Numerical Analysis of the Characteristics Inside Pre-reduction Shaft Furnace and Its Operation Parameters Optimization by Using a Three-Dimensional Full Scale Mathematical Model

Abstract

Numerical simulation is considered to be an important method to study the inner characteristics of metallurgical processes, thus providing effective strategies for practical smooth operations. A three-dimensional full scale mathematical model considering mass, momentum, energy transfers and chemical reactions under steady state is developed in the present work to describe the characteristics inside the pre-reduction shaft furnace of COREX smelting reduction ironmaking process. The uneven gas and solid flow distributions in both radial and axial directions not only restrain the gas utilization but also cause the difference in the solid metallization between the center and near wall to reach as high as 0.4. Predicted by the established model, the CO–CO₂–H₂–H₂O reducing gas with the temperature 1050–1100 K, the volume fraction of CO+CO₂ around 70%, the ratio of CO to CO₂ 5–7, as low as possible H₂O content, and the reasonably matched burden solid charging rate to control the top gas consumption per ton burden solid (TGC) in the range of 800–1000 Nm³/t, are the optimal operation conditions to further improve furnace efficiency.

1. Introduction

In recent decades, some alternative ironmaking processes have been developed around the world in order to reduce fuel consumption and CO₂ emission. COREX smelting reduction ironmaking process is one of them that have successfully realized industrial scale production. However, due to its short history in comparison with blast furnace process, related research combined with practical production is far from satisfactory, for instance, the upper pre-reduction shaft furnace of COREX-3000 process in Baosteel faces lots of problems, such as gas segregation distribution and burden sticking.^1,2) Therefore, it is of great importance to study the characteristics inside the furnace in order to provide some effective strategies for smooth operation.

In order to overcome the shortage of direct measurement methods, the mathematical models have been developed to simulate the transport phenomena inside the blast furnace process and they are proved to be useful for fully understanding the in-furnace characteristics.^3,4) As for the shaft type furnace process, both Hara et al.⁵⁾ and Tsay et al.⁶⁾ determined iron oxides gaseous reduction rate based on the shrinking unreacted core model, and then Valipour et al.⁷⁾ further developed the grain model to investigate the effects of pellet porosity and tortuosity factor on the reduction rate. Based on above works, the one-dimensional mathematical models of shaft furnace were established and mainly used to predict the gas and solid composition distributions in the axial direction.^{8,9,10,11,12,13)} The inner characteristic distributions, such as velocity and temperature in the radial direction, were described by introducing the two-dimensional mathematical models.^14,15,16) Furthermore, the development of threedimensional mathematical model is necessary to improve some drawbacks in the previous numerical simulation works of shaft furnace. For example, taking the gas volume flow rate in three-dimensional model instead of the gas velocity in two-dimensional model as the gas inlet boundary condition actually represents the practical operation condition.¹⁷⁾ On the other hand, although similar works related to the shaft type furnace of SC¹⁸⁾ or MIDREX¹⁹⁾ process had been reported, the COREX pre-reduction shaft furnace differs from others not only in the furnace structure, but also in operation conditions. As a result, the present work develops a three-dimensional full scale mathematical model combined with mass, momentum, energy transfers and chemical reactions under steady state to study the inner characteristics of the COREX pre-reduction shaft furnace. In addition, the influences of reducing gas temperature, composition, TGC and productivity on the gas utilization and solid metallization are further investigated to optimize operation conditions for practical production.

2. Model Formulation

Since both gas and solid phases are assumed to be fluids in the present work, the general conservation equation under steady state based on the computational fluid dynamics (Eq. (1)) is applied to simulate mass, momentum, energy and specie transfer behaviors.²⁰⁾

  

$$\nabla \cdot ({\epsilon }_p\cdot {\rho }_p\cdot \varphi \cdot {\overrightarrow{v}}_p)=\nabla \cdot ({\epsilon }_p\cdot {\Gamma }_{\varphi }\cdot \nabla (\varphi ))+S_{\varphi }$$(1)

where, p is the phase to be considered, both the effective diffusive transfer coefficient Γ_ϕ and the source S_ϕ vary with respect to the different variable ϕ, which are collected and explained in Table 1.

Table 1. Terms in Eq. (1).

Eq.	$\varphi$	${\Gamma }_{\varphi }$	$S_{\varphi }$
continuity	1	0	$M_O\cdot \mathrm{\sum }_{n=1}^6{R}_n$
			$-M_O\cdot \mathrm{\sum }_{n=1}^6{R}_n$
momentum	${\overrightarrow{v}}_g$	0	$\nabla \cdot {\overline{\overline{\tau }}}_g+{\epsilon }_g(-\nabla P+{\rho }_g\cdot \overrightarrow{g})+{\overrightarrow{F}}_{gs}$
	${\overrightarrow{v}}_s$		$\nabla \cdot {\overline{\overline{\tau }}}_{\text{s}}+{\epsilon }_s(-\nabla P+{\rho }_s\cdot \overrightarrow{g})-{\overrightarrow{F}}_{gs}+{\overrightarrow{F}}_w$
energy	Hg	Kg/Cp,g	$E_{gs}+M_O\cdot \mathrm{\sum }_{n=1}^7(R_n\cdot \mathrm{\Delta }{H}_n^T)$
	Hs	Ks/Cp,s	$-E_{gs}-M_O\cdot \mathrm{\sum }_{n=1}^6(R_n\cdot \mathrm{\Delta }{H}_n^T)$

R_n refer to Eqs. (2), (3), (4), (5), (6), (7), (8)

${\stackrel{=}{\stackrel{}{\tau }}}_p={\epsilon }_p{\mu }_p[\nabla {\overrightarrow{v}}_p+{(\nabla {\overrightarrow{v}}_p)}^{\text{T}}]-\frac{2}{3}{\epsilon }_p{\mu }_p(\nabla \cdot {\overrightarrow{v}}_p)\stackrel{=}{\stackrel{}{I}}$ ³⁾

${\overrightarrow{F}}_{gs}=-[150\frac{{\epsilon }_s(1-{\epsilon }_g){\mu }_g}{{\epsilon }_g{d}_s^2}+1.75\frac{{\rho }_g{\epsilon }_s\left|{\overrightarrow{v}}_s-{\overrightarrow{v}}_g\right|}{d_s}]({\overrightarrow{v}}_s-{\overrightarrow{v}}_g)$ ^21,22)

${\overrightarrow{F}}_w=\{\begin{array}{l}0\text{ above the gas inlet level}\\ f_s\cdot \frac{{\rho }_s{v}_s^2}{R_2-R_1}\cdot \frac{R_1^2-R_{max}^2}{2R_1}\text{ at inner wall below the gas inlet level }\\ f_s\cdot \frac{{\rho }_s{v}_s^2}{R_2-R_1}\cdot \frac{R_1^2-R_{max}^2}{2R_2}\text{ at outer wall below the gas inlet level }\end{array}$

where $R_{max}^2=\frac{R_2^2-R_1^2}{2ln(R_2/R_1)}$ ²³⁾

$E_{gs}=-\frac{6\cdot k_g\cdot {\epsilon }_g\cdot {\epsilon }_s}{d_s^2}\cdot (2.0+0.6\cdot {Re}_s^{1/2}\cdot {Pr}_g^{1/3})\cdot (T_g-T_s)$ ^24,25)

The species in the gas phase include CO, CO₂, H₂, H₂O and those in the solid phase include Fe₂O₃, Fe₃O₄, FeO, Fe, so the following successive reductions of iron oxides by CO and H₂ (Eqs. (2), (3), (4), (5), (6), (7)) are considered in the model, and the chemical reaction rates are calculated based on the three-interface unreacted core model^5,26) with the physical chemistry data of the species from Perry et al’s book²⁷⁾ and the reaction rate constants and the effective diffusion coeffients from Takahashi et al.’s work.²⁸⁾ Meanwhile, the importance of considering the water gas shift reaction (Eq. (8)) in the CO–CO₂–H₂–H₂O mixture reducing gas was proved by Negri et al.,²⁹⁾ so its reaction rate is calculated in the present model by introducing the rate equations and the rate constants determined by Takahashi et al. as well.²⁸⁾

  

$$\begin{array}{cc}{R}_1& 3F{e}_2{O}_3+CO\stackrel{}{⟶}2F{e}_3{O}_4+C{O}_2\end{array}$$(2)

  

$$\begin{array}{cc}{R}_2& F{e}_3{O}_4+CO\stackrel{}{⟶}3FeO+C{O}_2\end{array}$$(3)

  

$$\begin{array}{cc}{R}_3& FeO+CO\stackrel{}{⟶}Fe+C{O}_2\end{array}$$(4)

  

$$\begin{array}{cc}{R}_4& 3F{e}_2{O}_3+H_2\stackrel{}{⟶}2F{e}_3{O}_4+H_{2}O\end{array}$$(5)

  

$$\begin{array}{cc}{R}_5& F{e}_3{O}_4+H_2\stackrel{}{⟶}3FeO+H_{2}O\end{array}$$(6)

  

$$\begin{array}{cc}{R}_6& FeO+H_2\stackrel{}{⟶}Fe+H_{2}O\end{array}$$(7)

  

$$\begin{array}{cc}{R}_7& CO+H_{2}O\stackrel{}{⟶}C{O}_2+H_2\end{array}$$(8)

The schematic diagram of the COREX pre-reduction shaft furnace is shown in Fig. 1. The reducing gas is introduced through the circumferentially distributed inlets and discharged from the top, and its properties, such as viscosity, thermal conductivity, are calculated based on the previous researchers' works.^{30,31,32,33,34,35)} On the other hand, charged from the top, the burden solid gradually descends to the bottom, where a man-made deadman is established to facilitate the discharge of direct reduced iron (DRI) located in the central region. The average diameter of the solid particle is 15.4 mm and the voidage of the moving bed is fixed as 0.4 according to the measurement in the laboratory experiments. As for the boundary condition, the main operation conditions of the COREX pre-reduction shaft furnace in the base case are collected in Table 2. At the wall boundary, the free-slip condition is applied for the motion part of the gas phase by assuming to neglect the boundary layer under present mesh size. On the other hand, the wall friction is set to be 0 till the solids descending below the gas inlet level (Fig. 1), and Fanning’s equations are used to specify the friction between the solids and the outer wall or the inner man-made deadman.²³⁾ Besides, the convection with the heat transfer coefficient of 20 W/m²·K,²⁶⁾ the wall thickness of 1 m and the temperature of 300 K are chosen for the wall thermal condition.

Fig. 1.

The schematic diagram of the COREX pre-reduction shaft furnace.

Table 2. The operation conditions of the COREX pre-reduction shaft furnace in the base case.^17,37,38)

Parameters	Value
reducing gas temperature	1100 K
reducing gas composition in mole fraction
CO	67.6%
CO2	8.8%
H2	21.4%
H2O	2.2%
reducing gas volume rate	225654 Nm3/h
burden solid composition in mass fraction
TFe	66.4%
Fe2O3	94.4%
burden solid feed rate	212 t/h
hot metal productivity*	146 t/h
top gas pressure	300 kPa
TGC	1064 Nm3/t

*  data collected from the whole COREX process

Through the mesh work, there are 240362 cells, 634731 faces and 169465 nodes in the present model. The conservation equations are at first integrated over the control volume, the obtained finite volume equations are discretized in the first order upwind scheme and then solved by phase coupled SIMPLE method.³⁶⁾

3. Model Validation and Base Case Study

3.1. Model Validation

The established three-dimensional mathematical model of the COREX pre-reduction shaft furnace is verified against the measured results in the practical production as shown in Table 3, in which the relative error Δ is calculated by Eq. (9). The maximum relative error Δ between the measured and calculated results is 5.5%, so the present model is proved to be applicable to predict the characteristics inside the furnace.

  

$$\Delta =\frac{\left|\text{Measured Value}-\text{Calculated Value}\right|}{\text{Measured Value}}$$(9)

Table 3. Model validation with selective parameters.

Parameter	Value
	Measured	Calculated	Relative Error Δ
top gas composition in volume fraction
CO	43.6%	42.8%	1.8%
CO2	32.7%	31.2%	4.6%
H2	13.0%	13.6%	4.6%
gas pressure drop	52.7 kPa	55.6 kPa	5.5%
average gas temperature*	1028 K	1031 K	0.3%

*  in the circle with the radius of 2.15 m at the height of 15.5 m

3.2. Base Case Study

The characteristics inside the COREX pre-reduction shaft furnace are described in terms of velocity, temperature, species, gas utilization and burden metallization distributions. Five surfaces, namely X = 0, Z = 1, 6, 12, 18 m respectively, and several iso-surfaces of certain values are selected to fully display the characteristic distributions in the different regions of the furnace. It should be explained that due to the overlap among some iso-surfaces, the colors of influenced iso-surfaces look a little different from that demonstrated in the legend.

3.2.1. Velocity Distributions

The velocity distributions of the gas and solid phases are demonstrated in Fig. 2. The gas is injected at about 19 m/s, then the velocity sharply decreases to the value below 6 m/s because of the enlarged space. Since the gas inlets are circumferentially distributed at the outer wall, most reducing gas prefers to ascend to the top directly, thus leaving a diminishing inactive zone in the center. By contrast, less gas reaches the region below the gas inlet level, so the gas velocity near the bottom becomes as low as 0.1 m/s. As for the solid phase, it is obviously to find that its descending velocity gradually decreases with the increasing furnace diameter, and then greatly increases to the value above 0.0006 m/s when the solids approach the bottom outlet. Besides, there are two regions worthy to be noted. Due to the different slopes of the furnace walls above and below the gas inlet level, a slow solid descending region is formed around the gas inlets, which would results in the longer retention time. In consideration of high temperature and reducing potential there, which will be demonstrated later, it is likely to cause the high metallization solids to stick together. The other region is located above the man-made deadman. Due to the location of the annulus-shape outlet as shown in Fig. 1, the solids accelerate to escape from the center when approaching the man-made deadman.

Fig. 2.

The velocity distributions of the gas and solid phases inside the pre-reduction shaft furnace.

3.2.2. Temperature Distributions

Under the present operation conditions, there is little difference in the temperature distribution between the gas and solid phases in the lower part of the furnace as shown in Fig. 3. However, in the upper part, the gas temperature is some higher than the solid as a result of continuously charging burden from the top. Besides, the insufficient gas development and the fast solid descending velocity as discussed above makes the central temperature much lower at the same horizontal level.

Fig. 3.

The temperature distributions of the gas and solid phases inside the pre-reduction shaft furnace.

3.2.3. Specie Distributions

The mass fraction distributions of CO, CO₂, H₂ and H₂O in the gas phase are described in Fig. 4 while that of Fe₂O₃, Fe₃O₄, FeO and Fe in the solid phase are demonstrated in Fig. 5. Firstly, the mass fraction distribution of CO or H₂ show an opposite tendency to that of CO₂ or H₂O respectively due to the reduction reactions. Secondly, two great mass fraction gradients of the species in the gas phase indicate the rapid rates of the reductions from Fe₃O₄ to FeO and from FeO to Fe respectively, which is in accordance with the products' mass fraction distributions in the solid phase. Thirdly, in comparison with CO, the region for the reductions participated by H₂ is smaller, especially for the final stage reduction to produce metal. Lastly, the uneven flow behaviors of both phases in the radial direction strengthen the reduction from FeO to Fe around the gas inlet, specifically in the lower part of the furnace, so the largest mass fraction of Fe locates near the wall while that of FeO mainly distributes in the center.

Fig. 4.

The mass fraction distributions of the specie (CO, CO₂, H₂, H₂O) in the gas phase inside the pre-reduction shaft furnace.

Fig. 5.

The mass fraction distributions of the species (Fe₂O₃, Fe₃O₄, FeO, Fe) in the solid phase inside the pre-reduction shaft furnace.

3.2.4. Gas Utilization and Solid Metallization Distributions

Both the gas utilization rate (UR) and the solid metallization rate (MR) are important production indices to evaluate the operation and working status of pre-reduction shaft furnace. Based on Eqs. (10) and (11), the gas UR and solid MR inside the furnace are calculated and described in Fig. 6.

Fig. 6.

The gas UR and solid MR distributions inside the prereduction shaft furnace.

  

$$\text{Gas UR}=\frac{\left({\phi }_{CO,inlet}-{\phi }_{CO}\right)+\left({\phi }_{H_2,inlet}-{\phi }_{H_2}\right)}{{\phi }_{CO,inlet}+{\phi }_{H_2,inlet}}$$(10)

  

$$\text{Solid MR}=\frac{{\omega }_{Fe}}{\frac{112}{160}\cdot {\omega }_{F{e}_2{O}_3}+\frac{168}{232}\cdot {\omega }_{F{e}_3{O}_4}+\frac{56}{72}\cdot {\omega }_{FeO}+{\omega }_{Fe}}$$(11)

It is found that as the gas ascends from the inlets, the accompanying reductions participated by CO and H₂ gradually increase the gas UR from 0 to above 0.36 on the top under the operation conditions of the base case. Meanwhile, the regions with greater gas UR gradient are obviously consistent with where the rapid reductions happen as mentioned above. On the other hand, the ‘migration’ of O atom from the solid to the gas, thus increasing the gas UR, definitely makes positive contributes to the solid MR as well. The metallic iron is firstly produced in the upper part and then gradually to increase as the solid descends. As is analyzed in above sections, the over-development gas flow near the wall and the relatively slow solid descending velocity cause the difference in the solid MR between the center and near wall to reach as high as 0.4.

4. Operation Condition Optimization for the Pre-Reduction Shaft Furnace

In this section, four operation conditions, namely reducing gas temperature, reducing gas composition, TGC and productivity, are selected to investigate the effects of operation conditions on the pre-reduction shaft furnace, thus providing optimization suggestions for practical production. The URs of CO and H₂ at the top level and the maximum and average MRs at the gas inlet level are selected to facilitate the expressions of the modeling results.

4.1. Reducing Gas Temperature

The effects of the reducing gas temperature at the gas inlet on the UR and MR are depicted in Fig. 7. Under the present operation conditions, when the temperature increases by 200 K, the UR of H₂ greatly increases from 0.27 to about 0.50 while that of CO only improves by 0.06. Therefore the increasing temperature favors to promote the reductions participated by H₂ because of its strong endothermic characteristic. At the same time, either the maximum or average MR at the gas inlet level generally increases by 0.12 at least.

Fig. 7.

The effect of the gas temperature at the gas inlet on the UR and MR.

In a word, although the increase on the gas temperature helps to accelerate the reduction rate, especially to improve the performance of the reductions participated by H₂, the reducing gas temperature should be controlled between 1050–1100 K due to the following reasons. 1) obtain economic UR of CO because of its high proportion (67.6% in mole fraction) in the mixture gas, 2) avoid solids sticking under high metallization level.

4.2. Reducing Gas Composition

According to the operation conditions collected in Table 2, three schemes concerning the reducing gas composition are proposed. As for Scheme I, the volume fractions of H₂ and H₂O are fixed while the ratio of CO to CO₂ in the mixture gas varies from 3 to 11 by 2 step. As for Scheme II, the volume fraction of CO and CO₂ are fixed while the ratio of H₂ to H₂O in the mixture gas varies in the same range. As for Scheme III, under each fixed ratio, the volume fraction of CO+CO₂ increasing from 10 to 90% by 20% step while that of H₂+H₂O decreasing accordingly. The results of the three schemes are summarized in Fig. 8.

Fig. 8.

The effect of the gas composition at the gas inlet on the UR and MR.

The effects of the gas composition on the UR and MR are analyzed from the following five aspects. 1) Since the location with the maximum MR is always close to the gas inlet, the increase on reducing potential makes limited effect on the maximum MR. 2) Under the present conditions, the increase on the ratios of CO to CO₂ and H₂ to H₂O improves the average MR by 0.14 and 0.05 respectively. 3) Increasing the proportion of CO+CO₂ from 10 to 90% while decreasing that of H₂+H₂O accordingly improves the average MR at the gas inlet level by as high as 0.26, which indicates that CO still plays an important role in the reduction process. 4) As the ratio of reductant to oxidant in the mixture gas increases, the UR of CO or H2 reaches its peak value when the ratio is 7 or 11. 5) When increasing the proportion of CO+CO₂ to about 70%, the decreasing UR of CO and the rising UR of H₂ would reach a dynamic balance to obtain the highest quality product.

In a word, the optimal proportion of CO+CO₂ in the CO–CO₂–H₂–H₂O mixture gas is around 70%, and the optimal ratio of CO to CO₂ is between 5–7, while the ratio of H₂ to H₂O should be as high as possible.

4.3. TGC

The increasing TGC directly accelerates the update of the reducing gas on the solid surface, thus making positive effect on the reduction inside the furnace. However, once the gas volume flow exceeding some degree would cause the decrease on the gas UR. As is shown in Fig. 9, although the increase on the TGC from 600 to 1400 Nm³/t promotes both the maximum and average MRs at the gas inlet level by 0.13 and 0.35 respectively, the top URs of CO and H₂ decrease when the TGC exceeds 800 Nm³/t and 1000 Nm³/t respectively. In other words, there is optimal TGC in consideration of both UR and MR, which has a close relationship with the mixture gas composition.

Fig. 9.

The effect of the TGC on the UR and MR.

In a word, accelerating the update of reducing gas on the solid surface benefits iron oxide reduction, thus promoting the burden solid metallization level. On the other hand, the short contact period between the gas and the solid makes negative effect on the gas utilization. Base on the simulation results, the optimal TGC should be between 800–1000 Nm³/t, and the lower limit would be closed to when the proportion of CO in the mixture gas is above 67.6% in volume fraction while the upper limit would be closed to when the proportion of H₂ is above 21.4%.

4.4. Productivity

In this section, two kinds of schemes are proposed with the results shown in Fig. 10. As for the Scheme I, the injected gas volume flow is fixed at 225654 Nm³/h when the hot metal productivity of the whole process increases from 90 to 210 t/h, resulting in the decrease on the TGC, so either the trend of MR or UR with respect to the productivity is similar to the results in section 4.3. On the contrary, the injected gas volume flow in the Scheme II increases with the increasing productivity in order to keep the TGC fixed at 1064 Nm³/t. Both the maximum and average MRs are almost stable while the URs of CO and H₂ decline due to the short retention time between gas and solid phases inside the furnace. In addition, since the concentration of CO in the reducing gas is much higher than that of H₂, the increasing gas volume introduces more quantity of CO, thus causing a greater drop in its UR in comparison with H₂.

Fig. 10.

The effect of the productivity on the UR and MR.

In a word, when the productivity of the pre-reduction shaft furnace is required to increases, it is better to increase the accompanying gas volume flow simultaneously in order to obtain high quality products. It should be noticed that compared with the relative low productivity, the high productivity under the same TGC significantly reduce the gas utilization. So it is recommended to increase the gas volume flow but properly reduce the TGC with regard to the increase on the productivity.

5. Conclusions

The characteristics inside the COREX pre-reduction shaft furnace are numerically analyzed by establishing a three-dimensional full scale mathematical model combined mass, momentum, energy transfers and chemical reactions under steady state. The present model is also used to predict the influences of operation conditions, such as reducing gas temperature, composition, TGC and productivity on the gas UR and solid MR. Since the gas inlets circumferentially distributed around the outer wall, the injected gas prefers to ascend directly to the top rather than flow toward the center and bottom, while the solids descend with relatively high velocity when approaching the man-made deadman and the bottom outlet. Such uneven characteristic distributions directly make negative effect on the heat transfers as well as the iron oxides reduction inside the furnace, thus not only limiting the improvement on gas UR but also causing the difference of MR between the wall and the center to reach as high as about 0.4. On the other hand, some suggestions, such as adjusting the gas temperature between 1050–1100 K, controlling the volume fraction of CO+CO₂ around 70% with the ratio of CO to CO₂ between 5–7, lowering the H₂O content in the mixture gas as far as possible, reducing the TGC with increasing productivity to the range of 800–1000 Nm³/t, are proposed to optimize the operation conditions of the COREX pre-reduction shaft furnace, thus saving energy consumption and reducing CO₂ emission.

Nomenclature

C_P,p: heat capacity of phase p (J/kg·K)

d_s: solid particle diameter (m)

E_gs: volumetric enthalpy flux from gas to solid phase (W/m³)

f_s: friction coefficient (–)

${\overrightarrow{F}}_{gs}$ : gas-solid drag vector term

${\overrightarrow{F}}_w$ : wall friction vector term

$\overrightarrow{g}$ : acceleration of gravity vector term (m/s²)

H_n: enthalpy of reduction reaction n (J/kg)

H_p: enthalpy of phase p (J/kg)

I: identity tensor (–)

k_g: thermal conductivity of gas phase (W/m·K)

M_i: molecular weight of specie i (kg/kmol)

P: pressure (Pa)

Pr_g: the Prandtl number of the gas phase (–)

R₁, R₂: diameter of inner and outer wall of annular pipe (m)

Re_s: the relative Reynolds number based on the diameter of the solid particle (–)

R_n: rate of reduction reaction n (kmol/m³·s)

$S_{\varphi }$ : source term for variable $\varphi$ in Eq. (1)

T_p: Temperature of phase p (K)

${\overrightarrow{v}}_p$ : physical velocity vector of phase p (m/s)

 

Greek Symbols

ε_p: volume fraction of phase p (–)

ρ_p: density of phase p (kg/m³)

ϕ: general dependent variable in Eq. (1)

${\Gamma }_{\varphi }$ : diffusion coefficient for variable $\varphi$ in Eq. (1)

μ_p: viscosity of phase p (kg/m·s)

ϕ_i: volume fraction of specie i (–)

${\stackrel{=}{\stackrel{}{\tau }}}_p$ : stress tensor of phase p (Pa)

w_i: mass fraction of specie i (–)

Acknowledgement

The authors would like to thank the Fundamental Research Funds for the Central Universities (No. 0903005203199) for the financial support, Professor Emeritus Jun-ichiro Yagi from Tohoku University and Professor Chenguang Bai, Professor Liangying Wen, Dr. Guibao Qiu, Dr. Xuewei Lv, Dr. Shengfu Zhang, Dr Meilong Hu from Chongqing University for lots of suggestions on this work, Ms. Yan Fang for language review and anonymous reviewers and editors for the improvement of this work.

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