# A Self-Consistent Physical Model of the Bubbles in a Gas Solid Two-Phase Flow
## 1 Introduction  
Gas solid two-phase flow is a complex flow system composed of gas and solid particles, and represents an important branch of fluid dynamics. Gas solid two-phase flows exist widely in nature and industrial productions. In this context, gas solid two-phase fluidization separation, for example, has been extensively researched and developed, with a wide range of applications and high selection efficiency. It is an efficient separation method that has been gradually applied to industrial production [1; 2]. With gas article fluidization separation, the system is filled with uniform air flows in the fine particulate matter media bed, which creates particle medium fluidization and forms the gas solid suspended matter with certain density and fluid properties. The theoretical analysis of the two-phase flow is much more difficult than for single-phase flow because the general differential equations describing the two-phase flow have not yet been established. Generally, a simplified two-phase theoretical model can be applied in such systems. One of the models considers that the system is a continuous medium which is a mixture of two phases, where the concepts and methods of single-phase flow are still suitable for two-phase flow. The other is called the separated model, which considers that the concept and method of the single-phase flow can be respectively used in each phase of the two-phase system, while the interaction between the two phases is included [3; 4]. The second type of model has been widely associated with physical applications [5]. In the gas solid two-phase flows, the mixture of the gas and the particles is not uniform and time-dependent, with the two phases existing in the gas solid two-phase flows (see Figure 1). One is a continuous phase, composed of gas and particles, and is called the dense phase or emulsion phase; the other is a discontinuous phase appearing in the bubble state, and is called the bubble phase. When the quantity of gas flowing in the gas solid system is greater than the air flow required for the critical fluidized state, the excess gas can be in the form of bubbles. Bubble dynamic behavior is responsible for the most basic characteristics and phenomena in the gas solid two-phase flow system. This is significant for the understanding and study of the properties of gas solid two-phase flow systems [4]. The study of bubbles in gas solid two-phase flow forms an important part of the field of fluidization state [6]. The pressure drop, height, swelling degree, uniform mixing of the gas-solid two-phase flow, and stratified separation of particles in the gas-solid two-phase system strongly depend on the bubbles. The bubble size and rising velocity are important factors which affect the fluidization stability, the dynamics of particles, and the separation densities of fluidized beds [7]. The bubble phase and the dense phase transmit the heat and mass to each other, and then the gas-solid two-phase flows are not uniform. The growth of bubbles in the gas-solid two-phase flow can seriously affect the uniform stability of fluidization, affect the distribution of solid phase particles, and affect the state of the whole two-phase flow. As a result, research into bubble properties in the gas-solid two-phase flow system represent some of the most important and basic studies in the field [8]. [PERSON] et al. developed a statistical theory to study the properties of bubbles in gas-solid systems using capacitance probe measurement experiments, and obtained consistent results between the experiments and theories [9]. [PERSON] et al. studied the bubble phase and slugging phase in a fluidized bed by numerical simulations. It was found that the convective motion reveals a stronger effect on the bubble phase than on the slugging phase [10]. [PERSON] et al. calculated the size of bubbles in the gas-solid fluidized bed, and found that the Mori-Wen-Rowe equation is a better choice for the study of bubble size, as compared with the other theories [11].