Modeling of Global and Individual Kinetic Parameters in Wheat Straw Torrefaction: Particle Swarm Optimization and Its Impact on Elemental Composition Prediction

Abstract

With the growing demand for sustainable energy solutions, biomass torrefaction has emerged as a crucial technology for converting agricultural waste into high-value biofuels. This work develops dual kinetic modeling using global and individual parameters combined using particle swarm optimization (PSO) to predict energy densification based on elemental composition (CHNO) and high heating values (HHVs). The global parameters are calculated from experiments conducted at 250 °C, 275 °C, and 300 °C, and the individual parameters are obtained by adjusting experimental points at each temperature. A two-step kinetic model was used and optimized to achieve exceptional adjustment accuracy (98.073–99.999%). The experiments were carried out in an inert atmosphere of nitrogen with a heating rate of 20 °C/min and a 100 min residence time. The results obtained demonstrate a crucial trade-off: while individual parameters provide superior accuracy (an average fit of 99.516%) for predicting degradation by weight loss, global parameters offer better predictions for elemental composition, with average errors of 2.129% (carbon), 1.038% (hydrogen), 9.540% (nitrogen), and 3.997% (oxygen). Furthermore, it has been found that by determining the kinetic parameters at a torrefaction temperature higher than the maximum peak observed in the derivative thermogravimetric (DTG) curve (275 °C), it is possible to predict the behavior of the process within the 250–325 °C range with an R-squared value corresponding to an error lower than 3%. This approach significantly reduces the number of required experiments from twelve to only four by relying on a single isothermal condition for parameter estimation.

1. Introduction

Currently, clean energy sources are being utilized to reduce the environmental challenges caused by the extensive use of fossil fuels. Therefore, biomass residues have high potential as a sustainable alternative [1,2]. There are different processes employed to transform biomass into energy, including biochemical and thermochemical processes [3]. The first process is carried out through the degradation of organic matter by microorganisms. In contrast, the second is carried out under different pressure and temperature conditions to obtain solids, liquids, and gases through thermal decomposition. The main thermochemical technologies for preheating biomass are torrefaction and pyrolysis; in these processes, the material is degraded in an oxygen-free environment, at atmospheric pressure, and at reaction temperatures between 200 and 350 °C in the case of torrefaction [4,5], and between 350 and 700 °C for pyrolysis [6]. These processes not only remove moisture from the lignocellulosic biomass but also include the devolatilization, depolymerization, and carbonization of cellulose, hemicellulose, and lignin [7,8], as well as increase their calorific value. Different heating rates are usually established before reaching the treatment temperature, depending on the objective. Lin et al. [9] reported and used the different heating rates in torrefaction. Kinetic analysis is used to study the decomposition behavior of biomass under various thermal treatments, where high temperatures accelerate the kinetic rate of the reaction by increasing the synthesis gas and decreasing tar formation [10]. To measure weight loss characteristics, thermogravimetric analysis (TGA) is used [11], which allows the percentage of weight loss due to decomposition and dehydration of the biomass to be determined [12]. Basically, TGA measures the total content of solids and volatiles generated during the thermochemical process. Isothermal and non-isothermal decomposition are the two processes of biomass degradation [13]. Depending on these two factors, different approaches or models are used, including single-step, two-step, multi-step, and multicomponent models [14]. Kinetics and thermochemical models show that, at high temperatures, product evolution increases, the heating rate influences the thermochemical process evolution, and the mass and energy yield of the products are significantly influenced by both temperature and residence time [15]. This generally improves the properties of the solid fuel by removing moisture from the biomass and increasing the energy [16]. The literature presents studies that predict the devolatilization processes and the elemental content of carbon (C), hydrogen (H), nitrogen (N), and oxygen (O) in the biomass during thermal treatment using the two-step model [14,17]. Furthermore, several authors introduce the indices of decarbonization (DC), dehydrogenation (DH), and deoxygenation (DO) [18] to analyze the weight loss of C, H, N, and O [19,20], as well as the evolution of the high heating value (HHV) of biomass during the treatment [21]. The quality of the product that has been through a thermal process is given in terms of the HHV and a lower atomic ratio of H/C and O/C [22,23].

During thermochemical processes, complex chemical reactions, thermal degradation, and heat and mass transfer coexist [24]. This complexity requires precision in reactor design modeling, as well as techno-economic and environmental assessment [25]. Conventional techniques for modeling pyrolysis and torrefaction consist of thermodynamic analyses [26], kinetic analyses, computational fluid dynamics (CFD) [27], and process simulation such as Aspen Plus [4]. For theoretical studies, thermodynamic analysis is simple and efficient but has limitations. It is ineffective for designing a reactor for real-world applications, as it requires several assumptions [28]. In kinetic analysis, the complexity of the reaction rates requires many assumptions [14]. To perform an accurate simulation, fluid dynamics (CFD) models require a high computational load [29]. Therefore, a mathematical model is required to design and optimize the thermal conversion process reliably and accurately, given its complexity, the involvement of nonlinear parameters, and the high-dimensional conversion processes [30].

To address these disadvantages, Machine Learning (ML) is used, which is a subclass of artificial intelligence (AI), a technology that allows a machine to learn and is used in the optimization of thermochemical conversion processes and the prediction of their efficiency [31]. Advantages of ML-based models include that they do not require extensive knowledge and information about the process, but they only require a dataset of experimental measurements [32]. In solid yield prediction, real-time monitoring, and control, ML has recently been shown to be a solution for thermochemical conversion processes. Among the most common ML algorithms in the literature on optimization and prediction for thermochemical analysis, elemental analysis, and high calorific value problems, the following stand out: least squares regression (LSR) [33], Sum of Squared Error (SSE) [34], Nelder–Mead [35,36,37], CFD simulation [27], and particle swarm optimization (PSO) [13,38,39]. A powerful advantage of these algorithms is the ability to make predictions about global and individual Arrhenius parameters. The first algorithm consists of obtaining a set of Arrhenius parameters for each study temperature, while the second one obtains a set of Arrhenius parameters from a set of temperatures. In both cases, the experimental data are adjusted using a two-step kinetic model during a non-isothermal torrefaction process [40]. In this research, the main goal is to determine the impact of studying Arrhenius constants under the individual and global parameters scheme for elemental analysis, HHVs, atomic O/C and H/C ratios, and energy density in the wheat straw torrefaction process. In [33], the LSR method is used to predict a set of global Arrhenius parameters over a temperature range of 200–300 °C for different types of biomass, adjusting 31 experimental data points. In [17], the SSE algorithm performs individual parameter adjustment with 25 experimental data points in a temperature range of 220–300 °C for spruce and birch biomass. The Nelder–Mead algorithm is the most common kinetic parameter adjustment algorithm. It is used in [10] to predict a set of global Arrhenius parameters in a temperature range of 210–380 °C for juniper, adjusting nine experimental data points with a two-step kinetic model with an error of less than 2%. In [19], the Nelder–Mead algorithm is used to perform global parameter adjustment with 15 experimental data points in a temperature range of 210–290 °C in Eucalyptus grandi; in [14,21], the Nelder–Mead algorithm is used to perform a global parameter adjustment for poplar and fir with 30 experimental data points in a temperature range of 200–230 °C; and in [41], the Nelder–Mead algorithm is used with the Matlab function lsqcurvefit to perform individual parameter adjustment with a set of 48 experimental data points. In [42], a study of global and individual parameter adjustment for poplar and xylan is carried out in a temperature range of 200–240 °C using the Nelder–Mead algorithm with a set of 24 experimental points. In [8], individual parameter adjustment is performed for wood with 48 experimental data points in a temperature range of 200–300 °C using CFD software solved with the fourth-order Runge-Kutta method implemented in FORTRAN. The PSO algorithm is implemented in [13,40] to perform an individual parameter adjustment of 28 and 80 experimental data points for each study of xylan and sorghum distillation residues in a temperature range of 200–300 °C in both cases. Several studies of different biomasses at isothermal torrefaction temperatures have been analyzed using the two-step kinetic model to predict the solid biofuel yield using different algorithms.

Table 1. Summary of global and individual kinetic parameter studies on two-step isothermal torrefaction models.

Biomass	Temperature and Residence Time	Optimization Algorithm	Analysis of Kinetic Parameters	Pseudo-Comp. Description	Ref.
Ashe Juniper	210–380 °C, for 160 min	Nelder–Mead	Global	Solid yield and elemental composition prediction	[10]
Eucalyptus Grandis	210–290 °C, for 80 min	Nelder–Mead	Global	Solid yield and elemental composition prediction	[19]
Xylan	200–300 °C, for 120 min	PSO	Individual	Solid yield	[40]
Sorghum	200–300 °C, for 60 min	PSO	Individual	Solid yield	[13]
Poplar and Fir	200–300 °C, for 1740 min	Nelder–Mead	Global	Solid yield and elemental composition prediction	[14]
Poplar	200–230 °C, for 1740 min	Nelder–Mead	Global	Solid yield and elemental composition prediction	[21]
Poplar and Xylan	200–240 °C, for 600 min	Nelder–Mead	Global and Individual	Solid yield	[42]
Beech, Pine, Wheat, and Willow	200–300 °C, for 300 min	Least Squares Regression	Global	Solid yield and elemental composition prediction	[33]
Spruce and Birch	220–300 °C, for 120 min	Sum of Squares Errors	Individual	Solid yield and elemental composition prediction	[17]
Wheat Straw	250–300 °C, for 100 min	Nelder–Mead lsqcurvefit	Individual	Solid yield	[41]

lists some case studies of yield prediction under different parameter schemes.

Therefore, this research contributes to the assessment using the two-step kinetic model coupled with the PSO algorithm to predict biomass yield by making a comparison between two strategies: (1) individual parameters based on a single temperature and (2) global parameters considering four temperatures with the aim of reducing the amount of experimental data required to obtain a high-certainty predictive model of the energy densification that occurs during biomass torrefaction.

2. Materials and Methods

2.1. Experimental Procedure

This section describes the materials used in this research work and their physicochemical characterization: the proximate analysis, elemental composition, and the higher heating value (HHV) of raw wheat straw, as shown in Figure 1.

2.1.1. Feedstock and Preparation

The wheat harvest in Guanajuato has maintained one of the highest yields in the 2014–2024 decade, with 6.4 tons per hectare, achieving an average harvest of more than 400,000 tons annually, making it the third highest producing state in Mexico. This has led government institutions to promote and incentivize innovative initiatives to utilize the 30% of wheat straw that is not used for livestock in sustainable technologies within the circular economy. This is the main reason why this biomass was selected as the raw material for the development of this study. The biomass sample was collected in the city of Pénjamo (the largest wheat producer in Guanajuato), with crop fields located at the geographic coordinates 20°25′52″ N 101°43′20″ W during the autumn–winter harvest cycle [43].

2.1.2. Property Characterization

The wheat straw was ground to a uniform powder and sieved with a No. 20 mesh sieve. A thermogravimetric analysis (TGA) study of wheat straw was carried out using an SDT Q600 V8.3 device (manufactured by TA Instruments, New Castle, DE, USA-United States of America). The proximate analysis was based on the standard procedure of the American Society for Testing and Materials. The elemental composition was measured with a Thermo Scientific iCAP™ 7400 ICP-OES analyzer (manufactured by PerkinElmer, Waltham, MA, USA). The HHV was measured in a bomb calorimeter (IKA C3000, manufactured by Parr Instrument Company, Moline, IL, USA). The results of the analysis of the raw wheat straw are detailed in

Table 2. Properties of raw wheat straw (dry basis *).

Biomass	Wheat Straw	Analytic Method
Proximate analysis (%wt)
Humidity	8.590 ± 0.568	ASTM E871-82
Volatile matter (VM)	69.930 ± 0.256	ASTM E872-82
Fixed carbon (FC)	18.250 ± 1.670	ASTM E1775
Ash	3.220 ± 0.351
Elemental composition (%wt)
Carbon	46.205	Thermo Scientific iCAP 7400 ICP-OES analyzer
Hydrogen	6.275
Oxygen (by difference)	43.908
Nitrogen	3.612
HHV (MJ·kg−1, *)	18.535 ± 0.436	ASTM D5865

.

2.1.3. Thermogravimetric Analysis

Figure 1c shows the thermogravimetric analysis (TGA) and derived thermogravimetric analysis (DTG) of wheat straw. A TGA study was carried out on wheat straw using an SDT Q600 V8.3 device. TGA and TDG curves of pyrolysis were obtained with 35.555 mg of wheat straw at a heating rate of 10 °C/min in air, at temperatures of 25–800 °C, with a residence time of 80 min. Figure 1d shows that the pyrolysis process of wheat straw ranged between 25 and 800 °C. In the TGA curve (blue), the dehydration stage occurs from 25 and 150 °C, the decomposition of proteins and carbohydrates occurs between 150 and 400 °C, the decomposition of lipids occurs from 400 to 600 °C, and finally, the decomposition of other components occurs from 600 to 800 °C. After 600 °C, there are no changes in the wheat straw, leaving only fixed carbon. Abrupt changes in biomass occur during the protein and carbohydrate decomposition stage. The DTG curve (red) shows pronounced thermal decomposition, marking the limits of the wheat straw torrefaction process between 250 and 325 °C.

2.1.4. Torrefaction Experiments

Figure 2 shows the proposed experimental torrefaction methodology, which is based on three phases: (a) raw material, (b) torrefaction process, and (c) numerical modeling. First, the raw material used is wheat straw biomass, which has been ground into a fine powder to be sieved, weighed, and standardized into 50 g samples (%wt_i) for each test. Then, the conditions of the experiment are established, such as the duration of the test or residence time, the heating rate, and the maximum isothermal temperatures. To eliminate any trace of air (oxidizing agent) from entering the reactor, nitrogen (N₂) is injected at a flow rate of 50 mL/min before starting the experiment for 5 min. Once the residence time is completed, after the experiment is carried out, the sample is removed from the reactor and weighed to obtain the final percentage of weight loss of the sample (%wt_f). Four isothermal torrefaction temperatures were established for the design of the experiments proposed in this work, ranging from 250 to 275, 300, and 325 °C, at a heating rate of 20 °C/min. A residence time of 100 min was established. For the torrefaction process, a scale reactor was used, which has a K thermocouple that is used as a sensor to control and monitor the temperature. An Arduino UNO board (manufactured by Smart Projects, Scarmagno, Italy), a MAX6675 module (manufactured by Maxim Integrated Products, Sunnyvale, CA, USA) as an instrumentation stage between Arduino and the sensor, a 3500 W 220 VAC clamp resistor, an Autonics SPC1-50 power controller (manufactured by Autonics, Seoul, Republic of Korea), and a first-order Smith predictive control system with a delay (at 225 s) implemented in Matlab^® software were used version 2021a. The experimental results were stored in a database for further numerical modeling, which will be discussed in the next section.

2.2. Numerical Modeling

For this research, a heating rate of 20 °C/min, torrefaction temperatures of 250, 275, 300, and 325 °C, and a 100 min residence time were established. Three weight loss (wt) points were obtained for each temperature. The weight loss data were unified according to the %wt = m/m_o ratio. Solid yield kinetic modeling was performed using a two-step kinetic model and the PSO algorithm, through two Arrhenius kinetic parameter approaches: (1) global and (2) individual. To predict the solid yield, a set of Arrhenius parameters is obtained: for the first case, it is obtained from the four torrefaction temperatures, and in the second case, it is obtained for each temperature. Using the wheat straw yield fitting data and elemental composition data (of raw and torrefied biomass), the elemental component prediction analysis is performed during the 100 min residence time, as well as the prediction of the HHV and densification energy (DE). In the following subsections, the following mathematical models are explained: (1) the two-step kinetic model, (2) the elemental composition model, (3) HHV prediction, (4) densification energy, and (5) the particle swarm optimization (PSO) mathematical model.

2.2.1. Two-Step Kinetic Model

To describe the mechanism of biomass decomposition during the isothermal pyrolysis and torrefaction process, a two-step kinetic model was developed [44]. This model is presented in Equation (1). The pseudo-components represent the raw biomass (m_A), the intermediate product (m_B), and the torrefied biomass (m_C). The volatile compounds are represented by the pseudo-components m_VA and m_VB. At the beginning of the treatment process (at t = 0), m_A represents 100% of the solid yield, i.e., the raw biomass. As the temperature increases, the intermediate product m_B appears and the volatile component m_V1 is released. Subsequently, the solid residue m_C is generated, and the volatile compound m_V2 is released. The sum (m_T = m_A + m_B + m_C) of the pseudo-components m_A, m_B, and m_C describes the solid yield during the residence time of the pyrolysis and torrefaction process [42,45,46]. The scheme of the model is presented below:

Where, the constants k₁, k₂, k_V1, and k_V2 (at min⁻¹) obey the Arrhenius law of Equation (1) and are also called “kinetic parameters”. Where the pre-exponential parameter is A₀ (min⁻¹), the activation energy is E_a (J·mol⁻¹), the absolute temperature T (K), and the universal gas constant R = 8.314 J·K⁻¹·mol⁻¹, respectively.

$$k=A_0\mathit{exp}\left({\displaystyle \frac{-E_a}{RT}}\right)$$

(1)

Assuming that the reactions are first order, the Arrhenius constants for solid and volatile yields can be described as follows:

$$Y_A^{\left(T\right)}\left(t\right)={\displaystyle \frac{d{m}_A}{dt}}=-m_A\cdot \left(k_1+k_{V1}\right)$$

(2)

$$Y_B^{\left(T\right)\left(t\right){{{{\displaystyle \frac{d{m}_B}{dt}}}_1}_A}_B\left(k_2+k_{V2}\right)}$$

(3)

$$Y_C^{\left(T\right)\left(t\right){{{\displaystyle \frac{d{m}_C}{dt}}}_2}_B}$$

(4)

$$Y_{V1}^{\left(T\right)}\left(t\right)={\displaystyle \frac{d{m}_{V1}}{dt}}=k_{V1}\cdot m_A$$

(5)

$$Y_{V2}^{\left(T\right)}\left(t\right)={\displaystyle \frac{d{m}_{V2}}{dt}}=k_{V2}\cdot m_B$$

(6)

The kinetic parameters of Equation (2) are adjusted according to a range of 0 to −20 to 0 dB [10]. This aspect will be discussed in greater detail below. The Arrhenius law in Equation (1) describes the thermal degradation of wheat straw during the heating process. The model considers two zones: the heating rate and the pyrolysis or torrefaction temperature. Four experimental weight loss points are obtained for each treatment temperature in order to adjust them to the two-step kinetic model and the particle swarm optimization (PSO) algorithm in the case of individual parameters. Sixteen experimental data points were used for the global parameters.

Solving the ordinary differential equation provides the evolution of the weight of each pseudo-component as a function of time. The instantaneous solid yield, Y^(T)(t) for any temperature T, can be calculated according to Equation (7):

$$Y^{\left(T\right)}\left(t\right)={\displaystyle \frac{m_A\left(t\right)+m_B\left(t\right)+m_C\left(t\right)}{m_0}}\times 100$$

(7)

2.2.2. Elemental Composition Model

In this study, the matrix (of the form AX = b) of Equation (8) is used to calculate the elemental composition (C, H, N, and O):

$$\begin{array}{c}{\left[\begin{array}{ccc}\begin{array}{c}{Y_A}^{\left(250\right)}\\ {Y_A}^{\left(275\right)}\\ {Y_A}^{\left(300\right)}\\ {Y_A}^{\left(325\right)}\end{array}& \begin{array}{c}{Y_B}^{\left(250\right)}\\ {Y_B}^{\left(275\right)}\\ {Y_B}^{\left(300\right)}\\ {Y_B}^{\left(325\right)}\end{array}& \begin{array}{c}{Y_C}^{\left(250\right)}\\ {Y_C}^{\left(275\right)}\\ {Y_C}^{\left(300\right)}\\ {Y_C}^{\left(325\right)}\end{array}\end{array}\right]}_j\times {\left[\begin{array}{cccc}\begin{array}{c}{C}_A\\ C_B\\ C_C\end{array}& \begin{array}{c}{H}_A\\ H_B\\ H_C\end{array}& \begin{array}{c}{N}_A\\ N_B\\ N_C\end{array}& \begin{array}{c}{O}_A\\ O_B\\ O_C\end{array}\end{array}\right]}_j\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\left[\begin{array}{cccc}\begin{array}{c}{C_S}^{\left(250\right)}\\ {C_S}^{\left(275\right)}\\ {C_S}^{\left(300\right)}\\ {C_S}^{\left(325\right)}\end{array}& \begin{array}{c}{H_S}^{\left(250\right)}\\ {H_S}^{\left(275\right)}\\ {H_S}^{\left(300\right)}\\ {H_S}^{\left(325\right)}\end{array}& \begin{array}{c}{N_S}^{\left(250\right)}\\ {N_S}^{\left(275\right)}\\ {N_S}^{\left(300\right)}\\ {N_S}^{\left(325\right)}\end{array}& \begin{array}{c}{O_S}^{\left(250\right)}\\ {O_S}^{\left(275\right)}\\ {O_S}^{\left(300\right)}\\ {O_S}^{\left(325\right)}\end{array}\end{array}\right]}_j\end{array}$$

(8)

where A, B, and C represent the pseudo-components, and S is the torrefied biomass. Y^(T) is the weight fraction of the solid yield, and X^(T) is the elemental content at temperature T.

With information on the initial raw biomass and the final product of the torrefied biomass, both the weight loss fraction and the elemental analysis, it is possible to estimate the pseudo-components of the elemental composition for any study temperature and time point. The sum of the pseudo-components (A, B, and C) must be 100% at any given time [17,19]. The restrictions of Equations (9) and (10) help the algorithm of Equation (9) to converge better.

$$C_B>C_A\hspace{1em}\hspace{1em}{H}_B<H_A\hspace{1em}\hspace{1em}{O}_B<O_A$$

(9)

$$C_C>C_B\hspace{1em}\hspace{1em}{H}_C<H_B\hspace{1em}\hspace{1em}{O}_C<O_B$$

(10)

2.2.3. Higher Heating Value (HHV) Prediction

Knowing the instantaneous element, the dynamics of HHV behavior can be determined. The correlation of Equation (11) [47,48] is used to establish the HHV during the torrefaction process. The concentrations of carbon, hydrogen, and oxygen represent the weight percentages of carbon, hydrogen, and oxygen on an ash-free dry basis [21]; the concentration of nitrogen is small enough to be considered zero. Predicting the HHV (MJ/kg) is important because it provides insight into the potential of torrefied biomass as a solid fuel, as well as the quality of the process.

$$HHV=-1.3675+0.3137\times C+0.7009\times H+0.0318\times O$$

(11)

2.2.4. Energy Densification (ED)

Energy densification (ED) is calculated according to Equation (12), which relates the HHV of biomass at any point of the torrefaction process to the HHV of the raw biomass [11,33,49]. ED is considered an index of energy density improvement that examines the effectiveness of torrefaction.

$$ED={\displaystyle \frac{HH{V}_{torrefied}}{HH{V}_{raw}}}$$

(12)

2.2.5. Particle Swarm Optimization (PSO) Mathematical Model

The particle swarm optimization (PSO) algorithm is a metaheuristic technique commonly used in high-dimensional search spaces for nonlinear problems [40,50]. Based on the Arrhenius Equation (1), it is assumed that the factors A₀ and Ea are independent of temperature. The calculation of Arrhenius kinetic parameters in a thermal degradation process using a two-step kinetic model is a complex problem that depends on the integrals of Equations (1)–(6), making the PSO algorithm a robust strategy for finding global optima for different isothermal temperatures. The PSO algorithm is initialized with a random number of particles defined within a search space. This search space will be defined in a range from −20 to 0 dB for the Arrhenius kinetic constants for wheat straw. The selection of this range for the kinetic constants is established to give greater importance to the most significant reactions occurring within the torrefaction process. In linear terms, according to Equation (1), by applying the natural logarithm on both sides of the equation, this range represents a variation of two orders of magnitude, which is consistent with the influence of different reaction stages observed experimentally and with typical values reported in the literature [10,14,19,42] for similar reactions within the proposed torrefaction temperature range. For each iteration, the PSO finds an optimal solution, and the particles are updated to their best local position (pbest) and their best global position (gbest). When these optimal values are found, the particles update their position and velocity according to Equations (13) and (14).

$${\overrightarrow{v_i}}^{l+1}=\omega \times \Delta {\overrightarrow{v_i}}^l+c_{1}rand\left(\right)\left(pbes{t}_i-{\overrightarrow{x_i}}^l\right)+c_{2}rand\left(\right)\left(gbes{t}_i-{\overrightarrow{x_i}}^l\right)$$

(13)

$${\overrightarrow{x_i}}^{l+1}={\overrightarrow{x_i}}^l+{\overrightarrow{v_i}}^{l+1}$$

(14)

Here, v_i and x_i are the velocity and position of the ith particle. The variable ω indicates the adjustment rate of the position change, j is the number of iterations performed by the algorithm, and the cognitive and social learning factors are the variables c₁ and c₂. Finally, rand() is a random number between 0 and 1.

As the objective function (OF) of the PSO algorithm, the least squares method is applied. This OF is used to predict the weight loss curve, according to Equation (15).

$$O{F}^{\left(T\right)}=\sum _{j=1}^n{\left[{\left(TGA\right)}_{\mathit{exp},j}^{\left(T\right)}-{\left(TGA\right)}_{\mathit{cal},j}^{\left(T\right)}\right]}^2$$

(15)

Here, T is the isothermal torrefaction temperature; ${\left(TGA\right)}_{\mathit{exp},j}^{\left(T\right)}$ is the jth experimental data point for a sample of the percentage weight loss of wheat straw at a given temperature; ${\left(TGA\right)}_{cal,j}^{\left(T\right)}$ is the calculated percentage weight loss of the sample; and n symbolizes the number of data points measured experimentally.

To determine the degree of similarity between the weight loss curve calculated using the PSO algorithm and the actual weight loss curve obtained from experimental data, Equation (16), known as the quality of fit (fit), is used.

$$fit\left(\%\right)=\left[1-{\displaystyle \frac{\sqrt{\frac{OF}{n}}}{\left(TG{A}_{exp}({)}_{max}\right)}}\times 100\%\right]$$

(16)

Here, $\left(TG{A}_{exp}({)}_{max}\right)$ is the maximum weight loss percentage of the sample in the experiment.

2.2.6. Model Validation Metrics

To calculate relative errors, global errors (E_Global) and individual errors (E_Individual) are implemented in Equations (17) and (18), respectively. Here, P_REF represents the reference of real parameters for each elemental component at different temperatures. P_Global and P_Individual are the elemental components calculated after the torrefaction process.

$$\%E_{Global}={\displaystyle \frac{\left|P_{Global}-P_{REF}\right|}{P_{REF}}}\cdot 100\%$$

(17)

$$\%E_{Individual}={\displaystyle \frac{\left|P_{Individual}-P_{REF}\right|}{P_{REF}}}\cdot 100\%$$

(18)

2.3. Mathematical Modeling Methodology

Figure 3 shows the numerical prediction section. To set up the PSO algorithm, the initial parameters selected were set to 500 particles, 500 iterations, and the search space was for the lower bound of A₀ = 9 × 10⁴ and Ea = 9 × 10³ and at the upper bound of A₀ = 9 × 10⁸⁰ and Ea = 9 × 10⁶, for each of the Arrhenius k_i constants. To obtain the adjustment curves for each proposed temperature, the PSO algorithm is studied in terms of accuracy by estimating the objective function (1.551 × 10⁻¹²), estimating computational efficiency by measuring the algorithm’s execution time (10.145 min), and estimating convergence efficiency with the adjustment (99.99996887%), and this methodology is shown in Figure 3. The initial search range of the PSO was adjusted to generate real values of k₁, k_V1, k₂, and k_V2 for wheat straw between −20 and 0 dB (0 > k_i > −20 dB, where i = 1, 2, V1, and V2).

The numerical prediction section addresses four cases in this work: searching for Arrhenius parameters that help to establish degradation curves by weight loss at different isothermal torrefaction temperatures, finding the elemental composition (CHNO), predicting the higher heating value (HHV), and determining the densification energy (DE) of wheat straw.

In one approach, using “global Arrhenius parameters”, which consist of adjusting sixteen experimental values of the percentage weight loss for the study temperatures (250, 275, 300, and 325 °C), four experimental values for each isothermal temperature are determined, using the particle swarm optimization (PSO) algorithm. Considering the weight loss yields and the initial and final elemental analysis of wheat straw, it is possible to calculate the elemental component matrix X^(T) as described in (8) for any instant of time, obtain the yields per temperature, and, consequently, perform predictions of the elemental components of the matrix and the HHV of Equation (11). A relationship is found between the atomic ratios of H/C and O/C and the torrefaction temperature.

On the other hand, when using “individual Arrhenius parameters”, the process is similar to that of global parameters, with the difference that, with the PSO algorithm, the Arrhenius parameters are obtained for each isothermal study temperature. Once a weight loss degradation curve is obtained for each temperature, the predictions of elemental components, the HHV, and ED are calculated.

Once the parameters and the Arrhenius kinetic constants are determined through individual or global analysis, the percentage weight loss functions, determined from Equation (8), are calculated for each isothermal torrefaction temperature. Using the percentage weight loss data and the initial and final values of the elemental analysis, the percentages of CHNO present in the wheat straw during the torrefaction process are predicted using Equation (9), and then the HHV and ED are predicted using Equations (11) and (12).

3. Results

3.1. Isothermal Torrefaction

An isothermal torrefaction process was carried out to study wheat straw from room temperature to 250, 275, 300, and 325 °C. Figure 4 shows the data from the experiment. Figure 4a shows the graphs of the heating curves at a heating rate of 20 °C/min, while Figure 4b shows the weight data measured at different time intervals of an initial mass of 50 g. The following conditions were used: at 250 °C, blue line: 41.680 g at 33 min, 37.320 g at 53 min, and 31.380 g at 83 min; at 275 °C, red line: 37.375 g at 40 min, 32.262 g at 60 min, and 27.150 g at 80 min; at 300 °C, black line: 30.190 g at 48 min, 25.220 g at 68 min, and 23.470 g at 88 min; and at 325 °C, magenta line: 23.910 g at 57 min, 23.540 g at 77 min, and 22.480 g at 97 min. The m/m_o ratio was used to normalize weight loss performance, representing the final weight (m) and the initial weight (m_o). This indicates that there is a close relationship between temperature and weight loss; in other words, the higher the temperature, the greater the weight loss.

3.2. Properties of Torrefied Biomass

Table 3. Elemental analysis of wheat straw torrefaction.

Temperature (°C)	Ultimate Analysis (%wt), Dry-Ash-Free				HHV (MJ·kg−1)	O/C	H/C
	C	H	N	O a
raw	46.205	6.275	3.612	43.908	18.535 ± 0.436	0.713	1.623
250	49.573	5.715	3.721	40.991	19.689 ± 0.759	0.620	1.383
275	57.089	4.893	3.806	34.212	21.373 ± 0.044	0.450	1.028
300	60.127	4.413	4.761	30.699	22.239 ± 0.087	0.383	0.881
325	68.596	4.007	7.922	22.475	23.341 ± 0.156	0.246	0.701

^a by difference.

shows the experimental data from the elemental analysis of raw and torrefied wheat straw at different isothermal temperatures. This is important for evaluating the quality of the process through energy densification. The elemental analysis shows a reduction in the proportion of oxygen in the biomass. Due to the change in elements during torrefaction, the table shows that the atomic ratios of O/C and H/C decreased.

3.2.1. Van Krevelen Diagram

Atomic ratios H/C and O/C are indices that provide insight into the energy density of a torrefied biomass. Figure 5 shows a Van Krevelen diagram of the atomic ratios H/C (aromaticity) versus O/C (polarity) for each torrefaction temperature. Both atomic ratios decrease as the temperature increases, which means a degradation of carboxyl and hydroxyl groups. Some standard regions are illustrated to make a comparison between the product of torrefied wheat straw and some solid fuels; this provides information for understanding the reaction pathways. In this case, a process of dehydration occurs, combined with the cleavage of weak linkages between small substituents and the leading polymer chains [51].

The temperature ratio datasets are correlated, giving a coefficient of determination of 0.987. The slope of the correlation is 2.001, indicating that at higher torrefaction temperatures, the impact on the H/C and O/C ratio is greater by a factor of 2.0.

3.2.2. Energy Densification Results

The energy density of wheat straw increases as the isothermal torrefaction temperature increases. The O/C and H/C atomic ratios are low as the temperature increases. In addition, it has a low moisture content. Figure 6 shows the energy density ratios at torrefaction temperatures of 250, 275, 300, and 325 °C. The black represents the data obtained experimentally, with 1.034–1.259%; the blue curves were obtained from the analysis of the global parameters, with 1.034–1.226%; and the red curves were obtained from the individual parameters, with 1.009–1.198%. There is a close relationship between the torrefaction temperature and energy density. From the curves, the analysis used individual parameter deviations from the experimental data to a greater degree, with a mean relative error of 7.788, while when using global parameters, the mean relative error is 2.945. This is mainly due to the dependence of energy density on HHVs. At 275 °C, the error is significant due to errors in the prediction based on the analysis of the elemental components.

3.3. Torrefaction Kinetics Using PSO Algorithm

Global and individual parameters were obtained from the experimental results to compare their suitability, as shown in

Table 4. Global and individual Arrhenius kinetic parameters obtained from PSO.

Algorithm PSO	Temperatures (°C)	Arrhenius Constants (s−1)	Arrhenius Parameters		Fit (%)
			A (s−1)	Ea (kJ·mol−1)
Global Parameters	250, 275, 300, and 325	k1	1.141 × 1011	1.310 × 105	99.99996887
		kV1	9.789 × 105	8.087 × 104
		k2	9.724 × 106	9.762 × 104
		kV2	6.763 × 105	8.557 × 104
Individual Parameters	250	k1	2.401 × 1011	1.350 × 105	99.99996187
		kV1	9.199 × 105	8.066 × 104
		k2	4.426 × 107	1.036 × 105
		kV2	7.338 × 107	1.075 × 105
	275	k1	9.058 × 1010	1.432 × 105	98.07280000
		kV1	2.785 × 103	5.713 × 104
		k2	9.706 × 104	8.003 × 104
		kV2	4.218 × 104	9.595 × 104
	300	k1	5.178 × 1010	1.289 × 105	99.99999997
		kV1	5.490 × 104	6.870 × 104
		k2	3.770 × 109	1.297 × 105
		kV2	2.222 × 104	7.035 × 104
	325	k1	9.283 × 105	7.746 × 104	99.99048931
		kV1	2.651 × 105	7.451 × 104
		k2	7.643 × 107	1.104 × 105
		kV2	1.603 × 102	4.830 × 104

. The global parameter fitted is 99.99996887%, indicating an acceptable fit of the sixteen experimental data points to the two-step kinetic model. This means there is good predictive behavior for the torrefaction process at 250, 300, and 325 °C. However, for 275 °C, a major error is found, as shown in Figure 7a. For individual parameters, the error decreases significantly, as shown in Figure 7b; the fit is 99.99996187%, 98.07280000%, 99.99999997%, and 99.99048931% for 250, 275, 300, and 325 °C, respectively. Therefore, there is a better fit for individual Arrhenius parameters than for the global ones. However, the cumulative error when combining the individual parameters could be greater than that of the global one.

The model is validated by correlating the calculated torrefied mass with the experimental data, as shown in Figure 8. The global parameters maintained an R² of 0.9998. For the individual parameters, an R² of 0.9963 is obtained. This indicates that the correlation between the predicted weight loss and the experimental data provides better results for the global parameters.

3.4. Prediction of Properties of Torrefied Biomass

This section discusses how the two-step kinetic model and experimental results were used to predict the final properties of the torrefied biomass.

3.4.1. Elemental Composition Prediction

Based on the elemental composition data in Table 3, the composition of the torrefied biomass can be predicted for any isothermal temperature and point in time using Equation (8). The Matlab function fmincon was used to perform the prediction, as well as the restriction Equations (9) and (10).

Figure 9 shows the results of elemental composition profiles predicted according to the two-step kinetic model during the torrefaction of wheat straw for the case of global parameters. The blue line represents the torrefaction temperature of 250 °C, the red line, 275 °C, the black line, 300 °C, and the magenta line, 325 °C. The carbon content is enriched, the hydrogen and oxygen contents are reduced, and the nitrogen content increases. It can be seen that at higher torrefaction temperatures, the slopes of the elemental contents of the torrefied biomass increase or decrease sharply. This is due to the close relationship between temperature and the elemental content present in the biomass. In Figure 9d, corresponding to the nitrogen estimation, a reduction can be observed around minute 20, during the transition from the heating rate to the isothermal temperature. This is mainly because the PSO algorithm, when finding global parameters, presents errors in some areas of the solid yield fit due to experimental values obtained, as well as to the solution itself found by the algorithm at different temperatures. Despite these inconsistencies, the solution provides an approximate estimate of the behavior of nitrogen throughout the torrefaction process. Furthermore, these estimated inconsistencies in the nitrogen prediction at different torrefaction temperatures, according to Equation (11), do not influence the HHV prediction since nitrogen is not considered in the equation.

Using the data in

Table 5. Elemental analysis prediction results.

Analysis Elemental	PREF	PGlobal	PIndividual	EGlobal	EIndividual	PREF	PGlobal	PIndividual	EGlobal	EIndividual
Temp (°C)	Carbon (%wt)					Hydrogen (%wt)
Raw	46.205	46.090	50.042	0.249	8.304	6.275	6.300	5.857	0.393	6.655
250	49.573	49.951	51.889	0.763	4.671	5.715	5.641	5.334	1.297	6.674
275	57.089	55.737	50.808	2.369	11.002	4.893	4.979	5.645	1.761	15.367
300	60.127	62.963	60.352	4.717	0.375	4.413	4.357	4.504	1.264	2.063
325	68.596	66.849	68.499	2.547	0.141	4.007	4.026	3.963	0.476	1.095
	Nitrogen (%wt)					Oxygen (%wt)
Raw	3.612	3.648	3.589	0.989	0.650	43.908	43.963	40.512	0.124	7.733
250	3.721	3.613	3.929	2.908	5.599	40.991	40.795	38.848	0.478	5.227
275	3.806	3.952	3.727	3.842	2.074	34.212	35.332	39.820	3.273	16.392
300	4.761	4.622	4.561	2.909	4.204	30.699	28.057	30.583	8.606	0.379
325	7.922	4.987	5.015	37.052	36.693	22.475	24.139	22.522	7.402	0.207

, the relative error of each elemental component is calculated: 2.129% for carbon, 1.038% for hydrogen, 9.540% for nitrogen, and 3.977% for oxygen.

Figure 10 shows the results of the predicted elemental composition profiles for individual parameters. The blue line represents the torrefaction temperature of 250 °C, the red line, 275 °C, the black line, 300 °C, and the magenta line, 325 °C. The carbon and nitrogen content increases, while the hydrogen and oxygen content decreases. According to Equations (18) and (19), the prediction of the elemental composition of the final analysis has a relative error of 4.899% for carbon content, 6.371% for hydrogen content, 9.844% for nitrogen content, and 5.988% for oxygen content.

According to Table 5, considering the initial state of the wheat straw and the four isothermal torrefaction temperatures, the results provide the following predictions: for the global parameters, there is a maximum error of 37.052% for nitrogen at a temperature of 325 °C, a minimum error of 0.124% for oxygen in the raw biomass, and an average error of 4.171%, with a standard deviation of 7.871%; for the individual parameters, there is a maximum error of 36.693% for hydrogen at a temperature of 275 °C, a minimum error of 0.141% for carbon at a temperature of 325 °C, and an average error of 6.775%, with a standard deviation of 8.308%, for relative errors. As can be seen, both analyses show similar results. This is expected in the case of individual parameter analysis, as it presents better fits in the weight loss curves, which directly affects the prediction of the elemental components. However, the results show that global parameter adjustment is a valid option for estimating the elemental components in wheat straw.

However, in some cases, the errors obtained through global adjustments are larger than those obtained by individual adjustments. One advantage of the proposal is that with a small amount of experimental data at different temperatures, a good fit is achieved with the two-step kinetic model. In contrast, for example, in the case of global parameters, the adjustments were made in [19] with 30 data points, in [21] with 28 data points, and in [33] with 31 data points. In the case of individual parameters, the adjustments were implemented in [17] with 25 data points, in [14] with 48 data points, and in [41] with 124 data points. It was demonstrated that the prediction of a set of global parameters that satisfy the two-step kinetic model and the adjustment of the Arrhenius curves between −20 to 0 dB reduces the propagation of errors.

3.4.2. HHV Prediction

The results of the HHV prediction of wheat straw at different torrefaction temperatures (250, 275, 300, and 325 °C) are presented in Figure 11. The HHV is obtained from the elemental composition using the empirical Equation (10). The torrefaction treatment shows a dependence between temperature and time in the HHV analysis from raw material to the torrefied biomass. Global parameters are shown in Figure 11a, where the results range from 18.905 to 23.191 MJ·kg⁻¹. Regarding individual parameters, in Figure 11b, the HHV results are between 19.724 and 23.615 MJ·kg⁻¹. In both cases, the interval from 0 to 17 min corresponds to the time in which the heating rate (20 °C/min) reaches the torrefaction temperature. Subsequently, the HHV began to increase due to changes in the elemental composition of the wheat straw as the temperature increased. The results show that the carbon content increases while the oxygen content decreases.

Table 6. HHV analysis prediction results.

	HHV (MJ·kg−1)			Relative Errors (%)
Temp (°C)	Ref	Global	Individual	Global	Individual
raw	18.535	18.905	19.724	1.995	6.417
250	19.689	19.552	19.884	0.695	0.989
275	21.373	20.732	19.794	3.001	7.389
300	22.239	22.331	21.694	0.415	2.449
325	23.341	23.191	23.615	0.642	1.172

shows the values obtained from experimental data for raw biomass (18.535 MJ·kg⁻¹) and torrefied biomass at the different study temperatures. In addition, the predicted HHVs are shown using global and individual parameter analysis. The relative errors between the experimental data and those obtained with the global and individual parameters were estimated to validate the prediction results. The results showed that the maximum relative error occurs at 275 °C in the individual parameters, with a value of 7.389. In contrast, the minimum relative error corresponds to the global parameters, with a value of 0.415 at 300 °C. The average relative error for the global parameters is 1.350, and for the individual parameters, it is 3.683. This is because the predictions for the global parameters show lower errors in elemental composition analysis.

It was found that by determining the kinetic parameters at a torrefaction temperature higher than the maximum peak observed in the derivative thermogravimetric (DTG) curve (275 °C), it is possible to predict the behavior of the process within the 250–325 °C range with an R-squared value corresponding to an error lower than 3%. This approach significantly reduces the number of required experiments from twelve to only four by relying on a single isothermal condition for parameter estimation.

4. Conclusions

Torrefaction has been proposed as an effective thermal pretreatment process for the thermochemical conversion of biomass. To optimize this process and predict the quality of biochar, a dual kinetic model based on Arrhenius law was developed to describe the torrefaction of wheat straw at different temperatures. The schemes using global and individual parameters were compared and used to predict the evolution of the elemental composition (CHNO) and the higher heating value (HHV) of the resulting biochar.

The results demonstrate that the proposed model accurately reproduces the experimental profiles, showing strong agreement between the predicted and observed values of elemental composition and HHV. During the torrefaction process, the O/C and H/C atomic ratios decreased as the temperature increased, followed by a dehydration process, causing an increase in energy densification when the temperature is higher.

Furthermore, it was found that by determining the kinetic parameters at a torrefaction temperature higher than the maximum peak observed in the derivative thermogravimetric (DTG) curve (i.e., 275 °C), it is possible to predict the behavior of the process within the 250–325 °C range with an R-squared value corresponding to an error lower than 3%. This approach significantly reduces the number of required experiments—from twelve to only four—by relying on a single isothermal condition for parameter estimation. Such a strategy enhances experimental efficiency and reinforces the model’s robustness for process design and optimization.

Biomass torrefaction can have environmental benefits in relation to carbon sequestration. Furthermore, the reduction in experiments to obtain the kinetic model, as was developed in the present study, drives economic benefits because the experimentation costs could be up to 75% less.

Limitations and Future Prospects

For future work, the proposed methodology will be applied to different types of biomass to establish a comparison with this and the new results obtained. Experiments will be conducted under different case studies in pyrolysis and co-pyrolysis, including heating rates and a wide range of temperatures (300–600 °C).