# A Self-Consistent Physical Model of the Bubbles in a Gas Solid Two-Phase Flow
## 2 Theoretical Model
### Gas Velocity Equation in the Gas-Solid Two-Phase Flow System  
In the gas-solid two-phase flow system, the bubbles slowly rise from the bottom of the whole system as shown in Figure 1. This is similar to the motions of bubbles in the water [23]. Consequently, our research focuses on the characteristics of bubbles in the direction \(Z\) of the motion with a steady state in the X-Y plane. Based on the results of the previous results and the experiments, [PERSON] obtained the comprehensive expression that the pressure drop \(\Delta P\) of the gas-solid system decreases with the change of \(Z\), which reads  
\[\frac{\Delta P}{Z}=150\frac{(1-\varepsilon)^{2}\mu_{g}u}{\varepsilon^{3}(\phi _{s}d_{p})^{2}}+1.75\frac{(1-\varepsilon)\rho_{f}u^{2}}{\varepsilon^{3}\phi_{ s}d_{p}}. \tag{9}\]  
\(\phi_{s}\) is the spherical degree of solid particles. The first item of the Equation (9) at the right is the viscosity term, and it takes the leading role when the flow rate is low. The second item of Equation (9) at the right is the inertia term, which plays a major role when the flow rate is higher and the flow is turbulent. \(\varepsilon\) is the void content in the gas-solid two-phase flow which represents the proportion of space occupied by the gas and the space occupied by the system. \(u\) is the gas velocity of the gas-solid two-phase flow. \(d_{p}\) is the solid particle diameter. \(\rho_{f}\) is the density of the gas-solid flow. \(\mu_{g}\) is the viscosity of this flow. The critical fluidization velocity \(u_{f}\) is the fluidized velocity where the gas-solid flow pressure drop \(\Delta P\) is equal to the weight of the solid particles, which can be derived from the Ergun equation for the minimum fluidization conditions. The Ergun Equation (9) can be written as  
\[1.75\frac{Re_{f}^{2}}{\phi_{s}\varepsilon^{3}}+150\frac{1-\varepsilon}{\phi_{ s}^{2}\varepsilon^{3}}Re_{f}=A_{r}. \tag{10}\]  
Here, \(Re_{f}=d_{p}u_{f}\rho_{f}/\mu_{g}\), and \(A_{r}=d_{p}^{3}\rho_{f}(\rho_{p}-\rho_{f})g/\mu_{g}^{2}\). As a result, we can obtain the equation, which is  
\[\frac{(1-\varepsilon)\rho_{f}d_{p}u_{f}}{\varepsilon^{3}\phi_{s}^{2}\mu_{g}}+ \frac{1.75\rho_{f}^{2}d_{p}^{2}u_{f}^{2}}{150\phi_{s}\varepsilon^{3}\mu_{g}^{ 2}}=\frac{\rho_{f}d_{p}^{3}(\rho_{p}-\rho_{f})g}{150\mu_{g}^{2}}, \tag{11}\]  
The critical fluidization velocity is obtained as  
\[u_{f}=\frac{\mu_{g}}{\rho_{f}d_{p}}[\sqrt{1.84\times 10^{3}\frac{(1-\varepsilon )^{2}}{\phi_{s}^{2}}+\frac{\phi_{s}\varepsilon^{3}}{1.75}\frac{gd_{p}^{3}\rho_ {f}}{\mu_{g}^{2}}(\rho_{p}-\rho_{f})}-42.85\frac{1-\varepsilon}{\phi_{s}}], \tag{12}\]where \(g\) is the acceleration of gravity, and \(\rho_{p}\) is the solid particle density. The relationship of the solid particle density \(\rho_{p}\), the density of the entire system \(\rho_{f}\), and the gas/air density \(\rho_{0}\) is \(\rho_{f}=(1-\varepsilon)(\rho_{p}-\rho_{0})\). There are several drag models proposed based on analytical [24,25], experimental [26], and numerical [27] approaches. In the current study we use [PERSON]'s model [28] which has been widely used in fluidized beds investigations. Moreover, the average bubble diameter of the gas-solid two-phase flow in the [PERSON] and Wen models can be expressed as a function of the bubble rise height \(Z\), with the maximum bubble diameter of \(d_{bm}=1.64[(u-u_{f})]^{0.4}\). In the whole two-phase flow system, bubbles rise slowly from low to high, and the size diameter rising along the height of \(Z\) is [28]  
\[d_{b}(Z)=0.54(u-u_{f})^{0.4}(Z+4\sqrt{A_{0}})^{0.8}g^{-0.2}, \tag{13}\]  
\(A_{0}\) is the average area of each circular pinhole.