# Thermogravimetric Study of Refuse Derived Fuel Produced from Municipal Solid Waste of Kazakhstan
## 2 Materials and Methods
### Kinetics of RDF Samples  
Due to the heterogeneous phases and complexity of the reaction steps in RDF samples, combustion characteristics of RDF and its fractions, isoconversional kinetics analysis were conducted. The rate of equation for solid phase reactions for heterogeneous RDF could be expressed as Equation (1):  
\[\frac{d\alpha}{dt}=k(t)f(\alpha)=Aexp\bigg{(}-\frac{E}{RT}\bigg{)}(1-a) \tag{1}\]  
where \(\frac{d\alpha}{dt}\) is the rate of conversion, \(k(t)\) is the rate of constant, \(f(\alpha)\) is depends on the reaction model, \(A\) is the pre-exponential Arrhenius factor (s\({}^{-1}\)), \(E\) is the apparent activation energy (kJ/mol), R is the universal gas constant, and T is the reaction temperature (K). The conversion rate (\(\alpha\)) was defined from the following Equation (2):  
\[\alpha=\frac{m_{o}-m_{t}}{m_{o}-m_{f}} \tag{2}\]  
where \(m_{o}\) is the initial weight of the sample; \(m_{t}\) is the weight at time t, and \(m_{f}\) is the final weight of the samples. Reaction kinetics was applied to the thermal data determined under non-isothermal conditions with a constant heating rate. For non-isothermal measurements linear heating rate, \(\beta\) is defined as \(\beta=\text{dT}/\text{dt}\), thus Equations (1) and (2) could be given as [29; 38; 39]:  
\[\frac{d\alpha}{dT}=\bigg{(}\frac{A}{\beta}\bigg{)}\exp\bigg{(}-\frac{E_{a}}{RT }\bigg{)}f(\alpha) \tag{3}\]  
Two different approaches were found as most commonly used models for integral isoconventional (free model) methods, which are FWO [25] and Kissinger-Akahira-Sunose (KAS) methods [22]. FWO method assumes the change in the apparent activation energy is constant during the thermal process and uses [PERSON]'s approximation [40; 41]. Thus,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Proximate (wt\%, as Received)** & **RDF** & **Plastic** & **Paper** & **Textile** \\ \hline Moisture & 1.5 & 0.8 & 2.6 & 1.2 \\ Volatile matter & 86.7 & 87.3 & 85.7 & 86.7 \\ Ash & 8.2 & 8.7 & 8.5 & 4.0 \\ Fixed carbon & 3.6 & 3.2 & 3.2 & 8.1 \\ \hline
**Ultimate (wt\%, and dry ash free)** & & & & \\ Carbon & 66.5 & 83.5 & 46.8 & 47.8 \\ Hydrogen & 9.9 & 13.0 & 6.4 & 6.1 \\ Nitrogen & 0.6 & 0.5 & 0.6 & 1.3 \\ Sulfur & 0.2 & 0.3 & 0.1 & 0.2 \\ Oxygen \({}^{*}\) & 22.8 & 2.7 & 46.1 & 44.6 \\ \hline
**Gross Calorific value (MJ/kg) \({}^{**}\)** & 23.4 & 32.6 & 16.4 & 20.1 \\ \hline \hline \end{tabular} \({}^{*}\) by difference; \({}^{**}\) measured by bomb calorimeter.  
\end{table}
Table 2: Proximate analysis, ultimate analysis, and calorific values of the samples.  
rearranging and integrating both sides of Equation (3) and with the Doyle's approximation, Equation (3) can be modified as [13; 38]:  
\[log\beta=\log\left(\frac{AE_{a}}{Rg(\alpha)}\right)-2.315-0.4567\frac{E}{RT} \tag{4}\]  
where the integration function \(g(\alpha)\) is considered as a constant value for a given conversion \(\alpha\) and it does not affect the activation energy. The activation energy \(E\) can be determined by the linear relationship between \(log\beta\) and 1/T at different heating rates but same conversion.  
The expression of KAS method applies approximation of p(u) = x\({}^{-2}\)e\({}^{-\mathrm{u}}\) to Equation (4). After rearranging, the expression becomes as [38]:  
\[ln\left(\frac{\beta}{T_{A}^{2}}\right)=\ln\left[\frac{AR}{E_{a}g(\alpha)} \right]-\frac{E}{RT} \tag{5}\]  
As in the previous method, the activation energy \(E\) can be obtained by the slope of straight line \(ln\left(\frac{\beta}{T^{2}}\right)\) vs. 1/T and R is 8.314 J/mol K.  
In addition to these kinetic models, [PERSON] method is assumed as isoconversional differential method which is expressed as below [42]:  
\[ln\left(\frac{d\alpha}{dt}\right)=\ln[f(\alpha)A_{a}]-\frac{E}{RT} \tag{6}\]  
Conversional function \(f(\alpha)\) in Friedman method is constant. It describes reaction, which is independent from the temperature and depends on mass loss rate. The activation energy was calculated by using the slope of ln\(\left(\frac{da}{dt}\right)\) vs. 1/T plot.