*2.3. Thermal Analysis Kinetic*

The combustion process of the sintering fuel can be regarded as a gas–solid heterogeneous reaction. The total combustion reaction consists of two independent chemical reactions:

$$\text{Anthraite} + (\text{x}/2 + y + z/2)\text{O}\_2 \rightarrow \text{xCO} + y\text{CO}\_2 + z\text{H}\_2\text{O} \tag{4}$$

$$\text{Coke} + (\text{x}/2 + y + z/2) \text{O}\_2 \rightarrow \text{xCO} + y \text{CO}\_2 + z \text{H}\_2\text{O} \tag{5}$$

In order to further clarify the combustion reaction mechanism of the pure mixed fuel and the quasi-particle fuel, we introduced two kinetic models to study the combustion behavior of the sample:

$$\frac{\mathbf{d}\alpha}{\mathbf{d}t} = \sum\_{i=1}^{2} c\_i k\_i f(\alpha\_i) \tag{6}$$

where *t* is the reaction time, *ci* is the proportion of a reaction to the total response, *ki* is the combustion reaction rate constant, and *f*(α*i*) is a function of the differential reaction mechanism.

The relationship between apparent reaction rate and temperature can be derived from the Arrhenius equation,

$$k = A\varepsilon^{-E/RT} \tag{7}$$

where *E* is reaction activation energy, *A* is the pre-exponential factor, *R* is the universal gas constant, and *T* is the temperature.

Currently, volumetric models (VM) and random pore models (RPM) are widely used to describe various coal char combustion reactions and are used to calculate kinetic parameters,

$$\frac{d\alpha\_{\rm VM}}{dt} = A\_{\rm VM}e^{-E\_{\rm VM}/RT}(1 - \alpha\_{\rm VM})\tag{8}$$

$$\frac{d\alpha\_{\rm RPM}}{dt} = A\_{\rm RPM}e^{-E\_{\rm RPM}/RT}(1 - \alpha\_{\rm RPM})\sqrt{1 - \psi \ln(1 - \alpha\_{\rm RPM})}\tag{9}$$

where ψ is the parameter of particle structure,

$$
\psi = \frac{4\pi L\_0 (1 - \varepsilon\_0)}{S\_0^2} \tag{10}
$$

where *S*<sup>0</sup> is the pore surface area, *L*<sup>0</sup> is the pore length, and ε<sup>0</sup> is the porosity of particles.

*Processes* **2020**, *8*, 475

Since the experimental materials use two types of fuels with different combustion performances, it is necessary to optimize the VM and RPM. The expressions of DVM and DRPM are obtained by combining Equations (6)–(9),

$$\frac{d\alpha\_{\rm DNM}}{dt} = \sum\_{i=1}^{2} c\_i A\_i e^{-Ei/RT} (1 - \alpha\_i) \tag{11}$$

$$\frac{d\alpha\_{\rm DRPM}}{dt} = \sum\_{i=1}^{2} c\_i A\_i e^{-E\_i/RT} (1 - \alpha\_i) \sqrt{1 - \psi\_i \ln(1 - \alpha\_i)} \tag{12}$$

In the non-isothermal analysis experiment, in order to determine the kinetic parameters and improve the calculation accuracy, three or more types of heating rates are usually selected. Thus, this experiment adopts four different heating rates to calculate kinetic parameters. Under the constant heating rate of the experiment, the reaction temperature can be obtained from the initial temperature and reaction time,

$$T = T\_0 + \beta t\tag{13}$$

where β is the heating rate, and *T*<sup>0</sup> is the starting temperature of 25 ◦C. After *t* = (*T* − *T*0)/β is substituted into Equations (11) and (12), the formulas can be integrated to give

$$\alpha\_{\rm DVM} = \sum\_{i=1}^{2} c\_i \left( 1 - \exp\left( -\frac{A\_i RT^2}{\beta E\_i} \cdot \exp\left(\frac{-E\_i}{RT}\right) \right) \right) \tag{14}$$

$$a\_{\rm DRPM} = \sum\_{i=1}^{2} c\_i \left( 1 - \exp\left( -\exp\left(\frac{-E\_i}{RT}\right) \frac{A\_i RT^2}{\beta E\_i} \cdot \left(1 + \exp\left(\frac{-E\_i}{RT}\right) \frac{\psi i A\_i RT^2}{4\beta E\_i}\right) \right) \right) \tag{15}$$

The combustion kinetic parameters were calculated by the above two kinetic models at different heating rates. The experimental data of the reaction rate (dα/d*t*) and conversion rate (α) were fitted in 1stop software using a nonlinear least-squares method. Then, the parameters *A*, *E* and ψ that are obtained are substituted into Equations (14) and (15) to obtain the relationship between the sample conversion rate (α) and the temperature (*T*) during combustion. At the same time, due to the possible deviation between the actual value and the calculated value of the model, the root mean square error (*RMSE*) is introduced to evaluate the error between the fitted data and the actual value of the DVM and DRPM models,

$$RMSE(a) = \frac{\sqrt{\sum\_{i=1}^{N} \left(a\_{\text{exp}}^{i} - a\_{\text{cal}}^{i}\right)^{2}}}{N} \times 100\% \tag{16}$$

$$RMSE \left(\frac{\text{d}\alpha}{\text{dt}}\right) = \frac{\sqrt{\sum\_{i=1}^{N} \left(\frac{\text{d}\alpha^{i}}{\text{dt}\,\text{exp}} - \frac{\text{d}\alpha^{i}}{\text{dt}\,\text{cal}}\right)^{2}}}{N} \times 100\% \tag{17}$$

where α*<sup>i</sup>* exp and α*<sup>i</sup>* cal are the experimental and calculated values of the conversion rate at points *i* = 1, 2, 3, ... ; <sup>d</sup><sup>α</sup> dt *i* exp and <sup>d</sup><sup>α</sup> dt *i* cal are the experimental and calculated values of reaction rate at some points, and *N* is the number of data points.
