# Detailed Modeling of the Direct Reduction of Iron Ore in a Shaft Furnace

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}emissions from the steel industry. The shaft furnace is divided into three sections (reduction, transition, and cooling), and the model is two-dimensional (cylindrical geometry for the upper sections and conical geometry for the lower one), to correctly describe the lateral gas feed and cooling gas outlet. This model relies on a detailed description of the main physical–chemical and thermal phenomena, using a multi-scale approach. The moving bed is assumed to be comprised of pellets of grains and crystallites. We also take into account eight heterogeneous and two homogeneous chemical reactions. The local mass, energy, and momentum balances are numerically solved, using the finite volume method. This model was successfully validated by simulating the shaft furnaces of two direct reduction plants of different capacities. The calculated results reveal the detailed interior behavior of the shaft furnace operation. Eight different zones can be distinguished, according to their predominant thermal and reaction characteristics. An important finding is the presence of a central zone of lesser temperature and conversion.

## 1. Introduction

_{2}emissions, which are 40 to 60% lower for the DR-electric arc furnace route, compared to the blast furnace, basic oxygen route [2]. Among DR processes, shaft furnaces represent over 82% of the world’s DR iron production, with the two main processes being MIDREX (65%), as shown in Figure 1, and HYL-ENERGIRON (17%) [3].

_{2}, CO, or H

_{2}-CO mixtures [4,5,6,7,8,9]. Subsequent studies developed models that simulated the reduction zone of the shaft furnace in one dimension [10,11]. With the aim of correctly describing the lateral gas feed, some studies have introduced two-dimensional models [12,13,14]; however, these models did not consider the presence of methane, which is responsible for important reactions in the process. More recently, several authors introduced other reactions [15] and accounted for the cooling zone [16,17]. Some even developed plant models [18]; however, these works were limited to one-dimensional models.

_{2}-CH

_{4}reducing gas, and accounted for transition and cooling sections. The present model, named REDUCTOR, is 2-dimensional in the steady-state regime. The model includes a sophisticated, pellet sub-model. We consider eight heterogeneous and two homogeneous chemical reactions. These features represent a more advanced and detailed model, compared to previous studies. Moreover, the results were validated against two sets of plant data.

## 2. Mathematical Model

#### 2.1. Principle

_{0.95}O [19]), and by two gaseous reactants, namely, H

_{2}and CO. The following six reduction reactions were therefore considered:

- Methane decomposition reaction$$\text{}{\mathrm{CH}}_{4\left(\mathrm{g}\right)}\rightleftharpoons {\mathrm{C}}_{\left(\mathrm{s}\right)}+2{\mathrm{H}}_{2\left(\mathrm{g}\right)}\text{}$$
- Carbon monoxide disproportionation (inverse Boudouard reaction)$$\text{}2{\mathrm{CO}}_{\left(\mathrm{g}\right)}\text{}\rightleftharpoons {\mathrm{C}}_{\left(\mathrm{s}\right)}+{\mathrm{CO}}_{2\left(\mathrm{g}\right)}\text{}$$

_{p}) is assumed to be unique and unchanging during the reduction reaction, and the initial pellet composition is known. The gas phase is composed of six species: H

_{2}, CO, H

_{2}O, CO

_{2}, N

_{2}, and CH

_{4}. The reducing gas is injected from the sidewall, at a height of $\mathrm{z}={\mathrm{H}}_{\mathrm{Feed},\mathrm{gas}}$which then moves upward, against the solid flow, before finally exiting the furnace at the top. The temperature and composition of this reducing gas are known. A secondary feed gas—the cooling gas—which is introduced from the bottom of the furnace ($\mathrm{z}=-{\mathrm{H}}_{\mathrm{inf}}$), is also considered. This cooling gas exits the furnace, through the wall in the upper part of the conical section. The temperatures of the solid and gas are different and vary, according to their position (r, z) within the furnace. The solid temperature is assumed to be uniform in the interior of the pellets. Thus, this model is based on a faithful description of the physical-chemical and thermal phenomena, from the reactor scale to the crystallite scale, as shown in Figure 2. In the pellet sub-model, the pellet is assumed to be initially comprised of dense grains; these grains later fragment into smaller crystallites at the wüstite stage, in agreement with microscopic observations [19]. Thus, from the reactor to the crystallites, we have a 4-scale model.

#### 2.2. Equations

#### 2.2.1. Gas Phase

_{i}, given in Table 1.

#### 2.2.2. Solid Phase

_{j}, given in Table 2.

#### 2.3. Transport Coefficients

#### 2.4. Reaction Rates

#### 2.4.1. Iron Oxide Reduction

_{2}-CO. The reaction rate was used as a function of the local reduction conditions (temperature and gas composition), inside the reactor. We used the law of additive reaction times [23], which considers the different resistances (chemical reaction, diffusion, external transfer) involved in series. Therefore, the time required to attain a certain conversion is approximately the sum of the characteristic times: ${\tau}_{i}$ [14,23]. This sub-model was initially developed for simulating reduction by H

_{2}only, as detailed previously [14]; we extended this model for reduction by CO. The characteristic times and the reaction rates are listed in Appendix B.

#### 2.4.2. Methane Reforming and Water Gas Shift Reactions

_{7}, is given in Table 3. Because the reforming of CH

_{4}was hardly observed on the iron oxide catalysts, as reported in the literature [25], it was considered that such reforming only occurs with iron as a catalyst. We assumed that sufficient iron was formed on the outside of the pellet, when the reduction degree exceeded 50%.

_{0.95}O, and by Equation (22)

_{2}O

_{3}or Fe

_{3}O

_{4}. Here, besides iron, various iron oxides also catalyze the reaction. The corresponding expressions for ${k}_{8}$ and ${k}_{8}^{\prime}$ are given in Table 3, according the literature [24,25].

#### 2.4.3. Carbonization Reactions

_{3}C in the solid, with both being considered as C.

#### 2.5. Boundary Conditions

#### 2.6. Meshing and Numerical Solution

## 3. Results and Discussion

#### 3.1. Pressure Field, Velocity of Gas and Temperature Field

#### 3.2. Solid Mass Fractions

_{2}and CO and the temperature high—the conversion to iron was completed, in approximately 7 m. In the central part of the reactor, where the temperature was lower and the gas, lower in H

_{2}and CO, the conversion was not completed and some wüstite remained in the cooling zone. Though the average metallization degree was approximately 94%, metallization was not uniform, with most pellets being completely reduced, whereas others were not.

#### 3.3. Gas Mole Fractions

_{2}and CO contents. Above the gas inlet, the H

_{2}and CO contents decreased, while H

_{2}O and CO

_{2}were formed, as a result of the reduction reactions. In the central zone, with less reduction, lower amounts of H

_{2}O and CO

_{2}were formed, and part of the cooling gas, rich in CH

_{4}, was present.

#### 3.4. Overall Picture

_{2}and CO, involved in each reaction, and the molar percentage of methane, reformed by H

_{2}O or CO

_{2}, or decomposed to carbon and H

_{2}. This diagram is an illustration of how modeling work can help one to understand the detailed behavior of a reactor. Clearly, these results could not be obtained from other means.

#### 3.5. Validation

## 4. Conclusions

## Author Contributions

## Funding

_{2}in industry’, 2014–18, VALORCO, No 1382C0245; the authors thank Mrs. Nathalie Thybaud and Aïcha El Khamlichi, and the coordinator, Eric de Coninck; and (ii) one operated by the National Research Agency (ANR) and referenced by ANR-11-LABX-0008-01 (LabEx DAMAS).

## Acknowledgments

## Conflicts of Interest

## Appendix A. Notation

Latin | |

a_{b} | specific area of the bed (m^{2}/m^{3}) |

a_{c} | activity of carbon |

c_{t} | total molar concentration of the gas (mol m^{−3}) |

c_{pg} | molar specific heat of the gas (J mol^{−1} K^{−1}) |

c_{ps} | mass specific heat of the solid (J kg^{−1} K^{−1}) |

d_{p} | pellet diameter (m) |

D | diffusion or dispersion (D_{a}, D_{r}) coefficient (m^{2}/s) |

h | heat transfer coefficient (W m^{−2} K^{−1}) |

H | height of the cylindrical section of the shaft (m) |

H_{feed gas} | height of the reducing gas inlet (m) |

H_{inf} | height of the conical section of the shaft (m) |

K_{eq} | equilibrium constant |

K | permeability coefficient (kg m^{−3} s^{−1}) |

k | mass transfer coefficient, or reaction rate constant |

M | molar weight (kg mol^{−3}) |

p | gas pressure |

P_{i} | partial pressure of component i (bar) |

r | radius (m) |

R | ideal gas constant (J mol^{−1} K^{−1}) |

S | source term |

T | temperature (K) |

u | velocity (m s^{−1}) |

v | reaction rate (mol m^{−3} s^{−1}) |

w_{j} | mass fraction of solid j |

X | degree of conversion |

x_{i} | molar fraction of i in the gas |

z | height (m) |

Greek | |

Δ_{r}H | heat of reaction (J mol^{−1}) |

ε | porosity |

τ | characteristic time (s) |

λ | thermal conductivity (W m^{−1} K^{−1}) |

µ_{g} | viscosity of the gas (Pa s) |

ρ_{g} | mass density of the gas (kg m^{−3}) |

ρ_{b} | mass density of the bed (kg ${\mathrm{m}}_{\mathrm{bed}}^{-3}$) |

${\tilde{\rho}}_{j}$ | molar density of species j in the bed (${\mathrm{mol}\text{}\mathrm{m}}_{\mathrm{bed}}^{-3})$ |

Subscripts | |

b | bed |

c | catalyst |

cryst | crystallite |

chem | chemical |

diff | diffusional |

interg | intergranular |

ini | initial |

intrac | intra-crystallite |

interc | inter-crystallite |

∞ | in the bulk gas |

eff | effective (for the bed) |

eq | at equilibrium |

g | gas |

grain | grain |

p | pellet |

r | radial |

s | solid |

z | axial |

## Appendix B. Characteristic Times and Reaction Rates

**Table A1.**Kinetic sub-model of a single pellet. Expressions of the characteristic times. i: reaction number (see Section 2.1), k: H

_{2}or CO.

Hematite → Magnetite | Magnetite → Wüstite | Wüstite → Iron | |
---|---|---|---|

External transfer | ${\tau}_{ext,i}=\frac{{\tilde{\rho}}_{F{e}_{2}{O}_{3},ini}{d}_{p}}{18{k}_{g}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ | ${\tau}_{ext,i}=\frac{8{\tilde{\rho}}_{F{e}_{3}{O}_{4},ini}{d}_{p}}{57{k}_{g}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ | ${\tau}_{ext,i}=\frac{{\tilde{\rho}}_{F{e}_{0.95}O}{d}_{p}}{6{k}_{g}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ |

Intergranular diffusion | ${\tau}_{diff,interg\left(i\right)}=\frac{{\tilde{\rho}}_{F{e}_{2}{O}_{3},ini}{\left({d}_{p}\right)}^{2}}{72{\left({D}_{k,eff}\right)}_{interg,i}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ | ${\tau}_{diff,interg\left(i\right)}=\frac{2{\tilde{\rho}}_{F{e}_{3}{O}_{4},ini}{\left({d}_{p}\right)}^{2}}{57{\left({D}_{k,eff}\right)}_{interg,i}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ | / |

Intragranular diffusion | / | ${\tau}_{diff,intrag\left(i\right)}=\frac{2{\tilde{\rho}}_{F{e}_{3}{O}_{4},ini}{\left({d}_{grain,ini}\right)}^{2}}{57{\left({D}_{k,eff}\right)}_{intrag,i}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ | / |

Inter-crystallite diffusion | / | / | ${\tau}_{diff,interc\left(i\right)}=\frac{{\tilde{\rho}}_{{e}_{0.95}O}{\left({d}_{p}\right)}^{2}}{24{\left({D}_{k,eff}\right)}_{interc,i}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ |

Intra-crystallite diffusion (solid phase) | / | / | ${\tau}_{diff,intrac,\left(i\right)}=\frac{{\tilde{\rho}}_{F{e}_{0.95}O}{d}_{cryst,ini}^{2}}{24{D}_{sol}\left({c}_{ox,eq}-{c}_{ox,\infty}\right)}$ |

Chemical reaction | ${\tau}_{chem,i}=\frac{{\tilde{\rho}}_{F{e}_{2}{O}_{3}}{d}_{grain,ini}}{6{k}_{i}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ | ${\tau}_{chem,i}=\frac{{\tilde{\rho}}_{F{e}_{3}{O}_{4}}{d}_{grain,ini}}{2{k}_{i}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ | ${\tau}_{chem,i}=\frac{{\tilde{\rho}}_{F{e}_{0.95}O}{d}_{cryst,ini}}{2{k}_{i}{c}_{t}\left({x}_{k,\infty}-{x}_{k,eq\left(i\right)}\right)}$ |

**Table A2.**Kinetic sub-model of a single pellet. Expressions of the reaction rates. i: reaction number (see Section 2.1).

Reaction i | Reaction Rate mol m^{−3} s^{−1} |
---|---|

i = 1 and 4 | ${v}_{i}=\frac{1}{3}{\tilde{\rho}}_{F{e}_{2}{O}_{3},ini}{\left\{{\tau}_{ext,i}+2{\tau}_{diff,interg\left(i\right)}\left[{\left(1-{X}_{i}\right)}^{-\frac{1}{3}}-1\right]+\frac{{\tau}_{chem,i}}{3}{\left(1-{X}_{i}\right)}^{-\frac{2}{3}}\right\}}^{-1}$ |

i = 2 and 5 | ${v}_{i}=\frac{1}{3}{\tilde{\rho}}_{F{e}_{2}{O}_{3},ini}{\left\{{\tau}_{ext,i}+2\left({\tau}_{diff,interg\left(i\right)}+{\tau}_{diff,intrag\left(i\right)}\right)\left[{\left(1-{X}_{i}\right)}^{-\frac{1}{3}}-1\right]+\frac{{\tau}_{chem,i}}{3}{\left(1-{X}_{i}\right)}^{-\frac{2}{3}}\right\}}^{-1}$ |

i = 3 and 6 | ${v}_{i}=\frac{1}{3}{\tilde{\rho}}_{F{e}_{2}{O}_{3},ini}{\left\{{\tau}_{ext,i}+2\left({\tau}_{diff,interc\left(i\right)}+{\tau}_{diff,intrac,\left(i\right)}\right)\left[{\left(1-{X}_{i}\right)}^{-\frac{1}{3}}-1\right]+\frac{{\tau}_{chem,i}}{3}{\left(1-{X}_{i}\right)}^{-\frac{2}{3}}\right\}}^{-1}$ |

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**Figure 2.**Schematic representation of the REDUCTOR model, from the reactor scale to the crystallite scale (see Appendix A for notations).

**Figure 5.**(

**a**) Pressure field and velocity streamlines of gas flow inside the bed, (

**b**) temperature distribution of the gas phase, and (

**c**) temperature distribution of the solid phase.

Species i | S_{i} mol m^{−3} s^{−1} |
---|---|

${\mathrm{H}}_{2}$ | ${S}_{{H}_{2}}=-{v}_{1}-\frac{16}{19}{v}_{2}-{v}_{3}+3{v}_{7}+{v}_{8}+2{v}_{9}$ |

CO | ${S}_{CO}=-{v}_{4}-\frac{16}{19}{v}_{5}-{v}_{6}+{v}_{7}-{v}_{8}-2{v}_{10}$ |

${\mathrm{H}}_{2}\mathrm{O}$ | ${S}_{{H}_{2}O}={v}_{1}+\frac{16}{19}{v}_{2}+{v}_{3}-{v}_{7}-{v}_{8}$ |

${\mathrm{CO}}_{2}$ | ${S}_{C{O}_{2}}={v}_{4}+\frac{16}{19}{v}_{5}+{v}_{6}+{v}_{8}+{v}_{10}$ |

${\mathrm{CH}}_{4}$ | ${S}_{C{H}_{4}}=-{v}_{7}-{v}_{9}$ |

Species j | S_{j} kg m^{−3} s^{−1} |
---|---|

${\mathrm{Fe}}_{2}{\mathrm{O}}_{3}$ | $-3{M}_{{\mathrm{Fe}}_{2}{\mathrm{O}}_{3}}\left({v}_{1}+{v}_{4}\right)$ |

${\mathrm{Fe}}_{3}{\mathrm{O}}_{4}$ | ${M}_{{\mathrm{Fe}}_{3}{\mathrm{O}}_{4}}\left(2{v}_{1}-{v}_{2}+2{v}_{4}-{v}_{5}\right)$ |

${\mathrm{Fe}}_{0.95}\mathrm{O}$ | ${M}_{{\mathrm{Fe}}_{0.95}\mathrm{O}}\left(\frac{60}{19}{v}_{2}-{v}_{3}+\frac{60}{19}{v}_{5}-{v}_{6}\right)$ |

$\mathrm{Fe}$ | $0.95{M}_{\mathrm{Fe}}\left({v}_{3}+{v}_{6}\right)$ |

$\mathrm{C}$ | ${M}_{\mathrm{c}}\left({v}_{9}+{v}_{10}\right)$ |

Reactions | Reaction Rate Constants k_{i} | References | |
---|---|---|---|

7 | ${k}_{7}=392\mathrm{exp}\left(\frac{6770}{RT}\right)\left({\mathrm{mol}\text{}\mathrm{cm}}^{-3}{\mathrm{s}}^{-1}\right)$ | [25] | |

8 | $\mathrm{Fe}$ | ${k}_{8}=93.3\mathrm{exp}\left(-\frac{7320}{RT}\right)$$\left({\mathrm{mol}\text{}\mathrm{cm}}^{-3}{\mathrm{s}}^{-1}\right)$ | [25] |

${\mathrm{Fe}}_{0.95}\mathrm{O}$ | ${k}_{8}=1.83\times {10}^{-5}\mathrm{exp}\left(\frac{7.84}{RT}\right)$$\left({\mathrm{mol}\text{}\mathrm{cm}}^{-3}{\mathrm{s}}^{-1}\right)$ | [25] | |

${\mathrm{Fe}}_{3}{\mathrm{O}}_{4}$ | ${k}_{8}^{\prime}=2.683372\times {10}^{5}\mathrm{exp}\left(-\frac{112000}{RT}\right)$$\left({\mathrm{mol}\text{}\mathrm{kg}}_{\mathrm{cat}}^{-1}{\mathrm{s}}^{-1}\right)$ | [24] | |

${\mathrm{Fe}}_{2}{\mathrm{O}}_{3}$ | ${k}_{8}^{\prime}=4.56\times {10}^{3}\mathrm{exp}\left(-\frac{88000}{RT}\right)$$\left({\mathrm{mol}\text{}\mathrm{kg}}_{\mathrm{cat}}^{-1}{\mathrm{s}}^{-1}\right)$ | [24] | |

9 | ${k}_{9}=16250\mathrm{exp}\left(-\frac{55000}{RT}\right)$$\left({\mathrm{mol}\text{}\mathrm{m}}^{-3}{\mathrm{s}}^{-1}\right)$ | [16,26] | |

10 | ${k}_{10}=1.8\mathrm{exp}\left(-\frac{27200}{RT}\right)$$\left({\mathrm{mol}\text{}\mathrm{m}}^{-3}{\mathrm{s}}^{-1}\right)$ ${k}_{10}^{\prime}=2.2\mathrm{exp}\left(-\frac{8800}{RT}\right)$$\left({\mathrm{mol}\text{}\mathrm{m}}^{-3}{\mathrm{s}}^{-1}\right)$ | [16,26] |

Plant A | Plant B | Unit | ||||
---|---|---|---|---|---|---|

Plant Data | Reductor Results | Plant Data | Reductor Results | |||

Outlet solid | Composition (%) | |||||

Fe_{2}O_{3} | 0 | 0 | 0 | 0 | wt % | |

Fe_{3}O_{4} | 0 | 0 | 0 | 0 | wt % | |

FeO | 7.47 | 7.1 | n.a. | 4.3 | wt % | |

Fe | 85.72 | 85.9 | n.a. | 87.77 | wt % | |

C | 2 | 2.2 | 2 | 0.91 | wt % | |

Gangue | 4.71 | 4.8 | 6.3 | 7.02 | wt % | |

Production | 119.2 | 119.8 | 26.4 | 27.33 | t/h | |

Metallization | 93.8 | 94 | 93 | 95.3 | % | |

Outlet gas | Flow rate | 193 | 200 | n.a. | 54 | kNm^{3}/h |

Composition (%) | ||||||

H_{2} | 40.28 | 40.41 | 37 | 37.72 | vol % | |

CO | 19.58 | 19.89 | 18.9 | 20.87 | vol % | |

H_{2}O | 19.03 | 19.52 | 21.2 | 20.61 | vol % | |

CO_{2} | 17.09 | 14.69 | 14.3 | 13.13 | vol % | |

CH_{4} | 2.95 | 3.91 | }8.6 | 7.67 | vol % | |

N_{2} | 1.02 | 1.55 | vol % | |||

Temperature | 285 | 284 | n.a. | 285 | °C |

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**MDPI and ACS Style**

Hamadeh, H.; Mirgaux, O.; Patisson, F. Detailed Modeling of the Direct Reduction of Iron Ore in a Shaft Furnace. *Materials* **2018**, *11*, 1865.
https://doi.org/10.3390/ma11101865

**AMA Style**

Hamadeh H, Mirgaux O, Patisson F. Detailed Modeling of the Direct Reduction of Iron Ore in a Shaft Furnace. *Materials*. 2018; 11(10):1865.
https://doi.org/10.3390/ma11101865

**Chicago/Turabian Style**

Hamadeh, Hamzeh, Olivier Mirgaux, and Fabrice Patisson. 2018. "Detailed Modeling of the Direct Reduction of Iron Ore in a Shaft Furnace" *Materials* 11, no. 10: 1865.
https://doi.org/10.3390/ma11101865