## 1 Introduction  
Fluidization is understood as a process where an upward flow of gas or liquid (a liquefying medium, hereinafter referred to as a medium) suspends a bed of solid particles of various chemical compositions, most often catalyst particles (hereinafter referred to as particles or a particle bed) which brings them into a mobile or \"liquefied\" state [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The particle bed in this state has some liquid properties: particles move freely relative to each other; objects placed in the particle bed can sink or float depending on their specific gravity; the surface of the particle bed always takes a horizontal position; particles flow out as liquid if there is any side hole in the reactor wall, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Figure 1 shows the dependence of the fluidization regime on the velocity of the gaseous fluidizing agent. The particle bed passes from the compacted state (Figure 1A) to the fluidized state (in Figure 1, it is an intermediate position between A and C) when the drag force and the volumetric forces affecting the particles from the side of the medium flow exceed the gravitational force of the particles [1, 12, 13, 14, 15, 16]. The ratio of the flow rate to the cross-sectional area of the reactor where fluidization occurs is called the critical or minimum fluidization velocity \(U_{mf}\). It depends on the properties of the particles and the liquefying medium, but does not depend on the height of the particle bed [6]:  
\[U_{mf}=\frac{\varepsilon_{mf}^{3}}{5\Big{(}1-\varepsilon_{mf}\Big{)}}\frac{( \rho_{p}-\rho)g}{S^{2}\mu}, \tag{1}\]  
where \(\varepsilon_{mf}\) is the porosity of the particle bed at the fluidization point; \(\rho_{p}\) is the particle density; \(\rho\) is the density of the medium; \(g\) is the acceleration of gravity; \(S\) is the area of the horizontal section of the particle bed; and \(\mu\) is the dynamic viscosity of the medium. This dependence is empirical, and is used for a preliminary assessment of \(U_{mf}\) in the absence of experimental data. The error is 15 to 20% on average [6]. When the fluidization velocity \(U_{mf}\) is reached, the difference in the pressure of the medium before and after the particle bed, proportional to the weight of all particles, ceases to change [1, 12, 13, 14, 15, 16]. The behavior of the particle bed with a further increase in the velocity of the medium depends on the properties of the particles and the nature of the medium [1, 12, 13, 14, 15, 16]. If the medium is a liquid, the particle bed will expand uniformly  
Figure 1: Dependence of the fluidization regime on the relative gas velocity and on the volume fraction of particles. Relative gas velocity is the ratio of the gas velocity to the minimum fluidization velocity. Vertical dashed lines indicate the relative gas velocities range in which one or another fluidization regime exists. Schematic images of fluidization regimes are located near their corresponding ranges of relative gas velocities. until a certain porosity threshold of the \(\varepsilon_{p}\) bed is reached, which is called homogeneous fluidization [1; 12; 13; 14; 15; 16]. A further increase in the velocity of the medium will lead to the gradual removal of all particles from the reactor; therefore, no other fluidization regimes occur during liquid fluidization [1; 12; 13; 14; 15; 16]. If the medium is gas, the behavior of the particle bed is determined by how much the minimum fluidization velocity is exceeded, and what the density and average diameter of the particles are [2; 3]. Regardless of these conditions, it will not be possible to achieve uniform fluidization--gas bubbles will begin to form in the particle bed; therefore, such fluidization is called inhomogeneous (Figure 1B), and the fluidization becomes bubbling fluidization (Figure 1C) [3; 4; 5; 9; 10; 11].