

CONFLICT OF INTEREST STATEMENT

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Open Research

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available by request.