Particle adhesion and powder flow behaviour
J. Tomas
Mechanical Process Engineering, The Otto-von-Guericke-University Magdeburg
Universitätsplatz 2, D – 39 106 Magdeburg, Germany
The fundamentals of powder consolidation and flow behaviour using a reasonable
combination of particle and continuum mechanics are explained. By means of the model “stiff
particles with soft contacts” the combined influence of elastic, plastic and viscous repulsion in
particle contacts is demonstrated. On this physical basis, the stationary, instantaneous and
time yield loci as well as uniaxial compressive strength are derived and shown for a very
cohesive sub-micron titania powder. Finally, these models are used to assess the powder flow
behaviour by means of flow function ffc.
1
Introduction
The well-known flow problems of cohesive particulate solids in storage and transportation
containers, conveyors or process apparatuses - mainly mentioned by Jenike [1] - leads to
bridging, channelling, oscillating mass flow rates and particle characteristics with feeding and
dosing problems. Taking into account this selected technical problems, it is really necessary to
deal with the fundamentals of particulate solids consolidation and flow behaviour, namely
from using a reasonable combination of particle and continuum mechanics.
2
Particle contact Constitutive Model
The well-known failure hypotheses of Tresca, Coulomb-Mohr and the yield locus concept
of Jenike [1] and Schwedes [2] as well as the Warren-Spring-Equations [3 to 7] were
specified from Molerus [8, 9] by the cohesive steady-state flow criterion. The consolidation
and non-rapid, frictional flow of fine and cohesive particulate solids was explained by acting
adhesion forces in particle contacts [8].
On principle, there are four essential mechanical deformation effects in particle surface
contacts and their physical behaviour can be distinguished as follows:
(1) reversible elastic (Hertz [10], Huber [11], Mindlin [12], Dahneke [13], DMT theory from
Derjagin et al. [14] and JKR theory from Johnson et al. [15], Thornton [16]) without
hysteresis, deformation rate and consolidation time effects;
(2) irreversible plastic (Krupp [17], Schubert [18], Molerus theory [8] Maugis [19] and
Thornton [20]), strain rate and consolidation time invariant;
(3) reversible viscoelastic (Hsuin [21] and Rumpf et al. [22]), strain rate and consolidation
time variable;
(4) irreversible viscoplastic (Rumpf et al. [22]), strain rate and consolidation time variable.
If an compressive normal force FN is acting on a soft contact of two isotropic, stiff, linear
elastic, mono-disperse spherical particles the previous contact point is deformed to a small
contact area and the adhesion force between these two partners is increasing, see Rumpf et al.
[22] and Molerus [8].
Generalising these findings, the adhesion force FH0 without any additional consolidation can
be approached as a single rough sphere-sphere-contact. Herewith, FH0 considers a
characteristic hemispherical micro-roughness height or radius hr < d instead of particle size
[24]:
FH 0 =
 C H ,sls ⋅ h r
C H ,sls ⋅ (2 ⋅ h r ) 
d / hr
⋅ 1 +
≈
2
2
2
24 ⋅ a F=0
 2 ⋅ (1 + h r / a F=0 )  12 ⋅ a F=0
( 1)
The characteristic adhesion distance in Eqs.( 1) and ( 4) lies in a molecular scale a = aF=0 ≈ 0.3
- 0.4 nm. It depends mainly on the properties of liquid-equivalent packed adsorbed layers and
is to be estimated for a molecular interaction potential minimum. Provided that these
molecular contacts are stiff enough compared with the soft particle contact behaviour
influenced by mobile adsorption layers due to molecular rearrangement, this separation aF=0 is
assumed to be constant during loading and unloading in the interesting macroscopic pressure
range of σ = 0.1 - 100 kPa. The Hamaker constant solid-liquid-solid CH,sls acc. to Lifschitz
theory is related to continuous media dependent on their permittivities (dielectric constants)
and refractive indices, see Israelachvili [23].
40
FN
FH(FN)
t=0 h
FHt(FN)
t = 24 h
d
30
FH
a
20
10
0.0
- 4.0
d
adhesion force FH in nN
FHtot(FN) t = 24 h
FN
0.0
5.0
10.0
κ = 0.77 for ffc = 2, ϕi = 30°
15.0
20.0
normal force FN in nN
25.0
Fig. 1. Re-calculated particle contact forces of titania acc. to Fig. 3, median particle size d50 =
0.61 µm, moisture XW = 0.4 % accurately analysed with Karl Fischer titration.
Now this soft particle contact is flattening to a plate-plate-contact by acting of the adhesion
force FH0 itself and by an external normal force FN. Thus, the total adhesion force consists of
an instantaneous FH and a contact time influenced component FHt, see Fig. 1:
FHtot = FH + FHt = (1 + κ + κ t ) ⋅ FH 0 + (κ + κ t ) ⋅ FN
( 2)
This Eq.( 2) can be interpreted as a general linear particle contact constitutive model, i.e.
linear in forces, but non-linear concerning material characteristics [28]. The elastic-plastic
contact consolidation coefficient κ
κ = κ p / (κ A − κ p )
( 3)
includes a plastic repulsion coefficient κp describing a dimensionless ratio of attractive Van
Der Waals pressure pVdW to repulsive particle micro-hardness pf for a plate-plate model:
κp =
C H ,sls
p VdW
=
pf
6 ⋅ π ⋅ a 3F=0 ⋅ p f
( 4)
The elastic-plastic contact area coefficient κA represents the ratio of plastic deformation Apl to
total contact area A C = A pl + A el :
κ A = 2 / 3 + A pl /(3 ⋅ A C )
( 5)
The pure elastic contact deformation Apl = 0, κA = 2/3, replacing pf → pmax(E), has no
relevance for fine cohesive particles and should be excluded here. Commonly, for pure plastic
contact deformation Ael = 0 or AC = Apl, κA = 1 is obtained. This dimensionless strain
characteristic κ is read here as the slope of adhesion force, see Fig. 1 and details in [29],
κp
κA
FH =
⋅ FH 0 +
⋅ FN = (1 + κ ) ⋅ FH 0 + κ ⋅ FN
( 6)
κA − κp
κA − κp
influenced by predominant plastic contact failure. This contact consolidation coefficient κ is a
measure of irreversible particle contact stiffness or softness as well, see Fig. 1. A small slope
stands for low adhesion level FH ≈ FH0 because of stiff particle contacts, but a large inclination
means soft contacts or consequently, a cohesive powder flow behaviour, see Fig. 3 as well.
Additionally, a term for contact volume strain rate ε&V influence on adhesion force in a
particle contact was inserted with a viscous contact consolidation coefficient κt =
attraction/repulsion force ratio as a dimensionless combination of attractive contact strength
σa ≡ pVdW and repulsive particle contact viscosity ηV/dt ≡ pf, i.e. viscous stiffness, equivalent
to plastic deformation, Fig. 1.
κt =
σa
σ d ⋅ dt
= a ⋅
ηV ⋅ ε&V η V drvis
( 7)
The tensile strength σa of viscous flowing material is created by means of liquid-equivalent
adsorption layer bridges with Van Der Waals or hydrogen bondings. In opposition to time
invariable plastic contact deformation, all parameters depend on a time increment dt ≈ ∆t.
The intersection of function FH = (1 + κ ) ⋅ FH 0 + κ ⋅ FN with abscissa (FH = 0) in the negative
extension range of consolidation force FN, Fig. 1, is surprisingly independent of the Hamaker
constant CH,sls:
π
FN ,Z = − ⋅ a F=0 ⋅ h r ⋅ p f
2

 2 A pl  
d / hr
π
 ⋅ 1 +
≈ − ⋅ a F= 0 ⋅ h r ⋅ p f
⋅  +
2
2
 3 3 ⋅ A C   2 ⋅ (1 + h r /a F=0 ) 
( 8)
Considering the model prerequisites for cohesive powders, this minimum normal (tensile)
force limit FN,Z combines the opposite influences of a particle stiffness, micro-yield strength
pf ≈ 3⋅σf or resistance against plastic deformation and particle distance distribution The lastmentioned is characterised by roughness height hr as well as molecular centre distance aF=0 for
− dU / da = F = 0 = Fattraction + Frepulsion force equilibrium. It corresponds to an abscissa
intersection σ1,Z of the constitutive consolidation function, Fig. 3.
Generally, this adhesion force level, see Fig. 1, amounts up to 105 - 106 fold of particle weight
for very cohesive fine particles.
3
Particle contact Failure and cohesive Powder Flow criteria
Obviously, concerning the formulation of failure conditions at the particle contacts we can
follow the Molerus theory [8, 9], but here with a general supplement for the particle contact
constitutive model Eq.( 2). It should be paid attention that the stressing pre-history of a
cohesive powder flow is stationary (steady-state) and delivers significantly a cohesive
stationary yield locus in radius-centre-stresses of a Mohr circle or in a τ-σ-diagram [28], see
Fig. 2,
σ R ,st = sin ϕ st ⋅ (σ M ,st + σ 0 )
( 9)
τ st = tan ϕst ⋅ (σ st + σ 0 )
( 10)
with isostatic tensile strength σ0 obtained from the adhesion force FH0 and ε 0 = 1 − ρ b ,0 / ρ s ,
Eq.( 1).
σ0 =
1 − ε 0 FH 0
⋅ 2
ε0
d
( 11)
From it, the stress dependent effective angle of internal friction ϕe acc. to Jenike [1] as a slope
of cohesionless effective yield locus follows obviously [28], see Fig. 4:


σ1 + σ 0

sin ϕ e = sin ϕst ⋅ 
 σ1 − sin ϕst ⋅ σ 0 
( 12)
If the major principal stress σ1 reaches the stationary uniaxial compressive strength σc,st,
σ1 = σ c,st =
2 ⋅ sin ϕst ⋅ σ 0
1 − sin ϕst
( 13)
the effective angle of internal friction amounts to ϕe = 90° and for σ1 → ∞ follows ϕe → ϕst,
Fig. 4. For the relation between the angle of internal friction ϕi (slope of yield locus) and the
stationary angle of internal friction ϕst following definition is used, see Molerus [8] or [29]:
tan ϕ st = (1 + κ ) ⋅ tan ϕ i
( 14)
Therefore considering Eq.( 2), the new relation between the time dependent angle of internal
friction ϕit (slope of time yield locus) and the time invariable stationary angle of internal
friction ϕst (slope of stationary yield locus) is defined as [26, 27], see Fig. 2:
tan ϕ st = (1 + κ + κ t ) ⋅ tan ϕ it = const. ≠ f ( t )
( 15)
Now, with Eq.( 7) the angle of internal friction of a time consolidation ϕit is to be expressed:
FN
consolidation stress
σ1
uniaxial compressive strength
σc, σct
ϕi, ϕst, ϕit angles of internal friction
FS
s
Time Yield Locus
ϕit
Stationary Yield Locus
plastic yielding dV=0
shear stress τ = FS / A
shear force FS
t >> 0
Yield Locus
ϕi
τc
»
ϕst
shear displacement s
σ0
σTan
σ2 σ c
σTan, t σAn σM,st=(σ1+σ2)/2
normal stress σ = FN / A
Fig. 2. Characteristics of instantaneous, stationary and time yield locus.
σ1
σct
tan ϕ it =
tan ϕ i
tan ϕ i
tan ϕ i
=
=
κ
κ ⋅ tan ϕ i
tan ϕ i ⋅ σ a
1+ t
1+ t
1+
⋅t
1+ κ
tan ϕ st
tan ϕ st ⋅ η V ,e
( 16)
First of it, with this Eq.( 16) following physical sense predictions are possible [26, 28], Fig. 2:
(1) If no time consolidation occurs t = 0, both friction angles are equivalent ϕit = ϕi. The
linear
instantaneous
yield
locus
in
radius–centre-stresses
is
obtained:


 sin ϕst

sin ϕst
σ R = sin ϕi ⋅ [σ M + σ Z (σ VR , σ VM )] = sin ϕi ⋅ σ M + 
− 1 ⋅ σ M,st +
⋅ σ 0  ( 17)
sin ϕi
 sin ϕi



Per definition, only the tensile strength σZ and no inclination depends directly on the
consolidation pre-history obtained from a Taylor series linearisation of the yield locus
[28] near Mohr circle of cohesive stationary flow, see Eq.( 9). Now the simplest
formulation of the linear yield locus dependent on radius σVR ≈ σR,st and center stresses
(average pressure in the powder) σVM ≈ σM,st is given:


σ
σ R = sin ϕi ⋅ σ M + VR − σ VM 
( 18)
sin ϕi


The smaller the consolidation radius stress σVR < σR,st, the larger σVM > σM,st corresponding with larger σV2/σV1 ratio, see Schwedes [25] - and the smaller the powder
tensile strength σZ amounts.
(2) But if t > 0 the angle of internal friction during time consolidation decreases ϕit < ϕi and
the linear time yield locus is in τ-σ-coordinates:

  sin ϕst

sin ϕst
τ = tan ϕit ⋅ σ + 
− 1 ⋅ σ M,st +
⋅ σ0 
sin ϕit


  sin ϕit
( 19)
(3) For t → ∞ follows ϕit → 0, that means, the time yield locus is a parallel line to the σ axis, i.e. failure criterion of ideal plasticity by Tresca. The bulk material is hardening to a
complete solid state with plastic failure conditions as a limitation.
Consequently, with the derivation of time yield locus the uniaxial compressive strength σct
is found as function of the major principal stress σ1 being comparable with a linear
constitutive model:
σ ct =
2 ⋅ (sin ϕst − sin ϕit )
2 ⋅ sin ϕst ⋅ (1 + sin ϕit )
⋅ σ1 +
⋅σ = a ⋅σ + σ
(1 + sin ϕst ) ⋅ (1 − sin ϕit )
(1 + sin ϕst ) ⋅ (1 − sin ϕit ) 0 1,t 1 ct ,0
( 20)
The slope a1,t and the intersection of σc - axis σct,0 are time dependent, Eq.( 16). The
abscissa intersection σ1,Z of linear consolidation constitutive function σc(σ1), Fig. 3, corresponds to the FN,Z value of contact consolidation function acc. to Eq.( 8) and Fig. 1.
uniaxial compressive strength σc in kPa
25.0
σc(σ1) t = 0
σct(σ1) t = 24 h
20.0
ffc < 1
hardened
non flowing
1 < ffc < 2
very cohesive
15.0
2 < ffc < 4
cohesive
10.0
σ1
σc
10 < ffc < 4
easy flowing
5.0
free flowing
ffc = σ1/σc ≥ 10
0.0
-5.0
0.0
5.0
10.0
15.0
20.0
consolidation stress
25.0
30.0
35.0
40.0
σ1 in kPa
Fig. 3. Consolidation function of titania, d50 = 0.61 µm, moisture XW = 0.4 %
Again, the following physical sense predictions can be made:
(1) If no time consolidation occurs, both angles are equivalent ϕit = ϕi and the linear
constitutive model for plastic contact deformation is obtained σ c = a 1 ⋅ σ1 + σ c, 0
σc =
2 ⋅ (sin ϕst − sin ϕi )
2 ⋅ sin ϕst ⋅ (1 + sin ϕi )
⋅ σ1 +
⋅σ
(1 + sin ϕst ) ⋅ (1 − sin ϕi )
(1 + sin ϕst ) ⋅ (1 − sin ϕi ) 0
( 21)
(2) But if t > 0 the angle of internal friction during time consolidation decreases ϕit < ϕi and
the slope a1,t increases.
(3) For t → ∞ is ϕit → 0, that means, the slope follows a1,t → 1. This is the largest slope
considering the model prerequisites of an only viscous flow. If the first derivative is
greater then one a 1,t = dσ ct / dσ1 > 1 a non-linear relation should be considered.
(4) Notice that for t → ∞ the intersection of σct – axis σct,0 achieves a upper limit, which
depends only on surface energy σss and particle size and not on time and viscosity [28]:
σ ct , 0 =
4
2 ⋅ sin ϕst ⋅ (1 + sin ϕit )
1 − ε 0 sin ϕst σ ss
⋅ σ0 ≅ 4 ⋅ π ⋅
⋅
⋅
(1 + sin ϕst ) ⋅ (1 − sin ϕit ) ϕit →0
ε 0 1 + sin ϕst d
( 22)
Assessing Powder Flow Behaviour and Compressibility
Assessing the flow behaviour of a powder, Eq.( 20) shows that the flow function acc. to
Jenike [1] is not constant and depends on the consolidation stress level σ1:
ff ct =
(1 + sin ϕst ) ⋅ (1 − sin ϕit )
σ1 1
= ⋅
σ ct 2 sin ϕst − sin ϕit + sin ϕst ⋅ (1 + sin ϕit ) ⋅ σ 0 / σ1
( 23)
But roughly we can write for a small intersection with the ordinate σc,0, Fig. 4, i.e. isostatic
tensile strength σ0 → 0 near zero, the stationary angle of internal friction is equivalent to the
effective angle ϕst ≈ ϕe and the Jenike [1] formula is obtained in order to demonstrate the
general model validity:
ff c ≈
(1 + sin ϕe ) ⋅ (1 − sin ϕi )
2 ⋅ (sin ϕ e − sin ϕi )
( 24)
Thus, the semi-empirical classification by means of the flow function introduced by Jenike
[1] is adopted here with a certain physical sense completion, as shown in Table 1.
The class "non flowing" is characterised by the fact that the unconfined yield strength σct is
higher than the consolidation stress σ1 and thus in case of time consolidation, caking,
cementation or hardening the powder has been agglomerated to solid state [27].
Obviously, the flow behaviour is mainly influenced by the difference between the friction
angles, Eq.( 24), as a measure for the adhesion force slope κ in the general linear particle
contact constitutive model, Eq.( 2). Therefore we can re-calculate these coefficients from flow
function measurements:
κ=
1 + (2 ⋅ ff c − 1) ⋅ sin ϕi
⋅
tan ϕi ⋅ (2 ⋅ ff c − 1 + sin ϕi )
1
 1 + (2 ⋅ ff c − 1) ⋅ sin ϕi
1 − 
 2 ⋅ ff c − 1 + sin ϕi



2
−1
( 25)
A characteristic value κ = 0.77 for ϕi = 30° of a very cohesive powder is included in the
adhesion force diagram, Fig. 1, and shows directly the correlation between strength and force
increasing with pre-consolidation, see Table 1.
Due to the consolidation function, a small slope stands for a free flowing particulate solid
with very low adhesion level because of stiff particle contacts but a large inclination means a
very cohesive powder flow behaviour because of soft particle contacts, see Fig. 1.
Table 1: Flowability assessment and elastic-plastic contact consolidation coefficient κ(ϕi =
30°).
κ-values
ϕst in deg
evaluation
examples
flow function ffc
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,
moist powder,
∞
hardened (ffct)
hydrated cement
Obviously, the finer the particles the “softer” the contacts and the more cohesive the
powder [26, 28]. Köhler [30] has experimentally confirmed this thesis for alumina powders
(α-Al2O3) down to the submicron range (σc,0 ≈ const. = 2 kPa, d50 median particle size in µm):
0.62
ff c ≈ 2.2 ⋅ d 50
( 26)
Analogously to adiabatic gas law p ⋅ V κad = const. , a differential equation for isentropic
compressibility of a powder dS = 0, i.e. remaining stochastic homogeneous packing without a
regular order in the continuum, is to be derived:
dσ M ,st
dρ b
dp
= n⋅ = n⋅
ρb
p
σ M ,st + σ 0
( 27)
The total pressure including particle interaction p = σM,st + σ0 should be equivalent to a
pressure term with molecular interaction p + a VdW / Vm2 ⋅ (Vm − b ) = R ⋅ T in Van Der Waals
(
)
equation of state to be valid near gas condensation point. A “condensed” loose powder
packing is obtained ρb = ρb,0, if only particles are interacting without an external consolidation
stress σM,st = 0, e.g. particle weight compensation by a fluid drag, and Eq.( 27) is solved:
ρ b  σ 0 + σ M ,st
=
ρ b , 0 
σ0



n
( 28)
Therefore, this physically based compressibility index n ≡ 1/κad lies between n = 0, i.e.
incompressible stiff bulk material and n = 1, i.e. ideal gas compressibility, see Fig. 4 above.
Considering the predominant plastic and viscous particle contact deformation and
rearrangement in the stochastic homogeneous packing of a cohesive powder, following values
of compressibility index are to be suggested, see Table 2.
For hopper design purposes in powder mechanics, the major principle stress σ1 during
preconsolidation is to be used instead of the centre stress (average pressure) σM,st [29], see
Fig. 4.
These models are directly applied to evaluate the test data of a new oscillating shear cell
[31] and a press-shear-cell in the high-level pressure range from 50 to 2000 kPa of liquid
saturated, compressible filter cakes [32].
Table 2: Compressibility index of powders, semi-empirical estimation for σ1 = 1 – 100 kPa.
index n
0 – 0.01
0.01 – 0.05
0.05 - 0.1
0.1 - 1
evaluation
incompressible
low compressibility
compressible
very compressible
examples
gravel
fine sand
dry powder
moist powder
flowability
free flowing
cohesive
very cohesive
bulk density ρb
ρb,0
ρb = ρb,0 · (1 +
σ1 n
)
σ1,Z
0<n<1
angle of internal friction ϕe, ϕst, ϕi
σ1
90°
effektive angle of internal friction ϕe
(
ϕe = arc sin sin ϕst ·
σ1 + σ0
σ1 - sin ϕst · σ0
stationary angle of internal friction
ϕst = const.
angle of internal friction ϕi ≈ const.
σ1
effective wall stress σ1'
unconfined yield strength σc
σ1 = σc,st
σc = a1 · σ1 + σc,0
σ1' = σ1/ ff
bmin
θ
ff = 1
σ1'
σ1'
σc,st
bmin =
σc,0
σ1,Ζ
0
(m+1) · σc,0 · sin 2 (ϕw + θ)
ρb · g · (1 - a1 · ff)
bmin,st
major principal stress during consolidation σ1
Fig. 4. Consolidation functions of a cohesive powder for reliable hopper outlet design.
5
conclusions
Taking into consideration all the different properties of cohesive to very cohesive powders
tested (particle size distribution, moisture content, material properties etc.), the model fit can
be characterised as satisfactory to good. Thus, the model has proved its effectiveness and can
be accordingly applied in reliable silo design for flow and pressure calculation [26].
6 Symbols and Indices
a
separation
a1
slope of σc(σ1) consolidation function
A
area, particle contact area
b
outlet width
CH
Hamaker constant
d
particle size
E
modulus of elasticity
F
force
ff
flow factor acc. to Jenike
ffc
flow function acc. to Jenike
g
m
p
pf
r
γ&
ε
η
ηV
κ
θ
ϕ
ρ
σ
σ1
σ0
τ
gravity acceleration
mass, stress field or hopper shape factor
pressure
plastic yield strength of particle contact
contact radius
shear deformation rate gradient
porosity
viscosity
viscous yield strength of particle contact
contact consolidation coefficient
hopper angle
angle of friction
density
normal stress
major principal stress
isostatic tensile strength
shear stress
ad
An
b
c
C
e
el
H
i
l
m
M
min
N
p
pl
R
s
st
t
Tan
V
VdW
vis
W
Z
0
adiabatic
pre-shear
bulk
compressive
total contact
effective
elastic
adhesion (Haft-)
internal
liquid
molar
centre of Mohr circle
minimum
normal
pressure
plastic
radius of Mohr circle
solid
stationary
time dependent
tangential
volume
Van Der Waals
viscous
wall
tensile (Zug-)
initial, zero point
References
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