Experimental and Modeling Study on Reduction and Heat Transfer Characteristics of Single Iron Ore Pellet in H₂/CO Atmospheres

Abstract

Experimental and modeling study was carried out for the reduction of single iron ore pellet in the H₂/CO atmosphere with considering the porosity evolution and heat transfer characteristics of product layer, reaction layer, and unreacted layer. A mathematical model was proposed and validated with a good match to the experimental data. The experimental results showed that both porosity and effective diffusion coefficient increased with the reduction time, temperature, and H₂ content in the gas mixtures. With the increase of temperature, the pore size presented multi-level distribution characteristics. Adding 20 vol.% CO in H₂ as a threshold value (H₂/CO = 8:2) proved to achieve higher reduction degree than other conditions. It was found that the addition of CO in H₂ increased the temperature of the reaction layer by about 10 to 27°C higher than the initial temperature. Temperature profiles and temperature difference evolution of these three layers inside the pellet presented a first decrease, where the reduction degree was about 0.3, and then an increase during the reduction process. With more CO participated in the reaction, a delayed high-temperature reaction layer was presented from extra heat supply of CO reduction, owing to the low diffusion rate. The radiation heat was dominant for the heat transfer between the environment and iron ore pellet, while inside the pellet heat flux changed among three layers and verified with the temperature and gas atmosphere.

1. Introduction

The Blast furnace (BF) ironmaking technology, which is the main process of global steel production has achieved much improvements for the reduction of CO₂ emission in last few decades, while dominated energy consumption and CO₂ emission still limit the development of a low-carbon economy. Comparatively, gas-based direct reduction ironmaking technology (DRI), such as Midrex and HYL/Energiron, using gaseous reactants (e.g. CO, H₂, and H₂/CO mixtures) to reduce iron ores can omit the coke process and decrease bulk of CO₂ emission.¹⁾ Unlike the carbothermal reaction of solid carbon with iron ores (e.g. iron ore pellet, sinter, and lump), the heat transfer and reaction kinetic characteristics of the gas-based reduction process are essential to final reduction degree and metallization degree.

Since iron ore pellet was widely used in direct reduction ironmaking (DRI), studies of basic experiment, mathematical modeling, and numerical simulation on the reduction kinetics of iron ore pellet ore were reported in many literatures.^2,3,4,5) As a review of Zare Ghadi, et al.²⁾ for the gaseous reduction of iron ore pellet, previous experimental studies focused on the factors of reduction temperature,³⁾ pellet porosity,⁴⁾ multiple gaseous reactants,⁵⁾ and pellet material,⁶⁾ etc. All these factors were devoted to determine the controlling step in the heat/mass transfer and reaction processes. Furthermore, reduction ability also strongly depended on the initial mineral phase. For instance, it was found that an increase of Al₂O₃ and SiO₂ contents would decrease this ability and resulted in an increasing activation energy.⁷⁾ The increase in basicity, namely the ratio of the lime to silica (C/S), of which the value was less than 1.6, was found to accelerate the degree of reduction (DOR) and degree of metallization (DOM).⁸⁾ However, a further increase in C/S ratio resulted in a decreasing reduction rate due to the formation of calcium ferrite phases. Therefore, an accurate analysis of the reduction process of iron oxide pellets requires a clear understanding of the interactions between various rate steps and factors.

To quantitively describe the reaction characteristics of the iron ore pellet, numerous mathematical models for gas-solid reaction were proposed in the literature, from simple unreacted shrinking core model to porous solid model and grain model.^2,9,10) The unreacted shrinking core model includes one interface shrinking core model and multi-interface shrinking core model.¹⁰⁾ Iron ore pellet is a porous particle with its porosity strongly depended on the material and reduction condition.⁴⁾ Among these models, the one interface shrinking core model was considered to be applicable to a pellet with dense solid phase but limited for porosity, tortuosity, and local reduction degree.²⁾ The grain model used by Valipour et al.¹⁰⁾ and Khoshandam et al.¹¹⁾ was proved to match well with the grainy structure of iron ore pellets, and further considering simultaneously reactions in the non-isothermal transient condition. Nonetheless, the grain model required a numerical solution of the diffusion-reaction equations and some while was incompatible for computation due to its integration in a multi-particle reactor model.⁹⁾ Ghadi et al.¹²⁾ proposed a mathematical time-dependent and non-isothermal model from the grain model to study the reduction of wüstite pellet. Their results were compared with the unreacted shrinking core model (USCM), and it was found that the later model cannot properly predict the impact of gas mixture parameters and local reduction characteristics. Peters and Hoffmann¹³⁾ used a discrete approach to predict the reduction behavior of a single porous particle, which served to be applied to packed bed reactor. Recently, E¹⁴⁾ developed a DEM-based model of the reduction process of a single iron ore pellet with the consideration of heat and mass transfer.

Since heat transfer happened when the reduction process started through convection, radiation, and conduction between solid and gas, which were influential on the temperature, pellet natural property, and gaseous reactant agents. Zare Ghadi et al.¹⁵⁾ investigated the heat and mass transfer of a wüstite pellet. The behavior of temperature distribution altered at a special point, which lay within the reduction area where wüstite converted to iron. This alteration of material caused the change of properties, depending on the reaction of different gas atmosphere (H₂ or CO). Furthermore, the reduction degree was directly related to the local temperature of the pellet with different porosities and pellet sizes. Milandia et al.¹⁶⁾ analyzed the heat transfer characteristics of iron ore spherical pellet during the reduction process. The temperature distribution within the pellet inside was proved to be uneven, and it was found that the heat transfer rate differed from the diameter of the iron ore pellet. Therefore, the heat transfer and porosity evolution were essential to the local or overall reduction of the iron ore pellet, of which is seldom referred in detail.

To investigate the reduction and heat transfer characteristics of a single iron ore pellet in the H₂/CO atmosphere, experimental and modeling studies were carried out. Reduction degree, porosity, and diffusion coefficient were measured and used as model input parameters. The reduction degree and reduction rate predicted from the proposed model were compared to the experimental data. Temperature profile, temperature difference, and heat transfer characteristics inside the pellet were presented and compared under different gas atmospheres and reduction temperatures.

2. Materials and Methods

2.1. Materials

The iron ore pellet from Xinjiang Bayi Iron & Steel Co., Ltd., China, was used in this study. The average size of the iron ore pellet is 14 mm. The chemical compositions of the iron ore pellet were analyzed via an X-ray fluorescence spectrometer (ARL Advant’X Intellipower^TM 3600, ThermoFisher Scientific, America), and the results are given in Table 1.

Table 1. Chemical composition of the raw iron ore pellet.

Sample	chemical composition (wt.%)
	TFe	SiO2	MgO	CaO	Al2O3
Iron ore pellet	65.01	1.81	1.397	1.52	0.79

Note: TFe denotes the total content of iron in the sample.

2.2. Experimental Method

The experiment of the reduction of a single iron ore pellet carried out in the high-temperature tube furnace. The tube furnace was used to simulate the reduction characteristics of the iron ore pellet in the direct reduction ironmaking process, which has been applied in previous studies.^17,18,19) The diagram of the experimental apparatus is shown in Fig. 1. The tube furnace was manufactured by Shanghai Hongtong Heat Treatment Equipment CO., LTD, China, which was made of stainless steel. The gas transportation system included gas cylinders (e.g. CO, H₂, and N₂), regulators, values, mass flow controllers (MFCs), and a computer. The temperature of the tube furnace was controlled by a temperature controller, with the connection of heating elements inside the furnace to create an isothermal zone. The gas flow, of which the flowrate was controlled by MFC, entered into the tube furnace and flowed up to exist from the top.

Fig. 1.

Schematic diagram of the high-temperature tube furnace.

Before the reduction experiment, the iron ore pellet was dried in the muffle furnace for 6 hours at 105°C to remove the moisture. Then, the iron ore pellet was placed in the basket and held close to the top flange, where was far away from the high-temperature zone. The furnace was then sealed by two flanges and a gas flow of nitrogen was injected into the sealed chamber. The flowrate of N₂ was set to 2.0 L/min as a protective gas to remove the air. The temperature of the sealed chamber was heated up to the setting temperature at a heating rate of 40°C/min. When the temperature rose to the set value and held for 10 min, the iron ore pellet in the basket was lifted down to the high-temperature zone in the middle of the sealed chamber, manually with a rod connected to the basket. When the temperature was stable, the nitrogen flow was replaced to a gas flow of H₂ and CO mixtures. Before the experiment, external gas diffusion effect was removed or minimized to be ignored with the increasing gas flowrate. The effect of the mixing gas flow rate was considered and the flow rate was set to 600 mL/min. The iron ore pellet reacted with the gas mixture for more than 60 min. The volume ratios of H₂ to CO were set as 10:0, 9:1, 8:2, and 6:4, respectively, to study the effect of the mixing ratio (H₂/CO) on the reduction of the iron ore pellet at constant temperature and residence time. The effect of the temperature was studied when the mixing ratios of H₂ to CO were 10:0 and 8:2, respectively, at a constant residence time. When the temperature and gas mixing ratio were set to constant, the variable was the residence time period to get the reduction degree. Each experiment of the reduction of the iron ore pellet was set only one of the three variables (temperature, residence time, and gas mixing ratio). After that, the gas flow transferred from the gas mixture back to N₂. The isothermal process was stopped and the sample was cooled down for further analysis. When the tube furnace cooled down in the N₂ atmosphere, the iron ore pellet was taken out rapidly and weighed on a scale for the weight loss measurement. Then, the reduced or partially-reduced iron ore pellet was stored in a nitrogen-sealed container. This pellet was cut by a low speed cutter to observe the reduced cross section. One part was used for pore analysis. The pore size was measured by using an AutoPore 9510 mercury intrusion porosimeter, which was manufactured by Mirocmeritics instrument corporation. The pore size, cumulative pore volume, and incremental pore volume was measured based on the Washburn equation.²⁰⁾ After measuring the weight loss, the reduction degree from the raw sample to metallic iron (Fe) was defined in terms of removable oxygen:   

$$\alpha =\frac{m_0-m_t}{m_0}=\frac{\mathrm{\Delta }m}{m_0}$$(1)

where Δm was the mass loss of oxygen during the reduction, m₀ was the initial mass of the pellet and m_t was the mass of the iron ore pellet at time t.

3. Mathematical Model

The shrinking core model of the iron ore reduction was used and validated in Refs.^1,2,4,6) Owing to the high reduction rate in H₂, products from Fe₂O₃ to Fe₃O₄ and Fe₃O₄ to FeO were mainly reduced by CO and H₂, and the final form FeO to metallic Fe was reduced by CO,²¹⁾ which covered the range of conversion from 1/9 to 1.0.²²⁾ Besides, it was found that once a thin layer of lower iron oxides (magnetite or wüstite) was formed on the surface from 700°C to 900°C, the reaction mechanism shifted to diffusion control.²³⁾ Due to the rapid reduction rate of H₂, Fe₂O₃ can be rapidly reduced to FeO, followed by the diffusion and reduction of CO. The one interface shrinking core model was used in this study.

Before establishing the mathematical model, some of the general required assumptions, including spherical particle with uniform porosity, remained size and shape, no cracks, irreversible and first order reaction, are made. The reaction mode of the iron ore pellet is shown in Fig. 2. The temperatures of the product layer and unreacted core are different but uniform in each layer, due to the different physical properties of the product and unreacted part. Based on the one interface shrinking core model, the reaction layer was considered as a relatively thin front surface. Therefore, these values of the reaction layer were not considered currently in this study. However, for the product layer and unreacted layer, the porosities changed less below the melting temperature, which were mainly affected by temperature. In this study, parameter values of the thermal conductivity (λ), specific heat (C_p), and density of the pellet (ρ_s) were considered in the model but different from the product layer and unreacted layer, which has been used in many researches.^3,24) Parameters are given in Table 2.

Fig. 2.

The schematic diagram of pellet reduction reaction process. (Online version in color.)

Table 2. Parameters used in this study.

Parameters	Value	Parameters	Value
Cp, pl (J kg−1 K)24)	820	ρs (kg/m3)24)	2600
Cp, ul (J kg−1 K)3)	975.6	ρ0 (kg/m3)3)	4222
hg (W m−2 K−1)29)	35.66	λs (W·m−1·K−1)24)	2.13
r0 (mm)	14	λ0 (W·m−1·K−1)3)	0.5
Tw (°C)	750–950	τp	3
Ra (J·mol−1·K−1)	8.3145	σ (W m−2 K−4)	5.67×10−8
vH2 (cm−3 mol)	6.12	φ (wt.%)	27.95
vCO (cm−3 mol)	18	ε	0.85

Ono-Nakazato et al.²⁵⁾ found that the contribution of the water–gas shift reaction (WGST) was estimated to be 3% against its equilibrium at a low reduction temperature below 1200 K, and similar results found by T. Kon et al.²⁶⁾ This mean that at low temperature the influence of the water gas shift on the reduction was small but at high temperature (>1273 K) the total water gas shift was largely changed.²⁷⁾ In our current study, the reduction ranged from 1023 to 1223 K (750–950°C) where the contribution of WGST was small on the equilibrium based on the results found in the literature. However, based on the different reduction activities between H₂ and CO below and above 810°C, the effect of WGST on the reduction and the heat transfer analysis was added as well.

Gases are considered as ideal mixtures. In this study, a single reaction interface mode was considered as a quasi-static reaction process. The main reaction equations of the iron ore pellet on the reaction interface in the H₂/CO atmosphere were:   

$$3\mathrm{C}\mathrm{O}+\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_3=3\mathrm{C}{\mathrm{O}}_2+2\mathrm{F}\mathrm{e}$$(2)

  

$$3{\mathrm{H}}_2+\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_3=3{\mathrm{H}}_2\mathrm{O}+2\mathrm{F}\mathrm{e}$$(3)

From the above two reactions, it was found that the stoichiometric numbers were same. Thus, when considering the mass balance, hydrogen and carbon monoxide were considered together as a whole body, and the proportion reacted with iron ore in accordance with the proportion of mixed gas.

3.1. Mass and Reaction

From the ambient environment to the particle surface, the diffusion rate (r_g) of the H₂ and CO was expressed as:   

$$r_g=-\frac{d{n}_g}{dt}=4\pi r_0^2{k}_g\left(C_{Ra}-C_{Rs}\right)$$(4)

During the reduction process, gas flowed through the iron ore pellet. Thus, the mass transfer characteristics was determined by   

$$Sh=\frac{k_{g}d}{D}=2.0+0.6R{e}^{1/2}S{c}^{1/3}$$(5)

On the interface of the unreacted core and reactant gases, the total reaction rate was   

$$r_c=-\frac{d{n}_g}{dt}=4\pi r_t^2{k}_{rea}{C}_{Rt}$$(6)

Herein, the interface reaction rate was equal to the sum of the reaction rate with CO and reaction rate with H₂ based on the mixing ratio. The reaction rate with CO or H₂ was   

$$k_{rea,i}=A_{0}exp\left(-\frac{E_{A,i}}{R_a{T}_{rea}}\right)P_i,i=\mathrm{C}\mathrm{O},{\mathrm{H}}_2$$(7)

The kinetic data of the iron ore reduction was referred in the work of Valipour et al.²⁸⁾ Other parameters used in this study are given in Table 2.

Then, the internal diffusion rate of the gas phase was expressed as   

$$r_D=-\frac{d{n}_g}{dt}=4\pi r^2{D}_{eff}\frac{d{C}_g}{dr}$$(8)

According to the assumption, the internal diffusion rate r_D was regarded as a constant under steady or quasi steady diffusion conditions, thus   

$$r_D=4\pi D_{eff}\frac{r_0{r}_t}{r_0-r_t}\left(C_{Rs}-C_{Rt}\right)$$(9)

The porosity evolution and pore size of the iron ore pellet were measured with experiment and used as a parameter in this model. In addition, with the measurement of porosity and pore size, the effective diffusion coefficient (D_eff) can be calculated by:   

$$D_{eff}={\left(\frac{1}{D_m}+\frac{1}{D_k}\right)}^{-1}$$(10)

In this equation, D_m is molecular diffusion and D_k is Knudsen diffusion, which are given as   

$$D_m=\frac{D_i{\epsilon }_P}{{\tau }_P}$$(11)

  

$$D_i=\frac{0.0101T^{1.75}{\left(\frac{1}{M_{CO}}+\frac{1}{M_{H_2}}\right)}^{1/2}}{p{\left[{\left(\sum v_{CO}\right)}^{1/3}+{\left(\sum v_{H_2}\right)}^{1/3}\right]}^2}$$(12)

and   

$$D_k=\frac{2}{3}{r}_m\cdot v_i$$(13)

  

$$v_i =\sqrt{\frac{8R_{a}T}{\pi M_i}}$$(14)

From Eqs. (10), (11), (12), (13), (14), the effective diffusion coefficient can be calculated with the experimental data, such as porosity, pore size, and temperature. Thus, the experimental data was used to be parameter input for the mathematical model. On the interface between the solid and gas phases, the solid reaction rate is:   

$$v_c=-\frac{d{n}_B}{bdt}=-\frac{4\pi r_t^2{\rho }_0}{b{M}_B}\frac{d{r}_t}{dt}$$(15)

where b was the ratio of the stoichiometric coefficient of gas to solid reactants, ρ₀ was the density of the iron ore pellet, kg/m³, and M_B was the molecular weight, g/mol.

In the experiment, the amount of the gas phase was set to be more than the theoretical reaction amount. This theoretical amount was calculate based on the Fe₂O₃ concentration in the iron ore pellet, which can be found in Table 1. The reduction degrees of the iron ore pellet at different gas flowrates were tested before this experiment. The gas flowrate was chosen 600 mL/min in the experimental section, which was more than the theoretical amount. Thus, the gas concentration in the ambient atmosphere was equal to the gas concentration on the particle surface. Therefore, the external diffusion effect on the reduction of the iron ore pellet can be ignored. Then, when the reaction gets to a steady state, with the combination of Eqs. (6), (9) and (15), the relationship between reduction degree X and reduction time t can be obtained as   

$$\begin{array}{l}{\displaystyle \frac{k_{rea}{D}_{eff}{r}_0{C}_{Ra}b{M}_B}{{\rho }_0}}t={\displaystyle \frac{1}{6}}{k}_{rea}\left(r_0^3-3r_0{r}_t^2+2r_t^3\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-r_0{r}_t{D}_{eff}+r_0^2{D}_{eff}\end{array}$$(16)

The density of the product layer was different from the density of the unreacted layer. The mass loss (Δm) can be denoted as   

$$\mathrm{\Delta }m=m_0-m_t$$(17)

and   

$$m_0={\rho }_0\cdot \frac{4}{3}\pi r_0^3$$(18)

  

$$m_t={\rho }_0\cdot \frac{4}{3}\pi r_t^3+{\rho }_s\cdot \frac{4}{3}\pi \left(r_0^3-r_t^3\right)$$(19)

In Eqs. (17), (18), (19), m₀ was the initial mass of the iron ore pellet, kg, and m_t was the mass at time t, kg. The reduction degree (X) can be calculated as   

$$X=\frac{\mathrm{\Delta }m}{m_{O2}}$$(20)

The mass of the oxygen can be calculated through   

$$m_{O2}=m_0\cdot \phi$$(21)

where φ was the mass fraction of oxygen atoms bound to iron in the iron ore pellet, wt.%, which can be calculated from Table 1.

Thus, substituting Eqs. (17), (18), (19), and (21) to Eq. (20), the reduction degree can be calculated as the following equation.   

$$X=\frac{1}{\phi }\left(1-\frac{{\rho }_s}{{\rho }_0}\right)\left[1-{\left(\frac{r_t}{r_0}\right)}^3\right]$$(22)

Then, with the combination of Eqs. (15) and (21), the relationship of the reduction time and reduction degree was achieved. Due to the different bond oxygen abilities of CO and H₂ with oxygen at different temperatures, the properties of chemical reactions and the heat absorption and release of reactions were different. Thus, the equilibrium of the reaction in terms of the molar ratio of the gas mixtures.   

$$\begin{array}{l}t={\displaystyle \frac{r_0^2{\rho }_0}{6b{D}_{eff}{C}_{Ra}{M}_B}}\left[1+2\left(1-x\right)-3{\left(1-x\right)}^{{\displaystyle \frac{2}{3}}}\right]\\ \hspace{1em}+{\displaystyle \frac{r_0{\rho }_0}{b{k}_{rea}{C}_{Ra}{M}_B}}\left[1-{\left(1-x\right)}^{{\displaystyle \frac{1}{3}}}\right]\end{array}$$(23)

where   

$$x=\frac{\phi {\rho }_0}{{\rho }_0-{\rho }_s}\cdot X$$(24)

3.2. Heat Transfer

The concentrations of gaseous reactants (e.g. CO and H₂) on the particle surface was equal to the value in the ambient atmosphere. Thus, the convection heat transfer (Q_con.) between the gas flow and iron ore pellet was considered herein. The heat transfer mode of the iron ore pellet was considered and given in Fig. 3. The temperature of the gaseous reactant on the particle surface was supposed to be equal to the temperature of the particle surface. For the reduction of iron ore pellet, the heat exchanges included convective heat transfer with surrounding fluid, conductive heat transfer to other pellets or walls, and radiative heat transfer to its nearby environment.³⁰⁾ In addition, heat transfer mode among pellets mainly was radiation, which was used in Ref.^17,30) The heat conduction among pellets was weaker than the radiation because the contact area between pellet and pellet was limited. in the study, while the radiative heat exchange between particles was considered in the model. Therefore, the energy balance can be   

$$Q_p=Q_{rad.}+Q_{con.}+Q_{rea.}$$(25)

where Q_p was the total heat change of the iron ore pellet, Q_rea. was the reaction heat generated from the reduction of iron ore with gases. Q_rad. was the radiation heat between the wall and iron ore pellet, and Q_con. was the convection heat between the iron ore pellet and gaseous reactants.

Fig. 3.

The schematic diagram of the heat transfer process during the reduction process. (Online version in color.)

The heat changes of the product layer and unreacted layer were different and hence the total heat change of the particle (Q_p) was   

$$Q_p=Q_{pl}+Q_{ul}=C_{p,pl}{m}_{pl}\frac{d{T}_{pl}}{dt}+C_{p,ul}{m}_{ul}\frac{d{T}_{ul}}{dt}$$(26)

where Q_pl and Q_ul were the heat changes of the product layer and unreacted layer, respectively. m_pl and m_ul were the masses of product layer and unreacted layer.   

$$m_{pl}={\rho }_s\cdot \frac{4}{3}\pi \left(r_0^3-r_t^3\right)$$(27)

  

$$m_{ul}={\rho }_0\cdot \frac{4}{3}\pi r_t^3$$(28)

The radiation heat (Q_rad.), reaction heat (Q_rea.), and convection heat (Q_con.) were given as   

$$Q_{rad.}=\epsilon \sigma A_p\left(T_w^4-T_{pl,t}^4\right)$$(29)

  

$$Q_{rea}=n_B\cdot \mathrm{\Delta }{H}_i$$(30)

and   

$$Q_{con.}=h_g{A}_p\left(T_{g,0}-T_{pl,t}\right)$$(31)

4. Results and Discussion

4.1. Reduction Process and Pore Evolution

Figure 4 is the optical view of the cross section of a single iron ore pellet at different residence time periods from 10 to 60 min. The evolution of the shrinking core as the black part was found on the cross section of the iron ore pellet. It is well known that the reaction mode of the iron ore pellet would perform as an unreacted shrinking core, which was found in this study as well. Pores were found on the cross section of the iron ore pellet.

Fig. 4.

The cross-sectional area of the iron ore pellet at different reduction time. (Online version in color.)

The results of the porosity evolution versus reduction time, cumulative porosity increment with radius, and its changing rate with reduction time at the reduction temperature of 850°C in the H₂ atmosphere are shown in Figs. 5(a) to 5(c). From Fig. 5(a), the initial porosity of the iron ore pellet was about 21.69%. Then, the porosity increased to 45.13% when the reduction time was 60 min. Along the particle radius, the cumulative porosity increment was increased from the outer layer to the center of the particle, as shown in Fig. 5(b), indicating that the porosity changed and affected the gas diffusion. The changing rate of the porosity with the reduction time was calculated and shown in Fig. 5(c), which first increased and then decreased with the reduction time. The evolutions of the porosity and its changing rate was related to reaction sequences as Fe₂O₃→Fe₃O₄→FeO→Fe. It was found that the former two reaction steps were fast than the final step from FeO to Fe.³¹⁾ However, the oxygen weight loss in the former two reaction steps was one third of the total amount but the final step occupied about two thirds. When the reduction time was below 10 min, the porosity increased slowly. The dramatical increase of the porosity occurred when the reduction time was from 10 to 50 min, where the main reaction step was from FeO to Fe.³¹⁾ Thus, with the increasing reduction time, the porosity increased rapidly. Conversely, the changing rate of the porosity decreased when the reduction time was above 30 min. With the consumption of oxygen, the amount of the remaining ferrous oxide decreased and thus the porosity changing rate decreased.

Fig. 5.

The porosity evolution versus reduction time (a), cumulative porosity increment with radius (b) and porosity evolution rate (c) at the temperature of 850°C in the H₂ atmosphere. v denotes the porosity evolution rate and t means the reduction time. (Online version in color.)

The pore sizes and porosities of iron ore pellet samples at different reduction temperatures in different H₂/CO atmospheres were given in Fig. 6. Figure 6(a) shows the pore sizes under different temperatures in H₂ atmosphere, when the reduction time was set to 40 min. The initial pore size of the iron ore pellet ranged from 3.6 to 10 μm. When the temperature increased, the pore size decreased and displayed two distributions at 750°C and 800°C, respectively. One ranged from 0.13 to 0.56 μm and the other size ranged from 2.0 to 5.0 μm. When the temperature increased from 850 to 950°C, the smaller pore size distribution moved up from 0.15 to 1 μm. Besides, more larger pore size distributions were found at 950°C. The result indicated that the porosity of the iron ore pellet increased with the temperature. In Fig. 6(b), the porosity of the iron ore pellet increased with the temperature at the same volume ratio of H₂ to CO. Nonetheless, when the temperature was constant, the increasing CO content in the gas mixtures would reduce the porosity. The pellet reduced in H₂ had a higher porosity than the sample reduced in the gas mixtures of H₂ and CO, and similar findings were proved that a pellet reduced in the CO atmosphere contained a tight grain structure and a few channels formed inside, while the pellet reduced in H₂ had loose structures and more small pores.

Fig. 6.

Pore sizes (a) and porosities (b) of iron ore pellet samples at different temperatures. H₂/CO is the volume ratio of H₂ to CO in the mixing gas atmospheres. (a) H₂/CO = 10:0 (b) The reduction time was 40 min. (Online version in color.)

4.2. Effective Diffusion Coefficient

With the calculation from Eqs. (9), (10), (11), (12), (13), the effective diffusion coefficient (D_eff) was calculated and the results are given in Fig. 7. At a low temperature (750°C), less change of the effective diffusion coefficient was found. Nevertheless, when the temperature increased from 750 to 950°C, D_eff increased sharply by 3 to 5 times in the early and late reaction stages, respectively. D_eff increased with the increasing reduction time and attributed to the increase of pore size and porosity, which was shown in Figs. 5 and 6. The results also showed that D_eff decreased with the increasing content of CO in H₂ while increased with the temperature in the same atmosphere. The pore size and porosity of the iron ore pellet during the H₂ reduction process was larger than the results tested in CO.

Fig. 7.

Evolution of the effective diffusion coefficient at different temperatures in the same atmosphere (a) and under different mixing gas atmospheres with H₂/CO ratios (b). (Online version in color.)

4.3. Model Prediction and Validation

The reduction degrees of iron ore pellets with different H₂/CO ratios were given in Fig. 8(a). The results of model prediction for the reduction degree versus time matched well with the experimental data, which validated the applicability of the model in this study. The reduction degree (X) at the H₂/CO ratio of 8:2 was higher than those of other H₂/CO ratios. The reduction of the iron ore pellet with CO was an exothermal reaction while it was an endothermal reaction in H₂. From the experimental results, it was found that adding 20 vol.% CO to H₂ can accelerate the reaction rate at 850°C, indicating that the reduction with CO provided extra energy for the endothermal reaction in H₂. However, when the addition amount of CO was excessive, owing to the lower reaction rate with CO, the total reduction rate decreased. Figure 8(b) shows the relationship between the reduction rate (dX/dt) and time predicted by the proposed model. At the same temperature, the reduction rate decreased gradually with time. The reduction rate for the H₂/CO ratio of 8:2 was higher than the rates at other conditions. All reduction rate at different H₂/CO ratios decreased with the reduction time, due to the increasing pathway and diffusion resistance for the reducing gases.

Fig. 8.

Comparison of the model prediction and experimental measurement for the reduction degree (X) and the reduction rate (dX/dt) of iron ore pellet samples with the effect of H₂/CO ratio at 850°C. (a) reduction degree and (b) reduction rate. (Online version in color.)

Figure 9 gives the evolution of the reduction degree and reduction rate with the effect of temperature. The results from model prediction were validated with the experimental data as well, which further verified the accuracy of the model. Higher temperature promoted the reduction of the iron ore pellet since the endothermic reaction was dominant in the H₂-rich atmosphere. The reduction rate (dX/dt), calculated and given in Fig. 9(b), decreased with the reduction time. At the early reaction stage, when the reduction time was below 22 min, the reduction rate was higher at high temperature because of higher reduction rate in H₂ at higher temperature and the short diffusion distance of the gas reactant close to the surface of the pellet. With the reduction reaction layer moving further inside the pellet, the diffusion distance of the gas increased and the diffusion resistance also increased. Besides, the temperature of the reaction layer decreased due to the endothermic reaction with H₂. Then, the reduction rate would decrease if no more heat generated from the reduction of iron ore in CO. The reduction rate changed to be lower than the value at lower temperature in the later stage. Furthermore, model prediction of the reduction rate showed similar tendencies with the experimental measurements. The results from Fig. 9 indicated that the reduction was inhibited in the later stage of the reduction process. Below 810°C, the ability of bond oxygen of CO was higher than H₂, and the dominant reduction occurred between iron ore with CO and more heat was generated from the reduction. The increasing temperature of the reaction layer increased the reduction rate and even to be higher than other conditions. In addition, with more gas product moving outward, the amount of the gaseous reactant diffused into the pellet was reduced per unit time, owing to the increasing diffusion resistance caused by the porous structure, comparing to the reaction at beginning (before 22 min). Therefore, the reaction rate decreased during the reduction process and reversed, and the reduction degree line in Fig. 9(a) showed a dramatically and then slow increasing tendency.

Fig. 9.

Comparison of the model prediction and experimental measurement for the reduction degree (X) and reduction rate (dX/dt) of a single iron ore pellet at different reduction temperatures from 750 to 950°C with a H₂/CO ratio of 8:2. (Online version in color.)

However, when the reduction time was above 40 min and above 0.9, there was an error under the condition of high temperature (950°C). The reason may be due to the difference of the reduction rate between H₂ and CO reduction rate. Meanwhile, due to the gas diffusion and internal pore changed in the experimental process, it was difficult for the gas to diffuse into the interior at the final stage of the reaction, resulting in a reduction degree close to that of 40 min. Thus, there are errors in the experimental measurement process, resulting in errors in the final stage of the reaction when the time was above 40 min.

4.4. Temperature Distribution and Evolution

The temperature profile and their differences of the product layer (T_pl and ΔT_pl), reaction layer (T_rea and ΔT_rea), and unreacted layer (T_ul and ΔT_ul) with the reduction time and particle radius were predicted, and the results with the effect of H₂/CO ratio on are shown in Figs. 10(a) to 10(f). The yellow short dot line means the temperature difference was 0°C. The temperature difference below 0°C means the actual temperature of the layer decreased and was below the initial temperature. From all three temperature profiles of Figs. 10(a), 10(c) and 10(e), the results showed that the temperature of each layer first decreased and then increased with the reduction time. Along the particle radius, the temperature of the reaction layer in Fig. 10(c) showed same tendency with the reduction time but higher temperature in the final stage. With heat exchange occurred among layers, the temperature in both product layer and unreacted layer decreased and then increased as well, as shown the evolution of T_pl and T_ul in Figs. 10(a) and 10(e). Note that temperature profiles of the product layer and unreacted layer were corresponded to the temperature where the reaction layer reached at time t. Comparatively, the temperature for each layer at the H₂/CO ratio of 8:2 was higher than other gas mixing ratios, which can be explained to illustrate the higher reduction rate and reduction degree in Fig. 8.

Fig. 10.

Temperature profiles (a, c, and e) and their difference (b, d, and f) of product layer, reaction layer, and unreacted layer as a function of reduction time and particle radius with the effect of H₂/CO ratio. The yellow short dot line means the temperature difference was 0°C. The ambient temperature is 850°C. (Online version in color.)

From the heat transfer analysis and the results shown in Fig. 10, it showed that the temperature of the reaction layer was lower than other conditions. The reduction temperature of the iron ore pellet was set to 850°C, where the reactivity of H₂ was higher than CO. The reduction of iron ore pellet with H₂ would absorb lots of heat from the ambient, product layer and unreacted layer. At the initial time 10 min, the location of the reaction front was close to the pellet surface, where can be found from the 3D plot in Fig. 10(c). Bulk of H₂ diffused to this layer and reacted with iron ore. Owing to its endothermic reaction, more heat flowed from the unreacted layer and product layer, which causing a sharp decrease of the temperature of the reaction layer.

Comparing other H₂:CO conditions of 10:0, 9:1 or 6:4, the reduction (H₂:CO=10:0) was an endothermic reaction thus the temperature reduced and the temperature differences of all three layers were below. When H₂:CO was 9:1 and the temperature was 850°C, the endothermic reaction was dominant and the reaction layer needed to absorb heat from the product layer and unreacted layer. However, when CO started to react with iron ore and heat generated from the reaction layer, the temperatures of three layers were higher than the condition of H₂:CO=10:0. Compared to the condition of H₂:CO=8:2, at the initial time 10 min, the endothermic reaction of H₂ with iron ore was dominant as well. The temperatures of three layers decreased, which were same with the conditions of H₂:CO=9:1 and 10:0. The lowest temperature of the reaction layer at the condition of H₂:CO=8:2 was speculated that less H₂ diffused and reacted with iron ore, comparing to the condition of H₂:CO=9:1. Meanwhile, less CO diffused to the reaction layer and the low reaction rate of CO with iron ore would cause a further decreasing temperature of the reaction layer. Thus, from Fig. 10(c), the time point when the minimum temperature difference of the reaction layer (H₂:CO = 8:2) reached was longer than other conditions. However, after that, with more heat generated from the reaction of CO with iron ore, the temperature of the reaction increased rapidly for the condition of H₂:CO=8:2.

For the condition of H₂:CO=6:4, less H₂ diffused to the reaction layer at the beginning (<10 min) and thus the temperature difference of each layer was higher than others. When CO started with react with iron ore, the temperature increased. This may be related to the gas mixture of H₂ and CO. However, owing to the slow reaction rate of CO at this temperature (850°C), the reaction of H₂ was dominant, and the temperature was lower than the condition of H₂:CO=8:2.

The temperature difference (ΔT) between the instant temperature and initial temperature was calculated and shown in Figs. 10(b), 10(d), and 10(f). ΔT first decreased to be negative and then increased to be positive with the reduction time in all three layers, corresponding to the temperature profile. At the beginning, hydrogen was mainly involved in the reaction with a higher reduction rate than in CO, resulting in a decrease of the temperature in the reaction layer. When the reduction occurred in pure H₂ (H₂/CO=10:0), temperature differences as shown the black line in Figs. 10(b), 10(d) and 10(f) were all below 0 throughout the reduction process. In Fig. 10(d), the temperature difference of the reaction layer (ΔT_rea) decreased to −21.1°C at the H₂/CO ratio of 10:0 at 10 min. This means the temperature of the reaction layer was 21.1°C smaller than the initial temperature of 850°C. When the temperature of the reaction layer decreased during the reduction process, the reduction rate decreased both in CO and H₂. When the temperature of the reaction layer was 21.1°C smaller than the initial temperature, the results mean the heat was absorbed by the reaction layer and no enough heat was supplied in time for next reduction layer. Thus, if the temperature of the reaction layer continued to decrease, the reduction of the iron ore pellet needed longer time and the reduction degree would be lower in H₂, compared to the results in CO/H₂ mixtures.

The increase of ΔT_rea was due to CO involved in the reduction process and more heat generated for H₂ reduction, resulting in the temperature of each layer (T_pl, T_rea, and T_ul) rose up and the temperature differences of product layer (ΔT_pl) and unreacted layer (ΔT_ul) increased to be positive (> 0°C). Besides, the reduction time that reached the temperature difference point of 0°C was reduced with adding CO in H₂ and to minimum at the H₂/CO of 8:2. This gave an information that the temperature of the reaction layer increased and was larger than the initial temperature and more heat was generated during the reduction process. For the temperature evolution of the unreacted layer, similar results were found with the product layer. Differently, the minimum temperature difference reached to −25°C in H₂, attributing to no more heat was supplied inside the iron ore pellet while heat was generated in the gas mixtures of H₂ and CO.

The evolution of the temperature difference for the product layer, reaction layer, and unreacted layer with the effect of temperature are shown in Fig. 11. The yellow short dot line means the temperature difference was 0°C. The temperature difference below 0°C means the actual temperature of the layer decreased and was below the initial temperature. Similarly, with the comparison of the effect of H₂/CO ratio, the temperature difference first decreased and then increased with time for all layers at different temperatures. Nevertheless, the condition at higher temperature had a higher negative temperature difference in all layers (dash and double dot lines in Fig. 11) at the beginning. This was due to the reduction in H₂ and more heat needed in the endothermal reaction.

Fig. 11.

Temperature difference profiles as a function of reduction time inside single iron ore pellet with the effect of temperature from 750 to 950°C. (a) product layer, (b) reaction layer, and (c) unreacted layer. The yellow short dot line means the temperature difference was 0°C. (Online version in color.)

The different point was the evolution of the temperature difference in the regions from 750 to 800°C and 850 to 950°C from Figs. 11(a) to 11(c). When the reduction temperature was from 750 to 800°C, the temperature difference at former temperature in both production layer (ΔT_pl) and reaction layer (ΔT_rea) were lower than the results at 800°C before 30 min. Nevertheless, when the reduction temperature increased from 850 to 950°C, the temperature difference for both ΔT_pl and ΔT_rea decreased before 30 min. The minimum value of the temperature difference at each temperature was found when the reduction was 5 min for ΔT_pl and ΔT_rea and 10 min for ΔT_ul. The time difference was speculated to be related to the heat transfer coefficient in Table 2. The thermal conductivity of the product layer was higher than the unreacted layer, and thus more heat transferred between the reaction layer and product layer. When the temperature difference rose to be above 0°C, its value increased with both reduction time and reduction temperature from 750 to 950°C. This result was found in all three temperature differences (ΔT_pl, ΔT_rea, and ΔT_ul). When the reduction temperature was below 810°C, the ability to bond oxygen atom for CO was higher than H₂.³²⁾ Hence, the reduction at 800°C was accelerated and more heat generated, resulting a lower negative temperature difference than the condition at 750°C. However, when the temperature was higher than 810°C, the ability of H₂ to bond oxygen atoms was higher than CO, leading to the dominant H₂ reduction reaction. Bulk of heat was absorbed by the reduction of H₂, resulting in lower temperature and higher negative temperature difference. When CO started to involve in the reaction, more heat generated and increased the temperature of three layers. Consequently, the higher temperature at the later stage of the reaction forwarded higher positive temperature difference.

4.5. Reaction Temperature Difference

Figure 12(a) shows the maximum positive temperature difference (ΔT_max) of all three layers predicted by the model with the effect of H₂/CO ratios. In pure H₂, where the ratio of H₂ to CO was 10:0, ΔT_max was not shown because of the endothermal reaction in H₂ and the actual temperature of each layer was lower than the initial value. In the H₂/CO atmosphere, the maximum positive temperature difference (ΔT_max) increased for all three layers, when the H₂/CO ratio changed from 9:1 to 8:2. After that, ΔT_max decreased at the H₂/CO ratio of 6:4. The results of ΔT_max at different reduction temperatures and H₂/CO ratios were given in Fig. 12(b). At a low CO concentration (H₂/CO = 10:0 and 9:1), ΔT_max first decreased and then decreased, and the temperature difference was below 0°C during the reduction process. The different abilities of depriving oxygen atoms of CO and H₂ below and above 810°C and the transition of endothermic and exothermic reaction resulted in these changes when the H₂/CO ratio was 10:0 and 9:1, respectively, which has been discussed in detail in the section 4.4 for the results of Fig. 11. Nevertheless, with adding more CO in H₂, the maximum positive temperature increased from 10 to 27°C with H₂/CO ratio of 8:2 and from 6 to 23°C with O ratio of 9:1 from 750 to 950°C. More CO addition (> 20 vol.%) in the gaseous reactant conversely resulted in a lower temperature, which was mainly because of the lower reduction rete in CO and the delayed heat transfer to increase the temperature.

Fig. 12.

Maximum positive temperature difference (ΔT_max) for different layers inside the iron ore pellet (a) and reduction processes at different H₂/CO ratios and temperatures (b). The ambient temperature is 850°C. (Online version in color.)

4.6. Heat Transfer Characteristics

Figure 13 shows the evolution of the heat flow and specific heat during the reduction process with the effect of H₂/CO ratio. In Figs. 13(a) and 13(b), both radiation heat flow (Q_rad.) and convection heat flow (Q_con.) increased and then decreased with the reduction time. When the reduction time was below 6 min, the initial evolution of the heat flow for both Q_rad. and Q_con. performed as a similar increasing tendency. Then, Q_rad. in the pure H₂ was positive and higher than other conditions, owing to the lower temperature of the particle surface in H₂ than other conditions shown in Fig. 10. The maximum Q_rad. in H₂ was about 0.055 J/s while it decreased to 0.040 J/s with the H₂/CO ratio of 8:2. Compared to the radiation heat, the convection heat was smaller, indicating the radiation heat was dominant during the reduction process. With adding CO in H₂, Q_rad. was first positive and then decreased to be negative in the later reaction stage in consequence of increased the particle temperature higher than the initial temperature. The positive part means the heat flow direction was from the wall and gas phase to the particle surface while the negative reflected the direction was reversed. Less radiation heat flow when the H₂/CO ratio was 8:2 was found in Fig. 13(a), corresponding to the higher temperature of the product layer and less heat transfer with the wall.

Fig. 13.

Evolutions of the heat flow and specific heat with the effect of H₂/CO ratio. (a) radiation heat (Q_rad.), (b) convection heat (Q_con.), (c, d) evolution of specific heat for product layer (Q_pl) and unreacted layer (Q_ul). The ambient temperature is 850°C. (Online version in color.)

In Fig. 13(c), the results showed that the specific heat of the product layer (Q_pl) decreased dramatically at the beginning and then increased after 10 min. The sharp decrease of Q_pl was caused by the endothermal reaction with H₂ and the later increase attributed to energy supply from the exothermal reaction with CO, radiation heat, and convection heat, which can be proved with the comparison for the reaction in H₂. The specific heat (Q_pl) was negative because of the decreasing temperature of the product layer at the beginning and the heat flow direction was from the unreacted layer to reaction layer. Then, Q_pl shifted to be positive with more heat supplied from the reduction of CO. The reduction time where Q_pl was positive was higher at H₂/CO ratio of 8:2, corresponding to the temperature evolution. For the specific heat evolution of the unreacted layer (Q_ul), Fig. 13(d) showed that Q_ul in pure H₂ first decreased and then increased but the value was negative throughout the reduction process. Herein, the heat flow direction was from the unreacted layer to the reaction layer. In the H₂/CO atmosphere, the specific heat increased to be positive during the production process with a higher value at H₂/CO ratio of 8:2 than other ratios. This indicated that less heat flow was from the unreacted layer to the reaction layer when CO started to reduce the iron ore pellet and generated heat as a supply for the unreacted layer.

Figure 14 gives the results of the evolution of the heat flow and specific heat during the reduction process with the effect of temperature at the H₂/CO ratio of 8:2. Similar profiles of the radiation heat flow (Q_rad.) and convection heat flow (Q_con.) were found in Figs. 14(a) and 14(b). Q_rad. and Q_con. both increased and then decreased to be negative with time, regardless of the condition at 750°C. This attributed to a lower pellet temperature than the initial temperature for the reduction of H₂ and the heat flow was from the environment to the particle surface. For the heat flow evolution in the H₂/CO atmosphere, the positive part means the heat flow direction was from the wall and gas to the particle surface while the negative reflected the direction was reversed. The radiation heat flow was dominant as an order of magnitude higher than the convection heat. At the beginning, the higher reduction rate in H₂ than CO lead the reaction to be endothermal and the pellet temperature reduced, as shown in Fig. 11(a). This phenomenon was more obvious at high temperature, causing more heat flow for both Q_rad. and Q_con.. With the participation of CO reduction, the exothermal reaction generated heat and the temperature of the product layer increased, and thus the radiation heat and convection heat decreased. When the temperature of the product layer was higher than the environmental temperature, the heat flow direction was shifted from the particle to the environment as negative values in Figs. 14(a) and 14(b). Higher reduction temperature benefitted the reduction both in H₂ and CO, and hence from 800 to 950°C the heat flow increased in both directions.

Fig. 14.

Evolutions of the heat flow and specific heat with the effect of reduction temperature. (a) radiation heat (Q_rad.), (b) convection heat (Q_con.), (c, d) evolution of specific heat for product layer (Q_pl) and unreacted layer (Q_ul). (Online version in color.)

For the evolution of the specific heat (Q_pl and Q_ul) as shown in Figs. 14(c) and 14(d), the results showed both negative at all tested temperatures when the reduction time was smaller than 30 min. Q_pl and Q_ul first decreased rapidly and then increased with the reduction time. The higher temperature caused lower Q_pl and Q_ul values below 30 min, which was due to the heat absorbed by the reduction in H₂. Q_pl at higher temperature increased to be positive and was higher than the value of lower temperature when the reduction time was more than 30 min. The later increase attributed to energy supplies from the exothermal reaction with CO, radiation heat, and convection heat. Comparatively, the sharp decrease of Q_ul in Fig. 14(d) was related to the endothermal reaction in H₂ which absorbed heat from the unreacted layer. A short decrease of Q_ul was found and then it was approaching 0 J/s before 30 min. In addition, the different tendency between Q_pl and Q_ul was related to the thermal conductivities of these two layers, which was given in Table 2. The thermal conductivity of the product layer was higher than the unreacted layer, providing an information that the heat transfer between the reaction layer and production layer was more sensitive than the transfer between the reaction layer and unreacted layer.

5. Conclusion

Experiments of the reduction of single iron ore pellet under different H₂/CO atmospheres and temperatures were carried out and a mathematical model with considering the heat transfer effect and porosity evolution was proposed in this study. The proposed model was validated to match well with reduction degree and reduction rate of the experimental results. Results showed that the porosity and effective diffusion coefficient increased with the temperature and H₂ content in H₂/CO. With the increase of temperature, the pore size of the pellet presented multi-level distribution characteristics. Comparing to the reduction in pure H₂, adding 20 vol.% CO in H₂ can increase the reduction degree and reaction rate, which proved to be a threshold value for the balance between the endothermic reaction with high reduction rate in H₂ and exothermic reaction with low reduction rate in CO. During the reduction process, the addition of CO in H₂ increased the temperature of the reaction layer by about 10 to 27°C higher than the initial temperature. Temperature profiles and their temperature difference of the product layer, reaction layer, and unreacted layer presented both first decrease and then increase during the reduction process. The shift point of the negative and positive value of temperature and heat flow was found to increase with the temperature and reached a maximum at adding 20 vol.% CO in H₂. The shifting mechanism was that the higher temperature of the reaction layer was from an extra energy supply from the reduction with CO. The radiation heat was dominant for the heat transfer between the environment and iron ore pellet, while inside the pellet heat flow would change among three layer and verifying with the temperature and atmosphere.

Acknowledgements

This study is supported by the National Natural Science Foundation of China (Grant No. 21908063), the Shanghai Pujiang Program (21PJ1402300), the Fundamental Research Funds of the Central Universities (JKB01211715), and the Open Research Fund of State Key Laboratory of Multiphase Complex Systems (No. MPCS-2021-D-07).

Nomenclature

A₀: Pre-exponential factor

A_p: Particle surface area (m²)

b: Stoichiometric coefficient of gas to solid reactant

C: Gas concentration (mol/m³)

Cp: Heat capacity (J kg⁻¹ K)

D_eff: Diffusion coefficient (m² s⁻¹)

D_m: Molecular diffusion coefficient (m² s⁻¹)

D_k: Knudsen diffusion coefficient (m² s⁻¹)

E_A,i: Activation energy (J mol⁻¹)

h_g: Convection heat coefficient (W m⁻² K⁻¹)

ΔH: Reaction heat (J mol⁻¹)

k_rea: Reaction rate (s⁻¹)

m_t: Particle mass at time t (kg)

m₀: Initial particle mass (kg)

m_O2: Total oxygen mass in iron ore pellet (kg)

m_p: Particle mass (kg)

Δm: Mass loss of oxygen atom in iron ore pellet (kg)

M: Molar mass (g mol⁻¹)

n: Mole number

n_B: Mole number of reduced iron oxide

P: Pressure (Pa)

Q: Heat flux (W)

Q_r: Radiation heat flux (W m⁻²)

r: Particle radius (m)

r₀: Initial radius (m)

r_c: Total reaction rate (mol/s)

r_g: Diffusion rate (mol/s)

r_t: Radius at time t (m)

r_D: Internal diffusion rate (m s⁻¹)

r_m: Mean radius of micropores (m)

R_a: Gas constant (J·mol⁻¹·K⁻¹)

t: Time (s)

T: Temperature (K)

T_pl: Temperature of product layer (K)

T_ul: Temperature of unreacted layer (K)

T_g: Gas temperature (K)

T_w: Wall temperature (K)

ΔT: Temperature difference (°C)

v: Molecular diffusion volume (m³/mol)

v_c: Solid reaction rate

v_i: Root mean square velocity of gas i

X: Reduction degree

ρ: Density (kg m⁻³)

τ: Tortuosity

ε: Emissivity or porosity

λ: Thermal conductivity (W m⁻¹ K⁻¹)

σ: Stefan-Boltzmann constant (W m⁻² K⁻⁴)

φ: Oxygen mass fraction in pellet

Subscripts

0: Initial time t=0

B: Iron oxide (Fe₂O₃)

con: Convection heat

eff: Effective coefficient

g: Gas

i: CO or H₂

O: Oxygen (O₂)

p: Particle

pl: Product layer

rea: Reaction

rad: Radiation heat

Ra: Gas concentration in ambient phase

Rs: Gas concentration on particle surface

Rt: Gas concentration on reaction interface

s: Product layer

t: Time

ul: Unreacted layer

w: Wall of furnace

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