上海理工大学学报  2020, Vol. 42 Issue (5): 453-459 PDF

Measurement of velocity and angle distribution in particle separation process
TAO Ziyin, LI Ran, CHEN Quan, XIU Wenzheng, HUA Yunsong, YANG Hui
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: Inspired by the mixing entropy of the particle system, a measuring device based on the plane array CCD camera was built in order to carry out a series of measurements for the particle system composed of spherical particles with mean particle sizes of 2.6 mm and 1.25 mm respectively. The average speed and the angle of slope on the surface of the particle system were calculated through the pyramid-layered optical flow algorithm and the image method. The following conclusions are drawn. The subtle changes in the particle separation process can be described by the total entropy of the particle system. The average velocity of the flow layer on the surface of the particle system is affected by the angle of slope, and those two show a direct proportional relationship. Within the rotation speed range where the movement state of mixed particles is continuous avalanche, the global entropy of particle system, the average velocity of surface flow layer and the angle of slope have similar trend, indicating that the separation degree of particle system can be judged by the aid of the average speed and the angle of slope.
Key words: pyramid-layered optical flow algorithm     particle flow     velocity distribution     mixing entropy

1 实验方法 1.1 实验装置

 图 1 二维滚筒实验装置图 Fig. 1 Experimental device diagram for two-dimensional drum

 图 2 表面流动层的区域划分示意图 Fig. 2 Schematic diagram of the regional division on the surface flow layer
1.2 流动层颗粒的速度测量算法

Harris角点检测算法是一种直接基于灰度图像的角点检测算法，其主要原理是对于强度信息为 $I\left(x,y\right)$ 的图像，在像素点 $\left(x,y\right)$ 处将一个局部的小窗口 $w\left(x,y\right)$ 沿任意方向移动一段微小位移 $\left(u,v\right)$ $w(x,y)$ 为滑动窗口函数，通过引入灰度变化函数 $E\left(x,y\right)$ 来表征图像的自相关性。Harris对于灰度变化函数 $E\left(x,y\right)$ 的定义如下：

 $E\left(x,y\right)=\sum \limits_{x,y}w\left(x,y\right){\left[I\left(x+u,y+v\right)-I\left(x,y\right)\right]}^{2}$ (1)

 $I\left(x+u,y+v\right)=I\left(x,y\right)+{I}_{x}u+{I}_{y}v+O\left({u}^{2},{v}^{2}\right)$ (2)

$O\left({u}^{2},{v}^{2}\right)$ 表示泰勒展开式中的高阶项。忽略高阶无穷小项，可将 $E\left(x,y\right)$ 化简为二次型：

 $E\left(x,y\right)=\left[u,v\right]{{M}}\left[\begin{array}{c} u \\ v \end{array}\right]$ (3)

 ${{M}} = \sum\limits_{x,y} w \left( {x,y} \right)\left[ {\begin{array}{*{20}{c}} {{I_x}^2}&{{I_x}{I_y}}\\ {{I_x}{I_y}}&{{I_y}^2} \end{array}} \right]$ (4)

 $CRF={\rm{Det}}\left({{M}}\right)-k{{\rm{T}}{\rm{r}}\left({{M}}\right)}^{2}$ (5)

 图 3 Shi-Tomasi角点检测算法处理混合颗粒体系示意图 Fig. 3 Schematic diagram of Shi-Tomasi corner detection algorithm for processing the mixed particle system

 $I\left(x+\Delta x,y+\Delta y,t+\Delta t\right)= I\left(x,y,t\right)$ (6)

 $\frac{\partial I}{\partial x}\frac{{\rm{d}}x}{{\rm{d}}t}+\frac{\partial I}{\partial y}\frac{{\rm{d}}y}{{\rm{d}}t}+\frac{\partial I}{\partial t}=0$ (7)

$u=\dfrac{{\rm{d}}x}{{\rm{d}}t},\;v=\dfrac{{\rm{d}}y}{{\rm{d}}t}$ ，可以得到光流的基本约束方程为

 ${I}_{x}u+{I}_{y}v+{I}_{t}=0$ (8)

 $\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{I_{x1}}}&{{I_{y1}}}\\ {{I_{x2}}}&{{I_{y2}}}\\ \vdots & \vdots \end{array}}\\ {\begin{array}{*{20}{c}} {{I_{x9}}}&{{I_{y9}}} \end{array}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} u\\ v \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {I_{t1}}}\\ { - {I_{t2}}}\\ {\begin{array}{*{20}{c}} \vdots \\ { - {I_{t4}}} \end{array}} \end{array}} \right]$ (9)

 $\left[\begin{array}{c}u\\ v\end{array}\right]={\left({{{A}}}^{{\rm{T}}}{{A}}\right)}^{-1}{{{A}}}^{{\rm{T}}}\left(-{{B}}\right)$ (10)

 ${{g}}^{L-1}=2\left({{g}}^{L}+{{d}}^{L}\right)$ (11)

 ${{d}}={{g}}^{0}+{{d}}^{0}= \sum \limits_{L=0}^{{L}_{m}}{2}^{L}{{d}}^{L}$ (12)

1.3 二元颗粒体系的倾斜角测量方法

 图 4 滚筒颗粒体系倾斜角示意图 Fig. 4 Schematic diagram of the angle of slope in the particle system
1.4 颗粒混合程度的熵

 图 5 二维滚筒的正视图 Fig. 5 Front view of the two-dimensional drum
 $s\left(k\right)=-\left({x}_{\rm a}\left(k\right)\ln{x}_{\rm a}\left(k\right)+{x}_{\rm b}\left(k\right)\ln{x}_{\rm b}\left(k\right)\right)$ (13)

 $S\left(t\right)=\frac{1}{N}\sum \limits_{k}s\left(k,t\right)n\left(k,t\right)$ (14)

2 结果与讨论 2.1 颗粒体系的总体熵

 图 6 二维滚筒内颗粒体系总体熵的变化曲线 Fig. 6 Variation curve of the total entropy of the particle system in the two-dimensional drum
2.2 表面流动层的速度特征

 图 7 区域Ⅱ内的速度分布直方图 Fig. 7 Histogram of the velocity distribution in region II

 图 8 表面流动层的速度空间分布随转速的变化曲线 Fig. 8 Spatial distribution of the velocity on the surface flow layer at different rotating speeds
2.3 颗粒速度、倾斜角与总体熵的关系

 图 9 颗粒运动平均速度与倾斜角随转速的变化曲线 Fig. 9 Average speed and angle of slope of the particle motion at different rotating speeds

3 结　论