将下面进行学术翻译：利用局部颗粒浓度的概率密度分布对变径段和等径段内固相分布进行定量比较分析有助于进一步认识变径提升管内气固两相流动复杂机制。如图13所示在扩径段和等径段内随着颗粒密度的增加颗粒浓度概率密度曲线均向高浓度区移动表明变径管内固体浓度随密度增加而增大的变化规律是一致的。不同的是扩径段内颗粒浓度分布随密度的变化幅度比在等径段内小说明等径段内固体浓度分布对颗粒密度的敏感性更高。而且沿边壁向

Utilizing the probability density distribution of local particle concentration, quantitative comparison and analysis of the solid-phase distribution in the variable-section and constant-section segments can help to further understand the complex mechanism of gas-solid two-phase flow in the riser. As shown in Figure 13, in the expanding and constant sections, the probability density curves of particle concentration move towards the high concentration region as the particle density increases, indicating that the variation law of solid concentration in the riser increases with the increase of particle density. However, the change in the distribution of particle concentration with density in the expanding section is smaller than that in the constant section, indicating that the sensitivity of solid concentration distribution to particle density is higher in the constant section. Moreover, the difference in probability density distribution between the two types of particles gradually decreases from the wall to the center in the expanding section, while the difference in distribution curves at each radial position in the constant section changes relatively small. This may be due to the larger diameter of the expanding section compared to the constant section, which prolongs the radial movement of particles and increases the range of radial changes in particle concentration.

For low-density FCC particles, there is no significant change in the probability density distribution at the same radial position in both the expanding and constant sections, while for high-density glass bead particles, the magnitude of the change in the probability density distribution is larger, showing an obvious characteristic of moving towards the high concentration region compared to FCC particles. In other words, under high-density particle conditions, the solid-phase concentration at the height of H2 in the expanding section is lower than that in the constant section, and the difference in concentration distribution between the constant section and the expanding section increases with the increase of particle density. This phenomenon reflects that the outlet type has a significant impact on the gas-solid flow in the riser. In this study, the constrained outlet hindered particles from leaving the riser in time, causing particle aggregation near the outlet, which may be the main reason for the increase in concentration in the constant section. Moreover, this situation is aggravated with the increase of particle density, resulting in higher concentration in the variable-section than in the expanding section. However, the concentration at the middle height of the expanding section shown in Figure 13 does not mean that the overall particle concentration in the expanding section is lower than that in the constant section, considering that particle accumulation mainly occurs at the bottom of the expanding section.