Introduction
Numerical Modeling
Governing Equations
Equations | Description |
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Mass conservation |
\( \nabla {\cdot} \left( {\varepsilon_{\text{i}} \rho_{\text{i}} \mathbf{u}_{\text{i}} } \right) = S_{\text{i}}, \)
\( {\text{where }}S_{\text{i}} = - \sum\limits_{\text{k}} {\beta_{\text{i,k}} R_{\text{k}}^{ *} } \)
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Momentum conservation | |
Gas |
\( \nabla {\cdot} \left( {\varepsilon_{\text{g}} \rho_{\text{g}} \mathbf{u}_{\text{g}} \mathbf{u}_{\text{g}} } \right) = \nabla{\cdot} {{\varvec{\tau}}_{\text{g}} - \varepsilon_{\text{g}}} \nabla p + \rho_{\text{g}} \varepsilon_{\text{g}} \mathbf{g} + \mathbf{F}_{\rm g}^{\rm s} + \mathbf{F}_{\text{g}}^{\rm l,d} \)
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\( {\varvec{\tau}}_{\text{g}} = \varepsilon_{\text{g}} \mu_{\text{g}} \left[ {\nabla \mathbf{u}_{\text{g}} + \left( {\nabla \mathbf{u}_{\text{g}} } \right)^{\text{T}} } \right] - {\frac{2}{3}}\varepsilon_{\text{g}} \mu_{\text{g}} \left( {\nabla {\cdot} \mathbf{u}_{\text{g}}} \right){\text{I}} \)
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Solid |
\( \nabla {\cdot} \left( {\varepsilon_{\text{s}} \rho_{\text{s}} \mathbf{u}_{\text{s}} \mathbf{u}_{\text{s}} } \right) = \nabla {\cdot} {{\varvec{\tau}}_{\text{s}} - \varepsilon_{\text{s}} }\nabla p_{\text{s}} + \rho_{\text{s}} \varepsilon_{\text{s}} \mathbf{g} \)
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\( {\varvec{\tau}}_{\text{s}} = \varepsilon_{\text{s}} \mu_{\text{s}} \left[ {\nabla \mathbf{u}_{\text{s}} + \left( {\nabla \mathbf{u}_{\text{s}} } \right)^{\text{T}} } \right] - {\frac{2}{3}}\varepsilon_{\text{s}} \mu_{\text{s}} \left( {\nabla {\cdot} \mathbf{u}_{\text{s}} } \right){\text{I}}\)
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Liquid |
\( \mathbf{F}_{\text{l,d}}^{\text{g}} + \mathbf{F}_{\text{l,d}}^{\text{s}} + \varepsilon_{\text{l,d}} \rho_{\text{l}} \mathbf{g} = 0 \)
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Heat and species conservation |
\( \nabla {\cdot} \left( {\varepsilon_{\text{i}} \rho_{\text{i}} \mathbf{u}_{\text{i}} \phi_{\text{i,m}} } \right) - \nabla {\cdot} \left( {\varepsilon_{\text{i}} {\Gamma_{\text{i}}}\nabla \phi_{\text{i,m}} } \right) = S_{{\phi_{\text{i,m}} }} \)
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if \( \phi_{\text{i,m}} \) is H
i,m, \({\Gamma}_{\text{i}} = {\frac{{k_{\text{i}} }}{{c_{\text{p,i}} }}} \), | |
\( S_{{\phi_{\text{i,m}}}} = \delta_{\text{i}} h_{\text{ij}} \alpha \left( { T_{\text{i}} - T_{\text{j}} } \right) + c_{\text{p,i}} T_{\text{i}} \delta_{\text{i}} \sum\limits_{\text{k}} {\sum\limits_{\text{l}} {\beta_{\text{k,l}} R_{\text{k}}^{*} + \eta_{\text{i}} \sum\limits_{\text{k}} {R_{\text{k}}^{*} } \left( { - \Updelta H_{\text{k}} } \right)} } \)
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if \( \phi_{\text{i,m}} \) is ω
i,m, \( {\Gamma}_{\text{i}} = \rho_{\text{i}} D_{\text{i}} \), \( S_{{\phi_{\text{i,m}}}} = \sum\limits_{\text{k}} {\alpha_{\text{i,m,k}} R_{\text{k}}^{*} } \), where | |
\( \phi_{\text{i,m}} = \omega_{\text{g,co}} ,\omega_{{{\text{g,co}}_{ 2} }} ,\omega_{{{\text{s,Fe}}_{2} {\text{O}}_{3} }} ,\omega_{{{\text{s,Fe}}_{3} {\text{O}}_{4} }},\omega_{\text{s,FeO}} ,\omega_{\text{s,flux}} \)
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Phase volume fraction |
\( \sum\limits_{\text{i}} {\varepsilon_{\text{i}} = 1} \)
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State equation |
\( p = \sum\limits_{\text{i}} {\left( {y_{\text{i}} M_{i} } \right)} {{{RT_{\text{g}} }}/{{V_{\text{g}} }}} \)
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Momentum Transfer, Chemical Reactions, and Transport Coefficients
Phases | Interaction Forces | Ref. |
---|---|---|
Gas–solid, G–S |
\( \mathbf{F}_{\text{g}}^{\text{s}} = - \mathbf{F}_{\text{s}}^{\text{g}} = - \left( {\alpha_{\text{f}} \rho_{\text{g}} \left| {\mathbf{u}_{\text{g}}^{\text{s}} } \right| + \beta_{\text{f}} } \right)\mathbf{u}_{\text{g}}^{\text{s}} \)
| [5] |
Gas–liquid, G–L | \( \mathbf{F}_{\text{g}}^{\text{l,d}} = - \mathbf{F}_{\text{l,d}}^{\text{g}} = - \left( {{\frac{{h_{\text{l,d}} }}{{d_{\text{l}} }}} + {\frac{{A_{\text{sl,d}} }}{6}}} \right)\left[ {150\left( {{\frac{{\varepsilon_{s} + h_{\text{l,t}} }}{{d_{\text{w}} }}}} \right)\mu_{\text{g}} + 1.75\rho_{\text{g}} \left| {{\mathbf{U}}_{\text{g}} } \right|} \right] {\frac{{{\mathbf{U}}_{\rm g} }}{{\varepsilon_{\rm g}^{3} }}} \) where | [35] |
\( d_{\text{l}} = \max \left\{ {d_{\rm l,g} ,d_{\text{l,h}} } \right\} \)
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\( d_{\text{l,g}} = {{{\left[ {\max \left\{ { - 6.828sign\left( {\sqrt {X_{\text{p}} } - 0.891} \right)\left( {\sqrt {X_{\text{p}} } - 0.891} \right)^{2} ,0} \right\} + 0.695} \right]}}/{{\sqrt {{{{\rho_{\text{l}} g}}/{\sigma }}} }}} \)
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\( d_{\text{l,h}} = \frac{{{\left[ {\max \left\{ {6.828sign(f_1)({f_1})^{2} ,0} \right\} + 0.695} \right]}}}{{\sqrt {{{{\rho_{\text{l}} g}}/{\sigma }}} }} \)
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\( f_{1} = \left[ {\max {{\frac{\left\{{\ln \left( {{\frac{{h_{\text{l,t}} }}{{h_{\text{l,to}} }}}} \right),0}\right\}}{0.513}}} ^{{}} } \right]^{{{\frac{1}{2.642}}}} - 0.891 \)
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\( X_{\text{p}} = {\frac{{\Updelta p_{\text{e}} }}{{\left( {\Updelta x\rho_{\text{l}} g} \right)}}}\left\{ {\rho_{\text{l}} g\varphi^{2} {\frac{{d_{\text{s}}^{2} }}{{{\frac{\sigma }{{\varepsilon_{\text{s}}^{2} }}}}}}} \right\}^{0.3} \left( {1 + \cos \theta } \right)^{ - 0.5} \)
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Liquid–solid, L–S |
\( \mathbf{F}_{\text{l,d}}^{\text{s}} = \left( {{\frac{150}{36}}\mu_{\text{l}} {\frac{{A_{\text{sl,d}}^{2} }}{{h_{\text{l,d}}^{ 3} }}} + {\frac{1.75}{6}}\rho_{\text{l}} {\frac{{A_{\text{sl,d}} }}{{h_{\text{l,d}}^{ 3} }}}\left| {\mathbf{U}_{l} } \right|} \right)\mathbf{U}_{l}\)
| [35] |
Reaction Formula | Reaction Rate | Reference |
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Fe2O3(s) + CO(g) → Fe(s) + CO2(g)
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\( R_{ 1}^{ *} = {\frac{{{{{12\xi_{\text{ore}} \varepsilon_{\text{ore}} P\left( {y_{\text{co}} - y_{\text{co}}^{*} } \right)}}/{{\left( {8.314T_{\text{s}} } \right)}}}}}{{{{{d_{\text{ore}}^{ 2} }}/{{D_{\text{g,co}}^{\text{e}} \left[ {\left( {1 - f_{\text{o}} } \right)^{{{-\frac{1}{3}}}} - 1} \right] + d_{\text{ore}} \left\{ {k_{\text{l}} \left( {1 + {({1}/{{K_{\text{l}} }})}} \right)} \right\}^{ - 1} }}}}}} \)
| [36] |
FeO(l) + C(s) → Fe(l) + CO(g)
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\( R_{2}^{*} = {\frac{{k_{2} A_{\text{c}} }}{{{\text{V}}_{\text{b}} a_{\text{FeO}} }}} \)
| [1] |
C(s) + CO2(g) → 2CO(g)
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\( R_{ 3}^{ *} = {\frac{{6\xi_{\text{coke}} \varepsilon_{\text{coke}} {{{py_{{{\text{co}}_{ 2} }} }}/{{\left( {8.314T_{\text{s}} } \right)}}}}}{{{{{d_{\text{coke}} }}/{{{{{k_{\text{f}} + 6}}/{{\left( {\rho_{\text{coke}} E_{\text{f}} k_{3} } \right)}}}}}}}}} \)
| [1] |
FeO(s) → FeO(l)
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\( R_{4}^{*} = \left\langle {{\frac{{T_{\text{i}} - T_{{\min ,{\text{sm}}}} }}{{T_{{\max ,{\text{sm}}}} - T_{{\min ,{\text{sm}}}} }}}} \right\rangle_{0}^{1} {\frac{{\oint {\omega_{\text{sm}} \mathbf{u}_{\text{i}} \rho_{\text{i}} \varepsilon_{\text{i}} dA} }}{{M_{\text{sm}} {\text{Vol}}_{\text{cell}} }}} \)
| [38] |
Flux(s) → slag(l)
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Effective Diffusion Coefficients
Effective Conductivity Coefficients
Heat Transfer Coefficients Between Phases
Heat Loss Through the Furnace Wall
CZ Treatments
Treatments | Nonlayered treatments | Layered Treatment | |
---|---|---|---|
Isotropic | Anisotropic | ||
Variables | |||
Solid volume fraction |
\( \varepsilon_{\text{s}} = \xi_{\text{ore}} \varepsilon_{\text{ore}} + \xi_{\text{coke}} \varepsilon_{\text{coke}} \)
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\( \begin{aligned} \phi_{\text{s}} = \left\{ \begin{aligned} \phi_{\text{ore}}\quad {\text{for ore layer}} \hfill \\ \phi_{\text{coke }} \quad{\text{for coke layer}} \hfill \\ \end{aligned} \right. \hfill \\ {\text{where\,}} \phi = \varepsilon ,d,k \hfill \\ \end{aligned} \)
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Solid particle size |
\( d_{\text{s}} = \left( {{\frac{{\xi_{\text{ore}} }}{{d_{\text{ore}} }}} + {\frac{{\xi_{\text{coke}} }}{{d_{\text{coke}} }}}} \right)^{ - 1} \)
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Solid heat conductivity |
\( k_{\text{s}} = \left( {{\frac{{\xi_{\text{ore}} }}{{k_{\text{ore}} }}} + {\frac{{\xi_{\text{coke}} }}{{k_{\text{coke}} }}}} \right)^{ - 1} \)
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Vertical to the layer | |||
Gas flow resistance in CZ |
\( \alpha_{\text{f}} = a{\frac{{1 - \varepsilon_{\text{s}} }}{{d_{\text{s}} }}} \)
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\( \alpha_{\text{f}} = \xi_{\text{ore}} \alpha_{\text{ore}} + \xi_{\text{coke}} \alpha_{\text{coke}} \)
| For ore layer \( \alpha_{\text{f}} = \alpha_{\text{ore}} , \)
\( \beta_{\text{f}} = \beta_{\text{ore}} \)
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\( \beta_{\text{f}} = \xi_{\text{ore}} \beta_{\text{ore}} + \xi_{\text{coke}} \beta_{\text{coke}} \)
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Parallel to the layer | |||
\( \beta_{\text{f}} = b{\frac{{\mu_{g} \left( {1 - \varepsilon_{\text{s}} } \right)^{2} }}{{d_{\text{s}}^{2} \varepsilon_{\text{s}} }}} \)
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\( \alpha_{\text{f}} = \left( {{\frac{{\xi_{\text{ore}} }}{{\sqrt {\alpha_{\text{ore}} } }}} + {\frac{{\xi_{\text{coke}} }}{{\sqrt {\alpha_{\text{coke}} } }}}} \right)^{ - 2} \)
| For coke layer \( \alpha_{\text{f}} = \alpha_{\text{coke}} , \)
\( \beta_{\text{f}} = \beta_{\text{coke}} \)
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\( \beta_{\text{f}} = \left( {{\frac{{\xi_{\text{ore}} }}{{ {\beta_{\text{ore}} } }}} + {\frac{{\xi_{\text{coke}} }}{{{\beta_{\text{coke}} } }}}} \right)^{ - 1} \)
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Numerical Technique
Simulation Conditions
Variables | BF |
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Gas: | |
Volume flux, Nm3 × tHM−1
| 1511 |
Inlet gas components, mole percentage | 34.95 pct CO |
0.0 pct CO2
| |
0.81 pct H2
| |
0.0 pct H2O | |
64.23 pct N2
| |
Inlet gas temperature, K | 2313.6 |
Top pressure, atm | 2.0 |
Solid: | |
Ore, t × tHM−1
| 1.64 |
Ore components, mass fraction | Fe2O3 0.656 |
FeO 0.157 | |
CaO 0.065 | |
MgO 0.024 | |
SiO2 0.059 | |
Al2O3 0.029 | |
MnO 0.006 | |
P2O5 0.008 | |
Average ore particle size, m | 0.03 |
Coke, t × tHM−1
| 0.5023 |
Coke components, mass fraction | C 0.857 |
Ash 0.128 | |
S 0.005 | |
H 0.005 | |
N 0.005 | |
Average coke particle size, m | 0.045 |
Flux, t × tHM−1
| 0.0264 |
Flux components, mass fraction | CaO 0.438 |
MgO 0.079 | |
SiO2 0.024 | |
Al2O3 0.033 | |
CO2 in CaO 0.344 | |
CO2 in MgO 0.082 | |
Ore voidage | 0.403(100dore)0.14
|
Coke voidage | 0.153logdcoke + 0.724 |
Average ore/(ore+coke) volume ratio | 0.5923 |
Liquid: | |
Hot metal rate, t × day−1
| 2034 |
components, mass fraction | C 0.04 |
Si 0.004 | |
Mn 0.0045 | |
P 0.0003 | |
S 0.0003 | |
Fe 0.9509 | |
density, kg × m−3
| 6600 |
viscosity, kg × m−1 × s−1
| 0.005 |
conductivity, w × m−1 × K−1
| 28.44 |
surface tension, N × m−1
| 1.1 |
Slag rate, t × tHM−1
| 0.377 |
components, mass fraction | CaO 0.324 |
MgO 0.120 | |
SiO2 0.324 | |
Al2O3 0.200 | |
FeO 0.016 | |
MnO 0.009 | |
S 0.007 | |
density, kg × m−3
| 2600 |
viscosity, kg × m−1 × s−1
| 1.0 |
conductivity, w × m−1 × K−1
| 0.57 |
surface tension, N × m−1
| 0.47 |