Effects of vertical vibration on hopper ﬂows of granular ...

PHYSICS OF FLUIDS VOLUME 14, NUMBER 10 OCTOBER 2002

Effects of vertical vibration on hopper flows of granular materialC. R. Wassgrena)

School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907-1288

M. L. Hunt, P. J. Freese, J. Palamara, and C. E. BrennenDivision of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125

~Received 30 August 2001; accepted 9 July 2002; published 3 September 2002!

The discharge of granular material from a hopper subject to vertical sinusoidal oscillations wasinvestigated using experiments and discrete element computer simulations. With the hopper exitclosed, side-wall convection cells are observed, oriented such that particles move up along theinclined walls of the hopper and down at the center line. The convection cells are a result of thegranular bed dilation during free fall and the subsequent interaction with the hopper walls. The massdischarge rate for a vibrating hopper scaled by the discharge rate without vibration reaches amaximum value at a dimensionless velocity amplitude just greater than 1. Further increases in thevelocity decrease the discharge rate. This decrease occurs due to a decrease in the bulk density ofthe discharging material when vibration is applied. ©2002 American Institute of Physics.@DOI: 10.1063/1.1503354#

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I. INTRODUCTION

Hoppers and bins are routinely used to transport, procor store bulk materials such as coal, ore, grain, and plasAlthough hoppers are common, the internal flow of the mterial is not well understood, relying heavily on empiricinformation to maintain operation. For example, whenbatch of material is introduced into a hopper for the fitime, the material at the exit may arch and prevent flow.remedy the situation, vibration may be used, sometimethe crude form of a hammer, to perturb the material ainitiate the flow. Alternatively, the hopper may be equippwith a bin activator to continuously shake part of the hopwall. These bin activators must be carefully designed tohance the flow and not result in further settling or cloggiof the material.

The focus of this study is to examine through expements and discrete element simulations the effect of vertvibration on flow from a planar hopper. This study is a copanion paper to the recent work by Huntet al.1 on the effectsof horizontal vibration on hopper discharge. The earlier wodemonstrated that horizontal vibration increased the mdischarge rate as compared with the discharge rate frohopper without vibration and that the increase dependedthe vibration velocity amplitude. In addition, the dischargigranular material flows from alternating sides of the hopproducing an inverted funnel pattern.

Over the last decade, the effect of vertical vibration obed of material has been studied extensively. For a boxis vibrated sinusoidally,z5a sin(vt) wherea is the ampli-tude of vibration andv is the radian frequency, the bed ehibits several different flow patterns depending on themensionless acceleration amplitude,G5av2/g where g isthe gravitational constant, and the frequency,f 5v/(2p).For G.1, side-wall convection cells appear where partic

a!Author to whom correspondence should be addressed.

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move down along vertical walls of the container andwithin the remainder of the bed. Standing waves, formingone-half the forcing frequency, appear on the free surfacthe bed for 2.2,G,3.5 and waves forming at one-quartthe forcing frequency occur forG.5.5. Neighboring regionsof the particle bed can oscillate out-of-phase, termed ‘‘kiwaves,’’ for G.3.5 with counter-rotating ‘‘kink convectioncells’’ bracketing each wave node. The paper by Wassget al.2 describes these phenomena in greater detail.

The effects of vertical vibration on flow from wedgeshaped hoppers and flat bottom bins were first examinedTakahashiet al.3 and Suzukiet al.4 These studies reportethe appearance of convection cells near the inclined wboundaries of the hopper. In addition, the discharge rateshown to increase with vibrational frequency at a fixedceleration level, but that at the highest accelerations thecharge rate decreased significantly. Vibration also induflow in bins that could not discharge under gravity alonLindemann and Dimon5 recently investigated flow from vertically oscillated funnels with small exit widths and waangles~as measured from the hopper centerline!. They foundthat the dimensionless acceleration amplitude,G, and wallangle significantly affect how particles ‘‘jam’’ or mechancally arch at the exit. Lindemann and Dimon5 also report thatthe flow rate from vibrated hoppers decreases with increaG. Knight et al.,6 although not investigating hopper flowspecifically, also reported the appearance of convection cin experiments using vertically oscillating conical containeMore recently, Evesque and Meftah7 examined the time re-quired to discharge a known quantity of sand from a seavertically vibrating hourglass and observed that the discharate decreases with increasing acceleration amplitude.

Without vibration, the mass discharge rate from a hoper,W, is proportional to the bulk density of the bed near thopper exit,rb , the square root of the acceleration actingthe bed,g ~the acceleration due to gravity!, and the hydraulicdiameter of the hopper exit,Dh , raised to the 5/2 power:

9 © 2002 American Institute of Physics

P license or copyright, see http://pof.aip.org/pof/copyright.jsp

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3440 Phys. Fluids, Vol. 14, No. 10, October 2002 Wassgren et al.

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To predict trends in the discharge rate with vibration, Suzet al.4 proposed a simple model that includes the variationthe ‘‘effective gravity’’ acting on the granular material ovean oscillation cycle. Since the hopper is oscillating, thefective gravity the bed experiences relative to the hopwalls, geff , will vary throughout an oscillation cycle asgeff

5g@12G sin(vt)# for G<1. If the acceleration amplitude othe oscillations is greater than one (G.1) the bed leaves thehopper walls during a portion of the oscillation cycle acontacts the walls at some later time. The equations ornally derived by Suzukiet al.4 also included an empiricallyderived expression for the bulk density of the bed as a fution of G.

This paper examines the particle trajectories andcharge rates from a vertically oscillating, wedge-shaped hper using both experiments and two-dimensional discreteement computer simulations. First the experimentsdescribed and qualitative and quantitative results aresented. Next, results from discrete element computer simtions are presented. Last, these results are compared wmodel originally proposed by Suzukiet al.4 to predict themass discharge rate from a vertically oscillating hopper.

II. EXPERIMENTS

The experimental apparatus consisted of a plawedge-shaped hopper filled with 1.3 mm diameter soda-lglass spheres. The hopper was mounted on an electronetic shaker that subjected the entire hopper apparatuvertical, sinusoidal oscillations. A schematic of the apparais shown in Fig. 1.

Two different hoppers~an MB Dynamics and a LingA-175! were used in the experiments. The front and rwalls of the first hopper, referred to hereafter as hoppewere lined with smooth window glass; the 45° side wawere made of Plexiglass with a smooth milled finish. Tdistance,C, between the front and back walls was 12.7 m~approximately 10 particle diameters! and the exit width,B,

FIG. 1. A schematic of the experimental apparatus.


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5(BC)1/2, of approximately 7.1 mm. The second hopper,hopper II, had a depth of 12.4 mm and an exit width of 9mm (Dh510.9 mm); the front and back walls were Plexglass. The experiments with the glass walls of hopper Inot appear to be affected by static electricity. With the Pleglass walls of hopper II, some particles did adhere towalls after the bed was emptied; however, the amountmaterial remaining on the walls was much less than the tamount discharged.

The vibration frequency and acceleration were motored using accelerometers mounted to the base plate oshakers. For hopper I, the frequencies,f , examined in theexperiments ranged from 20 to 60 Hz and the dimensionacceleration amplitude of the oscillations,G5av2/g, rangedfrom 0 to 4.0. For hopper II, the range was larger, with frquencies from 5 to 80 Hz, and dimensionless acceleratup to G58.0.

In all of the experiments, the hoppers were filled wi200–250 g of particles so that the initial height of the frsurface was at least 10Dh . The average discharge rate wdetermined by recording the time required for the hoppecompletely discharge a known mass of material. Withoutbration, the instantaneous discharge rate was approximaconstant except for a short transient when the hopper exfirst opened, and a final transient when the free surfheight of the material above the exit is approximately eqto the exit diameter.8

A. The flow field

To follow the trajectories of particles within the hoppesome of the clear glass spheres were dyed black. The nobrating hopper displayed a typical funnel flow pattern asdischarged where a region of stagnant material appearedjacent to each inclined hopper wall. The free surface ofmaterial formed a V-shaped valley with particles continously avalanching down the sloped surfaces toward the cter line of the hopper. As the height of the free surface frothe exit plane decreased, the stagnant regions became smuntil all of the material in the hopper flowed.

With vibration, the flow pattern depended on the magtude of the vibrational acceleration. With the hopper eclosed forG.1, two convection cells appeared with particlmoving up in a narrow boundary layer along the inclinwalls of the hopper and down at the center line of the hoppThese convection cells were similar, but in the oppositerection, to the convection cells observed in granular bsubject to vertical oscillations in containers with verticwalls.6,9 Surface waves forming at one-half the oscillatiofrequency~referred to asf /2 waves! also appeared on thfree surface of the bed when the dimensionless acceleraamplitude,G, was between 2.5 and 4; these waves are simto those observed in experiments of vibrating granubeds.2,10 Figure 2 shows photographs of the surface wawith the hopper illuminated from the back.

When the exit of the hopper was opened the circulatpatterns continued as the hopper discharged. Figure 3 sha sequence of images in which the hopper is filled with


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3441Phys. Fluids, Vol. 14, No. 10, October 2002 Effects of vertical vibration on hopper flows

ternating layers of clear and dyed beads. The imagestaken at G56 and f 510 Hz. At t510 s the hopper isopened and the material begins to discharge. The matcontinues to circulate as the hopper discharges untilt521.8 s the hopper is completely emptied. Thef /2 surfacewaves also continue to form as the material discharges fthe hopper until the height of the free surface of the bed frthe hopper exit is approximately five exit widths.

Note that the convection strength and direction depstrongly on the hopper wall angle.11,12 For wall angles lessthan approximately 10°~measured with respect to the vercal! the convection cells are oriented downward at the wwhile for angles greater than 10° particles move up at

FIG. 2. Photograph of a back-lit, vibrating hopper showing the formationf /2 surface waves forG53.4 andf 530 Hz.

FIG. 3. A sequence of photographs of a vibrating hopper~G56, f 510 Hz!showing the formation of side-wall convection cells near the inclined wof the hopper. The hopper is initially closed but is opened att510 s. Thehopper discharges completely att521.8 s.


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walls. Here only a single wall angle of 45°, for which thconvection cells were oriented upward at the walls, wasvestigated.

B. Mean discharge rates

The mean discharge rate from the hopper,W, was mea-sured over the range of frequencies and accelerations.discharge rates are normalized by the mean discharge ratthe nonvibrating hopper,W0 , and are presented in Fig. 4 asfunction of dimensionless oscillation acceleration amplituG5av2/g, and in Fig. 5 as a function of dimensionless ocillation velocity amplitude,av/(gDh)1/2. Each experimen-tal data point in the figures represents the average of fiveight measurements. The difference in discharge rate fexperiment to experiment was at most 3%; the uncertaintthe discharge rate was approximately 5% due to the tenique for measuring the discharge time and the loss ofterial due to static charge.

As shown in Fig. 4, there does not appear to be asignificant difference between the data obtained with the

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FIG. 4. The ratio of the mass discharge rate from an oscillating hopperW,divided by the mass discharge rate from a nonvibrating hopper,W0 , plottedas a function of the dimensionless oscillation acceleration amplitudeG5av2/g.

FIG. 5. Mass discharge rate from an oscillating hopper divided by the mdischarge rate from a nonvibrating hopper,W/W0 , plotted as a function ofthe dimensionless oscillation velocity amplitude,av/(gDh)1/2.


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different hoppers. The discharge rate,W/W0 , for frequenciesbelow 50 Hz decreases with increasing acceleration levea fixed acceleration, the decrease is most significant atlowest frequencies. For the highest frequencies~60 and 80Hz!, the dimensionless discharge rate is approximately eqto or slightly greater than unity.

Figure 5 indicates that the dimensionless dischargescales more closely with the velocity amplitude than withacceleration amplitude. This result was also found in thelier studies on horizontal vibration.1 Again, the dischargerates above unity correspond with the frequencies betw60 and 80 Hz. For frequencies lower than 60 Hz, the dshow a monotonic decrease with vibrational velocity withsecondary dependence on acceleration level. At the lardimensionless velocity amplitude investigated,av/(gDh)1/2

'2.0, the discharge rate ratio is approximately 0.5.

C. Velocities of exiting particles

The hopper exit velocity was measured for different fquencies and accelerations by recording the flow throughfront surface of the hopper at a rate of 2000 framessecond using a Redlake MotionScope camera. Partracking13 is used to determine the particle velocities nearexit plane. By maximizing the number of pixels per particthe error in the measurements is approximately 2%.

The velocity of the exiting particles was obtainedaveraging over particles within 2 diameters above the hopexit plane. Hence, the interrogation area moved with thenusoidally oscillating hopper exit with the neutral positiotaken at the beginning of the oscillation cycle. The data walso averaged over several cycles of vibration with the stial and temporal averaged velocity calculated in time intvals of 0.1/f . Each data point represents an average oapproximately 100–200 particle trajectories. The root-mesquare~rms! velocity fluctuations over an oscillation cyclwere also calculated.

Figure 6 presents the normalized velocity measuremas a function of the oscillation cycle atG51.0 for 10, 20,and 40 Hz. The values are normalized by the cycle-avera

FIG. 6. Discharge velocity of particles relative to the hopper exit normaliby the cycle-averaged velocity and rms velocity as a function of oscillacycle. Data are shown forG51.0 andf 510, 20, and 40 Hz.


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velocity. The results show that normalized velocities vasinusoidally with a phase lag that depends on the vibratiofrequency. At 40 Hz, the phase lag is almostp, but for 10Hz, the phase lag is smaller, approximately 0.6. The amtude of the normalized velocity also depends on frequenAt 10 Hz, the maximum velocity amplitude is approximate50% of the average value; at 40 Hz the amplitude is appromately 10% of the average value. Figure 6 also presentsnormalized rms velocity fluctuations as a function of the ocillation cycle. Although there is some variation over thcycle, the values remain at approximately 20–25% ofaverage velocity. Note that these fluctuations are averaover the width of the hopper exit. Similar trends are observfor G53.0 at 20, 40, and 60 Hz.

The velocity data forG56.0 for 20 and 40 Hz, shown inFig. 7, are plotted over two cycles of vibration since thacceleration is beyond the period-doubling bifurcation acceration observed in earlier studies.2 The variation in the dis-charge velocity between the two cycles is clearly observethe data for 20 Hz. The figure also shows that over part ofcycle, the hopper is moving down faster than the dischargmaterial; hence, the material actually re-enters the hop

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FIG. 7. Discharge velocity of particles relative to the hopper exit normalizby the cycle-averaged velocity and rms velocity as a function of oscillatcycle. Data are shown forG56.0 andf 520 and 40 Hz.

FIG. 8. Cycle-averaged discharge and rms velocities normalized bynonvibrating hopper discharge velocity as a function of dimensionlesslocity amplitude.



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TABLE I. The parameters used in the computer simulation for the vertically oscillating closed hopper.

Parameter Value

Particle diameter distribution 0.8–1.2 mm~uniform distribution!Particle density 2500 kg/m3

Exit width/mean particle diameter 25Wall angle measured from the centerline 65°

Number of particles 513Coefficient of restitution for particle/particle contacts 0.9

Friction coefficient for particle/particle contacts 0.5Coefficient of restitution for particle/wall contacts 0.9

Friction coefficient for particle/wall contacts 1.0

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The fluctuation velocities are also significantly larger ththose observed at the smaller accelerations. At 40 Hz,differences over the two cycles of vibration are less evide

Figure 8 presents the cycle-averaged discharge andvelocities atG50, 1, 3, and 6 normalized by the discharvelocity measured for a nonvibrating hopper. The cycaveraged velocities are within 25% of the value for no vibtion. In addition, the cycle-averaged rms velocities are ofsame magnitude. Hence, vibration of the hopper doessignificantly affect the average discharge velocity of the mterial.

III. DISCRETE ELEMENT COMPUTER SIMULATIONS

A two-dimensional soft-particle discrete element simution was also used to study the vibrated hopper phenomThe granular material in the simulation consists of spherparticles with a uniform distribution of diameters betweenspecified minimum and maximum diameter. Each partihas two translational and one rotational degree of freedThe contact model uses a damped linear spring in the nodirection and a linear spring in series with a frictional slidielement in the tangential direction.14 The coefficient of res-titution ranges from 0.90 to 0.95 for both the particle-tparticle and the particle-to-wall contacts. The friction coecient for the particle-to-particle contacts is 0.5, and forparticle-to-wall contacts is either 0.5 or 1.0. The larger vues of the friction coefficient increased the net circulatwithin the bed. The particles have a density of 2500 kg/3

and an average diameter of 3 mm. The details of the silation parameters can be found in Wassgren.9

FIG. 9. Side-wall convection cells in a simulated vertically oscillating hoper with a closed exit. The vectors indicating the net displacement ofticles per oscillation cycle are shown. The displacements are averaged20 oscillation cycles and the circles are the vector arrowheads.

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A. Convection in a closed hopper

The first series of simulations were performed for a cotainer with inclined walls and a closed exit using 513 pticles atf 52.0 and 5 Hz. The parameters for this simulatiare listed in Table I. Two convection cells similar to thoobserved in the experiments appear as shown in Fig. 9.ticles move up along the inclined walls of the container adown at the center line. When the particle bed leaveshopper floor, few particle collisions occur with the hoppwalls since the walls slope away from the bed and theremains closely packed as shown in Fig. 10~a!. Thus, thewalls apply no force to the particle bed as the bed movesrelative to the hopper. While the bed is in flight it dilatesince it is no longer constrained by the container waWhen the dilated bed falls back toward the hopper floocontacts the inclined walls of the container prior to impactithe base as shown in Fig. 10~b!. Due to friction, the wallsretard the downward motion of particles near the wallcompared to the remainder of the bed. The resulting nettion of particles near the container walls over an oscillatcycle is upwards along the inclined walls. This same mecnism can also explain the downward motion of particalong vertical walls in vertically vibrated granular beds.9

B. Discharge from a stationary hopper

The flow of granular material from the simulated hoppwas also examined. Since the particles are not ‘‘recycledthe simulations, a particle is eliminated from the calculationce it reaches a distance of five hopper widths belowexit plane. The simulation uses a sufficient number of p

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FIG. 10. Snapshots from a simulation of a vertically oscillating hopper~G52.0, f 55 Hz! with a closed exit at a phase angle of~a! f/(2p)50.30~bed moving up relative to the hopper walls! and ~b! f/(2p)50.80 ~bedmoving down relative to the walls!.



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TABLE II. The parameters used in the computer simulations for the vertically oscillating open hopper.

Parameter Value

Particle diameter distribution 2.8–3.2 mm~uniform distribution!Particle density 2500 kg/m3

Exit width/mean particle diameter 11Wall angle measured from the centerline 45°

Number of particles 10 000Coefficient of restitution for particle/particle contacts 0.95

Friction coefficient for particle/particle contacts 0.5Coefficient of restitution for particle/wall contacts 0.95

Friction coefficient for particle/wall contacts 0.5

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ticles (N510 000) to ensure that a steady flow is achievThe remainder of the simulation parameters are givenTable II.

When a particle crosses the horizontal plane of the hper exit, the time, the horizontal position of the particle, tparticle’s translational and rotational velocities, and the pticle’s radius and mass are recorded. The mean dischargefrom the hopper is measured by counting the total numbeparticles discharged from the hopper between a specstart and end time. The start and end times are chosenthat the initial and final transient periods are avoided. Tflow properties are computed across the exit of the chanby dividing the exit into 10 equal length segments. If a pticle lies in two adjacent sections, the properties are weighby the fraction of the particle mass within that section. Nothat the exit is 11 particle diameters in width.

Figure 11 presents the mass flow rate, the velocity,the bulk density as a function of position across the exit ononvibrating hopper. The values are normalized by the vues averaged over the width. The bulk density of a sectiocalculated from the mass flow rate divided by the avervelocity in that section and the corresponding section widNote that the maximum solid fraction of a monolayerparticles across the hopper exit isp/6. The solid fractioncalculated from the simulations is 84% of the maximu

FIG. 11. Spatial distribution of the mass flow rate,W, particle velocity,V,and bulk density,rb , across the exit plane of a simulated nonvibratihopper. The quantities are normalized by their average over the exit p

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value. The average discharge velocity is 1.3(gDh)1/2, so thatthe mass flow rate isW051.3rb(gDh)1/2A whereDh is takenas the width of the hopper exit (11d) andA511d2.

C. Discharge rates with vibration

The ratio of the mass discharge rate to the masscharge rate for the nonvibrating hopper,W/W0 , is plotted inFig. 12 as a function of amplitude oscillation velociav/(gDh)1/2 for 0,G,5 and for frequencies from 5 to 6Hz. The simulation data show the same trends as thegathered from the experiments. If the data are plotted afunction of the acceleration amplitude, the simulation resuappear to follow the experimental trends indicating thatdimensionless discharge rate was closer to unity for higfrequencies~50 and 60 Hz! at a fixed value of the acceleration amplitude.

Besides the average discharge rate, the simulations wused to determine other properties of the flow. In Fig. 13discharge rate as a function of phase angle (Df52p/10) ispresented at a frequency of 20 Hz. The data are obtainemass-averaging over 70 oscillation cycles. The valuesnormalized by the discharge rate without vibration. As tacceleration amplitude increases, the ratio of the discharates show deviations from unity over the cycle of vibratioAs the hopper begins its upward motion@0,f/(2p),0.25#, the ratio is greater than unity and the materialdischarging faster than it would without vibration. As th

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FIG. 12. Mass discharge rate from an oscillating hopper divided by the mdischarge rate from a nonvibrating hopper,W/W0 , plotted as a function ofthe dimensionless oscillation velocity amplitude,av/(gDh)1/2. The plotonly includes simulation data.


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hopper moves [email protected],f/(2p),0.75#, the ratiois less than unity. ForG50.5 and 1.0, the discharge ratincreases to values above unity as it moves [email protected],f/(2p),1.0#. For these two accelerations the discharate appears to roughly follow a cosine pattern with an aplitude that increases with vibrational velocity. ForG52.0and 3.0, the discharge ratio deviates from a cosine wave,the ratio remains below unity for a greater portion of tcycle.

In Fig. 14, similar results are shown for the normalizdischarge rate at 60 Hz atG50.5, 1.0, and 2.0. The ratioappear to be slightly larger than unity as the hopper moupwards, and at or slightly below unity for the remainderthe cycle. The cycle-averaged discharge rates are all wi10% of the discharge rate without vibration.

In addition to the variation of the mass discharge rwithin the vibration cycle, it is also possible to determine tvelocity of the particles exiting the hopper. Figure 15 p

FIG. 13. The ratio of the mass discharge rate for a hopper oscillatingfrequency of 20 Hz to the mass discharge rate from a nonvibrating hopW/W0 , plotted against fraction of an oscillation cycle,f/(2p), for variousdimensionless acceleration amplitudes.

FIG. 14. The ratio of the mass discharge rate for a hopper oscillatingfrequency of 60 Hz to the mass discharge rate from a nonvibrating hopW/W0 , plotted against fraction of an oscillation cycle,f/(2p), for variousdimensionless acceleration amplitudes.


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sents the vertical exit velocity relative to the hopper enormalized by the cycle-averaged velocity as a functionphase angle for a frequency of 20 Hz. The discharge veloand mass flow rate show the same phase dependence. Hparticles are leaving the hopper faster as the hopper is ming upwards and slower as the hopper is moving downwaFigure 16, however, shows that the bulk density of the eing material varies little over the oscillation cycle.

The average relative discharge velocity and bulk denfor an oscillating hopper normalized by the average dcharge velocity and bulk density for a stationary hopperplotted in Figs. 17 and 18, respectively, as a function ofoscillation velocity amplitude. These figures indicate thatvariations in the average mass flow rate due to vibrationinfluenced more by the variations in the bulk density thanthe variations in the discharge velocity. For the parameinvestigated the ratio of the bulk density for a vibrating hoper to the nonvibrating bulk density remains less than 1 wthe smallest deviation occurring at small velocity amplitud

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FIG. 15. The discharge velocity of particles relative to the hopper enormalized by the cycle-averaged velocity,Vrel /Vrel,avg, as a function ofphase angle,f, for a hopper oscillating at a frequency of 20 Hz.

FIG. 16. The bulk density of particles at the hopper exit normalized bycycle-averaged bulk density,rb /rb,avg, as a function of phase angle,f, fora hopper oscillating at a frequency of 20 Hz.


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The discharge velocity ratio, however, is less than 1 for fquencies less than approximately 20 Hz but is greater thafor frequencies greater than 20 Hz.

Measurements of the horizontal position of particles atheir velocities as they discharge from the hopper are amade with the simulations. The vertical velocity of particlrelative to the hopper exit are shown in Fig. 19 for a hoposcillating at f 520 Hz and G52.0. The data points armass-averaged over 70 oscillation cycles. The dischargefile for a nonvibrating hopper is also included in the plot ais indicated by the thicker line. The vertical velocity relativto the hopper at discharge fluctuates in a manner consiswith the measurements in Fig. 15. Furthermore, it is oserved that while the granular bed is in flight, the dischaprofiles are more uniform than when the bed is in contwith the hopper walls. This occurs because the walls shthe particle bed at the discharge plane.

IV. DISCUSSION

The simulation and experimental data show simitrends although the absolute values of the data are nosame. This discrepancy may be due to the fact that the s

FIG. 17. Average discharge velocity~relative to the hopper exit! from anoscillating hoppper normalized by the average discharge velocity frostationary hopper,Vrel,avg/V0 , plotted as a function of the dimensionlesoscillation velocity amplitude,av/(gDh)1/2. The plot only includes simula-tion data.

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lations were two dimensional while the experiments wethree dimensional. For a particular set of vibration paraeters,G and f , the instantaneous mass flow rate from thopper closely follows the instantaneous velocity of exitiparticles relative to the hopper~Figs. 13 and 15! while thebulk density of particles leaving the hopper varies little ovan oscillation cycle~Fig. 16!. However, for differing vibra-tion conditions the average relative velocity remains neaconstant, less than 10% variation for the parameters invegated~Fig. 17!, and instead it is the variation in the averabulk density that most affects the mass flow rate~Fig. 18!.Note that although the strength of the convection cells vawith velocity amplitude,9 there is little influence on the discharge velocity from the hopper. Hence, the convection cseem to play only a minor role on the mass discharge ra

The data gathered in this study is also compared withmodel proposed by Suzukiet al.4 in which the discharge rateis assumed to be a function of the instantaneous ‘‘effecgravity’’ acting on the particle bed. In their model, Suzuet al.4 suggest that the instantaneous discharge rate fromhopper is given by

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FIG. 18. Average discharge bulk density from an oscillating hopper normized by the average bulk density from a stationary hopper,ravg/r0 , plottedas a function of the dimensionless oscillation velocity amplitudav/(gDh)1/2. The plot only includes simulation data.

geff5H g@12G sin~vt !# when the bed rests on the hopper walls,

0 when the bed is in flight,

dybed/dt when the bed just impacts the hopper walls,

~3!

doen-den-ation

where the acceleration acting on the bed at impact,dybed/dt,is approximately equal to the difference between the flvelocity at impact and the particle bed free fall velocity juprior to impact divided by the duration of the impact perioSuzukiet al.4 also propose that the only vibration parameon which the bulk density of the material,rb , should dependis the dimensionless acceleration,G, with the bulk density

rt.r

increasing when vibration is applied.The data found in the experiments and simulations

not agree with the previously described model. The bulk dsity measurements presented here indicate that the bulksity has a dependence on both the frequency and acceleramplitude and, furthermore, the bulk densitydecreaseswhenvibration is applied. Suzukiet al.4 claim that the bulk density


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of the materialincreaseswhen the hopper is subject to osclations. In addition to the differences with the bulk densobservations, the relative velocity of the material leavinghopper is different than what is proposed in the Suzukiet al.4

model. In their model the velocity of the exiting materialproportional to the square root of the effective gravity,Vrel

}Ageff, where the effective gravity is given in Eq.~3!. Asshown in Fig. 15 where the velocity normalized by the cycaveraged velocity is plotted as a function of oscillation phangle forf 520 Hz, the effective gravity model proposed bSuzukiet al.4 is inaccurate in predicting the velocity of paticles exiting the hopper. In particular, the assumptions tthere is no discharge during the period when the bed is nocontact with the hopper walls and that there is a sudden sof material when the bed impacts the hopper walls are clein error. In fact, the measurements reported here indicatethe effective gravity acting on the bed remains essenticonstant and that the variation in the mass flow rate fromhopper is due to the movement of the hopper exit. Asshown in Fig. 17, the average discharge velocity varies liover the range of oscillation conditions investigated heThe variations in the average mass flow rate are thus duvariations in the average bulk density~Fig. 18!.

The mechanism causing the slight increase in the expmental discharge rate at low velocity amplitudes remainsclear. One possibility is that the relative mobility of the mterial increases in this region while the density remarelatively unaffected. Indeed, the experimental measuments by Ziket al.15 show that the mobility of a particle in avibrating bed is much greater whenG.1 than forG,1.

It is interesting to compare the flow behaviorhorizontally1 and vertically oscillated hoppers. Both typesvibration affect the hopper flow but in very different wayFor horizontal oscillations, small side-wall convection ceappear at the free surface of the particle bed near the howalls. Particles fall down along the hopper walls in the gthat opens up when the hopper moves away from the parbed. The convection cells do not extend along the lengththe wall as they do in vertically vibrated hoppers. A mostriking difference between horizontally and vertically osclated hoppers occurs in the discharge patterns and flow r

FIG. 19. The discharge velocity profile relative to the hopper exit at variphase angles for a hopper oscillating atG52 and f 520 Hz.


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While the discharge pattern in a vertically vibrated hoppdischarges more readily from the central core of the hopa horizontally oscillated hopper has an ‘‘inverted’’ pattewhere material discharges more readily at the hopper waIn addition, the mass discharge rate in a horizontally oslated hopper increases with applied vibration in contrasthe decrease in mass flow rate observed with verticallycillated hoppers. For both cases the discharge rate scalesoscillation velocity amplitude.

V. CONCLUSIONS

Granular flow from a vertically vibrating hopper wainvestigated. Particles within a closed hopper exhibit convtion cells and surface waves similar to those observedstudies of horizontal beds. The convection cells are oriensuch that particles move up along the inclined hopper wand down at the hopper’s center line. Observations fromcrete element simulations indicate that the convective mois a result of the dilation of the particle bed during free fand interaction with the hopper walls.

The ratio of the discharge rate from a vibrating hopperthe discharge rate from a nonvibrating hopper decreasesvibration except at the highest frequencies. The discharate scales with the vibration velocity amplitude with a seondary dependence on the acceleration amplitude. The vtion in the mass discharge rate during an oscillation cyoccurs primarily due to the relative velocity between the dcharging material and the oscillating container. The bulk dsity of the material changes little during the oscillation cycThe variations of the mass discharge rate for different oslation parameters, however, is due to variations in the dcharging material’s bulk density. These findings do not sport the model proposed by Suzukiet al.4 that accounts forthe mass discharge rate trends using a variable effecgravity acting on the material within the hopper.

Although subjecting an entire hopper to vertical vibrtions may not be practical in many applications, these resserve to demonstrate the significant effects that vibrationhave on flow from a hopper. The paper also demonstrateseffect of vibration on the internal flow field. Although nostudied in this paper, vertical vibration may prevent archof a cohesive material due to the internal circulation of tmaterial.5

1M. L. Hunt, R. C. Weathers, A. T. Lee, C. E. Brennen, and C. R. Wassg‘‘Effects of horizontal vibration on hopper flows of granular materialsPhys. Fluids11, 68 ~1999!.

2C. R. Wassgren, C. E. Brennen, and M. L. Hunt, ‘‘Vertical vibration ofdeep bed of granular material in a container,’’ J. Appl. Mech.63, 712~1996!.

3H. Takahashi, A. Suzuki, and T. Tanaka, ‘‘Behavior of a particle bed infield of vibration. I. Analysis of particle motion in a vibrating vesselPowder Technol.2, 65 ~1968!.

4A. Suzuki, H. Takahashi, and T. Tanaka, ‘‘Behavior of a particle bed infield of vibration. II. Flow of particles through slits in the bottom ofvibrating vessel,’’ Powder Technol.2, 72 ~1968!.

5K. Lindemann and P. Dimon, ‘‘Two-dimensional granular flow in a vbrated small-angle funnel,’’ Phys. Rev. E62, 5420~2000!.

6J. B. Knight, H. M. Jaeger, and S. R. Nagel, ‘‘Vibration induced si

s


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s,’

Sc

li-

or

ular

n,’’

dy,

u-

u-


separation in granular media: the convection connection,’’ Phys. Rev.70, 3728~1993!.

7P. Evesque and W. Meftah, ‘‘Mean flow of a vertically vibrated hourglasInt. J. Mod. Phys. A7, 1799~1993!.

8R. M. Nedderman, U. Tu¨zun, S. B. Savage, and G. T. Houlsby, ‘‘The flowof granular materials—I. Discharge rates from hoppers,’’ Chem. Eng.37, 1597~1982!.

9C. R. Wassgren, ‘‘Vibration of granular materials,’’ Doctoral thesis, Cafornia Institute of Technology, Pasadena, CA, 1997.

10F. Melo, P. Umbanhower, and H. Swinney, ‘‘Hexagons, kinks, and disder in oscillated granular layers,’’ Phys. Rev. Lett.75, 3838~1995!.


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’

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-

11J. B. Knight, ‘‘External boundaries and internal shear bands in granconvection,’’ Phys. Rev. E55, 6016~1997!.

12E. L. Grossman, ‘‘Effects of container geometry on granular convectioPhys. Rev. E56, 3290~1997!.

13A. Karion, ‘‘Couette flows of granular materials: Mixing, rheology, anenergy dissipation,’’ Doctoral thesis, California Institute of TechnologPasadena, CA, 2000.

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