## 1 Introduction
[13] argued that replacing the number 33.7 with 27.2 in Equation (2) is most suitable for the gas-solid system. [PERSON]. [14] et al. also did an in-depth discussion of [PERSON] and [PERSON]'s equation. For a conical fluidized bed, the original Ergun equation cannot be directly used because of the gas velocity gradient in the axial direction, which makes the pressure drop rate vary in the axial direction. To solve this problem, [PERSON] and [PERSON] [15] developed a model to calculate \(\Delta P\) of a conical liquid-solid fluidized bed by integrating the pressure drop rates of different layers. [PERSON] et al. [16; 17] proposed a correlation based on the Ergun equation to calculate \(U_{0\rm mf}\) and \(\Delta P_{\rm mf}\) of a conical fluidized bed using a mathematical processing method similar to [PERSON] and [PERSON]'s model. [PERSON] et al. [18] used [PERSON] and [PERSON]'s model to predict \(U_{0\rm mf}\) and \(\Delta P_{\rm mf}\) for air-sand conical beds, and the effects of the sand particle size, cone angle, and static bed height on the flow behavior were discussed. [PERSON] et al. [19] put forward a method of modifying the coefficients \(C_{1}/C_{2}\) in Equation (1) to increase the prediction accuracy, which is adopted and improved in this paper. In this study, we use regression analysis to try to obtain a better combination of \(C_{1}\) and \(C_{2}\) that is more suitable for the working conditions of this study. Dimensional analysis is a widely used modeling method that only pays attention to the mathematical relationships between independent variables and dependent variables while ignoring the physical laws between physical parameters. A dimensionless model has a unified and straightforward equation form that is composed of a series of dimensionless groups. The general equation form is as follows:  
\[\pi_{0}=\pi_{1}^{p_{1}}\cdot\pi_{2}^{p_{2}}\cdot\pi_{3}^{p_{3}}\cdots\pi_{n}^{P_{ n}} \tag{3}\]  
where \(\pi_{0}\), \(\pi_{1}\),..., \(\pi_{n}\) are all dimensionless groups and \(P_{1}\), \(P_{2}\),..., \(P_{n}\) are the exponents of these dimensionless groups. Since the dimension of a dimensionless group is 1, no matter what the exponents are, the equation can satisfy the principle of dimensional harmony. The main difficulty of this method is determining the dimensionless groups, which requires knowing which factors affect the dependent variables. The value of each component can be determined by linear regression or nonlinear regression of the experimental data. Based on selecting appropriate dimensionless groups, a dimensionless model generally achieves higher accuracy under specific working conditions. [PERSON] et al. [20; 21; 22] used the dimensional analysis method to develop modeling correlations for \(U_{0\text{mf}}\), \(\Delta P_{\text{mf}}\), and the bed expansion ratio of conical beds for different kinds of fluidization materials. [PERSON] [23] proposed a dimensionless model using the term \(\cos(\alpha)\) instead of \(\sin(\alpha)\) compared with [PERSON]'s model [20], considering that \(\sin(\alpha)\) approaches zero for mini-tapered beds, which may lead to an incorrect prediction result. [PERSON] et al. [24] thoroughly considered the effect of PSD (particle size distribution) on \(U_{0\text{mf}}\) and proposed different correlations for different PSDs (Gaussian/flat/binary). Because the dimensional analysis method has strong applicability, it has been used by many scholars to derive modeling correlations in the field of fluidized bed [25; 26; 27; 28; 29; 30; 31]. Some of the correlations proposed by other scholars are listed in Table 1.  
However, in the literature related to the modeling of \(U_{0\text{mf}}\) and \(\Delta P_{\text{mf}}\), most of the research equipment that was used was positive pressure cylindrical fluidized beds. Due to the differences in the gas path structure and the fluidization chamber shape, the previous research results cannot be directly applied to negative pressure conical fluidized beds.