# A Simple Direct Method to Obtain Kinetic Parameters for Polymer Thermal Decomposition
## 1 Introduction
The direct differential [12], Freeman-Carroll [13], and Coats-Redfern [14] approaches are the most commonly used of these methods. These methods consider the activation energy and pre-exponential factor to be constant throughout the entire reaction. However, some practitioners [12] limit \(\alpha\) to an experimental range between 0.01 and 0.11 at the low limit and between 0.67 and 0.98 at the high limit, depending on the material, reaction order, and method. This is completed because extreme curvature is observed in experiments at large \(\alpha\), which leads to deviation from Arrhenius behavior, especially for the direct differential method. There are isoconversional and non-isoconversional methods of analysis. Isoconversional methods evaluate the kinetic parameters at successive values of \(\alpha\) rather than applying a reaction model [10]. These methods typically require kinetic curves at different heating rates because they assume that the activation energy is the same at each degree of conversion. However, some studies show large differences in the thermal decomposition of certain materials at different heating rates [3; 15]. The most popular isoconversional approaches are the standard isoconversional [16], Ozawa-Flynn-Wall [17], Friedman [18], Vyazovkin [19], and Kissinger-Akahira-Sunose [20] methods. These methods are controversial according to the study [21] that claim that the activation energy obtained using integral isoconversional methods is mathematically incorrect and recommend avoiding them. Model-free non-isoconversional methods allow direct calculation of the activation energy and pre-exponential factors using a single equation. These methods are generally easy to apply, but can be sensitive to data noise [22]. The most commonly used examples are the generalized direct (GDM) [22], Friedman [23], McGrattan [24], Lyon [25], and Kissinger [26] methods. The differences among these methods lie in the definition of the reference point used to obtain the kinetic parameters. The reference point in the Lyon method is chosen at the maximum decomposition rate, whereas the GDM considers the reference point to be the position where the third derivative of the decomposition rate is zero. The dependence on the definition of a single point requires a separate analysis to be conducted for each method. Finally, curve-fitting optimization methods are very efficient for complex processes involving multiple reactions, but require significant computational effort [27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. In summary, isoconversional model-free methods require kinetic curves at different heating rates. However, the definition of kinetic parameters as a function of the degree of conversion does not allow their direct use in computational pyrolysis models. Non-isoconversional model-free methods can be directly applied to computational models because it is necessary to have only a single kinetic curve at a certain heating rate. However, these methods employ a unique reference point in defining the reaction and are, therefore, very sensitive to competing reactions [9]; these equations do not necessarily follow a perfect exponential Arrhenius response. Curve-fitting optimization methods are more precise, because they require only an individual kinetic curve at a certain heating rate and can be directly applied to computational models. However, their complexity makes their utilization difficult for non-experts. Therefore, the challenge is to develop a method with a suitable balance between accuracy and complexity. A modification of the direct differential method [12] is proposed to address this issue. This approach allows simple analytical equations to be used for calculating the kinetic parameters of thermal decomposition while assigning equal weight to the decomposition behavior at the beginning, middle, and end of the process.