**2. Mathematical formulation**

The microwave heating process is modeled using a 1D heat transfer equation given below.

$$\frac{\partial T}{\partial t} = \frac{1}{\rho c\_p} \frac{\partial k}{\partial \mathbf{x}} \frac{\partial T}{\partial \mathbf{x}} + \frac{k}{\rho c\_p} \frac{\delta T^2}{\delta \mathbf{x}^2} + \frac{P(\mathbf{x})}{\rho c\_p} \tag{1}$$

$$P(\mathbf{x}) = \frac{P\_0}{D\_p} \times \left(\exp\left(\mathbf{X} - \mathbf{x}\right) / D\_p\right) \tag{2}$$

$$\mathbf{D}\_{\mathbb{P}}\left(\text{depth of penetration}\right) = \frac{\lambda\_0}{2 \ast \pi \sqrt{\left(2 \ast \mu\_r'\right)}} \ast \left\{ \left[1 + \left(\frac{\mathcal{E}\_r''}{\mathcal{E}\_r'}\right)^2\right]^{1/2} - 1 \right\}^{-\lambda\_0} \tag{3}$$

Eq. (1) was discretized to study the heat distribution in each node with respect to time. The slab was discretized into 50 � 50 nodes using the equations given below.

$$\frac{\partial T}{\partial \mathbf{x}} = \frac{T\_{i+1}^{n+1} - T\_{i-1}^{n+1}}{2\delta \mathbf{x}} \tag{4}$$

$$\frac{\partial k}{\partial \mathbf{x}} = \frac{k\_{i+1}^{n+1} - k\_{i-1}^{n+1}}{2\delta k} \tag{5}$$

$$\frac{\delta T^2}{\delta \mathbf{x}^2} = \frac{T\_{i+1}^{n+1} - 2\, T\_i^{n+1} + T\_{i-1}^{n+1}}{\delta \mathbf{x}^2} \tag{6}$$

The schematic representation of a square slab with node points is shown in **Figure 1**. Here, i represents the horizontal domain, and i+1 and i�1 represent forward and backward nodes, respectively. Where i, varying from 1 to M, *i* = 1 represents the left boundary, and *i=M* represents the right boundary. The time-domain [0, t] was divided into n segments, each of duration δ*t* = *t*/*n.*

The implicit finite difference approximation method was used in this study, where the governing equations were discretized (**Figure 1**) into different domains in the form of node points.

Model assumptions are as follows:

1.Heat source is uniformly distributed.


#### **Figure 1.**

*Schematic representation of a square slab with node points i representing the horizontal domain, and i+1 and i*�*1 represent forward and backward nodes, respectively.*

#### **2.1 Modeling and Process Parameter**

**Figure 2.**

*Schematic representation of a square slab where P0 is microwave power flux (MW/m<sup>2</sup> ) and 2X is the length of the slab.*

#### **2.2 Simulation parameter calculation process**

Proximate analyses and ultimate analyses of coal were taken from Ref. [19]. Now, the amount of carbon monoxide required to convert hematite to magnetite was estimated by using equations given below (**Tables 1** and **2**) [21].

$$\text{2 }\text{C} + \text{O}\_2 = \text{2 }\text{CO} \tag{7}$$

$$
\Delta G\_f^0 = -111700 - 87.65T \tag{8}
$$

$$2\,Fe\_2O\_3 + CO = 2\,Fe\_3O\_4 + CO\_2 \tag{9}$$

$$
\Delta G\_f^0 = -44300 - 39.89T \tag{10}
$$

The fraction of coal needed is estimated using proximate and ultimate analyses, and we found that 7.5% of coal is required. Iron ore and coal powder were mixed

### *Simulation Study of Microwave Heating of Hematite and Coal Mixture DOI: http://dx.doi.org/10.5772/intechopen.106312*


#### **Table 1.**

*Proximate analyses of the coal sample.\**


#### **Table 2.**

*Ultimate analyses of the coal sample.\**

properly according to a stoichiometric calculation for the reduction of hematite into a desirable amount of magnetite.

Initial and final boundary conditions are given below:

$$\mathbf{t} = \mathbf{0}, T = T\_0, \mathbf{0} \le \mathbf{x} \le \mathbf{X} \tag{11}$$

$$\mathbf{x} = \mathbf{0}, -\mathbf{k}\ \frac{\partial T}{\partial \mathbf{x}} = \mathbf{0}, \mathbf{t} > \mathbf{0} \tag{12}$$

$$\mathbf{x} = \mathbf{X}, -\mathbf{k}\ \frac{\partial T}{\partial \mathbf{x}} = \mathbf{h} \ (\mathbf{T} - \mathbf{T}\_{\infty}) + \mathbf{\mathcal{E}}\ \sigma \left(\mathbf{T}^{4} - \mathbf{T}\_{\infty}{}^{4}\right) \tag{13}$$
