# A Self-Consistent Physical Model of the Bubbles in a Gas Solid Two-Phase Flow
## 3 Results and Discussion  
Equations (8) and (11)-(16), form self-consistent systems of equations. For a given \(Z\), initial pressure \(P_{0}\), and initial volume \(V_{0}\), the gas velocity \(u\), bubble size \(d_{b}(Z)\), and the change law of the bubble rising velocity \(u_{b}(Z)\) are simulated through the self-consistent calculations in the gas-solidtwo-phase flow systems as shown in Figure 1. Figure 2 shows the heat capacity ratio \(\gamma\) of the bubble systems in the gas-solid two-phase flow as the function of the pressure \(P\) for the different temperatures \(T\). It can be seen from Figure 2 that the heat capacity ratio of the bubbles increase with the increasing of pressure \(P\) at a fixed temperature \(T\). It shows that our numerical result is obviously different from that in the ideal gas systems, which is a function of the temperature and the pressure for our mode. The heat capacity ratio in ideal gas system is a constant (1.4) and the value does not change with the state parameters of the systems. In the past, when studying the characteristics of bubbles, the heat capacity ratio has usually been regarded as the ideal gas constant [23]. We obtain the expression of specific heat capacity of the gas-solid two-phase flow system through establishing a non-ideal gas model. With the increasing temperature, the heat capacity ratio also increases clearly. Meanwhile, it can be clearly seen that Equation (16) is a complicated exponential root and the index variety can lead to the fundamental changes of the equation. Moreover, the small changes of the heat capacity ratio also lead to extensive changes of \(u\), \(d_{z}\), and \(u_{b}\). As a result, the bubble system heat capacity \(\gamma\) indicates an important effect on the properties of the overall bubbles in the gas-solid two-phase flow. In this paper, three different parameters for solid particles are selected respectively in order to compare them with each other. These parameters are black triangles (\(\phi_{s}=0.8\), \(\rho_{p}=1.3\times 10^{3}\) kg/cm\({}^{3}\), \(d_{p}=0.03\) m), black squares (\(\phi_{s}=0.8\), \(\rho_{p}=1.4\times 10^{3}\) kg/cm\({}^{3}\), \(d_{p}=0.02\) m) and black dots (\(\phi_{s}=0.8\), \(\rho_{p}=1.5\times 10^{3}\) kg/cm\({}^{3}\), \(d_{p}=0.15\) m) as shown in Figures 3-5. In Figure 3, the air velocity \(u\) as the function of \(Z\) in three sets of parameters is obtained through the self-consistent calculations in the gas-solid two-phase flow. As can be seen in Figure 3, with the increase of the height \(Z\), the air flow rate \(u\) in the gas-solid two-phase flow shows a downward trend, with a smaller and smaller decline rate with increasing \(Z\), gradually stabilizing and reaching a stable airflow state. This shows that the smaller the particle density, the greater the air flow rate in the gas-solid two-phase flow. The greater the diameter of solid particles, the greater the airflow velocity at the same height. The physical reason behind this is that the large particle diameter results in a large void ratio \(\varepsilon\), which leads to a large flow velocity. Due to the differences in the particle size, the pressure, and the temperature in the gas-solid two-phase flow, the whole system becomes very different and the environment around the bubbles also becomes very different. As shown in Figure 2, this effect is reflected in the specific heat capacity of the bubbles, which changes significantly, resulting in changes of gas speed and solid speed in the whole gas-solid two-phase flow. It is clear that the air velocity gradually stabilizes, which is beneficial  
Figure 2: Heat capacity ratio versus the pressure for the different temperatures. to the stability of the gas-solid two-phase flow. The results indicate that such a system can be used as a medium for particle separations. Figure 4 shows the bubble diameter \(d_{b}\) as the function of the gas flow height \(Z\) by the self-consistent calculations in the gas-solid two-phase flow.