01.04.2010  Ausgabe 2/2010 Open Access
Modeling of Blast Furnace with Layered Cohesive Zone
 Zeitschrift:
 Metallurgical and Materials Transactions B > Ausgabe 2/2010
Introduction
Numerical Modeling
Governing Equations
Equations  Description 

Mass conservation 
\( \nabla {\cdot} \left( {\varepsilon_{\text{i}} \rho_{\text{i}} \mathbf{u}_{\text{i}} } \right) = S_{\text{i}}, \)
\( {\text{where }}S_{\text{i}} =  \sum\limits_{\text{k}} {\beta_{\text{i,k}} R_{\text{k}}^{ *} } \)

Momentum conservation  
Gas 
\( \nabla {\cdot} \left( {\varepsilon_{\text{g}} \rho_{\text{g}} \mathbf{u}_{\text{g}} \mathbf{u}_{\text{g}} } \right) = \nabla{\cdot} {{\varvec{\tau}}_{\text{g}}  \varepsilon_{\text{g}}} \nabla p + \rho_{\text{g}} \varepsilon_{\text{g}} \mathbf{g} + \mathbf{F}_{\rm g}^{\rm s} + \mathbf{F}_{\text{g}}^{\rm l,d} \)

\( {\varvec{\tau}}_{\text{g}} = \varepsilon_{\text{g}} \mu_{\text{g}} \left[ {\nabla \mathbf{u}_{\text{g}} + \left( {\nabla \mathbf{u}_{\text{g}} } \right)^{\text{T}} } \right]  {\frac{2}{3}}\varepsilon_{\text{g}} \mu_{\text{g}} \left( {\nabla {\cdot} \mathbf{u}_{\text{g}}} \right){\text{I}} \)
 
Solid 
\( \nabla {\cdot} \left( {\varepsilon_{\text{s}} \rho_{\text{s}} \mathbf{u}_{\text{s}} \mathbf{u}_{\text{s}} } \right) = \nabla {\cdot} {{\varvec{\tau}}_{\text{s}}  \varepsilon_{\text{s}} }\nabla p_{\text{s}} + \rho_{\text{s}} \varepsilon_{\text{s}} \mathbf{g} \)

\( {\varvec{\tau}}_{\text{s}} = \varepsilon_{\text{s}} \mu_{\text{s}} \left[ {\nabla \mathbf{u}_{\text{s}} + \left( {\nabla \mathbf{u}_{\text{s}} } \right)^{\text{T}} } \right]  {\frac{2}{3}}\varepsilon_{\text{s}} \mu_{\text{s}} \left( {\nabla {\cdot} \mathbf{u}_{\text{s}} } \right){\text{I}}\)
 
Liquid 
\( \mathbf{F}_{\text{l,d}}^{\text{g}} + \mathbf{F}_{\text{l,d}}^{\text{s}} + \varepsilon_{\text{l,d}} \rho_{\text{l}} \mathbf{g} = 0 \)

Heat and species conservation 
\( \nabla {\cdot} \left( {\varepsilon_{\text{i}} \rho_{\text{i}} \mathbf{u}_{\text{i}} \phi_{\text{i,m}} } \right)  \nabla {\cdot} \left( {\varepsilon_{\text{i}} {\Gamma_{\text{i}}}\nabla \phi_{\text{i,m}} } \right) = S_{{\phi_{\text{i,m}} }} \)

if \( \phi_{\text{i,m}} \) is H
_{i,m}, \({\Gamma}_{\text{i}} = {\frac{{k_{\text{i}} }}{{c_{\text{p,i}} }}} \),  
\( S_{{\phi_{\text{i,m}}}} = \delta_{\text{i}} h_{\text{ij}} \alpha \left( { T_{\text{i}}  T_{\text{j}} } \right) + c_{\text{p,i}} T_{\text{i}} \delta_{\text{i}} \sum\limits_{\text{k}} {\sum\limits_{\text{l}} {\beta_{\text{k,l}} R_{\text{k}}^{*} + \eta_{\text{i}} \sum\limits_{\text{k}} {R_{\text{k}}^{*} } \left( {  \Updelta H_{\text{k}} } \right)} } \)
 
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  
\( \phi_{\text{i,m}} = \omega_{\text{g,co}} ,\omega_{{{\text{g,co}}_{ 2} }} ,\omega_{{{\text{s,Fe}}_{2} {\text{O}}_{3} }} ,\omega_{{{\text{s,Fe}}_{3} {\text{O}}_{4} }},\omega_{\text{s,FeO}} ,\omega_{\text{s,flux}} \)
 
Phase volume fraction 
\( \sum\limits_{\text{i}} {\varepsilon_{\text{i}} = 1} \)

State equation 
\( p = \sum\limits_{\text{i}} {\left( {y_{\text{i}} M_{i} } \right)} {{{RT_{\text{g}} }}/{{V_{\text{g}} }}} \)

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] 
\( d_{\text{l}} = \max \left\{ {d_{\rm l,g} ,d_{\text{l,h}} } \right\} \)
 
\( d_{\text{l,g}} = {{{\left[ {\max \left\{ {  6.828sign\left( {\sqrt {X_{\text{p}} }  0.891} \right)\left( {\sqrt {X_{\text{p}} }  0.891} \right)^{2} ,0} \right\} + 0.695} \right]}}/{{\sqrt {{{{\rho_{\text{l}} g}}/{\sigma }}} }}} \)
 
\( d_{\text{l,h}} = \frac{{{\left[ {\max \left\{ {6.828sign(f_1)({f_1})^{2} ,0} \right\} + 0.695} \right]}}}{{\sqrt {{{{\rho_{\text{l}} g}}/{\sigma }}} }} \)
 
\( f_{1} = \left[ {\max {{\frac{\left\{{\ln \left( {{\frac{{h_{\text{l,t}} }}{{h_{\text{l,to}} }}}} \right),0}\right\}}{0.513}}} ^{{}} } \right]^{{{\frac{1}{2.642}}}}  0.891 \)
 
\( X_{\text{p}} = {\frac{{\Updelta p_{\text{e}} }}{{\left( {\Updelta x\rho_{\text{l}} g} \right)}}}\left\{ {\rho_{\text{l}} g\varphi^{2} {\frac{{d_{\text{s}}^{2} }}{{{\frac{\sigma }{{\varepsilon_{\text{s}}^{2} }}}}}}} \right\}^{0.3} \left( {1 + \cos \theta } \right)^{  0.5} \)
 
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 

Fe_{2}O_{3(s)} + CO_{(g)} → Fe_{(s)} + CO_{2(g)}

\( R_{ 1}^{ *} = {\frac{{{{{12\xi_{\text{ore}} \varepsilon_{\text{ore}} P\left( {y_{\text{co}}  y_{\text{co}}^{*} } \right)}}/{{\left( {8.314T_{\text{s}} } \right)}}}}}{{{{{d_{\text{ore}}^{ 2} }}/{{D_{\text{g,co}}^{\text{e}} \left[ {\left( {1  f_{\text{o}} } \right)^{{{\frac{1}{3}}}}  1} \right] + d_{\text{ore}} \left\{ {k_{\text{l}} \left( {1 + {({1}/{{K_{\text{l}} }})}} \right)} \right\}^{  1} }}}}}} \)
 [36] 
FeO_{(l)} + C_{(s)} → Fe_{(l)} + CO_{(g)}

\( R_{2}^{*} = {\frac{{k_{2} A_{\text{c}} }}{{{\text{V}}_{\text{b}} a_{\text{FeO}} }}} \)
 [1] 
C_{(s)} + CO_{2(g)} → 2CO_{(g)}

\( R_{ 3}^{ *} = {\frac{{6\xi_{\text{coke}} \varepsilon_{\text{coke}} {{{py_{{{\text{co}}_{ 2} }} }}/{{\left( {8.314T_{\text{s}} } \right)}}}}}{{{{{d_{\text{coke}} }}/{{{{{k_{\text{f}} + 6}}/{{\left( {\rho_{\text{coke}} E_{\text{f}} k_{3} } \right)}}}}}}}}} \)
 [1] 
FeO_{(s)} → FeO_{(l)}

\( R_{4}^{*} = \left\langle {{\frac{{T_{\text{i}}  T_{{\min ,{\text{sm}}}} }}{{T_{{\max ,{\text{sm}}}}  T_{{\min ,{\text{sm}}}} }}}} \right\rangle_{0}^{1} {\frac{{\oint {\omega_{\text{sm}} \mathbf{u}_{\text{i}} \rho_{\text{i}} \varepsilon_{\text{i}} dA} }}{{M_{\text{sm}} {\text{Vol}}_{\text{cell}} }}} \)
 [38] 
Flux_{(s)} → slag_{(l)}

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 
\( \varepsilon_{\text{s}} = \xi_{\text{ore}} \varepsilon_{\text{ore}} + \xi_{\text{coke}} \varepsilon_{\text{coke}} \)

\( \begin{aligned} \phi_{\text{s}} = \left\{ \begin{aligned} \phi_{\text{ore}}\quad {\text{for ore layer}} \hfill \\ \phi_{\text{coke }} \quad{\text{for coke layer}} \hfill \\ \end{aligned} \right. \hfill \\ {\text{where\,}} \phi = \varepsilon ,d,k \hfill \\ \end{aligned} \)
 
Solid particle size 
\( d_{\text{s}} = \left( {{\frac{{\xi_{\text{ore}} }}{{d_{\text{ore}} }}} + {\frac{{\xi_{\text{coke}} }}{{d_{\text{coke}} }}}} \right)^{  1} \)
 
Solid heat conductivity 
\( k_{\text{s}} = \left( {{\frac{{\xi_{\text{ore}} }}{{k_{\text{ore}} }}} + {\frac{{\xi_{\text{coke}} }}{{k_{\text{coke}} }}}} \right)^{  1} \)
 
Vertical to the layer  
Gas flow resistance in CZ 
\( \alpha_{\text{f}} = a{\frac{{1  \varepsilon_{\text{s}} }}{{d_{\text{s}} }}} \)

\( \alpha_{\text{f}} = \xi_{\text{ore}} \alpha_{\text{ore}} + \xi_{\text{coke}} \alpha_{\text{coke}} \)
 For ore layer \( \alpha_{\text{f}} = \alpha_{\text{ore}} , \)
\( \beta_{\text{f}} = \beta_{\text{ore}} \)

\( \beta_{\text{f}} = \xi_{\text{ore}} \beta_{\text{ore}} + \xi_{\text{coke}} \beta_{\text{coke}} \)
 
Parallel to the layer  
\( \beta_{\text{f}} = b{\frac{{\mu_{g} \left( {1  \varepsilon_{\text{s}} } \right)^{2} }}{{d_{\text{s}}^{2} \varepsilon_{\text{s}} }}} \)

\( \alpha_{\text{f}} = \left( {{\frac{{\xi_{\text{ore}} }}{{\sqrt {\alpha_{\text{ore}} } }}} + {\frac{{\xi_{\text{coke}} }}{{\sqrt {\alpha_{\text{coke}} } }}}} \right)^{  2} \)
 For coke layer \( \alpha_{\text{f}} = \alpha_{\text{coke}} , \)
\( \beta_{\text{f}} = \beta_{\text{coke}} \)
 
\( \beta_{\text{f}} = \left( {{\frac{{\xi_{\text{ore}} }}{{ {\beta_{\text{ore}} } }}} + {\frac{{\xi_{\text{coke}} }}{{{\beta_{\text{coke}} } }}}} \right)^{  1} \)

Numerical Technique
Simulation Conditions
Variables  BF 

Gas:  
Volume flux, Nm^{3} × tHM^{−1}
 1511 
Inlet gas components, mole percentage  34.95 pct CO 
0.0 pct CO_{2}
 
0.81 pct H_{2}
 
0.0 pct H_{2}O  
64.23 pct N_{2}
 
Inlet gas temperature, K  2313.6 
Top pressure, atm  2.0 
Solid:  
Ore, t × tHM^{−1}
 1.64 
Ore components, mass fraction  Fe_{2}O_{3} 0.656 
FeO 0.157  
CaO 0.065  
MgO 0.024  
SiO_{2} 0.059  
Al_{2}O_{3} 0.029  
MnO 0.006  
P_{2}O_{5} 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  
SiO_{2} 0.024  
Al_{2}O_{3} 0.033  
CO_{2} in CaO 0.344  
CO_{2} in MgO 0.082  
Ore voidage  0.403(100d_{ore})^{0.14}

Coke voidage  0.153logd_{coke} + 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  
SiO_{2} 0.324  
Al_{2}O_{3} 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 