2.1. Methanol Determination
Since the first method for the analysis of methanol in transformer oil became available in 2007 [
1], a handful of publications have presented analytical methods for its determination [
4,
9,
18,
19,
20,
21]. The characteristics of these methods are presented in
Table 1. The availability of sensitive methodologies (ng/g) for alcohol analysis in transformer oil has contributed to their use as cellulose degradation indicators.
All of these methods are based on the principle of static headspace gas chromatography, as initially reported by Jalbert et al. [
1]. Headspace (HS) is essentially an equilibration technique in which volatiles can be separated from a complex sample matrix in the gas phase. The benefits of using automated HS for sample introduction include the minimization of injector and column contamination and less analyte loss since there is less sample manipulation [
19]. The oil sample (7–13 mL) is heated at temperatures between 85 and 100 °C for about 40 min to allow the volatile oil-dissolved species (e.g., methanol) to reach equilibrium in the gas phase. Then, the vapor is transferred by the carrier gas into the gas chromatography (GC) column for separation. Molavi et al. [
20] present an HS method where gas samples are manually injected into the GC. A fused silica capillary column of mid-polarity phase (6% cyanopropylphenyl and 94% dimethylpolysiloxane) for separation of volatile compounds has frequently been used for this application [
18,
19,
21]. The first reported method [
1] required the use of liquid nitrogen to cool the GC column (T < 20 °C) in order to avoid chemical interference, present in the oil matrix, with methanol. The newly available methods [
4,
8,
18,
19,
21] do not require cryogenic focusing and the separation is typically done by temperature programming starting at 40 °C. The GC separation running time varies between 20 and 55 min depending on the method parameters.
For component identification, two types of detectors have been coupled with gas chromatography: flame ionization detection (FID) [
18] and mass spectrometry (MS) [
4,
8,
18,
19,
20,
21]. While FID is the most widely used detector for GC, MS is considered the most powerful [
22]. The initial separation method (HS-GC) is similar for the two detectors. In addition, MS allows masses to be scanned repeatedly during the chromatogram experiment. Therefore, in addition to obtaining a chromatogram that contains information about the sum of all ion abundances, known as a total-ion chromatogram, it is possible to display the mass spectrum at a particular time during the chromatogram. In addition, a single mass-to-charge (
m/
z) value can be selected and monitored through the entire GC run, a technique known as selected-ion-monitoring (SIM) [
22]. A
m/
z = 31 (CH
2OH) is associated with methanol and ethanol fragmentation.
An ethanol isotopomer, ethanol-d6, with the structure CD
3CD
2OD, has been the internal standard of choice for the alcohol methodology with MS detection [
8,
18,
19,
21]. Jalbert et al. [
19] proposed the use of ethanol-d6 to compensate for random instrumental and method fluctuations that could affect quantification. The five deuterium atoms bonded to the carbons are not solvent exchangeable. However, the deuterium bonded to the oxygen is subject to exchange. It has been experimentally observed that, once ethanol-d6 is added to oil, the deuterium on the oxygen is exchanged for hydrogen from the oil matrix. Therefore,
m/
z = 33 (CD
2OH) is associated with ethanol-d6 fragmentation in a mineral oil matrix instead of the expected
m/
z = 34 (CD
2OD) for its pure form [
19]. It is worth mentioning that the fragment of
m/
z = 33 (CD
2OH) also corresponds to the fragmentation of the molecule ethanol-d5, CD
3CD
2OH, which indicates that this molecule is also suitable as an internal standard. Nevertheless, the higher cost of ethanol-d5, which costs about twice as much as ethanol-d6, favored the use of d6 as the internal standard for alcohol determination in oil. Deuterated ethanol was chosen because it has not been identified in ageing experiments and it resembles the analytes of interest. The selected chromatographic conditions allowed adequate separation from the analytes, methanol and ethanol [
19]. In addition, this internal standard has been found useful to compensate for matrix effects observed in oxidized, high-acidity mineral oils and even ester-based oils [
23]. A prepared ethanol-d6 solution is added in a constant amount to all samples, blanks and calibration standards for quantitative analysis. Another internal standard, 2-propanol, has been proposed in the literature [
20]. Nevertheless, the proposed method was not validated to ageing experiments or field samples to confirm that 2-propanol is not generated during mineral oil ageing.
The analytical performance of the available methods for methanol detection is presented in
Table 1. The method sensitivity, expressed as the limit of detection (LOD), is approximately 1–144 ng/g. Sensitive LOD are critical for the identification of methanol in mineral oil. The majority of methods reported in
Table 1 have been validated with oil samples collected from in-service electrical equipment. All methods have an adequate linear dynamic range that covers the expected alcohol concentrations in field samples. For precision and accuracy, the values in parentheses in
Table 1 correspond to the sample concentration and the number of replicates, n. Precision expressed in terms of relative standard deviation (%RSD) is adequate (2–14%) for standards of low and high concentration. Accuracy expressed in terms of relative error (%RE), indicates that the calibration procedure for these methods is accurate (%RE < 10%). Moreover, the accuracy reported by Bruzzoniti et al. [
18] was obtained through a round robin test (CIGRE JWG A2/D1.46) involving seven international laboratories.
Ever since methanol and ethanol were identified as degradation markers [
1], there has been a need for a simple, robust, sensitive standard procedure for their determination and straightforward method translation. Two technical committees, CIGRE A2D1.46 and IEC TC10, have performed round robin tests with the participation of international laboratories.
2.2. Laboratory Ageing and Methanol Generation
The limitations on the use of 2-FAL, particularly when thermally upgraded Kraft (TUK) paper was used in power transformers, motivated the search for a more suitable chemical marker, independent of the type of paper. TUK paper is used in a high percentage of modern transformers. The first publication in this field appeared in 2007 in the journal Cellulose [
1]. In this paper, for the first time, the authors revealed a direct link between methanol formation and the scission of 1,4-β-glycosidic bonds. From laboratory ageing experiments, they revealed a linear relationship between these two parameters regardless of the type of paper: ordinary Kraft (Clupak HD75) or TUK (Manning 220) paper. In
Figure 1, the following curves were obtained from accelerated ageing of paper using sealed glass ampoules over a period of 14,000 h. The graphs in the left section in the figure correspond to the generation of methanol. These two graphs reveal the first part, which is mainly due to the depolymerization of the cellulose amorphous phase, and the second part, which has a less pronounced slope due to the depolymerization of the more stable crystalline phase. It is interesting to note that when methanol appears in the case of the TUK papers, for the same level of deterioration, its generation is higher than what is observed for standard Kraft papers. Moreover, the moisture levels seem not to influence methanol generation for an equivalent level of degradation. These data were compared with 2-FAL generation for both types of paper. On the right side of
Figure 1, the generation of 2-FAL in standard Kraft paper appears when the degree of polymerization (DP
v) is around 500 with exponential growth and is practically not observed when TUK paper is used. The term, NS, corresponds to the cellulose scission number calculated as DP
v(0)/DP
v(t)—1. DP
v(0) and DP
v(t) correspond to the degree of polymerization at an initial time (t = 0) and at a given time.
In 2011, Schaut et al. [
24] presented laboratory data showing the validity of methanol as an early paper degradation marker. The authors presented ageing data and stability studies in vials with different septum types. Despite overpressure buildup in some vials, they were able to obtain reliable data on methanol behavior. Other data obtained in glass ampoules at different temperatures and moisture contents reveal the sensitivity of methanol over 2-FAL for standard Kraft paper containing low levels of moisture. More recently, another lab performing ageing in bottles with caps reached the same conclusion: that methanol is an early-stage cellulose degradation marker [
5].
Another experimental technique to accelerate the degradation of transformer cellulose insulation consists of using an excimer laser coupled with an infrared camera to monitor the temperature reached by the samples [
25,
26]. Using this technique, the authors broke the chemical bonds of the cellulose and were able to measure the alcohol generation. Although this technique seems an unusual way of characterizing thermal ageing, the authors found that oxygen and moisture have a detrimental effect on ageing. These results suggest that the laser-induced cellulose degradation occurs through direct photolysis (i.e., direct breakage of C-C, C-O and C-H bonds), leading to radical formation, which, in turn, is believed to induce the acid hydrolysis degradation mechanism, a moisture-dependent process. Moreover, as shown in
Figure 2, scanning electron microscopy (SEM) mapping shows the deterioration of the fibers using this technique.
2.3. Kinetics of Cellulose Depolymerization and Methanol Formation
For cellulose materials used as solid transformer insulation, kinetic studies provide information about the ageing mechanism and the possibility of using estimated rate constants for lifetime prediction. As shown in
Figure 3, a relationship exists between methanol formation and the breaking of 1,4-β-glycosidic bonds in cellulose [
1]. Therefore, studies have been performed to confirm the kinetic correspondence between these two processes [
27,
28,
29,
30].
The kinetics of cellulose degradation has been extensively studied in the last 80 years, and ageing models have changed substantially. The first kinetic depolymerization model was introduced by Ekenstam [
31]; based on his early model, a pseudo-zero-order Ekenstam’s approximation, Equation 1, has been used by many researchers to track the degradation of cellulose:
DP
v(0) and DP
v(t) correspond to the degree of polymerization at an initial time (t = 0) and at a given time that approaches the LODP (leveling-off degree of polymerization); k
0 is the rate constant and t, the time. Calvini [
32] provides a thoughtful explanation of the meaning of this equation. The left-hand side (1/DP
v(t) − 1/DP
v(0)) is the number of broken bonds per anhydroglucose unit and should not be confused with the “fraction” of broken bonds. Calvini [
32] proposed to multiply the right-hand side by a concentration-dependent parameter to guarantee internal coherence. This correction factor n
0 = 617(1/LODP − 1/DP
v(0)) allows one to estimate the ageing time necessary to perform kinetic analysis by measuring the value of LODP and the initial degradation of the Ekenstam domain. In the last 20 years, three further derivations of the Ekenstam equation have been developed [
33]: (i) the Emsley model (1997) [
34], (ii) the Ding and Wang model (2008) [
35], and (iii) the Calvini model (2008) [
36]. The Emsley equation reads
in which k
a is the initial degradation rate constant, and k
2, the rate at which k
a decreases. Emsley et al. [
34] concluded that this model provided greater accuracy in the prediction of the time required to reach low DP
v values. This model was successfully applied by other authors to the study of the depolymerization of standard wood Kraft [
27] and TUK insulating papers [
28] under accelerated ageing conditions (T between 60 and 130 °C). For methanol (CH
3OH) generation during the opening of the glycosidic bonds, the authors proposed the following equation, by analogy with Equation (2).
The kinetic results revealed that the production of chain-end groups and the formation of methanol require about the same activation energy (Ea), with a similar frequency factor (lnAa). This confirms that both reactions (depolymerization and methanol formation) are governed by the same degradation pattern regardless of the type of insulating material in a transformer. The authors also observed that the severity of the ageing conditions has no effect on Ea while Aa is affected by the ageing medium. It was also noticeable that the length of the ageing test was not sufficient to allow an accurate determination of the parameters at lower temperatures (<100 °C). A significant error was observed when modeling the kinetic data for cases where the LODP was not reached, a situation that was predominant during the ageing of TUK papers. For example, over three years of ageing at 70 °C, there were no significant changes in DPv.
In 2008, Ding and Wang [
35] proposed a single first-order evolution equation in terms of the percentage DP
v loss for the cellulose degradation:
ωDP is the accumulated DPv loss of cellulose. Different values of the degradation variable correspond to different states of the cellulose: (a) ωDP = 0 corresponds to the undegraded state; (b) ωDP = 1 corresponds to the fully degraded state (i.e., failure); and (c) 0 < ωDP < 1 corresponds to the degraded state. corresponds to the capacity of the DPv degradation reservoir as defined by the constraint condition ωDP (t = tf) = 1, where tf is the time to failure of the specimen under specific experimental conditions. Ding and Wang validated this equation by fitting data from several experiments under different experimental conditions. In their experiments, they performed accelerated ageing studies of Kraft and TUK papers in mineral oil at different temperatures and various conditions including dry, moist and sealed vessel ageing.
However, Calvini [
33] emphasizes that the equation of a scientific model should satisfy at least three constraints:
internal coherence, tested through dimensional analysis;
boundary conditions, tested through their limits at t = 0 and t = ∞;
physical meaning.
Calvini analyzed the Ding and Wang Equation (4) [
33,
37] based on the above constraints; he disagreed with their kinetic approach. For the Emsley Equation (2), Calvini proposed an upgrade to the original version [
33], which met only the first two of the three constraints.
The modified Emsley Equation (5) satisfies all three constraints. On the right-hand side of Equation (5), k
a is the initial rate constant of the degradation; k
2, the rate constant at which k
a decreases; and t, the elapsed time. The internal coherence constraint allows the use of scissions per monomer (1/DP
v(t) − 1/DP
v(0)) or scissions per chain (DP
v(0)/DP
v(t) − 1), without changing the values of the constants k
a and k
2. The constant parameter (DP
v(0)/LODP − 1) does not influence the activation energy estimated by the Arrhenius law. In 2008, Calvini et al. [
36] proposed a model that applies first-order kinetic laws to weak, amorphous and crystalline regions of cellulose.
In the Calvini Equation (6), the left-hand side keeps the same meaning as in Equation (5), scissions per chain. On the right-hand side, , and correspond to the initial number of weak bonds, amorphous region bonds, and crystalline region bonds, respectively. The constants , and are the corresponding reaction rate constants, and t is the elapsed time. Equation (6) satisfies all the constraints.
To go further in the assessment of methanol for the monitoring of cellulosic condition in transformers, Jalbert et al. [
29] extended two previous studies (standard Kraft paper [
27] and TUK paper [
28]) to accelerate the opening of the bonds at higher temperatures (T between 130 and 210 °C). The goal of their study was to extend the degradation beyond the LODP and confirm if the same reaction pattern was present over a range of 60–210 °C, which would then allow the estimation of the rate constants of lower temperatures by extrapolation of the high-temperature values. They tested some of the mathematical models listed above to track the experimental data measured under accelerated conditions for depolymerization and methanol formation. Jalbert et al. [
29] explored the applicability of Equation (6) to the data obtained from the accelerated ageing studies of cellulosic insulation by looking first at the contribution of the weak links. As Calvini et al. [
38] pointed out, if weak links really exist, they should not resist a preliminary alkaline/acidic purification treatment. Therefore, the number of weak links (preoxidized groups in the β-D-glucopyranosyl units of the cellulosic material) before ageing was estimated. This was achieved by comparing the specimens’ DP
v before and after the reduction of these groups after treatment with a solution of TBAB (tert-butylamine borane) in phosphate buffer (pH 7) for 24 h. The results of these experiments [
29] show that the contributions of weak bonds on standard Kraft and TUK paper were negligible. Calvini et al. [
36] had determined that if the number of weak links can be neglected and k
a >> k
c, the kinetics of degradation of the amorphous region is affected by a deceleration corresponding to the LODP; therefore,
. If it is also considered that the opening of the bonds in the crystalline region is at its beginning (linear function from Ekenstam’s approximation [
36]), Equation (6) changes as follows [
28,
29]:
More recently, Calvini [
33,
39] showed that
might be influenced by a limitation of viscometric analysis during DP
v determination. This limitation consists of the alkaline degradation of the viscometer of oxidized spots (potentially degraded anhydroglucose units). Therefore, Equation (7) can be updated to
where
is the linearized contribution of other slower reactions that affect the kinetics of the amorphous regions. The term J gathers all the constant parameters of the unknown mechanisms and kinetics and
is the rate constant.
The applicability of the above equations to the accelerated ageing of cellulosic insulation was explored by Jalbert et al. [
29], who investigated the effect of the potential viscometric limitation mentioned by Calvini (
versus
). Changes in weight and degree of polymerization of samples from accelerated ageing at 190 °C and from a 40-year-old decommissioned transformer were measured before and after inducing alkaline β-alkoxy fragmentation (treatment with 0.01 M NaOH). Even though the authors induced the accelerated ageing at DP
v values below the papers’ LODP, there were no differences between the weight and measured DP
v after the NaOH treatment; the same results were also obtained for the transformer paper samples. Thus, the effect of the oxidation of the OH sites along the cellulosic chains and the formation of acid-stable but alkali-labile cross-links that could affect the viscometric determinations of DP
v were considered negligible. Consequently, Jalbert et al. [
29] applied Equation (7) instead of Equation (8) to their results. An example is shown in
Figure 3, which presents the two-branch patterns of the amorphous and crystalline regions. The authors concluded that solid insulation could be depolymerized to a level beyond the LODP by a random opening of the glycosidic bonds in both the amorphous and crystalline regions of the material. The main mechanism of degradation corresponds to acid hydrolysis, especially for the amorphous region. In the crystalline region, the predominant mechanism is a pyrolysis-like mechanism that lowers the DP
v below the LODP.
To track methanol generation, Jalbert et al. could not apply the three-segment model due to a fluctuation of methanol concentrations observed at longer ageing periods when the insulation has already reached its end of life (DP
v < 200). Therefore, the original Emsley version, Equation (3) was used. As observed in previous studies [
27,
28], the kinetic results revealed that the production of chain-end groups and the formation of methanol require about the same activation energy (E
a), with a similar frequency factor (lnA
a).
Table 2 shows the Arrhenius parameters obtained in that study [
29].
The Arrhenius relationship applied to the data for standard Kraft and TUK paper shows a very good linearity with 1/T for the full range of temperatures 60–210 °C. The E
a and lnA
a were in agreement with the values shown in
Table 2. The authors emphasized that, to achieve this correlation, it is important to perform ageing tests at least until LODP to avoid misleading interpretations of kinetic data. These results confirm that the degradation reaction is governed by a unique mode of degradation, which opens up the possibility of using accelerated ageing tests at higher temperatures to estimate insulation conditions at transformer operating temperatures.
Jalbert et al. [
29] also observed that the nitrogenous substances present in TUK paper significantly modified the reaction mechanism by limiting the action of water and hydronium ions coming from the oil.
Figure 4 shows the degradation scheme for standard Kraft and TUK paper in mineral oil.
2.4. Paper’s Mechanical Properties and Methanol
Paper is not a uniform material; it is essentially a mat of cellulose fibers [
40]. During the manufacturing process, a high percentage of fibers are aligned with the machine direction. This process produces an anisotropic material. As an example, the mechanical strength of paper is higher in the machine direction than in the cross-machine direction [
41]. Currently, there is no unanimously accepted parameter to establish the end of life of paper based on its degree of depolymerization or remaining mechanical strength. In some cases, a DP
v of 200 [
42,
43] or even 100 has been mentioned [
44], while in other studies 50% [
42,
45] or 20% [
46] of the original mechanical strength is considered.
Several analytical models [
47,
48,
49] are used to describe paper’s mechanical strength and elongation. These models assume a homogeneous (perfectly uniform) fiber network of cellulose, whose strength is based on the endurance of the individual fibers. Other models assume a perfect composite material reinforced with fibers [
47]. However, paper strength results based on these models present a certain divergence from the laboratory results, due mainly to the assumption that the paper has no anomalies or defects. In addition to the analytical models, other models based on numerical modeling [
41] have been developed to describe the mechanical strength of paper.
The structural hierarchy of paper is a theory developed by Kortschot [
50]. It explains the complex structures and connections that compose paper as a material. The mechanical properties of paper are based on the strengths of the different structural levels. These levels are interconnected, and diverse parameters define strength at different levels. At the molecular level, strength depends on the structural integrity of cellulose chains (cellulose’s molecular weight and degree of polymerization). At the microstructural level, the chains of cellulose form elementary fibrils, which subsequently form microfibrils, fibrils and fibers (hemicellulose and lignin). The individual strength of the different cellulose fibrils and fibers, in addition to the strength of interfiber bonds, determines the strength of paper at this level [
51,
52]. At the macrostructural level, mechanical strength is based on the uniformity of fiber orientation, distribution of mass, presence of impurities and manufacturing defects such as local changes in the grammage. Local defects in the paper produce zones called “mesostructures” where the paper presents lower-than-average mechanical strength. Mesostructures are where the paper is most likely to mechanically fail [
49,
50]. Thus, the loss of paper’s mechanical properties due to ageing starts at the molecular level with cellulose depolymerization and with subsequent reactions that produce a loss of strength at the different structural levels. Paper’s mechanical strength may also be affected by the hornification that occurs during drying. However, this mechanism may also be produced at the same time as the thermally accelerated ageing of paper [
53]. Hornification results in the formation of hydrogen bonds inside the cellulose fibers [
54]. This phenomenon decreases the fibers’ flexibility and increases their brittleness, with the consequent reduction in mechanical strength [
53,
54].
Different techniques are used to determine the changes in paper’s mechanical performance during ageing, such as tensile strength, flexural strength, folding endurance strength, edge tearing resistance and bursting strength [
55,
56,
57,
58,
59]. Tensile testing is the technique used in several studies to track the loss of mechanical strength during paper ageing [
30,
41,
45,
46,
55,
57,
59,
60,
61,
62,
63,
64,
65,
66,
67]. It has been shown that the condition of paper, when chain scission is involved, can be monitored using tensile strength as a reference [
41]. However, the negative aspects of tensile testing are its sensitivity to the water content in paper and the measurement temperature [
55,
57] and its high standard deviation of the results.
Tensile strength is also used as a reference because there are internal radial and axial forces in the interior of a power transformer [
68], such as axial Lorentz magnetic force. This force produces a contraction and stretching (vibration) of the coils, with the consequent elongation of the paper that wraps the conductors [
41]. In addition, the electrostatic force (friction) created by oil circulation also affects the strength of the external paper layers [
69]. For this reason, some papers are produced by the creping process (crepe paper) to improve their tensile elongation properties. In the pulp and paper industry, the tensile strength of paper is measured using the tensile index (Tidx). This index is defined as the breaking force per width and divided by the grammage of paper [
70]. Grammage is defined as the mass per unit area of paper, expressed as grams per square meter [
56].
Studies of thermally accelerated ageing of paper impregnated with mineral oil revealed a quasilinear, temperature-independent correlation between the mechanical strength of paper (using the zero and wide span modes) and the cellulose’s degree of polymerization [
30,
41,
45,
46,
52,
57,
61,
62,
71], as seen in
Figure 5. In the first part of the master curve, up to around DP
v = 700, for different temperatures and types of paper, the mechanical strength of paper does not decrease with the decline in degree of polymerization [
41]. This is because, at that level of ageing, paper’s mechanical strength depends on the degree of polymerization but also on other parameters such as individual fiber strength, extent of chain entanglements, degree of crystallinity and interfiber bond strength [
30,
41]. In this first part of the curve, depolymerization is also faster. This is due to easier access to the amorphous regions of cellulose, which are less well organized, to break glycosidic bonds. In the second part of the master curve, a linear correlation between the two parameters is clear. Both parameters decrease at the molecular and macrostructural levels. The mechanical strength of paper is driven by the rate of depolymerization of cellulose and the molecular weight of cellulose [
41]. The increase in ageing temperature acts as a catalyst to accelerate ageing. However, the same path is followed by the paper independent of temperature. In addition, the accumulation of points at the end of the curve marks the end of the paper’s service life at around a DP
v of 150 or 10–20% of the initial mechanical strength [
30]. Based on experimental results and this linear correlation, some equations were developed to correlate the two parameters [
46,
72]. Nevertheless, it is practically impossible to measure both parameters for power transformers in service.
Several research studies have investigated the relationship between chemical markers and degree of polymerization. Some studies found correlations between the concentration of carbon oxides and 2-FAL in oil with DP
v [
40,
66,
73,
74,
75]. These correlations present advantages, disadvantages and, in some cases, lack sensitivity in TUK papers [
74], as shown in
Figure 1. As described in
Section 2 of this manuscript, Jalbert et al. [
1] showed that there is a quasilinear correlation between methanol generation and DP
v.
Based on the previous two quasilinear correlations—the first between mechanical properties and degree of polymerization and the second between methanol and degree of polymerization—a third quasilinear correlation was deduced between the generation of methanol and the change in mechanical properties during paper ageing for a wide range of temperatures [
30,
61,
62,
76] (see
Figure 6). Again, two linear tendencies, one for Kraft and one for TUK paper, are generated between the two parameters due to the partition phenomenon (discussed in detail in
Section 5). Based on this correlation, the concentration of methanol in mineral oils can be used to estimate the level of degradation and remaining percentage of mechanical properties in Kraft and TUK papers. This correlation also connects two different structural levels of paper: the generation of methanol related to the depolymerization of cellulose chains at the molecular level and the mechanical strength of paper at the macrostructural level. The influence of temperature on this correlation is also direct. An increase in temperature accelerates the ageing of cellulose, but again cellulosic paper follows a single path during ageing. A similar correlation and similar behavior are expected at lower temperatures such as for in-service transformers. Thus, this correlation creates the possibility of developing tools to assess the state of transformers; however, it is important to emphasize that a model based on laboratory data needs to be validated with real transformer data such as during postmortem analysis.
The kinetics of the changes in mechanical strength has been studied using models that depend on cellulose depolymerization. From these kinetic studies, Arroyo et al. [
30], who used Equation (7) of this manuscript as their starting point, proposed a modified Calvini equation to track the changes in paper’s tensile strength during ageing:
= (DPvo/LODP–1), and are the initial quantity of glycosidic bonds in the amorphous and crystalline regions, respectively, and ka and kc are the reaction rate constants for the amorphous and crystalline regions.
Equation (9) was proposed based on the intrinsic and physical relationships among the molecular weight of cellulose, the formation of methanol and tensile strength. In all cases, a good regression was obtained when the kinetic models were applied using the results of accelerated ageing experiments in laboratory conditions. Using the kinetic models for the loss of mechanical strength and the Arrhenius equation, the models showed a linear dependence between the ageing temperature, even at high temperatures (190 °C), and the reaction rate constant of the change in tensile strength (for both Kraft and TUK paper) [
30]. The activation energy and frequency factor for each of three studied reactions—depolymerization, methanol formation, and loss of tensile strength—have similar orders of magnitude. The results support the intrinsic physical relationship between these parameters and the decision to use the same kinetics equation to model all three [
30].
As discussed in
Section 2.2 and
Section 2.3, acid hydrolysis, the main mechanism of ageing of cellulose impregnated in oil, randomly attacks and breaks the glycosidic bonds in cellulose chains. Each scission in the cellulose chains generates methanol and the depolymerization of the cellulose [
29]. At the same time, the depolymerization triggers different reactions that affect the integrity of the paper’s various structural levels that will result in a decrease in its mechanical strength at the macrostructural level [
30].