Jamming and Flow in 2D Hopper

1
Jamming and Flow in 2-D Hopper

• Sepehr Sadighpour (Duke University)
• Paul Mort (Proctor Gamble)
• R. P. Behringer (Duke University)
• Supported by IFPRI

2
Motivation

• How do grain size and outlet diameter influence
  jamming probability?
• What does the probability distribution of jams
  look like?

Grains exiting silo often jam. Above Typical
method of un-jamming a hopper
3
Anatomy of a Jam

• Grains self-support on container walls by forming
  stress chains
• Pressure maxes out with depth
• Hence, lower particles dont feel all the weight
• This is how a small arch near outlet can hold up
  an entire silo

2-D jam of photoelastic disks in our apparatus
4
Prelude to the Beverloo Equation The Variables
of Flow
5
Beverloo Equation in 3-D A Derivation
(Boundary Layer Effect)
The 3D Beverloo Equation
6
Beverloo Equation in 2D
7
Founding our Hypothesis

• The screening effect explains
• Macroscopic effects
• Plateauing pressure
• Microscopic effects
• Jamming
• Free falling particles ? The Beverloo equation
• So what does the insulation of the outlet region
  suggest about the probability distribution of
  jamming?

8
Our Hypothesis
9
Sketch of our Apparatus
Flips like an hourglass to quickly run trials on
both inclines. Diameter of opening is variable.
Plug
10
Experimental Setup

• 2 sheets of Plexiglass
• 5000 bidisperse
• photoelastic disks in between
• (diameter 3mm 5mm)
• Polarizers taped to front and back for high speed
  video, in order to see arch formation dynamics.

11
Actual Experiment

• We measured duration of continuous flow for
  various outlet sizes on the two inclines
• 100s of runs per opening size gave probability
  distribution for survival time

2.3cm outlet, 45 incline. Slightly faster than
real time.
12
Results An Example
The survival probability exhibits expected
exponential decay.
13
A Useful Form of Ps(t)
14
Characteristic Time t What form could it have?
15
A Reasonable Guess at t
16
Verifying t
17
Verifying t
t2 varies linearly with opening size, as posited
form suggested.
18
Summary

• Photoelastic material connect jamming to arch
  formation
• The probability of continuous flow decays
  exponentially with time
• Reasonable results for t(D)