Numerical Analysis of Effect of Water Gas Shift Reaction on Flash Reduction Behavior of Hematite with Syngas

Abstract

The water gas shift reaction (WGSR) is the most important side reaction in direct iron reduction processes in syngas. In this study, an Euler-Lagrange model has been developed to simulate the flash reduction behavior of hematite with syngas in a drop tube reactor. Based on model validation, the effect of WGSR on the flash reduction is investigated by comparing results predicted by models with and without WGSR. Results indicate that the WGSR has a minor effect in CO–H₂ system while a major effect in H₂–CO–CO₂–H₂O system. The difference of gas composition caused by WGSR leads to a difference of gas reduction capacity, which results in different reduction behavior. The relationship between the composition of gas mixture and the equilibrium constant of WGSR determines the direction of WGSR and thus determines the positive or negative effect of WGSR on the reduction process. The higher oxygen partial pressure and temperature, the stronger influence of WGSR can be considered to have.

1. Introduction

Because of high energy consumption and CO₂ emission of the blast furnace, some alternative ironmaking processes have been developed in recent years, including the COREX process, FINEX process and ITmk3.¹⁾ Compared to the others, the flash ironmaking process has attracted much attention because of its advantages of low energy consumption, low emissions, and high rates of heat and mass transfer. It cancels the sintering/pelletization and coke-making steps, and aims at directly reducing fine iron ore concentrates (particle diameter < 100 μm) in an in-flight process at 1300–1550°C within a few seconds.²⁾ The process utilizes hydrogen, natural gas, coal gas, or a combination of them as both reducing agent and fuel. From the viewpoint of cost and kinetics, reformed natural gas or coal gasification, composed of the reducing gases H₂ and CO as well as the completely oxidized species H₂O and CO₂, is a better candidate than hydrogen for this process.

Flash reduction behavior of iron concentrate has been well-studied in a single reduced gas including H₂,^3,4,5) CO⁶⁾ and CH₄,^7,8) and in a synthetic gas.^9,10,11,12) When iron ore is reduced by CO–CO₂–H₂–H₂O system, the water gas shift reaction (WGSR) influences the reduction process.¹³⁾ This effect has been widely investigated by experiments, but most of which have focused on the pellets reduction process. Murayama et al.¹⁴⁾ investigated the reduction of wustite pellets in a fixed bed with H₂–CO and H₂–CO₂ gas mixtures. Results showed that the effect of WGSR on the reduction was not so significant in H₂–CO, but became significant in H₂–CO₂ with an increase of CO₂ composition in the inlet gas. In addition, Kemppainen et al.¹⁵⁾ investigated the WGSR in an olivine pellet layer in the upper part of blast furnace shaft. Ono-Nakazato et al.¹⁶⁾ studying the reduction of FeO_1.05 powder packed bed with H₂–CO mixture at 1093 K, found the contribution of the WGSR was estimated to be 3% against its equilibrium. Kon et al.¹⁷⁾ investigated the reduction behavior in a sinter bed, and found the WGSR did not appear as a significant effect in the reduction. Li et al.¹⁸⁾ reported that the WGSR counteracted the wustite sinter reduction in CO–H₂–CO₂–N₂ system including high content of CO₂ and H₂ at elevated temperature.

Numerical simulation is an effective method to reveal the inner characteristics of reaction process, so the effect of the WGSR also has been widely investigated via simulation. Negri et al.¹⁹⁾ established a mathematical model of a moving-bed reactor for the direct reduction of iron oxides. They found that the model with WGSR more closely represented the reducing furnace behavior at 1123–1273 K compared with no WGSR. Valipour and Mokhtari²⁰⁾ found that the effect of the WGSR was not considerable on the reduction rate and temperature distribution inside the pellet but affected the distribution of gaseous species considerably. Takahashi et al.²¹⁾ reported that side reactions including WGSR and reactions for formation and decomposition of methane should be included in the mathematical model to obtain the good simulation results for the reduction with gas mixture in the shaft furnace.

Although the effect of the WGSR in the pellets reduction has been reported, its influence on the flash reduction rarely been studied. Additionally, there are three reasons why the WGSR behavior may not be similar in these two processes. First, the catalysis of metallic iron on the WGSR can be ignored in the flash reduction due to very few concentrate particles loading, but it is significant in pellets reduction. Second, the flash reduction takes place in a few seconds which is far less than the reduction time of pellet reduction; therefore reactions may not reach equilibrium in the flash reduction process. Third, the temperature of pellets reduction is lower than the temperature expected to be used in the flash ironmaking process, which implies that the WGSR will have a faster reaction rate in the flash reduction. Thus, the role of the WGSR in the flash reduction must be investigated further, and it is essential to evaluate the possibility and efficiency of applying synthetic gas in the flash ironmaking technology.

In this research, a three-dimensional (3D) computational fluid dynamics (CFD) model is established to describe the flash reduction behavior of iron ore particles using syngas as the reductant in a drop tube reactor. Then, the influence of the WGSR on the flash reduction is discussed in H₂–CO–N₂ and H₂–CO–H₂O–CO₂–N₂ systems under different gas compositions and reduction temperatures.

2. Mathematical Model

The drop tube reactor has been adopted as an ideal laboratory reactor for flash reduction of iron ore particles because of the advantages of fast heating rate, concurrent gas-solids flow without back mixing and particle collision, and short resident time.^{3,4,5,6,7,8,9,10,11,12)} The chemical reactions in the flash reduction process involves heterogeneous reactions between particles and gas, as well as homogeneous reactions among gas. The interaction between particles in the reactor is neglected due to the low volume fraction of particles. Thus, an Eulerian–Lagrangian method is applied to solve a multiphase reactive flow. Because this study focuses on the stable trends of flash reduction behavior, the steady-state model is built to simulate the entire process.

2.1. Continuous Phase

The Eulerian method is adopted to simulate the continuous phase. The mixing and transport of different chemical species is calculated by the species transport model. The governing equations for the mass, momentum, energy and specie are shown below:   

$$\frac{\partial }{\partial x_i}\left({\rho }_g{u}_i\right)=S_{p,\mathrm{m}}$$(1)

  

$$\frac{\partial }{\partial x_i}\left({\rho }_g{u}_i{u}_j\right)=-\frac{\partial p}{\partial x_j}+\frac{\partial }{\partial x_i}\left({\tau }_{ij}-{\rho }_g\overline{u_i^{\prime }{u}_j^{\prime }}\right)+{\rho }_{g}g+S_{\mathrm{p},\mathrm{m}\mathrm{o}\mathrm{m}}$$(2)

  

$$\frac{\partial }{\partial x_i}\left({\rho }_{\mathrm{g}}{u}_{i}H\right)=\frac{\partial }{\partial x_i}\left({\lambda }_{eff}\frac{\partial T_g}{\partial x_i}\right)+S_{\mathrm{p},\mathrm{h}}+S_{\mathrm{r}\mathrm{a}\mathrm{d}}+S_{\mathrm{h}}$$(3)

  

$$\frac{\partial }{\partial x_i}\left({\rho }_{\mathrm{g}}{u}_i{Y}_i\right)=\frac{\partial }{\partial x_i}\left({\rho }_{\mathrm{g}}{D}_{eff}\frac{\partial Y_j}{\partial x_i}\right)+S_{\mathrm{p},{\mathrm{Y}}_{\mathrm{i}}}+S_{{\mathrm{Y}}_{\mathrm{i}}}$$(4)

where the viscous stress tensor in Eq. (2), τ_ij, can be expressed as   

$${\tau }_{ij}=\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}\frac{\partial u_l}{\partial x_l}{\delta }_{ij}\right)$$(5)

2.2. Turbulence Models

The high-speed jet in the reactor inlet is a fully developed turbulent flow, which must be taken into account. Wu et al.²²⁾ compared various turbulence models with experimental data for a cold flow in a Texaco gasifier test bed. It was found that the realizable k–ε model was slightly better than the standard k–ε and RNG k–ε models. Therefore, the realizable k–ε turbulence model with standard wall function is adopted in this model. The Reynolds stress term in Eq. (2) can be expressed as:   

$$-{\rho }_g\overline{u_i^{\prime }{u}_j^{\prime }}={\mu }_t\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)-\frac{2}{3}\left({\rho }_{g}k+{\mu }_t\frac{\partial u_l}{\partial x_l}\right){\delta }_{ij}$$(6)

The governing equations of the turbulence kinetic energy k and its dissipation rate ε are shown as follows:   

$$\frac{\partial }{\partial x_i}\left({\rho }_{g}k{u}_i\right)=\frac{\partial }{\partial x_j}\left[\left({\mu }_g+\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+G_k+G_b^{}-{\rho }_g\epsilon$$(7)

  

$$\begin{array}{l}{\displaystyle \frac{\partial }{\partial x_i}}\left({\rho }_g\epsilon u_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\mu }_g+{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]\\ +{\rho }_g{C}_{1}S\epsilon -{\rho }_g^{}{C}_2{\displaystyle \frac{{\epsilon }^2}{k+\sqrt{v\epsilon }}}+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}{C}_{3\epsilon }{G}_b\end{array}$$(8)

2.3. Discrete Phase

The mass of concentrate particles is changed owing to the heterogeneous reactions, which can be calculated by:   

$$\frac{d{m}_p}{dt}=\frac{d{m}_{{Fe}_2{\mathrm{O}}_3-{\mathrm{H}}_2}}{dt}+\frac{d{m}_{{Fe}_2{\mathrm{O}}_3-CO}}{dt}$$(9)

The Lagrangian method is adopted to track the motion of particles, and a stochastic particle-tracking model is adopted to consider the effect of turbulent fluctuation on particle dispersion. Hematite concentrate particles are assumed to be sphere particles with uniform diameters. The particle velocity is determined by the force acting on a particle, shown as:   

$$\frac{d{u}_p}{dt}=F_{\mathrm{D}}\left(u_g-u_p\right)+\frac{\left({\rho }_{\mathrm{p}}-{\rho }_g\right)g}{{\rho }_{\mathrm{p}}}$$(10)

where the two terms on the right-hand side represent the drag force and the total force of gravity and buoyancy. The drag force is calculated as:   

$$F_{\mathrm{D}}=\frac{18{\mu }_g}{{\rho }_{\mathrm{p}}{d}_{\mathrm{p}}^2}\frac{C_{\mathrm{D}}R{e}_p}{24}$$(11)

The particle Reynolds number, Re_p, can be expressed as:   

$$R{e}_{\mathrm{p}}=\frac{{\rho }_g{d}_{\mathrm{p}}\left|u_{\mathrm{g}}-u_{\mathrm{p}}\right|}{{\mu }_g}$$(12)

The drag coefficient C_D for the sphere particle is a function of the particle Reynolds number:   

$$C_{\mathrm{D}}=a_1+\frac{a_2}{R{e}_p}+\frac{a_3}{R{e}_{\mathrm{p}}^2}$$(13)

where a₁, a₂, and a₃ are constants given by Morsi and Alexander.²³⁾ The concentrate particle temperature is determined by the convective heat transfer between concentrate particles and the surrounding gases, thermal radiation and the reactive heat, as shown below:   

$$\begin{array}{l}{m}_{\mathrm{p}}{c}_{p,\mathrm{p}}{\displaystyle \frac{d{T}_{\mathrm{p}}}{dt}}=h{A}_{\mathrm{P}}\left(T_{\text{g}}-T_p\right)+A_{\mathrm{P}}{\epsilon }_{\mathrm{P}}\sigma \left({\theta }_R^4-T_{\mathrm{P}}^4\right)+\\ f_{\mathrm{h}}\left({\displaystyle \frac{d{m}_{{Fe}_2{\mathrm{O}}_3-{\mathrm{H}}_2}}{dt}}\right)H_{{Fe}_2{\mathrm{O}}_3-{\mathrm{H}}_2}+f_{\mathrm{h}}\left({\displaystyle \frac{d{m}_{{Fe}_2{\mathrm{O}}_3-CO}}{dt}}\right)H_{{Fe}_2{\mathrm{O}}_3-CO}\end{array}$$(14)

where the convective heat transfer coefficient, h, can be obtained as:   

$$Nu=\frac{h{d}_p}{{\lambda }_g}=2.0+0.6{Re}_{\mathrm{p}}^{1/2}{Pr}^{1/3}$$(15)

2.4. Chemical Reactions

2.4.1. Heterogeneous Reactions

The reduction process of hematite goes through a stepwise procedure. However, it is extremely difficult to measure the intrinsic kinetics of each step reaction (Fe₂O₃→Fe₃O₄→FeO→Fe) in a few seconds. Thus, this study only considers the overall reduction process (Fe₂O₃→Fe).   

$${Fe}_2{\mathrm{O}}_3+3{\mathrm{H}}_2(\mathrm{g})=2\mathrm{F}e+3{\mathrm{H}}_2\mathrm{O}(\mathrm{g})$$(16)

  

$$F{e}_2{\mathrm{O}}_3+3\mathrm{C}\mathrm{O}(\mathrm{g})=2\mathrm{F}\mathrm{e}+3\mathrm{C}{O}_2(\mathrm{g})$$(17)

Chen et al.^5,6) measured the flash reduction kinetics of hematite concentrate particles in H₂–N₂ and CO–N₂ at 1473–1623 K by experiments. Considering the experimental error, Fan et al.²⁴⁾ modified the reduction kinetics expression by a CFD approach based on the experiments conducted by Chen et al.^5,6) To improve the accuracy of the model, the kinetic expressions modified by Fan et al.²⁴⁾ are adopted to represent the flash reduction rate of hematite particles.   

$$\frac{d{m}_{{Fe}_2{\mathrm{O}}_3-{\mathrm{H}}_2}}{dt}=-w_{O,\mathrm{i}}\cdot m_{\mathrm{p},\mathrm{i}}\cdot {\left(\frac{dX}{dt}\right)}_{H_2}$$(18)

  

$${\left(\frac{dX}{dt}\right)}_{H_2}=8.47\times 10^7\times e^{\left(-\frac{218\mathrm{\hspace{0.17em} }000}{RT}\right)}\cdot \left(p_{{\mathrm{H}}_2}-\frac{p_{{\mathrm{H}}_2\mathrm{O}}}{K_{H_2}}\right)\cdot \left(1-X\right)$$(19)

  

$$\frac{d{m}_{{Fe}_2{\mathrm{O}}_3-CO}}{dt}=-w_{O,\mathrm{i}}\cdot m_{\mathrm{p},\mathrm{i}}\cdot {\left(\frac{dX}{dt}\right)}_{\mathrm{C}\mathrm{O}}$$(20)

  

$${\left(\frac{dX}{dt}\right)}_{\mathrm{C}\mathrm{O}}=5.18\times 10^7\times e^{\left(-\frac{241\mathrm{\hspace{0.17em} }000}{RT}\right)}\cdot \left(p_{\mathrm{C}\mathrm{O}}-\frac{p_{\mathrm{C}{\mathrm{O}}_2}}{K_{\mathrm{C}\mathrm{O}}}\right)\cdot \left(1-X\right)$$(21)

In reference to the above equations, it is noted that the reduction rate of H₂ is faster than that of CO. The reduction degree X is the mass percentage of oxygen that is reduced from iron ore, and is calculated by the following formula:   

$$X=\frac{m_{\mathrm{p},\mathrm{i}}-m_{\mathrm{p}}}{m_{\mathrm{p},\mathrm{i}}\cdot w_{\mathrm{O},\mathrm{i}}}$$(22)

The variation of particle heat capacity should be taken into account because the composition of particles changes with the reduction process.   

$$c_{p,\mathrm{p}}=X\cdot c_{p,\mathrm{F}\mathrm{e}}+\left(1-X\right)\cdot c_{p,\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_3}$$(23)

The heat capacity of Fe and Fe₂O₃ are obtained by HSC 6.0 software.²⁵⁾

2.4.2. Homogeneous Reactions

The homogeneous reaction rate is determined by the minimum value of the chemical reaction rate and the turbulent mixing rate. The Eddy-dissipation model is used to determine the turbulent mixing rate. It assumes that the chemical reaction is faster than the time scale of the turbulence eddies.   

$$R_{i,\text{gas}}=min\left(R_{i,chem},R_{i,\text{tur}}\right)$$(24)

  

$$R_{i,\text{tur}}=min\left[{\nu }_{i,r}^{\prime }{M}_{w,i}A\rho \frac{\epsilon }{k}\underset{R}{min}\left(\frac{Y_R}{v_{R,r}^{\prime }{M}_{w,R}}\right),{\nu }_{i,r}^{\prime }{M}_{w,i}AB\rho \frac{\epsilon }{k}\left(\frac{{\mathrm{\Sigma }}_p{Y}_R}{{\sum }_j^N{v}_{j,r}^{″}{M}_{w,j}}\right)\right]$$(25)

The finite-rate kinetic model is used to determine the chemical reaction rate, and the net source of chemical species i due to the reaction is calculated as the sum of the Arrhenius reaction sources over N_R reactions. The net reaction rate of reversible reaction is the difference between the forward reaction rate and the reversed reaction rate.   

$$\begin{array}{l}{R}_{i,\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}=\\ M_{w,i}\sum _{r=1}^{N_R}\mathrm{\Gamma }\left({\nu }_{i,r}^{″}-{\nu }_{i,r}^{\prime }\right)\left(R_{f,r}{\prod _{j=1}^{N_r}\left[C_{j,r}\right]}^{{\eta }_{j,r}^{\prime }}-R_{b,r}{\prod _{j=1}^{N_r}\left[C_{j,r}\right]}^{{\eta }_{j,r}^{″}}\right)\end{array}$$(26)

  

$$R_{b,r}=\frac{R_{f,r}}{K_r}$$(27)

  

$$K_r=exp\left(-\frac{\mathrm{\Delta }{G}_r^0}{RT}\right){\left(\frac{p_{atm}}{RT}\right)}^{\sum _{i=1}^N\left({\nu }_{i,r}^{″}-{\nu }_{i,r}^{\prime }\right)}$$(28)

where R_f,r, R_b,r and K_r are the forward reaction rate, backward reaction rate, and equilibrium constant for the rth reaction, respectively.

This study only considers the homogeneous reaction of WGSR as follows:   

$$CO\left(\mathrm{g}\right)+{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)\leftrightarrow \mathrm{C}{\mathrm{O}}_2\left(\mathrm{g}\right)+{\mathrm{H}}_2\left(\mathrm{g}\right)$$(29)

The WGSR is a reversible reaction with low temperature favoring the conversion of CO and H₂O to CO₂ and H₂. It is not affected by pressure because there is no change in the volume from reactants to products. The WGSR can be catalyzed by both iron and iron oxides.²⁶⁾ However, the particle volume fraction in the reactor is in the order of 10⁻⁶, and therefore the catalysis of iron and iron oxides can be negligible. Considering the noncatalytic condition and short reaction time in the reactor, the forward reaction rate R_f and backward reaction rate R_b of the WGSR obtained by Bustamante et al.^27,28) are employed in this model.   

$$R_f=7.4\times {10}^{8}exp\left(-\frac{2.883\times {10}^5}{8.314T}\right){[CO]}^{0.5}[{\mathrm{H}}_2\mathrm{O}]$$(30)

  

$$R_b=1.09\times {10}^{7}exp\left(-\frac{2.22\times {10}^5}{8.314T}\right)[{CO}_2]{[{\mathrm{H}}_2]}^{0.5}$$(31)

The equilibrium constant of WGSR is obtained from HSC 6.0 software:²⁵⁾   

$$K_{\text{WGSR}}=2.89\times {10}^{-2}exp\left(\frac{3.21\times {10}^4}{8.314T}\right)$$(32)

2.5. Radiation Model

In Fluent, only the P-1 and discrete ordinates (DO) radiation models allow for the particle radiation. The P-1 radiation model is used when the optical thickness is larger than 1, while the DO radiation model can be adopted in a wide range of optical thicknesses. Thus, the DO model is used for solving the radiative heat-transfer involving both gas and particles. The absorption coefficient for the gas phase is calculated by using the weighted sum of gray gases model (WSGGM), and the gas-phase scattering is ignored.

2.6. Solution Method

The CFD model is established and computed using the finite-volume-based commercial CFD software, FLUENT 17.0. The SIMPLE algorithm is used to couple the pressure and velocity. The second order scheme is chosen for pressure discretization. The second-order upwind scheme is adopted to evaluate the momentum, energy, and species, and the first-order upwind scheme is used for other terms. The reaction rates of heterogeneous reactions are developed and implemented by user-defined functions.

3. Model Setup

In this study, a 3D model is established on the basis of the settings of the experiment.^5,6) A drop tube reactor with an isothermal zone of 1.2 m and an inner diameter of 0.056 m is used as the reactor. Hematite particles carried by 0.3 L/min gas mixture are injected into the reactor from the feeder tube with 1.2 mm inner diameter in the reactor center. The main gas mixture is introduced into the reactor from the gas inlet at the top of reactor. Both the initial temperatures of particles and gas are kept at 300 K. The flow rate of gas is adjusted to change the particle resident time and the gas partial pressure. The operating pressure is 0.85 atm. The properties of hematite are listed in Table 1.

Table 1. Physical properties of hematite concentrate particles.

Parameters	Value
Density	5240 kg/m3
Diameter	21.3 μm
Emissivity	0.8
Scattering factor	0.8
Absorbed reaction heat fraction	30%
Swelling coefficient	1

The computation geometry and mesh of this reactor are shown in Fig. 1. The mesh is locally refined in the center injection zone to obtain the accurate solutions. After the grid-independent study, the grid consisting of 153600 cells is employed. The mass flow inlet boundary is used at the inlets of gas and concentrate particle, and the pressure outlet is defined at the reactor outlet. The reactor wall is considered to be no-slip, and the inhomogeneous distribution of wall temperature is optimized until the calculated particle temperature profile agrees with the particle temperature profile calculated by the literature.²⁴⁾

Fig. 1.

(a) Schematic diagram and computational domain of the drop tube reactor. (b) Three-dimensional computational mesh. (Online version in color.)

4. Results and Discussion

The effect of WGSR on the flash reduction behavior of hematite concentrate particles in syngas will be explored in the following subsections. Results are predicted by two models. The first model ignores the WGSR, and is denoted by NWGSR. The second takes the WGSR into account, and is denoted by WWGSR. The relative deviation δ between the reduction degrees predicted by NWGSR and WWGSR is used to describe the impact of WGSR.   

$$\delta =\frac{X_{NWGSR}-X_{WWGSR}}{X_{NWGSR}}\times 100\%$$(33)

4.1. Model Validation

The simulation results are validated with the experimental data^5,6) according to the reduction degree at the reactor exit. Table 2 shows the relative error ∆ under different operating conditions. The maximum relative error Δ between the simulated and experimental data is 9.30%, which indicates a good agreement between the experiment and simulation. Thus, this CFD model can reasonably predict the flash reduction behavior of hematite particles.   

$$\mathrm{\Delta }=\frac{\left|X_{experimental}-X_{simulated}\right|}{X_{experimental}}\times 100\%$$(34)

Table 2. Comparison of final reduction degree between simulation and experiment.

Atmosphere	T (K)	PH2/CO (atm)	VH2/CO (L/min)	Reduction degree (–)		Relative error Δ (%)
				Experimental	Simulated
H2–N2	1473	0.2	0.9	0.54	0.56	3.70
	1473	0.2	1.5	0.37	0.39	5.41
	1473	0.2	1.9	0.33	0.31	6.06
	1523	0.2	0.9	0.75	0.76	1.33
	1523	0.2	1.5	0.57	0.57	0
	1523	0.2	1.9	0.43	0.47	9.30
	1623	0.1	0.4	0.89	0.89	0
	1623	0.1	0.7	0.67	0.7	4.48
	1623	0.1	0.9	0.61	0.6	1.64
CO–N2	1473	0.6	1.5	0.34	0.34	0
	1473	0.6	3	0.18	0.17	5.56
	1473	0.85	2.2	0.42	0.4	4.76
	1573	0.45	0.7	0.73	0.76	4.11
	1573	0.45	1.1	0.64	0.64	0
	1623	0.45	0.7	0.85	0.91	7.06
	1623	0.45	1.1	0.8	0.82	2.50
	1623	0.45	1.7	0.71	0.67	5.63

4.2. Reduction Behavior in H₂–CO–N₂ System

4.2.1. Effect of WGSR in H₂–CO–N₂ System

In this section, the effect of WGSR on the flash reduction process is investigated in the CO–H₂–N₂ system, as shown in Table 3. Note that the flow rates of CO and N₂ can be obtained by their partial pressure, and are not detailed in the table for simplicity. In addition, the change of particle residence time resulting from the WGSR can be ignored. The reduction degree increases with increasing partial pressure of the reductive gas and temperature in both NWGSR and WWGSR. Moreover, on account of the faster reduction rate of H₂ than CO, the reduction degree increases more with increasing H₂ than with CO. No significant differences in the reduction degree can be detected, which indicates that the WGSR has a small or even a negligible influence on the reduction process in the H₂–CO system. It is reasonable that because of few oxide gases produced by reduction reactions and the lower kinetic rate of the WGSR, the WGSR proceeds at a very slow rate. Comparing case A3 with case A10 reveals that with the increase of particle flow, the effect of the WGSR becomes more significant.

Table 3. Effect of WGSR on the reduction behavior of hematite in the H₂–CO–N₂ system.

Case	T (K)	PH2 (atm)	PCO (atm)	VH2 (L/min)	mparticle (kg/s)	t (s)	Reduction degree (%)
							NWGSR	WWGSR
A1	1573	0.1	0.05	0.4	10−6	3.031	67.47	67.48
A2	1573	0.1	0.1	0.4	10−6	3.030	69.29	69.3
A3	1573	0.1	0.2	0.4	10−6	3.026	72.62	72.64
A4	1573	0.2	0.1	1.0	10−6	2.594	86.02	86.03
A5	1573	0.2	0.2	1.0	10−6	2.588	87.27	87.28
A6	1573	0.2	0.3	1.0	10−6	2.582	88.41	88.43
A7	1473	0.1	0.2	0.4	10−6	3.197	33.8	33.8
A8	1523	0.1	0.2	0.4	10−6	3.104	51.2	51.21
A9	1623	0.1	0.2	0.4	10−6	3.015	87.28	87.31
A10	1573	0.1	0.2	0.4	10−5	2.029	49.73	49.86

4.2.2. Effect of WGSR Reaction Rate

In order to clarify the effect of the reaction rate of WGSR on the reduction process, several kinetic rates of WGSR presented in Table 4 are tested in our model. These kinetic rates are either very low or very fast, which leads to the WGSR already being at thermodynamic equilibrium, or very high levels of CO/H₂ conversion.

Table 4. Different reaction rates of WGSR.

No.	Rate expression ((kmol/m3)/s)	Comment	Reference
Rate 1	$R_f=7.4\times {10}^{8}exp\left(-\frac{2.883\times {10}^5}{8.314T}\right){[CO]}^{0.5}[{\mathrm{H}}_2\mathrm{O}]$	slower	27, 28)
	$R_b=1.09\times {10}^{7}exp\left(-\frac{2.22\times {10}^5}{8.314T}\right)[{CO}_2]{[{\mathrm{H}}_2]}^{0.5}$	slower
Rate 2	$R_f=2.75\times {10}^{3}exp\left(-\frac{8.36\times {10}^4}{8.314T}\right)[CO][{\mathrm{H}}_2\mathrm{O}]$	faster	29)
	$R_b=\frac{R_f}{K}[{CO}_2][{\mathrm{H}}_2]$	faster
	$K=2.65\times {10}^{-2}exp(3\mathrm{\hspace{0.17em} }956/T)$
Rate 3	$R_f=7.68\times {10}^{10}exp\left(-\frac{36\mathrm{\hspace{0.17em} }640}{T}\right){[CO]}^{0.5}[{\mathrm{H}}_2\mathrm{O}]$	faster	30, 31)
	$R_b=6.4\times {10}^{9}exp\left(-\frac{39\mathrm{\hspace{0.17em} }260}{T}\right)[{CO}_2]{[{\mathrm{H}}_2]}^{0.5}$	slowest
Rate 4	$R_f=2.34\times {10}^{10}exp\left(-\frac{2.883\times {10}^5}{8.314T}\right){[CO]}^{0.5}[{\mathrm{H}}_2\mathrm{O}]$	faster	32, 33)
	$R_b=2.2\times {10}^{7}exp\left(-\frac{1.9\times {10}^5}{8.314T}\right)[{CO}_2]{[{\mathrm{H}}_2]}^{0.5}$	faster
Rate 5	$R_f=2.78\times {10}^{3}exp\left(-\frac{1.26\times {10}^4}{8.314T}\right)[CO][{\mathrm{H}}_2\mathrm{O}]$	fastest	34)
	$R_b=9.59\times {10}^{4}exp\left(-\frac{4.66\times {10}^4}{8.314T}\right)[{CO}_2][{\mathrm{H}}_2]$	fastest

Table 5 illustrates the comparison between the gas temperatures and species mole fractions at the reactor exit predicted by different reaction rates of WGSR. The results indicate that gas composition is slightly sensitive to the reaction rate of the WGSR, which is consistent with the result reported by Gómez-Barea and Leckner.³⁴⁾ The minor differences in gas composition among the four cases indicates that even though the kinetic rate of the WGSR is very fast, this reaction still proceeds so slowly that the gas temperature and particle resident time undergo no change. The slight change in reduction degree can be attributed to the change in gas mixture. The Rate 3 model has a faster forward rate and the slowest backward rate of the WGSR, so the amount of hydrogen is the highest. Because the reaction rate of H₂ is higher than that of CO, the reduction degree increases with the enhancement of H₂, which leads to the relatively higher reduction degree in the Rate 3 model. It can be inferred that a faster rate of forward WGSR enhances the reduction process. However, the too fast reaction rate in the Rate 5 model becomes increasingly unstable and difficult to converge, so the results for the Rate 5 model are not shown in Table 5. Thus, the WGSR rates must be carefully considered before they are applied to the simulation process, because each WGSR reaction rate only works for a specific condition. Moreover, further research is needed to investigate the kinetic of the WGSR in a drop tube reactor for the improvement in accuracy.

Table 5. Effect of reaction rates of WGSR on the reduction behavior of case A5.

No.	Rate 1	Rate 2	Rate 3	Rate 4
Mole fraction (%)
H2	23.03	23.02	23.11	23.02
CO	23.47	23.48	23.39	23.32
H2O*102	49.85	51.00	30.91	39.57
CO2*102	6.57	5.41	13.97	9.36
Tg (K)	1106.07	1106.08	1105.60	1106.00
t (s)	2.588	2.588	2.588	2.588
X (%)	87.28	87.27	87.39	87.29

4.2.3. Relationship between Synergistic Effect and WGSR

Fan et al.³⁵⁾ also investigated the flash reduction kinetics of hematite concentrate by H₂–CO mixture in a drop tube reactor through a CFD approach. They analyzed the global reduction rate of hematite in H₂–CO mixture by using the reduction rates of hematite in single gases (Eqs. (19) and (21)), as follows:   

$$\begin{array}{c}{\left({\displaystyle \frac{dX}{dt}}\right)}_{H_2+\mathrm{C}\mathrm{O}}=\left[1+\left(-0.004T+7.004\right)\cdot {\displaystyle \frac{p_{CO}}{p_{CO}+p_{{\mathrm{H}}_2}}}\right]\\ \cdot {\left({\displaystyle \frac{dX}{dt}}\right)}_{H_2}+{\left({\displaystyle \frac{dX}{dt}}\right)}_{CO}\end{array}$$(35)

As can be deduced from Eq. (35), the global reduction rate of hematite in H₂–CO mixture is not the simple summation of reduction rates of component gases, but rather a linear combination, which implies a synergistic effect in H₂–CO mixture. In order to investigate whether the synergistic effect is caused by the WGSR, the reduction degree calculated from the global reduction rate without WGSR (Method 1) is compared with that calculated from the single reduction rate of the component gas with WGSR (Method 2). The reduction conditions in Table 3 are adopted. As shown in Table 6, the significant difference of reduction degree predicted by Method 1 and Method 2 demonstrates that the synergistic effect cannot be completely reproduced only by the WGSR. Kon et al.¹⁷⁾ also found the synergetic effect in the reduction process of a sinter packed bed by H₂–CO–H₂O–CO₂ mixture. They believed that the synergistic effect may be a result of the improvement of sample structure, rather than the influence of WGSR. Because the influence of WGSR did not appear as a significant difference in the small packed bed, it can be inferred that the synergetic effect may be a result of the improvement of the sample structure. Therefore, the synergistic effect in the flash reduction process by H₂–CO mixture may be related to the structural change of hematite, which needs to be further clarified.

Table 6. Comparison of the reduction behavior predicted by different methods.

Case	Method 1		Method 2
	t (s)	X (%)	t (s)	X (%)
A1	3.043	75.43	3.031	67.48
A2	3.046	79.68	3.030	69.30
A3	3.046	84.06	3.026	72.64
A4	2.604	91.47	2.594	86.03
A5	2.601	93.86	2.588	87.28
A6	2.596	95.12	2.582	88.43

4.3. Reduction Behavior in H₂–CO–H₂O–CO₂–N₂ System

In this section, the influence of the WGSR on the flash reduction process is investigated in the H₂–CO–H₂O–CO₂–N₂ system. Based on the above research results, this impact is closely related to the gas composition. Here, the composition of the gas mixture is characterized by the reaction quotient Q_r which is calculated as follows:   

$$Q_{\mathrm{r}}=\frac{P_{H_2}\cdot P_{{CO}_2}}{P_{H_2\mathrm{O}}\cdot P_{CO}}$$(36)

On account of HSC 6.0 software,²⁵⁾ the equilibrium constant of the WGSR, K_WGSR, is 0.338 at 1573 K. To explore the different behavior of WGSR in different gas compositions, the initial reaction quotient of the gas mixture is employed as 0.1, 0.338, and 1.0 by fixing the partial pressures of H₂ and CO at 0.2 atm and changing the partial pressures of H₂O and CO₂. The flow rate of H₂ is set as 1.0 L/min. The flow rates of other gases can be derived from their partial pressure, and are not detailed in the following tables for the sake of simplicity. The particle mass rate is set as 10⁻⁵ kg/s for highlighting the influence of the WGSR.

4.3.1. Effect of WGSR at Q_r<K_WGSR

In this subsection, the flash reduction behavior of hematite concentrate particles is investigated in the H₂–CO–H₂O–CO₂–N₂ system in which the initial reaction quotient (Q_r=0.1) is lower than the equilibrium constant of the WGSR (K_WGSR=0.338 at 1573 K). As shown in Table 7, because introducing a quantity of CO₂ and H₂O into H₂–CO mixture limits the reducing power of gas mixture, the reduction degree decreases as the partial pressures of CO₂ and H₂O increase in both WWGSR and NWGSR. Moreover, the reduction degree predicted by WWGSR is slightly higher than that predicted by NWGSR, implying that the WGSR slightly enhances the reduction degree.

Table 7. Effect of WGSR on the reduction behavior of hematite at Q_r=0.1 (Q_r<K_WGSR).

Case	T (K)	PH2O (atm)	PCO2 (atm)	t (s)	Reduction degree (%)		Relative deviation (%)
					NWGSR	WWGSR
B1	1573	0.02	0.002	1.857	68.68	68.91	−0.33
B2	1573	0.04	0.004	1.847	65.09	65.46	−0.57
B3	1573	0.08	0.008	1.821	56.69	57.46	−1.36
B4	1573	0.1	0.01	1.805	51.73	52.71	−1.89
B5	1473	0.1	0.01	1.784	21.12	21.27	−0.71
B6	1523	0.1	0.01	1.789	33.81	34.22	−1.21
B7	1623	0.1	0.01	1.821	67.35	69.12	−2.63

To explain the favorable impact of WGSR, case B4 is chosen from cases B1–B7 as a representative to illustrate. Figure 2 illustrates the flash reduction behavior of case B4 predicted by WWGSR and NWGSR. Note that for comparison, the legends in figures have been set in the same range. The reaction rates of WGSR in Figs. 2(a) and 2(b) demonstrate that the WGSR direction tends to shift to the right at Q_r<K_WGSR, so the mole fractions of H₂ and CO₂ predicted by WWGSR are higher than those predicted by NWGSR, while CO and H₂O in WWGSR are less than in NWGSR, as demonstrated in Figs. 2(c)–2(f). Because H₂ has a faster reduction rate than CO, H₂ produced by WGSR accelerates the reduction rate of hematite, and then increases the reduction degree. Zuo et al.³⁶⁾ also found that the increase of the H₂ content in the H₂–CO mixture gas led to a more rapid increase in the reaction rate of hematite pellets.

Fig. 2.

Typical flash reduction behavior (case B4 in Table 7) predicted by WWGSR and NWGSR: (a–b) reaction rate of WGSR; (c–f) species mole fraction distributions. (Online version in color.)

Because the change of gas concentration in the center is more sensitive than that in the near-wall region, the gas temperature and species mole fraction distributions along the reactor centerline are used to display some qualitative results, as presented in Fig. 3. As expected, the conversion of CO and H₂O to CO₂ and H₂ caused by the forward WGSR leads to the increases of H₂ and CO₂ in gas mixture, accompanied by the decreases of H₂O and CO. A consistent temperature curve means that the gas conversion has quite limited effect on the heat transfer. It can be inferred that the difference of reduction degree in Table 7 is mainly owing to the gas composition resulting from the WGSR.

Fig. 3.

Profiles of gas temperature and species along the reactor centerline in case B4. (Online version in color.)

Difference in gas composition caused by WGSR will lead to difference in reducing strength of gas mixture, which in turn leads to different reduction behavior. To quantitatively describe the effect of WGSR on the gas reducing strength, the reduction capacity F is introduced in this study. In terms of the reduction rate expressions of Eqs. (19) and (21), the reduction capacities of H₂ and CO are defined as:   

$$F_{H_2}=8.47\times 10^7\times e^{\left(-\frac{218\mathrm{\hspace{0.17em} }000}{RT}\right)}\cdot \left(p_{{\mathrm{H}}_2}-\frac{p_{{\mathrm{H}}_2\mathrm{O}}}{K_{H_2}}\right)$$(37)

  

$$F_{\mathrm{C}\mathrm{O}}=5.18\times 10^7\times e^{\left(-\frac{241\mathrm{\hspace{0.17em} }000}{RT}\right)}\cdot \left(p_{\mathrm{C}\mathrm{O}}-\frac{p_{\mathrm{C}\mathrm{O}}}{K_{\mathrm{C}\mathrm{O}}}\right)$$(38)

According to the above equations, the reducing capacity increases with the partial pressure of H₂/CO and decreases with the partial pressure of H₂O/CO₂. The total reduction capacity of gas mixture can be calculated by:   

$$F_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=F_{{\mathrm{H}}_2}+F_{\mathrm{C}\mathrm{O}}$$(39)

Figure 4(a) displays the gas reduction capacity profiles along the reactor centerline predicted by WWGSR and NWGSR in case B4 where the forward WGSR prefer to occur. As expected, the WWGSR model predicts higher reduction capacity of H₂ and lower reduction capacity of CO than dose the NWGSR model due to the higher content of H₂ and CO₂ and the lower content of CO and H₂O. The total reduction capacity in the WWGSR model is higher than that in the NWGSR model, which indicates that the reduction capacity of gas mixture is increased by the forward WGSR. Furthermore, Fig. 4(b) presents the difference between reduction capacities predicted by WWGSR and NWGSR, i.e., the change of reduction capacity caused by the WGSR. The increase in H₂ reduction capacity is much greater than the decrease in CO reduction capacity, which leads to the difference of total reduction capacity resulting from the WGSR being greater than zero. This further indicates that the forward WGSR improves the reduction ability of gas mixture.

Fig. 4.

In case B4 (a) gas reduction capacity profiles along the reactor centerline, (b) difference between reduction capacities predicted by WWGSR and NWGSR. (Online version in color.)

As shown in cases B1–B4 in Table 7, the increase in reduction degree caused by the forward WGSR is more significant at higher oxygen partial pressures. To explore the reason for this change, Fig. 5(a) shows the profile of the difference between reduction capacities predicted by WWGSR and NWGSR in cases B1 and B4. It is evident that the improvement of total reduction force due to the forward WGSR enlarges with the increase of oxygen partial pressure. This is because the higher oxygen partial pressure can accelerate the forward WGSR rate, and results in the greater change in gas mixture composition.

Fig. 5.

Difference between reduction capacities predicted by WWGSR and NWGSR: (a) at different oxygen partial pressures (cases B1 and B4); (b) at different temperatures (cases B4 and B7). (Online version in color.)

Although the forward WGSR is thermodynamically favored at low temperature, high temperature can increase the reaction rate of the forward WGSR, causing greater positive influence on the total reduction capacity of gas mixture, as shown in Fig. 5(b). Thus, in cases B4–B7 the absolute value of the relative deviation of reduction degree between WWGSR and NWGSR increases from 0.71% to 2.63% when the temperature increases from 1473 K to 1623 K.

In conclusion, the favorable impact of the WGSR on reduction degree is detected at Q_r<K_WGSR where the WGSR goes forward. This impact increases with increasing oxygen partial pressure and temperature. Moreover, it is worth mentioning that the similar results can also be drawn in the H₂–CO–H₂O–N₂ system because the WGSR moves forward in both cases. In order to take advantage of the forward WGSR, the reaction quotient of gas mixture should be controlled less than the equilibrium constant of the WGSR.

4.3.2. Effect of WGSR at Q_r=K_WGSR

Although the initial reaction quotient of gas mixture Q_r is equal to the equilibrium constant of the WGSR K_WGSR, the WWGSR model predicts a slightly higher reduction degree than dose the NWGSR model, as shown in Table 8. This discrepancy is caused by the oxidation gas produced in the reduction process. Due to the faster reaction rate of H₂ than CO, the amount of the produced H₂O is greater than the produced CO₂, causing the reaction quotient of gas mixture to be slightly less than K_WGSR. From the perspective of reaction equilibrium, the WGSR goes in the forward direction, leading to the decrease of CO and enhancement of H₂, as shown in Fig. 6. As previously mentioned, the forward WGSR can improve the reduction ability of the gas mixture and thus the reduction degree. Similarly, the impact of the WGSR on reduction process increases with the increase of partial pressure of oxidized gas.

Table 8. Effect of WGSR on the reduction behavior of hematite at Q_r=0.338 (Q_r=K_WGSR).

Case	T (K)	PH2O (atm)	PCO2 (atm)	t (s)	Reduction degree (%)		Relative deviation (%)
					NWGSR	WWGSR
C1	1573	0.02	0.0068	1.857	68.27	68.49	−0.32
C2	1573	0.04	0.0135	1.843	64.27	64.64	−0.58
C3	1573	0.08	0.027	1.816	54.63	55.41	−1.43
C4	1573	0.1	0.0338	1.803	48.91	49.94	−2.11

Fig. 6.

Typical flash reduction behavior (case C4 in Table 8) predicted by WWGSR and NWGSR: (a–b) reaction rate of WGSR; (c–f) species mole fraction distributions. (Online version in color.)

4.3.3. Effect of WGSR at Q_r>K_WGSR

As shown in Table 9, when the initial reaction quotient (Q_r=1.0) is greater than the equilibrium constant of the WGSR (K_WGSR=0.338), the reduction degree in WWGSR is lower than that in NWGSR. This means that the WGSR decreases the reduction degree of hematite at Q_r>K_WGSR, and a similar result was found in the research of Valipour and Mokhtari.²⁰⁾

Table 9. Effect of WGSR on the reduction behavior of hematite at Q_r=1.0 (Q_r>K_WGSR).

Case	T (K)	PH2O (atm)	PCO2 (atm)	t (s)	Reduction degree (%)		Relative deviation (%)
					NWGSR	WWGSR
D1	1573	0.02	0.02	1.856	68.02	67.81	0.31
D2	1573	0.04	0.04	1.847	62.93	62.20	1.19
D3	1573	0.08	0.08	1.816	51.4	48.94	4.89
D4	1573	0.1	0.1	1.806	46.10	42.62	7.71
D5	1473	0.1	0.1	1.795	18.94	18.29	3.42
D6	1523	0.1	0.1	1.796	30.20	28.65	5.09
D7	1623	0.1	0.1	1.820	60.31	54.54	9.56

As can be seen Fig. 7, at Q_r>K_WGSR, the WGSR proceeds in reverse direction, accompanied by the consumption of H₂ and CO₂ as well as the enhancement of CO and H₂O. Thus, the reverse WGSR decreases the reduction capacity of H₂ and increases the reduction capacity of CO, as shown in Fig. 8. However, the total reduction capacity of gas mixture is weakened by reverse WGSR, resulting in the decreases in both of the reduction rate and reduction degree of hematite.

Fig. 7.

Typical flash reduction behavior (case D4 in Table 9) predicted by WWGSR and NWGSR: (a–b) reaction rate of WGSR; (c–f) species mole fraction distributions. (Online version in color.)

Fig. 8.

In case D4 (a) gas reduction capacity profiles along the reactor centerline, (b) difference between gas reduction capacities predicted by WWGSR and NWGSR. (Online version in color.)

As shown in cases D1–D4 in Table 9, the decline in reduction degree caused by WGSR amplifies with increasing oxygen partial pressures. The reason for this that, the higher amount of CO₂ in the gas mixture facilitates the reverse WGSR, resulting in the greater negative influence on the total reduction power of gas mixture, exhibited in Fig. 9(a).

Fig. 9.

Difference between reduction capacities predicted by WWGSR and NWGSR: (a) at different oxygen partial pressures (cases D1 and D4); (b) at different temperatures (cases D4 and D7). (Online version in color.)

Because the reverse WGSR is thermodynamically and dynamically favorable at a higher temperature, the high temperature speeds up the reverse WGSR, leading to an increase of the negative influence on total reduction force of gas mixture, as shown in Fig. 9(b). Thus, in cases D4–D7 in Table 9, when the temperature increases from 1473 K to 1623 K, the relative deviation of reduction degree between NWGSR and WWGSR increases from 3.42% to 9.56%.

In conclusion, the reverse WGSR plays a harmful role in the flash reduction process at Q_r>K_WGSR where the WGSR tends to backward. This impact is more significant at higher oxygen partial pressure and temperature. Additionally, the similar results also can be obtained in the H₂–CO–CO₂–N₂ system because the WGSR moves backward in both cases. To avoid the reverse WGSR, the reaction quotient of gas mixture should be less than the equilibrium constant of the WGSR.

5. Conclusions

In this study, a 3D Euler-Lagrange model including heat and mass transfer, and heterogeneous and homogeneous reactions, is developed to simulate the flash reduction behavior of hematite with syngas in a drop tube reactor. After validation, the influence of the WGSR on the reduction behavior is investigated in the H₂–CO and H₂–CO–CO₂–H₂O systems by comparing the reduction degree predicted by the WWGSR and NWGSR models. The gas species distributions, reaction rate distributions, and reduction capacities profiles are demonstrated, with the purpose of understanding the influence mechanism of WGSR.

(1) The influence of the WGSR is weak in the H₂–CO system, but it increases with increasing particle mass rate. The reaction rate of the WGSR can affect the reduction process by influencing the gas composition.

(2) The synergistic effect of H₂–CO mixture on the reduction degree in the experiment cannot be completely reproduced by only WGSR in this model. Deeper investigations are needed to explore the relationship between the synergistic effect and the structural change of iron ore.

(3) In the H₂–CO–CO₂–H₂O system, the influence of the WGSR is concerned with the relationship between the initial reaction quotient of gas mixture, Q_r, and the equilibrium constant of WGSR, K_WGSR. When Q_r<K_WGSR or Q_r=K_WGSR, the WGSR proceeds in the forward direction, which leads to the increase of the total reduction capacity of gas mixture, and in turn promotes the hematite reduction. While at Q_r>K_WGSR, the WGSR moves in the reverse direction, causing the decrease of the total reduction capacity, which in turn weakens the hematite reduction.

(4) The impact of the WGSR on reduction behavior becomes more significant with increasing reduction temperature and partial pressure of oxidizing gas.

Acknowledgment

This work is financially supported by the National Natural Science Foundation of China (51904066), Fundamental Research Funds for the Central Universities (N182503032), Postdoctoral Foundation of Northeastern University (20190201) and Postdoctoral International Exchange Program (Dispatch Project, 20190075).

Nomenclature

A_p: particle surface area, m²;

C_D: drag coefficient, kg/(m³·s);

c_p,g, c_p,p: heat capacity of gas and particle, J/(kg·K);

D_eff: effective diffusivity coefficient, m²/s;

d_p: particle diameter, m;

F_D: drag force per unit particle mass;

f_h: particle reaction heat absorption ratio;

g: gravitational acceleration, (m²/s);

G_k: generation term for turbulence kinetic energy;

h: convective heat transfer coefficient, W/(m²·K);

H_Fe2O3-H2, H_Fe2O3-CO: reaction heat of Fe₂O₃ reduced by H₂ and CO, J/kg;

k: turbulence kinetic energy, m²/s²;

K_H2, K_CO: reaction equilibrium constant of FeO reduced by H₂ and CO;

K_WGSR: equilibrium constant of WGSR;

m_p: instantaneous particle mass, kg;

m_p,i: initial particle mass, kg;

Nu: Nusselt number;

p: gas pressure, Pa;

P_H2, P_H2O, P_CO, P_CO2: partial pressure of H₂, H₂O, CO and CO₂, atm;

Pr: Prandtl number;

Q_r: initial reaction quotient of gas mixture;

R: universal gas constant, J/(kmol·K);

Re_p: particle Reynolds number;

S_p,m, S_p.mon, S_p,h, S_pYi: interphase exchange terms for mass, momentum, enthalpy and species;

S_h, S_Yi: source terms due to the homogeneous gas-phase reactions;

S_rad: radiation source term;

Sc_t: turbulent Schmidt number;

t: particle resident time, s;

T_g, T_p: temperature of gas and particle, K;

θ_R: radiation temperature, K;

u_g, u_p: velocity of gas and particle, m/s;

w_o,i: initial oxygen mass fraction in particle, %;

X: reduction degree, %;

Y_i: mass fraction of chemical species i;

Z: distance from the particle inlet, m;

τ_ij: viscous stress tensor;

ε: dissipation rate of turbulence kinetic energy, m²/s³;

ε_p: particle emissivity;

λ_eff: effective thermal conductivity, W/(m·K);

μ_g, μ_t: dynamic viscosity and turbulent viscosity, kg/(m·s);

ρ_g, ρ_p: density of gas and particle, kg/m³;

σ: Stefan-Boltzmann constant, W/(m²·K⁴);

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