1 of 26

Problem 4

STUCK METALLIC SPHERES

Reporter: Stanisław Rakowski Authors: Stanisław Rakowski, Kamil Dutkiewicz

2 of 26

Problem statement

  • Fill a bottle with small metal/plastic spheres with diameters of the same order of magnitude as the size of the opening. Try to pour the spheres out of the bottle by turning it upside down. Similar to pouring salt from small openings, one can see that after a certain time the spheres become stuck and stop pouring out.

2

  1. Investigate the phenomenon.
  2. What is the average time it takes before the system becomes stuck?
  3. What bottle shapes can prevent the system from getting stuck?

3 of 26

Relevant parameters

  • diameter
  • mass
  • friction/material

Ball’s parameters

  • diameter
  • mass
  • friction/material

Nozzle parameters

  • orifice diameter
  • Shape/angle

Initial conditions

  • number of balls
  • Initial height
  • Opening method (velocity)
  • orifice diameter
  • Shape/angle
  • number of balls
  • Initial height
  • Opening method (velocity)

3

0.6 mm

0.2 g

60°

v

h

4 of 26

Relevant parameters

  • mass of a ball
  • material of a ball
  • angle
  • number of balls
  • initial height
  • opening velocity

Variables

Fixed

  • orifice diameter
  • ball’s diameter

4

[1] Jamming of Granular Flow in a Two-Dimensional Hopper, K. To et al, Phys. Rev. Lett. 86, 71�[2] Jamming in granular matter, A. Garcimartin et al, AIP Conference Proceedings 742, 279 (2004)

​

5 of 26

Experimental setup

5

0 g

scale

cup

0.58 mm

0.2 g

500 balls

As fast as possible

60°

6 of 26

Arch formation

6

7 of 26

Modeling the flow as a probabilistic process

7

8 of 26

Probability of arch formation model

8

9 of 26

Measuring probability distribution

9

10 of 26

Measuring probability distribution

10

11 of 26

How it depends on the diameter of the orifice

11

 

12 of 26

Calculating expected number of balls

12

 

 

 

13 of 26

Measuring time

13

14 of 26

Final solution to nº 2

14

15 of 26

Designing the optimal nozzle

  • 3. What bottle shapes can prevent the system from getting stuck?

​

​

15

Optimization problem:

​

Given a cylinder of diameter L and balls of radius r, find the shortest (in terms of h) nozzle, that has an orifice of diameter d and never or almost never gets stuck.

​

​

For simplicity, we limit our solutions to cylindrically symmetrical ones.

L = 10 cm

d

h

16 of 26

Sequence of nozzles

16

17 of 26

Polishing our nozzle

17

After 16 iterations…

Simulation

​

LIGGGHTS – open source discrete element method particle simulation software

https://www.engineerdo.com/2019/10/04/liggghts/

18 of 26

18

Polishing our nozzle

19 of 26

Optimal nozzle comparison

19

100% stuck

5,5% stuck

20 of 26

Summary

​

  • We stated that spheres in a bottle get stuck because of arch formation
  • We developed a theory that explained the distribution of the results and the mechanism behind the phenomenon
  • We calculated how much time it take for a system to get stuck
  • We designed the shape of a nozzle that prevents the system from jamming in 94.5 % of cases

​

​

​

​

20

21 of 26

Angle dependence

21

[1]Jamming of Granular Flow in a Two-Dimensional Hopper, K. To et al, Phys. Rev. Lett. 86, 71�[2]Jamming in granular matter, A. Garcimartin et al, AIP Conference Proceedings 742, 279 (2004)

​

22 of 26

Friction dependence

22

[1]Jamming of Granular Flow in a Two-Dimensional Hopper, K. To et al, Phys. Rev. Lett. 86, 71�[2]Jamming in granular matter, A. Garcimartin et al, AIP Conference Proceedings 742, 279 (2004)

​

23 of 26

Initial conditions dependence

23

24 of 26

Finding the optimal nozzle

24

2 diameters of the ball

desired d

25 of 26

Base experimental and simulated results

25

26 of 26

But

26

This is why flat nozzles require more balls to get stuck