Plug Conveying in a Horizontal Tube

0.27 MB·English

by  Martin Strauss

Granular Matter manuscript No.
(will be inserted by the editor)
5
Martin Strauß · Sean McNamara · Hans J. Herrmann
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Plug Conveying in a Horizontal Tube
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Abstract Plug conveying along a horizontal tube has layers,the slowly moving bed at the bottom of the pipe
.
t been investigated through simulation, using a discrete and the traveling ripples or the “slugs” [36; 34; 22; 45;
a
m element simulation approach for the granulate particles 39], where slugs are accumulations of particles which fill
and a pressure field approach for the gas. The result is up the cross-sectionof the tube andmove quickly in the
-
d compared with the simulation of the vertical plug con- direction of transportation. Slug conveying occurs when
n veying.Thedynamicsofaslugaredescribedbyporosity, the ratio of grainflux to gas flux is high. This transport
o velocity and force profiles. Their dependence on simula- mode is sometimes also called “plug conveying”, in this
c
tion parameters provides an overall picture of slug con- paper this term is used in reference to vertical convey-
[
veying. ing. Currently plug conveying is gaining importance in

1 industry,becauseit causeslessproductdegradationand
v Keywords slug conveying · slug · dense phase · pipeline erosion than dilute phase conveying.
9 pneumatic transport · granular medium
Unfortunately,currentmodels[16;31]ofslugconvey-
1
PACS 47.55.Kf · 45.70.Mg ing disagree even on the prediction of such basic quan-
7
1 tities as the pressure drop and the total mass flow, and
0 thesequantitieshaveagreatimpactintheindustrialap-
5 1 Introduction plication. One of the reasons for the lack of valid mod-
0 els is that it is difficult to study slugs experimentally in
/
t A quite commonmethod for the transportationof gran- a detailed way. Usually experimental setups are limited
a
to the measurement of the local pressure drop, the to-
m ular media is pneumatic conveying, where grains are
tal mass flux and the velocity of the slugs. The most
driven through pipes by air flow. Practical applications
- promising experimental studies have been performed by
d for pneumatic conveying can be found in food indus-
electrical capacity tomography [14; 45; 2] and stress de-
n try and in civil and chemical engineering. One distin-
o guishes two modes of pneumatic conveying: dilute and tectors [27; 26; 39]. Plenty of experimental research has
c dense phaseconveying.Dilute phaseconveyinghas been been done on the prediction of the total pressure drop
: along a horizontal conveying line [9; 28; 25; 29; 22; 17]
v studied in much detail [1; 7; 30; 24; 21; 3; 41; 20] and is
i well understood. The grains are dispersed and dragged Simulational studies are handicapped by the high com-
X putationalcost for solving the gasflow and the particle-
individually by the gasflow andthe interactionbetween
r grainsisweak.Thisisnottruefordensephaseconveying, particle interaction, and are therefore mostly limited to
a
twodimensions.Forthedensephaseregime,simulations
where the particle interaction is important and where
have been done for bubbling fluidized beds [42; 37; 15;
particle density waves can be observed. The behavior
13;8;44],whichshowathighgasvelocitiesthefirstsigns
of these density waves depends strongly on the orienta-
of pneumatic transport [19; 40; 12], for the strand type
tion of the tube. In vertical plug conveying, most of the
ofconveying[10;35;38;43],andforslugconveying.The
particles are located in “plugs” (dense regions) that are
simulation results on slugs published by Tsuji et al [36],
pushed up the tube by the pressure gradient [32]. In a
Tomitaetal[34]andLevy[18]arediscussedlaterinthis
horizontaltube the granular medium separates into two
paper.
M. Strauß Thegoalofthispaperistoprovideadetailedviewof
Institut fu¨r Computerphysik,Universit¨at Stuttgart,
slugs, by using a discrete element simulation combined
70569 Stuttgart,GERMANY
Tel.: [phone] with a solver for the pressure drop. This approach pro-
Fax: [phone] vides access to important parameters like the porosity
E-mail: [email protected] andvelocityofthe granulateandthe shearstressonthe
2
wall at relatively low computational cost. Contrary to the Carman-Kozeny relation [4] was chosen, which pro-
the experiments, it is possible to access these parame- vides a relation between the porosity φ, the particle di-
terswithhighspatialresolutionandwithoutinfluencing ameter d and the permeability of a granular medium of
the process of transportation. Additionally to slug pro- monodisperse spheres,
files, characteristic curves of the pressure drop and the
d2φ3
influence of simulation parameters are measured. The κ(φ)= . (4)
simulation results are compared to the results for verti- 180(1−φ)2
cal plug conveying [32].
After linearizing around the normal atmospheric pres-
sure P0 the resulting differential equation only depends
′ ′
ontherelativepressureP (P =P0+P ),theporosityφ
2 Simulation Model andthe granularvelocityu,whichcanbeobtainedfrom
theparticlesimulation,andsomematerialconstantslike
Plug conveyingis a special case of the two phase flow of the viscosity η:
grains and gas. It is therefore necessary to calculate the
′
motionofbothphases,aswellastheinteractionbetween ∂P = P0∇(κ(φ)∇P′)− P0∇u. (5)
them. In the following, we explain how our algorithm ∂t ηφ φ
treats each of these problems.
This differential equation can be interpreted as a dif-
fusion equation with a diffusion constant D = φκ(φ)/η.
Theequationissolvednumerically,usingaCrank-Nickelson
2.1 Gas Algorithm approach for the discretization. Each dimension is inte-
grated separately.
Themodelforthegassimulationwasfirstintroducedby The boundary conditions are imposed by adding a
McNamara and Flekkøy [23] and has been implemented term ∓S on the right hand side of equation (5) at the
for the two dimensional case to simulate the rising of top and the bottom of the tube, where S ∝ vgP0. This
bubbleswithinafluidizedbed.Forthesimulationofslug mimicsaconstantgasflux withvelocityv atapressure
g
conveying we developed a three dimensional version of P0 into and out of the tube.
this algorithm.
The algorithm is based on the mass conservation of
thegasandthegranularmedium.Conservationofgrains 2.2 Granulate Algorithm
impliesthatthedensityρ ofthegranularmediumobeys
p
Themodelforthegranularmediumsimulateseachgrain
individually using a discrete element simulation (DES).
∂ρ
p +∇·(uρ )=0, ρ =ρ (1−φ), (1) For the implementation of the discrete element simula-
p p s
∂t tionweusedaversionofthemoleculardynamicsmethod
described by Cundall and Strack [5]. The particles are
where the specific density of the particle material is ρ ,
s
approximatedby monodisperse spheres and rotations in
the porosity of the medium is φ (i.e. the fraction of the
three dimensions are taken into account.
space available to the gas),and the velocity of the gran-
ulate is u. The equation of motion for an individual particle is
The mass conservation equation for the gas is ∇P
mx¨ =mg+F − , (6)
c
ρ (1−φ)
∂ρ s
g +∇·(v ρ )=0, (2)
∂t g g wherem is the mass ofa particle,g the gravitationcon-
stant and F the sum over all contact forces. The last
where ρ is the density of the gas and v its veloc- c
g g term, the drag force, is assumed to be a volume force
ity. This equation can be transformed into a differential given by the pressure drop ∇P and the local mass den-
equationforthegaspressureP usingtheidealgasequa-
sity of the granular medium ρ (1−φ).
tion ρ ∝ φP, together with the assumption of uniform s
g The interaction between two particles in contact is
temperature.
given by two force components: a normal and a tan-
For small Reynolds numbers the velocity v of the
g gential component with respect to the particle surface.
gas is related to the granulate velocity u through the
The normal force is the sum of a repulsive elastic force
d’Arcy relation:
(Hooke’s law) and a viscous damping. The tangential
η forceisproportionaltotheminimumofthenormalforce
−∇P = φ(v −u), (3)
κ(φ) g (sliding Coulomb friction) and a viscous damping. The
viscousdamping is used when the relativesurface veloc-
where η is the dynamic viscosity of the air and κ is the ity of the particles in contact is small. The same force
permeability of the granular medium. This relation was laws areconsideredfor the interactionbetween particles
first given by d’Arcy in 1856 [6]. For the permeability κ and the tube wall.
3
2.3 Gas-Grain Interaction pressureissettoP0 =1013.25hPa,Simulationsarepre-
formed for gas viscosities η from 0.045cP to 0.085cP
The simulation method uses both a continuum and a and gas flows V˙ between 1.1l/min and 9.2l/min. The
discreteelementapproach.While the gasalgorithmuses gas flow is usually given by the superficial gas veloc-
fields,whicharediscretizedonacubicgrid,thegranulate ity v = φv = 4V˙/πD2 [11], where v is the equivalent
s g t g
algorithmdescribesdiscreteparticlesinacontinuum.An gas velocity for an empty tube.
interpolationisneededforthealgorithmstointeract.For Theflowinthecorrespondingexperimentforthever-
the interpolation a tent function F(x) is used: tical conveying [32] is turbulent (particle Re ≈ 65). In
the simulation, an effective gas viscosity is used to ac-
1−|x/l|, |x/l|≤1, count for the effect of turbulence. For an effective gas
F(x)=f(x)f(y)f(z), f(x)= viscosity η =0.0673cP,a gas flow V˙ =2.3l/min and a
(0, 1<|x/l|,
Coulomb coefficient µ = 0.5 slug conveying is observed
(7) as shown in figure 1. The measured pressure drop is
10hPa/m, the slug velocity is 0.42m/s, and the slug
where l is the grid size used for the discretization of the
length is 4-8cm. An image of a corresponding slug is
gas simulation.
shown in figure 2.
For the gas algorithm the porosity φ and the gran-
j As pointedoutinthe introduction,somesimulations
ular velocity u must be derived from the particle posi-
j ofhorizontalslugconveyinghavealreadybeenpublished.
tions x and velocities v , where i is the particle index
i i The horizontal transport of a slug has been studied by
andj istheindexofthegridnode.Thetentfunctiondis-
Tsujietal[33].Thetube withdiameter5cmandlength
tributes the particle properties aroundthe particle posi-
80cm contained 1000 particles with a diameter of 1cm,
tion smoothly on the grid:
whichthesamediameterratioasusedinthisstudy.Con-
trary to our simulations, the slug was created by the
1
φ =1− F(x −x ), u = v F(x −x ), initial conditions, where a given range of the tube was
j i j j i i j
1−φj filled with particles. Nevertheless the qualitative behav-
i i
X X
ior of the particles is the same as shown in figure 2 and
(8)
17.
where x is the positionof the gridpoint andthe sumis Tomita et al [34] assumed the slugs to be indivisible
j
taken over all particles. objects. The computed trajectories of the slugs are sim-
For the computation of the drag force on a particle ilar to the trajectories observed in our studies, however
thepressuredrop∇P andtheporosityφ attheposition he neglects the possibility of slugs to grow or even to
i i
of the particle are needed. These can be obtained by a dissolve which is seen in our results.
linear interpolation of the fields ∇P and φ from the Levy [18] applied a two fluid approach on the hor-
j j
gas algorithm: izontal conveying. He observes the break up of a large
artificially build slug into smaller slug. Such behavior is
φ = φ F(x −x ), ∇P = ∇P F(x −x ), observed neither in our simulation nor in Tsuji’s.
i j j i i j j i
j j
X X
(9)
3.1 Characteristic curves
where the sum is taken over all grid points.
The“characteristiccurves”ofapneumatictransportsys-
temareplotsofthepressuredropagainstthe superficial
3 Simulational Results gasvelocityv =φv fordifferentmassflowsofthegran-
s g
ulate. This kind of diagram is highly dependent on the
The setup for the simulation consists of a horizontal material characteristics of the tube wall and the gran-
tube of length L = 52.5cm and of internal diameter ulate and can be used to predict the overall transport
t
D = 7mm. The air and the granular medium is in- performance for given parameter sets. Such a diagram,
t
jected at a constant mass flow rate at on end of the from data of our simulation, is shown in figure 3.
tube.Atthebeginningofasimulationthetubeisempty. The diagram provides the typical qualitative behav-
The default parameters were chosen to be the same as ior for pneumatic transport. From top left to bottom
in an earlier paper [32] for vertical plug conveying. As right with increasing superficial gas velocity, three re-
default the mass flow rate of the granular medium is gionscanbedistinguished.First,forsmallsuperficialgas
2.49kg/h.Defaultparametersfortheparticlesare:diam- velocities (v < 0.38m/s), one has bulk transport. The
s
eter d = 1.4mm, density ρ = 937kg/m3, Coulomb co- tube is completely filled with granulate, so the pressure
s
efficient µ = 0.5 and restitution coefficient e = 0.5. The drop is high. Nevertheless the drag force on the bulk is
gasvolumehasbeendiscretizedinto150x2x2gridnodes, too small to overcome the friction between the granular
which corresponds to a grid constant of 3.5cm. The gas medium and the wall. In this case the transport comes
4
ulate between the slugs is negligible and causes only a
40
2.49 kg/h small deviation. At high gas velocities this is no longer
9.95 kg/h
35 true, because the plugs no longer contain the majority
of the particles.
30 Inthefollowing,thedependenceofthepressuredrop
m) onthemassflux ofthegranulate,theairviscosityη,the
a/ 25
hP atmosphericpressureP0andtheCoulombcoefficientµis
p ( discussed. For the parameter studies the superficial gas
o 20
dr velocityhasbeenfixedto1m/s.Forhighervelocitiesthe
e
essur 15 sensitivity to the parameter values decreases.
pr
10
30
5
0
25
0 0.5 1 1.5 2 2.5 3 3.5 4
superficial gas velocity v (m/s)
s m)
Fig. 3 Total pressure drop against superficial gas velocity Pa/ 20
for different granular mass flows. Plotted are characteris- p (h
tic curves from the simulation for the granular mass flows o
2.49kg/hand9.95kg/h.Formassflow2.49kg/hbulktrans- e dr
port is observed when the superficial gas velocity is below ur 15
0.25m/s. ess
pr
10
through the enforced granular mass flow at the inlet of
the tube.
5
For moderate velocities (0.38m/s ≤ v < 4m/s),
s 1 2 3 4 5 6 7 8 9 10
slugconveyingisobserved(fig.1).Theparticlesinjected
mass flux (kg/h)
attheinletorganizeintoslugs.Afterashortacceleration,
Fig. 5 Dependenceof thepressuredrop on themass fluxof
the slugs move forward with a constant velocity. A slug
thegranulate at a superficial gas velocity 1m/s.
always leaves particles behind it and usually maintains
its length by collecting particles in front of it. A slug
disintegrateswhenitgetstoosmall.Theparticlesbehind Asonecanseeinfigure5thepressuredropincreases
the slug rest at the bottom of the tube. The amount of linearly with mass flux of the granulate. The increase in
particles left behind a slug is independent of the slug the pressure drop is associated with an increase of the
size. Largerparticle amounts left by a dissolvedslug are number of slugs within the tube.
collectedby the followingslugandcauseitto grow.The The gas flow can be influenced by changing the at-
porosity of the granular medium inside a slug is close to mospheric pressure P0 or the diffusion constant D. An
the minimum porosity, and the slug edges are smooth. increase in background pressure P0 combined with an
For high superficial gas velocities (vs > 4m/s) the increase in superficial velocity leaves the pressure drop
tube is almost empty (fig. 4); in the simulation the par- unchanged. This can be deduced directly from equation
ticlesarepushedouteitherassmallslugsorasindividual (5)bynoting thatbothterms onthe righthandside are
particlesslidingonthebottomofthetube.Theporosity proportional to P0.
increases with the superficial gas velocity. In this region The diffusion constant D ∝ d2/η can be changed
thesimulationmethodunderestimatesthepressuredrop, throughtheparticlediameterdandtheviscosityη.There-
becauseitdoesnotconsiderthe increasingdragforceon foreitissufficienttoanalyzetheparameterspaceforthe
single particles, which in the experiment dominates in viscosityataconstantdiameterasshowninfigure6.The
this region. The boundaries between the described re- pressure drop is decreasing with increasing viscosity.
gions depend on the simulation parameters. The parameter of the particle simulation with most
A nearly proportional relation is observed between influence on the transport of the granular medium is
thetotalpressuredropandthetotalnumberofparticles theCoulombcoefficientµ.Therestitutioncoefficienthas
in the tube. This can be explained by the observation onlyasmalleffect,exceptwhenitisunrealisticallylarge.
that the amount of particles between the slugs is small Asonecanseeinfigure7thepressuredropincreases
and in the slugs most particles are densely packed at a with µ. For low Coulomb coefficients (µ < 0.1) the in-
well defined porosity. Through d’Arcy’s law, the total jected particles are sliding individually along the tube
pressure drop depends linearly on the tube length filled with accelerating speed. As most of the tube is empty,
with this porosity, since the pressure drop on the gran- thetotalpressuredropissmall.Asµincreases,theparti-
5
3.2 Slug statistics
13
12.5 Thespatio-temporalimageoftheporosityalongthehor-
izontaltube(fig.1)providesaroughpictureofslugsand
12
theirmovementalongthetube.Astatisticalapproachis
m) 11.5 necessaryto getmoreprecisevalues.Propertiesofinter-
a/
P est are the porosity and the granular velocity within a
h
p ( 11 slug, the slug length, and their dependence on the hori-
o
e dr 10.5 zontalpositionxoftheslugwithinthetube.Togetsome
ur averagevaluesfortheporosityandthegranularvelocity,
press 10 the tube was segmented into horizontal slices of length
3.5mm.Foreachslicetheaverageporosityandgranular
9.5
velocity was computed every 0.01s. The contribution of
9 a particle was weighed by the volume occupied by that
particle within a given slice.
8.5
0.04 0.05 0.06 0.07 0.08 0.09 The resulting vertical porosity was used to identify
viscosity h (cP) slugs. Every regionwith a porosity lower than 0.6 is de-
fined to belong to a slug.
Fig.6 Dependenceofthepressuredroponthedynamicvis-
cosity η of thegas at a superficial gas velocity 1m/s.
12
12
10
10
z) 8
H
m) 8 me (
hPa/ er ti 6
op ( 6 gs p
e dr plu 4
ur
ess 4
pr 2
2
0
0 0.1 0.2 0.3 0.4 0.5
0 x (m)
0 0.1 0.2 0.3 0.4 0.5 0.6
Fig. 8 Number of slugs per time as a function of tube po-
Coulomb coefficient m
sition x. The corresponding spatio-temporal image is shown
Fig. 7 Dependence of the pressure drop on the Coulomb in figure 1. Thetotal tubelength is 52.5cm,and default pa-
coefficient µ. The superficial gas velocity is 1m/s. For low rameters are used. The data is averaged over 22.5s.
Coulombcoefficients(µ<0.1)evensmallparticlegroupsare
able to slide within the tube,so slugs donot occur.
Figure 8 shows the slug rate, or number of slugs per
time asa function ofthe positionalongthe tube. At the
inlet of the tube the incoming granular medium frag-
clesstartfirsttoslideasonelayer.Thentheparticlelayer ments into manysmallslugs.Eventhoughtheir velocity
gets slower and thicker and therefore causes an higher is low, the resulting slug rate is high. As can be seen in
pressure drop. At a Coulomb coefficient of µ = 0.1, a figure1,manyslugsdissolvealongthetube,thereforethe
peak in the pressure drop coincides with the transition slug rate decreases. Most slugs dissolve within the first
fromtheconveyingbyaslidingparticlelayertoslugcon- 0.3m along the tube. Beyond x = 0.3m the number of
veying. Slugs first occur when the cross-section of the slugs remains stable until they leave the system.
tube is filled locally somewhere along the tube. They Foreachslug,the centerofmass,the minimalporos-
aremoreefficientintransportingparticles,thereforethe ity, the maximal granular velocity and the slug length
pressure drop first decreases until a pure slug convey- have been computed.
ing is reached (0.1 < µ < 0.15). For higher coefficients Figure9 showsthe minimum slugporosityas afunc-
(µ > 0.15) the slugs become slower due to the growing tion of the horizontalposition x of a slug. At eachvalue
friction and cause therefore a rising pressure drop. of x, the mean porosity and its uncertainty (standard
6
0.6 0.45
0.58 0.4
0.56 0.35
0.54 m/s)
y ( 0.3
fporosity 0 0.5.52 ar velocit 0.25
ul 0.2
n
a
0.48 gr
0.15
0.46
0.1
0.44
0.05
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
x (m) x (m)
Fig. 9 Minimal porosity in slugs at tube position x, corre- Fig. 10 Maximal granularvelocity inslugs at tubeposition
spondingtotheslug rateshown in figure8.Thebarsdenote x,correspondingtotheslugrateshowninfigure8.Thebars
the uncertainty of the average, and the dotted lines indicate and dotted lines have the same meaning as in figure 9. The
the width of the distribution; at each value of x, half of the granular velocity saturates in the middle of the tube, the
observed slugs have a porosity lying between the upper and increase of thegranular velocity at thebeginning and at the
thelower dotted lines. The slug porosity decreases when the end of the tube is due to the boundary effects described in
granulate enters the tube (x <0.2m) and increases when it figure9.
leaves thetube (x>0.45m).
deviation divided by the square root of the number of
Onsmall time scales the gasvelocity v can be assumed
slugs) were calculated. These quantities are shown by g
constant. Only the drag force depends on the granular
the bars in figure 9. To show the distribution of poros-
velocity u through equation (3). For a certain granular
ity aboutthe mean,the twodotted lines wereadded.At
velocity the drag force balances the friction forces, and
each position x, half of the slugs have a porosity lying
thus u remains constant. The solution is stable under
between these two lines. The same analysis was carried
small fluctuations in u.
out for the data in Figures 10 and 11. As one can see in
figure 9 at the left of the graph, the granular medium is
inserted at the inlet of the tube with a porosity of 0.57.
From there the porosity decreases quickly to about 0.47
at a height x=0.2m and then remains almost constant 0.09
until x = 0.45m. At the end of the tube (x > 0.45m)
0.08
the porosity increases again until the grains leave the
simulation space at x=0.525m.
0.07
Figure 10 shows the corresponding particle velocity.
The change in porosity comes along with an increase of 0.06
m)
the granular velocity within the slugs. The granulate is h ( 0.05
inserted with an initial velocity of 0.04m/s. The gran- gt
n
ular velocity saturates to a final velocity (0.3m/s) at a g le 0.04
positionof0.2m.Starting atabout0.45m,the granular plu
0.03
medium accelerates until the grains leave the tube.
An explanation for the constant slug velocity in the 0.02
middleofthetubecanbederivedusingthebalanceequa-
tion for the forces on a slug: 0.01
0
F =−Fc+α(φ)(vg −u). (10) 0 0.1 0.2 0.3 0.4 0.5
x (m)
whereF is the forceonthe slug due tothe frictionwith
c
Fig. 11 Mean sluglengthat tubeposition x,corresponding
the wall. The last term α(φ)(v −u) is the drag force
g totheslug rate shown in figure 8. Thebars and dotted lines
ontheslug,whichisproportionaltotherelativevelocity havethesamemeaningasinfigure9.Theaveragesluglength
betweentheparticlesandthegaswithintheslug.Atthe increases along the tube, consistent with the decrease in the
middle ofthe tube theporosityoftheslugφisconstant. slug rate (fig. 8).
7
Figure11showsthemeansluglengthalongthetube.
1
The averagesluglength increasesalongthe tube, due to
the collection of particles of dissolved slugs by the fol-
0.9
lowing ones. Beyond x = 0.3m a stable slug length is
reached. The increase of the slug length combined with
thedecreaseofthenumberofslugspertime(fig.8)con- 0.8
serves the mass flux of the granulate.
aryAcosncdaintioonneastexen=fr0o.m525Fmigsa.ff9e,c1t0s aonndly1a1,smthaellbroeugniodn- forosity 0.7
p
close to the end of the tube. In vertical plug convey-
0.6
ing,this boundary regionis much larger.This difference
arisesbecausethegrainsbetweentheplugsorslugshave
different behaviors. In vertical conveying, particles are 0.5
falling downwards, so that when a plug is removed at
the top of the tube, the flux of particles onto the next
0.4
plug is reduced shortly thereafter. On the other hand, −40 −30 −20 −10 0 10 20
inhorizontalconveying,thegrainsbetweentheslugsare D x/d
simply lying on the bottom of the tube. When a slug
Fig. 12 Horizontal porosity profile along an averaged slug,
is removed at the end of the tube, this information is containing about 470 particles, positioned at the middle of
nottransmitted to the next plug.The succeedingslug is the tube. The profile was averaged over six slugs with gran-
only influenced by the thickness of the particle layerleft ular velocity 0.29±0.1m/s and slug length 0.05±0.03m.
Thedatabelong tothesimulation displayed in figure1. The
behind. Another consequence of this difference is that,
horizontal axis denotes the relative horizontal position ∆x
contraryto the verticalconveying,the distance between
along the tube with respect to the center of mass. The hori-
slugs has no effect on their interaction. zontal position is given in multiples of the particle diameter
The slug profiles (Fig. 12-16) indicate that the slugs d=1.4mm.
only interact through the thickness of the particle layer
left behind them. Defining everything with a porosity
side of the slug, on the right hand side of figure 12, the
lower then 0.6 as belonging to a slug, the interaction
porosity decreases from 85% to 50%. In the middle the
range with the particle layer is of the size of the slug
porosity of the slug remains almost constant, and in-
length.
creases at the backside of the slug. The high porosity
The diagramsin figures 9 to 11 imply that there is a
before andafter the slug correspondsto a dense particle
typicalporosity,granularvelocityandlengthofslugsand
layer at the bottom of the tube and a region devoid of
acharacteristicslugprofileforagivenpositionalongthe
particles above this layer (fig. 17).
tube exists.Inthe followingaveragedverticalandradial
Figure13displaysthe velocityprofileofthegranular
profilesofslugsatthepositionof0.26marediscussed.To
mediumfortheslugsshowninFigure12.Onecandistin-
getsomesensibleprofiles,slugswithlengthandgranular
guish four different regions: Ahead of the slug (∆x/d ≥
velocity close to the mean values (L = 0.05±0.03m,
p 30), the high porosity corresponds to the few particles
u=0.29±0.1m/s) were selected.
resting at the bottom of the tube, averaged over the
cross-section of the tube. These particles originate from
the backside of a preceding slug. The friction outweighs
3.3 Horizontal slug profiles the drag force on the particles. At the front side of the
slug (10≥∆x/d>30), the particles accelerate, as they
Whileinexperimentsrecognizingandmeasuringparam- are pushed by the low porosity region. In the following,
eters for global slug conveying are rather simple, the thisregionatthe frontofthe slugwillbe calledcollision
measurement of profiles for individual slugs remains a region.Insidetheslug,wheretheporositysettlestoalow
nearly impossible task. So one of the reasons for simu- value, the average granulate velocity is almost constant
lating slug conveying is to provide a detailed picture of (|∆x/d|>10).Atthebackside(−10<∆x/d≤−40)of
what happens within a slug, and how parameters like the slug the granular medium slows down until it rests
porosity, granular velocity, or shear stress change along again. Thus the slug is always loosing material at the
the slug. back side. This region will be called disintegration re-
Figure 12 to 16 present averages of different quan- gion.
tities over six slugs. These slugs were taken from the The trajectory of a single particle through the slug
middle of the tube x = 0.26m with granular velocity can be sketched by a snapshot of the horizontal forces
0.29±0.1m/sandsluglength0.05±0.03m.Thecoordi- acting on it. Figure 14 displays the drag force F and
d
nate ∆x denotes the relative vertical position along the the sum over the interparticle and friction forces on a
tube with respect to the center of mass. The porosity particle (here called particle forces)F . These forces are
p
profile of the slugs is shown in Figure 12. At the front averaged over the cross-section of the tube. Before en-
8
becomes negligible andthe particles aresloweddownby
0.3
the friction with the wall until they come to rest.
0.25
m/s) 0.2 60
y (
cit 50
o 0.15
el
v
e
articl 0.1 2m) 40
p N/
0.05 all stress ( 30
w 20
0
−40 −30 −20 −10 0 10 20
D x/d 10
Fig.13 Averagedvelocityofthegranularmediumalongthe
averaged slug of figure 12. Inside the region around the slug
(∆x/d<−40&∆x/d>30)thegranularmediumismoving 0 −40 −30 −20 −10 0 10 20
forward, outside it is resting.
D x/d
Fig. 15 Stress between the wall and the granular medium
alongtheaveragedslugoffigure12.Thelowwallstressbefore
2
particle forces and after the slug corresponds to the weight of the layer of
drag force particlesrestingatthebottomofthetube,whichisgivenby
thedotted curve.
1
g
m The normal wall stress corresponding to the slug in
ce / 0 Figure12 is shownin Figure15. Inhorizontalconveying
al for thenormalwallstressisthe sumofthenormalinterpar-
nt ticle forces and the weight of the particles. The low wall
o −1
oriz stressbeforeandaftertheslugcorrespondstotheweight
h of the layer of particles left behind by the slugs. Higher
wall stresses are produced by the slug including the col-
−2
lisionanddisintegrationregion.Thewallstressincreases
within the collisionregionandthen decreasesin the dis-
−3 integration region. As one can see, the wall stresses are
−40 −30 −20 −10 0 10 20 much higher than can be accounted for by the weight
D x/d of the material. Therefore, the particle in the slug must
Fig. 14 Forcesinthedirectionofmotionactingonparticles have a high granular temperature.
along the averaged slug in figure 12. Interparticle forces and Figure 16 shows the granular temperature along a
friction are summed up and displayed as particle force. The
slug. The granular temperature is the average kinetic
drag force corresponds to the pressure drop of the gas. Out-
sideoftheslug(∆x/d<−40&∆x/d>30)theparticleforce energy of the particles minus the kinetic energy of the
is zero. The fluctuations in the particle force (x/d≈20) are motion of their center of mass. The granular tempera-
strongest in the collision region, where the moving particles ture risesrapidly inthe collisionregion.Due to the high
within the slug collide with the resting particles before the
damping(e=0.5)thesetemperaturesdecreasealongthe
slug.
slug and vanish behind the slug.
tering the slug, the drag force on the particles is small,
because the air is able to pass above the particles. The
large fluctuations in the particle force arise when the 3.4 Cross-sections of a slug
moving particles within the slug collide with the resting
particlesbeforetheslug.Withinthisregiontheparticles Inhorizontalslug conveying,the radialsymmetry ofthe
are piled up, until the cross-section of the tube is com- systemisbrokenbythegravitationalforce.Theparticles
pletely filled. In the slug the drag force and the friction tend to segregate and settle on the bottom of the tube
forcebalanceeachother.Behindthe slug,the dragforce (fig. 17). In the middle of the slug, the complete cross-
9
to the high friction with the wall. The uppermost par-
1.4e−09 ticles within the tube, moving freely between the upper
tubewallandtheotherparticlesbelow,havethehighest
1.2e−09 velocity.
ur (J) 1e−09
at
per 8e−10 4 Comparison of horizontal and vertical
m
e transport
ar t 6e−10
ul
n
gra 4e−10 The same simulationmethod as usedfor horizontalcon-
veying has also been applied to vertical conveying. A
parameter study and a detailed analysis of the plugs for
2e−10
theverticalplugconveyinghasbeenpublishedseparately
[32],withtubedimensionsanddefaultparametersforthe
0
−40 −30 −20 −10 0 10 20 particles and the gas being identical.
D x/d
Fig. 16 Granular temperature along the averaged slug of
(a)
figure 12. The granular temperature increases rapidly at the
frontoftheslug.Duetothelargedamping(e=0.5)thegran-
ular temperatureis dissipated. Behind thecenterof the slug
the temperature decreases linearly until it vanishes behind
theslug.
sectionofthetube isfilledwithparticles.Atthebottom
of the tube, the particles form layers along the wall. (b)
Before the slug the particles are resting (fig. 18). At
the front side of the slug the particles gain velocity in
direction of motion. This velocity decreases behind the
slug until the particles are again resting.
0.4 Fig. 20 Spatio-temporal images of the porosity along the
tube for (a) the horizontal and (b) the vertical plug convey-
ing. All the parameters, except for the direction of gravity
0.35 are thesame in both simulations.
m/s) 0.3 As expected fromexperiments,plug conveyingis ob-
y (
cit servedinboththehorizontalandtheverticalcases.How-
elo 0.25 everthedetailsoftheflowpatternsandthequantitative
v
e properties are different.
cl
arti 0.2 The most conspicuous difference between the flow
p
patterns is that in the horizontal transport most slugs
dissolvewhile travelingalongthe tube. Succeeding slugs
0.15
grownthroughcollectingtheremnantsofprecedingslugs.
Contrary to the vertical transport, there is a final slug
0.1 length. After reaching certain length slugs do not dis-
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
solve any more. In the vertical transport the growth of
height D z/d
plugscomesthroughthemergingofsmallerplugs.Verti-
cal plugs are slower (0.17m/s) then in horizontal trans-
Fig. 19 Vertical velocity profile through the slug shown in
port (0.3m/s), as the drag force acting on the plugs is
figure17
partly compensated by the gravitational force. The ini-
tialnumberofplugsatthebeginningofthetubeislower
Within the slug the lower particle layers are slower in the horizontal conveying, because the minimal num-
than the upper ones, as shown in figure 19. The parti- ber of particles needed to fill up the cross-section of the
cle layer at the bottom of the tube is slowed down due tube is higher.
10
Another feature also observedin experiments is that (a)
in horizontal transport the particles between the slugs 12
formarestinglayeratthebottomofthetube.Invertical
case the corresponding particles are accelerating down- 10
wards,theimpactoftheseparticlesonasucceedingplug
is considerable higher then in the horizontal case. m) 8
Thecharacteristiccurvesshowsomesignificantdiffer- a/
P
h
ences between the transport modes. Generally the pres- p (
sure drop in the tube filled with plugs for the vertical dro 6
transportisaboutfourtimeshigherthanforthehorizon- ure
tal transport. The additional pressure drop is needed to ess 4
carrytheplugsagainstthegravitationalforce.Thequal- pr
itativebehaviorofthe pressuredropwiththe superficial
2
gasvelocityand the viscosityis the same for bothcases.
The pressure drop in slug conveying decreasesabout six
times more rapidly with increasing superficialgas veloc- 0
0 0.1 0.2 0.3 0.4 0.5 0.6
ity. It is decreasing less with increasing gas viscosity. In
Coulomb coefficient m
both cases the pressure drop shows a linear dependency
(b)
on the mass flux of the granulate. An increase of the
45
mass flux results in more plugs along the tube.
The dependence on friction is different between hor-
40
izontal and vertical conveying, as can be seen from fig-
ure21.While inthe horizontalcaseforlowCoulombco-
35
efficients µ < 0.05 the granular material is transported m)
a/
as single particles, which are rolling on the bottom of P
h 30
the tube, in the vertical case the granular material is p (
o
transported as plugs even with no friction at all. Here dr
e 25
the pressure drop in the horizontal case is considerable ur
lower then anywhere else. A peak in the pressure drop press 20
(µ=0.1)for the horizontaltransportdenotes transition
fromthetransportofparticlelayersatthebottomofthe
15
tube byslidingorrollingtoslugtransport.Ontheother
hand, the pressure drop increases monotonically for all
10
values of µ in vertical conveying. For high Coulomb co- 0 0.1 0.2 0.3 0.4 0.5 0.6
efficients (µ > 0.3) in the horizontal case, the pressure Coulomb coefficient
drop increases almost linearly. In the vertical case, the
Fig. 21 Dependency of the pressure drop on the Coulomb
increase of the pressure drop is not linear. As one can
coefficient for (a) the horizontal and (b) the vertical plug
seefromFig21b,thereisachangeinslopenearµ=0.5. conveying.
This changeseems to be associatedwith the appearance
of sticking plugs.
port. The increase and decrease of the stress is spread
Adetailedviewontheplugprofilesexhibitsmoredif-
overthe wholeplug length.The biggestdifference inthe
ferences. Contraryto the verticalplugs, horizontalslugs
examined plug profiles is found for the granular tem-
havesmoothboundariesdue to the slopes atthe ends of
perature. While in the vertical transport high granular
the slug.Within the verticalplug a slightly lowerporos-
temperatures are limited to the upper front of the plugs
ity is reached (vert. 0.47, horiz. 0.49). The slugs do not
and vanish rapidly within the plug due to damping, in
have a constant granular velocity. The average granular
the horizontal transport the temperature decreases lin-
velocity along the horizontal slug increases from both
earlyalongthe slugandreacheszerofarbehind the slug
sidesuntilthemiddleofthe slug.Alsoashearingofpar-
(fig. 16).
ticle layers within the slug is observed, where the upper
particlesarethe fastest.The granularvelocityis aligned
in the direction of transportall along the tube, which is
not true for the particles between the plugs in the verti- 5 Conclusion
calcase.Intheverticalcaseonlyasmalldifferenceinthe
granularvelocityoftheradialparticlelayersisobserved, In this paper, a simple model [23] with coupled grain
within the plug these layers have a constant velocity. and gas flow has been applied to pneumatic transport.
The magnitude of the maximal wall stress within a The implementation is three-dimensional; rotation and
plugisofthesameorderforhorizontalandverticaltrans- Coulomb friction are taken into account. The fluxes of