Fig. 6.1 Prof. Dr. J. Tomas, chair of Mechanical Process Engineering Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.1 6. Particle interactions, powder storage and flow 6.1 Dynamics of a flowing particle packing 6.2 Fundamentals of particle adhesion and adhesion forces 6.3 Mechanics of particle adhesion 6.4 Testing methods of particle adhesion 6.5 Flow properties of cohesive powders 6.6 Testing devices and techniques of powder flow properties 6.7 Applications in silo hopper design 6.8 Evaluation of residence time distribution of processes
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Fig. 6.1
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.1
6. Particle interactions, powder storage and flow 6.1 Dynamics of a flowing particle packing 6.2 Fundamentals of particle adhesion and adhesion forces 6.3 Mechanics of particle adhesion 6.4 Testing methods of particle adhesion 6.5 Flow properties of cohesive powders 6.6 Testing devices and techniques of powder flow properties 6.7 Applications in silo hopper design 6.8 Evaluation of residence time distribution of processes
Fig. 6.2
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.2
Fig. 6.3
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.3
Survey of constitutive functions, processing and handling problems of cohesive powders Property, problems
Physical principle
Physical assessment of product quality Particle size
d in µm Physical law Assessment char-
acteristic Value range
Evaluation
Large adhe-sion poten-tial1) FG
FH0
2
s20
sls,H
G
0H
dag2C
FF
⋅ρ⋅⋅π=
2
2
d)µm100(
WeightAdhesion
≈
1 - 100 100 - 104 104 - 108
slightly adhesive adhesive
very adhesive
10 - 100 1 - 10
0.01 - 1 Large in-tensification of adhe-sion2)
FN
FH(FN)FN
FH(FN)
( )0H
0HNH
FFFF
++⋅κ=
Contact consolida-tion coefficient κ
by flattening
0.1 – 0.3 0.3 – 0.77
> 0.77
soft very soft
extreme soft
< 10 < 1
< 0.1
Poor flow-ablity2)
σ1 σc
c
1cff
σσ
= Flow function ffc 2 - 4
1 - 2 < 1
cohesive very cohesive non-flowing
< 100 < 10 < 0.1
Large com-pres-sibility2)
σ1
∆h
n
0
st,M
0,b
b 1
σ
σ+=
ρρ
Compressibility
index n 0.05 – 0.1
0.1 - 1 compressible
very compressi-ble
< 100 < 10
Small per-meability3,4)
∆hW
∆hb
u b
Wf h
hku∆∆
⋅= Permeability
kf in m/s < 10-9
10-9 - 10-7 10-7 - 10-5
non-permeable very low
low
< 1 1 - 10
10 - 100 Poor fluidi-sation5,6)
( ))d(ufp P=∆
Channelling Group C, non-fluidising
< 10
1) Rumpf, H.: Die Wissenschaft des Agglomerierens. Chem.-Ing.-Technik, 46 (1974) 1-11. 2) Tomas, J.: Product Design of Cohesive Powders - Mechanical Properties, Compression and Flow Behavior. Chem. Engng. & Techn., 27 (2004) 605-618. 3)Förster, W.: Bodenmechanik - Mechanische Eigenschaften der Lockergesteine, 4. Lehrbrief, Bergakademie Freiberg 1986. 4) Terzaghi, K., Peck, R. B., Mesri, G.: Soil mechanics in engineering practice, Wiley, New York 1996.
5) Geldart, D.: Types of Gas Fluidization, Powder Techn. 7 (1973) 285-292. 6) Molerus, O.: Fluid-Feststoff-Strömungen, Springer, Heidelberg 1982.
Fig. 6.4
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.4
Fig. 6.5
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.5
Interactions of Polar Molecule Pair
Interaction pair potential due to MIE (1903) and e.g. the LENNARD-JONES potential:
UAa
Ban m= − + integer exponents n < m (1) U U
aa
aaB
U U= ⋅ ⋅
−
= =4 0
120
6
(2)
Pot. equilibrium separ.: aBAU
m n
=
−=
0
1
(3) equilibrium separation: am Bn AF
m n
=
−=
⋅⋅
0
1
(4)
Bond energy: Um n
mA
aBFn= −
−⋅
=0 (5) potential ratio:
UU
m nm
B
an aF=
=−
<0
1 (6)
Maximum attraction force: d Uda
dFda
2
2 0= − = : Fm nm
n Aa F
nmaxmax
= −−+
⋅⋅
+1 1 (7)
Separation ratios: 111
0
0
1
0
1
< =
< =
⋅ +⋅ +
=
=
−
=
−aa
mn
aa
m mn n
F
U
m n F
U
m nmax ( )
( ) (8)
Strain: 1aa
01aa
aa
0F
FF0F
0F
0U
00U
max
max−=ε<=ε<−=
∆=ε
==
=
== (9)
Modulus of elasticity: ( )
Ea
d Uda
m n nA
an m
UaF a F
nB
FF
= − ⋅ = − ⋅ ⋅ = ⋅ ⋅−
= =+
==
1
0
2
203
03
0
( ) (10)
Pull-off strength: σ Z
nm n
E mnm
,max =+
⋅++
+−1
111
1
(11)
-20
-15
-10
-5
0
5
10
15
20
0,00 0,10 0,20 0,30 0,40 0,50
atomic centre separation a in nm
inte
ract
ion
pair
pot
entia
l U in
10
-21 J
-20
-15
-10
-5
0
5
10
15
20
pote
ntia
l for
ce F
in 1
0-11 N
repulsion potential Uab
repulsion force Fab
attraction force Fanattraction potential Uan
aF=0aU=0
+ repulsion
- attraction aFmax
bond energy UB
total force F
total potential U
Fig. 6.6
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.6
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.9
2. Liquid bridge at direct contact (a = aF=0) of two equal-sized spheres
a) Pendular state (liquid bridges)
b) Funicular state (bridges + filled pores)
c) Capillary state (filled pores)
for a real packing:
for cubic packing of monodisperse spheres:
Fs
FH
α
d/2
R1
R'2
h
R2 Fs
d/2
σlg
σlg
1. Bond typesMoisture Bonding in a Particle Packing
Fig. 6.10
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.10
wat
er c
onte
nt X
W
XWK
Desorption
Adsorption
capillarycondensation
multimolecular layersmonolayer
relative partial pressure ϕ = pi/pSiϕK 1
2. Sorption isotherme of capillary-porous particles and packings
dewatering
moisten
satu
ratio
n
capi
llary
con
dens
atio
n
adso
rptio
n
pKe
XWC water content XWXWS
capi
llary
pre
ssur
e p
K 1. Capillary pressure hysteresis of a particle packing
Moisture Bonding in a Particle Packing
Fig. 6.11
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.11
Crystallisation Bridge between KCl 99 Particles d = 100 - 600 µm
Bulk caking and hardening in store house:
Fig. 6.12
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.12
σct(t) Atot
σDsfAsf(t)
(1)
(2)
(3)
dt)t(dA
dt)t(dA sf
Dsfct
tot ⋅σ=σ⋅
dtV)t(dV
dt)t(d
tot
sfDsf
ct ⋅σ=σ
dtdtm
)t(dm)1()t(Lt
0 s
sf
sf
sDsfct ∫⋅
ρρ
⋅ε−⋅σ=σ
Stress Transmission at Time Consolidation (Caking)
Fig. 6.13
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.13
rrK,el << 1
rK,plr << 1
material data: E* effective modulus of elasticity, pf micro-yield strength, ηΚ contact viscosity
hK,el
FN
FN
rrK,el
FN
hK,pl
FN
rK,pl
runloadingyield
ing
loading
WD = ∫ FR (hK) dhK
particle centre approach hK
cont
act n
orm
al fo
rce
FN
3π pf E*hK,f = ( )2r
2
kN = dFNdhK
elastic plastic andviscoplastic
force
response FR =π · r · pf · hK,pl
π · r · ηK · hK,vis·
13 E* ·√d · hK,el
3
stiffness kN = π · r · pf12 E* ·√d · hK,el
deformation
work WD = 215 E*·√d · hK,el
5 ·r ·pf ·(hK,pl - hK,f)
π2
2 2
π2 · r · ηK · hK,vis·t
2·
Particle Contact Deformation in Normal Direction without Adhesion
Fig. 6.14
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.14
Fig. 6.15
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.15
Testing the Adhesion Force between Particle and Surface
H. Masuda and K. Gotoh, Adhesive Force of a Single Particle, pp.141, in K. Gotoh, M. Masuda, K. Higashitani, Powder Technology Handbook, Marcel Dekker, New York 1997
FHFN FH
c) Vibration method d) Impact separation method
e) Hydrodynamic method
a) Spring balance method b) Centrifugal method
u
Pressing Detachment
FC
Fig. 6.16
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.16
Fig. 6.17
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.17
Fig. 6.18
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.18
Fig. 6.19
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.19
Fig. 6.20
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.20
(1) shear and dilatancy dV > 0
cohesionτc
ϕi
0 normal stress σ
shea
r st
ress
τ
τc
yield locus
angle of internal friction
σc σc
uniaxial pressure
ϕi
0normal stress σ
shea
r st
ress
τ
yield locus
−σZ1−σZ
τc
σc
σZ1
σZ1
uniaxial tension
σZσZ
σZ σZ
isostatictension
τσ
∆h→
angle ofdilatancy ν (+)
ϕi
0normal stress σ
shea
r st
ress
τ
yield locus
σc
τc
τ c
Biaxial Stress States of Sheared Particle Packing
Fig. 6.21
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.21
σ0 σ0
σ0σ0
no deformation
isostatic tensilestrength
σiso
σisoσiso
σiso
isostatic pressure,compression dV < 0
(3) shear and compression dV < 0
(2) stationary shear dV = 0
τσ
∆h→
ν (-)angle ofdilatancy
Biaxial Stress States of Sheared Particle Packing
0normal stress σ
shea
r st
ress
τyieldlocus
−σ0σ1σ2
ϕst
stationaryyield locus
σM,st
σR,st
stationary angle of internal friction
σ στ
ϕi
0normal stress σ
shea
r st
ess τ
yield locus
−σZσ1σ2 σM,st
σR,st
σiso
ϕi
consolidationlocusϕi ϕi
Fig. 6.22
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.22
shea
r st
ress
τ
normals stress σ
Yield Locus
−σ0
0
ϕi
σM,st
Stationary Yield Locus
End point
ϕst
ϕi angle of internal friction,ϕst stationary angle of internal friction,σ0 isostatic tensile strength of unconsolidated packing; andσM,st centre stress for steady-state flow
shea
r st
ress
τ
σ1normal stress σ
τc
Yield Locus
σVRσVM
ϕi
0 σc−σZσiso
Stationary Yield Locus
−σZ1 σ2
Consolidation Locus
σM,st
σR,st
σ1 major principal stress,σ2 minor principal stress,σc uniaxial compressive strength,σZ1 uniaxial tensile strength,σZ isostatic tensile strength,σiso isostatic pressure;
a) The three flow parameters
b) Stress states
c) Stress states at Mohr circle of steady-state flow:
shea
r st
ress
τ
normal stress σ
Yield Locus
−σ00
ϕi
σM,st
End point
ϕst
σ1σst
ϕst σR,st
τst
Stationary Yield Locus:
τst = cosϕst.σR,st
σst = σM,st - sinϕst.σR,st
σR,st = sinϕst.(σM,st + σ0)
Tangential point:
Yield Locus:τ = tanϕi
.(σ + σΖ)−σZ
Stationary Yield Locus
Biaxial Stress States of Sheared Particle Packing
Fig. 6.23
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.23
σz(1) uniaxial tensile strengthσz(3) isostatic tensile strengthτc cohesionϕst stationary angle of internal frictionϕi angle of internal friction
ϕi = 0τ
τ = f ( γ ).
c) a wet-mass viscoplastic powder without Coulomb friction
σ
A preshear pointE end pointγ shear rate gradientρb bulk densityσ1 major principal stressσc uniaxial compressive strength
Yield Loci and Powder Flow Parameters for:
ϕi = ϕstϕi
Eτ
σ
a) a dry, cohesion-less or free flowing particulate solid
τ
ρb = const. EA
τc
−σZ(3) -σZ(1) σc σ1σ
ϕst
ϕi
b) a general case of moist or fine cohesive powder
.
Fig. 6.24
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.24
Fig. 6.25
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.25
Fig. 6.26
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.26
Fig. 6.27
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.27
displacement s
shea
r fo
rce
FS
shea
r st
ress
τ =
FS /
A
σpre
normal stress σ = FN / A
Incipient Yield and Steady-State Flow
preshearplastic yielding dV=0
instantaneousyield locus
steady-state flow
σ<σpreσpre
FN
FS
s
preshear FN
FS
s
shear
incipientyielding
0
σpre
shear dV>0
σ
Fig. 6.28
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.28
displacement s
shea
r fo
rce
FS
shea
r st
ress
τ
= F S
/ A
σc
σ1
σctnormal stress σ = FN / A
ϕit
ϕi
τc
Instantaneous, Stationary, Time Yield Locus and Wall Yield Locus
preshear
ϕst
t >> 0
−σ0
ϕW
steady-state flow
end point
σM,st
incipientyieldingshear
σ<σpreσpre
stationaryyield locus
−σZ
FN
FS
s
preshearFN
timeconsolidationt >> 0 FN
FS
s
shear
FN
FS
s
wall shear
time yield locus
wall yield locusyield locus
time t (or displacement s = vS.t)
FS
FN
σpre σ>σpre
Fig. 6.29
Prof. Dr. J. Tomas, chair of Mechanical Process Engineering
Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.29