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 (H_{2}) as reducing agents and a considerable fraction of nitrogen (N_{2}) as a heat carrier for heating the solid burden. It has been identified that the presence of inert N_{2} can curb the reduction, and the prevailing mechanism is attributed to the dilution effect of interfacial chemical reaction, i.e., N_{2} 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, N_{2} 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 N_{2}. Thus, the amount of the reducing agent finally participating in the reaction is affected or even determined by the amount of N_{2}. 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 H_{2}) 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., _{AB}_{AB}_{AB}_{2}. 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 _{AB}

Focussing on gaseous reduction of iron ores, Szekely and co-authors [11] were among the first who evaluated the applicability of _{AB}_{2} and H_{2}-N_{2} 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 N_{2} along the diffusion path was ignored in the improved approach by Murayama and co-authors, and thus, the influence of N_{2} 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 N_{2} on the reduction was assessed in a quantitative way. The results showed that, for CO reduction of iron ores, the use of _{AB}_{2} reduction, particularly under conditions of high N_{2} fraction and reduction degree, the use of _{AB}_{2} 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 CO_{2} 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 H_{2}-N_{2} atmosphere. This is one of the motivations behind the current study. Further studies regarding the more practical H_{2}-CO-N_{2} 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 H_{2}-N_{2} 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 _{AB}

The current model takes into account the reversible gaseous reduction of wüstite, Fe_{x}_{2} = _{2}O, 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.

In the internal shell (cf. Figure 1), H_{2} diffuses inwards to the unreacted (shrinking) core surface through a mixture of H_{2}, H_{2}O and N_{2}, while an identical amount of H_{2}O diffuses outwards along the same path. As a result, the molar flow rates of H_{2} and of H_{2}O are equal and opposite. For the inert N_{2}, 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 N_{2} 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 H_{2} 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:
_{2}, H_{2}O, N_{2}, initial state, interface and pellet surface, respectively. _{AB}_{AC}_{BC}_{2} due to interfacial chemical reaction:
_{AB}_{AC}_{BC}_{ABC}

Alternatively, when neglecting the influence of N_{2}, i.e., using _{AB}_{2} along the diffusion path, Δ, is introduced as follows:
_{ABC}

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% CO_{2} atmosphere at 1273 K. After that, the wüstite pellets were reduced at 1273 K in a H_{2}-N_{2} atmosphere with different N_{2} 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].

Y_{A,s} |
0.721 | 0.387 | 0.210 |

_{C,s} |
0.279 | 0.613 | 0.790 |

_{B,s} |
0 | ||

1273 | |||

101325 | |||

_{0}, mm |
6.2 | ||

^{3} |
4.88 × 10^{4} |
||

_{AB}^{2}/s |
12.48 (at 1273 K) | ||

_{AC}^{2}/s |
11.43 (at 1273 K) | ||

_{BC}^{2}/s |
4.45 (at 1273 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 _{AB}_{ABC}

In the USCM with _{AB}_{AB}

Using the corresponding _{AB}_{2} 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 N_{2} 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 N_{2} fraction in bulk stream.

As for the deviation shown in Figure 3, the emphasis can be put on inspecting the mathematical model where _{AB}_{AB}_{AB}_{AC}_{BC}_{AB}_{2}-H_{2}O-N_{2} system is smaller than _{AB}

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 _{ABC}_{AB}

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 _{AB}_{2} is ignored i.e., Δ = 0. Therefore, the sum of mole fractions of H_{2} and H_{2}O at reaction interface equals the one in bulk stream, i.e., _{A,i}_{B,i}_{A,s}_{B,s}_{2} in this ternary diffusion system involving equimolar counter-diffusion owing to the noticeable difference in molecular size between H_{2} and H_{2}O. As for N_{2} molecules, the total force exerted by H_{2} molecules differs from that by H_{2}O molecules. In turn, N_{2} 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 N_{2} is zero under the conditions of equimolar counter diffusion of H_{2} and H_{2}O. Therefore, a concentration gradient of N_{2} and the corresponding Fickian diffusion flow rate must be built up to cancel out the bulk flow rate.

The concentration difference of N_{2} along its diffusion path was calculated using the USCM with _{ABC}

As can be seen in Figure 5, the N_{2} concentration difference is positive and increases with an increase in the reduction degree. This implies that the mole fraction of N_{2} at reaction interface is smaller than that in the bulk stream. In other words, the sum of mole fractions of H_{2} and H_{2}O at reaction interface is bigger than the sum in bulk stream, i.e., _{A,i}_{B,i} > Y_{A,s}_{B,s}_{2} and H_{2}O at reaction interface for the two models. As illustrated in Figures 6 and 7, the concentrations of H_{2} and H_{2}O at reaction interface corresponding to the USCM with _{ABC}_{AB}

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 _{ABC}_{AB}_{ABC}_{2}.

Using the two models with the macro-kinetic parameters estimated based on the data of Exp. 1, the influence of N_{2} 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 N_{2} 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 _{ABC}_{2}-N_{2} atmosphere.

An USCM with multicomponent gas diffusion (i.e., USCM with _{ABC}_{2}-N_{2} atmosphere has been developed based on theoretical expressions established in a previous publication. The present model and the original USCM with _{AB}_{AB}_{AB}_{2}-H_{2}O-N_{2} 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 N_{2} fraction is low.

In summary, the main findings and conclusions are drawn as follows:

Using _{AB}

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 H_{2}-N_{2} atmosphere.

The present model can be adopted to shed more light on the process of gaseous reduction in H_{2}-N_{2} atmosphere. After appreciable algebraic manipulation, it can be used to further characterise a more complicated scenario regarding the gaseous reduction in CO-H_{2}-N_{2} 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.

#### Main parameters of experiments regarding hydrogen reduction of wüstite pellets [13].

Y_{A,s} |
0.721 | 0.387 | 0.210 |

_{C,s} |
0.279 | 0.613 | 0.790 |

_{B,s} |
0 | ||

1273 | |||

101325 | |||

_{0}, mm |
6.2 | ||

^{3} |
4.88 × 10^{4} |
||

_{AB}^{2}/s |
12.48 (at 1273 K) | ||

_{AC}^{2}/s |
11.43 (at 1273 K) | ||

_{BC}^{2}/s |
4.45 (at 1273 K) | ||

0.673 (at 1273 K) |

_{2} mineralization potential of ironmaking and steelmaking slags under direct gas-solid reactions in flue gas. International Journal of Greenhouse Gas Control, 2019, 87, pp. 100–111._{2} mineralization potential of ironmaking and steelmaking slags under direct gas-solid reactions in flue gas

_{2}-N_{2} and CO-N_{2} by unreacted-core model. Tetsu-to-Hagane, 1982, 68(15), pp. 2253–2262._{2}-N_{2} and CO-N_{2} by unreacted-core model

_{2} on gaseous reduction of iron ores with CO or H_{2}. ISIJ International, 2018, 58(2), pp. 370–372._{2} on gaseous reduction of iron ores with CO or H_{2}

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underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise 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Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leader-follower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model A study of local smoothness-informed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H _{2}-N_{2}atmosphereAlgorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model