A Theoretical Model for Assessing the Influence of N₂ on Gaseous Reduction of Iron Ores with CO or H₂

Abstract

The influence of N₂ on gaseous reduction of iron ores has yet to be investigated more rigorously. A generalized theoretical model for iron ore reduction that is featured with equimolar counterdiffusion of gaseous reactant and product in the presence of an inert component is derived in the current paper, where the influence of N₂ is quantitatively assessed in terms of a normalized overall reduction rate. The results show that N₂ impacts CO reduction of iron ores mainly via the prevailing mechanism of dilution effect and the sole use of D_AB in the corresponding ternary system brings merely minor errors. In contrast, especially under conditions of high N₂ fraction and reduction degree, the effect of mass diffusion must be borne in mind for H₂ reduction and the sole use of D_AB can lead to marked errors. The main novelty of the present work is the derivations of the theoretical equations fully following the Maxwell-Stefan relation for multicomponent diffusion and the well-known concept of topochemical reaction for gaseous reduction of iron ores. It is hoped that the brief discussion will stimulate further application of the theoretical equations to the development of more versatile models and to studies of the engineering type.

1. Introduction

Gaseous reduction of iron ores has been extensively studied over decades^1,2,3,4,5,6) and it is nowadays commonly believed that three elementary sub-steps, i.e., mass convection through external film, mass diffusion within internal shell and chemical reaction at reaction interface, take place as the reduction proceeds.

In the ironmaking reactors, a notable fraction of N₂ is usually present together with a reducing agent, i.e., CO, H₂ or their mixture. As a result, N₂ is typically mixed in the bulk stream in most of the laboratory-scale experiments. It has been recognized that the presence of N₂ can retard the reduction and the prevailing mechanism has been attributed to the dilution effect, i.e., N₂ lowers partial pressure of the reducing agent and consequently leads to a decrease in thermodynamic driving force for the reduction reaction. This implies that the dilution effect stems solely from a standpoint of interfacial chemical reaction. According to the theories of multicomponent mass transfer,⁷⁾ however, N₂ can also provide its influence via (both external convective and internal diffusive) mass transfer. This is probably the main motivation behind the exceptional work by Murayama and co-authors,⁸⁾ who derived N₂-dependent pseudo-binary diffusion coefficients for the sub-step of internal mass diffusion and then determined rate parameters of both CO–N₂ and H₂–N₂ reduction systems based on experimental data. It should be stressed that in the derivations of the pseudo-binary diffusion coefficients, concentration gradient of N₂ along the diffusion path was ignored and thus the influence of N₂ was underestimated. It is therefore desirable to conduct more rigorous studies in order to shed more light on this fundamental and considerably important issue.

The emphasis of the current paper is on deriving a theoretical model for iron ore reduction that is featured with equimolar counterdiffusion of gaseous reactant and product in the presence of an inert component. The underlying concepts and key deriving procedures are firstly introduced. After that, the model is demonstrated by quantitatively assessing the influence of N₂ on gaseous reduction of iron ores considering a scenario of more industrial interest.

2. Model Description

2.1. Assumptions and Simplifications

It has been identified that the transformation from wustite to metallic iron limits the reduction of iron ores under conditions of ironmaking practice. The current model thus copes only with the reversible gaseous reduction of wustite in the presence of N₂, Fe_xO+A=xFe+B, where A is the gaseous reactant (CO or H₂) and B is the gaseous product (CO₂ or H₂O). In the model derivations, the well-known concept of topochemical reaction is applied and based on an individual wustite (bearing) pellet, the gaseous reduction is illustrated schematically in Fig. 1. In conformity with the literature, furthermore, the pseudo steady state is postulated and Knudsen diffusion is ignored.

Fig. 1.

Schematic description of the gaseous reduction based on a sphere object.

2.2. Model Formulation

2.2.1. Mass Convection through External Film

It is evident that, if the mass transfer coefficient is available, the calculation of mass convection rate through external film is straightforward. However, the mass transfer coefficient concerning multicomponent gas systems is still far from a satisfactory evaluation. In the current work, instead, mass convection through external film is characterized based on the film model.⁹⁾ In such model, the entire partial pressure difference between the bulk stream and the pellet surface is considered to be localized in an external thin film (cf. Fig. 1), where transport of gaseous components is governed by molecular diffusion.

In the external film, gaseous reactant A diffuses inwards to the pellet surface through a mixture of A, B and N₂, while an identical amount of gaseous product B diffuses outwards through the same path. This kind of diffusion can be termed as equimolar counterdiffusion in the presence of an inert component. In such ternary diffusion system, the diffusive rates of A inwards and of B outwards are equal and opposite. Since it is neither consumed nor generated, the mass flow rate of N₂ is zero. Nevertheless, this is not to say that the partial pressure gradient of N₂ along the diffusion path is zero. It is therefore natural to equate the diffusive rate of A inwards with the mass convection rate through external film, which is also identical to the overall reduction rate under pseudo steady state.

Following the Stefan-Maxwell relation for multicomponent diffusion,⁸⁾ three linear ordinary differential equations, which encompass the overall reduction rate that is equal to the diffusive rate of A inwards, can be written.   

$$-\frac{4\pi P}{RT}{r}^2\frac{d{Y}_{\mathrm{A}}}{dr}=Q_{\mathrm{T}\mathrm{e}\mathrm{r}}\left(\frac{Y_{\mathrm{A}}}{D_{\mathrm{A}\mathrm{B}}}+\frac{Y_{\mathrm{B}}}{D_{\mathrm{A}\mathrm{B}}}+\frac{Y_{\mathrm{C}}}{D_{\mathrm{A}\mathrm{C}}}\right)$$(1)

  

$$\frac{4\pi P}{RT}{r}^2\frac{d{Y}_{\mathrm{B}}}{dr}=Q_{\mathrm{T}\mathrm{e}\mathrm{r}}\left(\frac{Y_{\mathrm{A}}}{D_{\mathrm{A}\mathrm{B}}}+\frac{Y_{\mathrm{B}}}{D_{\mathrm{A}\mathrm{B}}}+\frac{Y_{\mathrm{C}}}{D_{\mathrm{B}\mathrm{C}}}\right)$$(2)

  

$$\frac{4\pi P}{RT}{r}^2\frac{d{Y}_{\mathrm{C}}}{dr}=Q_{\mathrm{T}\mathrm{e}\mathrm{r}}\left(\frac{Y_{\mathrm{C}}}{D_{\mathrm{A}\mathrm{C}}}-\frac{Y_{\mathrm{C}}}{D_{\mathrm{B}\mathrm{C}}}\right)$$(3)

where Q_Ter, P, R, T, Y and r are the overall reduction rate involving the present ternary system, total pressure, gas constant, temperature, gas mole fraction and radial coordinate, respectively. Subscripts A, B and C stand for gaseous reactant, gaseous product and N₂, respectively. D_AB, D_AC and D_BC are the binary diffusion coefficients of each component pair.

After considerable algebraic manipulation, Eqs. (1) and (2) can be combined into   

$$-\frac{4\pi P}{RT}{r}^2\frac{d(Y_{\mathrm{A}}+\kappa Y_{\mathrm{B}})}{dr}=Q_{\mathrm{T}\mathrm{e}\mathrm{r}}\frac{1-\kappa }{D_{\mathrm{A}\mathrm{B}}}$$(4)

where the dimensionless κ is   

$$\kappa =\frac{1/D_{\mathrm{A}\mathrm{B}}-1/D_{\mathrm{A}\mathrm{C}}}{1/D_{\mathrm{A}\mathrm{B}}-1/D_{\mathrm{B}\mathrm{C}}}$$(5)

On integration between the two ends of the gas film (i.e., r=r₀+δ and r=r₀) followed by rearrangement, the overall reduction rate is given as   

$$Q_{\mathrm{T}\mathrm{e}\mathrm{r}}=-\frac{4\pi P}{RT}\frac{D_{\mathrm{A}\mathrm{B}}}{1-\kappa }\frac{r_0(r_0+\delta )}{\delta }\left\{(Y_{\mathrm{A},0}-Y_{\mathrm{A},\mathrm{s}})+\kappa (Y_{\mathrm{B},0}-Y_{\mathrm{B},\mathrm{s}})\right\}$$(6)

where r₀ and δ are the pellet radius and gas film thickness. Subscripts 0 and s represent the bulk stream and pellet surface.

The overall reduction rate can also be obtained by integrating Eq. (3).   

$$Q_{\mathrm{T}\mathrm{e}\mathrm{r}}=\frac{4\pi P}{RT}\frac{D_{\mathrm{A}\mathrm{C}}{D}_{\mathrm{B}\mathrm{C}}}{D_{\mathrm{B}\mathrm{C}}-D_{\mathrm{A}\mathrm{C}}}\frac{r_0(r_0+\delta )}{\delta }ln\frac{Y_{\mathrm{C},0}}{Y_{\mathrm{C},\mathrm{s}}}$$(7)

Noting the summation of mole fractions of A, B and N₂ is unity at any specific location along the diffusion path, Eq. (7) thus becomes   

$$Q_{\mathrm{T}\mathrm{e}\mathrm{r}}=\frac{4\pi P}{RT}\frac{D_{\mathrm{A}\mathrm{C}}{D}_{\mathrm{B}\mathrm{C}}}{D_{\mathrm{B}\mathrm{C}}-D_{\mathrm{A}\mathrm{C}}}\frac{r_0(r_0+\delta )}{\delta }ln\frac{1-Y_{\mathrm{A},0}-Y_{\mathrm{B},0}}{1-Y_{\mathrm{A},\mathrm{s}}-Y_{\mathrm{B},\mathrm{s}}}$$(8)

2.2.2. Mass Diffusion within Internal Shell

Comparing with mass convection through external film, the only difference of mass diffusion within internal shell is linked with the porous medium for diffusion. Since molecular diffusion through a porous medium is typically treated by introducing the approach of effective diffusion coefficient, the overall reduction rate that is identical to the diffusive rate within internal shell can be obtained by applying similar procedures.   

$$Q_{\mathrm{T}\mathrm{e}\mathrm{r}}=-\frac{4\pi P}{RT}\frac{\epsilon }{\tau }\frac{D_{\mathrm{A}\mathrm{B}}}{1-\kappa }\frac{r_0{r}_{\mathrm{i}}}{r_0-r_{\mathrm{i}}}\left\{(Y_{\mathrm{A},\mathrm{s}}-Y_{\mathrm{A},\mathrm{i}})+\kappa (Y_{\mathrm{B},\mathrm{s}}-Y_{\mathrm{B},\mathrm{i}})\right\}$$(9)

  

$$Q_{\mathrm{T}\mathrm{e}\mathrm{r}}=\frac{4\pi P}{RT}\frac{\epsilon }{\tau }\frac{D_{\mathrm{A}\mathrm{C}}{D}_{\mathrm{B}\mathrm{C}}}{D_{\mathrm{B}\mathrm{C}}-D_{\mathrm{A}\mathrm{C}}}\frac{r_0{r}_{\mathrm{i}}}{r_0-r_{\mathrm{i}}}ln\frac{1-Y_{\mathrm{A},\mathrm{s}}-Y_{\mathrm{B},\mathrm{s}}}{1-Y_{\mathrm{A},\mathrm{i}}-Y_{\mathrm{B},\mathrm{i}}}$$(10)

where ε and τ are the porosity and tortuosity of the internal shell. Subscript i denotes the reaction interface and therefore, r_i is the radius of the unreacted core.

2.2.3. Chemical Reaction at Reaction Interface

The overall reduction rate is also equal to the interfacial chemical reaction rate (inwards),   

$$Q_{\mathrm{T}\mathrm{e}\mathrm{r}}=-\frac{4\pi P}{RT}k{r}_{\mathrm{i}}^2(Y_{\mathrm{A},\mathrm{i}}-Y_{\mathrm{A},\mathrm{e}})$$(11)

where k is the reaction rate constant.

In Eq. (11), Y_A,e is the mole fraction of A at reaction equilibrium and can be obtained on the basis of mass conservation, i.e.,   

$$Y_{\mathrm{A},\mathrm{e}}=\frac{Y_{\mathrm{A},\mathrm{i}}+Y_{\mathrm{B},\mathrm{i}}}{1+K}$$(12)

where K is the equilibrium constant of chemical reaction.

Substituting Eq. (12) into Eq. (11), it gives   

$$Q_{\mathrm{T}\mathrm{e}\mathrm{r}}=-\frac{4\pi P}{RT}\frac{K}{1+K}k{r}_{\mathrm{i}}^2\left(Y_{\mathrm{A},\mathrm{i}}-\frac{Y_{\mathrm{B},\mathrm{i}}}{K}\right)$$(13)

The generalized model in the current work consists of the five theoretical equations derived above, i.e., Eqs. (6), (8), (9), (10) and (13), which can be employed to solve for five unknowns providing the other parameters are available.

2.3. Results and Discussion

In spite of its complexity, the theoretical model can take into full account of the aforementioned external mass convection, internal mass diffusion and interfacial chemical reaction. Under conditions of ironmaking practice, however, the process is commonly recognized to be controlled by both internal mass diffusion and interfacial chemical reaction especially after the middle stage of reduction. In the current work, therefore, the theoretical model is applied to this scenario of more industrial interest. The model is hence simplified by precluding Eqs. (6) and (8) as well as setting Y_A,s = Y_A,0 and Y_B,s = Y_B,0. Then, the remaining unknowns are given Y_A,i and Y_B,i together with Q_Ter.

The overall reduction rate can also be derived assuming that the influence of N₂ stems solely from the dilution effect. It follows that N₂ exerts no influence on the internal diffusion, suggesting that the binary diffusion coefficients concerning N₂ (i.e., D_AC and D_BC) can be precluded in the calculation of overall reduction rate and only D_AB is required. This explains why D_AB has been used in most of the related studies and a portion of them even involves multicomponent gas systems. In the same scenario, the overall reduction rate using only D_AB can be obtained by implementing the conventional procedures outlined, for example, by Szekely et al.¹⁰⁾   

$$Q_{\mathrm{A}\mathrm{B}}=-\frac{4\pi P}{RT}\frac{K}{1+K}{r}_{\mathrm{i}}^2\left(Y_{\mathrm{A},\mathrm{s}}-\frac{Y_{\mathrm{B},\mathrm{s}}}{K}\right){\left\{\frac{1}{\frac{\epsilon }{\tau }{D}_{\mathrm{A}\mathrm{B}}}\left(r_{\mathrm{i}}-\frac{r_{\mathrm{i}}^2}{r_0}\right)+\frac{1}{k}\right\}}^{-1}$$(14)

A normalized overall reduction rate, Q_Norm, is thus defined as   

$$Q_{\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}}=\frac{Q_{\mathrm{A}\mathrm{B}}}{Q_{\mathrm{T}\mathrm{e}\mathrm{r}}}$$(15)

For the sake of demonstration, the pertaining parameters in the calculations below are given typical values, i.e., T = 1173 K, ε = 0.35, τ = 2.0 and r₀ = 0.006 m. k is calculated with the correlations proposed by Tsay and co-authors.¹¹⁾ K is given using the equation in a reliable source.¹²⁾ The binary diffusion coefficients D_AB, D_AC and D_BC are computed using the Fuller’s equation.¹³⁾ Moreover, the unreacted core radius r_i is related to the reduction degree f as   

$$r_{\mathrm{i}}=r_0{(1-f)}^{1/3}$$(16)

On the basis of the normalized overall reduction rate, the influence of N₂ on gaseous reduction of iron ores after the middle stage is demonstrated in Figs. 2 and 3. It should be pointed out that similar curves are plotted when altering the pertaining parameters including T, ε, τ and r₀. Therefore, the representative results shown in the figures are analyzed and discussed.

Fig. 2.

Influence of N₂ on CO reduction of iron ores after the middle stage of reduction.

Fig. 3.

Influence of N₂ on H₂ reduction of iron ores after the middle stage of reduction.

The influence of N₂ on CO reduction of iron ores is depicted in Fig. 2, where the normalized overall reduction rates corresponding to different reduction degrees are generally smaller than unity and decrease as the mole fraction of N₂ increases. At a specific N₂ mole fraction, furthermore, the normalized overall reduction rate becomes smaller with a higher reduction degree. The results indicate that the theoretical overall reduction rate is bigger than the one only using D_AB. The primary reason lies in the fact that the binary diffusion coefficients concerning N₂, i.e., both D_AC and D_BC, are bigger than D_AB and more CO can be transported inwards to the interface, leading to a higher thermodynamic driving force for the reduction reaction. However, since D_AC and D_BC are bigger than D_AB to only a lesser extent, the negative deviation of the overall reduction rate from its theoretical value is negligible. As can be seen in Fig. 2, when the mole fraction of N₂ approaches unity, the deviation is still less than 2% (i.e., the normalized overall reduction rate is around 0.98) although the reduction degree is as high as 0.9. It is therefore concluded that N₂ exerts its influence on CO reduction mainly via the dilution effect and more interestingly, the sole use of D_AB in the ternary system brings merely minor errors.

The influence of N₂ on H₂ reduction differs and is illustrated in Fig. 3, where the normalized overall reduction rates are bigger than unity and increase as the mole fraction of N₂ increases. Also, a higher reduction degree leads to a bigger normalized overall reduction rate at a specific N₂ mole fraction. The theoretical overall reduction rate is therefore smaller than the one using only D_AB. This can also be explained by the order of the diffusion coefficients, i.e., both D_AC and D_BC are smaller than D_AB and consequently, some portion of H₂ is pushed away outwards from the interface. It can be found that when H₂ is utilized as the reducing agent, the differences between the three binary diffusion coefficients become discernable especially for D_AC, which is almost 65% lower than D_AB. Therefore, the positive deviation of the overall reduction rate from its theoretical value is pronounced. As displayed in Fig. 3, the deviation rises up to 10% when both mole fraction of N₂ and reduction degree barely exceed 0.5. It is concluded that except for the dilution effect, the influence of mass diffusion must be borne in mind for H₂ reduction of iron ores especially under conditions of high N₂ fraction and reduction degree. Correspondingly, the sole use of D_AB in the ternary system can bring marked errors either in parameter regression based on data fitting or in model prediction.

3. Conclusions

A generalized theoretical model for iron ore reduction that is featured with equimolar counterdiffusion of gaseous reactant and product in the presence of an inert component has been derived based on the characteristic phenomena and underlying theories. The theoretical model has been adopted to quantitatively assess the influence of N₂ on gaseous reduction of iron ores with CO or H₂. As for further model validation and application, experimental work is being undertaken and the results will be deferred to a subsequent publication.

Acknowledgement

The authors are grateful for the financial support offered by the National Science Foundation of China (Grants 51574064, 51604068) and the Fundamental Research Funds for the Central Universities (Grant N150203004).

Nomenclature

Symbols

D: molecular diffusion coefficient [m²/s]

f: reduction degree [-]

k: reaction rate constant [m/s]

K: equilibrium constant of chemical reaction [-]

P: pressure [Pa]

Q: mass flow rate [mol/s]

r: radius [m]

R: gas constant [J·mol⁻¹·K⁻¹]

T: temperature [K]

Y: mole fraction [-]

Greek Symbols

δ: gas film thickness [m]

ε: shell porosity [-]

κ: dimensionless quantity [-]

τ: shell tortuosity [-]

Subscripts

0: bulk for gas stream or initial for pellet radius

A: gaseous reactant

B: gaseous product

C: inert component

AB: pair of gaseous reactant and gaseous product

AC: pair of gaseous reactant and inert component

BC: pair of gaseous product and inert component

e: equilibrium

i: interface

Norm: normalized quantity

s: surface

Ter: ternary system

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