Numerical Analysis of Carbon Monoxide–Hydrogen Gas Reduction of Iron Ore in a Packed Bed by an Euler–Lagrange Approach

Abstract

In recent years, various methods to decrease carbon dioxide emissions from iron and steel making industries have been developed. The latest blast furnace operation design is intended to induce the low reducing agent operation, highly reactive material is considered a promising way to improve reaction efficiency. Another method utilizes hydrogen in the blast furnace process for highly efficient reduction. Mathematical modeling may help to predict complex in-furnace phenomena, including momentum, heat, and mass transport. However, the current macroscopic continuum model gives no information on the individual particles. In this work, a new approach based on the discrete element method was introduced to consider the interaction between particles under fluid flow in accordance with the arrangement and properties of individual particles. We used an Euler–Lagrange method to precisely understand the influence of the reaction conditions on the behavior of coke and ore particles in three dimensions. The heterogeneity of the reaction rate and temperature distribution was observed to be influenced by the particle arrangement. The endothermic and exothermic reactions influenced each other in the packed bed. Temperature distributions nearly correlated with the gas velocity distribution because convection processes greatly affected the reaction rate. Although convection heat transfer was not a dominant issue in the packed bed, promotion of the reaction by a gas flow was effective.

Appendix

DEM–CFD Combining Model for Particle–Fluid Motion

The DEM–CFD model is briefly described here for completeness. The same procedure was used in previous research.[15,16,44] The governing equations for the translational and rotational motion of particle i can be described as

$$ m_{\text{p}} \frac{{d\varvec{v}_{\text{p}} }}{dt} = \mathop \sum \limits_{j \ne i}^{{N_{\text{ct}} }} \varvec{F}_{\text{c}} + \varvec{F}_{{f}} + \varvec{F}_{\text{g}} , $$

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$$ I_{\text{p}} \frac{{d\varvec{\omega}_{\text{p}} }}{dt} = r_{\text{p}} \mathop \sum \limits_{j \ne i}^{{N_{\text{ct}} }} \left( {\varvec{F}_{{cs}} - \varvec{F}_{\text{r}} } \right), $$

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where $m_{\text{p}}$ is the particle mass, $\varvec{v}_{\text{p}}$ is the particle velocity, $t$ is time, $\varvec{F}_{\text{c}}$ is the particle–particle contact force, $N_{\text{c}}$ is the contact particle number, $\varvec{F}_{{f}}$ is the particle–fluid interaction force, $\varvec{F}_{\text{g}}$ is the gravity force, $I_{\text{p}}$ is the moment of inertia, $\varvec{\omega}_{\text{p}}$ is the angular velocity, $r_{\text{p}}$ is the particle radius, $\varvec{F}_{\text{cn}}$ is the normal component of the contact force, $\varvec{F}_{\text{cs}}$ is the tangential component of contact force, and $\varvec{F}_{\text{r}}$ is the rolling friction force. The interparticle contact force was calculated by the “soft sphere” assumption based on Hertz’s contact theory:

$$ \varvec{F}_{\text{c}} = \varvec{F}_{\text{cn}} + \varvec{F}_{\text{cs}} , $$

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$$ \varvec{F}_{\text{cn}} = - \tau_{n} l_{n} - \eta_{n} \varvec{v}_{{{\text{p}},n}} , $$

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$$ \varvec{F}_{\text{cs}} = \left\{ {\begin{array}{*{20}l} { - \tau_{\text{s}} l_{\text{s}} - \eta_{\text{s}} \varvec{v}_{{{\text{p}},{\text{s}}}}\quad \left( {\left| {\varvec{F}_{\text{cs}} } \right| < \mu \left| {\varvec{F}_{\text{cn}} } \right|} \right)} \\ { \mu \varvec{F}_{\text{cn}} \quad \left( {\left| {\varvec{F}_{\text{cs}} } \right| \ge \mu \left| {\varvec{F}_{\text{cn}} } \right|} \right)} \\ \end{array} ,} \right. $$

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where $\tau$ is the spring coefficient, $\eta$ is the dashpot coefficient, $l$ is the relative displacement increment at a discrete time $\Delta t$, and $\mu$ is the frictional coefficient. Refer to previous article for further information to calculate $\tau$, $\eta$, and $\varvec{F}_{r}$ [15].

The motion of the continuum fluid is calculated from the equation of continuity and Navier–Stokes equation based on local mean variables over a calculational cell.

$$ \frac{\partial }{\partial t}\left( {\varepsilon \rho_{\text{g}} } \right) + \nabla \cdot \left( {\varepsilon \rho_{\text{g}} \varvec{v}_{\text{g}} } \right) = 0, $$

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$$ \frac{\partial }{\partial t}\left( {\varepsilon \rho_{\text{g}} } \right) + \nabla \cdot \left( {\varepsilon \rho_{\text{g}} \varvec{v}_{\text{g}} \varvec{v}_{\text{g}} } \right) = - \varepsilon \nabla p_{\text{g}} + \varepsilon \mu_{\text{g}} \nabla^{2} \varvec{v}_{\text{g}} + \varepsilon \rho_{\text{g}} \varvec{g} + \varvec{F}_{\text{p}} , $$

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where $\varepsilon$ is the void fraction, ρ _g is the fluid density, v _g is the fluid velocity, $p_{\text{g}}$ is the fluid pressure, and $\mu_{\text{g}}$ is the fluid viscosity. The void fraction $\varepsilon$ was calculated from the proportion of the solid existing within a local cell volume (Natsui et al.[15]). To determine $\varvec{F}_{\text{p}}$, Ergun’s equation (1952)[42] was applied in the low-void fraction region $\left( {\varepsilon < 0.8} \right)$, and Wen-Yu’s equation (1966)[43] was applied in the high-void fraction region $\left( {\varepsilon > 0.8} \right)$:

$$ \varvec{F}_{\text{p}} = \alpha \left( {\varvec{v}_{\text{p}} - \varvec{v}_{\text{g}} } \right), $$

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where $\alpha$ is the particle–fluid momentum exchange coefficient. The drag force coefficient was derived from the Schiller–Naumann equation expressed as a function of the particle Reynolds number[44]:

$$ \alpha = \left\{ {\begin{array}{*{20}l} {\frac{{\mu_{\text{g}} \left( {1 - \varepsilon } \right)}}{{\rho_{\text{g}} \varepsilon d_{\text{p}}^{2} }}\left[ {150\left( {1 - \varepsilon } \right) + 1.75 {Re}_{\text{p}} } \right]\quad\left( {\varepsilon \le 0.8} \right)} \\ {\frac{3}{4}B\frac{{\mu_{\text{g}} \left( {1 - \varepsilon } \right)}}{{\rho_{\text{g}} \varepsilon^{2.7} d_{\text{p}}^{2} }} {Re}_{\text{p}}\qquad\left( {\varepsilon > 0.8} \right)} \\ \end{array} } \right., $$

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$$ B = \left\{ {\begin{array}{*{20}l} {\frac{24}{{{Re}_{\text{p}} }}\left( {1 + 0.15 {Re}_{\text{p}}^{0.687} } \right) \qquad \left( {{Re}_{\text{p}} \le 1000} \right)} \\ {0.43\qquad \left( {{Re}_{\text{p}} > 1000} \right)} \\ \end{array} } \right., $$

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$$ {{Re}}_{\text{p}} = \frac{{\left| {\varvec{v}_{\text{p}} - \varvec{v}_{\text{g}} } \right|\rho_{\text{g}} \varepsilon d_{\text{p}} }}{{\mu_{\text{g}} }}. $$

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F _f is distributed to particles as a reaction force of F _p:

$$ \varvec{F}_{\text{d}} = \frac{{\pi d_{\text{p}} \beta_{\text{t}} }}{6(1 - \varepsilon )}(\upsilon_{\text{g}} - \upsilon_{\text{p}} ). $$

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Rate Coefficient of the Multi-interface Unreacted Core Model

A gas reduction reaction occurs at the interface of a particle, and thus iron ore can be applied to the unreacted core model.[34,35] In this research, the temperature range was assumed to be T > 848 K (575 °C), and the composition of iron oxide changes with the degree of oxidation; i.e., Fe₂O₃ (hematite), Fe₃O₄ (magnetite), FeO (wüstite), Fe (iron). Since different reduction reactions occur on the inside and outside of a particle, it was necessary to set up a multi-reaction interface (see Figure 2). Table II shows the rate coefficients of the multi-interface unreacted core model in each interface. Here, in this section, A is the chemical reaction resistance, k _c is the chemical reaction rate coefficient, B is the inner particle diffusion resistance, D _e is the inner particle effective diffusion coefficient, F is the surface mass transfer resistance, k _F is the surface mass transfer coefficient, R is the reduction degree, W is the ore flux, Y is the mol fraction of gaseous component, and K is the equilibrium constant.

Table II Rate Coefficients of the Multi-interface Unreacted Core Shrinking Model for Iron Ore Reduction