2.1. The mathematical model for the continuous phases

The widely used Eulerian volumetric average [39] in the porous media concept was applied to the governing equations of the continuum phases denoted as phasei. In this regard, the conservation of mass, momentum and energy can be summarised as

$$\frac{\partial}{\partial t} \left( \epsilon \alpha\_i \rho\_i \phi\_i \right) + \nabla \cdot \left( \epsilon \alpha\_i \rho\_i \vec{v}\_i \phi\_i \right) = \nabla \cdot \left( \epsilon \alpha\_i \Gamma \nabla . \left( \phi\_i \right) \right) + \mathcal{S} \tag{1}$$

where E, α, r, v, Γ and S are void space, volume fraction, density, velocity, diffusion term and source term, respectively. In order to derive each equation, the variable, ϕ, should be replaced by the values reported in Table 1.


Table 1. Values for the governing equations.

The m<sup>0</sup>, F and H on the right hand side of each equation are the source terms representing mass, momentum and energy transfer between phases. The very well-known Ergun and Ranz-Marshall correlations are used to consider momentum transfer and energy exchange between phases.

The void space and particle diameter must also be known to solve the governing equations for the fluid phases. To this purpose, the XDEM calculates the volumetric average value of void space for each CFD cell by knowing the number of particles in each CFD cell based on a kernelbased interpolation procedure proposed by Xiao et al. [40].

### 2.2. The mathematical model for discrete entities

The XDEM predicts both dynamics and thermochemical conversion of a particulate system. The thermochemical conversion of a particle includes not only the internal temperature distribution but also transport of species through diffusion and/or convection and chemical conversion via reactions in a porous media. The velocity, position and acceleration are calculated in the dynamics module via the DEM method while the temperature and species distribution in the conversion module of the XDEM via the DPM method.

#### 2.2.1. Dynamics module

The proved DEM is based on the soft sphere model applied in the dynamic module. In this method, the particles are assumed to be deformable and allowed to overlap. The motion of each separate particle is tracked using the equations of classical mechanics. Newton's and Euler's second law for translation and rotation of each particle are integrated over time, and the particle positions are updated accordingly during time integration.

• Equation of motion:

$$m\_i \frac{d\overrightarrow{v\_i}}{dt} = m\_i \frac{d^2\overrightarrow{x\_i}}{dt^2} = \overrightarrow{F\_i}^c + \overrightarrow{F\_i^g} + \overrightarrow{F\_i}^{\text{ext}} \tag{2}$$

$$I\_i \frac{d\overrightarrow{\omega}\_i}{dt} = \sum\_{j=1}^{n} \overrightarrow{M}\_{i,j} \tag{3}$$

and tangential Fc,t

Table 2. Particle equations.

3. Shaft

2.2.2. Conversion module

three equilibrium reactions:

i,j !

Hertz-Mindlin model [41, 42] is used to calculate the contact forces.

Mass conservation <sup>∂</sup>

Momentum conservation � <sup>∂</sup>

Energy equation <sup>∂</sup>

Species equation <sup>∂</sup>

Boundary conditions �λeff <sup>∂</sup><sup>T</sup>

Reaction ri <sup>¼</sup> kre� <sup>E</sup>

collision forces generated while colliding with the neighbouring bodies.

<sup>∂</sup><sup>t</sup> εpr<sup>f</sup> � �<sup>þ</sup> <sup>∇</sup> ! : εpr<sup>f</sup> vf

The eXtended Discrete Element Method (XDEM): An Advanced Approach to Model Blast Furnace

<sup>∂</sup><sup>t</sup> rhp � � <sup>¼</sup> <sup>1</sup> rn ∂ <sup>∂</sup><sup>r</sup> rnλeff <sup>∂</sup><sup>T</sup> ∂r � � <sup>þ</sup> <sup>q</sup>

q 000 <sup>¼</sup> <sup>P</sup><sup>l</sup>

q 000

m� ¼

�Di ∂r<sup>f</sup> ,i ∂r � �

<sup>∂</sup><sup>t</sup> εpr<sup>f</sup> ,i � �<sup>þ</sup> <sup>∇</sup> ! : εpr<sup>f</sup> ,ivf

> ∂r � �

0 @

<sup>∂</sup><sup>x</sup> <sup>ε</sup>pp � � <sup>¼</sup> <sup>ε</sup>pμ<sup>f</sup>

<sup>r</sup> <sup>h</sup> ð Þ p,T �hp,m Lf <sup>Δ</sup><sup>t</sup> hp,T <sup>≥</sup> hp,m 0 hp,T<hp,m

! Þ¼m\_ s, <sup>f</sup>

000

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<sup>k</sup>¼<sup>1</sup> <sup>ω</sup>\_ kHk in case of chemical reaction

¼ �mL � <sup>f</sup> � m h � l,m � hl,ref � � in case of melting

1 A

!Þ ¼ <sup>1</sup> rn ∂

<sup>R</sup> ¼ αð Þþ TR � T<sup>∞</sup> q\_rad þ q\_ cond

<sup>R</sup> <sup>¼</sup> <sup>β</sup><sup>i</sup> <sup>r</sup><sup>f</sup> ,i,R � <sup>r</sup><sup>f</sup> ,i,<sup>∞</sup> � �

<sup>∂</sup><sup>r</sup> rnεpD <sup>∂</sup>r<sup>f</sup> ,i ∂r � � <sup>þ</sup> <sup>ε</sup>pm\_ s,f ,i

�

<sup>K</sup> vf !

�

RT T<sup>m</sup> Q<sup>n</sup> <sup>j</sup>¼<sup>i</sup> <sup>r</sup><sup>j</sup> γj

The current approach accounts for thermal conversion of a single particle via heat and mass transfer and chemical reactions. A discrete particle may be considered to consist of a solid, inert, liquid and gaseous phase, whereby local thermal equilibrium between the phases is assumed. The distribution of temperature and species within a single particle is accounted for by a system of one-dimensional and transient conservation equation as shown in Table 2.

A blast furnace is a vertical shaft furnace that reduces iron ores to the so-called pig iron for subsequent processing into steels of various qualities. Metallic ore, coke and flux, containing elements such as limestone, are charged in layers into the top of a blast furnace, while usually air is introduced under pressure through the tuyeres (nozzles) at the bottom of the blast furnace. The air is preheated to temperatures between 900� and 1250� C and reacts vigorously with preheated coke to form reducing gas (carbon monoxide). It streams upwards through the void space between the solid materials at high temperatures of app. 1650� C. Hence, a significant reduction of the iron ores takes place in the upper shaft or stack between the top and the cohesive zone of the blast furnace. During indirect reduction with carbon monoxide as opposed to direct reduction with carbon, the iron-bearing materials are stepwise reduced from haematite (Fe2O3) to magnetite (Fe3O4) and wustite (FeO) to produce liquid iron following

Here, Fext i ! is the summation of external forces acting on a particle such as drag and buoyancy forces from surrounding fluids. The contact force F<sup>c</sup> i ! of a particle is the sum of all normal Fc,n i,j !


Table 2. Particle equations.

The m<sup>0</sup>, F and H on the right hand side of each equation are the source terms representing mass, momentum and energy transfer between phases. The very well-known Ergun and Ranz-Marshall correlations are used to consider momentum transfer and energy exchange between

! μ ∇P þ g þ F

Equation ϕ Γ S Continuity 1 0 m<sup>0</sup>

Energy h κ H

The void space and particle diameter must also be known to solve the governing equations for the fluid phases. To this purpose, the XDEM calculates the volumetric average value of void space for each CFD cell by knowing the number of particles in each CFD cell based on a kernel-

The XDEM predicts both dynamics and thermochemical conversion of a particulate system. The thermochemical conversion of a particle includes not only the internal temperature distribution but also transport of species through diffusion and/or convection and chemical conversion via reactions in a porous media. The velocity, position and acceleration are calculated in the dynamics module via the DEM method while the temperature and species distribution in

The proved DEM is based on the soft sphere model applied in the dynamic module. In this method, the particles are assumed to be deformable and allowed to overlap. The motion of each separate particle is tracked using the equations of classical mechanics. Newton's and Euler's second law for translation and rotation of each particle are integrated over time, and

> d2 xi ! dt<sup>2</sup> <sup>¼</sup>Fc i ! <sup>þ</sup> <sup>F</sup><sup>g</sup> i ! <sup>þ</sup> <sup>F</sup>ext i !

is the summation of external forces acting on a particle such as drag and buoyancy

i !

Ii dω<sup>i</sup> ! dt <sup>¼</sup> <sup>X</sup><sup>n</sup> j¼1 Mi,j !

based interpolation procedure proposed by Xiao et al. [40].

the conversion module of the XDEM via the DPM method.

the particle positions are updated accordingly during time integration.

mi dvi ! dt <sup>¼</sup> mi

forces from surrounding fluids. The contact force F<sup>c</sup>

2.2. The mathematical model for discrete entities

Momentum v

128 Iron Ores and Iron Oxide Materials

Table 1. Values for the governing equations.

phases.

2.2.1. Dynamics module

• Equation of motion:

Here, Fext i !

and tangential Fc,t i,j ! collision forces generated while colliding with the neighbouring bodies. Hertz-Mindlin model [41, 42] is used to calculate the contact forces.

#### 2.2.2. Conversion module

The current approach accounts for thermal conversion of a single particle via heat and mass transfer and chemical reactions. A discrete particle may be considered to consist of a solid, inert, liquid and gaseous phase, whereby local thermal equilibrium between the phases is assumed. The distribution of temperature and species within a single particle is accounted for by a system of one-dimensional and transient conservation equation as shown in Table 2.

#### 3. Shaft

(2)

(3)

i,j !

of a particle is the sum of all normal Fc,n

A blast furnace is a vertical shaft furnace that reduces iron ores to the so-called pig iron for subsequent processing into steels of various qualities. Metallic ore, coke and flux, containing elements such as limestone, are charged in layers into the top of a blast furnace, while usually air is introduced under pressure through the tuyeres (nozzles) at the bottom of the blast furnace. The air is preheated to temperatures between 900� and 1250� C and reacts vigorously with preheated coke to form reducing gas (carbon monoxide). It streams upwards through the void space between the solid materials at high temperatures of app. 1650� C. Hence, a significant reduction of the iron ores takes place in the upper shaft or stack between the top and the cohesive zone of the blast furnace. During indirect reduction with carbon monoxide as opposed to direct reduction with carbon, the iron-bearing materials are stepwise reduced from haematite (Fe2O3) to magnetite (Fe3O4) and wustite (FeO) to produce liquid iron following three equilibrium reactions:

$$3\text{ Fe}\_2\text{O}\_3 + \text{CO} \to 2\text{ Fe}\_3\text{O}\_4 + \text{CO}\_2$$

$$\text{Fe}\_3\text{O}\_4 + \text{CO} \to 3\text{ FeO} + \text{CO}\_2$$

$$\text{FeO} + \text{CO} \to \text{Fe} + \text{CO}\_2$$

These reactions were experimentally investigated by Murayama et al. [43] under different CO-CO2 gas mixtures and were employed to predict and validate the individual reaction rate of the abovementioned reduction scheme for a single ore particle. The reduction degree was measured and is defined as the fractional removal of oxygen related to the initially available oxygen as

$$F = \frac{m\_O^0 - m\_O}{m\_O^0} = 1 - \frac{m\_O}{m\_O^0} \tag{4}$$

Figure 1. Comparison between experimental data and predictions for isothermal reduction [44] of haematite to magnetite.

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Figure 2. Comparison between experimental data and predictions for isothermal reduction of magnetite to wustite [44].

where the total mass of removable oxygen is defined as the oxygen mass to reach the next lower oxide in the sequence of haematite (Fe2O3), magnetite (Fe3O4), wustite (FeO) and iron (Fe) so that the reduction degree always varies between 1 and 0 for each individual reduction reaction. The reaction rate is expressed by an Arrhenius equation for which the relevant parameters are listed in Table 3.

The predicted results were compared to experimental data at different temperatures and are shown in the Figures 1–3. In general, a very satisfactory agreement between predicted and experimental data is achieved taking into account the temperature spread and experimental uncertainties such as material data. Hence, deviation occurrence is attributed to morphological changes such as porosity or crystal structure. In particular the latter was excluded from the study since crystal and grain structures address a length scale outside the scope of these investigations. Therefore, deviations of this scale are in general acceptable as pointed out by Peters [44].

The validated reaction mechanism was applied to a non-isothermal packed bed of iron ore particles [45] as encountered in the stack of a blast furnace, and the predicted reduction degree was compared to measurements as depicted in Figure 4. Both experiments and predictions were carried out for a packed bed consisting of pellets and pellets/nut coke. For both setups, the predicted integral reduction degree agrees well with experimental data and thus proves the superiority of the current DEM-CFD approach for predicting thermal conversion of packed beds.


Table 3. Kinetic parameters used in the XDEM simulation for reduction reactions.

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3 Fe2O3 þ CO ! 2 Fe3O4 þ CO2

Fe3O4 þ CO ! 3 FeO þ CO2

FeO þ CO ! Fe þ CO2

These reactions were experimentally investigated by Murayama et al. [43] under different CO-CO2 gas mixtures and were employed to predict and validate the individual reaction rate of the abovementioned reduction scheme for a single ore particle. The reduction degree was measured and is defined as the fractional removal of oxygen related to the initially available

> <sup>O</sup> � mO m0 O

where the total mass of removable oxygen is defined as the oxygen mass to reach the next lower oxide in the sequence of haematite (Fe2O3), magnetite (Fe3O4), wustite (FeO) and iron (Fe) so that the reduction degree always varies between 1 and 0 for each individual reduction reaction. The reaction rate is expressed by an Arrhenius equation for which the relevant

The predicted results were compared to experimental data at different temperatures and are shown in the Figures 1–3. In general, a very satisfactory agreement between predicted and experimental data is achieved taking into account the temperature spread and experimental uncertainties such as material data. Hence, deviation occurrence is attributed to morphological changes such as porosity or crystal structure. In particular the latter was excluded from the study since crystal and grain structures address a length scale outside the scope of these investigations. Therefore, deviations of this scale are in general acceptable as pointed out by

The validated reaction mechanism was applied to a non-isothermal packed bed of iron ore particles [45] as encountered in the stack of a blast furnace, and the predicted reduction degree was compared to measurements as depicted in Figure 4. Both experiments and predictions were carried out for a packed bed consisting of pellets and pellets/nut coke. For both setups, the predicted integral reduction degree agrees well with experimental data and thus proves the superiority of the current DEM-CFD approach for predicting thermal conversion of packed

Reduction E (J/mol) Kr (�) Fe2O3 ! Fe3O4 7.5010 � <sup>10</sup><sup>4</sup> 8.58 Fe3O4 ! FeO 7.8102 � <sup>10</sup><sup>4</sup> <sup>25</sup> FeO ! Fe 1.4601 � 105 <sup>2300</sup>

Table 3. Kinetic parameters used in the XDEM simulation for reduction reactions.

<sup>¼</sup> <sup>1</sup> � mO m0 O

(4)

<sup>F</sup> <sup>¼</sup> <sup>m</sup><sup>0</sup>

oxygen as

130 Iron Ores and Iron Oxide Materials

Peters [44].

beds.

parameters are listed in Table 3.

Figure 1. Comparison between experimental data and predictions for isothermal reduction [44] of haematite to magnetite.

Figure 2. Comparison between experimental data and predictions for isothermal reduction of magnetite to wustite [44].

and generate two different liquids, molten iron and slag. The softening and melting process makes the CZ to form impermeable ferrous layers forcing gas to flow horizontally through the permeable coke slits. This means that the CZ has remarkable effects on the operation of the blast furnace and the description of the characteristics of the CZ is extremely important. Since it is not possible to interrupt the BF to investigate details of the phenomena occurred inside, the

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The iron-bearing materials start softening at a temperature depending on the chemical composition, reduction degree and the load of burden. In this study, the softening range is considered to be from 1200 K to 1400 K according to Yang et al. [46, 47]. To describe the softening process in the CZ, a packed bed is randomly filled with particles assuming pellet which is shown in Figure 5. The radius of pellets varies in the range of 5.4–6.6 mm in a cylindrical vessel with 50 mm in diameter and height. The particles are coloured by their sizes. The value of load acting on the top of particles is determined 100 KPa according to the standard conditions in the softening test [48, 49]. For this test case, the conversion and dynamic modulus of the XDEM are used. The conversion module allows to heat up the particles from 1200 K to 1400 K, and the other module is responsible for the movement and the deformation of particle with the assistance of the relationship of Young's modulus (E) and temperature. It was assumed that this relationship is linear within the softening range of pellets. Figure 6 shows the influence of the changes of Young's modulus (E) within XDEM simulation on the rearrangement of pellets in the packed bed. It is observed that to reach %40 in bed shrinkage, it is needed that the value

numerical simulation becomes more practical.

of Young's modulus (E) is decreased approximately by two orders.

Figure 5. The geometry used in XDEM simulation for softening behaviour of some pellets.

Figure 3. Comparison between experimental data and predictions for isothermal reduction of wustite to iron [44].

Figure 4. Comparison of model prediction and experiments for reduction of a bed of pellets (red triangles) and pellets mixed with nut coke (green squares) [45].

#### 4. Cohesive zone

Cohesive zone (CZ) is formed between shaft and dripping zone. In this region, the reduced iron-bearing particles start softening because of the weight of burden above and the high temperature in the CZ. As particles soften, the porosity of the ore layer decreases causing a high pressure drop. As the temperature increases further, the softened particles start melting and generate two different liquids, molten iron and slag. The softening and melting process makes the CZ to form impermeable ferrous layers forcing gas to flow horizontally through the permeable coke slits. This means that the CZ has remarkable effects on the operation of the blast furnace and the description of the characteristics of the CZ is extremely important. Since it is not possible to interrupt the BF to investigate details of the phenomena occurred inside, the numerical simulation becomes more practical.

The iron-bearing materials start softening at a temperature depending on the chemical composition, reduction degree and the load of burden. In this study, the softening range is considered to be from 1200 K to 1400 K according to Yang et al. [46, 47]. To describe the softening process in the CZ, a packed bed is randomly filled with particles assuming pellet which is shown in Figure 5. The radius of pellets varies in the range of 5.4–6.6 mm in a cylindrical vessel with 50 mm in diameter and height. The particles are coloured by their sizes. The value of load acting on the top of particles is determined 100 KPa according to the standard conditions in the softening test [48, 49]. For this test case, the conversion and dynamic modulus of the XDEM are used. The conversion module allows to heat up the particles from 1200 K to 1400 K, and the other module is responsible for the movement and the deformation of particle with the assistance of the relationship of Young's modulus (E) and temperature. It was assumed that this relationship is linear within the softening range of pellets. Figure 6 shows the influence of the changes of Young's modulus (E) within XDEM simulation on the rearrangement of pellets in the packed bed. It is observed that to reach %40 in bed shrinkage, it is needed that the value of Young's modulus (E) is decreased approximately by two orders.

Figure 5. The geometry used in XDEM simulation for softening behaviour of some pellets.

4. Cohesive zone

132 Iron Ores and Iron Oxide Materials

mixed with nut coke (green squares) [45].

Cohesive zone (CZ) is formed between shaft and dripping zone. In this region, the reduced iron-bearing particles start softening because of the weight of burden above and the high temperature in the CZ. As particles soften, the porosity of the ore layer decreases causing a high pressure drop. As the temperature increases further, the softened particles start melting

Figure 4. Comparison of model prediction and experiments for reduction of a bed of pellets (red triangles) and pellets

Figure 3. Comparison between experimental data and predictions for isothermal reduction of wustite to iron [44].

variations with time agree well with the experimental results. Figure 9 shows that the ability of the XDEM to fully describe solid and liquid flows in a four-way CFD-DEM coupling considering particle-particle, particle-wall and particle-fluid interaction in terms of dynamics and thermal conversion. As it can be observed, the particles are risen and pushed downwards due to the buoyancy and drag force. Moreover, the particles which are faced the warm liquid earlier are heated up and melted faster than the others. It can be also seen that liquid flows downwards gradually because of the flow resistance from the particles and its temperature decreases by exchanging convection heat with the ice particles and mixing with produced melt. Moreover,

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Although we modelled solid–liquid flows under the melting process, the present technique is adequately robust and efficient to be applied to the complex melting process of granular packed bed by a hot gas which is a solid-gas-liquid system appropriate for many industrial

The liquid iron and slag produced in the cohesive zone trickle down through the packed bed of coke particles towards the hearth in a zone known as the dripping zone. It is located deep inside the blast furnace where measurements are not easy to perform. It highly affects the production rate, the quality of hot metal and process efficiency [53]. In this region, the two liquid phases descend slowly and hot gas introduced through the tuyeres ascends upwards

Figure 9. The evaluation of temperature and velocity of liquid and the size distribution of particles at t = 20 s.

The XDEM has been validated for one single ice particle that undergoes melting [52].

applications particularly blast furnace modelling.

5. Dripping zone and hearth

Figure 6. The effect of Young's modulus on structural rearrangement of the packed bed of pellets.

Due to the lack of appropriate experimental data for the melting of iron-bearing material by a hot gas in a real condition occurring in a CZ, the melting of a packed bed of ice particles in a solid– liquid system is selected. The simulation setup is shown in Figure 7. Figure 8 compares the predicted results with experimental data [51] of the mass loss history for a packed bed exposed to an water mass flow of a velocity of 0.1 m/s. The comparison shows that the predicted mass

Figure 7. Schematic representation of the packed bed of particles used in the simulation [50].

Figure 8. Comparison between experiments and predictions of the mass loss history of a melting packed bed.

variations with time agree well with the experimental results. Figure 9 shows that the ability of the XDEM to fully describe solid and liquid flows in a four-way CFD-DEM coupling considering particle-particle, particle-wall and particle-fluid interaction in terms of dynamics and thermal conversion. As it can be observed, the particles are risen and pushed downwards due to the buoyancy and drag force. Moreover, the particles which are faced the warm liquid earlier are heated up and melted faster than the others. It can be also seen that liquid flows downwards gradually because of the flow resistance from the particles and its temperature decreases by exchanging convection heat with the ice particles and mixing with produced melt. Moreover, The XDEM has been validated for one single ice particle that undergoes melting [52].

Although we modelled solid–liquid flows under the melting process, the present technique is adequately robust and efficient to be applied to the complex melting process of granular packed bed by a hot gas which is a solid-gas-liquid system appropriate for many industrial applications particularly blast furnace modelling.

Figure 9. The evaluation of temperature and velocity of liquid and the size distribution of particles at t = 20 s.
