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\begin{document}

\title{Shelf-life of Biodiesel by Isothermal Oxidation Induction Period at
Variable Temperatures}



\author[1]{R. O. Dunn}%
\affil[1]{USDA-ARS National Center for Agricultural Utilization Research}%


\vspace{-1em}



  
  \date{May 24, 2024}


\begingroup
\let\center\flushleft
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\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Biodiesel (fatty acid methyl esters {[}FAME{]}) is a renewable
biomass-based diesel (BBD) fuel made from plant oils, animal fats and
waste greases. Its production continues to increase globally, especially
with more countries moving to increase the use of renewable fuels. One
of the main disadvantages of biodiesel is its relatively poor oxidative
stability. During longer periods of storage at fuel terminals or in
vehicle tanks and fuel systems, biodiesel can react with oxygen in
ambient air, leading to degradation that can adversely affect its
viscosity and ignition quality. The shelf-life of biodiesel is an
important property that can determine the conditions such as how long it
can be stored at low temperatures. The objective of this work is to
develop mathematical equations to estimating the shelf-life of biodiesel
at T = 25 \selectlanguage{ngerman}°C. This was done by measuring the induction period by
Rancimat instrument (IP \textsubscript{R}) at different temperatures and
extrapolating the results using two types of linear models. Model
equations were developed from regression analysis performed on results
for canola, palm and soybean oil-FAME (CaME, PME and SME), methyl oleate
(MeC18:1) and methyl linoleate (MeC18:2). The best results from
confirmation analysis were obtained for \textbf{Model B} type equations.
This model type correlated ln(IP \textsubscript{R}) with inverse T (T
\selectlanguage{english}\textsuperscript{-1}) data and provided the most realistic estimates of
the shelf-life (SL \textsubscript{B}) of the five FAME studied in this
work.%
\end{abstract}\selectlanguage{ngerman}%



\sloppy


\textbf{INTRODUCTION}

Biodiesel, renewable diesel (``green'' diesel) and sustainable aviation
fuel (SAF) are three types of biomass-to-liquid fuels that can be made
from vegetable oils, animal fats or waste cooking oils. Biomass-based
diesel (BBD) such as biodiesel and renewable diesel are suitable for
combustion in compression-ignition (diesel) engines. Production of these
BBD has increased in the U.S. and throughout the world in recent years
as efforts to develop renewable alternatives continue to be ramped up.

The Clean Fuels Alliance of America (CFAA) reported that 1.6 billion gal
of total BBD were produced in the U.S. in 2022 (CFAA, 2022). This
industry trade group has published its goal of ramping up production to
6 billion gal of BBD by 2030, a quantity that could eliminate 35 MMT of
CO\textsubscript{2}-equivalent greenhouse gas emissions per year. As of
January 2023, the U.S. Energy Information Administration estimated that
biodiesel plant production capacity was 2.1 billion gal annually in the
U.S. (EIA, 2023).

While renewable diesel may be used in blend ratios up to 100 vol\%, this
fuel is more expensive to produce than biodiesel (Lane, 2023).
Production of renewable diesel is a multi-step continuous process that
requires high-purity hydrogen gas, expensive metal catalysts, high
pressures and temperatures and the liquid fuel needs to be separated by
condensation from light fractions in the product stream
(Guti\selectlanguage{ngerman}érrez-Antonio et al., 2015). In contrast, biodiesel is typically
produced in one step under relatively mild conditions (ambient pressures
and low temperatures {[}\textasciitilde{} 60 °C{]}) and uses inexpensive
base catalysts (Amin, 2019). Conversion time is relatively short (1-2 h)
ameliorating the common use of batch or semi-batch process equipment.

The production of renewable diesel in the U.S. has increased
significantly in the past five years. Partially as a result, increases
in biodiesel production have slowed down. However, biodiesel is still
relevant in many parts of the world where refining capacity for
renewable diesel production is limited or nonexistent. Countries such as
Argentina, Brazil and Indonesia are continuing to increase their
national production levels of biodiesel.

Currently, the ASTM fuel specification D6751 for biodiesel as a blending
stock for mixing with conventional diesel fuel (petrodiesel) limits its
content to 20 vol\% (`B20') (ASTM, 2021). In recent years, governmental
incentives favoring greener fuels as well as consumer demand have been
the main driving forces for expanding biodiesel blend ratios in
petrodiesel to `B50' and higher. As a result, one of the major concerns
with biodiesel, its relatively poor oxidative stability, is receiving
increased attention.

When stored at the terminal or in fuel tanks and systems, biodiesel can
be exposed to oxygen present in ambient air, making it susceptible to
oxidative degradation, especially if it is exposed periodically to heat.
Biodiesel is a fatty derivative that is primarily composed of alkyl
esters of fatty acids obtained from the feedstock lipid. The most common
form of biodiesel is fatty acid methyl esters (FAME). The FAME are
composed of saturated, monounsaturated and polyunsaturated esters
(SFAME, MUFAME and PUFAME). Reactivity of biodiesel with oxygen
increases when its degree of unsaturation increases. While unsaturated
FAME are significantly more reactive to oxygen than SFAME, linoleates
are 40 times more reactive than oleates in neat systems without an added
initiator (Frankel, 2005a).

Oxidative degradation can adversely affect the fuel quality of biodiesel
by altering its kinematic viscosity (KV), acid value (AV) or peroxide
value (PV). As a safeguard, the ASTM biodiesel fuel specification D6751
includes a minimum limit for oxidative stability as measured by
induction period (IP) under accelerated conditions (ASTM, 2021). The IP
is generally defined as the time period where the oxidation of biodiesel
shifts from the formation of primary products (hydroperoxides) to the
degradation of primary products to secondary products (acids, alcohols,
aldehydes, ketones, lactones and aromatics) (Frankel, 2005a; Kamal-Eldin
and Yanishlieva, 2005; White, 2000). Once this level of degradation is
present, it is likely that its KV and AV increase above their respective
maximum limits (6.0 mm² s\selectlanguage{english}\textsuperscript{-1} at 40 degC and 0.50 mg KOH
g\selectlanguage{english}\textsuperscript{-1}), diminishing the overall fuel quality.

When a Rancimat instrument is used to analyze oxidative stability, the
result is term `IP\textsubscript{R}' (in h). According to ASTM
specification D6751, this parameter is measured at T = 110 degC while the
fuel sample is sealed in a test tube with 10 L h\selectlanguage{english}\textsuperscript{-1} air
steadily bubbled through it (ASTM, 2021). If the sample yields
IP\textsubscript{R} = 3 h or longer, it is within specification, with
respect to the oxidative stability. If not, then the biodiesel should be
treated with antioxidants until it can pass the specification.

The IP\textsubscript{R} is measured under tight laboratory conditions
that include steadily bubbling the sample with air and an elevated
measurement temperature 110 degC (ASTM, 2021). However, most realistic
storage conditions will have lower temperatures, generally between 25
and 45 degC, and little if any constant contact with fresh air (oxygen).
One common method used by the fats and oils industry is to measure
IP\textsubscript{R} data for the product at three or more temperatures
and extrapolate the results to obtain ``shelf-life'' data at lower
temperatures.

A simple formula used by the fats and oils industry to estimate the
shelf-life of products is described by the \textbf{Model A} linear type
equation (Dunn, 2008; Farhoosh, 2007; Frankel, 2005c; Nakatani et al.,
2001):

ln(IP\textsubscript{R}) = A\textsubscript{0} + A\textsubscript{1}(T) (1)

where IP\textsubscript{R} = induction period measured isothermally by a
Rancimat instrument (in h), A\textsubscript{0} and A\textsubscript{1}are
constants determined by linear regression analysis of
ln(IP\textsubscript{R}) versus T data and T = temperature (in K). For a
given FAME mixture such as biodiesel, measuring IP\textsubscript{R} for
a series of elevated temperatures. Once the experimental data is
analyzed to obtain coefficients A\textsubscript{0} and
A\textsubscript{1}, then \textbf{Model A} type equations for a given
FAME can be extrapolated to T = 25 degC (298.15 K) to calculate shelf-life
(SL\textsubscript{A}) as follows:

ln(SL\textsubscript{A}) = A\textsubscript{0} +
A\textsubscript{1}(298.15) (2)

where SL\textsubscript{A} is obtained by taking an inverse natural
logarithm of the results from Eq. 2.

An alternative method for estimating the shelf-life of biodiesel and
other fatty derivatives is to employ \textbf{Model B} type equations
(Eq. 3) to extrapolate the results:

ln(IP\textsubscript{R}) = B\textsubscript{0} +
B\textsubscript{1}(T)\selectlanguage{english}\textsuperscript{-1} (3)

where B\textsubscript{0} and B\textsubscript{1} are constants determined
by linear regression analysis of ln(IP\textsubscript{R}) versus
T\selectlanguage{english}\textsuperscript{-1} data (Dunn, 2008). Analogous to \textbf{Model
A}type equations, the IP\textsubscript{R} of FAME are measured for a
series of elevated temperatures, and the results analyzed to complete
Eq. 3 and extrapolate it to calculate shelf-life (SL\textsubscript{B})
at 25 degC (298.15 K) using the following equation:

ln(SL\textsubscript{B}) = B\textsubscript{0} +
B\textsubscript{1}(298.15)\selectlanguage{english}\textsuperscript{-1} (4)

where SL\textsubscript{B} is also obtained by taking an inverse natural
logarithm of the outcome.

Another advantage of using the \textbf{Model B} approach is that it may
be used to calculate the activation energy (E\textsubscript{a}) and
frequency factor (Z\textsubscript{0}) for oxidation of biodiesel (Dunn,
2008). These parameters are useful in determining the reaction rate
constant (k\textsubscript{T}):

k\textsubscript{T} =
Z\textsubscript{0}{[}exp\{\selectlanguage{english}-E\textsubscript{a}(R\textsubscript{g}T)\selectlanguage{english}\textsuperscript{-1}\}{]}
(5)

where R\textsubscript{g} = gas constant (8.3144 J
mol\selectlanguage{english}\textsuperscript{-1} K\selectlanguage{english}\textsuperscript{-1}). The
E\textsubscript{a}can be estimated from the following equation:

E\textsubscript{a} = R\textsubscript{g}(B\textsubscript{1}) (6)

where coefficient B\textsubscript{1} is defined in Eq. 3 and
E\textsubscript{a} is in kJ mol\selectlanguage{english}\textsuperscript{-1}.

The collision frequency factor can be estimated if the reaction kinetic
model is known. For example, if the kinetics follow a first order model:

f(\selectlanguage{greek}α\selectlanguage{english}) = (1 - \selectlanguage{greek}α\selectlanguage{english}) (7)

where \selectlanguage{greek}α \selectlanguage{english}= degree of conversion, then Z\textsubscript{0} can be
calculated as follows:

Z\textsubscript{0} = -ln(1 - \selectlanguage{greek}α\selectlanguage{english}){[}exp\{-B\textsubscript{0}\}{]} (8)

Equation 8 can be generalized for different kinetic models by
substituting g(\selectlanguage{greek}α\selectlanguage{english}) for -ln(1 - \selectlanguage{greek}α\selectlanguage{english}) where g(\selectlanguage{greek}α\selectlanguage{english}) is defined as:

g(\selectlanguage{greek}α\selectlanguage{english}) = [?]f(\selectlanguage{greek}α\selectlanguage{english})\textsuperscript{-1}d\selectlanguage{greek}α \selectlanguage{english}(9)

where the boundary limits 0 and \selectlanguage{greek}α\selectlanguage{english}* are defined as \selectlanguage{greek}α \selectlanguage{english}at t = 0 and IP
(Dunn, 2008; Khawam and Flanagan, 2006; Zhang et al., 2013).

The main objective of the present work is to develop model equations for
estimating the shelf-life of biodiesel at T = 25 \selectlanguage{ngerman}°C (298.15 K) from the
experimental IP\textsubscript{R}-T data. Regression analyses were
performed to obtain coefficients A\textsubscript{0} and
A\textsubscript{1} in Eq. 1 for \textbf{Model A} type equations, and
B\textsubscript{0} and B\textsubscript{1} in Eq. 3 for \textbf{Model
B}type equations. One model equation of each type was developed for each
of five FAME mixtures: canola, palm and soybean oil-FAME (CaME, PME and
SME), methyl oleate (MeC18:1) and methyl linoleate (MeC18:2).
Experimental IP\textsubscript{R} data were measured at five temperatures
for the first four FAME and at six temperatures for MeC18:2. Once the
data were gathered and the \textbf{Model A} and \textbf{B} type
correlations determined, the results were extrapolated according to Eqs.
2 and 4 to calculate SL\textsubscript{A} and SL\textsubscript{B} data
for each FAME at 298.15 K. The final results were compared with each
other and shelf-life data for biodiesel from the literature to assess
how realistic the predictions from SL\textsubscript{A} and
SL\textsubscript{B} were.

\textbf{EXPERIMENTAL METHODS}

\textbf{Materials}

Biodiesel samples were acquired from the following sources: 1) CaME was
from Archer Daniels Midland (Decatur, IL, USA); 2) PME produced by Sime
Darby Biodiesel Sdn. Bhd. (Selanger, Malaysia) with assistance from the
Malaysian Palm Oil Board (Washington, DC, USA); and 3) SME was from Ag
Environmental Products (Lenexa, KS, USA). According to the fuel
producers, these biodiesel samples were not treated with performance
enhancing additives including antioxidants. Fatty acid concentration
profiles for these biodiesel fuels are presented in \textbf{Table S1} in
the supporting information.

Pure MeC18:1 (99.5 ± 0.1 \% 9(Z)-octadecenoic acid methyl ester) and
MeC18:2 (99.0 ± 0.1 \% 9(Z),12(Z)-octadecadienoic acid methyl ester)
were supplied by Nu Chek Prep (Elysian, MN, USA). The purities of
MeC18:1 and MeC18:2 were verified by analysis with a PerkinElmer
(Waltham, MA, USA) Clarus 500 gas chromatograph (GC) with a flame
ionization detector (FID) using an HP-88 column (30 m x 0.250 mm x 0.20
\selectlanguage{greek}μ\selectlanguage{english}m) from Agilent (Santa Clara, CA, USA). The carrier gas was hydrogen
and injector and detector temperatures were 240 \selectlanguage{ngerman}°C and 280 °C. The
temperature program was: 1) hold at 100 °C for 5 min; 2) ramp at 10 °C
min\selectlanguage{english}\textsuperscript{-1} to 220 degC; 3) hold at 220 degC for 15 min. Three
replicate analyses were performed and results for the pure FAME were
reported as average values.

\textbf{Induction Period (IP\textsubscript{R})}

A model 743 Rancimat instrument from MetrOhm (Riverview, FL, USA) was
employed to measure IP\textsubscript{R} at five different temperatures
for CaME, PME, SME and MeC18:1 and six temperatures for MeC18:2.
Depending on the FAME, measurement temperatures ranged between 50 and
140 degC (323.15 and 313.15 K). The test method used to measure
IP\textsubscript{R}, EN 15751 (CEN, 2009), was conducted using the
short-type test tubes on oil sample masses of 3 +- 0.1 g. Two replicate
measurements were performed and the results reported as average values.

Mean absolute deviation (MAD) and root-mean-squared mean deviation
(RMSD) data were calculated using the following two equations:

MAD = (\selectlanguage{greek}Σ \selectlanguage{english}\textbar{} IP\textsubscript{R,i} --
IP\textsubscript{i}\textbar{})/\emph{n} \textsubscript{i} (10)

RMSD = {[}(\selectlanguage{greek}Σ\selectlanguage{english}(IP\textsubscript{R,i} -- IP\textsubscript{i})\selectlanguage{ngerman}²/\emph{n}
\textsubscript{i}{]}\textsuperscript{½}(11)

where IP\textsubscript{R,i} = experimental IP\textsubscript{R} value at
temperature T\textsubscript{i}, IP\textsubscript{i} = IP value
calculated from a model equation at T\textsubscript{i} and\emph{n}
\textsubscript{i} = total number of data pairs (IP\textsubscript{R,i},
IP\textsubscript{i}). The calculation of MAD and RMSD, least-squares
linear regression analyses, analysis of variance (ANOVA) and statistical
analysis operations were conducted in Microsoft (Redmond, WA, USA)
Excel® spreadsheets.

\textbf{RESULTS AND DISCUSSION}

\textbf{Composition and Fuel Properties}

The fatty acid concentration profiles and fuel properties of CaME, PME
and SME are summarized in \textbf{Tables S1} and \textbf{S2} in the
supporting information. These results were discussed in full in an
earlier study (Dunn, 2020) and a summary of those findings are presented
herein.

Analysis of the FAME concentration profiles (\textbf{Table S1} ) showed
that the SME had a higher total PUFAME concentration (62.09 mass\%) and
a lower total MUFAME concentration (21.67 \%) than either CaME (26.8 and
65.74 \%) or PME (9.7 and 40.49 \%). CaME had the lowest total SFAME
concentration (7.40 \%) followed by SME (16.21 \%) and PME (49.82 \%).
SME had the highest calculated iodine value (IV) = 132.76 g
I\textsubscript{2}100\selectlanguage{english}\textsuperscript{-1}g\selectlanguage{english}\textsuperscript{-1}, followed
by CaME (109.6) and PME (51.8).

Selected fuel properties of the three biodiesel fuels were summarized
in\textbf{Table S2} . In descending order, IP\textsubscript{R} at 110 degC
was 9.34 h, 7.71 h and 2.745 h for CaME, PME and SME. The low value for
SME was expected given its high total PUFAME content and comparable with
data reported in other studies (Moser, 2008a, 2014, 2016; Yang et al.,
2013). The experimental IP\textsubscript{R} value for CaME was also
comparable with data reported in the literature (Joshi et al., 2009;
Moser, 2008b; Yang et al., 2013). However, the IP\textsubscript{R} of
PME was lower than that of CaME, despite PME having a lower total PUFAME
concentration. Generally higher IP\textsubscript{R} values (9.2-15.2 h)
were reported for PME in the literature (Moser, 2008b, 2016; Sarin et
al., 2010; Shahabuddin et al., 2012). In the earlier study (Dunn, 2020),
it was postulated that PME had undergone partial degradation before its
use in the study. PME had AV = 0.72 mg KOH
g\selectlanguage{english}\textsuperscript{-1}(\textbf{Table S2} ), a value that exceeded the
maximum limit (0.50 mg KOH g\selectlanguage{english}\textsuperscript{-1}) in the ASTM biodiesel
fuel standard specification D6751 (ASTM, 2021). This higher value may
have indicated that PME had undergone hydrolytic degradation. PME also
exhibited a PV = 11.3 meq kg\selectlanguage{english}\textsuperscript{-1} (\textbf{Table S2} ), a
value that was nearly twice the PV measured for CaME. Trace
concentrations of hydroperoxides are known to act as autoxidation
initiators (Frankel, 2005a; Kamal-Eldin and Yanishlieva, 2005). The
effects of high AV and PV suggested that PME may have experienced
autoxidation before being used in the study.

The pure MeC18:1 and MeC18:2 samples were also analyzed for
IP\textsubscript{R}. At 110 degC, IP\textsubscript{R} = 2.12 +- 0.05 h for
MeC18:1, a value that was comparable with data reported in other studies
(Knothe, 2008; Moser, 2009). Those studies also reported
IP\textsubscript{R} = 0.94 h and 1.0 h for MeC18:2. Nevertheless, no
reliable IP\textsubscript{R} data were obtained for MeC18:2 at 110degC in
the present study.

\textbf{IP\textsubscript{R} Results at Variable Temperatures}

Experimental IP\textsubscript{R} data for the five FAME at variable
temperatures are summarized in \textbf{Table 1} .

Also shown in the table are `Q10' ratios and response factors (RF).
These are defined as follows:

Q10 = (IP\textsubscript{R} at T)/(IP\textsubscript{R} at T + 10 degC) (12)

RF = (IP\textsubscript{R} of FAME)/(IP\textsubscript{R} of MeC18:1) (13)

where T = temperature. The RF was calculated using
IP\textsubscript{R}data measured at the same T for the FAME and MeC18:1,
which limited the results shown in the table to the range T = 90-130 degC
for measuring the IP\textsubscript{R} of MeC18:1.

\textbf{Table 1} Oxidation induction period measured by Rancimat
instrument (IP\textsubscript{R}) of fatty acid methyl esters (FAME)\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
FAME & T & T & IP\textsubscript{R}\textsuperscript{a} (h) &
Q10\textsuperscript{b} & Q10\textsuperscript{b} &
RF\textsuperscript{c}\tabularnewline
\midrule
\endhead
& \selectlanguage{ngerman}°C & K & & Ratio & Avg &\tabularnewline
CaME & 90 & 363.15 & 35.3 (0.3) & & & 2.0\tabularnewline
& 100 & 373.15 & 16.0 (0.2) & 2.2 & & 2.00\tabularnewline
& 110 & 383.15 & 9.34 (0.04) & 1.7 & & 4.42\tabularnewline
& 120 & 393.15 & 4.41 (0.01) & 2.12 & & 5.3\tabularnewline
& 130 & 403.15 & 2.295 (0.007) & 1.92 & 2.0 & 5.67\tabularnewline
PME & 100 & 373.15 & 16.5 (0.4) & & & 2.1\tabularnewline
& 110 & 383.15 & 7.71 (0.06) & 2.1 & & 3.65\tabularnewline
& 120 & 393.15 & 3.57 (0.07) & 2.16 & & 4.3\tabularnewline
& 130 & 403.15 & 1.79 (0.08) & 2.00 & & 4.41\tabularnewline
& 140 & 413.15 & 0.88 (0.05) & 2.04 & 2.1 &\tabularnewline
SME & 80 & 353.15 & 32.8 (0.3) & & &\tabularnewline
& 90 & 363.15 & 14.34 (0.01) & 2.3 & & 0.8\tabularnewline
& 100 & 373.15 & 6.50 (0.01) & 2.21 & & 0.81\tabularnewline
& 110 & 383.15 & 2.745 (0.007) & 2.37 & & 1.3\tabularnewline
& 120 & 393.15 & 1.56 (0.02) & 1.77 & 2.2 & 1.87\tabularnewline
MeC18:1 & 90 & 363.15 & 17.78 & & & 1.0\tabularnewline
& 100 & 373.15 & 8.03 (0.04) & 2.22 & & 1.00\tabularnewline
& 110 & 383.15 & 2.11 (0.05) & 3.79 & & 1.00\tabularnewline
& 120 & 393.15 & 0.8 (0.1) & 2.6 & & 1.0\tabularnewline
& 130 & 403.15 & 0.40 (0.04) & 2.0 & 2.7 & 1.00\tabularnewline
MeC18:2 & 50 & 323.15 & 19.5 & & &\tabularnewline
& 60 & 333.15 & 9.75 (0.09) & 2.00 & &\tabularnewline
& 70 & 343.15 & 4.65 (0.04) & 2.10 & &\tabularnewline
& 80 & 353.15 & 2.53 (0.01) & 1.84 & &\tabularnewline
& 90 & 363.15 & 1.72 (0.04) & 1.47 & & 0.1\tabularnewline
& 100 & 373.15 & 1.03 (0.02) & 1.68 & 1.82 & 0.13\tabularnewline
\bottomrule
\end{longtable}

T = Measurement temperature; CAME = canola oil-FAME; PME = palm
oil-FAME; SME = soybean oil-FAME; Q10 = oxidation reactivity ratio; Avg
= average value; RF = response factor.

\textsuperscript{a} Average values from \emph{n} = 2 replicate
measurements. Standard deviations (SD) in parentheses (blank if SD = 0).

\textsuperscript{b} Calculated from Eq. 12.

\textsuperscript{c} Calculated from Eq. 13.

The `Q10' ratio is known as the relative oxidation reactivity for fatty
derivatives (Frankel, 2005c; Ghosh et al., 2019; Pereira et al., 2015).
For oxidation, this ratio is typically between 2 and 3, indicating that
a 10 \selectlanguage{ngerman}°C increase in T decreases IP\textsubscript{R} by 50-66 \%. Four
FAME in \textbf{Table 1} exhibited average Q10 ratios of 2.0-2.7, while
MeC18:2 had an average Q10 = 1.82, a value that was only 9.2 \% below
the lower threshold. An examination of the individual Q10 data for
MeC18:2 shows that the ratio tended to decrease as T increased, with the
Q10 \textless{} 2.00 occurring at higher temperatures. This may indicate
that MeC18:2 was impacted by thermal decomposition as well as oxidative
degradation at higher temperatures.

The IP\textsubscript{R} of all five of the FAME studied could only be
ranked at T = 100 °C. This ranking, in descending order was:

PME \textgreater{} CaME \textgreater{} MeC18:1 \textgreater{} SME
\textgreater{} MeC18:2 (14)

where the IP\textsubscript{R} of PME was only slightly greater than that
of CaME, with a probability (\emph{p} ) = 0.298 that the two values were
equivalent. In general, when IP\textsubscript{R} values were similarly
ranked for two or more FAME (two FAME at 80 °C; four FAME at 90, 110 and
120 °C; three FAME at 130 °C), they followed this ranking. The only
exception was observed for CaME and PME, where CaME had higher
IP\textsubscript{R} values at T = 110-130 °C. In this T-range,
deviations between IP\textsubscript{R} values for CaME and PME were
significant (\emph{p} \selectlanguage{english}[?] 0.012). It was noted earlier that analysis of
the fuel properties of PME (\textbf{Table S1} in the supporting
information) suggested the sample experienced a small degree of
degradation before its use in the study. This is the most likely
explanation for the otherwise unexpected observation that the
IP\textsubscript{R} of CaME was greater than that of PME, especially at
T \selectlanguage{english}[?] 110 degC.

\textbf{Model A}

Plots of ln(IP\textsubscript{R}) versus T representing \textbf{Model
A}are presented in \textbf{Figure 1} . The straight lines drawn through
the data verify the linear correlation between IP\textsubscript{R} and T
for the five FAME studied in the present work. Results from least
squares regression analysis of the data are summarized in \textbf{Table
2} . These results yielded intercept and slope coefficients
(A\textsubscript{0} and A\textsubscript{1}) that were significant
(\emph{p} -value \textless{}\textless{} 0.05). The
A\textsubscript{0}coefficients for CaME, PME and SME were generally in a
narrow range, from 28.1 to 31. The same trend was observed for the
A\textsubscript{1}coefficients (\selectlanguage{english}-0.078 to \selectlanguage{english}-0.068) of these three FAME.
In contrast, the A\textsubscript{0} and A\textsubscript{1} coefficients
were significantly higher in magnitude for MeC18:1 (39 and \selectlanguage{english}-0.098) and
lower in magnitude for MeC18:2 (22 and \selectlanguage{english}-0.059).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Fig. 1} Graph of ln(IP\textsubscript{R}) as a function of T data
for five fatty acid methyl ester (FAME) mixtures. IP\textsubscript{R} =
oxidative induction period measured by Rancimat instrument (h); T =
temperature; CaME, PME and SME = canola, palm and soybean oil-FAME;
MeC18:1 = methyl oleate (99.5 \%); MeC18:2 = methyl linoleate (99.0 \%).
Lines drawn through the data points represent results from linear
regression analyses according to \textbf{Model A}.

The linear correlation from application of \textbf{Model A} to
IP\textsubscript{R}-T data was well established, with respect to
correlation coefficients (R\selectlanguage{ngerman}²) \selectlanguage{english}[?] 0.985. The results also exhibited low
standard

\textbf{Table 2} Results from regression analysis of
ln(IP\textsubscript{R}) \emph{versus} T data (\textbf{Model A} ).
ln(IP\textsubscript{R}) = A\textsubscript{0} + A\textsubscript{1}(T);
IP\textsubscript{R} in h, T in K\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Parameter & CaME & PME & SME & MeC18:1 & MeC18:2\tabularnewline
\midrule
\endhead
df & 3 & 3 & 3 & 3 & 4\tabularnewline
A\textsubscript{0} & 28.1 & 30.2 & 31 & 39 & 22\tabularnewline
Err{[}A\textsubscript{0}{]} & 0.7 & 0.4 & 1 & 2 & 1\tabularnewline
\emph{p}-value & 0.00004 & 0.000004 & 0.00009 & 0.0004 &
0.00004\tabularnewline
A\textsubscript{1} & -0.068 & -0.0734 & -0.078 & -0.098 &
-0.059\tabularnewline
Err{[}A\textsubscript{1}{]} & 0.002 & 0.0005 & 0.003 & 0.006 &
0.003\tabularnewline
\emph{p}-value & 0.00005 & 0.000005 & 0.0001 & 0.0004 &
0.00005\tabularnewline
R\selectlanguage{ngerman}² & 0.997 & 0.9993 & 0.995 & 0.987 & 0.985\tabularnewline
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} & 0.06 & 0.03 & 0.09 & 0.2 & 0.1\tabularnewline
F-ratio & 1,242 & 6,049 & 750 & 299 & 328\tabularnewline
\bottomrule
\end{longtable}

df = Degrees of freedom; A\textsubscript{0} = intercept coefficient;
A\textsubscript{1} = slope coefficient; Err{[}A\textsubscript{i}{]} =
standard error of coefficient `A\textsubscript{i}'; \emph{p} -value =
probability A\textsubscript{i} [?] 0 (\textless{} 0.05); R\selectlanguage{ngerman}² = adjusted
correlation coefficient; \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} = standard error of the
y-estimate; F-ratio = variance ratio (model/residuals). See\textbf{Table
1} for abbreviations.

errors of the y-estimate (\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} [?] 0.2) for calculating
ln(IP\textsubscript{R}) directly from T data. Results from ANOVA
demonstrated high variance (F-) ratios (model/residuals), indicating
that the goodness

of fit of the model equations was high. However, examination of the data
in \textbf{Figure 1} revealed small degrees of curvature in the data for
MeC18:2. \textbf{Figure S1} is a residual plot in the supporting
information (residuals = IP\textsubscript{R} -- IP\textsubscript{A}{[}IP
calculated from the corresponding \textbf{Model A} equation{]}) that
revealed a non-random pattern, suggesting that a linear model may be
insufficient for correlating IP\textsubscript{R} to T. When the data for
MeC18:2 were fitted to a second-order polynomial by regression analysis,
the results showed that R\selectlanguage{ngerman}² increased from 0.985 to 0.997 and F-ratio
from 328 to 850 relative to the linear model summarized in\textbf{Table
2} . All three regression coefficients had \emph{p} -value \selectlanguage{english}[?] 0.02 and
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} decreased from 0.1 to 0.06. Nevertheless, it was
desired to maintain consistency in the application of \textbf{Model A}
to correlate IP\textsubscript{R}-T data for the five FAME studied
herein. Therefore, the linear model equations summarized in\textbf{Table
2} were employed to calculate shelf-life data at 25 \selectlanguage{ngerman}°C for all FAME,
including MeC18:2.

Shown in \textbf{Figure 2} is a confirmation graph of
IP\textsubscript{A} values \emph{versus} experimental
IP\textsubscript{R} values. The dashed line drawn through the data
represents the IP\textsubscript{A} = IP\textsubscript{R} line. All data
points appear in\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Fig. 2} Confirmation analysis of IP data calculated
from\textbf{Model A} equations (IP\textsubscript{A})
\emph{versus}experimental IP\textsubscript{R} data. Dashed line:
IP\textsubscript{A}= IP\textsubscript{R}. See \textbf{Fig. 1} for
abbreviations

close proximity to the dashed line, suggesting the application
of\textbf{Model A} yielded good results for all five FAME. Consequently,
regression analysis of the data in \textbf{Figure 2} yielded linear

correlations (R\selectlanguage{ngerman}² \selectlanguage{english}[?] 0.989) between IP\textsubscript{A} and
IP\textsubscript{R}, (\textbf{Table 3} ). These results yielded
intercept coefficients (C\textsubscript{0}) that were low in value,
between \selectlanguage{english}-0.1 and +0.6 and ANOVA yielded \emph{p} -value \textgreater{}
0.05, suggesting that a value of zero (0) can be assumed for
C\textsubscript{0}. On the other hand, the slope

\textbf{Table 3} Results from confirmation analysis of IP calculated
from \textbf{Model A} equations (IP\textsubscript{A})
\emph{versus}experimental IP\textsubscript{R} data. IP\textsubscript{A}
= C\textsubscript{0} + C\textsubscript{1}(IP\textsubscript{R})\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Parameter & CaME & PME & SME & MeC18:1 & MeC18:2\tabularnewline
\midrule
\endhead
df & 3 & 3 & 3 & 3 & 4\tabularnewline
C\textsubscript{0} & 0.3 & 0.11 & 0.4 & -0.1 & 0.6\tabularnewline
Err{[}C\textsubscript{0}{]} & 0.6 & 0.09 & 0.2 & 0.5 &
0.3\tabularnewline
\emph{p}-value & 0.6 & 0.3 & 0.2 & 0.9 & 0.2\tabularnewline
C\textsubscript{1} & 0.97 & 0.97 & 0.95 & 0.99 & 0.87\tabularnewline
Err{[}C\textsubscript{1}{]} & 0.04 & 0.01 & 0.01 & 0.05 &
0.04\tabularnewline
\emph{p}-value & 0.0001 & 0.000003 & 0.000008 & 0.0003 &
0.00002\tabularnewline
R\selectlanguage{ngerman}² & 0.994 & 0.9996 & 0.9990 & 0.989 & 0.991\tabularnewline
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} & 1 & 0.1 & 0.4 & 0.8 & 0.6\tabularnewline
F-ratio & 710 & 8,904 & 4,136 & 353 & 552\tabularnewline
MAD\textsuperscript{a} & 0.63 & 0.12 & 0.43 & 0.42 & 0.62\tabularnewline
RMSD\textsuperscript{b} & 0.84 & 0.19 & 0.66 & 0.62 &
0.98\tabularnewline
\bottomrule
\end{longtable}

C\textsubscript{0} = Intercept coefficient; C\textsubscript{1} = slope
coefficient; Err{[}C\textsubscript{i}{]} = standard error of coefficient
`C\textsubscript{i}'; \emph{p} -value = probability C\textsubscript{i} [?]
0 (\textless{} 0.05); MAD = mean absolute deviation; RMSD =
root-mean-squared deviation. See \textbf{Table 1} and \textbf{2} for
abbreviations.

\textsuperscript{a} Deviations between IP\textsubscript{R} and
IP\textsubscript{A}; calculated from Eq. 10.

\textsuperscript{b} Deviations between IP\textsubscript{R} and
IP\textsubscript{A}; calculated from Eq. 11.

coefficients (C\textsubscript{1}) were significant with values for four
of the FAME being in the range 0.95-0.99, which was close to the desired
value of unity (1) for a model confirmation analysis. The results

for MeC18:2 yielded a lower C\textsubscript{1} coefficient (0.87),
suggesting that its model equation under predicted IP\textsubscript{A},
with respect to its corresponding experimental
IP\textsubscript{R}values. This was confirmed for four of the six data
pairs evaluated for MeC18:2, specifically the IP\textsubscript{R} data
at T = 50, 60, 90 and 100 \selectlanguage{ngerman}°C (323.15, 333.15, 363.15 and 373.15 K).

The MAD and RMSD results obtained from application of Eqs. 10 and 11 to
(IP\textsubscript{R}, IP\textsubscript{A}) data pairs are summarized
in\textbf{Table 3} . These data showed generally low values where MAD =
0.12-0.63 and RMSD = 0.19-0.98. MeC18:2 had the highest RMSD value, a
result that may have been influenced by the relatively low
C\textsubscript{1} coefficient in its model confirmation equation. On
the other hand, CaME yielded the highest MAD value (0.63), despite
having a relatively high C\textsubscript{1} coefficient (0.99) while the
MAD value of MeC18:2 was slightly lower.

The residuals (IP\textsubscript{R} \selectlanguage{english}- IP\textsubscript{A}) calculated
from applying the \textbf{Model A} type equations for the five FAME are
presented in \textbf{Figure 3} . In general, the residuals were \selectlanguage{english}[?] 0.51
for 20 of the 26 total data pairs. Of the remaining six data pairs, only
one residual exceeded 1.43, which was for\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Fig. 3} Residual data from application of \textbf{Models A}
and\textbf{B} equations versus experimental data. Residual =
IP\textsubscript{R} -- {[}IP\textsubscript{A} or value calculated
from\textbf{Model B} (IP\textsubscript{B}){]}. See \textbf{Figs. 1}
and\textbf{2} for abbreviations

MeC18:2 at T = 50 \selectlanguage{ngerman}°C (323.15 K; residual = 2.26). In summary, it appears
that the application of \textbf{Model A} to IP\textsubscript{R}-T data
did a good job in predicting IP\textsubscript{A} values within the range
of measurement temperatures for all five of the FAME studied in the
present work.

\textbf{Model B}

Experimental results plotted as ln(IP\textsubscript{R}) versus
T\selectlanguage{english}\textsuperscript{-1} curves for \textbf{Model B} are shown
in\textbf{Figure 4} . The lines drawn through the data demonstrate a
linear correlation (R2 \selectlanguage{english}[?] 0.989) between IP\textsubscript{R} and
T\selectlanguage{english}\textsuperscript{-1} for all five FAME, an observation that is verified
in the regression analysis results presented in \textbf{Table 4} . The
intercept and slope coefficients for \textbf{Model B}(B\textsubscript{0}
and B\textsubscript{1}) were significant (\emph{p} -\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Fig. 4} Graph of ln(IP\textsubscript{R} {[}h{]}) as a function
of T\textsuperscript{-1} data for CaME, PME, SME, MeC18:1 and MeC18:2.
Lines drawn through the data points represent results from linear
regression analyses according to \textbf{Model B}. See \textbf{Fig.
1}for abbreviations

value \textless{}\textless{} 0.05). Similar to the results
for\textbf{Model A} , the intercept and slope coefficients for CaME, PME
and SME were in narrow ranges, between -27.5 and -23.7 for
B\textsubscript{0}, and between 9,900 and 11,320 for B\textsubscript{1}.
Also correspondingly, the B\textsubscript{0} and
B\textsubscript{1}coefficients were significantly higher in magnitude
for MeC18:1 (-37 and 14,400) and lower in magnitude for MeC18:2 (-19.1
and 7,100).

\textbf{Table 4} Results from regression analysis of
ln(IP\textsubscript{R}) \emph{versus} T\textsuperscript{-1} data
(\textbf{Model B} ). ln(IP\textsubscript{R}) = B\textsubscript{0} +
B\textsubscript{1}(T\textsuperscript{-1}); IP\textsubscript{R} in h, T
in K\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Parameter & CaME & PME & SME & MeC18:1 & MeC18:2\tabularnewline
\midrule
\endhead
df & 3 & 3 & 3 & 3 & 4\tabularnewline
B\textsubscript{0} & -23.7 & -27.5 & -27.0 & -37 & -19.1\tabularnewline
Err{[}B\textsubscript{0}{]} & 0.8 & 0.2 & 0.7 & 2 & 0.7\tabularnewline
\emph{p}-value & 0.00008 & 0.000001 & 0.00005 & 0.0003 &
0.00001\tabularnewline
B\textsubscript{1} & 9,900 & 11,320 & 10,800 & 14,400 &
7,100\tabularnewline
Err{[}B\textsubscript{1}{]} & 300 & 90 & 300 & 800 & 300\tabularnewline
\emph{p}-value & 0.00006 & 0.000001 & 0.00004 & 0.0003 &
0.00001\tabularnewline
R\selectlanguage{ngerman}² & 0.996 & 0.9998 & 0.997 & 0.989 & 0.993\tabularnewline
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} & 0.07 & 0.02 & 0.06 & 0.2 & 0.09\tabularnewline
F-ratio & 1,059 & 16,019 & 1,518 & 368 & 750\tabularnewline
\bottomrule
\end{longtable}

B\textsubscript{0} = intercept coefficient; B\textsubscript{1} = slope
coefficient; Err{[}B\textsubscript{i}{]} = standard error of coefficient
`B\textsubscript{i}'; \emph{p} -value = probability B\textsubscript{i} [?]
0 (\textless{} 0.05). See \textbf{Tables 1} and \textbf{2} for
abbreviations.

The linear correlations obtained from application of \textbf{Model B} to
IP\textsubscript{R}-T\textsuperscript{-1} demonstrated in the high R\selectlanguage{ngerman}²
coefficients, low \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} ([?] 0.2) and high F-ratios ([?] 368),
all indicators that the model equations had relatively high goodness of
fit for each FAME. Similar to the results from applying \textbf{Model A}
, the \textbf{Model B} results for MeC18:2 in \textbf{Figure 4}exhibited
some curvature in the data. The residual plot for MeC18:2
(see\textbf{Figure S2} in the supporting information) revealed a
non-random pattern and fitting the data to a second-order polynomial
(data not shown) resulted in slight improvements for R\selectlanguage{ngerman}² (0.997), F-ratio
(848) and \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} (0.06). However, the three regression
coefficients had \emph{p} -value [?] 0.09, indicating they were not
significant to the correlation of the data. Given these results, the
linear model described by \textbf{Model B} was determined to be
sufficient for correlating IP\textsubscript{R-}T\textsuperscript{-1}
data for MeC18:2. Therefore, the linear correlation results in
\textbf{Table 4} were employed to calculate shelf-life results at 25 \selectlanguage{ngerman}°C
for all FAME studied in the present work.

A conformation graph showing IP\textsubscript{B} data calculated from
application of the \textbf{Model B} equations \emph{versus} experimental
IP\textsubscript{R} data is presented \textbf{Figure 5} . Analogous to
the results in \textbf{Figure 2} , all data points are in close
proximity to the dashed line representing IP\textsubscript{B} =
IP\textsubscript{R}. This suggested\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Fig. 5} Confirmation analysis of IP\textsubscript{B} data
calculated from \textbf{Model B} equations \emph{versus} experimental
IP\textsubscript{R} data. Dashed line: IP\textsubscript{B} =
IP\textsubscript{R}. See \textbf{Figs. 1} and \textbf{3} for
abbreviations

that the application of \textbf{Model B} yielded good results for all
five FAME studied in the present work. Results from regression analysis
of the data in \textbf{Figure 5} are summarized in \textbf{Table 5} .
These analyses yielded linear correlations (R\selectlanguage{ngerman}² \selectlanguage{english}[?] 0.982) between
IP\textsubscript{B} and IP\textsubscript{R}. The intercept

\textbf{Table 5.} Results from confirmation analysis of IP calculated
from \textbf{Model B} equations (IP\textsubscript{B})
\emph{versus}experimental IP\textsubscript{R} data. IP\textsubscript{B}
= D\textsubscript{0} + D\textsubscript{1}(IP\textsubscript{R})\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Parameter & CaME & PME & SME & MeC18:1 & MeC18:2\tabularnewline
\midrule
\endhead
df & 3 & 3 & 3 & 3 & 4\tabularnewline
D\textsubscript{0} & 0.0 & -0.06 & 0.0 & -0.3 & 0.3\tabularnewline
Err{[}D\textsubscript{0}{]} & 0.5 & 0.07 & 0.1 & 0.6 &
0.2\tabularnewline
\emph{p}-value & 0.9 & 0.4 & 0.98 & 0.7 & 0.2\tabularnewline
C\textsubscript{1} & 1.01 & 1.014 & 0.996 & 1.05 & 0.93\tabularnewline
Err{[}D\textsubscript{1}{]} & 0.03 & 0.008 & 0.008 & 0.07 &
0.02\tabularnewline
\emph{p}-value & 0.00004 & 0.000001 & 0.000001 & 0.0007 &
0.000001\tabularnewline
R\selectlanguage{ngerman}² & 0.997 & 0.9997 & 0.9998 & 0.982 & 0.998\tabularnewline
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{y} & 0.7 & 0.1 & 0.2 & 1 & 0.3\tabularnewline
F-ratio & 1,338 & 14,165 & 16,772 & 218 & 2,028\tabularnewline
MAD & 0.38 & 0.081 & 0.14 & 0.61 & 0.39\tabularnewline
RMSD & 0.57 & 0.12 & 0.16 & 0.86 & 0.54\tabularnewline
\bottomrule
\end{longtable}

D\textsubscript{0} = Intercept coefficient; D\textsubscript{1} = slope
coefficient; Err{[}D\textsubscript{i}{]} = standard error of coefficient
`D\textsubscript{i}'; \emph{p} -value = probability D\textsubscript{i} [?]
0 (\textless{} 0.05). See \textbf{Tables 1-3} for abbreviations.

coefficients (D\textsubscript{0}) that were very low (between -0.3 and
+0.3) and ANOVA indicated they were not significant (\emph{p} -value
\textgreater{} 0.05) allowing D\textsubscript{0} = 0 to be assumed. The
slope coefficients (D\textsubscript{1}) were significant (\emph{p}
-value \textless{}\textless{} 0.05) with values for four of the FAME
being close to unity (0.996-1.05) indicating good confirmation of these
model equations. Similar to the results for \textbf{Model A} ,
the\textbf{Model B} confirmation equation for MeC18:2 yielded the lowest
D\textsubscript{1} coefficient (0.93), suggesting that its model
equation under predicted IP\textsubscript{B}, with respect to
experimental IP\textsubscript{R} values. This was confirmed for four of
the six data pairs evaluated for MeC18:2, where
IP\textsubscript{R}\textgreater{} IP\textsubscript{B} at the same four
temperatures (50, 60, 90 and 100 \selectlanguage{ngerman}°C {[}323.15, 333.15, 363.15 and 373.15
K{]}) as observed above for its \textbf{Model A} equation.

The MAD and RMSD results calculated from (IP\textsubscript{R},
IP\textsubscript{B}) data pairs are presented in \textbf{Table 5} .
Similar to the results for \textbf{Model A} , the confirmation results
for \textbf{Model B} exhibited generally low values, MAD = 0.081-0.61
and RMSD = 0.12-0.86. MeC18:2 had the highest MAD and RMSD values,
suggesting its predicted IP\textsubscript{B} values were influenced by
the lower D\textsubscript{1} coefficient in its confirmation equation.

The results for the residuals (IP\textsubscript{R} \selectlanguage{english}-
IP\textsubscript{B}) calculated from application of the \textbf{Model
B}type equations to the data for the five FAME are presented
in\textbf{Figure 3} . Overlayed with these results are residuals
(IP\textsubscript{R} \selectlanguage{english}- IP\textsubscript{A}) obtained from
the\textbf{Model A} equations (discussed earlier). In general, the
residuals for \textbf{Model B} were \selectlanguage{english}[?] 1.47 for all 26 data pairs with 21
data pairs having residuals \selectlanguage{english}[?] 0.44. The highest residual (1.47) was for
MeC18:1 at T = 100 degC (373.15 K). Summarizing the \textbf{Model
B}results, the application of this type of model equation to
IP\textsubscript{R}-T\selectlanguage{english}\textsuperscript{-1} data did an excellent job
predicting IP\textsubscript{B} values within the range of measurement
temperatures corresponding to the five FAME studied in the present work.

Direct comparison of results from applying \textbf{Model A}
and\textbf{Model B} type equations to IP\textsubscript{R} data for the
five FAME showed that the calculation of IP\textsubscript{B} values from
the\textbf{Model B} equations yielded more accurate results. This is
best demonstrated by the deviation results obtained from confirmation
analyses performed on both models. First, comparing the data plotted
in\textbf{Figures 2} and \textbf{5} shows that IP\textsubscript{A} data
points tended to be more separated from the dashed line than the
IP\textsubscript{B} results, especially when
IP\textsubscript{R}\textgreater{} 15 h. This observation is supported by
the residuals data overlayed for \textbf{Models A} and \textbf{B} in
\textbf{Figure 3} . These results demonstrate that residuals were
generally larger in the IP\textsubscript{A} data than the
IP\textsubscript{B} data. Finally, comparing MAD and RMSD data in
\textbf{Tables 3} and \textbf{5} shows that values were consistently
lower for \textbf{Model B} results for four of the five FAME studied
herein. The exception was observed for MeC18:1, where the \textbf{Model
A} results yielded MAD = 0.42 and RMSD = 0.62, values that were slightly
lower than those for the \textbf{Model B} results (0.61 and 0.86).

\textbf{Shelf-life Results from Extrapolation of Models A and B
Equations}

Thermal degradation at temperatures above 60 degC can alter the mechanism
of oxidation due to the solubility of oxygen decreasing as temperature
increases (Frankel, 2005b). Measurement of the IP\textsubscript{R} works
by the evolution of volatile organic acids from the heated oil which can
be swept by the continuous flow or air into a test tube containing
deionized water (Frankel, 2005c). The water-organic acid mixture is
continuously monitored by a conductivity probe and increases are
recorded as a function of time. The IP\textsubscript{R} is then
calculated at the onset point where conductivity increases
exponentially. Decreasing the measurement temperature hinders the
formation of volatile organic acids. Therefore, elevated temperatures
(60-140 degC) are necessary to generate them for the analysis of
IP\textsubscript{R}. This condition is the main driving force for
estimating the shelf-life of fatty derivatives such as biodiesel by
extrapolating IP\textsubscript{R}-T data to lower temperatures.

\textbf{Model A} and \textbf{Model B} type equations obtained for the
five FAME studied herein were extrapolated to estimate shelf-life data
at T = 25 degC (298.15 K). These results are summarized under columns
showing calculated data for \textbf{Model A} (SL\textsubscript{A})
and\textbf{Model B} (SL\textsubscript{B}) in \textbf{Tables 6}
and\textbf{7} . The general procedure for both model types was to first
calculate ln(IP\textsubscript{A}) or ln(IP\textsubscript{B}) using
either Eq. 2 or Eq. 4, then take the inverse natural logarithm to obtain
the SL\textsubscript{A} or SL\textsubscript{B} results for each FAME.
Also presented in the tables are confidence intervals calculated at 95
\% confidence levels.

\textbf{Table 6} Shelf-life of FAME at T = 25 degC (298.15 K) calculated
from \textbf{Model A} type equations (SL\textsubscript{A})\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
FAME & SL\textsubscript{A}\textsuperscript{a} (h) & Confidence
Limits\textsuperscript{b} & Confidence
Limits\textsuperscript{b}\tabularnewline
\midrule
\endhead
& & Upper & Lower\tabularnewline
CaME & 2,760 & 3,190 & 2,390\tabularnewline
PME & 3,970 & 4,250 & 3,700\tabularnewline
SME & 2,220 & 2,750 & 1,800\tabularnewline
MeC18:1 & 10,700 & 16,400 & 7,000\tabularnewline
MeC18:2 & 74.8 & 99.8 & 56.0\tabularnewline
\bottomrule
\end{longtable}

See \textbf{Tables 1-3} for abbreviations.

\textsuperscript{a} Procedure for each FAME: 1) calculate
ln(IP\textsubscript{A}) = A\textsubscript{0} + A\textsubscript{1}(298.15
K) {[}A\textsubscript{0} and A\textsubscript{1} from \textbf{Table 2}
{]}; 2) SL\textsubscript{A} = inverse natural log of extrapolated
ln(IP\textsubscript{A}) value.

\textsuperscript{b} Confidence level = 95 \%.

\textbf{Table 7} Shelf-life of FAME at T = 25 \selectlanguage{ngerman}°C (298.15 K) calculated
from \textbf{Model B} type equations (SL\textsubscript{B})\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
FAME & SL\textsubscript{B}\textsuperscript{a} (h) & Confidence
Limits\textsuperscript{b} & Confidence
Limits\textsuperscript{b}\tabularnewline
\midrule
\endhead
& & Upper & Lower\tabularnewline
CaME & 13,400 & 15,700 & 11,500\tabularnewline
PME & 34,400 & 36,000 & 33,000\tabularnewline
SME & 9,080 & 10,500 & 7,830\tabularnewline
MeC18:1 & 108,000 & 159,000 & 74,000\tabularnewline
MeC18:2 & 116 & 140 & 95.5\tabularnewline
\bottomrule
\end{longtable}

See \textbf{Tables 1} , \textbf{4} and \textbf{5} for abbreviations.

\textsuperscript{a} Procedure for each FAME: 1) calculate
ln(IP\textsubscript{B}) = B\textsubscript{0} + B\textsubscript{1}(298.15
K) {[}B\textsubscript{0} and B\textsubscript{1} from \textbf{Table 4}
{]}; 2) SL\textsubscript{B} = inverse natural log of extrapolated
ln(IP\textsubscript{B}) value.

\textsuperscript{b} Confidence level = 95 \%.

The SL\textsubscript{B} data for CaME, PME and MeC18:1 were an order in
magnitude greater than the corresponding SL\textsubscript{A} data. For
these three FAME, the SL\textsubscript{B} values were 4.9, 8.7 and 10.0
times higher than the respective SL\textsubscript{A} values. The
SL\textsubscript{B} of SME was 4.1 times greater than its
SL\textsubscript{A}, while the SL\textsubscript{B} for MeC18:2 was only
55 \% higher than its SL\textsubscript{A}. It appears the inflation in
the SL\textsubscript{B} values relative to corresponding
SL\textsubscript{A} values was larger for the three FAME with high total
MUFAME concentrations ([?] 40.49 mass\%). In contrast, FAME with high
total PUFAME concentrations (62.09 \% for SME and 99.0 \% for MeC18:2)
exhibited smaller increases in SL\textsubscript{B}, with respect to
SL\textsubscript{A}. Nevertheless, SL\textsubscript{B} was consistently
greater than SL\textsubscript{A} for all five FAME studied in the
present work.

A comparison of SL\textsubscript{A} results with shelf-life data
reported in two earlier studies is presented in \textbf{Figure 6} . The
results presented in the present work compared well with the `SL(1)'
data reported in an earlier study (Dunn, 2020). In that study,
shelf-life data were obtained by\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Fig. 6} Comparison of shelf-life data (SL\textsubscript{A}) data
obtained from extrapolation of the \textbf{Model A} type equations to 25
\selectlanguage{ngerman}°C (298.15 K) with shelf-life data {[}SL(1) and SL(3){]} reported in
earlier studies (Dunn, 2008, 2020). Error bars for
SL\textsubscript{A}data are 95 \% confidence intervals taken from
\textbf{Table 6}. See\textbf{Fig. 1} for abbreviations

linear regression of IP\textsubscript{R}-T data and extrapolating the
resulting equations to T = 25 °C. However, the SL(1) data were obtained
from equations derived from experiments performed at three temperatures
in the ranges T = 90-110 °C (363.15-383.15 K) for CaME, PME, SME and T =
80-100 °C (353.15-373.15 K) for MeC18:2. In the present work,
SL\textsubscript{A} data were calculated from \textbf{Model A} type
equations derived from analysis of IP\textsubscript{R}-T data measured
at five or six measurement temperatures. As a consequence, variations
between SL\textsubscript{A} and SL(1) data from the earlier study were
observed. While SL\textsubscript{A} \textgreater{} SL(1) for CaME and
MeC18:2, the opposite trend was noted for PME, SME and MeC18:1. In the
cases where SL\textsubscript{A} \textgreater{} SL(1), deviations were
small, 200 h for CaME and 44 h for MeC18:2. In the remaining three
cases, deviations were larger with MeC18:1 demonstrating the largest
deviation (9,000 h), followed by PME (1,200 h) and SME (900 h).

Ranking the calculated SL\textsubscript{A} results in descending order
yielded the following:

MeC18:1 \textgreater{} PME \textgreater{} CaME \textgreater{} SME
\textgreater{} MeC18:2 (15)

This order generally agrees well with what would be expected based on
the degrees of unsaturation (DU) of PME, CaME, SME and MeC18:2, where DU
= 0.58, 1.28, 1.53 and 2.00 (calculated from results in \textbf{Table
S1} in the supporting information). Predicting the shelf-life of FAME
generally does not correlate well with DU (Knothe, 2002) and this is
exemplified when considering that according to SL\textsubscript{A},
MeC18:1 held the highest ranking above despite having DU = 1.00, a value
that was greater than the DU calculated for PME. Nevertheless, ranking
the SL(1) data reported earlier (Dunn, 2020) yielded similar results
with the remarkable exception that CaME {[}SL(1) = 2,560 h{]} and SME
(3,130 h) were switched in the rankings. The ranking from the earlier
work was curious since SME had a significantly higher total PUFAME
concentration (62.09 mass\%) than CaME (26.77 \%). Given the
aforementioned differences in the experimental method for acquiring and
analyzing IP\textsubscript{R}-T data to estimate SL\textsubscript{A} for
the present work and SL(1) in the earlier study, the change in the
ranking of CaME and SME suggests that the methodology can significantly
impact the outcomes of the analyses. Therefore, it was concluded that
the method employed in the present work (five to six
IP\textsubscript{R}-T data points over variable T-ranges) yielded more
reliable results than the method used in the earlier study, with respect
to the estimation of the shelf life of biodiesel at lower temperatures.

Two data points in \textbf{Figure 6} were represented as `SL(3)' results
reported in an earlier study (Dunn, 2008). This study was carried out
using a different instrument to measure the IP, which was generally
termed as the oil stability index (OSI). The experimental data were
measured taken at six temperatures in the range T = 60-80 °C
(333.15-353.15 K) for SME and eight temperatures in the range T = 60-95
°C (333.15-368.15 K) for MeC18:1. The SL\textsubscript{A} values for SME
and MeC18:1 were 12.9 and 5.0 times greater than their corresponding
SL(3) values. The most likely explanation for these poor comparisons was
the variation in instruments and experimental conditions employed to
measure the respective IP data for analysis.

The SL\textsubscript{B} data obtained from extrapolation of
the\textbf{Model B} equations are presented graphically in
\textbf{Figure 7} . These data are presented in a separate figure mainly
for convenience\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Fig. 7} Comparison of shelf-life data (SL\textsubscript{B}) data
obtained from extrapolation of the \textbf{Model B} type equations to 25
\selectlanguage{ngerman}°C (298.15 K) with shelf-life data {[}SL(2){]} reported in an earlier
study (Dunn, 2020). Error bars for SL\textsubscript{B} data are 95 \%
confidence intervals taken from \textbf{Table 7}. See \textbf{Fig. 1}for
abbreviations

and to avoid overcrowding the results presented in \textbf{Figure 6} .
Presented alongside the SL\textsubscript{B} data are shelf-life data
obtained from analysis of pressurized-differential scanning calorimetry
(P-DSC) data reported in an earlier study (Dunn, 2020), which are
denoted as `SL(2)'. The mathematical procedures for estimating the SL(2)
data are explained in more detail in the earlier work.

The SL\textsubscript{A} and SL(2) data were most comparable for SME,
MeC18:1 and MeC18:2, where the deviations were \selectlanguage{english}-1,400, 2,400 and 37 h,
respectively. On the other hand, deviations for CaME and PME were much
larger (\selectlanguage{english}-31,100 and \selectlanguage{english}-12,200 h). Similar comparison of
SL\textsubscript{B} and SL(2) data suggested that the values may be
comparable for MeC18:2 (deviation = 78 h). Otherwise, the deviations for
CaME, PME, SME and MeC18:1 were larger (\selectlanguage{english}-20,500, 18,200, 5,500 and
99,700 h). Analogous results were observed from comparison of
SL\textsubscript{B} and SL(1) results, where deviations for MeC18:2 were
lowest (85 h), while those for CaME, PME, SME and MeC18:1 were larger
(10,800, 29,200, 5,950 and 88,300 h).

Ranking the SL\textsubscript{B} values in \textbf{Table 7} in descending
order yielded the following results:

MeC18:1 \textgreater{} PME \textgreater{} CaME \textgreater{} SME
\textgreater{} MeC18:2 (16)

This ranking was identical to the ranking shown noted in Eq. 15 for
SL\textsubscript{A}. The calculated SL\textsubscript{B} of MeC18:1 was
between one and two orders in magnitude greater than the
SL\textsubscript{B} values for CaME, PME and SME, and three orders in
magnitude greater than the SL\textsubscript{B} of MeC18:2. Compared with
the SL\textsubscript{A} and SL\textsubscript{B} rankings in the present
work, the SL(2) data reported in (Dunn, 2020) exhibited some variations,
where CaME had the highest value, followed by PME, MeC18:1, SME and
MeC18:2. This ranking did not agree well with those shown in Eqs. 15 and
16 for SL\textsubscript{A} and SL\textsubscript{B}. It is possible that
thermal degradation occurred in the non-isothermal P-DSC scans conducted
for the earlier study (Dunn, 2020). These effects may have altered the
reaction kinetics as the P-DSC cell temperature was ramped to higher
temperatures.

Earlier, it was noted that SL\textsubscript{B} was consistently greater
than SL\textsubscript{A} for all five FAME studied in the present work.
For CaME, PME and MeC18:1, deviations between these two values were more
than one order in magnitude. There may be some question on the
reliability of SL\textsubscript{A} and/or SL\textsubscript{B} data as
estimates for the shelf-life of biodiesel at low temperatures. For
example, SL\textsubscript{A} = 2,760 h and SL\textsubscript{B} = 13,400
h were estimated for CaME (see \textbf{Tables 6} and \textbf{7} ). These
data convert to 115 d (16.4 wk) and 558 d (79.8 wk). Between the two
time periods, the latter appears to be more realistic for CaME, which
has a low total PUFAME concentration (26.77 mass\%), if it is stored in
a dark space absent from moisture. Similar arguments apply to
SL\textsubscript{B} results for PME (1,430 d {[}205 wk{]}; total PUFAME
= 9.7 \%), SME (378 d {[}54.0 wk{]}; total PUFAME = 62.09 \%) and
MeC18:1 (4,500 d {[}643 wk{]}; no PUFAME). Both sets of results could be
realistic for MeC18:2 (SL\textsubscript{A}: 3.12 d {[}0.45 wk{]};
SL\textsubscript{B}: 4.83 d {[}0.69 wk{]}; 99.0 \% PUFAME). Given these
considerations, it is concluded that the extrapolation of \textbf{Model
B} type equations to lower temperatures yielded more realistic
shelf-life results (SL\textsubscript{B}) than similar processing
of\textbf{Model A} type equations (SL\textsubscript{A}).

\textbf{Calculation of Activation Energy from Model B}

It was proposed in an earlier study (Dunn, 2008) that kinetic parameters
for the oxidation of biodiesel can be obtained from solutions
to\textbf{Model B} type equations. The rate of increase in the degree of
conversion of a substrate (\selectlanguage{greek}α\selectlanguage{english}) is expressed by:

d\selectlanguage{greek}α/\selectlanguage{english}dt = k\textsubscript{T}f(\selectlanguage{greek}α\selectlanguage{english}) (17)

where k\textsubscript{T} is the reaction rate constant, described
mathematically in Eq. 5, and f(\selectlanguage{greek}α\selectlanguage{english}) mathematically describes the kinetics
of the reaction. Rearranging and integration yields the following
relationship:

g(\selectlanguage{greek}α\selectlanguage{english}) = k\textsubscript{T}[?]t (18)

where g(\selectlanguage{greek}α\selectlanguage{english}) is determined from f(\selectlanguage{greek}α\selectlanguage{english}) using Eq. 9 and [?]t is the time
interval (t - t\textsubscript{0}). At the induction period of the
reaction, IP = [?]t where g(\selectlanguage{greek}α\selectlanguage{english}) is determined between \selectlanguage{greek}α \selectlanguage{english}= 0 at
t\textsubscript{0} and \selectlanguage{greek}α \selectlanguage{english}= \selectlanguage{greek}α\selectlanguage{english}* at t = t* ([?]t = t* - t\textsubscript{0}).
Substituting Eq. 5 for k\textsubscript{T} in Eq 18, taking the natural
log of both sides and rearranging yields:

ln([?]t) = ln{[}g(\selectlanguage{greek}α\selectlanguage{english})/Z\textsubscript{0}{]} +
(E\textsubscript{a}R\textsubscript{g}\textsuperscript{-1})(T)\textsuperscript{-1}(19)

When [?]t = IP\textsubscript{R}, Eq 19 is equivalent to Eq. 3,
the\textbf{Model B} type equations derived in the present study.
Therefore, kinetic parameters E\textsubscript{a} and Z\textsubscript{0}
for the oxidation reaction can be calculated from the slope and
intercept coefficients from model equations for each FAME. Earlier, an
expression was given for calculating E\textsubscript{a} from the slope
coefficients (B\textsubscript{1}) in Eq. 6. The following equation can
be used to obtain pre-exponential factor Z\textsubscript{0} from the
intercept coefficients (B\textsubscript{0}) assuming the kinetic models
are known:

Z\textsubscript{0} = g(\selectlanguage{greek}α\selectlanguage{english}){[}exp\{-B\textsubscript{0}\}{]} (20)

If the reaction kinetics are not known, then Eq. 20 can be rearranged to
yield a ratio of Z\textsubscript{0} over g(\selectlanguage{greek}α\selectlanguage{english}), which allows calculation
of Z\textsubscript{0} (and k\textsubscript{T}) for varying models:

{[}Z\textsubscript{0}/g(\selectlanguage{greek}α\selectlanguage{english}){]} = exp\{-B\textsubscript{0}\} (21)

The results presented in \textbf{Table 8} include E\textsubscript{a}data
and Z\textsubscript{0}/g(\selectlanguage{greek}α\selectlanguage{english}) ratios calculated from applying Eqs. 6 and
21 to the \textbf{Model B} equation data in \textbf{Table 4} . Also
shown are estimates for Z\textsubscript{0} factors determined for two
popular kinetic models for the oxidation of fatty derivatives, first
order (Kamal-Eldin and Yanishlieva, 2005; Litwinienko, 2005; Litwinienko
et al., 2000) and ``simple'' autocatalysis (Adachi et al., 1995; Ishido
et al., 2001; Luna et al., 2007). The g(\selectlanguage{greek}α\selectlanguage{english}) functions were determined at
\selectlanguage{greek}α\selectlanguage{english}* = 0.5 at the IP\textsubscript{R} of the FAME (Dunn, 2020).

\textbf{Table 8} Activation energy (E\textsubscript{a}) and
pre-exponential factor (Z\textsubscript{0}) calculated from\textbf{Model
B} equations\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
FAME & E\textsubscript{a}\textsuperscript{a} (kJ mol
\textsuperscript{-1}) & {[}Z\textsubscript{0}/g(\selectlanguage{greek}α\selectlanguage{english}){]}\textsuperscript{b}
& Z\textsubscript{0} & Z\textsubscript{0}\tabularnewline
\midrule
\endhead
& & & First-Order\textsuperscript{c} &
Autocatalysis\textsuperscript{d}\tabularnewline
CaME & 82 (3) & 1.92 \selectlanguage{ngerman}× 10\textsuperscript{10} & 1.33 \selectlanguage{ngerman}×
10\textsuperscript{10} & 8.26 \selectlanguage{ngerman}× 10\textsuperscript{10}\tabularnewline
PME & 94.1 (0.7) & 8.83 \selectlanguage{ngerman}× 10\textsuperscript{11} & 6.12 \selectlanguage{ngerman}×
10\textsuperscript{11} & 3.79 \selectlanguage{ngerman}× 10\textsuperscript{12}\tabularnewline
SME & 90 (2) & 5.36 \selectlanguage{ngerman}× 10\textsuperscript{11} & 3.72 \selectlanguage{ngerman}×
10\textsuperscript{11} & 2.30 \selectlanguage{ngerman}× 10\textsuperscript{12}\tabularnewline
MeC18:1 & 120 (6) & 8.96 \selectlanguage{ngerman}× 10\textsuperscript{15} & 6.21 \selectlanguage{ngerman}×
10\textsuperscript{15} & 3.85 \selectlanguage{ngerman}× 10\textsuperscript{16}\tabularnewline
MeC18:2 & 59 (2) & 1.91 \selectlanguage{ngerman}× 10\textsuperscript{8} & 1.32 \selectlanguage{ngerman}×
10\textsuperscript{8} & 8.20 \selectlanguage{ngerman}× 10\textsuperscript{8}\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}

See \textbf{Table 1} for abbreviations.

\textsuperscript{a} Calculated from Eq. 6. Numbers in parentheses are
standard errors.

\textsuperscript{b} Calculated from Eq. 21.

\textsuperscript{c} Model: f(\selectlanguage{greek}α\selectlanguage{english}) = (1 - \selectlanguage{greek}α\selectlanguage{english}); g(\selectlanguage{greek}α\selectlanguage{english}) = -ln(1 - \selectlanguage{greek}α\selectlanguage{english}) = 0.693 at
\selectlanguage{greek}α\selectlanguage{english}* = 0.5 at induction period (IP).

\textsuperscript{d} Model: f(\selectlanguage{greek}α\selectlanguage{english}) = \selectlanguage{greek}α\selectlanguage{english}(1 - \selectlanguage{greek}α\selectlanguage{english}); g(\selectlanguage{greek}α\selectlanguage{english}) = -ln{[}(1 - \selectlanguage{greek}α\selectlanguage{english})/\selectlanguage{english}\selectlanguage{greek}α{]} +
C = 4.30 at \selectlanguage{greek}α\selectlanguage{english}* = 0.5 at IP. C = integration constant {[}g(\selectlanguage{greek}α\selectlanguage{english}) estimated
by numerical integration{]}.

The E\textsubscript{a} data in \textbf{Table 8} for PME, SME and MeC18:2
were comparable to E\textsubscript{a} data obtained from analysis of
P-DSC results in an earlier study (Dunn, 2020). On the other hand,
results varied significantly for CaME and MeC18:1, with respect to the
earlier study. For these two FAME, the values in \textbf{Table 8} were
-23 kJ mol\textsuperscript{-1} (lower) and +26 kJ
mol\textsuperscript{-1} for the present work. One study (Litwinienko and
Kasprzycka-Guttman, 2000) reported E\textsubscript{a} = 95 kJ
mol\textsuperscript{-1} for ethyl oleate and 76.0 kJ
mol\textsuperscript{-1} for ethyl linoleate. A literature review in that
study suggested E\textsubscript{a} values for oxidation were in the
ranges 93-100 kJ mol\textsuperscript{-1} for MeC18:1 and 69-76 kJ
mol\textsuperscript{-1} for MeC18:2.

Results in \textbf{Table 8} for the Z\textsubscript{0} factors
calculated for the first order kinetic model showed that results for
CaME, PME and SME were in a relatively narrow range, between 1.33 \selectlanguage{ngerman}×
10\textsuperscript{10} and 6.12 × 10\textsuperscript{11}, compared to
results for MeC18:1 and MeC18:2. The same general trend was observed in
the Z\textsubscript{0} values calculated from the autocatalytic model
(range = 8.26 × 10\textsuperscript{10} to 3.79 ×
10\textsuperscript{12}). For both models, the Z\textsubscript{0} factors
for CaME, PME and SME had values that were between
Z\textsubscript{0}factors for MeC18:1 and MeC18:2. Studies with ethyl
esters reported Z\textsubscript{0} = 5.66 × 10\textsuperscript{12} for
ethyl oleate and 1.45 × 10\textsuperscript{11} for ethyl linoleate
(Litwinienko and Kasprzycka-Guttman, 2000).

In general, there is a paucity of E\textsubscript{a} and
Z\textsubscript{0} data in the scientific literature for pure FAME and
their mixtures (biodiesel). The comparison data described above were
obtained from analysis of P-DSC data using model-free kinetic methods.
As of this writing, no reliable data from analysis of isothermal
induction period (IP\textsubscript{R}, etc.) could be identified. Future
studies on this topic would justify taking these concerns under
consideration.

\textbf{CONCLUSIONS}

The shelf-life of biodiesel is an important property that determines how
long it can be stored at low temperatures at fuel terminals and in
vehicle tanks and fuel systems. The present work was organized with the
objective of developing mathematical equations for estimating the
shelf-life of biodiesel at T = 25 °C (298.15 K). To meet this objective,
two types of models, \textbf{Model A} and \textbf{Model B} , were
obtained from regression analysis where ln(IP\textsubscript{R}) was
expressed as a linear function of T and T\selectlanguage{english}\textsuperscript{-1} (Eqs. 1
and 3). While the experimental IP\textsubscript{R} data were acquired
with measurement temperatures in the range 50-140 degC (323.15-413.15 K),
the shelf-life data (SL\textsubscript{A} and SL\textsubscript{B}) were
calculated by extrapolating the model equations to T = 298.15 K (Eqs. 2
and 4).

Both \textbf{Model A} and \textbf{Model B} type equations demonstrated
good correlation between ln(IP\textsubscript{R}) and T (R2 \selectlanguage{english}[?] 0.985) and
T\selectlanguage{english}\textsuperscript{-1} (R2 \selectlanguage{english}[?] 0.989) and predicting IP\textsubscript{A}and
IP\textsubscript{B} values within corresponding measurement temperature
ranges for the five FAME (CaME, PME, PME, MeC18:1 and MeC18:2) studied
in this work. Confirmation analysis of these results showed that
calculated IP\textsubscript{B} data from the \textbf{Model B} equations
were more accurate than IP\textsubscript{A} (from\textbf{Model A}
equations), with respect to measured IP\textsubscript{R} data. Four of
the five FAME exhibited lower MAD and RMSD values for calculated results
from the \textbf{Model B} type equations (exception: MeC18:1).

The shelf-life results from extrapolation of the model equations showed
that results from \textbf{Model B} (SL\textsubscript{B}) were
consistently greater than results from \textbf{Model
A}(SL\textsubscript{A}) equations. The results for CaME, PME, MeC18:1
and MeC18:2 demonstrated that SL\textsubscript{B} values exceeded
SL\textsubscript{A} values by an order in magnitude. While \textbf{Model
A} is traditionally used to estimate shelf-life data in the fats and
oils industry, this work showed that extrapolating \textbf{Model B} type
equations appeared to yield more realistic shelf-life estimates for
CaME, PME, SME and MeC18:1. The estimated shelf-life results for MeC18:2
(SL\textsubscript{A} = 74.8 h and SL\textsubscript{B} = 116 h) seemed
too low for realistic storage at T = 298.15 K.

Future studies on this topic might include acquiring more experimental
data for E\textsubscript{a} and Z\textsubscript{0} factors for FAME
mixtures under conditions similar to those used to measure
IP\textsubscript{R} at variable temperatures. Acquiring such data would
be helpful in the development of kinetic modeling approaches to estimate
shelf-life data for biodiesel at low temperatures. For example, an
earlier study (Dunn, 2020) suggested that the autocatalysis model f(\selectlanguage{greek}α\selectlanguage{english}) =
\selectlanguage{greek}α\selectlanguage{english}\selectlanguage{ngerman}\textsuperscript{½}(1 \selectlanguage{english}- \selectlanguage{greek}α\selectlanguage{english}) may work well for FAME mixtures that are
high in MeC18:1 concentration (Dunn, 2020).

\textbf{AUTHOR CONTRIBUTIONS}

Robert O. Dunn conceived the study, designed the experiments, developed
the methodology, supervised technical assistants, organized the data,
analyzed the results and wrote the manuscript.

\textbf{ACKNOWLEDGEMENTS}

The author expresses gratitude to Kimberly L. Ascherl and Kevin R.
Steidley for providing technical assistance for experiments conducted in
the course of this study. This work was funded as part of the inhouse
research of the USDA Agricultural Research Service.

\textbf{CONFLICT OF INTEREST STATEMENT}

The author declares no conflicts of interest.

\textbf{ETHICS STATEMENT}

No humans or animals were used in this research.

\textbf{ORCID} 0000-0002-5886-7147

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