Appendix
Particle Reynolds number (\(\text{Re}_{p}\)) was calculated by:
\(\text{Re}_{p}=\frac{U_{g}d_{\text{ave}}\rho_{g}}{\mu}\) (A1)
where \(U_{g}\) is the superficial gas velocity, \(d_{\text{ave}}\) is the average particle diameter, \(\rho_{g}\) is the gas density and\(\mu\) is the gas viscosity.
Relative Reynolds number (\(\text{Re}_{i}\)) was calculated by:
\(\text{Re}_{i}=\frac{{(U}_{g}-U_{t})d_{\text{ave}}\rho_{g}}{\mu}\)(A2)
where \(U_{t}\) is the terminal velocity of the particle.
Reynolds number based on terminal velocity (\(\text{Re}_{i}\)) was calculated by:
\(\text{Re}_{t}=\frac{U_{t}d_{\text{ave}}\rho_{g}}{\mu}\) (A3)
Terminal velocity (\(U_{t}\)) was calculated by:
\(U_{t}=\frac{U_{t}^{*}}{\left(\frac{\rho_{g}^{2}}{\mu(\rho_{p}-\rho_{g})g}\right)^{\frac{1}{3}}}\)(A4)
where
\(U_{t}^{*}=\left(\frac{18}{{d_{\text{ave}}^{*}}^{2}}+\frac{2.335-1.744\varphi}{{d_{\text{ave}}^{*}}^{0.5}}\right)^{-1}\)(A5)
where \(\varphi\) is particle sphericity and \(d_{\text{ave}}^{*}\) was calculated by:
\(d_{\text{ave}}^{*}=d_{\text{ave}}\left(\rho_{g}(\rho_{p}-\rho_{g})\frac{g}{\mu^{2}}\right)^{\frac{1}{3}}\)(A6)
where g is gravitational acceleration.
Archimedes number (Ar) was calculated by:
\(\text{Ar}=\frac{gd_{\text{ave}}^{3}\rho_{g}(\rho_{p}-\rho_{g})}{\mu^{2}}\)(A7)
Minimum fluidization velocity (\(U_{\text{mf}}\)) was calculated by:
\(K_{1}\text{Re}_{p,mf}^{2}+K_{2}\text{Re}_{p,mf}=\text{Ar}\) (A8)
where \(\text{Re}_{p,mf}\) is the particle Reynolds number when\(U_{g}\)=\(U_{\text{mf}}\), and \(K_{1}\) and \(K_{2}\) were calculated by:
\(K_{1}=\frac{1.75}{\varepsilon_{\text{mf}}^{3}\varphi}\) (A9)\(K_{2}=\frac{150(1-\varepsilon_{\text{mf}})}{\varepsilon_{\text{mf}}^{3}\varphi^{2}}\)(A10)
where \(\varepsilon_{\text{mf}}\) is volumetric bed voidage when\(U_{g}\)=\(U_{\text{mf}}\).
The Froude number (Fr) was calculated by:
\(\text{Fr}=\frac{{{(U}_{g}-U_{t})}^{2}}{gd_{\text{ave}}}\) (A11)