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
Molecule - plate Sphere - plate Conductor Non-conductor
Interaction Forces and Potentials between Smooth and Stiff Model Bodies
Partner Dipol moment and dispersion Electrostatic (COULOMB) (VAN-DER-WAALS)
1 a 2Sphere - sphere Point charge sphere-sphere q = 2ε0εrUel/d
d 1
d2
a
d = 2d1d2
d1 + d2
EVdW =
FVdW = -
- CH · d 24 · a
CH · d 24 a²
a1 ρn,2
EVdW = -
FVdW = -
CH · d1
12 a
CH · d1
12 a²
d 1
aAS
Q1 = π d1 ε0 εr · EQ2 = AS ε0 εr · E
UVdW = -
FVdW = -
π ρn,2 A
6 a³π ρn,2 A
2 a4
EVdW = -
FVdW = -
CH · l · d 24 2 a
√¬
√¬ 3/2
Plate - plate Conductor Non-conductorQ = AS · ε0 εr · E
EVdW =
FVdW =
- CH · AS
12 π a²
- CH · AS
6 π a3
2 Crossed cylinders
a
d1
d²
EVdW =
FVdW =
- CH· d1d2
12 a√¬
- CH · d1d2
12 a²√¬
Molecule-molecule
√¬ CH · l · d
16 2 a√¬ 5/2
Eel= ε0 εr Uel d1 lnπ2
ad1
2
a
d Mll
Eel = · q1 q2 · a l · d2ε0 εr
Eel = · q1 q2 · a AS
2 ε0 εr
a
AS
FC = · q1 q2 l · d2ε0 εr
Eel = ε0 εr · Uel1a
AS2
2
FC = ε0 εr · Uel1a2
AS2
2FC = · q1 q2
AS
2 ε0 εr
Eel = · q1 q2 · a π d1
2 ε0 εr
2
FC = · q1 q2 π d1
2 ε0 εr
2
FC = ε0 εr Uel · π2
d1a
2
2
2 Parallel chain Cylinder - cylinder Conductor Non-conductor molecules Q = π d l · ε0 εr · E
UVdW = -
FVdW = -
Α a6
6 A a7
UVdW = -
FVdW = -
3 π A l 8 dM a5
15 π A l 8 dM a6
2
2
HAMAKERconstant = f(A):
CH = π2 ρn,1 ρn,2 A
d = 2d1d2
d1 + d2
q = Q/AS =
ρn = ρ·NA/M number densitye = 1.6·10-19 A·s electronic charge ε0= 8.854·10-12 A·s/(V·m) permittivity of vacuum
nQ·eAS
(1+2· )·E surface charge densityεr,s - 1εr,s + 2
ε0≈ E electric field strengthUel electrostatic potentialF = - dU/da potential (counter) forcez ion valencyεr permittivity of interstitial medium
ISRAELACH VILI, J.: Intermolekular & Surface Forces, Academic Press London 1992, p.177
z1 z2 e²4π ε0 a
z1 z2 e²4π ε0 a²
π q1 q2 · d1 d22 2
2 ε0 εr (d1 + d2 + 2a)Eel =
π q1 q2 · d1 d22 2
2 ε0 εr (d1 + d2 + 2a)2FC =
UC =
FC =
FC = ε0 εr · Uell·da2
12
2
Eel = ε0 εr · Uell·da
12
2
l
a
d1 d2
l
Fig. 6.7
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 6.7
Fig. 6.8
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 6.8
Adhesion Forces between Stiff Solid Particles
a) Smooth sphere - smooth plate b) Rough sphere - smooth plate
102
10-2
1
104
10-4
10-1 1 10 102 103
particle separation a in nm
adhe
sion
forc
e F
H0
in n
N
α = 20°
liquid bridgerel. humidity 50%
non-conductor
van der Waals conductor
a0 = 0.4 nm
10-7
10-9
10-5
10-10
adhe
sion
forc
e F
H0
in N
10-2 10-1 1 10 102 103 104
10-8
10-6
10-4
10-3
particle size d in µm
h r = 0
nm1 n
m5 n
m10
nm
1 µm
100 nm
FH0,VdW = . 1 + CH hr d / hr
6 a02 2.(1 + hr / a0)2
a0 = 0.4 nm molecular force equilibrium separation σlg = 72 10-3 N/m surface tensionα = 20° bridge angleθ = 0° wetting angleCH = 19 10-20 J Hamaker constant acc. to Lifschitz
.
.
qmax = 160 10-19 As/µm2 surface charge densityU = 0.5 V contact potential
CH,sls = ( CH,ss - CH,ll )2 Hamaker constant particle-water-particle
adhe
sion
forc
e F
H0
in N
particle size d in µm
10-5
10-6
10-7
10-8
10-9
non-
cond
ucto
r
liquid
bridg
e
van d
er W
aals,
dry
cond
uctor
van d
er W
aals,
wet
wei
ght o
f sph
ere
10-1 1 10 102 103
αd2
a0
hr
a0
d2
1 10 102 103
roughness height 2.hr in nm
10-5
10-6
10-7
10-8
10-9
10 µm
1 µm
adhe
sion
forc
e F
H0
in N
liquid bridge, d = 10 µm 10 µm, α = 2,5 °
d = 100 µm van der
Waals
conductor, d = 10 µ
m
non-conductor, d = 10 µm
acc. to H. Schubert (1979):
Models according to Rumpf et al. (1974):
Fig. 6.9
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
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 consolidation stress σ1 in kPa
unia
xial
com
pres
sive
str
engt
h σ
c in
kPa
20.0
15.0
10.0
5.0
0.0ffc = σ1/σc 10≥
free flowing
4 < ffc < 10easy flowing
1 < ffc < 2very cohesive
σc(σ1) t = 0 σct(σ1) t = 24 h
25.0
-5.0
σ1 σc
ffc < 1hardenednon flowing
2 < ffc < 4cohesive
Consolidation Function of Titaniaparticle size dS= 200 nm, moisture Xw= 0.4%, temperature = 20 °C
( )( )
( )( ) i
i22
i
i22
i
ic
sintan11
tan1tan11
tan11
2sin1ff
ϕ−ϕ⋅κ++
ϕ⋅κ+ϕ⋅κ++
ϕ⋅κ++
⋅ϕ−
=
Flowability assessment and contact consolidation coefficient κ(ϕi = 30°) flow function
ffc κ-values ϕst in deg evaluation examples
100 - 10 0.01006 – 0.107 30.3 – 33 free flowing dry fine sand 4 - 10 0.107 – 0.3 33 – 37 easy flowing moist fine sand 2 - 4 0.3 – 0.77 37 – 46 cohesive dry powder 1 - 2 0.77 - ∞ 46 - 90 very cohesive moist powder < 1 ∞ - non flowing,
hardened (ffct) moist powder
hydrated cement
Fig. 6.30
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 6.30
bulk
den
sity
ρb
ρb,0
ρb = ρb,0 · (1 + )σM,stσ0
n
centre stress during consolidationor steady-state flow σM,st
isostatictensile strength -σ0
0
n = 0 incompressible
0 < n < 1 compressible
n = 1 ideal gas compressibility index
Isentropic Powder Compression
Adiabatic gas compression:
pV1
dpdV
adκ=−
(1)
Isentropic powder compression:
∫∫σρ
ρ σ+σσ
⋅=ρρ st,Mb
0,b 0 0st,M
st,M
b
b dnd
(2)
Compressibility index of powders, semi-empirical estimation for σ1 = 1 – 100 kPa
index n evaluation examples flowability 0 – 0.01 incompressible gravel free flowing
0.01 – 0.05 low compressibility fine sand 0.05 - 0.1 compressible dry powder cohesive
0.1 - 1 very compressible moist powder very cohesive
Fig. 6.31
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 6.31
a) particle contact deformation b) particle adhesion
c) powder yield loci d) consolidation functions
Compliant and Stiff Particle Contact and Powder Behaviour
displace-ment hK
0
compliant
stiff
-FH0
forc
e F N
f) compression function
normal force FN0
compliant
stiff
adhe
sion
forc
e F H
consolidation stress σ1
0
compliantcohesive
stiff, free flowing
unia
xial
com
pres
sive
/te
nsile
stre
ngth
σc,
σ Z1
consolidation stress σ10
compliantcompressible
stiff, incompressible
bulk
den
sity
ρb
ρ b,0
−σ0
normal stress σ0
cohesive
free flowing
shea
r st
ress
τ
−σ0
SYL
SYLYLYL
e) powder constitutive models
average pressure σΜ0
cohesive
radi
us st
ress
σR
−σ0
SYL
SYL
YL
YL
free flowing
CLCL
σiso
Fig. 6.32
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 6.32
Fig. 6.33
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 6.33
Fig. 6.34
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 6.34
Fig. 6.35
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 6.35
Widely Spread Residence Time Distribution
Q3(tV)tv
Storage Time too small:
Deaeration Problems
tv
Q3(tV)
Storage Time too Large
tv
Q3(tV)
Time Consolidation Problems
Inflammation or Explosion Hazards
Deterioration Problems
Fig. 6.36
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 6.36
Fig. 6.37
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 6.37
(3)
(2)
(4)
µm ... cm ?
%o ...% ?
h ... d ?
% ?
FNFs
+ € ?
Solution for Silo Plant Design
Marketing
Cost-Benefit-Analysis
Investment Tasks
(1) Economic Goals
Layout
Flow Sheets
Storage Capacity
Selection of Main Dimensions
Moisture
Particle Size Distribution
Storage Time
Chem.-Min. Composition
Flow Behaviour Testing of Bulk Materials
Logistics
Area Requirements
Fig. 6.38
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 6.38
?
??
?(5) Design, Shaft and Hopper Dimension
(6) Handling Equipment Selection
Feeder (Filling)
Discharge Aids
Gate
Chute
Feeder (Discharging)
Conveyor
Mass Flow Rate
Dosage
Power (Consumption
(7) Apparatus Design and Adaption
Fig. 6.39
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 6.39
(8) Design of
(9) Structural Design
(10) Design and Construction, considering
Level Control Devices
Dust Collection
Safety Instrumentation
Plant and Building
Wall Thickness, SteelReinforcement
Silo Pressures
Environmental Protection
Manufacturing
Access and Cleaning
Safety
Manpower and Social Services
Environmental Protection
Repair and Maintenance
Measurement and Control Instrumentation
... etc
Fig. 6.40
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 6.40
Fig. 6.41
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 6.41
0 10 20 30 40 50 60
45
40
35
30
25
20
15
10
5
0
hopper angle versus vertical Θ in deg
angl
e of
wal
l fri
ctio
n ϕ
w in
deg
Mass Flow
Core Flow
effective angle of internal friction
ϕe = 70° 60° 50° 40° 30°
Θ ≤ 12 180° - arccos 1 - sin ϕe
2 sin ϕe
- ϕW - arc sin sin ϕW sin ϕe
Bounds between Mass and Core Flowaxisymmetric Flow
(conical hopper)
select Θ:= Θ − 3°
Fig. 6.42
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 6.42
50
45
40
35
30
25
20
15
10
5
0
angl
e of
wal
l fri
ctio
n ϕ
w in
deg
55
0 10 20 30 40 50 60 hopper angle versus vertical Θ in deg
Core Flow
effective angle of internal friction
ϕe = 70° 60° 50° 40° 30°
Mass Flow
Θ ≤ 60,5° +arc tan 50° -ϕe
7,73°15,07°
1-42,3° + 0,131° · exp(0,06 · ϕe)
ϕW
with ϕW < ϕ −3° and Θ ≤ e 60°
Bounds between Mass and Core FlowPlane Flow
(wedge-shaped hopper)
Fig. 6.43
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 6.43
Θmax
lmin > 3 · bmin
bminbmin
D
lmin>3·bmin
bmin
Θmax Θmax Θwall
bmin
- Conical Hopper (axisymmetric stress field)
Cone Pyramid
shape factor m = 1 [ 3a ]
- Wedge-shaped Hopper (plane stress field)
vertical front walls
shape factor m = 0
Θ=Θ
2tanarctan max
wall
Fig. 6.44
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 6.44
inclined front walls
1,5 bmin
bmin
lmin > 6 · bmin
3 bmin
B
L
Θ1max
Θ2max
1,5 bmin
( ) ( )[ ]( ) ( )
[ ]b3bBlL
lLorbBtantanarc22
maxwall
−+−
−−Θ=Θ
Fig. 6.45
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 6.45
Fig. 6.46
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 6.46
20 30 40 50 60 70
1,5
flow
fact
or
ff
effective angle of internal friction ϕe in deg
2
1
conical hopper
wedge-shaped hopper
Ascertainment of Approximated Flow Factor
(angle of wall friction ϕW = 10° - 30°)
Fig. 6.47
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 6.47
Fig. 6.48
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 6.48
Fig. 6.49
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 6.49
Calculation of Silo Pressures according to Slice-Element MethodForce Balance F = 0∑ ↑
Shaft (Filling F):
vFww
vFh
wF63
6363bv
bvwFv
pλtanp
pλptanλU
AHwith
HHexp1Hgρp
0gρpAUtanλ
dydp
⋅⋅ϕ=
⋅=ϕ⋅⋅
=
−−=
=⋅−⋅ϕ⋅+
H
HTr
pv
pv
pnpn pW
pWdA
y
ypW
pW
dydy
ph ph
H*
ΘΘ
ρb · g · dy
ρb · g · dy
pv + dpv
pv + dpv
Hopper:
( )
( ) ( )Θtanλtan
1m1Θtan
tan1k1mk
pktanpandpkp,tanUA2Hwith
H*HHp
H*HH
H*HH
1kHgρp
Hyforppand*HH ywith0gρypk
dydp
0gρAUtantanΘk
A1
dydAp
dydp
ww1
v1wwv1nTr
Tr
Trv0
k
Tr
Tr
Tr
TrTrbv
Trv0vTrbvv
bw1vv
⋅ϕ+=
−
ϕ++=
⋅⋅ϕ=⋅=Θ⋅
=
−+
−−
−−⋅⋅
=
==−==+⋅−
=+
ϕ+−⋅⋅+
Fig. 6.50
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 6.50
Fig. 6.51
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 6.51
Evaluation of Residence Time Distribution of Processes
1. Batch processa) Cummulative residence time distribution F(τ) b) Frequency distribution f(τ)
2. Continuous processes with ideal residence time distribution
Block flow chart:
b) ideal continuous mixer c) cascade of ideal continuous mixers
F(τ)
0 τR τ
1
f(τ)
ττR0
[ [ [ [
page 1
τ≥ττ<τ
=τΘ=τR
R
for1for0
)()(F
τ=τ∞τ≠τ
=τδ=τ=τR
R
forfor0
)()(F)(f
τ=τ
ddm
m1)(f
*
*0
*0
*
m)(m)(F
τ=τ
τ
*0m )(m* τ
0m m
1
0
τ0
tracer tracer
a) ideal plug flow channel (piston flow, ideal replacement)
3. 1st and 2nd moment of residence time distribution
( )∑∫=
−
∞
−⋅τ⋅≈ττ⋅τ===τN
1i1i,si,si,m
0,s03,1
Fillm cc
c1d)(fM
mm
a) mean residence time (complete initial moment)
b) variance of residence time distribution (2nd central moment)
( ) ( ) ( ) ( )∑∫∫=
−
∞
−⋅τ−τ⋅≈τ−τ⋅=ττ⋅τ−τ==τσN
1i1i,si,s
2mi,m
0,s
c
cs
2m
0,s0
2m3,2
2 ccc1dc
c1d)(fM)(
s
0,s
Fig. 6.52
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 6.52
5. Cummulative residence time distribution F(τ/τm) for cascades of ideal mixers with various stage numbers n
6. Flow channel (pipe) with axial redispersion
1
0,8
0,6
0,4
0,2
F(τ/
τ m)
0 1 2 3 4
n=1 2
∞→n3 612
τ/τm
v
x
v'
vx
m =τ
Evaluation of Residence Time Distribution of Processes page 2
1,5
1,0
0,5
0 0,5 1,0 1,5 2,0 2,5
τ m· f
(τ/τ
m)
τ/τm
3
1
∞→n
2
6n=12
∑=
ττ
−⋅
ττ
⋅−
−=τ−n
1iexp
)!1i(11)(F
n,m
1i
n,m
n = 1 ideal mixer
ideal replacement∞→n
Bodenstein number of particlesPDxvBo ⋅
=
2P 'vD ⋅Λ∝
axial dispersion (diffusion)coefficient of particles
0Bo → ideal mixer
∞→Bo ideal replacement
4. Normalized frequency distribution of residence time τm.f(τ/τm) for
cascades of ideal mixers with various stage numbers n at constant total mean residence time τm = n.τm,n = const.
with mean residence timeof one stage (unit):
nVnV
VV mn
n,mτ
=⋅
==τ
ττ
−⋅
ττ
⋅−
=τ⋅τ−
n,m
1n
n,mm exp
)!1n(n)(f