# A Self-Consistent Physical Model of the Bubbles in a Gas Solid Two-Phase Flow
## 1 Introduction
Through the optical experimental study, [PERSON] et al. found that bubbles begin to be produced after fluidization of the particles, and the interactions between the hydro-kinetic force and the particles can suppress the generation of bubbles for sufficiently large particles in the two-phase flow [12]. [PERSON] et al. studied the relationship between the bubbles and the clusters in the gas-solid two-phase flow and found that they are similar phenomenons in the s systems [4]. [PERSON] measured the bubble phenomenon in a two-phase system using a magnetic resonance imaging experiment in real time for the first time, and studied the bubble size, velocity, etc. in detail [13]. The nonlinear characteristics of the gas-solid two-phase flow and the relationship with bubbles were studied by [PERSON] et al. [14]. [PERSON] researched the nonlinear dynamic characteristics of the gas-solid system using the two phase theory [15]. [PERSON] and [PERSON] investigated the nonlinear dynamic characteristics of the fluidized states. Their results show that solitons, like waves, exist in the system and play an important role in the system. This indicates that the nonlinear characteristic has an important role and influence on the systems [16]. At the South Korean Energy Research Institute, the bubble properties of the gas-solid fluidized bed system were studied by [PERSON] et al. using electronic resistivity probe experiments, and a change in bubble size with height was obtained [17]. [PERSON] and [PERSON] established a simplified model to predict the average velocity of bubbles rising [18]. The bubble phenomenon represents one of the most basic physical phenomena of the gas-solid two-phase flow system, and relevant scholars worldwide have paid much attention to the research of bubbles in the gas-solid two-phase flow. It is noted that at present, studies mainly focus on the improvement and research of productive processes in the actual systems. However, basic theoretical research is relatively lacking [19]. The high-speed dynamic systems have been widely used to analyze the bubble behaviors of the gas-solid system in experiments. With limited experimental approaches, we can observe and analyze the behavior of the system at the boundary surface but we cannot go deep into the internal system for detailed analysis and research. Based on the experimental results, the relatively simplified models can be fitted out, and are usually based on fluid mechanics or continuum medium theory. The numerical simulation can also be used to analyze the motion characteristics of bubbles. Due to a certain gap between the numerical simulation models and the existing experimental conditions, the simulation results can only partly explain the experimental results. In this work, we hope to understand the the gas-solid two-phase flow system more from the point of view of physics. Using thermodynamics and fluid mechanics theory, we establish a new bubble physical model. The dynamic properties of bubbles in the gas-solid two-phase flow system as shown in Figure 1 are studied using the self-consistent theoretical method. Through establishing and researching the bubble thermodynamic equation, we examine the properties of bubbles in the gas-solid two-phase flow system on the basis of Ergun and bubble dynamics equations.