The behavior of the flow of glass spheres in a vertically vibrating hopper is examined.  A two-dimensional hopper is mounted on a shaker that provides sinusoidal, vertical vibrations.  Both the frequency and amplitude of the vibrations are adjustable.  Hopper discharge rates and flow patterns are measured as the acceleration amplitude of the vibrations is increased from 0 to 4g's.  Comparisons are made with unvibrated hopper flows and with a two-dimensional discrete element simulation model

Brennen, C. E.

Hunt, M. L.

Wassgren, C. R.

English

Caltech Authors - Main

Proceedings of loth ASCE Eng. Mech. Conf., Boulder, CO, 1995. 
Granular Flow In a Vertically Vibrating Hopper 
C.R. ~assgren ' .  C.E. ~rennen', andM.L. ~ u n t '  
The behav~or of the flow of glass spheres m a vertically vlhraung hopper IS 
exarmned A hvo-dxmens~onal hopper a mounted on a shaker that provuies 
smusouial, v e n d  v~bmons  Both the frequency and amplitude of the vlbmuons are 
adjustable Hopper discharge rates and flow patterns, are measured as the 
acceleration ampl~tude of the vlbrauons 1s increased from 0 to 4g's Compansons are 
made wlth unv~brated hopper flows and wlth a two-dunenslonal d~screte element 
sunulauon model 
Introduction 
Perhaps the most common devlces used for transporting and stomg granular 
matenal are hoppers These dev~ces are sometlmes subjected to localued vlbrauons 
to Induce flow that has stopped due to bndgmg, a condkon where pmc1es ~nterlock 
and form a brtdge that supports the matenal above ~t The apphed vlbratlon m th~s 
sltuatlon 1s e~ther a hammer blow to the slde of the hopper or the use of an out-of- 
balance motor attached to the hopper wall 
Few studles have exammed the behav~or of a granular matenal flow from a 
vlbratmg hopper Takahaslu et a/ (1968) studxed the tralectones and veloclues of 
glass spheresi a two-d~tnens~onal wedge-shaped hopper (w~th a closed em), wluch 
wac subjected to \enlcd. wuoldal v~brauon\ They found lhat pmcles movcd m a 
~lrculalorv mouon uhere omcles mvel U D  at the mched stde walls and down m the 
central region. For a vibrating container with vertical w d s  the circulation is in the 
opposite direction (Evesque et a/.,  1986, Knight er al., 1993). Suzuki et ol. (1968a) 
measured the d~scharge rate of glass spheres in the same hopper used by Takahashi et 
a/. They found that the discharge rate increases with frequency, f, at a constant 
dimensionless acceleration amplitude, r=4n2aflg, where a is the amplitude of the 
vibrations and g is gravitational acceleration. For larger values of l- the discharge 
rate was found to decrease. Further work by this same group (Suzulu et al., 1968b) 
examined panicle segregation in a vibrating hopper. 
Evesque and Meftah (1993) observed a &nilat discharge effect for a vertically 
vibrated hourelass filled with sand. Thev found that the flow rate in the hourelass 
Increased shghtly for vlbratlons levels between 0 and just less than Ig but mcreases 
rapdy for r greater than Ig They were able to stop the flow for vlbrauon 
frequencies between 40 and 60 Hz for large enough acceleration amplmdes 
It IS clear that the behawor of a vemcally v~braung hopper flow a qute 
complicated This paper exanunes the behavlor of glass spheres dachargmg from a 
vemcally vibrated two-dmensional wedge-shaped hopper Expenmental 
observations and dscharge rate measurements are compared wlth prevlous results 
and wlth results from two-dmenslonal d~screte partlcle dynanucs slmulauons 
Ex~enments 
The expenmental apparatus conslsted of a two-dmenslonal wedge hopper 
mounted on a shaker that provlded venlcal, smuso~dal vlbrauons wlth conaollable 
frequency, f, and amphNde, a. The Intenor walls of the hopper conslsted of smooth 
glass on the front and rear faces and smooth luc~te on the s~de faces The hopper 
depth was 1 27 cm. the exlt width was 4 mm, and the wall angle measured from the 
horizontal was 45 degrees These dunenslons were chosen to produce a funnel flow 
condlt~on where a reglon of stagnant matenal formed along the mched hopper s~de 
walls whde the core flow conunued to dwharge 
The granular matenal was of 1 28 mm dlameter glass spheres that were dyed 
black and fluorescent yellow to a d  m flow vlsualuatlon. The angle of repose of the 
matenal was roughly 30 degrees Enough matenal was added to the hopper so that 
the total tme to dscharge the particles was always more than 20 seconds 
D m g  durcharge w~thout v~bta&on the free surface of the matenal formed a 
'v' shaped valley Particles conunuously avalanched down the sloped surfaces Next 
to the mched walls two stagnant flow reglons were observed that &d not extend 
down to the exlt As the level of the matenal decreased, the stagnant flow regons 
became smaller untd all of the matenal In the hopper flowed. 
Wlth vlbrauon and a closed exlt, a circulatory pattern was found, wluch was 
svtlllar to that observed by both Takahash et a1 (1968) and Kmght er a1 (1993) A 
slngle layer of pamcles at the mched slde walls traveled up to the free surface, then 
avalanched down the surface to the centerlme of the hopper and W y  cuculated to 
the bottom oE the hopper The regloon close to the exit of the hopper always r e m e d  
stagnant 
Usmg a strobe hght, the same vlbraung flows were observed to form small 
amplitude stanmng waves on the free surface for levels of vlbrauon greater than 
roughly 2 5g These waves were sundar to those observed by Fauve et a1 (1989). 
Wassgren et a1 (1994). and Melo et a1 (1994) 
For dlschargmg vibrated flows, the stagnant reglons were not present 
Instead, particles at the walls flowed slowly down toward the exlt The slope along 
the free surface of the matenal decreased as r mcreased until the small amphtude 
waves formed at wtuch point the average slope of the surface was zero The waves 
on the surface of the heap conunued as the matenal discharged unul the free surface 
was a few exit wldths from the exlt 
Measurements were made of the mean dscharge rate by recordmg the hme 11 
took for the hopper to completely mscharge for a gwen mass of matenal The results 
are plotted m figure 1 For vihnuon levels below lg the flow rate mcreased shghtly 
for frequencies greater than 20 Hz whlle for flowrates between 1 and 4g the flow 
rates decreased and appeared to be approaclung a constant value These results are 
slmlar to those reported by both Takahash et a1 (1968) and Evesque et al (1989) 
0 I 2 3 4 
Dhnem~onless accelerat~on amplitude, r=4112af21g 
Figure I. Hopper discharge rate as a function of dimensionless acceleration 
amplitude, r=4IIVIs, for various frequencies. 
Discrete Element Simulations 
In addtuon to the prevlous expenmental measurements, a two-dunenstonal soft- 
parucle cbscrete element simulauon was performed to gam further mslght into the 
flow dynanucs The only forces acung on the parucles were gravity and contact 
forces The contact force model consisted of a hea r  spnng and dashpot m parallel 
achng m the normal and tangenual duecuons wtth an ad&uonal shdmg fncuon hnum 
m the tangenual &rechon AN the slmulatlons were performed with the parameten 
given m table 1 Due to the ]muted number of parucles used m the sunulauons, 
parucles that were dxharged were subsequently used to refa the hopper ' h s  
provided a steady flow condmon for the hopper and chd not affect the flow near the 
hopper exlt 
Table 1 Discrete element slmulauon parameters 
Flow patterns from the simulated hopper showed stagnant reglons of the type 
s d a r  to those observed m the expenments for the non-vlbraung case For the 
vlbratmg flows the stagnant regions slowly disappeared over ume Waves were also 
observed along the free surface of the heap for a slmulauon that d d  not have pmcles 
refdling the hopper 
Measurements of the sunulated hopper discharge rate for a frequency of 30 
Hz are dsplayed m figure 1 The dscharge rate decreases wlth mcreasmg r and has 
values less than the expenmentally observed values The latter effect is most Ltkely 
due to the two-dunenslonahty of the slmulauon Current snnulauons exanune the 
dscharge rates for other frequencies and acceleration amphtudes, and lnvesugate how 
the dscharge rate vanes as a funcuon of phase angle over an oscdlauon cycle 
References 
Evesque, P and Meftah W (1993) Mean flow of a vemcally vlbrated 
hourglass Intemanonai Journal of Modem Physrcs 13 7 (9&10) 1799-1805 
Fauve, S , Douady, S , and Laroche, C (1989) Collecuve behaviors of 
granular masses under vertical v~brauons Journal of Phys~cs France, 50 (3) 187 
191 
Kmght, J B . Jaeger, H M , and Nagel, S R (1993) Vlbrauon-mduced slze 
separation m granular meda the convection connection Physrcal Revrew Lerrers 70 
(24) 3728-3731 
Melo, F , Umbanhowar, P , and Swlnney, H L (1994) Transluon to 
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Suzula, A .  Takahash, H . and Tanaka, T (1968a) Behavlour of a pmcle 
bed m the field of vibrauon I1 Flow of pmcles through sllts m the bottom of a 
vibraung vessel Powder Technology 2 72-77 
Suzula, A .  Takahaslu, H and Tanaka, T (1968b) Behavlour of a pmcle 
bed m the field of v~bratlon 111 M~xmg of pmcles m a vlbratlng vessel Powder 
Technology 2 78-81 
Takahaslu, H , Suzula, A and Tanaka, T (1968) Behaviour of a partlcle bed 
m the field of vlbrauon I Analysis of particle motlon m a vlbratlng vessel Powder 
Technology 2 65-71 
Wassgren, C R , Brennen, C E , and Hunt, M L (1994) Verucal vibrauon of a 
bed of granular matenal m a contamer Submtted to the Journal of Appl~ed 
Mechanrcs 
4 Wassgren 


Granular Flow in a Vertically Vibrating Hopper

Mean flow of a vemcally vlbrated hourglass

https://authors.library.caltech.edu/165/1/WAS154.pdf

Granular Flow in a Vertically Vibrating Hopper

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