**4. Kinetic Analysis**

#### *4.1. Unreacted Core Model for Coke*

From the fractional reaction curves obtained from the combustion experiment, the combustion reaction rate constant was determined using the unreacted core model [7]. The combustion reaction has five processes [8].

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1. O2 transport from the gas phase to the particle surface through the gas film:

$$-\dot{n}\_{\mathbb{S}^\*O\_2} = 4\pi r\_0^2 k\_f \left(\mathbb{C}\_{O\_2} - \mathbb{C}\_{O\_2 \times s}\right) \tag{3}$$

2. O2 transport from the particle surface to the reaction interface through the alumina powder layer after coke combustion:

$$-\dot{n}\_{d \cdot O\_2} = (D\_{O\_2})\_{eff} \frac{4\pi r\_0 r\_i}{r\_0 - r\_i} \left(\mathbb{C}\_{O\_2 \cdot s} - \mathbb{C}\_{O\_2 \cdot i}\right) \tag{4}$$

3. The combustion reaction at the reaction interface:

$$-\dot{R} = 4\pi r\_i^2 k\_\varepsilon \left(\mathbb{C}\_{O\_2 \cdot i} - \frac{\mathbb{C}\_{CO\_2 \cdot i}}{K}\right) \tag{5}$$

4. CO2 transport from the reaction interface to the particle surface through the alumina powder layer after coke combustion:

$$\dot{m}\_{\rm d\cdot CO\_2} = \left(D\_{\rm CO\_2}\right)\_{\varepsilon f \, f} \frac{4\pi r\_0 r\_i}{r\_0 - r\_i} \left(C\_{\rm CO\_2 \cdot i} - C\_{\rm CO\_2 \cdot g}\right) \tag{6}$$

5. CO2 transport from the particle surface to the gas phase through the gas film:

$$
\dot{m}\_{\text{g-CO}\_2} = 4\pi r\_0^2 k\_f \left( \mathbf{C}\_{\text{CO}\_2\cdot s} - \mathbf{C}\_{\text{CO}\_2} \right) \tag{7}
$$

The overall rate equation can be described by the quasi-steady state analysis method below:

$$-\dot{n} = \frac{4\pi r\_0^2 \left(\frac{K}{1+K}\right) \left(C\_{O\_2} - C\_{CO\_2}\right)}{\frac{1}{k\_f} + \frac{1}{D\_c} \cdot \frac{r\_0 (r\_o - r\_i)}{r\_i} + \frac{1}{k\_c} \cdot \frac{K}{1+K} \left(\frac{r\_0}{r\_i}\right)^2} \tag{8}$$

$$\frac{1}{D\_{\ell}} = \frac{K}{1+K} \left( \frac{1}{\left(D\_{O\_2}\right)\_{eff}} + \frac{1}{K \left(D\_{\text{CO}\_2}\right)\_{eff}} \right) \tag{9}$$

Equation (8) can be expressed by the following equation, assuming that the combustion reaction of coke was an irreversible reaction and the equilibrium constant *K* infinite:

$$-\dot{n} = \frac{4\pi r\_0^2 C\_{O\_2}}{\frac{1}{k\_f} + \frac{1}{D\_r} \cdot \frac{r\_0(r\_0 - r\_i)}{r\_i} + \frac{1}{k\_c} \cdot \left(\frac{r\_0}{r\_i}\right)^2} \tag{10}$$

. *n* can be replaced by the following equation:

$$-\dot{m} = -\frac{d}{dt}(\frac{4}{3}\pi r\_i^3 \rho\_{\text{Cm}}) = -4\pi r\_i^2 \rho\_{\text{Cm}} \cdot \frac{dr\_i}{dt} \tag{11}$$

The reaction ratio *F* is expressed by Equation (12):

$$F = 1 - \left(\frac{r\_i}{r\_0}\right)^3 \tag{12}$$

When Equations (10)–(12) are combined and integrated under boundary conditions; *r* = *r0* at *t* = 0 and *r* = *ri* at *t* = *t*, Equation(13) is obtained:

$$t = \frac{\rho\_{\text{Cm}} r\_0}{\mathcal{C}\_{\text{O}\_2}} \cdot \left| \frac{F}{3k\_f} + \frac{d\_r}{16D\_\varepsilon} \left\{ 3 - 3(1 - F)^{\frac{2}{3}} + 2F \right\} + \frac{1}{k\_\mathbb{C}} \left\{ 1 - (1 - F)^{\frac{1}{3}} \right\} \right| \tag{13}$$

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The gas film mass transfer coefficient, *kf*, can be calculated from Ranz–Marshall's Equation [9]. The value of the effective diffusion coefficient in the alumina layer, *De*, and the interfacial reaction rate coefficient of coke, *kC*, was obtained by parameter-fitting using the nonlinear least-squares method to the fractional reaction curves.

*De*, and *kC* can be expressed by substituting the coefficients in Arrhenius' equation as shown:

$$k\_{\mathbb{C}} = A\_{(k\_{\mathbb{C}})} \exp\left(-\frac{E\_{\mathfrak{a}\left(k\_{\mathbb{C}}\right)}}{RT}\right) \tag{14}$$

$$D\_{\varepsilon} = A\_{\left(D\_{\varepsilon}\right)} \exp\left(-\frac{E\_{a\left(D\_{\varepsilon}\right)}}{RT}\right) \tag{15}$$

Equations (13) and (14) can be transformed into the following equations:

$$
ln k\_{\mathbb{C}} = -\frac{E\_{\mathfrak{a}(k\_{\mathbb{C}})}}{R} \cdot \frac{1}{T} + \ln A\_{(k\_{\mathbb{C}})} \tag{16}
$$

$$
\ln D\_{\varepsilon} = -\frac{E\_{\rm af}(D\_{\varepsilon})}{R} \cdot \frac{1}{T} + \ln A\_{(D\_{\varepsilon})} \tag{17}
$$

Figure 7 shows the Arrhenius plot of *kc*. The values of *kc* are at the same level in all samples. The temperature dependence of *kc* is expressed as

$$\begin{array}{llll} \text{Code} & \text{(-125 \,\mu m)} & k\_{\text{c}} = 6.02 \times 10^{-2} \,\text{exp}(-9.32 \times 10^{3}/\text{RT}) & \text{(m/s)}\\ & \text{(125-250 \,\mu m)} & k\_{\text{c}} = 4.51 \times 10^{-2} \,\text{exp}(-5.12 \times 10^{3}/\text{RT}) & \text{(m/s)} \end{array}$$

**Figure 7.** Temperature dependence of the reaction rate constants *kc*.

Figure 8 shows the Arrhenius plot of *De*. The temperature dependence of *De* can be expressed as


**Figure 8.** Temperature dependence of the effective diffusivities *De*.

### *4.2. Chemical Reaction Control Step for Charcoal*

Because the surface area of charcoal is larger than that of coke, when a combustion reaction takes place, the reaction area of charcoal will also be larger. Moreover, due to the low ash ratio in the charcoal, the O2 transportation rate in the alumina layer will be large. Therefore, the reaction is based on the chemical reaction control step.

For the charcoal samples, the chemical reaction control step can be used in the analysis method shown below:

The combustion reaction at the reaction interface can be expressed by Equation (5).

The combustion rate can be expressed as

$$-\dot{n} = \frac{4\pi r\_0^2 \left(\frac{K}{1+K}\right) \left(C\_{O\_2} - \frac{C\_{CO\_2}}{K}\right)}{\frac{1}{k\_c} \cdot \frac{K}{1+K} \left(\frac{r\_0}{r\_i}\right)^2} \tag{18}$$

Under boundary conditions, *r* = *r0* at *t* = 0 and *r* = *ri* at *t* = *t*, and this gives Equation (19):

$$\left[1 - (1 - F)^{\frac{1}{\tilde{\mathcal{I}}}}\right] = \frac{C\_{O\_2}}{\rho c\_m r\_0} k\_c t \tag{19}$$

Based on the reaction curves obtained by the experiments, *kc* was determined using the unreacted core model [3].

*kC* can be substituted in Arrhenius' equation as shown by Equation (14) which also can be transformed as Equation (16).

Figure 9 shows the Arrhenius plot of *kc*.

The temperature dependence of *kc* can be expressed as

$$\begin{array}{ll} \text{Code} & \text{(-125 \text{\textquotedblleft}nm)} & \text{k}\varepsilon = 8.24 \times 10^{-3} \exp\left(-10.6 \times 10^{3}/RT\right) \\\\ \end{array} \tag{m\text{\textquotedblleft}m}$$

$$\left(125 \text{--} 250 \,\upmu\text{m}\right) \qquad \qquad kc = 8.54 \times 10^{-3} \exp\left(-10.3 \times 10^{3}/RT\right) \tag{m/s}$$

**Figure 9.** Temperature dependence of the reaction rate constants *kc*.

#### **5. Sintering Simulation Model**

#### *5.1. Simulation Method*

The simulation condition was based on the study results using Ohno's model [10]. The model has S'-type, C-type and P-type quasi-particles as shown in Figure 10. The S' type was calculated using Hottel's equation [11–13], while the C and P types were calculated based the results obtained in this study.

**Figure 10.** Classification of the quasi-particles.

Our mathematical model is based on the Dwight–Lloyd sinter machine and the calculation range is from the ignition point on the pallet to the discharge of the sinter ore.

The numerical analysis was based on the control volume method shown in Figure 11. The control volume method is obtained by dividing the analysis target region into equal minute portions. Various basic equations, representing phenomena, such as the continuous and energy conservation equations, govern the inside of the analysis target area and are relational equations to be established in each control volume. Assuming that these control volumes are in a sufficiently small area, there would not

be a large error even if the changes in various quantities inside the control volumes were linear or approximated to be constant values. In other words, changes in various quantities in a certain control volume can be represented using values at a representative point in the control volume adjacent to the representative point. In our mathematical model, the sintering material layer was divided into minute control volumes in one dimension, the basic governing equations were discretized, and each difference approximation equation was solved using an explicit method.

**Figure 11.** Pattern diagram of the control volume.

In the model used, the temperature distribution was estimated considering the combustion of carbonaceous materials, the decomposition reaction of CaCO3, the evaporation and condensation of water, and the formation and solidification of calcium ferrite melt according to Ohno's model [10].

*De* depends on coke distribution.

The combustion reaction rate of the quasi-particles is expressed by Equations (20) and (21):

$$r\_{Quasi-particle}^{\*} = 4\pi r\_{Quasi-particle}^{2} k' \mathbb{C}\_{O\_2} \tag{20}$$

$$k'=1/\left(\frac{1}{k\_f} + \frac{r\_0(r\_0 - r\_i)}{D\_c r\_i} + \frac{r\_0^2}{k\_c r\_i^2}\right) \tag{21}$$

In this equation, *De* has a value of 108 because the resistance of the diffusion can be ignored. The material balance is calculated using Equation (22):

$$\frac{\left.\rho\_{\rm i\cdot x}\right|\_{\rm t+\Delta t} - \left.\rho\_{\rm i\cdot x}\right|\_{\rm t}}{\Delta t} = -\frac{\left.\left(\rho\_{Z+\Delta Z}u\_{\rm i}\right)\right|\_{Z+\Delta Z} - \left.(\rho\_Z u\_{\rm i-1})\right|\_Z}{\Delta Z} + r\_{i,x}^\* \tag{22}$$

The material balance equations of N2, O2, CO2 and H2 respectively, can be expressed as follows:

$$\frac{\partial \left(\rho\_{N\_2} u\right)}{\partial Z} = -\frac{\partial \rho\_{N\_2}}{\partial t} \tag{23}$$

$$\frac{\partial \left(\rho\_{\rm O\_2} u\right)}{\partial Z} = -\frac{\partial \rho\_{\rm O\_2}}{\partial t} - r\_{\rm Coke}^\* \tag{24}$$

$$\frac{\partial \left(\rho\_{\text{CO}\_2} u\right)}{\partial Z} = -\frac{\partial \rho\_{\text{CO}\_2}}{\partial t} + r\_{\text{Coke}}^\* + r\_{\text{CaCO}\_3}^\* \tag{25}$$

$$\frac{\partial \left(\rho\_{H\_2O}u\right)}{\partial Z} = -\frac{\partial \rho\_{H\_2O}}{\partial t} + r\_{H\_2O}^\* \tag{26}$$

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Because convection did not occur in the solid phase, the thermal budget of the phase considering heat transfer and reaction heat can be represented as

$$\begin{aligned} \rho\_3 \mathbf{C}\_{P\cdot S} \frac{\partial T\_b}{\partial \mathbf{t}} &= \frac{6(1 - \iota\_a)}{d} h \Big( T\_\mathbf{S} - T\_\mathbf{s} \Big) + H\_{\text{Cok}} \Big( r^\*\_{\text{Cok}} n\_{\text{Cok}} + r^\*\_{\text{Quasi-particle}} n\_{\text{Quasi-particle}} \Big) \\ &+ H\_{\text{CaCO}\_3} r^\*\_{\text{CaCO}\_3} n\_{\text{CaCO}\_3} + H\_{\text{H}\_2\text{O-}V} r^\*\_{\text{H}\_2\text{O-}V} + H\_{\text{CF-G}} r^\*\_{\text{CF-G}} + H\_{\text{CF-S}} r^\*\_{\text{CF-S}} = k \frac{\partial^2 T\_s}{\partial \mathbf{Z}^2} \end{aligned} \tag{27}$$

The thermal budget of the gas phase factoring the heat transfer and combustion reaction heat can be expressed as

$$\rho\_{\mathcal{J}} \mathbb{C}\_{P \cdot \mathcal{J}} \frac{\partial T\_{\mathcal{S}}}{\partial t} - \frac{6(1 - \varepsilon\_{a})}{\dot{d}} h \Big(T\_{\mathcal{s}} - T\_{\mathcal{S}}\Big) + \mathbb{C}\_{P \cdot \mathcal{J}} \frac{\partial \rho\_{\mathcal{K}} u T\_{\mathcal{S}}}{\partial \mathcal{Z}} = k \frac{\partial^{2} T\_{\mathcal{S}}}{\partial \mathcal{Z}^{2}} \tag{28}$$

The particles were charged in the control volume. Therefore, the pressure loss of the fluid also needed to be considered. The pressure loss of a laminar-turbulent transition area can be represented by Ergun's equation as shown:

$$\frac{\Delta P}{\Delta Z} = \frac{150(1 - \varepsilon\_a)^2}{\left(qd\right)^2 \varepsilon\_a^{-3}} \cdot \frac{\mu\_\mathcal{g}}{\rho\_\mathcal{g}} \mathcal{U} + 1.75 \frac{1 - \varepsilon\_a}{qd \varepsilon\_a^{-3}} \mathcal{U}^2 \tag{29}$$

#### *5.2. Calculation Conditions*

Table 2 lists the common calculation conditions for the sintering process. The composition of raw materials was set to simplify the calculation condition. The influence of MgO was not considered in this study. The particle size of hematite was set to 2.5 mm and 0.25 mm. It was assumed that 2.5 mm and 0.25 mm were the sizes of the nuclear particle and the adhering fine ores, respectively, in the quasi-particle. The 5.1 mass % charcoal calculation was compared with the 4 mass % coke calculation when the fixed carbon content is the same which means that the combustion heat of coke and charcoal during this process is the same. However, in this study, the effect of V.M. was not discussed. As a thought, the gas generated when V.M. is heated may improve the permeability and affect the temperature profile in the same way as a gas fuel injection, mentioned by Oyama [14]. Further research is needed to clarify this factor.


**Table 2.** Common calculation conditions.

Table 3 lists the state of the coke quasi-particles in the sinter bed for calculation using the date of the sinter pot test based on Hida's study [15].


**Table 3.** Existing state of the coke quasi-particles in the sinter bed.
