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Influence of air on the bouncing dynamics of shallow vibrated granular beds: Kroll's model predictions Bruno Guerrero, Vladimir Idler, and Iván Sánchez Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b04912 • Publication Date (Web): 15 Apr 2016 Downloaded from http://pubs.acs.org on April 17, 2016

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Industrial & Engineering Chemistry Research

Inuence of Air on the Bouncing Dynamics of Shallow Vibrated Granular Beds: Kroll's Model Predictions Bruno V. Guerrero,

†, ‡

Vladimir I. Idler,

¶

and Iván J. Sánchez

∗, †, §

†Centro de Física, Instituto Venezolano de Investigaciones Cientícas (IVIC), Caracas

1020-A, AP 20632, Venezuela. ‡Present address: Departamento de Física y Matemática Aplicada, Facultad de Ciencias, Universidad de Navarra, AP 31080 Pamplona, Spain. ¶Departamento de Física, Universidad Simón Bolívar, AP 1080-A, Venezuela. §Present address: Research and Development Direction, Castillomax Oil and Gas S.A., Caracas, Venezuela. E-mail: *[email protected];*[email protected]

Abstract

that have been explored in the literature. Grain diameter, grain density and vibration frequency

In the present work we explore the modica-

are the most determinant.

tion of the periodicity of the motion of shallow granular columns in the framework of Kroll's

Introduction

one dimensional model for the motion of a vibrated bed.

Within the model, bed dynamics Vibrated granular beds are commonly used

depend on two parameters, the dimensionless maximum acceleration rameter

Γ

in

and a dissipative pa-

industry

for

operations

tion, milling and transport.

α depending on air viscosity, grain den-

like

size

separa-

In many cases,

sity, bed static porosity, oscillation frequency

the interest for vibrated beds responds to the

and grain diameter. We show how the bifurca-

widespread use of vibrouidized beds, where

tion diagram for the ight time of the bed as a

mechanical

excitation

due

to

vibrations

is

α = 0 Kroll's

added to the uidizing eect of a passing uid.

prediction equals that of the inelastic bouncing

However the case of pure vibrations (with-

function of

Γ changes with α.

For

is increased, bifurcations

out aeration) is still complex and encompasses

up to a point where not even

many interesting transport phenomena. Theo-

Γ

retical modeling of vibrated beds has followed

where an inelastic bouncing ball displays sev-

several, sometimes combined paths, from the

eral bifurcations.

We also illustrate how the

description of the mechanical response of a vi-

ight time reduces non linearly with increasing

brated bed to the study of heat transfer capabil-

α

We introduce isoperi-

ities and the modeling of the ight dynamics of

odic maps to illustrate regions of single, double

the bed. The latter is of special interest because

or more periodicities in the phase space.

We

it can shed light on the time response of shal-

also show and discuss the dependence of the

low beds, where the column behaves as a solid

ight time on the parameters entering the def-

block.

ball model. When shift to higher

Γ

α

a single bifurcation is predicted in a range of

in a monotonic way.

inition of

α

Adequately tuning this time response

may enhance the result of bulk solids manage-

within ranges of those parameters

ment operations like vibratory transport

ACS Paragon Plus Environment 1

17

, vi-


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

brated bed dryers tors

12,13

811

and vibrated bed reac-

Page 2 of 20

with a model that considers experimental char-

.

acteristics of vibrated beds used for industrial

The simplest approximation to a vibrated bed

application design, in this article we explore

is the well known inelastic bouncing ball model

Kroll's model predictions over a wide range of

(IBBM).

The IBBM describes the motion

parameters, inspired by congurations used in

of a single point mass, bouncing on a sinu-

literature studies as well as in industrial ap-

soidally driven plate, considering completely in-

plications.

elastic collisions with the plate. Its application

machinery like vibratory separators represent a

to vibrated beds is widespread.

challenge for resizing and rescaling, and man-

1416

2,3,1721

One of

Still nowadays, apparently simple

the main features of the dynamics of the IBBM

ufacturers often rely on past experience.

is the period doubling scenario

22

45

In

observed for

the eld, specially when on a budget, chang-

the dependence of bouncing ball's ight time

ing a motor or a shaft, the grain size or the

with the maximum dimensionless acceleration

interstitial uid (to avoid explosions hazards or

Γ.

to improve drying) of an already working pro-

Period doubling is a common dynamical

feature between of both a single mass and vi-

cess is a tough question.

brated particle beds, at least in a regime of

limited information about parametric response

shallow, dense beds:

of vibrated beds.

of grains,

23

a single vertical column

The revisit presented in the

and three dimen-

following pages aims to provide a theoretical

display a period doubling sce-

framework within selected parameter ranges,

nario similar to the IBBM. In recent years sev-

information that was implicit in Kroll's model

eral studies have explored options for including

bed equation, but has not been explored in the

dissipation in the bouncing motion of a single

past.

sional beds

two dimensional

24

46

Handbooks provide

20,2527

object on a vibrating plate.

For example: al-

lowing the vibrating plate to be able to bend; considering either a spring, or a sand lled ball

31

29

a liquid droplet

28

Overview of Kroll's model

30

instead of a single point

mass and considering air drag.

Kroll's model uses a one dimensional equation

32,33

to describe the motion of a porous piston, as-

The main issue with using IBBM for vibrated

suming the piston moves through an incom-

beds is that the model only considers parame-

pressible uid and using Darcy's law to describe

ters describing the forcing, and there is no way

uid ow through the piston. An analytic so-

to compare between beds of dierent character-

lution for the position of the bed and for the

istics.

pressure drop across it can be obtained. Kroll's

One of the main factors aecting the mo-

model represents the starting point for incor-

tion of a vibrated granular bed is its inter-

porating other processes like bed expansion

action with interstitial air (having critical importance in segregation nomena

6,3840

).

3437

or aerated vibrated beds.

and transport phe-

and Han et al.

42

A

Those au-

sionless

experimental and theoretical ight times, at least for their specic parameter conguration.

equation of motion is given by:

Following the interest aroused by the exper-

41,42

t is time. The maximum dimen2 acceleration is Γ = Aω /g , with g the

acceleration of gravity. Take o occurs at a time t0 = ω1 arcsin(1/Γ). During bed's ight, a gap of size s separates it from the plate and the

obtaining a good agreement between

imental studies of references,

the amplitude,

oscillation and

predictions from Kroll's model for a vibrated

43,44

z(t) = A sin(ωt), with f = ω/2π the frequency of

direction according to

thors compared their results with theoretical bed,

The granular bed

moving on a plate oscillating along the vertical

period bifurcation of vibrated beds: the works

41

47

is assumed to behave as a solid porous block,

This has inspired a couple of

recent experimental studies on air eects on the of Pastorl et al.

48

and looking

forward to achieve a theoretical understanding

s¨(t) −

of the bouncing dynamics of a vibrated bed


B (∆P ) = −¨ z (t) − g, m

(1)

Page 3 of 20

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where

∆P

top and bottom of the bed, tion area of the container,

ψ0 = arcsin(Γ−1 ) is the dimensionless o time and α is a dimensionless dissipa-

where

is the pressure dierence between the

B is the cross secm the mass of the

take

tive parameter given by:

bed and dots indicate derivatives with respect to time. The bed is assumed to have a bulk den-

ρm = (1 − φ)ρg with ρg the density of a sinφ the porosity of the bed (assuming that uid density is much lower than ρg ). Relating the volumetric ow rate Q through the

sity

α=

gle grain and

G = ρf Bs

(uid density

ρf

parameters dictate the dynamics of the bed:

is constant be-

which characterizes the energy injection and

cause the uid is assumed to be incompressible) it is possible to deduce that last relation for ing equation is

Q = B s˙ .

Q and Darcy's law 49 36,43 obtained for s˙ :

(6)

Equation (5) shows how two dimensionless

bed with the mass of uid contained within the gap

Bhη η 180η(1 − φ) = = ωmK ωKρm d2 ωρg φ3

This parameter

Using this

α is the frictional factor control-

ling energy dissipation via air drag

the follow-

Γ α.

36,42

, closely

related to coecients previously dened in airdamped bouncing ball studies

32,33

.

Also

1/α

could be interpreted as the characteristic time needed for the column to reach a regime where

∆P , s(t) ˙ =k h

its motion is governed by the viscosity of the

(2)

uid

in air,

h is the bed's height k ≡ K/η is the relative permeability, η is the dynamic viscosity and K is the intrinsic permeability (depending only where

on geometrical properties of the solid matrix

50

characterize ow regimes of liquid-particle suspensions

).

K= where

d

, or Reynolds and Froude numbers

the parameter

Physically speaking,

α = η/ωKρm

is a ratio of viscos-

ity to the momentum of the bed.

given by:

φ3 d2 180 (1 − φ)2

Overall dependence of bed's ight dynamics on dissipative parameter α

(3)

is the diameter of a single grain. Us-

Figure 1 shows the bifurcation diagram for dif-

ing equations (3) and (2) in equation (1), one obtains the nal equation for

53

used in uid mechanics.

ow conditions the Carman Kozeny equation

50,51

. In the context of vibrated beds moving

α has the signicance of other dimension-

less numbers such as Bagnold's number used to

For a random pack of spheres, under laminar holds for the permeability,

52

s˙ ,

ferent values of the dissipative parameter

describing the

α.

The IBBM prediction is included as a refer-

dynamics of the granular column:

ence, and can be seen to match exactly the case with

α = 0

(essentially Kroll's model reduces

to the IBBM in the absence of dissipation). As

Bhη s¨(t) + s(t) ˙ = g [Γ sin[ω(t + t0 )] − 1] mK

the value of

(4)

α

is increased the main eect is a

drift from the IBBM behavior that is more pronounced at larger

Equation (4) can be written in terms of two 0 dimensionless variables s (ψ) ≡ s(t)/A and ψ ≡

for

α = 0.05,

Γ.

See for example the curve

it shows only 4 bifurcations while

the IBBM shows 6 bifurcations within the same

ωt:

range in range of

Γ. Or cases with α ≥ 4 Γ explored there is not

where for the even a single

bifurcation. Deviation from the IBBM behav-

1 s¨0 (ψ) + αs˙0 (ψ) = [Γ sin[ψ + ψ0 ] − 1] , Γ

ior is more pronounced at larger

(5)

α.

See for example the curve for


Γ for a xed α = 0.05, at


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1:

Left:

Bifurcation diagram for dierent

Dependence of ight time with

α

at xed

Γ.

α

Page 4 of 20

values compared with that of the IBBM.

Only a couple of low

Γ

Right:

cases with a single ight time

are shown to illustrate the trend.

Γ≈4

there is not a signicant deviation from

the IBBM but for

Γ ≈ 18

the order of 20%.

Larger

a period-n motion involves n dierent ights.

α

Γ

the dierence is of

The map illustrates zones in the

α

values display a

eter space having a single ight (displayed in

further departure of Kroll's model with respect

green), two ights (displayed in blue) and 3 or

to the IBBM. The departure can be appreci-

more ights (displayed in red). Each horizon-

ated in more detail if one takes a look at Figure

tal line from the isoperiodic map is obtained

sequence S1 of the supplementary material.

from a bifurcation plot like those illustrated in

For low values of

Γ,

before any bifurcation

Figure 1. Two bifurcation plots for

α = 0.5

takes place, it is possible to observe a smooth functional dependence of the ight time with at xed

Γ.

α

-

param-

α=0

and

are shown at the right side of Figure 2

to explain the color code used.

Figure 1 (right) illustrates this func-

The map of Figure 2 is basically a modied

tional dependence, resulting in a monotonically

and simplied version of the map presented by

decreasing function. Although it may resemble

Han et al.

an exponential or a power law decay, we tested

ied since their calculation is based on a mod-

both ts and obtained merely fair agreement.

ication of the IBBM model including an air

This plot gives us an idea of how important is

damping factor without including parameters

the deviation from the IBBM behavior when

related to a vibrated granular bed, besides we

increasing

α

α

with xed

Γ.

Within the range of

33

in Figure 5 of their work. It is mod-

explore a smaller

Γ

range and a larger

α

range.

explored we can see that a reduction of the

It is simplied, since we do not pay attention to

order of 30% of the ight time with respect to

the chaotic behavior (any chaotic zone will be

the ight time predicted by the IBBM can be

included in a red zone). An important remark

obtained. Again a more pronounced deviation

in our case is the fact that we do not look for

from the IBBM is observed at larger

Γ.

the asymptotic behavior, as has been usually

An alternative way to look at the information

done in previous works.

32,33,41

We took a dif-

from the bifurcation plots is to look at the cor-

ferent approach because industrial applications

responding isoperiodic maps".

For example

may include residence times that are small com-

isoperiodic map shown

pared to asymptotic times of several thousands

take a look at the

α-Γ

54

in Figure 2. Dierent colors indicate the periodicity according to the denition of Gilet et al.

22

of oscillation cycles used previously.

:

The map of Figure 2 could be an important


Page 5 of 20

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Figure 2:


Left:

Isoperiodic map

α−Γ

obtained from numerical determination of the ight time of

a vibrated granular bed, using Kroll's model. Green, blue and red color indicate regions with one, two and three or more periodic ights.

Right:

Illustration of the color code used for the isoperiodic

α = 0 corresponding with α = 0.5.

map at the left, for the bifurcation diagram of the IBBM (case with bottom horizontal line of the isomap at the left) and for the case

Figure 3:

Left:

Isoperiodic map

α−Γ

explored in some literature studies.

to the

with rectangles or lines indicating the range of parameters

Right:

Zoom of the

from experiments performed by Han et al.

42

α − Γ isoperiodic map showing two datasets

who used stainless steel spheres of dierent sizes.

White lled symbols corresponds to the dataset of gure 4 from reference,

42

taken at dierent

grain diameters, at 60 Hz in air. Grey lled symbols corresponds to the dataset of gure 5 from reference

42

0.35 mm, in air. 0.38 and air viscosity was taken to be 2×10−5 Pa·s.

taken at dierent frequencies for a xed grain diameter of

porosity was assumed to be equal to

Static bed Horizontal

error bars are taken from experimental data. Vertical error bars are propagated from experimental uncertainties reported by Han et al. and assuming a 0.1% uncertainty for the frequency, 1% for the viscosity and 5% for the static bed porosity.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 20

reference when dealing with the design of vi-

or more for the lowest frequencies and smaller

brated bed process machinery.

grains. It is worth recalling an observation re-

For example

25

during vibratory conveying, it may be impor-

ported by Douady et al.:

tant to assure a combination of parameters giv-

ter instead of air as interstitial uid, no period

ing a response of the bed with the same period-

bifurcation was observed for their glass spheres

icity of the forcing. A double or higher period

up to

scenario may represent a waste of energy since

ply

if one of the ights is the optimal, then the rest

eective mass corrections

of ights are not because the granular material

limiting case illustrating the large single period

does not reach the same dilation, or the same

zone predicted for large alpha ( α

when they used wa-

Γ = 13. This last conguration will imα ≈ 30 (without considering buoyancy and 58

), and represents a

> 2).

ight time. Of course there may be situations

In addition, in Figure 3 we compare the re-

where multiple periodicities are interesting, for

sults of Kroll's model with experimental data

example if the periodicity of the vibrating pro-

available from Han et al.

cess is to be avoided.

observation is that the qualitative agreement

Instead of inducing a

stochastic vibration, it could be enough to keep a sinusoidal vibration and tune red zone for your given

is remarkable.

Γ to the nearest

The rst important

Each data set indicates a bi-

furcation (a periodicity change).

α.

The rst bi-

furcation for both datasets coincides with the

Looking at the isoperiodic map of Figure 2

boundary between the rst red zone and the

there is a large green zone at the top and left

rst blue zone from the left. It is also notorious

region of the parameter space with a single pe-

how the concavity of the boundaries (bifurca-

α

values. Zones

tions) changes both in the theoretical map and

with 2 and 3 or more ights appear periodi-

riod which continues to higher

in the experimental data. Compare the theoret-

cally as one moves to the lower right corner of

ical green-red boundary beginning at

the map. A good rule of thumb will be to raise

with a decreasing derivative, with the green-red

α

boundary beginning at

and/or lower

Γ

in order to approach a single

ight zone, as it was expected.

For example,

Γ ≈ 12.8

Γ ≈ 3.1,

with an in-

creasing derivative. Experimental data at low

Γ

when dissipation is important (large

tends to lean towards right, while datasets with

gle period zone becomes

α) the sinindependent of Γ over

large

a wide range (in this case within the explored

tative agreement is good for the rst bifurcation

range), as was showed in a related study dealing

and worsens for higher

with ight dynamics of a grain lled ball.

pected due to the fact that assumptions like air

As we mentioned earlier, the limit

31

α −→ 0

is

Γ have increasing curvatures. Γ.

The quanti-

This worsening is ex-

uncompressibility and bed behaving as a con-

precisely the regime when it is known that the

densed block, are less suited at large

Γ.

IBBM describes the ight dynamics of granu-

Specic dependence of bed's ight dynamics on d, ρg , f , η and φ

lar beds in the limit of shallow and dense beds or when air is evacuated.

20,2527

of the order of magnitude of

To give an idea

α

we list in Ta-

ble 1 its value or range of values for some selected previous works. We also include in Figure 3 a graphical illustration of the parameter range explored by those works.

From a practical point of view, it may be possi-

On a coarse

ble to change only a single parameter at a time

approximation we can see how studies where

in order to change

the IBBM has been reported to hold (without vacuum)

20,26,27

have

α < 0.2.

Recent works on

the eect of air on vibrated bed dynamics have been done for

0.1 < α < 0.5.

of

Experi-

α,

together with plots depicting the devia-

tion from the IBBM with each parameter for some constants

where there is a strong and notorious eect of

α

Therefore we extend our

ve relevant parameters entering the denition

41,42,52

mental studies on the motion of vibrated beds interstitial air have

α.

research including isoperiodic maps for all the

Γ

where one period zone holds.

For each case, one parameter is varied within

ranges that can reach 10


Page 7 of 20

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Table 1: List of estimated

α

values or ranges used in some representative previous experimental 3 3 studies on vibrated beds. Here, it was assumed: ρglass = 2500 Kg/m , ρalumina = 3650 Kg/m , ρstainless stell = 7400 Kg/m3 ; ηair = 1.98 · 10−5 Pa·s, ηvacuum = 0 Pa·s, ηpropane = 8.71 · 10−6 Pa·s, ηhelium = 2.13 · 10−5 Pa·s; φ = 0.36 (random close packing).

Ref.

≈ d (mm)

2

26

Γ

α

40 - 70

Glass, calcite

Air

2 - 3.5

0.02 - 0.08

10 - 100

Glass, salt

Air - Vacuum

0-13

0 - 0.91

0.099 - 0.332

30 - 80

Glass, alumina, sand

Air

3 - 8

0.35 - 10.59

0.15 - .018

10 - 110

Bronze

Vacuum

1 - 8.5

0

0.2 - 2

7.78

Glass

Propane - Helium

1.3

0.04 - 4.40

1.28 - 3

15 - 40

Glass

Air

0 - 8

0.01 - 0.13

2.85

10

Glass

Air

0.5 - 2.5

0.04

0.088 - 0.707

25

Alumina

Air

2 - 7

0.17 - 11.01

0.5 - 1

30- 70

Stainless steel

Air

1 - 17.5

0.02 - 0.14

0.5

110

Glass

Air - Vacuum

1 - 6

0 - 0.11

0.35 - 1

40 - 80

Stainless steel

Air -Vacuum

1- 16

0 - 0.22

0.465

15 - 50

Stainless steel

Air

1 - 4

0.10 - 0.32

20 19 27

Interstitial gas

1 - 1.5

56

57

Grain

0.63 - 0.8

25 55

f (Hz)

41 42 52

Left:

Figure 4:

Isoperiodic map

d−Γ


a vibrated granular bed, using Kroll's model. Green, blue and red color indicate regions with one, two and three or more periodic ights (see Figure 2). diameter

d

at xed

Γ.


Γ

Right:

Dependence of ight time with grain

cases with a single ight time are shown to illustrate

the trend.



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Page 8 of 20

literature, while keeping the rest as indicated

≤ d ≤ 2 mm), however the eect of grain size is extremely weak above a grain size of 2 mm. The

in Table 2.

eect of grain size was also noted by Akiyama

a range spanning typical cases reported in the The particular set of parameters

59

in table 2 was chosen to lay around a value of

et al.

α ≈ 0.02, i.e., below the boundary where air ef-

bution along vessels of dierent materials ob-

fects begin to be important. The inuence of a

served at low grain diameters diminished when

change in each individual parameter its linked

the grain size was increased, and again this ob-

to the corresponding change in

α.

where dierences in the pressure distri-

Therefore we

servation is consistent with Kroll's picture in

α

the sense that air eects are negligible for large

for each case. Further illustration is provided in

diameters. If the interstitial uid is not air but

the supplementary material, where bifurcation

for example water, with much larger viscosity,

plots for each variable are shown (see Figure se-

it is possible to expect a grain size dependence

included the minimum and maximum value of

d > 2

quences S2, S3, S4, S5 and S6). In addition, and

of the vibrated bed dynamics for

in order to give an idea of how these trends are

Finally, Kroll's predictions are consistent with

modied when the set of parameters changes,

experimental results of Han et al.,

α ≈ 0.5

(see Table 3) where

air eect are more important.

when

It can be seen

Only the parameter indicated in the horizontal

time with respect to the IBBM.

axis is varied in each case.

Grain diameter d

Parameter is illustrated in Figure 4.

Symbol

Viscosity

tical structure except for low diameters, indi-

Frequency

cating how for diameters over 2 mm, the pe-

Porosity

riodicity of the bed's ight dynamics is almost

Grain diameter

d

for the

Γ

g η f φ d ρg

Gravity

The top left map shows an almost constant ver-

independent of

Γ

is reduced.

at the right side of Figures 4, 5, 6, 7 and 8.

giving stronger deviations of the ight

The sole eect of

d

Table 2: Parameters used to generate the plots

how the trends are similar, but with the case of

α ≈ 0.5

where it

is showed how bifurcations occur for higher

in Figure 9 we illustrate trends for a parameter set chosen around

42

mm.

range explored. For

•

Air

2

9.8 m/s × 10−5 Pa·s 60 Hz 0.44 1 mm

3 ? 2500 Kg/m at 300 K. • Experimentally measured 37 Sánchez et al. ? Soda-lime glass.

Grain density

d lower than 2 mm (bottom left) there is a clear change of the structure of the map with d, remarked at large Γ. To further illustrate the role of d we show on the right side of Figure

Value

by

4 the relative ight time predicted with Kroll's

Grain density

model with respect to the IBBM, for low val-

Γ (before the 1st bifurcation). For almost the whole d range, Kroll's prediction is the same

ues of

The isolated eect of

ρg

is explored in Figure 5,

that IBBM's. The inset shows a zoom for low

for a range covering values as low as the density

d,

where the deviation from the IBBM (corre-

of expanded polystyrene up to lead. Both the

sponding to a value of 1) can be appreciated in

isoperiodic map and plots at its right indicate

more detail.

how the eect of

For diameters around 1 mm the

dierence with the IBBM is lower than 5% (for the

Γ

values considered).

If

d

Γ.

The result for low

is appreciated at large

Γ

and for suciently low densities. The work of

is reduced the

Ze-Hui et al.

deviation becomes more pronounced, and increases with

ρg

27

is an example of a case where

a large density hinders interstitial air eects.

d agrees with

where it is re-

They used stainless steel beads, with a density 3 of ≈ 7400Kg/m and found that the value of Γ

ported that air eects depend strongly on grain

for the rst bifurcation was almost independent

the observation of Pak et al.

56

size (they explored precisely the range

0.1

mm

of grain size and similar to the values predicted


Page 9 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Left:

Figure 5:

Isoperiodic map

ρg − Γ


a vibrated granular bed, using Kroll's model. Green, blue and red color indicate regions with one, two and three or more periodic ights (see Figure 2). density

ρg

at xed

Γ.


Γ

Right:

Dependence of ight time with grain


the trend.

by the IBBM. The rst row of Table 1 from the work Ze-Hui et al.

27

of compares

Γ

an improvement with respect to the IBBM. The

for the 1st

improvement could have been more dramatic

bifurcation for their stainless steel beads, for

if lower frequencies, of the order of 10 Hz had

ceramic balls of similar size and for the glass

been used. An exception could be the work of

beads used by Douady et al.

Pastenes et al.

25

Heavier stain-

52

who vibrated at 15 Hz, how-

Γ

ever although their frequency was low, their

than ceramic beads (at 60 Hz), and both dis-

granular bed was rather dense (stainless steel

less steel beads show a lower 1st bifurcation play a lower 1st bifurcation used by

25

Γ

than glass beads

beads) thus reducing the value of

α,

and the

(at 30 Hz). Although 1st bifurcation

relevance of air on bed's ight dynamics. It is

agrees with Kroll's prediction, it is not the same

important to recall that many vibrated bed op-

for higher bifurcations (see the rest of Table 1

erations, like screening or sifting and vibratory

from

feeding, are performed at frequencies of even as

27

).

low as 300 RPM (5 Hz),

Vibration frequency The isolated eect of

f

sidered. Besides, many important eects of in-

f

terstitial air have been reported in the low fre-

appears both in

quency regime.

α and Γ (the parameters controlling dissipation cates how frequency aects bed dynamics spe-

Γ.

As

Γ

37,40,6164

Interstitial gas viscosity

and forcing within the model). The map indicially for large

thus studying the

low frequency regime is something to be con-

is explored in Figure 6.

It is of special interest since

46,60

decreases, the range of

The isolated eect of

η

is explored in Figure

variability with frequency shortens and connes

7.

to low frequencies. The right side of Figure 6

to viscosities as low as that of benzene up to

shows how Kroll's ight time compares to that

viscosities like that of oxygen, although rarely

of the IBBM. Recent studies about the bifurca-

gases dierent than air are used in the liter-

tion pattern of vibrated beds have been done at

ature (an example is the work of,

110 Hz

and even

1). The corresponding isoperiodic map has an

in those cases, the use of Kroll's model showed

almost uniform vertical structure that changes

41

and in the 40-80 Hz range,

42

The range of viscosity studied corresponds


56

see Table


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6:

Left:

Isoperiodic map

f −Γ

Page 10 of 20

obtained from numerical determination of the ight time

of a vibrated granular bed, using Kroll's model. Green, blue and red color indicate regions with one, two and three or more periodic ights (see Figure 2). oscillation frequency

f

at xed

Γ.


Γ

Right:


cases with a single ight time are shown

to illustrate the trend.

η

at dierences between a bed of elongated parti-

is rather small when compared to the eect of

cles like rice grains or pellets, in contrast with a

d

bed of spherical particles. The isoperiodic map

marginally at large or

f

Γ.

The relative eect of

and, like in those cases, it is more pro-

nounced at larger

Γ.

Figure 7 right shows how

of Figure 8 shows how the structure is almost

the larger the viscosity, the larger the deviation

unaected by changes in the porosity, except

of Kroll's model from the IBBM. Within our ex-

for large

Γ.

plored range deviations from the IBBM do not

Comments on applicability

overcome 2%.

Bed porosity

Besides the quantitative comparison seen at gure 3 left, the validity of our results is founded

The isolated eect of static bed porosity is explored in Figure 8.

The range of

φ

on the validity of Kroll's model.

studied

is rather exaggerated for a bed of monodis-

bed,

perse spherical particles for which the Kozeny-

φ.

One impor-

and even can be

66

. The

the lower left corner of all our isoperiodic maps, than for the upper right corner, were devia-

We must emphasize that under

tions due to higher speeds are expected. How-

Kroll's picture, the porosity is assumed as a

ever even with a frequency of 110 Hz and

constant during the whole vibration cycle (it is

between 1 and 6, Pastor et al.

important to recall that in the case of vibrated

41

Γ

obtain fair

agreement with their experimental results.

shallow beds, dilation comes from the top and

If

taller columns are to be considered, this anal-

bottom layers, with a central bulk having al-

ysis should have to be modied to include a

most constant density through the oscillation

52,65

57

assumptions of the model are better suited for

may be a possible change of behavior due to

cycle

it is a good approximation for the

applied to model transport phenomena

tant concern during a vibrated bed operation compaction.

4144,52

pressure drop along a bed

Karman relation (3) was conceived, however it oers an idea of the role of

References

show how it describes the ight dynamics of a

proper model of the viscoelastic response of

). In this section we study variations

the bed as well as corrections due to spatio-

of the static bed porosity, for example looking


Page 11 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Left:

Figure 7:

Isoperiodic map

η−Γ

obtained from numerical determination of the ight time

of a vibrated granular bed, using Kroll's model. Green, blue and red color indicate regions with one, two and three or more periodic ights (see Figure 2). interstitial gas viscosity

η

at xed

Γ.

Right:


Γ


cases with a single ight time are

shown to illustrate the trend.

Figure 8:

Left:

Isoperiodic map

φ−Γ


a vibrated granular bed, using Kroll's model. Green, blue and red color indicate regions with one, two and three or more periodic ights (see Figure 2). porosity

φ

at xed

Γ.


Γ

Right:

Dependence of ight time with bed's


the trend.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 20

temporal porosity changes. A natural next step

predicted by the model of Kroll. We briey re-

of analysis would be to introduce the compress-

view the model, emphasizing the fact that two

ibility of the interstitial uid into the picture,

parameters control bed dynamics within the

however in this case the solution is not as easy

model, namely the commonly used maximum

to obtain as in Kroll's model. Gutman

dimensionless acceleration

67

used

Γ and a parameter α

an iterative process and Fourier series expan-

dened in Eq. (6) quantifying dissipation due

sions, Akiyama and Naito

to air resistance.

68

used the ortogonal

discollocation method and Thomas et al.

57

used

We show the eects of

Γ and α on bed's ight

a semiempirical method where pressure proles

dynamics, compared with the behavior of the

were obtained from experiments and used to

inelastic bouncing ball model, where no air ef-

compute the gap height. One or several of these

fects are considered. The characteristic period

strategies could aid in a future work, however

doubling scenario seen for the inelastic bounc-

for the case of shallow beds, corrections should

ing ball model is modied (specially for large

be minor.

Γ)

g η f φ d ρg

Gravity Viscosity Frequency Porosity Grain diameter Grain density

range and, in The plot at

tionality of this ight time reduction, providing an important criterion to determine the signi-

Value

2

Γ

the right side of Figure 1 illustrates the func-

the horizontal axis is varied in each case.

Symbol

is increased, reducing the number

general, reducing the ight time.

Only the parameter indicated in

Parameter

α

of bifurcations seen at a given

Table 3: Parameters used to generate the plots of Figure 9.

when

cance of interstitial air on bed's ight dynamics. As an alternative way to look at the dynam-

9.8 m/s × 10−5 Pa·s

ics of a vibrated bed, with special interest in

60 Hz

practical industrial applications, we introduced

0.36

simplied isoperiodic maps, indicating combinations of parameters that give out a single

0.324 mm 2500 Kg/m

3

periodic motion, double period motion or motion with more than 2 ight times. These maps represent an important qualitative tool to aid the design of vibrated bulk materials process-

Recent experimental investigations on the ef-

ing equipment, or experimental protocols, or to

fect of interstitial air on ight dynamics of a vibrated bed

41,42,52

parameters giving

aid the decision making process when an un-

have used a combination of

α

expected change is to be applied to a working

near below the limit over

process. As a general rule of thumb, lowering

which air eects are dramatic (roughly 0.5). In

α

order to further test the applicability of Kroll's model predictions, it will be interesting for fu-

The parameter

ture experimental explorations to perform simeters resulting in a larger

α,

f,

for example us-

α

depends on ve quantities,

d, vibration frequency ρg , interstitial air viscosity η porosity φ. By looking at deviations

grain density

and bed's

52

of Kroll's model prediction of the ight time

but with glass beads of couple of hundreds of

from that of IBBM's within a typically reported

microns in diameter or lowering the frequency used by Pastor, Han and their co-workers

Γ will drive a vibrated bed

namely: grain diameter

ilar studies but with a combination of paraming the frequency range of Pastenes et al.

and/or increasing

to a zone with single period motion.

range of these parameters, we can conclude that

41,42

φ

to frequencies around 10 Hz.

and

while

η do not produce important deviations d, f and ρg can induce deviations of the

order of 25 % within the range explored in this

Conclusion

work (based on the range used by experiments reported in the literature). This deviation oers

We have oered a discussion on modication of

a reference frame to understand the functional-

the ight dynamics of a shallow vibrated bed as

ity of each parameter and how they determine


Page 13 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 9: Dependence of ight time with ve parameters at xed

Γ.


Γ

cases

with a single ight time are shown to illustrate the trend when the set of parameters is taken from Table 3.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

bed ight dynamics, information that is not ev-

Page 14 of 20

(5) Sloot, E.; Kruyt, N. Theoretical and Ex-

ident from the equation of motion consider by

perimental

Kroll's model. Combining the functionality of

Granular Materials by Inclined Vibratory

the deviations with the structure of isoperiodic

Conveyors.

maps, our analysis (within the context of pa-

203.

rameters indicated in Table 2) sustains how air

(6) Harada,

eects become important for grain diameters

Tomita,

under 1 mm, frequencies under 50 Hz and grain 3 densities below 500 kg/m , as have been exper-

from

the

Transport

Y.

Li,

H.;

Spouting

Vertically

2002,

Funatsu, of

Fine

Vibrated

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Chem.

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(7) Darias, J.; Sánchez, I.; Gutiérrez, G. Ex-

Acknowledgements

perimental Study on the Vertical Motion of Grains in a Vibrated U-tube.

Matter

We would like to thank G. Gutiérrez, S. McNamara, J.R. Darias, C. Colonnello and L. Reyes

2011,

13, 13 .

Granular

(8) Bukareva, M.; Chlenov, V.; Mikhailov, N.

for fruitful discussions and S. McNamara, C.

Drying Ferrite Oowder in a Vibratory Flu-

Colonnello, J. Petit and J. Renaud for revising the manuscript.

of

Powder Technol.

S.;

Eng. Sci.

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Study

Powder Metallurgy and Metal Ceramics 1967, 6, 664. idized Bed.

This work was funded

by projects IVIC-087 and DID-USB-S1-IN-CB004-10.

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C.;

Pakowski,

Z.

Drying

of

Granular Products in Vibrouidized Beds.

Drying '80,

Supporting Information Supporting information available.

Figure se-

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d, ρg , f , η

and

φ.

(10) Thomas,

α,

Liu,

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//pubs.acs.org.

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(61) Möbius, M.; Nagel, S.; Dust: tion.

Cheng, X.;

1

(62) Caballero,

L.;

Melo,

Droplets

xed

Phys. Rev. Lett.

2004,

α values compared with

Right:

De-

α at low Γ

Γ.

Only a couple of

cases with a single ight time are shown to illustrate the trend.

of

2

Fine Powders Running Uphill by Vertical Vibration.

for

pendence of ight time with

93, 198001.

F.

diagram

that of the IBBM.

Air-Driven Granular Size Separa-

2004,

Bifurcation

dierent

Karczmar, G.;

Jaeger, H. Intruders in the

Phys. Rev. Lett.

Left:

Left:

Isoperiodic

map

. .

obtained from numerical deter-

93,

mination of the ight time of

258001.

a vibrated granular bed, using Kroll's model.

(63) van Gerner, H.; van der Hoef, M.; van der

Green, blue and

Meer, D.; van der Weele, K. Interplay of

red color indicate regions with

Air and Sand:

one, two and three or more pe-

elled.

Faraday Heaping Unrav-

Phys. Rev. E

2007,

76, 051305.

riodic ights.

Right:

Illustration

of the color code used for the

(64) Liu, C.; Wu, P.; Zhang, F.; Wang, L. Par-

isoperiodic map at the left, for

ticle Climbing Induced by Reciprocating

the bifurcation diagram of the

Air Flow.

Appl. Phys. Lett.

2013,

102,

IBBM (case with

183507.

and for the case with

Dimensional Simulations of a Vertically Vibrated Granular Bed Including Intersti-

Phys. Rev. E

(66) Clement,

C.;

2009,

79, 051301.

Pacheco-Martinez,

H.;

Swift, M.; King, P. Partition Instability Water-Immersed

Phys. Rev. E

2009,

Granular

80, 011311.

Systems.

(67) Gutman, R. Vibrated Bed of Particles: A Simple Model for the Vibrated Bed.

Trans. Inst. Chem. Eng.

1976,

54, 174.

(68) Akiyama, T.; Naito, T. Vibrated Bed of Powders: A New Mathematical Formulation.

corre-

tal line of the isomap at the left)

rez, G.; Reyes, L. I.; Botet, R. Three-

in

α = 0

sponding to the bottom horizon-

(65) Idler, V.; Sánchez, I.; Paredes, R.; Gutiér-

tial Air.

4

α − Γ

Chem. Eng. Sci.

1987,

42, 1305.


α = 0.5.

. .

5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3

Left:

Isoperiodic map

α − Γ with

5

ρg − Γ

Isoperiodic map

rectangles or lines indicating the


range of parameters explored in


some literature studies.

a vibrated granular bed, using

Zoom of the

α−Γ

Right:

isoperiodic

Kroll's model.

Green, blue and

map showing two datasets from


experiments performed by Han

one, two and three or more peri-

et al.

odic ights (see Figure 2).

42

who used stainless steel

Right:

spheres of dierent sizes. White


lled symbols corresponds to the

grain density

dataset of gure 4 from refer-

a couple of low

ence,

single ight time are shown to il-

42

taken at dierent grain

diameters, at 60 Hz in air. Grey

ρg

at xed

Γ

Γ.

Only

cases with a

lustrate the trend. . . . . . . . . .

lled symbols corresponds to the

6

Left:

Isoperiodic


ence


42

taken at dierent frequen-

for

ter of

a

0.35

xed

grain

diame-

mm, in air.

Static

a vibrated granular bed, using Kroll's model.

bed porosity was assumed to be equal to

Green, blue and


0.38

and air viscosity −5 was taken to be 2 × 10 Pa·s.


Horizontal error bars are taken


from experimental data.

oscillation frequency

Right:


Ver-

f

at xed

tical error bars are propagated

Γ.

from experimental uncertainties


reported by Han et al.

and as7

the frequency, 1% for the vis-

Isoperiodic

map

a

couple

of

low

Left:

Isoperiodic

map

Γ . .

mination of the ight time of 5


d − Γ

Kroll's model.

Green, blue and




one,


periodic ights (see Figure 2).

Kroll's model.

two

Right:

Green, blue and

and

three

or

more

Dependence of ight time

η at low Γ


with interstitial gas viscosity


xed



Right:

d at xed Γ. Only low Γ cases with a

grain diameter

single ight time are shown to illustrate the trend. . . . . . . . . .

Γ.

Only a couple of


Dependence of ight time with a couple of

10

η − Γ


cosity and 5% for the static bed

Left:

Only


suming a 0.1% uncertainty for

porosity. . . . . . . . . . . . . . .

9

f − Γ

map

dataset of gure 5 from refercies

4

Left:

Page 18 of 20

7


. .

11

Page 19 of 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research 8

Left:

Isoperiodic

φ − Γ

map

obtained from numerical determination of the ight time of a vibrated granular bed, using Kroll's model.

Green, blue and

red color indicate regions with one, two and three or more peri-

Right:



φ at xed Γ. Only low Γ cases with a

bed's porosity a couple of

single ight time are shown to illustrate the trend. . . . . . . . . . 9

11

Dependence of ight time with ve parameters at xed a couple of low

Γ

Γ.

Only

cases with a

single ight time are shown to illustrate the trend when the set of parameters is taken from Table 3.

13

List of Tables 1

List of estimated

α

values or

ranges used in some representative previous experimental studies on vibrated beds.

Here, it

ρglass = 2500 = 3650 Kg/m3 , ρstainless stell = 7400 Kg/m3 ; ηair = 1.98 · 10−5 Pa·s, ηvacuum = 0 Pa·s, ηpropane = 8.71 · 10−6 −5 Pa·s, ηhelium = 2.13 · 10 Pa·s; φ = 0.36 (random close pack-

was

assumed: 3 Kg/m , ρalumina

ing). 2

. . . . . . . . . . . . . . . .

7

Parameters used to generate the plots at the right side of Figures 4, 5, 6, 7 and 8. Only the parameter indicated in the horizontal axis is varied in each case.

3

. . . .

8

Parameters used to generate the plots of Figure 9.

Only the pa-

rameter indicated in the horizontal axis is varied in each case.

. .

12



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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