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Introduction
Gaseous reduction of iron-bearing ores is of crucial importance for the production of iron via the dominant blast furnace (BF) route. In the shaft of the BF, the descending solid ores are gradually reduced by the ascending gas mixture that contains mainly carbon monoxide (CO) and hydrogen (H2) as reducing agents and a considerable fraction of nitrogen (N2) as a heat carrier for heating the solid burden. It has been identified that the presence of inert N2 can curb the reduction, and the prevailing mechanism is attributed to the dilution effect of interfacial chemical reaction, i.e., N2 lowers partial pressure of the reducing agent and consequently leads to a decrease in chemical driving force for the reduction reaction. According to the theory of multicomponent mass transfer [1], however, N2 is also likely to exert influence via mass transfer. This can be explained with the well-known concept of a topochemical reaction for an individual particle of iron ore: Before reaching the reaction site, the reducing agent must travel through an external boundary film and then an internal solid product shell, where both are occupied by a gas mixture containing N2. Thus, the amount of the reducing agent finally participating in the reaction is affected or even determined by the amount of N2. For a thorough understanding of the BF process, it is therefore necessary to shed more light on the complicated gas-solid system involving coupled phenomena of heterogeneous chemical reaction and multicomponent mass transfer.
There are various gas-solid systems in metals and materials processing industries, and gaseous reduction of iron ores has been paid extensive attention over the years, leading to numerous publications reporting experimental results and mathematical models [2,3,4,5,6,7]. In the field of ironmaking, the most frequently adopted mathematical models are the unreacted shrinking core model (USCM) and its refined variant, e.g., the grain model [8]. In the original derivation of these two topochemical models, only a single-component reducing agent (e.g., pure CO or H2) was taken into account, and the mass transfer either in the external film or in the internal shell was regarded as equimolar counter diffusion of the gaseous reactant (referred to as A) and product (B). The corresponding molecular diffusion coefficient was thus given the binary diffusion coefficient with respect to A and B, i.e., DAB. It should be stressed that the binary diffusion coefficient DAB is relatively simple and convenient because it is independent of gas composition and can be computed using the Fuller [9] or more sophisticated Chapman-Enskog Eq. [10]. This is probably the main reason why DAB has been employed in most of the subsequent studies. However, in order to represent the true atmosphere in the BF, the gas mixtures applied often contain some portion of N2. Therefore, the diffusion system is no longer binary, and the corresponding multicomponent diffusion coefficients become dependent of gas composition. The question hence arises as whether the (exclusive) use of DAB for the gas mixtures under consideration is computationally acceptable when describing the progress of iron ore reduction in the true BF atmosphere.
Focussing on gaseous reduction of iron ores, Szekely and co-authors [11] were among the first who evaluated the applicability of DAB in ternary reducing mixtures involving an inert gas and recommended a pseudo binary approach that takes into account the composition dependence of the multicomponent diffusion coefficient. It should be pointed out that in developing the pseudo binary approach, Wilke [12] assumed a rather simplified system where only one active gas diffuses through a stagnant mixture consisting of the remaining gas components. Apparently, this assumption is in contradiction to the scenario of iron ore reduction, where equimolar counter diffusion takes place. In order to interpret their experimental data of wüstite reduction in both CO-N2 and H2-N2 atmospheres, Murayama and co-authors [13] suggested an improved pseudo binary approach that allows for the characteristic of equimolar counter diffusion. Nevertheless, the concentration difference of N2 along the diffusion path was ignored in the improved approach by Murayama and co-authors, and thus, the influence of N2 can be underestimated to an unknown extent.
Starting with the Maxwell-Stefan relation for multicomponent mass transfer, Shao and co-authors developed a set of theoretical expressions for iron ore reduction characterised by equimolar counter diffusion of a gaseous reactant and a product in the presence of an inert component [14]. The diffusion sub-steps of both the external film and internal shell were described in a more accurate manner, and the influence of N2 on the reduction was assessed in a quantitative way. The results showed that, for CO reduction of iron ores, the use of DAB for the ternary diffusion system gives rise to minor errors only. In contrast, for H2 reduction, particularly under conditions of high N2 fraction and reduction degree, the use of DAB can result in significant errors. It is also worth noting that the fraction of H2 in the BF shaft gas is expected to increase because a (partial) replacement of carbonaceous reductants by hydrogen (bearing) injectants in the BF has been proven to be a feasible way to reduce CO2 emissions in the iron and steel industry [15, 16]. For more efficient BF operation and control, therefore, it is desirable to carry out more rigorous studies on gaseous reduction of iron ores especially in H2-N2 atmosphere. This is one of the motivations behind the current study. Further studies regarding the more practical H2-CO-N2 atmosphere in the BF will be deferred to the subsequent publications.
The present paper is focussed on developing a more fundamental and accurate USCM for estimating the progress of iron ore reduction in H2-N2 atmosphere based on the theoretical expressions established previously. The underlying concepts and main expressions are first introduced, followed by an outline of the present model and the original USCM with DAB, and then a comparison of these two models based on a series of pertinent experimental data collected from the literature.
Model description
The current model takes into account the reversible gaseous reduction of wüstite, FexO + H2 = xFe + H2O, since the transformation from wüstite to metallic iron is usually considered as the rate-limiting stage for iron ore reduction under BF conditions. In the model derivation, the concept of topochemical reaction is applied, and the sub-step of mass convection through the external film is neglected. In addition, a pseudo steady state is postulated, and the effect of Knudsen diffusion is ignored. The gaseous reduction of an individual wüstite (bearing) pellet is illustrated schematically in Figure 1.
Fig. 1
Schematic illustration of gaseous reduction based on a sphere object.
In the internal shell (cf. Figure 1), H2 diffuses inwards to the unreacted (shrinking) core surface through a mixture of H2, H2O and N2, while an identical amount of H2O diffuses outwards along the same path. As a result, the molar flow rates of H2 and of H2O are equal and opposite. For the inert N2, the molar flow rate is zero because it is neither consumed nor produced. However, this is not to say that the partial pressure/concentration difference of N2 along the diffusion path is zero since the local molar mean velocity of the three components is non-zero. Under pseudo steady state, the overall reduction rate can thus be equated with the molar flow rate of H2 through the internal shell. Starting with the Stefan-Maxwell relation for multicomponent mass transfer and after appreciable algebraic manipulation, the following two expressions for the overall reduction rate can be obtained:
{Q_{ABC}} = \varepsilon \xi {{4\pi P} \over {RT}}{{{D_{AB}}} \over {1 - \kappa }}{{{r_0}{r_i}} \over {{r_0} - {r_i}}}\left\{ {\left( {{Y_{A,s}} - {Y_{A,i}}} \right) + \kappa \left( {{Y_{B,s}} - {Y_{B,i}}} \right)} \right\},{Q_{ABC}} = \varepsilon \xi {{4\pi P} \over {RT}}{{{D_{BC}}{\kern 1pt} {D_{AC}}} \over {{D_{BC}} - {D_{AC}}}}{{{r_0}{r_i}} \over {{r_0} - {r_i}}}\ln {{1 - {Y_{A,s}} - {Y_{B,s}}} \over {1 - {Y_{A,i}} - {Y_{B,i}}}},
where ε and ξ are the porosity and labyrinth factor (i.e., reciprocal of tortuosity) of the solid product shell, respectively. Q, P, R, T, Y and r are the overall reduction rate, total pressure, gas constant, temperature, mole fraction and radius, respectively. Subscripts ABC, A, B, C, 0, i and s denote the ternary diffusion system, H2, H2O, N2, initial state, interface and pellet surface, respectively. DAB, DAC and DBC are the binary diffusion coefficients of each gas component pair. In addition, the dimensionless quantity κ is written as follows:
\kappa = {{1/{D_{AB}} - 1/{D_{AC}}} \over {1/{D_{AB}} - 1/{D_{BC}}}},
Under pseudo steady state, the overall reduction rate is also equal to the consumption rate of H2 due to interfacial chemical reaction:
{Q_{ABC}} = 4\pi r_i^2kF,
where k is the reaction rate constant and F is the chemical driving force which is formulated as follows:
F = {P \over {RT}}{K \over {1 + K}}\left( {{Y_{A,i}} - {{{Y_{B,i}}} \over K}} \right),
where K is the equilibrium constant of chemical reaction. Equations (1–5) were already derived elsewhere [11] to which the reader is referred for more detailed information. In the current work, the overall reduction rate is further linked to the reduction degree of an individual pellet:
{Q_{ABC}} = - {4 \over 3}\pi r_0^3M{{df} \over {dt}},
where M is the molar density of reducible oxygen in the pellet, and the reduction degree can be expressed as follows:
f = 1 - {\left( {{{{r_i}} \over {{r_0}}}} \right)^3},
An USCM including DAB, DAC and DBC simultaneously, here denoted as USCM with DABC for the sake of brevity, is now developed by integrating Eq. (6) (using the 4th order Runge-Kutta method), and the relationship between reduction degree and reaction time can be estimated if the pertaining model parameters are known. On the other hand, if the relationship between f and t is available, i.e., if the reduction progress of an individual pellet has been measured utilising the thermogravimetric method, the so-called macro-kinetic parameters can be estimated on the basis of regression analysis. In the present model, the macro-kinetic parameters are the reaction rate constant k and the product of the porosity and labyrinth factor εξ, which can be called a “modified porosity” reflecting the permeability of the solid product shell.
Alternatively, when neglecting the influence of N2, i.e., using DAB only, the overall reduction rate is written as follows:
{Q_{ABC}} = {{4\pi P} \over {RT}}{K \over {1 + K}}r_i^2\left( {{Y_{A,s}} - {{{Y_{B,s}}} \over K}} \right){\left\{ {{1 \over {\varepsilon \xi }}{1 \over {{D_{AB}}}}\left( {{r_i} - {{r_i^2} \over {{r_0}}}} \right) + {1 \over k}} \right\}^{ - 1}},
For this case, the original USCM can be derived as follows:
t = M{r_0}{\left\{ {{P \over {RT}}{K \over {1 + K}}\left( {{Y_{A,s}} - {{{Y_{B,s}}} \over K}} \right)} \right\}^{ - 1}}\left\{ {{{{r_0}} \over 6}{1 \over {\varepsilon \xi }}{1 \over {{D_{AB}}}}\left[ {1 - 3{{\left( {1 - f} \right)}^{2/3}} + 2\left( {1 - f} \right)} \right] + {1 \over k}\left[ {1 - {{\left( {1 - f} \right)}^{1/3}}} \right]} \right\},
In order to facilitate a comparison of the two models, the concentration difference of N2 along the diffusion path, Δ, is introduced as follows:
\Delta = {P \over {RT}}\left( {{Y_{C,s}} - {Y_{C,i}}} \right),
Considering the USCM with DABC is developed on the basis of theoretical expressions, a relative modelling “error”, E, is also defined to quantify the comparison.
E = \left( {{{{Q_{AB}}} \over {{Q_{ABC}}}} - 1} \right) \times 100\%
Results and discussion
Macro-kinetic parameters
In order to illustrate the two USCMs, experimental data reported by Murayama and co-authors [13] are used as a basis for the comparison. In the laboratory-scale experiments described in the work, wüstite pellets were first prepared by partial reduction of pure hematite pellets in a 50% CO-50% CO2 atmosphere at 1273 K. After that, the wüstite pellets were reduced at 1273 K in a H2-N2 atmosphere with different N2 fractions. A total gas flow rate of 2.0 NL/min was maintained throughout each experiment with the intent to eliminate the mass transfer resistance of the external film so the mole fraction of each component at the pellet surface can be considered equal to the corresponding one in the bulk stream. Furthermore, the authors demonstrated that the reduction process is in the mixed control regime of internal diffusion and interfacial chemical reaction. It is therefore implied that the two USCMs analysed in the present paper are applicable to the experimental data, of which the main parameters are listed in Table 1.
Main parameters of experiments regarding hydrogen reduction of wüstite pellets [13].
Parameter
Value
Exp. 1
Exp. 2
Exp. 3
YA,s, -
0.721
0.387
0.210
YC,s, -
0.279
0.613
0.790
YB,s, -
0
T, K
1273
P, Pa
101325
r0, mm
6.2
M, mol/m3
4.88 × 104
DAB, cm2/s
12.48 (at 1273 K)
DAC, cm2/s
11.43 (at 1273 K)
DBC, cm2/s
4.45 (at 1273 K)
K, -
0.673 (at 1273 K)
The two USCMs are first applied to the data of Exp. 1 as a reference, and the macro-kinetic parameters were estimated based on regression analysis. As can be seen in Figure 2 and its insert, the macro-kinetic parameters obtained by the USCM with DAB are (slightly) smaller than the ones obtained by the USCM with DABC.
Fig. 2
Macro-kinetic parameters obtained by the two USCMs based on regression analysis.
In the USCM with DAB, the biggest DAB among the three binary diffusion coefficients (cf. Table 1) is used only, and hence, the internal diffusion is considered overestimated. In order to compensate for this effect, both the reaction rate constant (k) and the modified porosity (ɛξ) need to be lowered at a specific overall reduction rate. Still, adopting the corresponding macro-kinetic parameters, the predicted curves by both models are in good agreement with the experimental data. Hereinafter, the macro-kinetic parameters presented in Figure 2 are used in their corresponding USCMs, and the remaining experimental data of Exp. 2 and Exp. 3 are employed for model validation and comparison.
Model validation, comparisons and discussion
Using the corresponding k and ɛξ values in Figure 2, the relationships between reduction degree and reaction time under the conditions of Exp. 2 and Exp. 3 were first estimated by the USCM with DAB. The experimental data and the estimated results are compared in Figure 3. From the figure, it is seen that the model's results for the systems with bigger N2 fractions in bulk stream (cf. Table 1) overestimate the reduction degree. It may be concluded that even at fixed temperature, pressure and total gas flow rate, the fraction of N2 in bulk stream imposes an influence on the results and therefore also on the macro-kinetic parameters. As a result, different macro-kinetic parameters need to be used for each case with a different N2 fraction in bulk stream.
Fig. 3
Comparison of experimental data and results estimated by USCM with DAB.
As for the deviation shown in Figure 3, the emphasis can be put on inspecting the mathematical model where DAB is used only. As mentioned before, DAB is independent of gas composition and is thus a constant if temperature and pressure are specified. In contrast, the ternary diffusion coefficient for the gas mixture under consideration is basically a function of DAB, DAC, DBC and gas composition. The use of DAB for the ternary diffusion process could hence transmit the composition dependence to the macro-kinetic parameters obtained by regression analysis. As the ternary diffusion coefficient for the H2-H2O-N2 system is smaller than DAB, it can be envisioned that the predicted reduction degree would be lower when the ternary diffusion process is represented mathematically in a more accurate way.
In order to substantiate the argument above, the relationships between reduction degree and reaction time under the conditions of Exp. 2 and Exp. 3 were estimated by the USCM with DABC using k and ɛξ based on the data of Exp. 1. The comparison of the experimental data and estimated results is depicted in Figure 4, where the two predicted curves are found to be in good agreement with the experimental data. It is therefore indicated that using DAB only can cause inaccuracies in the mathematical model, thus giving rise to misleading interpretation of experimental observations and macro-kinetic parameters.
Fig. 4
Comparison of experimental data and results estimated by USCM with DABC.
The noticeable difference in the performance of the two USCMs is explained in the following text, where calculations are made mainly under the conditions of Exp. 2 for the sake of brevity.
In the USCM with DAB, the concentration difference of N2 is ignored i.e., Δ = 0. Therefore, the sum of mole fractions of H2 and H2O at reaction interface equals the one in bulk stream, i.e., YA,i + YB,i = YA,s + YB,s. In fact, however, there should exist a concentration gradient and then difference of N2 in this ternary diffusion system involving equimolar counter-diffusion owing to the noticeable difference in molecular size between H2 and H2O. As for N2 molecules, the total force exerted by H2 molecules differs from that by H2O molecules. In turn, N2 experiences a net force in the system and tends to have a non-zero bulk flow rate. As mentioned above, nevertheless, the flow rate of inert N2 is zero under the conditions of equimolar counter diffusion of H2 and H2O. Therefore, a concentration gradient of N2 and the corresponding Fickian diffusion flow rate must be built up to cancel out the bulk flow rate.
The concentration difference of N2 along its diffusion path was calculated using the USCM with DABC and then depicted in Figure 5 under the conditions of Exp. 2.
Fig. 5
Variation of concentration difference of N2 along diffusion path with reduction degree under conditions of Exp. 2.
As can be seen in Figure 5, the N2 concentration difference is positive and increases with an increase in the reduction degree. This implies that the mole fraction of N2 at reaction interface is smaller than that in the bulk stream. In other words, the sum of mole fractions of H2 and H2O at reaction interface is bigger than the sum in bulk stream, i.e., YA,i + YB,i > YA,s + YB,s. This eventually results in different concentrations of H2 and H2O at reaction interface for the two models. As illustrated in Figures 6 and 7, the concentrations of H2 and H2O at reaction interface corresponding to the USCM with DABC are both higher than the ones from the USCM with DAB.
Fig. 6
Comparison of H2 concentration at reaction interface between different USCMs.
Fig. 7
Comparison of H2O concentration at reaction interface between different USCMs.
To further demonstrate the differences between the two models, the chemical driving forces were calculated and depicted in Figure 8, where the driving force representing the USCM with DABC (cf. dotted curve) is lower than that of the USCM with DAB (cf. solid curve). Since the overall reduction rate is proportional to the chemical driving force under pseudo steady state, it becomes smaller for the USCM with DABC (cf. Figures 3 and 4). It is therefore confirmed that in addition to the dilution effect, the chemical driving force is also influenced by the multicomponent diffusion process because not only the amounts of reducing agent but also its product at the reaction interface is affected by the presence of N2.
Fig. 8
Comparison of chemical driving force between different USCMs.
Using the two models with the macro-kinetic parameters estimated based on the data of Exp. 1, the influence of N2 on hydrogen reduction of iron ores is finally illustrated in Figure 9. Here, the relative modelling error is a positive value and increases with an increase in either the reduction degree or in N2 mole fraction of the bulk stream. As can be found in the figure, the relative modelling error is generally bigger than 5%, and it is thus suggested to adopt the USCM with DABC for modelling gaseous reduction of iron ores in H2-N2 atmosphere.
Fig. 9
Influence of N2 fraction in bulk stream on hydrogen reduction of iron ores.
Conclusions and future prospects
An USCM with multicomponent gas diffusion (i.e., USCM with DABC) for iron ore reduction in H2-N2 atmosphere has been developed based on theoretical expressions established in a previous publication. The present model and the original USCM with DAB have been outlined and compared using a series of pertinent experimental data collected from the literature. On the basis of the comparison, it is revealed that the new USCM with multicomponent gas diffusion is a more proper option than the original USCM where only DAB is taken into account. The main reason lies in the fact that using DAB only to characterise the ternary diffusion system of H2-H2O-N2 can overestimate the chemical driving force and hence the overall process rate. However, it should be kept in mind that the difference between the two USCMs may become minor if N2 fraction is low.
In summary, the main findings and conclusions are drawn as follows:
Using DAB only can result in inaccuracies in the mathematical model, thus giving rise to misleading interpretation of experimental observations and macro-kinetic parameters.
In addition to the dilution effect, the chemical driving force is also influenced by the multicomponent gas diffusion process.
Under the conditions considered in the present study, the relative modelling error is generally bigger than 5%, and it is thus suggested to adopt the USCM with multicomponent gas diffusion for modelling gaseous reduction of iron ores in H2-N2 atmosphere.
The present model can be adopted to shed more light on the process of gaseous reduction in H2-N2 atmosphere. After appreciable algebraic manipulation, it can be used to further characterise a more complicated scenario regarding the gaseous reduction in CO-H2-N2 atmosphere, where the effect of water gas shift reaction plays an importance role. For this, the model will be extended in the near future. In addition, attempts will be made to integrate the present USCM into a simulation framework for mimicking the complicated gas-solid countercurrent flow involving coupled heat and mass transfers and chemical reactions in the BF.
