将下面两段话进行学术翻译：为了深入了解颗粒密度对变径管内气固流动的影响本节通过快速傅里叶变换获得功率谱分析颗粒浓度数据包含的气固混合流动的周期性频率特征。图7a绘制了FCC颗粒在同一表观气速、不同循环通量操作条件下扩径段H2高度上各个径向位置的颗粒浓度信号功率谱密度分布图。可以观察到不同循环通量下浓度信号的功率谱分布均呈现相似的分布趋势具体表现为在低频处峰值较高随着频率的增加功率谱密度值逐渐衰减并

In order to investigate the influence of particle density on gas-solid flow in a variable diameter pipe, this section employs the fast Fourier transform to obtain power spectra and analyze the periodic frequency characteristics of the gas-solid mixture flow contained in the particle concentration data. Figure 7a shows the power spectral density distribution of the particle concentration signal at various radial positions on the height of section H2 in the expansion section under the same apparent gas velocity but different circulation flux operating conditions for FCC particles. It can be observed that the power spectral distributions of the concentration signal under different circulation fluxes exhibit similar distribution trends, which are characterized by higher peak values at low frequencies and gradually decreasing power spectral density values with increasing frequency until they tend to zero. Moreover, the power fluctuations in the concentration signal are mainly concentrated in the frequency range of 0-0.1 Hz. The power spectral amplitude decreases gradually from the pipe wall towards the pipe center, indicating that the periodic motion of particles in the wall region is higher than that in the center region. However, the main frequency position shifts towards high frequencies, which is caused by stronger gas turbulence in the center region. With the increase of circulation flux, the main peak becomes more prominent, the main frequency increases, and the power spectral amplitude gradually increases, with a more pronounced increase in the low-frequency region than in the high-frequency region. This is because with the increase of flux, the circulation motion of particles inside the pipe gradually strengthens, and the periodic flow structure in the expansion section increases, leading to an enhancement of the frequency characteristics in the power spectrum.

As shown in Figure 7b, the power spectral density distribution of glass beads changes with the circulation flux in a similar way to that of FCC particles, and the distribution characteristics of the power spectral density of particle concentration at various radial positions from the wall to the center during the process of moving inward are also similar to those of FCC particles. However, compared with FCC particles, the power spectrum of glass beads changes more significantly with the circulation flux. This is because under the same solid circulation flux conditions, the particle concentration change caused by glass beads is larger than that of FCC particles in the expansion section, leading to a more significant change in flow structure. In the wall and transition regions, the main frequency peak of the power spectrum of glass beads is significantly higher than that of FCC particles, and the frequency range of power spectral fluctuations is wider than that of FCC particles, indicating that the periodic motion of glass beads in the expansion section is more pronounced. This difference increases with the increase of circulation flux. On the contrary, in the center region of the pipe, the main frequency and amplitude of glass beads are lower than those of FCC particles, and the frequency range of power spectral fluctuations is narrower than that of FCC particles. These results indicate that the periodic flow structures of gas-solid two-phase flow in the center and wall regions of the expansion section exhibit opposite trends under the influence of particle density properties. When the apparent gas velocity is constant, the aggregation of glass beads with higher particle density in the wall region increases, and the periodic fluctuations caused by the clustering structure of particles are more pronounced than those of FCC particles, resulting in an increase in the amplitude in the low-frequency region of the power spectral diagram. On the other hand, the movement intensity of glass beads particles in the same height of the pipe center is reduced due to the smaller gas-solid interaction, and the corresponding spectral energy is decreased.