# Study of Cocoa Pod Husks Thermal Decomposition
## 3 Results and Discussion
### Kinetic Model and Fitting Strategy  
During the thermal decomposition of biomass, different regions of weight loss can be observed. Each region is considered a fraction that turns into volatile material and solid residue (except for the region identified as moisture). Combustion of solid material gives rise to ashes. In the literature, different kinetic models have been proposed that allow adjusting the thermal decomposition curves of different biomasses. However, only those models that describe thermal decomposition at different heating rates using the same kinetic constants are considered valuable. Here, a model of independent decomposition reactions has been considered, each one corresponding to a fraction of the biomass that decomposes in a different temperature range. In short, the model can be represented as:  
\[w_{i_{0}}W_{i}\rightarrow(w_{i_{0}}-v_{i_{\infty}})C_{i}+v_{i_{\infty}}V_{i} \tag{1}\]  
where \(W_{i}\) corresponds to each one of the original biomass fractions, \(V_{i}\) corresponds to the volatiles released during its decomposition, \(C_{i}\) is the carbonaceous residue formed as a product of the reaction, \(v_{i_{\infty}}\) corresponds to the maximum mass fraction of the volatiles and \(w_{i_{0}}\) is the initial mass fraction of each fraction. Considering m as the number of different fractions that make up the biomass, the following can be expressed:  
\[\sum_{i=1}^{m}w_{i_{0}}=1 \tag{2}\]The experimental data have been normalized to zero residue which, for practical purposes of the model, would be equivalent to considering \(v_{i\infty}=w_{i_{0}}\). When considering kinetics of order n, the thermal decomposition equation of each one of the biomass fractions can be represented as:  
\[-\frac{dw_{i}}{dt}=k_{i}w_{i}{}^{n_{i}} \tag{3}\]  
where \(w_{i}\) is the mass fraction of \(W_{i}\) at each moment, \(k_{i}\) is the kinetic constant of the decomposition reaction and \(n_{i}\) is the reaction order. Considering that \(k_{i}\) can be expressed using the Arrhenius equation  
\[k_{i}=k_{0_{i}}\ exp\bigg{(}-\frac{E_{i}}{RT}\bigg{)} \tag{4}\]  
the equation that describes the kinetics of m components is expressed as:  
\[-\frac{dw_{i}}{dt}=k_{0_{i}}\ exp\bigg{(}-\frac{E_{i}}{RT}\bigg{)}w_{i}{}^{n_{ i}}\ \text{where}\ i=1,2,3,\dots,m \tag{5}\]  
The most common strategy to solve this equation, finding values for \(k_{0_{i}}\), \(E_{i}\) and \(n_{i}\) that describe the kinetics of m components, consists in the successive iteration minimizing the error between the experimental value and the calculated one, using tools such as \"Solver\" of Microsoft Excel(r) (Microsoft Excel 2019 MSO (16.0.10389.20033), Microsoft Corporation, Redmon, WA, USA), or similar. Using a reference temperature to minimize the interrelation among the kinetic parameters is widely used [30]. A more powerful strategy would consist of obtaining the second derivative of the kinetic equation and setting it equal to zero, in order to obtain the mathematical expression that defines the maximum of the absolute value of the weight derivative with respect to time, thus:  
\[-\frac{d^{2}w_{i}}{dt^{2}}=k_{0_{i}}\ exp\bigg{(}-\frac{E_{i}}{RT}\bigg{)}w_{i}{ }^{n_{i}}\bigg{[}\bigg{(}\frac{E_{i}}{RT}\bigg{)}\frac{\beta}{RT^{2}}-k_{0_{i} }exp\bigg{(}-\frac{E_{i}}{RT}\bigg{)}n_{i}w_{i}{}^{n_{i}-1}\bigg{]}=0 \tag{6}\]  
\[\frac{E_{i}\ w_{i}\beta}{RT^{2}}-n_{i}\ k_{0_{i}}\ exp\bigg{(}-\frac{E_{i}}{RT} \bigg{)}\ w_{i}{}^{n_{i}}=0 \tag{7}\]  
This expression introduces the known variable \(\beta\) (heating rate) into the equation and allows establishing the temperature range in which the maximum for each component is visually identified, restricting the solution of each temperature to its respective range, and thus obtaining an improved convergence.