Introduction
Numerical models
Hydrodynamic models
Continuity equation for phase k (=c, d): | |
\( {\frac{\partial }{\partial t}}\left( {\alpha_{k} \rho_{k} } \right) + \nabla \cdot \left( {\alpha_{k} \rho_{k} {\mathbf{v}}_{k} } \right) = 0 \)
| (1) |
Momentum equation for phase k (=c, d): | |
\( {\frac{\partial }{\partial t}}\left( {\alpha_{k} \rho_{k} {\mathbf{v}}_{k} } \right) + \nabla \cdot \left( {\alpha_{k} \rho_{k} {\mathbf{v}}_{k} {\mathbf{v}}_{k} } \right) = - \alpha_{k} \nabla p + \nabla \cdot \bar{\tau }_{k} + \alpha_{k} \rho_{k} {\mathbf{g}} + {\mathbf{M}}_{k} \)
| (2) |
Granular temperature equation: | |
\( \frac{3}{2}\left[ {{\frac{\partial }{\partial t}}\left( {\alpha_{\text{d}} \rho_{\text{d}} \Theta } \right) + \nabla \cdot \left( {\alpha_{\text{d}} \rho_{\text{d}} {\mathbf{v}}_{\text{d}} \Theta } \right)} \right] = \bar{\tau }_{\text{d}} :\nabla {\mathbf{v}}_{\text{d}} + \nabla \cdot \left( {\kappa_{\text{d}} \nabla \Theta } \right) - \gamma - 3\beta \Theta + \beta \left\langle {{\mathbf{\tilde{v}^{\prime}}}_{\text{c}} \cdot {\mathbf{C}}_{\text{d}} } \right\rangle + \frac{3}{2}\Gamma_{k} \Theta \)
| (3) |
Molecular temperature equation for phase c and d: | |
\( \alpha_{\text{c}} \rho_{\text{c}} C_{{{\text{p}},{\text{c}}}} {\frac{{{\text{D}}T_{\text{c}} }}{{{\text{D}}t}}} = \nabla \cdot \left( {\alpha_{\text{c}} k_{\text{c}}^{\text{eff}} \nabla T_{\text{c}} } \right) + \sum\limits_{i} {\left( { - \Updelta H_{i}^{\text{SMR}} } \right)R_{i}^{\text{SMR}} } + Q_{\text{c}}^{\text{i}} \)
| (4) |
\( \alpha_{\text{d}} \rho_{\text{d}} C_{\text{p,d}} {\frac{{{\text{D}}T_{\text{d}} }}{{{\text{D}}t}}} = \nabla \cdot \left( {\alpha_{\text{d}} k_{\text{d}}^{\text{eff}} \nabla T_{\text{d}} } \right) + \left( { - \Updelta H^{\text{SP}} } \right)R^{\text{SP}} - Q_{\text{c}}^{\text{i}} \)
| (5) |
Species composition: | |
\( {\frac{\partial }{\partial t}}\left( {\alpha_{\text{c}} \rho_{\text{c}} \varpi_{{{\text{c}},j}} } \right) + \nabla \cdot \left( {\alpha_{\text{c}} \rho_{\text{c}} {\mathbf{v}}_{\text{c}} \varpi_{{{\text{c}},j}} } \right) = \nabla \cdot \left( {\alpha_{\text{c}} \rho_{\text{c}} D_{{{\text{c}},j}}^{\text{eff}} \nabla \varpi_{{{\text{c}},j}} } \right) + M_{j} R_{j}^{i} \)
| (6) |
Kinetic models
Results and discussion
Hydrodynamic flow regimes
Particle diameter | 500 μm |
Particle density | 1,500 kg/m3
|
Sorbent-to-catalyst mass ratio | 1:4 |
Reactor size |
R = 0.1 m, H
t
= 4 m |
Initial bed height | 2 m |
Grid cell number | 12 × 12 × 80 |
Time step | 1 × 10−4 s |
Convergence criterion |
\( \left\| {{\mathbf{r}}_{m} } \right\| < \varepsilon \left\| {\mathbf{b}} \right\|\quad \varepsilon = 10^{ - 10} \)
|