Esterification of Glycerol and Rosin Catalyzed by Irganox 1425: A Kinetic Comparison to the Thermal Process

Abstract

Rosin is a biomass-based chemical raw material employed in multiple industries: paper, polymers, coatings, adhesives, and more, while glycerol production has experienced a notable increment in recent decades due to it being an unavoidable by-product of the biodiesel industry. Rosin polyol esters are of high interest, and a potential route for the valorization of glycerol. In this work, we compare in detail the esterification routes of rosin triglycerides via classical, industrial thermal processes at 260–280 °C and similar processes catalyzed by Irganox 1425, a high-molecular-weight, multifunctional, phenolic, primary antioxidant produced by BASF and usually in rosin processes. Its chemical name is calcium bis(ethyl 3,5-di-tert-butyl-4-hydroxybenzylphosphonate). To this end, a novel RP-HPLC method provided us with a detailed description of the compositional evolution of the reacting media. These data have been the basis of a non-linear kinetic modeling procedure where we applied non-linear regression and numerical integration algorithms to determine the network of chemical reactions and the kinetic model of the rosin–glycerol esterification process. Furthermore, the comparison of such kinetic models and their parameters allows us to understand the kinetic effect of the addition of the homogeneous catalyst. The effect of Irganox 1425 results in a notable enhancement of the reaction rates, thus allowing for operation at lower temperatures and a reduction in side reactions as decarboxylation.

1. Introduction

Rosin is a product of the distillation of natural resins from different species of pinaceae plants, which presents in the form of a transparent amber-colored resinous mass, another bio-based source of chemicals within the framework of the second-generation biorefinery concept [1]. Therefore, rosin is a part, even if a small one, of the more than 180 million tons of lignocellulosic biomass created each year [2].

Resins or oleoresins are external secretions of the metabolism of plants, excreted, especially in coniferous plants, through intercellular channels when incisions are made in their bark. Different constituents have been characterized in them, of which approximately 90% are resin acids and the remaining 10% are neutral compounds, essentially diterpenic alcohols, sesquiterpenes, aldehydes, and hydrocarbons [1].

World rosin production in 2023 amounted to 1205 kT, with gum rosin as the most relevant product (66.7%), tall-oil rosin being the second most relevant one (31.5%), and wood rosin accounting for a small 0.8% [3]. Nowadays, China is the dominant producer of rosin (408 kT in 2023), producing mainly gum rosin, while North American countries are the second largest suppliers, with tall-oil rosin as the most relevant product (184 kT/2023) and wood rosin as a side product (10 kT/2023). Europe produces 16% of the world’s rosin (164 kT/2023 of tall-oil rosin and 28 kT/2023 of gum rosin), experiencing an important rise in the last decade (from 81 kT/2010) [3]. Rosin prices have seen many fluctuations over the years, reaching peaks of up to $3000 per ton. As of 1 January 2019, the price of gum rosin was as low as 1050 USD/ton, while in 2022, with notable fluctuations, the price of this commodity was 1205 USD per ton [4].

As rosin is composed of abietic and pimaric acids rich in conjugated and non-conjugated C-C double bonds and carboxylic acid moieties, it is possible to alter its structure, obtaining numerous derivatives. The carboxyl group is directly related to esterification reactions, salt formation, decarboxylation, and the formation of nitriles and anhydrides, etc. The olefinic system may be involved in oxidation, reduction, disproportionation, hydrogenation, and dehydrogenation reactions. Diels–Alder-type additions are the most relevant reactions as far as resin acids are concerned [5,6]. Esterification, polymerization, adduct formation, disproportionation, hydrogenation, and dehydrogenation are key reactions and processes to obtain stable rosin products less prone to oxidation [1].

One of the main polyols esterified with rosin is glycerol, which is a colorless, odorless, hygroscopic, sweet, and viscous liquid. This polyol is used for a large number of applications in different fields such as cosmetics, food, pharmaceuticals, and the chemical industry [7]. Since glycerol is a by-product of biodiesel and soap manufacturing, the supply of this compound depends on the demand for these primary products. As biodiesel production has increased in the last two decades from 3.1 to 52 million tons in the period 2005–2022, the price of the raw material in question has dropped sharply (the price of crude glycerol from biodiesel is in the 200–400 US$ per ton range; 99%+ glycerol ranks in the 860–930 US$ per ton range). Crude glycerol production was expected to reach 680,000 tons by 2024 [8].

Rosin is of notable importance in the polymer sector, via esterified products with simple alcohols and polyols such as glycerol and pentaerythritol [9] to obtain emulsifiers, with fatty acid derivatives to create novel plastizicers [10], or via polymerization to modify natural materials such as poplar wood or DMF [11,12]. Esterified rosin presents deproportionated abietates and pimarates with good oxidation resistance, scarce brittleness, a fine thermal stability and a light yellow color, together with a high softening point in the range of 85–90 °C. Apart from applications in hot-melt and pressure-sensitive adhesives, paper sizing, solder fluxes, chewing gums, and inks [13], novel applications as road marking paint ingredients [14], novel vitrimer synthesis [15], and in the controlled release of biomaterials for applications in biomedicine, the food industry and agriculture [16,17] have been developed in recent years. Traditional rosin medical applications are based on the antimicrobial, anti-inflammatory, insecticidal, anti-tumor, and anti-obesity properties of the more than 50 rosin acids in several rosins worldwide [18]. The nanoparticles of carboxymethylcellulose and gum rosin have shown to be stable and an adequate vehicle to release 5-ASA [16]. In odontology, Chuenbarn et al. studied vancomycin release from loaded rosin gels and nanoparticles, ensuring that the drug was released over 7 days, maintaining high and efficient antimicrobial activity [19]. Lin et al. synthesized a rosin-based ester tertiary amine (RETA) using Diels–Alder addition, acylation, and esterification, which was able to form emulsions with a rosin-based phosphoric acid, showing excellent stability and adequacy for the release of doxorubicin [20]. The capacity of rosin derivatives for controlled release has also been applied in agriculture for the application of insecticides via pH–temperature-responsive hydrogels [21] and in the effective control of the fungus Rhizoctonia solani using pH-responsive tannic acid–rosin nano- and microcapsules [22]. The capacity of rosin acids and their derivatives as reagents for polymerization, esterification, and other reactions, together with their antimicrobial properties, makes them ideal for bio-based food packaging [23,24].

Rosin esterification is catalyzed by a wide assortment of catalysts [25], for example, in recent years, complex homogeneous catalysts [26], heterogeneous or solid catalysts [27], and lipases [28]. Moreover, the kinetics of classical heat-driven esterification processes have been studied [29], as well as similar processes using subcritical CO₂-enriched high temperature compressed water (HTCW) to intensify the esterification [30]. In the case of heat-driven esterification or when a homogeneous catalyst is employed, simple potential expressions are common in the proposed kinetic models, usually considering elemental reactions so that the partial orders or exponents in the concentration terms of the reagents are identical to their molecularity (stoichiometric coefficient) [26,29,30]. Sun et al. studied the glycerol–rosin esterification process using an acidic ionic liquid—[BuMeim][Sc₂Cl₇]—and performed several DFT calculations to find that esterification proceeds through an in-series reaction mechanism of the acids to the triglycerides. When working with heterogeneous catalysts, the relevant data are obtained in the absence of hindrances due to mass transfer, so that they reflect the intrinsic kinetics of the reactions under study. In this sense, Zhou et al. synthetized an annealed Fe₃O₄/MOF-5 catalyst, explored its application in the esterification of rosin and glycerol, and proposed 38 possible kinetic models of the Langmuir–Hinselwood–Hougen–Watson (LHHW), Eley–Rideal (ER), and potential or pseudohomogeneous (PH) types. They found, after a complex non-linear regression analysis, that the ER model featuring a surface reaction of adsorbed glycerol was the one that fit better to a wide experimental dataset [27].

This work describes, for the first time, the detailed kinetic modeling of the esterification of rosin acids with glycerol driven by Irganox 1425, a homogeneous catalyst, and/or heat, based on detailed experimental data obtained via a novel RP-HPLC procedure and by observing water and glycerol stripping during the esterification runs. To this end, several kinetic runs were performed at various temperatures, reagent molar ratios, and catalyst concentrations, and several potential kinetic models were fitted to experimental data to select the most adequate. Afterwards, we applied in-depth statistical procedures to select the most adequate kinetic model, which included the reaction network and the kinetic equations and parameters (Figure 1).

2. Materials and Methods

2.1. Materials

Technical-grade rosin and Irganox 1435 were kindly supplied by LURESA (Coca, Segovia, Spain). The H₂SO₄ 98%, UPS-grade glycerol, far-UV HPLC gradient acetonitrile, HPLC-grade hexane, and HPLC-grade isopropanol were from Thermo Fisher Scientific (Waltham, MA, USA).

2.2. Experimental Conditions and Equipment

The objective of this work was to follow the composition of the system in detail, together with the condensate of the vapors generated during the esterification reactions between rosin and glycerol under thermal and catalytic conditions, with the idea of obtaining a detailed kinetic model arising from the results obtained using both approaches. The influence of the process variables that determine the rates of the esterification reactions of rosin with glycerol was also studied.

The variables studied to analyze thermal esterification were as follows: (i) initial glycerol concentration (C_G0) between 7% and 15% by weight relative to the initial rosin mass, and (ii) reaction temperature (T). Runs were performed between 260 and 280 °C. As the reactions proceed without the need for a solvent, the molar ratio between glycerol and rosin at zero time (M) is considered the concentration variable. The following Table 1 compiles the experimentation performed in heat-driven esterification processes:

In the case of esterification runs conducted under the catalytic action of Irganox 1425 (calcium bis(ethyl 3,5-di-tert-butyl-4-hydroxybenzylphosphonate), (i) runs were performed at 270 and 280 °C; (ii) the glycerol to rosin initial molar ratio (M) was 0.33 or 0.25; and (iii) the catalyst concentration was either 0.3 or 0.5% with respect to the weight of rosin. The following Table 2 compiles the experimentation performed in catalytic esterification processes:

The experimental set-up was composed of a 500 mL four-necked spherical flask heated with an electric mantle and overhead stirring to finely control the temperature and the stirring rate. The IKA electric motor drives a glass rod with a Teflon paddle; agitation HE maintains constant to a speed of 300 rpm. The temperature of the system was measured with a thermocouple and controlled with a PID controller that acted on the heating blanket. A cooling system was connected to the reactor in order to condense, with few losses, the vapors generated during the esterification. This part of the set-up consisted of a Liebig condenser connected to the spherical flask with a distillation head. The separation via the decantation of the hydrocarbons and water, condensed at the outlet of the cooling system and collected in the burette, allowed for the measurement of the volume of water generated and removed from the reaction medium. A Teflon tape was used for the connections to ensure greater tightness at high temperatures. The system was placed on a lifting platform that facilitated the cooling and cleaning of the reactor. Figure 2 shows a drawing of this experimental device.

2.3. Experimental Procedure

The experiments were carried out using the set-up described under isothermal conditions. First, we introduced 100 g freshly ground rosin, to prevent oxidation, through the central neck of the flask. It was heated to the desired temperature and, during the heating, stirring began when the temperature was above the resin softening point, i.e., around 85 °C.

Once the reaction liquid reached the working temperature, we withdrew a sample and considered it as the zero-time sample: pure rosin. Afterwards, the desired amount of glycerol, which we had weighed previously, was added with a glass syringe. The addition of the polyol reduced the temperature to about 250 °C, but the working temperature was recovered in a few minutes, so the effect of this decline was not deemed relevant in kinetic terms. The start of each reaction was defined by the introduction of glycerol when the working temperature was stable again.

During the run, we took liquid samples at several timepoints using a 6 mm internal diameter glass pipette and added them to a beaker containing liquid air. In this way, we obtained fragmented samples more suitable for subsequent analysis via chromatographic methods, avoiding, at the same time, any compositional modification of the sample from air oxidation. We observed that this method repeatedly proved its cleanliness and convenience, as we conveniently and easily washed the glass pipette with acetone, being ready for the next sampling without cross-contamination issues.

In addition, during the initial phase of each run, we collected the produced water and the compounds stripped by its action in a test tube placed at the end of the Liebig condenser, having weighed the graduated test tube beforehand. These condensate samples were employed to determine the amount of steam of water produced in the course of the reaction. The composition of these samples was analyzed because a certain amount of glycerin was stripped from the reaction liquid by the water vapor.

2.4. Sample Analysis

The samples of the reaction liquid collected also allowed for monitoring the reaction evolution in a qualitative way by observing the color and turbidity of the samples, as turbidity is due to the presence of the glycerol phase within the reaction medium. Dark color (brown to black) samples indicated the presence or absence of decarboxylation or oxidation products (since these reactions cause the rapid darkening of the sample).

For quantitative analyses, HPLC methods were employed using JASCO 2000 series HPLC modular equipment (JASCO Corporation, Tokyo, Japan). The preparation of the reaction samples for subsequent analysis via RP-HPLC (Reverse-phase High-performance liquid chromatography) consisted of weighing 10 mg of the sample and dissolving it in 1 mL of tetrahydrofuran in a 1.5 mL microtube. Using a 1 mL automatic pipette, we transferred the solution to vials suitable for HPLC. For condensate samples, we took 100 μL of sample and 900 μL of a 5% citric acid solution used as an internal standard in ion exclusion chromatography.

RP-HPLC analyses were conducted using a nonpolar stationary phase, a C18 250 × 4.6 Mediterranea Sea column, and a mobile phase initially containing 70% v/v acetonitrile and 30% v/v Milli Q water at zero time. We used a compositional gradient to ensure the elution of all relevant analytes [31]. In this case, the first reservoir contained water, the second one was filled with acetonitrile, while the third contained a mixture of isopropanol and hexane in a 5:4, v/v proportion. We employed a 25 min ternary gradient with two linear gradient steps starting, as stated, with a 30% A + 70% B mixture at 0 min. This eluent composition was changed to 100% B in 10 min, 50% B + 50% C from minute 10 to 20, maintaining this last eluent composition during the last five minutes. Signals at a wavelength of 254 nm obtained in a JASCO DAD MD-2015 detector (JASCO Corporation, Tokyo, Japan) were collected using Chromnav 2.0 HPLC software. A sample of chromatograms defining each analyte and its retention time is shown in Figure 3. It can be appreciated that, in this novel RP-HPLC analytical method, a certain discrimination between compounds in each compound group can be performed, though, for the purposes of this study, compounds have been pooled as rosin acid, monoglyceride, diglyceride, and triglyceride groups.

We used ion exclusion HPLC to quantify the glycerol contained in the distillate samples [32]. In brief, JASCO 2000 series equipment was employed to such effect with the following operating conditions: a constant flow rate of 0.5 mL/min of acid Milli-Q water (0.005 M H₂SO₄) of the mobile phase flowing through a Rezex ROA-Organic Acid H⁺ (8%) column (150 × 7.80 mm) at 60 °C. Here, we employed a JASCO RI-2031 detector.

2.5. Mathematical Analysis

The fitting of any of the proposed kinetic models was performed by applying a Levenberg–Marquardt algorithm for non-linear regression of the experimental data together with the numerical integration of the corresponding kinetic equations by means of a fourth-order Runge–Kutta method. First, each model was fitted to experimental data at a definite temperature. Subsequently, simultaneous correlation to all data was performed so that the parameters for the multivariable fitting would be retrieved. In this case, all kinetic constants were expressed as Arrhenius exponential equations to consider the temperature effect on such constants.

Models were discriminated on the basis of conventional goodness-of-fit statistical criteria, among which are Fischer’s F-value and Akaike’s information criterion (AIC) [33], both used as information criteria, and RMSE and PVE, as percentages, (similar to R²) to account for the goodness-of-fit of the tested models to experimental data. The value of the former is based upon a null hypothesis that considers that the model fits appropriately to the experimental data. It is defined as implemented in the Aspen Custom Modeler v14 software [34]:

(1)${\mathrm{F}}_{95}={\displaystyle \frac{{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{{\left({\mathrm{y}}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}\right)}^2}{\mathrm{K}}}}{{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\frac{\mathrm{S}\mathrm{Q}\mathrm{R}}{\mathrm{N}-\mathrm{K}}}}}$

where N is the total number of data, K is the number of parameters, and SQR is the sum of quadratic residues, defined as ${\left(y_{\exp }-y_{calc}\right)}^2$, with y_exp and y_calc referring to the experimental and calculated values of the dependent variables, respectively.

AIC links the total amount of data with the number of parameters, and is a measurement of how appropriately a statistical model fits. AIC has previously been used and applied for kinetic model discrimination and can be calculated using the following equation:

(2)$\mathrm{A}\mathrm{I}\mathrm{C}=\mathrm{N}·\ln \left({\displaystyle \frac{\mathrm{S}\mathrm{Q}\mathrm{R}}{\mathrm{N}}}\right)+2\cdot \mathrm{K}$

Additionally, the residual mean squared error (RMSE) is a customary criterion to select the most statistically appropriate model [35]. It takes into account the amount of data and parameters in the model according to the following equation:

(3)$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{\mathrm{S}\mathrm{Q}\mathrm{R}}{\mathrm{N}-\mathrm{K}}}}$

Finally, the percentage of variation explained (PVE) also gives information on the quality of fit for each measured variable [36]. It is quantified using Equation (4):

(4)$\mathrm{P}\mathrm{V}\mathrm{E}(\%)=100·\left(1-{\displaystyle \frac{{\displaystyle \sum _{\mathrm{l}=1}^{\mathrm{L}}\mathrm{S}\mathrm{S}{\mathrm{Q}}_{\mathrm{l}}}}{{\displaystyle \sum _{\mathrm{l}=1}^{\mathrm{L}}\mathrm{S}\mathrm{S}\mathrm{Q}\mathrm{m}\mathrm{e}\mathrm{a}{\mathrm{n}}_{\mathrm{l}}}}}\right)$

where SSQ_l and SSQmean_l are defined according to Equations (5) and (6), respectively:

(5)$\mathrm{S}\mathrm{S}{\mathrm{Q}}_{\mathrm{l}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{{({\mathrm{y}}_{\mathrm{i},\exp }-{\mathrm{y}}_{\mathrm{i},\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}})}^2}{{\mathrm{y}}_{\mathrm{i},\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathsf{\gamma }\mathrm{l}}}}$

(6)$\mathrm{S}\mathrm{S}\mathrm{Q}\mathrm{m}\mathrm{e}\mathrm{a}{\mathrm{n}}_{\mathrm{l}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{{({\mathrm{y}}_{\mathrm{i},\exp }-{\overline{\mathrm{y}}}_{\mathrm{i},\exp })}^2}{{\mathrm{y}}_{\mathrm{i},\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathsf{\gamma }\mathrm{l}}}}$

and

(7)${\overline{\mathrm{y}}}_{\mathrm{i},\exp }={\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{{\mathrm{y}}_{\mathrm{i},\exp }}{{\mathrm{y}}_{\mathrm{i},\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathsf{\gamma }\mathrm{l}/2}}}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{1}{{\mathrm{y}}_{\mathrm{i},\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}}^{\mathsf{\gamma }\mathrm{l}/2}}}}}$

In Equations (5)–(7), γ_j is the so-called heteroscedasticity parameter, which is a measure of the type of error in the measured variable. When the value of this parameter is not fixed, as was the case, the Aspen Custom Modeler considers γ_l = 1 by default.

The tendency of the proposed models to be more adequate rises as the value of F increases, both AIC and the RMSE decrease, and the PVE tends to 100%. In all cases, the sum of residual squares tends to zero, with the data estimated using the most adequate kinetic model being more similar to the experimental data.

3. Results

3.1. Esterification in the Absence of a Catalyst

3.1.1. Glycerol Stripping During the Esterification Process

During the esterification runs, as the process is a reactive distillation to ensure that all water is eliminated from the reaction liquid, some glycerol stripping occurs, which is the reason why glycerol is added in a certain excess in industrial practice (0.5 moles of glycerol per mol of rosin) [19]. We have followed the concentration of glycerol in the distillates or condensates recovered during each run to estimate the glycerol losses. Even though condensate volume (Table 3) is low in comparison to the overall reaction mass (up to 115 g), glycerol concentration in condensates is relatively high (Table 4), meaning that glycerol losses of up to 7.3% in comparison to the initial glycerol load occur depending on the glycerol initial concentration and esterification temperature. The cumulative losses are compiled in Table 5. The experiment M3 was repeated three times, concluding than the experimental error was lower than 5%.

3.1.2. Proposed Reaction Networks and Kinetic Models

During the esterification runs, glycerol, a very polar reactive, reacts with rosin, creating diverse glycerides and, ultimately, conducting to the desired triglyceride product given the excess of glycerol in the reaction mixture [37]. As we have observed previously [4], glycerol solubility in rosin increases with temperature till reaching a plateau at 260 °C, with 0.5 mol of glycerol per liter of liquid, which equals a glycerol/rosin molar ratio of 0.14, notably lower than the molar ratio of industrial relevance (M = 0.5) or stoichiometric molar ratios (M = 0.33) that are explored in this work. Therefore, at zero time and during the first moments of each run, a biphasic reaction medium exists, turning into a monophasic one as ester products accumulate in the liquid [4]. In this sense, we have hypothesized that esterification reactions can happen not only in series, of the acid to the lumped monoglycerides (r1 in Figure 3); of the monoglycerides to the diglycerides, taken together (r2 in Figure 3); and of these later ones to the final triglyceride (r3), but also in parallel. The parallel reactions could be r4, suggesting that two acid molecules combine with one glycerol molecule, probably in the 1 and 3 positions of the polyol, to render a monoglyceride. Reaction r5 suggests that, once the monoglycerides are formed—probably the ones featuring an acid moiety in position 2 as they solubilized easily in the rosin phase—they can react with two acid molecules to create the triglyceride molecule. These reactions and the corresponding main chemical compounds involved are shown in the following Figure 4:

3.1.3. Kinetic Model Discrimination

Given the possibilities that emanate from the general reaction network in Figure 3, several kinetic models were proposed. Model 1 is the simplest one and involves only in-series esterification of the acid to the triglyceride, with monoglycerides (considered as only one compound) and diglycerides (taken together) as reaction intermediates; thus, model 1 only considers reactions r1, r2, and r3. Model 2, instead, also considers the reaction between one glycerol molecule and two acid molecules, reaction r4. Model 3 suggests that glycerol and rosin acids react in such a way that they are limited by the presence of two phases, probably in or near the interface of the rosin phase and glycerol phase, while the monoglycerides solubilize rapidly in the rosin phase and react with two acid molecules to form the triglyceride (r5), apart from reacting with only one acid molecule and forming diglycerides (r2). Finally, model 4 considers all possible reactions.

The reaction rates are calculated via Equations (8)–(12). As a first approach, to discriminate between the suggested kinetic models, we have considered that the reactions are elemental and that, therefore, simple potential kinetic equations are enough to relate each reaction rate to the concentration of the relevant reagents with their partial reaction orders identical to their stoichiometric coefficient. It should be noted, however, that these kinetic models are empirical in nature, not a result of a deduction from the mechanism, in particular those kinetic equations where the global order is 3 (2 as the partial order for the acid, 1 as the partial order of the chemical species reacting with acid molecules). These empirical kinetic models are relevant to the industrial process conditions tested here.

(8)${\mathrm{r}}_1={\mathrm{k}}_1\cdot {\mathrm{C}}_{\mathrm{A}}\cdot {\mathrm{C}}_{\mathrm{G}}$

(9)${\mathrm{r}}_2={\mathrm{k}}_2\cdot {\mathrm{C}}_{\mathrm{MG}}\cdot {\mathrm{C}}_{\mathrm{A}}$

(10)${\mathrm{r}}_3={\mathrm{k}}_3\cdot {\mathrm{C}}_{\mathrm{DG}}\cdot {\mathrm{C}}_{\mathrm{A}}$

(11)${\mathrm{r}}_4={\mathrm{k}}_4\cdot {{\mathrm{C}}_{\mathrm{A}}}^2\cdot {\mathrm{C}}_{\mathrm{G}}$

(12)${\mathrm{r}}_5={\mathrm{k}}_5\cdot {\mathrm{C}}_{\mathrm{MG}}\cdot {\mathrm{C}}_{\mathrm{A}}^2$

where C_A, C_G, C_MG, and C_DG refer to the concentrations of rosin acids, glycerol, monoglycerides, and diglycerides in the reaction liquid, in mol/L. The kinetic constants for reactions r1, r2, r3, r4, and r5 are k₁, k₂, k₃, k₄, and k₅, respectively.

After a first experiment-to-experiment fitting of the proposed kinetic models, we obtained values for each relevant kinetic constant at each tested reaction temperature. The linearized form of the Arrhenius equation for each of them allowed us to compute the activation energy and the preexponential factor using linear regressions. For the subsequent multi-temperature kinetic model fitting to all experimental data at one time, we expressed each kinetic constant in terms of the Arrhenius Equation (13).

(13)${\mathrm{k}}_{\mathrm{i}}={\mathrm{k}}_{\mathrm{i}0}\cdot \exp \left[-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{a}\mathrm{ki}}}{\mathrm{R}\mathrm{T}}}\right]=\exp \left[{\mathrm{lnk}}_{\mathrm{i}0}-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{a}\mathrm{ki}}}{\mathrm{R}}}\cdot {\displaystyle \frac{1}{\mathrm{T}}}\right]$

where k_i refers to any kinetic constant, k_i0 is the preexponential factor of such a constant, and E_{a ki} is the activation energy of the constant.

Table 6 compiles the values of the retrieved kinetic constants, in terms of their respective neperian logarithms of the preexponential factors and activation energies, for each of the models, together with their standard errors. Table 7, likewise, depicts the goodness-of-fit parameters of the proposed kinetic models. We can appreciate that the most complex model, model 4, fits better to the experimental data considering the RMSE, AIC, and SQR. However, model 3, despite being simpler, has only slightly worse values for these goodness-of-fit parameters, and a much higher value for Fisher’s F at 95% confidence (F-value), showing a fine fit to all datasets with a reduced complexity. Therefore, we select this model, which is in bold typesetting in Table 6, as the most adequate from a statistical point of view. Moreover, all kinetic parameters in this model show a lower standard error in comparison to the kinetic parameters of the other models. Figure 5 and Figure 6 show the fine fit of model 3 to all datasets together.

Regarding the absolute values of k₂ and k₄ for this model, it can be observed that the ratio k₄/k₂ is approximately 5, so even if r4—the reaction of monoglycerides to the triglyceride—cannot be shunned, the in-series reaction of monoglycerides to diglycerides (r2) is more relevant, as expected. As for the activation energies of the kinetic constants of reactions r1 to r4, as indicated in Table 6, they are in order, 86.16 kJ/mol (r1), 81.10 kJ/mol (r2), 88.49 kJ/mol (r3), and 88.51 kJ/mol (r4). These values indicate that, as to be expected, all these thermal esterification reactions are notably dependent on temperature.

3.2. Esterification in the Presence of Irganox 1425

Once we defined the reaction network of the heat-driven esterification of rosin and glycerol supported on a novel reverse-phase HPLC analytical method adapted from biodiesel process analytics and based on non-linear statistics, we studied the catalytic activity of Irganox 1425 (calcium bis[monoethyl(3,5-di-tert-butyl-4-hydroxylbenzyl)phosphonate]) in this esterification process. This compound is manufactured by BASF. It is a high-molecular-weight and multifunctional phenolic primary antioxidant able to offer high thermal and oxidative stability in polymerization processes including, but not limited to, those conducting to polyolefins, cross-linked elastomers, and polyesters. Experimentally, we observed that in similar conditions, the addition of this compound in relatively low percentages relative to rosin (0.3 and 0.5%) led to a 100–150% increase in the reaction rates and process times, thus revealing its notable capacity as a low-cost homogeneous catalyst for this process.

Esterification runs were performed in industrially relevant conditions: at temperatures in the 270–280 °C range and catalyst–antioxidant concentrations of up to 0.5% relative to rosin mass. The kinetic model 3 was fitted to data retrieved at each temperature and to all data at the same time (8 runs). Before conducting the model fitting, we slightly modified the kinetic equation of the first reaction so that it followed simple hyperbolic kinetics. The proposed reaction rate is empirical (Michaelis–Menten assimilation). This is explained by the relatively low solubility of glycerol in rosin. The constant K present in the denominator of the reaction rate equation for r1 indicates that, at a high glycerol concentration, it forms a second liquid phase, and this polar compound migrates into the rosin phase to be esterified, probably in the nearby region of the interphase, given the amphiphilic nature of the catalyst. At a low glycerol total concentration, which happens during the esterification process [4], only one liquid phase is present. Therefore, as probed by us in a previous work [4], the reaction system turns from a monophasic one into a biphasic one.

Preliminary non-linear regressions conducted with data at a constant temperature and at all temperatures indicated that K_T and K_cat were almost independent of temperature, so constant values could be used for both. Surprisingly, the kinetic constant of the parallel reaction (r5) did not change with the catalytic amount, and depended only on temperature, suggesting the non-catalytic nature of this particular reaction. Therefore, reaction r5 was considered to be only a heat-driven reaction. This fact could be explained by the large volume of the catalyst molecule that can lead to steric hindrances in the reaction of hydroxyl moieties of glycerol in positions C1 and C3 with rosin acid carbocations, which probably could not be attached to the glycerol molecule at the same time with a bulky catalyst molecule nearby. The constant K is split into a K_T term for the thermal part of the kinetic model and K_cat for the catalytic term. After certain simplifications during the preliminary statistical analysis of the catalytic data, the kinetic model contains the following equations:

(14)${\mathrm{r}}_1={\displaystyle \frac{\left[{\mathrm{k}}_{1\mathrm{T}}+{\mathrm{k}}_{1\mathrm{c}\mathrm{a}\mathrm{t}}\cdot {\mathrm{C}}_{\mathrm{c}\mathrm{a}\mathrm{t}}\right]\cdot {\mathrm{C}}_{\mathrm{A}}\cdot {\mathrm{C}}_{\mathrm{G}}}{1+\left[{\mathrm{K}}_{\mathrm{T}}+{\mathrm{K}}_{\mathrm{c}\mathrm{a}\mathrm{t}}\cdot {\mathrm{C}}_{\mathrm{c}\mathrm{a}\mathrm{t}}\right]\cdot {\mathrm{C}}_{\mathrm{G}}}}$

(15)${\mathrm{r}}_2=\left[{\mathrm{k}}_{2\mathrm{T}}+{\mathrm{k}}_{2\mathrm{c}\mathrm{a}\mathrm{t}}\cdot {\mathrm{C}}_{\mathrm{c}\mathrm{a}\mathrm{t}}\right]\cdot {\mathrm{C}}_{\mathrm{MG}}\cdot {\mathrm{C}}_{\mathrm{A}}$

(16)${\mathrm{r}}_3=\left[{\mathrm{k}}_{3\mathrm{T}}+{\mathrm{k}}_{3\mathrm{c}\mathrm{a}\mathrm{t}}\cdot {\mathrm{C}}_{\mathrm{c}\mathrm{a}\mathrm{t}}\right]\cdot {\mathrm{C}}_{\mathrm{DG}}\cdot {\mathrm{C}}_{\mathrm{A}}$

(17)${\mathrm{r}}_5={\mathrm{k}}_{5\mathrm{T}}\cdot {\mathrm{C}}_{\mathrm{MG}}\cdot {\mathrm{C}}_{\mathrm{A}}^2$

Results from the fit of the model to all datasets together are collected in Table 8 (the kinetic constants as their preexponential terms and activation energies) and Table 9 (the goodness-of-fit parameters). The information collected in this last table indicates that the model is able to fit with success to all data together, with low values of SQR, AIC, and RMSE and high values of Fisher’s F at 95% confidence, much higher than this parameter threshold at this confidence level, which is 1.75. This fitting bounty can also be appreciated in Figure 7 and Figure 8, where the points are the experimental data, and the lines indicate the values simulated with the kinetic model.

4. Discussion

In this research, we firstly studied the esterification of glycerol and rosin at temperatures in the industrially relevant range: from 260 to 280 °C, avoiding higher temperatures where decarboxylation happens (from 285 to 300 °C) [29], so that byproducts such as carbon dioxide and rosin oil (a mixture of decarboxylated abietic and pimaric acids) were avoided. This is reasonable, as industrial conditions are set to avoid these byproducts, which worsen both product color—turning it much darker—and softening point—reducing it before standard values around 80 °C, as measured by the ball test. In fact, decarboxylation is relevant at high temperatures due to its high activation energy: 231 kJ/mol [29].

Before selecting the appropriate reaction network, the stripping of glycerol due to the existing water product has been analyzed, in terms of condensate volume and glycerol concentration in samples withdrawn during each esterification process. We observed that, even when glycerol is in excess and the process temperature is high, glycerol withdrawal during the process is less than 8% in mass, with temperature being the process variable most relevant for glycerol stripping. Therefore, to operate at lower temperatures (for example, 260 or 270 °C [30], or even a lower temperature—220 °C—in the presence of the adequate catalyst [26]) reduces the energy input while avoiding the need for glycerol excess, especially if some reflux is implemented in the reactor.

In this sense, it is logical to consider kinetic models that only take into account the reactions in the network, starting with a series of consecutive reactions of the acid—and glycerol—to the triglyceride via mono- and diglyceride(s), r1–r3 [26,31]. However, Wang et al. suggest the occurrence of side reactions of the diglycerides to diglyceride dimers via etherification reactions of C2-C2 of adjacent glycerol molecules [9], although these authors do not fit the corresponding kinetic model to the experimental data, only suggesting the reaction network.

In a previous work [5], we observed that two phases coexist at a low reaction time, with glycerol being a polar phase dispersed in the rosin hydrophobic phase. This biphasic system turns into a monophasic one as the reactions proceed and the glycerol concentration reduces. In this work, we proposed a network with three simple in-series esterification reaction where the first reaction took place at a constant glycerol concentration (a so-called critical concentration) till a certain glycerol conversion, where the pseudo-first-order kinetic equation turned into a second-order kinetic equation with first-order exponents for both rosin acid and glycerol concentrations. However, in that previous work, we only considered data from rosin acid, as its concentration was measured via a SEC-HPLC method, with a much lower peak resolution (except for the acid peak at the end of each chromatogram). Here, studying again the thermal process at several temperatures and glycerol/rosin molar ratios, we proposed to choose between diverse reaction networks, also considering side reactions of glycerol and acid to a 1,3-diglyceride (r4) and of 2-monoglycerides dissolved in the rosin phase to the triglyceride. As stated previously, model 3, containing r1, r2, r3, and r5, was selected based on its goodness-of-fit parameters and low errors in the kinetic constants (both the neperian logarithms of the preexponential factors of the kinetic constants and their activation energies).

The most complex kinetic model, model 4, presented high error values for kinetic parameters linked to k₄, the kinetic constant of r4, suggesting it to be negligible. These activation energies from r1 to r4 were 86.16 kJ/mol (r1), 81.10 kJ/mol (r2), 88.49 kJ/mol (r3), and 98.30 kJ/mol (r4). These values are notably higher than the values obtained in other works: 50–64 kJ/mol [30], though in the range of activation energies for kinetic constants for the thermal esterification of rosin and pentaerythritol: from 65 to 233 kJ/mol depending on the reaction and the reference [29]. Finally, we have calculated their value in the 260–280 °C temperature interval via the Arrhenius equations of the kinetic constants in model 3. We appreciated that the r5 rate (from MG to TG) is 4.8–5.0 lower than the r2 rate (from MG to DG), thus suggesting a much lower extension of the side reaction compared to its consecutive counterpart. Even so, the presence of this side reaction under dehydrating conditions (reactive distillation), as we have performed in this research, is a possibility observed in esterification–acylation reaction systems producing triacetylglycerol [38].

Regarding the catalytic runs, our results using Irganox 1425 as a homogeneous catalyst led to the conclusion than this antioxidant, employed on an industrial scale for rosin–polyol esterifications, increases reactions rates by two- to threefold. Irganox 1425’s chemical name is calcium bis[monoethyl(3,5-di-tert-butyl-4-hydroxylbenzyl)phosphonate] and it has been used, at least since the first few years of this century, as a catalyst for rosin esterification [39]. Metal hybrid phosphonates, which are usually solid catalysts and, therefore, prone to be recycled and used in flow, have shown a wide applicability to diverse oxidation, esterification, dehydration, and coupling reactions [40]. In rosin esterification with glycerol, and in terms of turnover frequency (TOF), Irganox 1425 activity, using abietic acid concentration decreases in the linear region (till 25 min) as a starting point for calculations, is 0.124 s⁻¹ at 270 °C and 0.170 s⁻¹ at 280 °C. This activity is relatively low compared to natural zeolite (2.13 s⁻¹) and zeolite ZSM-5 (1.02 s⁻¹) at 250 °C [41], or with LaZSM-5 (0.76 s⁻¹) at 260 °C [42]. FC3R (spent fluid cracking catalyst) alone or doped with ZnO increases reaction rates by 2 and 5 times, respectively, at catalyst dosages of 1.6% with respect to rosin (about 5 times more than Irganox 1425 in this work) [9]. Thus, the activity of these solid catalysts seems to be lower in terms of catalyst mass when compared to Irganox 1425, though mass transfer limitations can be expected in FC3R, whose conical particles have an equivalent diameter of 47 mm.

When assessing the catalytic activity of Irganox 1425, the kinetic modeling of diverse datasets retrieved at various process temperatures, the catalyst concentrations and glycerol to rosin molar ratio considered all together was performed via a non-linear regression (Marquardt–Levenberg) coupled to a fourth-order Runge–Kutta integration of the kinetic equations. In model 3, in this case, the reaction r1 rate was considered to follow a simple hyperbolic equation similar to that of the Michaelis–Menten model, where order 1 for glycerol is the limit at a low glycerol concentration, while an asymptotic maximal value for r1 is accepted at high glycerol concentrations (initial time of the esterification process). All other reaction rates were second-order potential kinetic equations, with partial first order for each of the reagents. Furthermore, the kinetic modeling suggested that reaction r5 (MG→TG) was only heat-driven, so it was not catalyzed by the bulky Irganox 1425 (694.83 g/mol), which is probably due to steric hindrances [26]. A revision of the kinetic values at temperatures from 270 to 280 °C indicates that the catalytic term doubles the heat-driven contribution in reactions r1, r2, and r3, thus featuring the importance of adding Irganox 1435 to accelerate the in-series reactions of the reaction network.

Taking into account the kinetic constants, it is interesting to appreciate that the activation energy values for the thermal terms of the reaction rates varied from 56 to 114 kJ/ mol, while the catalytic action of Irganox 1435 was highly dependent on temperature: the catalytic terms of the kinetic constants presented activation energies in the 122–205 kJ/mol interval. With other catalysts, slightly lower activation energies could be appreciated. For example, they varied from 89.55 ± 13.33 to 103.28 ± 10.84 kJ/mol with an annealed Fe₃O₄/MOF-5 [28]. When using carbon dioxide as a homogeneous catalyst in a super/subcritical CO₂-enriched high-temperature liquid water (HTLW), whose reaction media is slightly acidic (pH from 3.49 to 3.70), the observed activation energy was 62.45 kJ/mol [43]. Zhang et al. observed similar values for the activation energies when using subcritical CO₂ systems for rosin and glycerol esterification: values ranged from 44.42 to 54.92 kJ/mol [31].

5. Conclusions

This work is devoted to the detailed study of the kinetics of esterification involving rosin acids and glycerol in the absence (thermal conditions) and presence of Irganox 1425 (calcium bis(ethyl 3,5-di-tert-butyl-4-hydroxybenzylphosphonate). All the experimental conditions are industrially relevant: 260–280 °C, a glycerol to rosin ratio of 0.33–0.5, and Irganox 1425 concentrations from 0.3 to 0.5% g/g rosin. A novel RP-HPLC method was developed to allow for the detailed quantitative analysis of all compounds and possible reactions involved, while glycerol and water stripping during the reactive distillation runs were analyzed to understand their effects on the intrinsic kinetics, which were negligible. After statistically selecting between four possible kinetic models, the best one featured a classical three-esterification reaction in-series scheme of the acids to the triglycerides, with an added parallel reaction of the monoglycerides to the final triglycerides via the simultaneous esterification of the two remaining hydroxyl moieties. This reaction proceeded at a lower rate than the other three. The addition of 0.3–0.5% Irganox 1425 resulted in a two- to threefold increase in all the reaction rates apart from the one related to the reaction of monoglycerides to triglycerides, which seemed not to be catalyzed by Irganox 1425, probably due to its high molecular volume.

Author Contributions

Conceptualization, F.G.-O. and M.L.; data curation, J.G.M., N.R.-A., and M.L.; formal analysis, M.E.G. and M.L.; funding acquisition, F.G.-O., M.E.G., and M.L.; investigation, J.G.M. and N.R.-A.; methodology, F.G.-O. and M.L.; project administration, F.G.-O., M.E.G., and M.L.; resources, M.L.; software, J.G.M.; supervision, F.G.-O., M.E.G., and M.L.; validation, M.E.G. and M.L.; writing—original draft, J.G.M., N.R.-A., and M.L.; writing—review and editing, F.G.-O., M.E.G., and M.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data on the concentrations of the relevant compounds are included in all figures, while the experimental conditions and calculation results are compiled in tables, but any original data are available upon request.

Acknowledgments

The authors also want to express their deepest gratitude to LURESA for the generous gift of industrial rosin and the antioxidant Irganox 1425.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Reaction network depicting all possible esterification reactions between abietic acid (as the main component of gum rosin) and glycerol (in blue), and a side reaction taking place at the highest process temperature, rosin acid decarboxylation (in black). As can be noted, 1- and 2-abietic monoglycerides coexist, while 1,2 and 1,3-diglycerides can be formed in a network comprising in-series and parallel esterification reactions. In this work, both monoglycerides are lumped together, and both diglycerides are considered together, to define simpler reaction networks.

Figure 2 Experimental set-up composed of a spherical flask and a heating mantle connected to a PID temperature controller and an overhead stirring system to control the agitation speed. Vapors are collected during the reaction with a Liebig condenser and a graduated test tube to measure the volume of condensates during the reaction.

Figure 3 Samples from RP-HPLC chromatograms corresponding to an experiment conducted at 270 °C with a glycerol to rosin molar ratio (M) of 0.33 in the absence of a catalyst. Yield of monoglycerides is 2–5%, of diglycerides is 8–15%, and of triglycerides is 80–90%, as expected in a reactive process coupled with water withdrawal via distillation.

Figure 4 Potential reaction networks explaining the irreversible esterification reactions of rosin and glycerol in a reactive distillation process with the removal of water.

Figure 5 Fit of model 3, featuring two in-series reactions of glycerol and rosin acids to the triglycerides via mono- and diglycerides, and a parallel reaction of monoglycerides to triglycerides at several glycerol/rosin acid molar ratios at 270 °C: (A) M = 0.23 (M6); (B) M = 0.33 (M3); and (C) M = 0.5 (M7). Experimental data are shown as points, while the lines in the same color indicate the fit of the selected kinetic model: ■ rosin acids; ▲ monoglycerides; ◆ diglycerides; and ▶ triglycerides.

Figure 6 Model 3 fit to thermal esterification runs at M = 0.33 and several temperatures: (A) T = 260 °C (M1); (B) T = 265 °C (M2); (C) T = 270 °C (M3); (D) T = 275 °C (M4); and (E) T = 280 °C (M5). Experimental data are shown as points, while the lines in the same color indicate the fit of the selected kinetic model: ■ rosin acids; ▲ monoglycerides; ◆ diglycerides; and ▶ triglycerides.

Figure 7 Fit of the final kinetic model (model 3) with both the thermal and catalytic terms to experimental runs conducted at 270 °C. (A) M = 0.25; C_cat = 0.3% (C1); (B) M = 0.25; C_cat = 0.5% (C2); (C) M = 0.33; C_cat = 0.3% (C3); and (D) M = 0.33; C_cat = 0.5% (C4). Experimental data are shown as points, while the lines in the same color indicate the fit of the selected kinetic model: ■ rosin acids; ▲ monoglycerides; ◆ diglycerides; and ▶ triglycerides.

Figure 8 Fit of the final kinetic model (model 3) with both the thermal and catalytic terms to experimental runs conducted at 280 °C. (A) M = 0.25; C_cat = 0.3% (C5); (B) M = 0.25; C_cat = 0.5% (C6); (C) M = 0.33; C_cat = 0.3% (C7); and (D) M = 0.33; C_cat = 0.5% (C8). Experimental data are shown as points, while the lines in the same color indicate the fit of the selected kinetic model: ■ rosin acids; ▲ monoglycerides; ◆ diglycerides; and ▶ triglycerides.

Table 1

Experimental conditions of esterification runs performed without the addition of any catalyst.

Experiment	T (°C)	M (CG0/CC0)
M1	260	0.33
M2	265	0.33
M3	270	0.33
M4	275	0.33
M5	280	0.33
M6	270	0.23
M7	270	0.5

Table 2

Experimental conditions of esterification runs driven by the catalyst Irganox 1425 at diverse temperature, molar ratio, and catalyst concentration values.

Experiment	T (°C)	M (CG0/CC0)	Ccat (%)
C1	270	0.25	0.3
C2	270	0.25	0.5
C3	270	0.33	0.3
C4	270	0.33	0.5
C5	280	0.25	0.3
C6	280	0.25	0.5
C7	280	0.33	0.3
C8	280	0.33	0.5

Table 3

Condensate volume recovered during the rosin–glycerol thermal esterification runs over the whole reaction time divided into time periods (for example, from time 0 to time 10, from time 10 to time 20, etc.).

Time (min)	Vcondensate (μL)
	M1	M2	M3	M4	M5	M6	M7
10	430.2	112.7	102.0	566.8	602.9	111.0	154.9
20	321.7	426.8	343.9	311.8	852.3	98.9	251.3
30	217.4	344.7	228.5	212.8	450.9	93.1	466.7
60	761.4	769.9	893.9	788.9	856.6	467.8	918.9
120	655.9	430.9	180.8	521.4	279.0	288.8	597.0
420	124.0	162.1	30.1	156.0	282.3	91.4	174.0

Table 4

Glycerol concentration, in grams per liter, of the condensate samples reflected in Table 3 for certain time periods along with the reaction times for thermal rosin–glycerol esterification runs.

Time (min)	CG (g/L)
	M1	M2	M3	M4	M5	M6	M7
10	79.87	103.81	129.13	144.17	241.37	91.86	236.47
20	61.70	126.07	161.99	206.74	213.92	118.92	260.16
30	117.68	158.27	209.98	190.38	234.86	218.53	231.90
60	131.20	142.47	142.04	185.57	239.94	170.47	257.56
120	127.04	142.77	180.61	165.81	197.38	144.70	256.17
420	120.09	113.57	136.14	129.77	162.47	49.35	216.16

Table 5

Glycerol cumulative losses in the condensates in percentage compared to the initial mass of glycerol in each esterification run.

Time (min)	CG/CG0 (%)
	M1	M2	M3	M4	M5	M6	M7
10	0.081	0.145	0.444	0.813	1.448	0.146	0.241
20	0.292	0.833	0.848	1.455	3.262	0.314	0.671
30	0.560	1553	1.190	1.858	4.316	0.604	0.925
60	1.727	2.644	2.270	3.314	6.361	1.744	2.482
120	1.955	3.419	3.201	4.175	6.909	2.341	3.489
420	1.991	3.638	3.342	4.376	7.365	2.405	3.613

Table 6

Results of the fitting of the proposed kinetic models to the whole dataset of runs performed in the absence of catalysts (thermal process). Here, k₁₀, k₂₀, and k₃₀ are expressed in [L/(mol·min)], while k₄₀ and k₅₀ units are [L²·/(mol²·min)], and E_{a k1,k2,k3} units are in [kJ/mol].

Parameter	Value	Std. Error	Parameter	Value	Std. Error
Model 1
Ln (k10)	13.62	1.61	Ea k1/R	10285	641.9
k10	8.22·105	5.00	Ea k1	85.47	5.33
Ln (k20)	13.82	1.72	Ea k2/R	9026	5229
k20	1.00·106	5.58	Ea k2	75.01	43.45
Ln (k30)	13.81	0.53	Ea k3/R	9998	20.95
k30	9.95·105	1.70	Ea k3	83.08	0.17
Model 2
Ln (k10)	13.81	8.08	Ea k1/R	10616	214.5
k10	9.95·105	3229	Ea k1	88.22	1.78
Ln (k20)	14.22	2.23	Ea k2/R	9794	81.08
k20	1.50·106	9.30	Ea k2	81.39	0.67
Ln (k30)	14.5	5.92	Ea k3/R	10353	388.6
k30	1.98·106	372.4	Ea k3	86.03	3.23
Ln (k40)	14.75	1.62	Ea k4/R	11829	1586
k40	2.55·106	5.05	Ea k4	98.30	13.18
Model 3
Ln (k10)	13.81	0.24	Ea k1/R	10368	9.603
k10	9.95·105	1.27	Ea k1	86.16	0.08
Ln (k20)	14.22	1.77	Ea k2/R	9759	67.39
k20	1.50·106	5.87	Ea k2	81.10	0.56
Ln (k30)	14.5	0.91	Ea k3/R	10649	33.91
k30	1.98·106	2.48	Ea k3	88.49	0.28
Ln (k50)	14.73	1.61	Ea k5/R	10651	59.48
k50	2.50·106	5.00	Ea k4	88.51	0.49
Model 4
Ln (k10)	13.81	6.4	Ea k1/R	10461	25.18
k10	9.95·105	601.8	Ea k1	86.93	0.21
Ln (k20)	14.22	2.36	Ea k2/R	9993	90.14
K20	1.50·106	10.6	Ea k2	83.04	0.75
Ln (k30)	14.5	1.02	Ea k3/R	10635	38.13
k30	1.98·106	2.8	Ea k3	88.38	0.32
Ln (k40)	14.73	10.88	Ea k4/R	12234	9984
k40	2.50·106	53103	Ea k4	101.66	82.97
Ln (k50)	14.51	1.82	Ea k5/R	10607	68.31
k50	2.00·106	6.2	Ea k3	88.14	0.57

Table 7

Goodness-of-fit parameters for the discrimination of proposed kinetic models fitted to all data in thermal esterification runs. Here, N refers to the number of data, while K is the number of kinetic parameters in the respective kinetic model.

Parameter	Model 1	Model 2	Model 3	Model 4
N	308	308	308	308
K	6	8	8	10
SQR	1.087	0.844	0.727	0.661
RMSE	0.594	0.520	0.486	0.463
F-value	30247	28838	44369	28840
AIC	−1727	−1801	−1847	−1872
PVE	97.78	98.45	99.17	99.24

Table 8

Kinetic parameters for the selected kinetic model (model 3) fitted to all data in esterification runs catalyzed by Irganox 1425 at one time.

Parameter	Value	Std. Error	Parameter	Value	Std. Error	Parameter	Value	Std. Error
ln (k10CAT)	28.30	3.540	Ea k1CAT/R	15,393	1937	KCAT	1.795	0.393
ln (k10T)	13.32	3.615	Ea k1T/R	10,128	1970	KT	0.024	0.010
ln (k20CAT)	27.70	5.391	Ea k2CAT/R	14,759	2953
ln (k20T)	17.03	4.450	Ea k2T/R	11,286	2436
ln (k30CAT)	44.67	5.085	Ea k3CAT/R	24,685	2792
ln (k30T)	20.02	5.417	Ea k3T/R	13,706	2968
ln (k50T)	7.289	3.100	Ea k5T/R	6737	2428
k10 CAT	1.95·1012		34.47	Ea k1 CAT		127.9	16.10
k10 T	6.07·105		37.15	Ea k1 T		84.16	16.37
k20 CAT	1.07·1012		219.4	Ea k2 CAT		122.7	24.54
k20 T	2.48·107		85.63	Ea k2 T		93.78	20.24
k30 CAT	2.43·1019		161.6	Ea k3 CAT		205.1	23.20
k30 T	4.96·108		225.2	Ea k3 T		113.9	24.66
k50 T	1.46·103		22.20	Ea k5 T		55.98	20.18

Table 9

Goodness-of-fit parameters relative to the final kinetic model for the esterification of rosin and glycerol catalyzed by Irganox 1425.

Parameter	N	K	SQR	RMSE	F-Value	AIC	PVE (%)
	406	16	1.315	0.0513	10,097	−2163	99.48

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