# A Self-Consistent Physical Model of the Bubbles in a Gas Solid Two-Phase Flow
## 3 Results and Discussion
It is shown that with increasing the height \(Z\), bubble sizes show a variation of decreasing and then increasing, just like the \(U\) type, in the gas-solid two-phase flow. This indicates that the gas velocity, the pressure, the particle density, and the particle diameter are combined together and collectively influence the diameter of bubbles in the gas-solid two-phase flow. When the bubbles rise with the air flow, the external pressure decreases, since the external airflow rate is relatively large. At the same time, the solid particles suppress the increase of bubble volume, resulting in the bubble sizes being smaller in the initial stage. With increasing height \(Z\), the external air velocity, solid phase particle density, particle diameter, spherical degree, and other factors cannot suppress the increase of air bubbles because the pressure \(P\) is so weak. At this time, the reduction of external pressure plays a key role. Consequently, the bubble sizes increase with the increase of the rising speed. It is found that the bubble phase depends strongly on the solid phase in the gas-solid two-phase flow. The greater the density of solid phase particles, the smaller the bubble sizes in the two-phase flow, which reveals that the high density of the particles can inhibit the increasing of the bubble sizes. Meanwhile, the particle size is smaller, and the bubble size is smaller at the same height \(Z\). This is because the particle size and void fraction are smaller in the gas-solid flow, which can limit the growth of the bubble. We find that our theoretical results can be compared with the recently reported results. It is experimentally found that the size of gas bubble increases gradually from the bottom of the systems (see e.g., Figure 7a of Ref. [29]). [PERSON] et al. have found that the bubble size enlarges by increasing the height position base of the numerical simulations (see e.g., Figure 10 of Ref. [30]). However, we should notice that a local minimum of the bubble sizes is observed for our self-consistent physical model, which are not demonstrated experimentally at present. Our theoretical results are in agreement with the reported results after the local minimum sizes. Moreover, we hope our theoretical findings can be verified by experiments. Figure 3: The speed of the flow versus the height \(Z\) for the different conditions. Figure 5 shows the change of the bubble rising velocity \(u_{b}\) with the height \(Z\) by the self-consistent calculations in the gas-solid two-phase flow. In Figure 4, the velocity of bubbles \(u_{b}\) is very different from the outside air velocity \(u\) in Figure 3 in the gas-solid two-phase flow. It is obvious that the bubble velocity is much faster than the air flow velocity. With the increasing of the height \(Z\), the bubble velocity increases and tends to be quasi-stable in gas-solid two-phase flows. In the gas-solid two-phase flow, the smaller the particle density, the larger the bubble velocity is. The diameter of solid particles is larger, and the bubble velocity is greater at the same height. This is because particle diameter becomes larger on increasing the void fractions in the two-phase flow, which gives rise to the large bubble phase velocity. The bubble velocity is influenced by the air velocity and the bubble size. Therefore, the properties of the bubble velocity are very similar to those of the gas flow velocity in gas-solid two-phase flow. Moreover, the bubble size firstly decreases and then increases. With the effect of the both aspects, the velocity of the bubble gradually increases and eventually stabilizes. We should also point out that the gradual stabilization of the bubble velocity is beneficial to the stability of gas-solid two-phase flow systems and the separations of different particles in this system. The tracer testing technology has been widely applied to study the properties of bubbles in multi-phase flows.