上海理工大学学报  2019, Vol. 41 Issue (5): 409-416 PDF

Discrete Element Method Simulation on the Immigration of Granular Matter in a Vertically Vibrating U-Tube
GUO Yu, FAN Fengxian, BAI Pengbo, LIU Ju
School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: Using the discrete element method (DEM), three-dimensional numerical simulations were performed to investigate the immigration of granular matter in a vertical vibrating U-tube. The numerical simulation results were compared with the experiment ones and good agreement was found. On this basis, the effects of friction on the dynamical behavior of granular matter were examined. The results show that with the presence of friction, the granular immigration occurs, causing a difference between the granular column heights in the two branches of the U-tube, and the granular immigration is accompanied by granular convective phenomenon. When the friction coefficient between the particle and the tube wall is 0, the granular column heights difference fluctuates around 0, and similar granular convective phenomena appear in the two branches. When the friction coefficient among particles is 0, the granular column heights difference almost keeps 0 and the granular convective phenomena disappear. It is also found that the vertical average particle velocities in the two branches of the U-tube differ largely when frictions exist. Moreover, there are some consistency in between the vertical average velocities of particles in the two branches of the U-tube when either the friction coefficient between particles and the tube wall or the friction coefficient among particles is 0.
Key words: granular matter     vertical vibration     U-tube     immigration     discrete element method

1 DEM模型与数值计算方法 1.1 DEM模型

 ${m_i}\frac{{{\rm{d}}{{{v}}_i}}}{{{\rm{d}}t}} = {m_i}{{g}} + \sum\limits_{j = 1}^N {\left( {{{{F}}_{n,ij}} + {{{F}}_{t,ij}}} \right)}$ (1)
 ${I_i}\frac{{{\rm{d}}{{{\omega} }_i}}}{{{\rm{d}}t}} = \sum\limits_{j = 1}^N {\left( {{{{M}}_{t,ij}} + {{{M}}_{r,ij}}} \right)}$ (2)

 ${{{F}}_{n,ij}} = \min\; \left(0{\rm{,}} - \rho {\delta _{n,ij}}^{3/2} - \frac{3}{2}{A_n}\rho \sqrt {{\delta _{n,ij}}} {\dot \delta _{n,ij}}\right){{{e}}_n}$ (3)
 $\begin{split} &\quad{{{F}}_{t,ij}} = - \min\, \left[ {{\mu _s}\left| {{{{F}}_{n,ij}}} \right|{\rm{,}}\displaystyle\int\limits_{{\rm{path}}}\! {\dfrac{{4G}}{{2 - \upsilon }}\!\sqrt {{R_{\rm{eff}}}{\delta _{n,ij}}} {\rm{d}}s +}}\right.\\ &\qquad{{{A_t}\sqrt {{R_{\rm{eff}}}{\delta _{n,ij}}} {v_{t,ij}}} } ] {{{e}}_t} \end{split}$ (4)

 $\rho = \frac{{2Y}}{{3(1 - {\upsilon ^2})}}\sqrt {{R_{\rm{eff}}}}$ (5)

 ${{{M}}_{t,ij}} = {R_i}{{{e}}_n} \times {{{F}}_{t,ij}}$ (7)

 ${{{M}}_{r,ij}} = - {\mu _{\rm{r}}}\rho {\delta _{n,ij}}^{3/2}\frac{{{{{{\omega}} }_{ij}}}}{{\left| {{{{{\omega}} }_{ij}}} \right|}}{R_{\rm{eff}}}$ (8)

1.2 数值计算方法

 图 1 U形管示意图 Fig. 1 Schematic diagram of the U-tube

 图 2 施加振动前颗粒总动能随时间的变化关系 Fig. 2 Total kinetic energy of the particles as a function of time before applying vibration

 ${t_{\rm{c}}} \approx 3.21{\left( {{m_{\rm{eff}}}/\rho } \right)^{2/5}}{v_{\rm{imp}}}^{ - 1/5}$ (9)

2 结果与讨论 2.1 竖直振动U形管中颗粒物质的行为模式

 图 3 不同情况下竖直振动U形管中颗粒物质的行为模式 Fig. 3 Behavior modes of granular matter in the vertically vibrating U-tube under different conditions
2.2 竖直振动U形管中两分支颗粒柱高度差随时间的演变

 $\phi \left( z \right) = {N_{\rm{p}}}\left( z \right){V_{\rm{p}}}/{V_{\rm{c}}}$ (9)

 图 4 不同摩擦系数条件下U形管两分支颗粒柱高度及高度差随时间的变化关系 Fig. 4 Granular column heights in two branches of the U-tube and the height difference as a function of time under different friction coefficients
2.3 颗粒竖直方向平均速度分布

 $\left\langle {{v_z}} \right\rangle = \sum\limits_{i = 1}^{N'} {({{\bar v}_{zi}}{{N'}_{\rm{p}}}_i)} \left/\sum\limits_{i = 1}^{N'} {{{N'}_{{\rm{p}}i}}} \right.$ (10)

 图 5 不同摩擦系数条件下颗粒竖直方向平均速度随相对位置的变化关系 Fig. 5 Average vertical velocity of particles as a function of relative position under different friction coefficients
3 结　论

a. 利用基于DEM方法的三维数值模拟，再现了实验中得到的U形管两分支颗粒柱高度差随时间演变的历程，展现了颗粒的对流现象等颗粒尺度动力学信息。

b. 当颗粒与管壁间摩擦系数为0时，U形管两分支颗粒柱高度交替增减，两分支内颗粒对流现象类似；而当颗粒与颗粒间摩擦系数为0时，U形管两分支颗粒柱高度差几乎为0，颗粒对流现象消失。

c. 摩擦系数均不为0时，U形管两分支颗粒的竖直方向平均速度分布差异很大；当颗粒与管壁间或颗粒与颗粒间摩擦系数为0时，U形管两分支内颗粒的竖直方向平均速度存在一定的一致性。

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