# A Thermogravimetric Analysis of Biomass Conversion to Biochar: Experimental and Kinetic Modeling
## 3 Results and Discussion
### Thermal Decomposition Kinetic Modeling
#### 3.2.1 Model-Free Methods  
Different model-free methods were proposed in the published literature by [PERSON] [60] ([PERSON] method), [PERSON] [61] and [PERSON] and Wall [62] (Flynn--Wall--Ozawa/FWO method) and [PERSON] [63] (Kissinger-Akahira-Sunose/KAS method), Starink methods [64] as well as Vyazovkin method [65; 66]. These are based on an isoconversional approach, with the conversion assumed to be constant and the degradation rate constant only depending on the temperature.  
The KAS method is a commonly used approach for calculating the kinetic parameters. The general equation of the KAS method is [56]:  
\[\ln\left(\frac{\beta}{T_{m}^{2}}\right)=-\frac{E}{R}\left(\frac{1}{T_{m}} \right)-\ln\left[\left(\frac{E}{A\cdot R}\right)\int_{0}^{\alpha}\frac{d\alpha }{f(\alpha)}\right] \tag{7}\]  
where \(T_{m}\) is the temperature at maximum decomposition rate. By plotting \(\ln\left(\frac{\beta}{T_{m}^{2}}\right)\) with respect to \(\frac{1}{T_{m}}\) and evaluation of the slope, one can calculate the apparent activation energy.  
According to the Starink method, the expressions proposed by FWO and KAS approaches can be transformed into a general equation, having the form (8) [67] that is an order of magnitude more accurate than the other two isoconversional methods (FWO and KAS).  
\[\ln\left(\frac{\beta}{T^{1.92}}\right)=const-1.0008\left(\frac{E_{\alpha}}{R \cdot T_{\alpha}}\right) \tag{8}\]  
One of the most used isoconversional methods is the one proposed by [PERSON], in which by taking the natural logarithm on both sides of relation (5), one can obtain [58]:  
\[\ln\left(\frac{d\alpha}{dt}\right)_{\alpha}=\ln\left[\beta\left(\frac{d\alpha }{dT}\right)_{\alpha}\right]=\ln\left((k_{0})_{\alpha}f(\alpha)\right)-\frac{ E_{\alpha}}{R\cdot T} \tag{9}\]  
In this approach, the conversion function \(f(\alpha)\) is considered independent of temperature, and the degradation process depends only on the mass loss rate. The evaluation of the activation energy can be carried out by plotting \(\ln\left(\frac{d\alpha}{dt}\right)\) with respect to \(\frac{1}{T}\) (usually a straight line) and determining the slope of this line.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Model**} & \multicolumn{2}{c}{**Differential Form—_f(a)_**} & \multicolumn{1}{c}{**Integral Form—_g(a)_**} \\ \hline
2D diffusion & D\({}_{2}\) & \(-[1/\ln(1-\alpha)]\) & \([(1-\alpha)\cdot\ln(1-\alpha)]+\alpha\) \\ \hline
3D diffusion (Jander) & D\({}_{3}\) & \(\left[3\cdot(1-\alpha)^{\frac{2}{3}}\right]/\left[2\cdot\left(1-(1-\alpha)^{ \frac{1}{3}}\right)\right]\) & \(\left[1-(1-\alpha)^{\frac{3}{3}}\right]^{2}\) \\ \hline
3D diffusion (Zhuravlev- & & \\ Lesokhin-Tempelman) & D\({}_{4}\) & \(3/\left[2\cdot\left((1-\alpha)^{\frac{1}{3}}-1\right)\right]\) & \(1-(2/3)\cdot\alpha-(1-\alpha)^{\frac{2}{3}}\) \\ \hline \multicolumn{4}{c}{Chemical transformation models} \\ \hline Zero order & F\({}_{0}\) & 1 & \(\alpha\) \\ \hline First order & F\({}_{1}\) & \((1-\alpha)\) & \(-\ln(1-\alpha)\) \\ \hline Second order & F\({}_{2}\) & \((1-\alpha)^{2}\) & \([1/(1-\alpha)]-1\) \\ \hline Third order & F\({}_{3}\) & \((1-\alpha)^{3}\) & \(1/2\left[(1-\alpha)^{-2}-1\right]\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._