powder mechanics

8/3/2019 Powder Mechanics

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THE MECHANICS OF DRY, COHESIVE POWDERS1

Jrgen Tomas

Mechanical Process Engineering, The Otto-von-Guericke-University Magdeburg

Universittsplatz 2, D 39 106 Magdeburg, GermanyPhone: ++49 391 67 18 783, Fax: ++49 391 67 11 160

e-mail:[email protected]

Abstract

The fundamentals of cohesive 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 influence of elastic-plastic repulsion in particle contacts is

demonstrated. With this as the physical basis, universal models are presented which include the

elastic-plastic and viscoplastic particle contact behaviours with adhesion, load-unload hysteresis

and thus energy dissipation, a history dependent, non-linear adhesion force model, easy to

handle constitutive equations for powder elasticity, incipient powder consolidation, yield and

cohesive steady-state flow, consolidation and compression functions, compression and preshear

work. Exemplary, the flow properties of a cohesive limestone powder (d50 = 1.2 m) are shown.

These models are also used to evaluate shear cell test results as constitutive functions for

computer aided apparatus design for reliable powder flow. Finally, conclusions are drawn

concerning particle stressing, powder handling behaviours and product quality assessment in

processing industries.

Keywords: Particle mechanics, adhesion forces, van der Waals forces, constitutive models,

powder mechanics, cohesion, powder consolidation, powder flow properties, flow behaviour,

powder compressibility, compression work, shear work, hopper design, limestone powder.

1 Paper at Bulk India 2003, 9 11 Dec. 2003 Mumbai, review version
mailto:[email protected]:[email protected]


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1. Introduction .............................................................................................................................3

2. Slow Frictional Flow of Cohesive Powder .............................................................................4

2.1 Particle contact constitutive models ..........................................................................................6

2.1.1 Normal force - displacement functions of particle contact ........6

2.1.2 Energy absorption in a contact with dissipative behaviour......11

2.1.3 Adhesion force - normal force model ......................................12

2.1.4 Viscoplastic contact behaviour and time dependency .............14

2.1.5 Tangential contact force ...........................................................15

2.2 Biaxial stress states in a sheared particle packing .................................................................16

2.2.1 Shear force - displacement relation..........................................16

2.2.2 Shear stress normal stress diagram .......................................17

2.3 Cohesive powder flow criteria .................................................................................................20

2.3.1 Elasticity of pre-consolidated powder......................................20

2.3.2 Cohesive steady-state flow.......................................................22

2.3.3 Incipient yield...........................................................................25

2.3.4 Incipient consolidation.............................................................26

2.3.5 The three flow parameters........................................................26

2.3.6 Consolidation functions ...........................................................27

2.4 Powder consolidation and compression functions .................................................................28

2.4.1 Powder Flowability ..................................................................28

2.4.2 Powder Compressibility ...........................................................292.4.3 Powder compression and preshear work..................................30

3. Design Consequences for Reliable Flow ..............................................................................33

4. Conclusions ...........................................................................................................................35

5. Acknowledgements ...............................................................................................................36

6. Symbols.................................................................................................................................37

7. Indices ...................................................................................................................................38

8. References .............................................................................................................................39


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1. INTRODUCTION

There are many industrial branches at which bulk powders are produced, handled, stored,

processed and used. Particulate solids are manufactured or used as raw or auxiliary materials,

by-products or final products by mechanical unit operations as separation or mixing, size

reduction or agglomeration, but also by thermal processes as precipitation, crystallisation, drying

or by particle syntheses in process industries (chemical, pharmaceutical, building materials,

food, power, textile, material, environmental protection or waste recycling industries,

biotechnology, metallurgy, agriculture) as well as electronics. The number of particulate

products in high-developed economics can amount to millions and is permanently increasing day

by day because of diversified requirements of various clients and consumers of the global

arket.m

Fig. 1. Storage in containers - mechanical behaviours of solid, liquid, gas and bulk solid according to

Kalman [4].

Solids or parcels and fluid products are comparatively easy to handle. But the mechanical behav-

iours of powders or granulates [1 - 3] depend directly on pre-stressing history. This can be dem-

onstrated by a simple tilting test of storage containers [4]. Depending on how to fill the con-

tainer, tilt and bring it back different shapes of the bulk surface will be generated, Fig. 1. A co-

hesive powder behaves as an imperfect solid, flows sometimes as a liquid or can be compressed

like a gas. Often it shows those properties which are expected at least and creates the most prob-

lems in powder processing and handling equipment. These well-known flow problems of cohe-

sive powders in storage and transportation containers, conveyors or process apparatuses includebridging, channeling and oscillating mass flow rates. In addition, flow problems are related to


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particle characteristics associated with feeding and dosing, as well as undesired effects such as

widely spread residence time distribution, time consolidation or caking, chemical conversions

and deterioration of bioparticles. Finally, insufficient apparatus and system reliability of powder

processing plants are also related to these flow problems. The rapid increasing production of

cohesive to very cohesive nanopowders, e.g. very adhering pigment particles, micro-carriers in

biotechnology or medicine, auxiliary materials in catalysis, chromatography or silicon wafer

polishing, make these problems much serious. Taking into account this list of technical problems

and hazards, it is essential to deal with the fundamentals of particle adhesion, powder consolida-

tion and flow, i.e. to develop a reasonable combination of particle and continuum mechanics.

This method appears to be appropriate to derive constitutive functions on physical basis in the

context of micro-macro transition of particle-powder behaviour.

Fig. 1 shows also that the powder has a memory concerning its physical-chemical product prop-

erties. In terms of mineral genesis it can be of global historic periods. These peculiarities of co-

hesive powders connected with its strong individualism to flow or not to flow, we try now to

understand it from a fundamental point of view:

2. SLOW FRICTIONAL FLOW OF COHESIVE POWDER

A comparatively low consolidation in a pressure range of about 0.1 - 100 kPa and slow

frictional flow with shear rates vS < 1 m/s - and thus shear s skPa12/v2 1 kPa (hS

tress contribution

zheight of shear zone, b apparent bulk viscosity, b bulk density):

1kPa1s

m/kg1000m1vhvRe

2

32b

2S

b

bSzSb =

=

= (1)

The powder flows laminar and the shear stress contribution by particle particle collisions as

turbulent momentum transfer is negligible. Interactions between particles and fluids, e.g.

interstitial pore flow, are not considered. Generally, this shear resistance of a cohesive powder is

caused by Coulomb friction between preferably adhering particles.

The well-known failure or yield hypotheses of Tresca, Coulomb and Mohr, Drucker and Prager

(in [5, 6]) are the theoretical basis to describe the slow powder flow using plasticity. Next, the

yield locus concept of Jenike [7, 8] and Schwedes [9 - 12], and the Warren-Spring-Equations [13

- 16], Birks [18 - 20], and the approach by Tzn [21] etc., were supplemented by Molerus [23 -

25] to describe the cohesive steady-state flow criterion. Forces acting on particles under stress in

a regular assembly and its dilatancy were considered by Rowe [26] and Horne [27]. Parallel to it,

Nedderman [28, 29], Jenkins [30], Savage [31] and others discussed the rapid and non-rapid par-

ticle flow as well as Tardos [32, 33] the slow and the so-called intermediate, frictional flow of

compressible powders without any cohesion from the fluid mechanics point of view.


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Fig. 2: Force displacement diagram of constitutive models of contact deformation of smooth spherical

particles in normal direction without (compression +) and with adhesion (tension -). The basic models for

elastic behaviour were derived by Hertz [49], for viscoelasticity by Yang [67], for constant adhesion by

Johnson et al. [58] and for plastic behaviour by Thornton and Ning [66] and Walton and Braun [65] and

for plasticity with variation in adhesion by Molerus [22] and Schubert et al. [63]. This has been expanded

stepwise to include nonlinear plastic contact hardening and softening. Energy dissipation was considered

by Sadd et al. [61] and time dependent viscoplasticity by Rumpf et al. [68]. Considering all these theories,

one obtains a general contact model for time and rate dependent viscoelastic, elastic-plastic, viscoplastic,

adhesion and dissipative behaviours, Tomas [43, 44, 45] which is explained in the next figures.


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Additionally, the simulation of particle dynamics is increasingly used, see, e.g., Cundall [34],

Campbell [35, 36], Walton [37, 38], Herrmann [39], Thornton [48].

The consolidation and non-rapid flow of fine and cohesive powders was explained by the adhe-

sion forces at particle contacts, Molerus [22, 23]. His advanced theory is the physical basis of

universal models which includes the elastic-plastic and viscoplastic particle contact behaviours

with hysteresis, energy dissipation and adhesion, a history dependent, non-linear adhesion force

model, constitutive equations for powder elasticity, incipient consolidation, yield and cohesive

steady-state flow, consolidation and compression functions, compression and preshear work [40

to 47]:

2.1 Particle contact constitutive models

In principle, there are four essential mechanical deformation effects in particle-surface contacts

and their force-response behaviour can be explained as follows:(1) elastic contact deformation (Hertz [49], Huber [50], Cattaneo [51], Mindlin [52, 53], Green-

wood [54], Dahneke [55], Derjaguin (DMT theory) [57], Johnson (JKR theory) [58], Thorn-

ton [59] and Sadd [61]) which is reversible, independent of deformation rate and consolida-

tion time effects and valid for all particulate solids;

(2)plastic contact deformation with adhesion (Derjaguin [56], Krupp [62], Schubert [63], Mol-

erus [22, 23], Maugis [64], Walton [65], Thornton [66] and Tomas [43]) which is irreversible,

deformation rate and consolidation time independent, e.g. mineral powders;

(3) viscoelasticcontact deformation (Yang [67], Rumpf [68] and Sadd [61]) which is reversibleand dependent on deformation rate and consolidation time, e.g. soft particles as bio-cells;

(4) viscoplastic contact deformation (Rumpf [68] and Tomas [44, 45]) which is irreversible and

dependent on deformation rate and consolidation time, e.g. nanoparticles fusion.

2.1.1 Normal force - displacement functions of particle contact

These normal force - displacement models are shown as characteristic constitutive functions in

Fig. 2. Based on these theories, a general approach for the time and deformation rate dependent

and combined viscoelastic, elastic-plastic, viscoplastic, adhesion and dissipative behaviours of aspherical particle contact was derived [43, 44] and is briefly explained here - the comprehensive

review [46] comprises all the derivations in detail:

First, two isotropic, stiff, linear elastic, mono-disperse spherical particles may approach with

decreasing separation in nm-scale a aF=0 to form a direct contact, see Fig. 3 panel a). Conse-

quently, a long-range adhesion force is created because of van der Waals interactions of both

surfaces. This adhesion force FH0 can be modelled as a single rough sphere-sphere-contact [54],

additionally, with a characteristic hemispherical micro-roughness height or radius hr < d instead

of particle size d [73, 74]:


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( ) 2 0F

rsls,H

2

0Fr

r2

0F

rsls,H0H

a12

hC

a/h12

h/d1

a12

hCF

===

++

= (2)

Fig. 3: Particle contact approach, elastic, elastic-plastic deformation and detachment. After approaching a

aF=0, panel a), the spherical contact is elastically compacted to a partial plate-plate-contact and shows

the Hertz [49] elliptic pressure distribution, panel b). As response of this adhesion force F H0 and an in-creasing normal load FN, the contact starts at the yield point pmax = pf with plastic yielding, panel c). The

micro-yield surface is reached and this maximum pressure has not been exceeded. A hindered plastic field

is formed at the contact with a circular constant pressure pmax and an annular elastic pressure distribution

dependent on radius rK,el, full lines in panel c). This yield can be intensified by mobile adsorption layers,

panel c) above, If one applies a (negative) pull-off force FN,Z then the contact plates fail and detach with

the increasing distance a > aF=0, panel d).

After loading with an external compressive normal force FN the previous contact point is de-

formed to a small contact area. As the contact deformation response, a non-linear function be-tween this elastic force and the centre approach (indentation height or overlap) hK is obtained

according to Hertz [49], demonstrated in Fig. 3 and Fig. 4


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Fig. 4. Force - displacement diagram of recalculated characteristic contact deformation of cohesive lime-

stone particles as spheres, median diameter d50 = 1.2 m, surface moisture XW = 0.5 %. Pressure and

compression are defined as positive but tension and extension are negative, above panel. The origin of

this diagram hK = 0 is equivalent to the characteristic adhesion separation for direct contact aF=0. After

approaching from an infinite distance - to this minimum separation aF=0 the sphere-sphere-contact with-

out any contact deformation is formed by the attractive adhesion force F H0 (the so-called jump in). As

the response, from 0 Y the contact is elastically compacted, forms an approximated circular contact

area, Fig. 3 panel b) and starts at the yield point Y at p max = pf with plastic yielding, Fig. 3 panel c). This

yield point Y is located below the abscissa, i.e. contact force equilibrium FN = 0 includes a certain elastic-

plastic deformation as response of adhesion force FH0. The combined elastic-plastic yield boundary or

limit of the partial plate-plate contact is achieved as given in Eq. (8). This displacement is expressed by

annular elastic Ael (thickness rK,el) and circular plastic Apl (radius rK,pl) contact area, Fig. 3 panel c). After

unloading between the points U A the contact recovers elastically according to Eq. (14) to a displace-

ment hK,A. The reloading curve runs from point A to U to the displacement hK,U, Eq. (15). If one applies a

certain pull-off force FN,Z = - FH,A as given in Eq. (16) but here negative, the adhesion boundary line at

failure point A is reached and the contact plates fail and detach with the increasing distance, Fig. 3panel d). This actual particle separation is considered for the calculation byKA,K0F hhaa += =


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a hyperbolic adhesion force curve FN,Z = - of the plate-plate model Eq. (18). This hysteresis

behaviour could be shifted along the elastic-plastic boundary and depends on the pre-loading or, in other

words, on pre-consolidation level F

3A,H aF

N,U. Thus, the variation in adhesion forces FH,A between particles de-

pend directly on this frozen irreversible deformation, the so-called contact pre-consolidation history

FH(F

N), see next Fig. 5.

repat F+

The plastic repulsion coefficient p describes a dimensionless ratio of attractive van der Waals

pressure pVdW (adhesion force per unit planar surface area) to repulsive particle micro-hardness

pffor a plate-plate model (e.g. pVdW 3 600 MPa):

f0F

sls

f3

0F

sls,H

f

VdWp

pa

4

pa6

C

p

p

=

====

(10)

This attraction term pVdW can also be expressed by surface tension, e.g. sls 3.10-4 0.06 J/m,

20F

sls,H

a

VdWslsa24

Cda)d(p

2

1

0F =

==

=

(11)

fist introduced by Bradley [69] and Derjaguin [56]. The characteristic adhesion distance in Eqs.

(2) and (10) 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 can be estimated for a molecular interaction potential minimum F0Fda/dU === or force equilibrium [72,87]. Provided that these

molecular contacts are stiff enough compared with the soft particle contact behaviour, this sepa-

ration aF=0 is assumed to be constant. The particle surface behaviours are influenced by mobileadsorption layers due to molecular rearrangement. The Hamaker constant CH,sls [70] includes

these solid-liquid-solid interactions of continuous media. Thus CH,sls can be calculated due to

Lifshitz theory and depends on dielectric constants and refractive indices [71, 72].

The elastic-plastic contact area coefficient A represents the ratio of plastic particle contact de-formation area Apl to total contact deformation area elplK AAA += which includes a certain

elastic displacement [43]

3

K

f,K

K

pl

A h

h

3

11

A

A

3

1

3

2=+= , (12)

with the centre approach hK,ffor incipient yielding at point Y in Fig. 4, pel(rK= 0) = pmax = pf:

2

ff,K

*E2

pdh

= (13)

Constant mechanical bulk properties provided, the finer the particles the smaller is again the

yield point hK,f which is shifted towards zero centre approach. Thus, an initial pure elastic con-

tact deformation Apl = 0, A = 2/3, has no relevance for cohesive nanoparticles and should be

excluded. But after unloading beginning at point U along curve U E, Fig. 4, the contact recov-


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ers elastically in the compression mode and remains with a perfect plastic displacement hK,E. For

this pure plastic contact deformation Ael = 0 and AK= Apl, A = 1 is obtained.

Below point E left the tension mode begins. Between U E A the contact recovers probably

elastically along a supplemented Hertzian parabolic curvature up to displacement hK,A:

( ) A,H3A,KK2,1unload,N Fhhr*E32F = (14)

Consequently, the reloading runs along the symmetric curve from point A to point U:

( ) U,N3

KU,K2,1reload,N Fhhr*E3

2F += (15)

If one applies a certain pull-off force FN,Z = - FH,A, here negative,

A,KVdW2,10HA,H hprFF += (16)

the adhesion (failure) boundary at point A is reached and the contact plates are failing and de-taching with the increasing distance KA,K0F hhaa += = . The displacement hK,A at point A of

contact detachment is calculated from Eqs. (8), (14) and (16) as an implied function (index (0)

for the beginning of iterations) of the displacement history point hK,U:

( )3 2)0(,A,KU,Kplel,f,KU,K)1(,A,K hhhhh += (17)

The actual particle separation a can be used by a long-range hyperbolic adhesion force curvewith the displacement h3Z,N aF

K,A for incipient contact detachment by Eq. (17):

3

0F

K

0F

A,K

A,KVdW2,10HKZ,N

a

h

a

h1

hprF)h(F

+

+=

==

(18)

This hyperbolic force - separation curve is shown in Fig. 4 bottom panel d).

2.1.2 Energy absorption in a contact with dissipative behaviour

Additionally, if one considers a single elastic-plastic particle contact as a conservative mechani-

cal system without heat dissipation, the energy absorption equals the lens-shaped area between

both unloading and reloading curves A - U in Fig. 4:

=U,K

A,K

U,K

A,K

h

h

KKunload,N

h

h

KKreload,Ndiss dh)h(Fdh)h(FW (19)

With Eqs. (14) and (16) for FH,A and (15), (8) for FN,U, one obtains finally the specific or massrelated energy absorption Pdissdiss,m m/WkW = , which includes the averaged particle mass

. In addition, the resultant Eq. (20) includes a characteristic contact number

in the bulk powder (coordination number k/ [22]):

s3

2,1P r3/4m =


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( )([ ]A,KU,KpU,KA

s22,1

A,KU,Kf

2/5

2,1

A,KU,K

s

diss,m hhhr32

hhp3

r

hh

20

*EW

+

= ) (20)

This specific energy of 1.6 to 31 J/g for the limestone powder example mentioned was dissi-

pated during one unloading - reloading - cycle in the bulk powder with an average pressure of

only M,st = 3.3 to 25 kPa (or major principal stress 1 = 5.9 to 41 kPa).

2.1.3 Adhesion force - normal force model

The slopes of elastic-plastic yield and adhesion boundaries in Fig. 4 are characteristics of irre-

versible particle contact stiffness or compliance. Consequently, if one eliminates the centre ap-

proach hK of the loading and unloading functions, Eqs. (8) and (14), an implied non-linear func-

tion between the contact pull-off force FH,A = - FN,Z at the detachment point A is obtained for the

normal force at the unloading point FN = FN,U:

( )( )

3/2

0HN

0H)0(,A,H

22,1

0HNfp

22,10HN0H)1(,A,H

FF

FF1

*Er2

FF3prFFFF

+

+

+

++= (21)

This unloading point U is stored in the memory of the contact as pre-consolidation history. This

general non-linear adhesion model, dashed curve in Fig. 5, implies the dimensionless, elastic-

plastic contact consolidation coefficient and, additionally, the influence of adhesion, stiffness,

average particle radius r1,2, average modulus of elasticity E* in the last term of the equation. It is

worth to note here that the slope of the adhesion force function is reduced with increasing radius

of surface curvature r1,2.Practically, a linear function FH = f(FN) is used to evaluate the correlation between adhesion and

normal force [43] which is more complex than the ideal plastic model of Molerus [23], Fig. 5:

( ) N0HNpA

p0H

pA

AH FF1FFF ++=

+

= (22)

The dimensionless elastic-plastic contact consolidation coefficient (strain characteristic) is

given by the slope of adhesion force FH influenced by predominant plastic contact failure.

pA

p

=(23)

This elastic-plastic contact consolidation coefficient is a measure of irreversible particle con-

tact stiffness or softness as well. A shallow slope implies low adhesion level FH FH0 because of

stiff particle contacts, but a large slope means soft contacts, or i.e., a cohesive powder flow be-

haviour. This model considers, additionally, the flattening of soft particle contacts caused by the

adhesion force FH0. Thus, the total adhesion force consists of a stiff contribution FH0 and a con-tact strain influenced component ( N0H FF )+ , Fig. 5.

This Eq. (22) can be interpreted as a general linear particle contact constitutive model, i.e. linearin forces, but non-linear concerning material characteristics. The intersection of function (22)


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with abscissa (FH = 0) in the negative extension range of consolidation force FN is surprisingly

independent of the Hamaker constant CH,sls, Fig. 5:

( ) fr0F20Frr

K

pl

fr0FZ,N pha2a/h12

h/d1

A3

A

3

2pha

2F

++

+

= =

=

= (24)

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 3f

or resistance against plastic deformation and particle distance distribution. The last-mentioned is

characterised by roughness height hr as well as molecular centre distance aF=0. It corresponds to

an abscissa intersection 1,Z of the constitutive consolidation function c(1), which is shown by

Eq. (53) and Fig. 13 in section 2.3.

Fig. 5. Adhesion force normal force diagram of recalculated particle contact forces of limestone (me-

dian diameter d50 = 1.2 m, surface moisture XW = 0.5 %, specific surface area AS,m = 9.2 m/g) according

to the linear model Eq. (22) and non-linear model Eq. (16) for instantaneous consolidation t = 0 as well as

a linear function for time consolidation t = 24 h [45, 46, 104] using data of Fig. 13. The points character-

ise the pressure levels of YL 1 to YL 4 according to Fig. 13. A characteristic line with the slope = 0.3 of

a cohesive powder is included and shows directly the correlation between strength and force enhancement

with pre-consolidation, Eq. (56). The powder surface moisture XW = 0.5 % is accurately analysed with

Karl-Fischer titration. This is equivalent to idealised mono- to bimolecular adsorption layers being in

equilibrium with ambient air temperature of 20C and 50% humidity.

Generally, the linearised adhesion force equation (22) is used first to demonstrate comfortably

the correlation between the adhesion forces of microscopic particles and the macroscopicstresses in powders [44, 47, 94]. Additionally, one can obtain a direct correlation between the


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micromechanical elastic-plastic particle contact consolidation and the macro-mechanical powder

flowability expressed by the semi-empirical flow function ffc according to Jenike [8].

It should be pointed out here that the adhesion force level in Fig. 5 is approximately 10 5 - 106

times the particle weight for fine and very cohesive particles. This means, in other words, that

one has to apply these large values as acceleration ratios a/g with respect to gravity to separate

these pre-consolidated contacts or to remove mechanically such adhered particles from surfaces.

2.1.4 Viscoplastic contact behaviour and time dependency

An elastic-plastic contact may be additionally deformed during the indentation time, e.g., by

viscoplastic flow. Thus, the adhesion force increases with interaction time [27, 41, 62, 68]. This

time dependent consolidation behavior (index t) of particle contacts in a powder bulk, see Fig. 5

above line, was previously described by a parallel series (summation) of adhesion forces [40, 41,

42, 44]. This previous method refers more to incipient sintering or contact fusion of a thermally

sensitive particle material [68] without interstitial adsorption layers. This micro-process is very

temperature sensitive [40, 41, 42].

Table 1: Material parameters for characteristic adhesion force functions FH(FN) in Fig. 5

Instantaneous contact consolidation Time dependent consolidation

Constitutive models

of contact deforma-

tion

plastic viscoplastic

Repulsion coeffi-

cient f3

0F

sls,H

f

VdWp

pa6C

pp

==

=

tp

K

VdWt,p =

Constitutive models

of combined contact

deformation

elastic-plastic elastic-plastic and viscoplastic

Contact area ratio( )elpl

plA

AA3

A

3

2

++=

( )elvisplvispl

t,AAAA3

AA

3

2

++

++=

Contact

consolidation

coefficient

pA

p

=

t,ppt,A

t,pp

vis

+=

Intersection with FN-

axis (abscissa)

( )sls,Hf2,1r0FZ,N CfphaF = ( )sls,HKf

f2,1r0Ftot,Z,N Cf

/tp1

phaF

+

=

Additionally, the increasing adhesion may be considered in terms of a sequence of rheological

models as the sum of resistances due to plastic and viscoplastic repulsion p + p,t, 5th line in

Table 1. These are characterized by the micro-yield strength pf, apparent contact viscosity and

time K/t. Hence the repulsion effect of cold viscous flow of comparatively strongly-bondedadsorption layers on the particle surface is taken into consideration [45, 46, 104]. Hence with the


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total viscoplastic contact consolidation coefficient vis, which includes both the elastic-plastic

and the viscoplastic repulsion, the linear correlation between adhesion and normal force FH(FN)

from Eq. (22) can be written as:

( ) Nvis0HvisNt,ppt,A

t,pp

0H

t,ppt,A

t,Atot,H FF1FFF ++=

++

= (25)

This rheological model is only valid for a short term indentation of , here ap-

proximately t < 60 h for the high-disperse (ultra-fine), cohesive limestone powder with a certain

water adsorption capacity (specific surface area A

( fK p/t < )

S,m = 9.2 m/g). All the essential material pa-

rameters are collected in Table 1 and the total adhesion force FH,tot is demonstrated in Fig. 5

above line.

2.1.5 Tangential contact force

The influence of a tangential force in a normal loaded spherical contact was considered by Cat-

taneo [51] and Mindlin [52, 53]. About this and complementary theories as well as loading,

unloading and reloading hysteresis effects, one can find a detailed discussion by Thornton [59].

He has expressed this tangential contact force as [59, 60]:

( ) NiK2,1T Ftan1hr*G4F = (26)

Here is the tangential contact displacement, the loading parameter dependent on loading,unloading and reloading, i the angle of internal friction, ( )+= 12EG the shear modulus, and

the averaged shear modulus is given as:1

2

2

1

1

G

2

G

22*G

+

= (27)

Thus, with = 1 the ratio of the initial tangential stiffness

KT

T r*G4d

dFk =

= (28)

to the initial normal stiffness according to Eq. (6) is:

( )=

212

kk

N

T (29)

Hence this ratio ranges from unity, for = 0, to 2/3, for = 0.5 [53], which is different from the

common linear elastic behaviour of a cylindrical rod.

The force displacement behaviours during stressing and the breakage probability, especially at

conveying and handling, are useful constitutive functions to describe the mechanics of primary

particles [75, 76, 77] and, additionally, particle compounds [78] and granules to assess their

physical product quality [79, 80, 81].


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2.2 Biaxial stress states in a sheared particle packing

After this introduction into the fundamentals we have to look at what a volume element of parti-

cles used to do during its flow. In contrast to another engineering fields, in process engineering

we are strongly interested in reliable flow and do not be so happy about stable arches, domes or

wall adhesion effects in our apparatuses. Obviously, we have to know exactly this flow limit.

2.2.1 Shear force - displacement relation

At a shear test after a certain elastic shear displacement, we can distinguish between

(1) incipient consolidation,

(2) incipient yield and

(3) steady-state flow

of a particle packing. This is demonstrated in a shear force - displacement diagram FS(s), Fig. 6.

If we apply a certain shear force FS then the powder shows an elastic distortion with reversibledisplacement selastic after unloading, Fig. 6.

Fig. 6. Shear force - displacement diagram of incipient consolidation and yield of a particle packing at

direct shear test. When the sample is critically consolidated steady-state flow is measured. The partial

expansion of the shear zone is also known as dilatancy h = h(s) h0.


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Using increasing shear stress beyond the elastic displacement selastic the packing generates a

shear zone, which can be ellipsoidal for a Jenike-type shear cell [9, 11]. The powder flow or ir-

reversible shear effect correlates directly with the dislocations of particles in a comparatively

narrow shear zone, drawn in the middle panel of Fig. 6. Simultaneously, this results in a certain

compression (-) or expansion (+) of the shear zone which can be expressed by the so-calledvolumetric strain 0V h/)s(h= . This dependent variable is measured by the cover height h(s)

and related to the initial height h0.

2.2.2 Shear stress normal stress diagram

Now the essential parameters of cohesive powder flow are explained in a shear stress normal

stress diagram for a biaxial stress state, Fig. 7. Only positive values of the stress pairs () are

taken into consideration, the negatives mean opposite directions. Mainly, compressive stresses

(pressures) occur and are defined here as positive. Tensile stresses are negative.

Fig. 7. Shear stress normal stress diagram of biaxial stress states of sheared particle packing (1) shear

and dilatancy (expansion) of the shear zone, cohesion, uniaxial pressure and tension, isostatic tension.


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First we turn to incipient yield. This state can be measured point by point with an overconsoli-

dated sample which reduces the shear resistance after obtaining a peak stress . During this shear

the shear zone expands dV > 0, Fig. 6 bottom panel. This dilatancy hr

can be microscopically

explained by both contact unloading, by particle rearrangement and structural expansion of the

shear zone. During yield, the macroscopic shear plane do not coincide with the tangential direc-

tions of shear forces of particle contacts, Fig. 7 above. A downhill particle sliding effect into

packing voids can be responsible for this dilatancy. This can be expressed by a positive, i.e.

counter-clockwise, direction of the angle of dilatancy .

If we connect all stress pairs () we may obtain a straight line which is called as yield locus.

The slope of this line is the angle of internal friction i. The intersection with the ordinate = 0

represents the cohesion c a shear resistance caused solely by particle adhesion effects without

any external normal stress . These adhesion forces in the particle contacts are drawn as arrows

for the normal components FN. To avoid too much confusion we have cut out to draw the tangen-

tial force components FT for every contact. The black colour at all contacts shall demonstrate the

irreversible deformation. The shear resistance c is directly caused by this internal contribution

of adhesion forces FH(FN) and depends on the stressing pre-history as discussed in section 2.1.

The intersections of Mohr (semi-)circle with abscissa are the so-called principal stresses 1 and

2, i.e. the largest (major) and the smallest (minor) normal stress without applying any shear

stress = 0. The Mohr circle which intersects the origin, i.e. minor principle stress 2 = 0, gives

us the uniaxial compressive strength c as the cohesive strength characteristic of the powder, see

Fig. 7 middle. As mentioned before, the negative intersection of Mohr circle with abscissa gives

us the uniaxial tensile strength Z,1 for2 = 0, see left in Fig. 7 below. The intersection of theyield locus with abscissa, the isostatic tensile strength Z represents the internal contribution of

adhesion forces FH(FN) to the total stress, i.e. the sum of external normal stress plus Z. Thus,

this characteristic Z depends directly on the stressing or pre-consolidation history, see section

2.1. For higher pre-consolidation or packing density we obtain a group of yield loci (not drawn

here).

For all yield effects in a shear zone one may reach an equilibrium state after a certain irreversible

displacement. This steady-state flow is also observed here for no volume change dV = 0 and is

characterised by a dynamic equilibrium of simultaneous contact shearing, unloading and failing,creating new contacts, loading, reloading, unloading and shearing again, Fig. 8 above. It is char-

acterised by an endpoint and the largest Mohr circle with the major and minor principle stresses

1 and 2 and equivalent to these the radius and centre stresses R,st and M,st, Fig. 8 (and Fig.

11). For higher pre-consolidation and various yield loci we obtain a group of Mohr-circles for

steady-state flow. The envelope of all the Mohr-circles is defined as the stationary yield locus

and may also approximated by a straight line. The slope of this line is defined as the stationary

angle of internal friction st. To extrapolate the stationary yield locus, the isostatic tensile

strength 0 of very loose packing density is obtained, Fig. 8. This is typically for an unconsoli-

dated powder, i.e. direct particle contacts but without any contact deformation. Thus, the funda-


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mental characteristic 0 does not depend on pre-consolidation. This is equivalent to the adhesion

FH0 without any contact deformation. The black colour is missed between these virgin contacts.

Fig. 8. Shear stress normal stress diagram of biaxial stress states of sheared particle packing (2) sta-tionary shear (steady-state flow), (3) shear and compression (negative dilatancy) of the shear zone,

isostatic pressure and tension. In general, the steady-state flow of a cohesive powder is cohesive. Hence,

the total normal stress consists of an external contribution , e.g. by weight of powder layers, plus (by

absolute value) an internal contribution by the pre-consolidation dependent adhesion, the isostatic tensile

stress Z.

The incipient consolidation is described by the so-called consolidation locus which lies at the

right hand side of Fig. 8 (and Fig. 11) represents the envelope of all stress states with plasticfailure which leads to a consolidation of the particle packing dV < 0. This line may have the


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same inclination as the slope of yield locus, the angle of internal friction i, and intersects the

abscissa at an isostatic normal stress iso. This isostatic stress state means that all principal

stresses have the same value in all three spatial directions iso = 1 = 2 = 3. It is equivalent to

the hydrostatic pressure state in fluid dynamics. Obviously, this characteristic depends also on

the stressing pre-history as discussed before. The dilatancy hr

is here negative and can be mi-

croscopically explained by both contact loading, particle rearrangement and structural compres-

sion. During yield the macroscopic shear plane do not coincide with the tangential directions of

shear forces of particle contacts, Fig. 8 below. A uphill particle sliding effect into packing

voids may be responsible for this compaction.

2.3 Cohesive powder flow criteria

2.3.1 Elasticity of pre-consolidated powder

Before we turn to the irreversible powder flow, first a tangent bulk modulus of elasticity for a

cohesive powder is derived at the kPa-stress level of powder loading, if a characteristic uniaxial

normal strain hK/d is assumed [82 - 84]. For that purpose, the micro/macro-transition [47] with

the normal stress - force relation Eq. (40) and the contact stiffness due to Eq. (6) are applied to a

packing of smooth spheres. We have to consider the total normal force FN,tot of a characteristic

particle contact which includes the contribution of pre-consolidation dependent adhesion

FH(FN,V):

( ) NV,N0HNV,NHtot,N FFF1F)F(FF +++=+= (30)

By the first derivative near FN 0 we can write for the bulk modulus of elasticity (contact ra-dius according to and E* per Eq. (5))K2,1

2K hrr =

( )( )

0FK

tot,N

0K

Zb

Ndh

dF

d

1

d/hd

dE

=+

= (31)

Using a micro/macro-transition Eq. (40) we obtain finally:

( )3/1

Z

23/1

22,1

V,N0H2

b*E

614*E

r2FF)1(*E3

41E

=

++

= (32)

For a unconsolidated loose packing of a cohesive powder Eb,0 follows [47, 45]:

3/1

0

2

0

0

3/1

22,1

0H2

0

00,b

*E

61

4

*E

r2

F*E3

4

1E

=

= (33)

A free-flowing powder is unable to sustain a tensile stress FH 0 and Eb refers solely to com-

pression in a mould with stiff walls [83].


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Fig. 9. Bulk shear modulus - centre stress diagram for load and unload of limestone powder (d 50 = 1.2

m). The physical model Eq. (35) was multiplied by a fit factor of 3 .10-3 to obtain the full line for unload

after steady-state flow which is now equivalent to test data measured by Medhe [85]. This bulk shear

modulus Gb is about 3 orders of magnitude smaller than the shear modulus of particle material assumed tobe G = 60 kN/mm and Poisson ratio = 0.28.

Consequently, the initial shear stiffness (shear modulus) for elastic shear displacement can be

derived from Eq. (26), provided that the shear displacements at a characteristic particle contact

and in the bulk are equivalent /d s/hSz (hSz characteristic height of the shear zone):

( ) 0FT

0Sz

b

Td

dF

d

1

h/sd

dG

=

= (34)

( )3/1

Z

23/1

22,1

V,N0Hb

*E

61*G

r*E2

FF)1(3*G

1G

=

++

= (35)

3/1

0

2

0

0

3/1

22,1

0H

0

00,b

*E

61*G

r*E2

F3*G

1G

=

= (36)

This simple model, Eq. (35), overestimates the shear modulus Gb = 60 - 120 N/mm for lime-

stone (particle size d50 = 1.2 m) compared to the shear modulus obtained from direct shear testsGb,load = 180 - 270 kN/m for loading and Gb,unload = 220 - 340 kN/m for unloading [85]. Obvi-


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22

ously, the shear modulus Gb depends on the pre-consolidation of the isostatic tensile strength Z

= f(M,st), see Eq. (48), which is demonstrated in Fig. 9.

It is worth to note here that the ratio of the shear stiffness given in Eq. (35) to the normal stiff-

ness Eq. (32) of the bulk equals the contact stiffness ratio kT/kN as in Eq. (29):

( ) === 212kkEGEG NT

0,b

0,b

b

b (37)

Approximately for cohesive powders, the shear stiffness is equivalent to the normal stiffness,

e.g. Gb/Eb = 0.82 for a common Poisson ratio = 0.3 of the particle material.

2.3.2 Cohesive steady-state flow

Using the elastic-plastic particle contact constitutive model Eq. (22) the failure conditions of

particle contacts are formulated [47]. It should be noted here that the stressing pre-history of a

cohesive powder flow is stationary (steady-state) and results significantly in a cohesive station-

ary yield locus in radius stress = f(centre stress) - coordinates

( )0st,Mstst,R sin += (38)

or in the ()-diagram of Fig. 11 [43]:

)(tan 0ststst += (39)

This shear zone is characterised by a dynamic equilibrium of simultaneous contact shearing,

unloading and failing, creating new contacts, loading, reloading, unloading and shearing again.

The stationary yield locus is the envelope of all Mohr-circles for steady-state flow (critical state

line) with a certain negative intersection of the abscissa

20H

0

00

d

F1

= . (40)

This isostatic tensile strength 0 of an unconsolidated powder without any particle contact de-

formation is obtained from the adhesion force FH0, Eq. (2), with the initial porosity of very loosepacking s0,b0 /1 = and b,0 according to Eq. (59).

In some cases one may observe cohesionless steady-state flow, i.e. 0 = 0 in Eq. (38), which isdescribed by the effective yield locus according to Jenike [8] with the effective angle of internal

friction e as slope:

st,Mest,R sin = (41)

Replacing the radius stresses in Eqs. (38) and (41) and we obtain a simple correlation between

the stress-dependent effective angle of internal friction e, the stationary angle of internal fric-

tion st and the centre stress M,st:

+=st,M

0ste 1sinsin (42)


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Fig. 10: Friction angle consolidation stress diagram and shear stress - normal stress diagram to show the

correlation between the cohesive stationary yield locus [40] as envelope of all Mohr circles for steady-

state flow and the cohesionless effective yield locus according to Jenike [8]. Using the practical hopperdesign method, the latter is necessary for the calculation of flow factor ff )),(( W1e [8,24,90] and

effective wall stress , Eq. (65), with respect to the radial stress field during discharging, see chapter 3.

The termination locus [21, 29] is an auxiliary line to the end point of yield locus, or approximately, to

the centre of end Mohr circle and describes only the cohesionless steady-state flow in agreement with

normality and co-axiality of shear zone and geometrical plane of shear cell. Both effective yield and ter-

mination loci are directly dependent on stress history, Eq. (44).

'1

The centre stress M,st can be replaced by the major principle stress 1 during steady-state flow

st

st01st,M

sin1

sin

+

= , (43)

and one obtains

+==0st1

01ste

sinsintansin (44)


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24

which is in accordance with the daily experimental experience in shear testing, Fig. 10 upper

diagram [43]. If the major principal stress 1 reaches the stationary uniaxial compressive

strength c,st, Fig. 10 diagram below,

st

0stst,c1

sin1

sin2

== (45)

the effective angle of internal friction amounts to e = 90 and for1 follows est.

In soil mechanics [86] an effective angle of friction is used as slope of an auxiliary line which

connects the preshear points of yield loci pre, or approximately, the maxima of end Mohr circles

at M,st. This so-called termination locus [21, 29] is directly dependent on stress history and

describes only the cohesionless steady-state flow.

Fig. 11: Combination of shear force displacement diagram with shear stress normal stress diagram to

obtain the shear points. The shear cell testing technique with pre-consolidation (twisting), consolidationby preshear as far as steady-state flow, shear and incipient yield is also included. The testing technique

for any yield locus j is as follows: The cell is filled with a fresh sample, loaded by a comparatively large

normal stress V (1) and pre-consolidated by twisting the cover. Than a smaller normal stress pre < V (2)

for preshear is applied. The cell is presheared as far as the steady-state flow (3) is obtained for a constant

volume dV = 0 of the shear zone. Than the cell is unloaded and loaded by a smaller normal stress < pre

(4) for preshear is applied on the shear cover. The cell is sheared to the peak stress (5) is obtained for a

expanding volume dV > 0 of the shear zone. The shear zone relaxes to steady-state flow at the given

small normal stress level and unloaded to = 0 (4). The cell is weighed, opened and the shear zone is

observed to evaluate it as a suitable good test. All these steps (1) (5) are repeated n-times (generally 2 x

4) for fresh and identically prepared powder samples.


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25

2.3.3 Incipient yield

To combine the angle of internal friction i for incipient contact failure (slope of yield locus)

with the stationary angle of internal friction st following relation is used [22, 47]:

( ) ist tan1tan += (46)

The softer the particle contacts, the larger are the differences between these friction angles and

consequently, the more cohesive is the powder response.

The instantaneous yield locus describes the limit of incipient plastic powder deformation or

yield. A linear yield locus, Fig. 11, is obtained from resolution of a general square function [47],

is simply to use (M,st, R,st centre and radius of Mohr circle for steady-state flow as parameter of

powder pre-consolidation):

( )

+=+= st,M

i

st,RiZi

sintantan (47)

It is worth to note here that only the isostatic tensile strength Z for incipient yield depends di-

rectly on the consolidation pre-history and is given by:

0

i

stst,M

i

stst,M

i

st,RZ

sin

sin1

sin

sin

sin

+

=

= (48)

The smaller a radius stress for pre-consolidation VR < R,st, the larger is the centre stress VM >

M,st right of largest Mohr circle for steady-state flow in Fig. 11, and the smaller can be the pow-

der tensile strength Z.

Fig. 12: Shear stress - normal stress diagram of yield loci (YL) and stationary yield locus (SYL) of lime-

stone powder, straight line regression fit 0.98, d50 = 1.2 m, solid density s = 2740 kg/m3, shear rate vS

= 2 mm/min, surface moisture XW = 0.5 %.


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26

2.3.4 Incipient consolidation

The consolidation locus represents the envelope of all Mohr circles for consolidation stresses,

i.e. the radius VRand centre VM stresses between the Mohr circle for steady-state flow and the

isostatic stress iso, Fig. 12. Provided that the particle contact failure is equivalent to that be-

tween incipient powder flow and consolidation, one can write for a linear consolidation locuswith negative slope -sini which is symmetrically with the linear yield locus, Eq. (52):

)(sin isoVMiVR += (49)

Due to this symmetry between yield and consolidation locus, one can directly estimate the

isostatic powder compression 1 = 2 = VM = iso from Fig. 8 for the radius stress VR= 0:

0

i

stst,M

i

stst,M

i

st,RZst,Miso

sin

sin1

sin

sin

sin2

+

+

=+

=+= (50)

2.3.5 The three flow parameters

Generally, when we use these radius Rand centre stresses M, the essential flow parameters are

compiled as one set of linear constitutive equations, i.e. for instantaneous consolidation, the con-

solidation locus (CL),

( ) st,Rst,MMiR sin ++= , (51)

for incipient yield, the yield locus (YL),

( ) st,Rst,MMiR sin += (52)

and for steady-state flow, the stationary yield locus (SYL):

( )0st,Mstst,R sin += (38)

These yield functions are completely described only with three material parameters plus the

characteristic pre-consolidation stress M,st or average pressure influence, see Tomas [47]:

(1) i incipient particle friction of failing contacts, i.e. Coulomb friction;

(2) st steady-state particle friction of failing contacts, increasing adhesion by means of flat-

tening of contact expressed with the contact consolidation coefficient , or by friction an-gles ( ist sinsin ) as shown in the next Eqs. (53) and (54). The softer the particle con-

tacts, the larger are the difference between these friction angles the more cohesive is the

powder;

(3) 0 extrapolated isostatic tensile strength of unconsolidated particle contacts without any

contact deformation, equals a characteristic cohesion force in an unconsolidated powder;

(4) M,st previous consolidation influence of an additional normal force at particle contact,

characteristic centre stress of Mohr circle of pre-consolidation state directly related to

powder bulk density. This average pressure influences the increasing isostatic tensilestrength of yield loci via the cohesive steady-state flow as the stress history of the powder.


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27

2.3.6 Consolidation functions

These physically based flow parameters are necessary to derive the uniaxial compressive

strength c which is simply found from the linear yield locus, Eq. (52) and Fig. 11, forc = 2.R

(2 = 0 and R= M) as a linear function of the major principal stress 1, Fig. 13, [43]:

( )( ) ( )

( )( ) ( ) 0,c110ist

ist1

ist

istc a

sin1sin1sin1sin2

sin1sin1sinsin2 +=

+++

+= (53)

Equivalent to this linear function of the major principal stress 1 and using again Eq. (52), the

absolute value of the uniaxial tensile strength Z,1is also found forZ,1 = 2.R(1 = 0 and R= -

M):

( )( ) ( ) 011, sin1

sin2

sin1sin1

sinsin2

+

+++

=

st

st

ist

istZ (54)

Fig. 13. Powder strength - consolidation stress diagram of constitutive consolidation functions of lime-

stone, straight line regression fit = 0.98, median particle size d50 = 1.2 m, surface moisture XW = 0.5 %

accurately analysed by Karl Fischer titration. Additional flow properties according to the basic Eqs. (53)

and (54) are the averaged angle of internal friction i = 37, stationary angle of internal friction i = 43,

isostatic tensile strength of the unconsolidated powder0 = 0.65 kPa.


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28

Both flow parameters candZ,1 depend on the pre-consolidation level of the shear zone which

is expressed by the applied consolidation stress for steady-state flow 1. A considerable time

consolidation under this major principal stress 1 after one day storage at rest is also shown in

Fig. 13. Equivalent linear functions are also used to describe these time consolidation effects

[40-47].

2.4 Powder consolidation and compression functions

These comfortable models of yield surface are easy to handle and to describe the consolidation

and compression behaviours of cohesive and compressible powders on physical basis [45, 104].

2.4.1 Powder Flowability

In order to assess the flow behaviour of a powder, Eq. (53) shows that the flow function due to

Jenike [7, 8] ff c1c /= is not constant and depends on the pre-consolidation level 1. Ap-proximately, one can write for a small intercept with the ordinate c,0, Fig. 13, the stationary

angle of internal friction is equivalent to the effective angle ste and Jenikes [8] formula is

obtained:

( ) (( )

)

ie

iec

sinsin2

sin1sin1ff

+

(55)

Thus, the semi-empirical classification by means of the flow function introduced by Jenike [8] is

adopted here with considerations for certain particle behaviour, Table 2:

Table 2: Flowability assessment and elastic-plastic contact consolidation coefficient (i = 30)flow function ffc -values st 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 moist powder

Obviously, the flow behaviour is mainly influenced by the difference between the friction an-

gles, Eq. (55), as a measure for the adhesion force slope in the general linear particle contact

constitutive model, Eq. (22). Thus one can directly correlate with flow function ffc [47]:

( )1

sin1ff2

sin)1ff2(11

1

sin1ff2tan

sin)1ff2(12

ic

icici

ic

+

+

+

+= (56)

A characteristic value = 0.3 fori = 30 of a cohesive powder is included in the adhesion force

diagram, Fig. 5, and shows directly the correlation between strength and force increasing with

pre-consolidation, Table 2. Due to the consolidation function, a small slope designates a free


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29

flowing particulate solid with very low adhesion level because of stiff particle contacts but a

large slope implies a very cohesive powder flow behaviour because of soft particle contacts, Fig.

13. Obviously, the finer the particles the softer are the contacts and the more cohesive is the

powder [40, 43]. Khler [88] has experimentally confirmed this thesis for alumina powders (-

Al2O3) down to the sub-micron range (c,0const.= 2 kPa, d50 median particle size in m):62.0

50c d2.2ff (57)

2.4.2 Powder Compressibility

A survey of uniaxial compression equations was given by Kawakita [89]. Thus in terms of a

moderate cohesive powder compression, to draw an analogy to the adiabatic gas law, a differential equation for isentropic compressibility of a powder dS = 0, i.e.

remaining stochastic homogeneous (random) packing without a regular order in the continuum,

is derived, beginning with:

.constVp ad =

0st,M

st,M

b

bd

np

dpn

d

+

==

(58)

The total pressure including particle interaction p = M,st + 0 should be equivalent to a pressureterm with molecular interaction ( ) TRbVV/ap m

2mVdW =+ in van der Waals equation of

state to be valid near gas condensation point. A condensed loose powder packing is obtained

b = b,0, if only particles are interacting without an external consolidation stress M,st = 0, e.g.

particle weight compensation by a fluid drag, and Eq. (58) is solved:n

0

st,M0

0,b

b

+=

(59)

Therefore, this physically based compressibility index n 1/ad lies between n = 0, i.e. incom-

pressible stiff bulk material and n = 1, i.e. ideal gas compressibility. Considering the predomi-

nant plastic particle contact deformation in the stochastic homogeneous packing of a cohesive

powder, following values of compressibility index are recommended in Table 3:

Table 3: Compressibility index of powders, semi-empirical estimation for1 = 1 100 kPaindex n evaluation examples flowability

0 0.01 incompressible gravel

0.01 0.05 low compressibility fine sand

free flowing

0.05 - 0.1 compressible dry powder cohesive

0.1 - 1 very compressible moist powder very cohesive

Our limestone powder shows a compressible behaviour with the index n = 0.051 (b,0 = 720

kg/m). Both functions are shown in Fig. 14. Obviously, for the loose packing near the originM,st = - 0, the compression rate (slope of bulk density) is maximum by particle rearrangement

andincipient contact deformations, Fig. 14 dashed line.


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30

2.4.3 Powder compression and preshear work

The mass related or specific compression work Wm,b of a cohesive powder is obtained by an ad-

ditional integration of the reciprocal compression function Eq. (59) for n 1:

( )

+

=

=

11n1nd

nW

n1

0

st,M

0,b

0

0 st,Mb

st,M

b,m

st,M

(60)

Fig. 14. Bulk density centre stress diagram of compression function and compression rate of limestone

powder according to Eqs. (58) and (59), curve regression fit = 0.94, median particle size d50 = 1.2 m,

surface moisture XW = 0.5 %.

It describes the correlation between the external work (lower limit M,st = 0) as the function of

average pressure for steady-state flow M,st only for compression.

The specific compression work starts at the origin, Fig. 15, and comprises only the contributionof normal and shear stresses for pre-consolidation up to the bulk density for stationary flow

within the shear zone of height hSz. Additionally, the energy input during this steady-state flowfor constant bulk density b of shear zone is obtained as ( Szprepre h/s= preshear distortion, spre

preshear displacement):

( )n1

0

st,M

0,b

0

Sz

stpre

0

preprepre

b

pre,b,m 1h2

2sinsd

1W

pre

+

= (61)

To compare this energy consumption in handling practice, e.g. Wmb,pre = 2 J/kg is equivalent to

the specific kinetic energy of a shear rate of vS,eq = 2 m/s, see Fig. 15,

s/m2kg/J22W2v pre,b,meq,S === (62)


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31

and to a lift height of bulk powder Hb = 0.2 m of the equivalent potential energy:

m2.0s/m81.9

kg/J2

g

WH

2

pre,b,m

b == (63)

From the specific preshear work, Eq. (61), we can derive the mass related power consumption of

steady-state flow (vS = dspre/dt preshear rate):n1

0

st,M

0,b

0

Sz

stSpre,b,mpre,b,m 1

h2

2sinv

dt

dWP

+

== (64)

Fig. 15: Specific work centre stress diagram of mass related preshear and compression work and mass

related power consumption of limestone powder according to Eqs. (60), (61) and (64), curve fit = 0.97,

median particle size d50 = 1.2 m, surface moisture XW = 0.5 %. The mass related preshear work is essen-

tially larger than the specific work which is necessary to compress the powder.

This work or power input is converted into inelastic contact deformations, lattice dislocations at

surfaces, heat dissipation, particle asperity abrasion or particle-wall abrasion and micro-crackingup to particle breakage. For example, this should be considered to evaluate problems with fugi-

tive dust during handling.

Generally, the influence of micro-properties as particle contact stiffness on the macro-behaviour

as powder flow properties, i.e. cohesion, flowability and compressibility, is shown in Fig. 16.

Increasing contact compliance determine decreasing slope of the elastic-plastic yield boundary

(limit) and increasing inclination of the adhesion boundary or limit. As the result, the slope of

the normal force-adhesion force function increases. Next, the difference between the stationary

angle and angle of internal friction of the powder becomes larger. Consequently, the slope of thepowder consolidation function increases and the powder is more compressible, Fig. 16.


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Fig. 16: Characteristic constitutive functions of stiff and compliant particle contact behaviours, free flow-

ing and cohesive powder behaviours, and finally, stiff incompressible and soft compressible powders

[45]:

a) Force - displacement diagram of characteristic contact deformation according to Fig. 4,

b) Adhesion force - normal force diagram of particle contact forces according to Fig. 5,

c) Shear stress normal stress diagram of yield and consolidation loci (YL, CL) and stationary yield

locus (SYL) according to Fig. 12,

d) Powder strength - consolidation stress diagram of consolidation functions acc. to Fig. 13,

e) Radius stress centre stress diagram of yield and consolidation loci (YL, CL) and stationary yield

locus (SYL) according to Eqs. (38), (51) and (52),

f) Bulk density - consolidation stress diagram of compression function according to the following Fig.17 above panel.


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3. DESIGN CONSEQUENCES FOR RELIABLE FLOW

Mainly for functional silo design purposes in Mechanical Process Engineering, these funda-

mental models can be applied by means of a characteristic apparatus dimension function, here a

minimum hopper opening width bmin,t avoiding cohesive powder arches. The effective wall stress

of an arch or bridge is calculated with the dimensionless flow factor ff according to Jenike [8]:

ff1'

1

= (65)

The prime () denotes an effective stress. This effective wall stress 1 of the cohesive bridge has

to be larger than the uniaxial compressive strength c. Than the desired bridge failure condition

is obtained by Eqs. (53) and (65):

ffa1 1

0,ccrit,c

'1

= , (66)

and finally for the outlet width b bmin:

( )

( )ffa1g

2sin)1m(b

1b

W0,cmin

++= (67)

Inserting the coefficients a1 and c,0, the hopper opening width bmin is calculated (g gravitational

acceleration, m = 1 conical hopper, m = 0 wedge hopper, W angle of wall friction, hopper

angle versus vertical, b bulk density):

( ) ( )( ) ([ )]1ff2sinsinsinsin1g

sinsin12sin)1m(2b

ististb

0stiWmin

+++= (68)

The essential consolidation functions necessary for reliable design are provided in Fig. 17. For

the cohesive steady-state flow 1 = c,st, Fig. 8 and Eq. (45), the flow factor is ff = 1 and a mini-

mum outlet width bmin,st < bmin(instantaneous flow) is obtained which prevents bridging during

the stationary hopper operation:

( )( )stb

0stWstmin,

sin1g

sin2sin)1m(2b

++

= (69)

As starting value, bmin,st should be used for numerical calculation of both functions e(1) andb(1) as well as the flow factor )),((ff W1e [7, 8, 24, 90]. However, the hopper angle

has to be properly pre-selected for a cone or wedge [7,91]:)),(( W1e

e

Ww

e

econe

sin

sinarcsin

sin2

sin1arccos180

2

1(70)

( )

+

+

e

Wewedge

06.0exp131.03.421

73.7

50arctan

7.15

15.60

2

1(71)


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Fig. 17. Consolidation functions of a cohesive powder for the purpose of hopper design for reliable flow.

The compression function b(1) Eq. (59) with Eq. (43), the effective angle of internal friction e(1) Eq.

(44) and the uniaxial compressive strength c(1) Eq. (53) are necessary to calculate numerically the

minimum outlet width bmin according to Eq. (67).

For the cohesive limestone powder (d50,3 = 1.2 m, 0 = 0.65 kPa, b 770 kg/m3, friction an-

gles i = 37, st = 43, W = 30, maximum hopper angle 13 and flow factor ff 1.3) prac-tically reasonable, minimum outlet width bmin,st = 0.74 m for steady-state flow, Eq. (69), but bmin


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= 0.81 m for incipient yield, Eq. (68), and incredible bmin,t = 4.4 m after time consolidation at rest

t = 24 h are determined for a conical hopper to ensure mass flow driven by gravitation without

any flow promotion. This example should express the enormous problems concerning reliable,

gravitational flow of powders which are prone to time consolidation, hardening and caking. Con-

sequently, various discharging aids should be applied in handling practice [12, 40, 96].

This conservative design method using linear consolidation function c(1), Eq. (53), is practi-

cally preferred when the particle properties of the powder are expected to change gradually dur-

ing a certain processing period. Both, shear test results - accurate measurements provided -

evaluated with these combined particle and continuum mechanical approach, has been used as

constitutive functions for computer aided silo design for reliable flow for more than 20 years

[40].

Also a supplemented slice-element standard method [92] is used for pressure calculations [93,

94]. Considering the reliable physical basis of the b(1) and e(1) functions for example, these

can be suitably extrapolated using pressure calculations for large silos with more than 1000 m3

storage capacity [94].

4. CONCLUSIONS

A complete set of physically based equations for steady-state flow, incipient powder consolida-

tion and yielding, compressibility and flowability has been shown. Using this, the yield surfaces

due to theory of plasticity may be described with very simple linear expressions:

( )( )

( )

+

+

==

)CL(locusionconsolidat

)SYL(locusyieldstationary

)YL(locusyield

sin

sin

sin

0

st,Rst,MMiR

0st,Mstst,R

st,Rst,MMiR

CL,SYL,YL (72)

The consolidation and yield loci and the stationary yield locus are completely described only

with three material parameters, i.e., angle of internal friction i, stationary angle of internal fric-

tion st, isostatic tensile strength of an unconsolidated powder0 plus the characteristic pre-con-

solidation (average pressure) influence M,st. The compressibility index n as an additional consti-

tutive bulk powder parameter was introduced and the classification 0 n < 1 recommended. Adirect correlation between flow function ffc and elastic-plastic contact consolidation coefficient

was derived.

This approach has been used to evaluate the powder flow properties concerning various particle

size distributions (nanoparticles to granules), moisture contents (dry, moist and wet) and material

properties (minerals, chemicals, pigments, waste, plastics, food etc.), which have been tested and

evaluated for more than the last 20 years [40]. Thus, these models are directly applied to evaluate

the test data of a new oscillating shear cell [95, 96, 97] and a press-shear-cell in the high-level

pressure range from 50 to 2000 kPa for liquid saturated, compressible filter cakes [98, 99, 100]

and for dry powders [101]. Additionally, the force displacement behaviour during stressing and

the breakage probability, especially at conveying and handling, are useful constitutive functions


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36

to describe the mechanics of granules to assess the physical product quality [80]. These contact

models are also needed to simulate the shear dynamics of cohesive powders using the discrete

element method (DEM) and to calibrate these simulations by shear cell measurements [102].

Detectable mechanoluminescence effects during intensive shear stressing of high-disperse parti-

cles are also related to these fundamentals [103].

The influence of particle surface properties, e.g. as contact stiffness, on the powder flow proper-

ties can be directly interpreted, Fig. 16, and practically used to design stress resistant, dust-free,

free flowing and non-caking particulate products in process industries [1, 2, 3, 104, 105].

5. ACKNOWLEDGEMENTS

The author would like to acknowledge his co-workers Dr. S. Aman, Dr. T. Grger, Dr. W. Hintz,

Dr. Th. Kollmann and Dr. B. Reichmann for providing relevant information and theoretical tips.

The advises from H.-J. Butt [106] and S. Luding [107] with respect to the fundamentals of parti-

cle and powder mechanics were especially appreciated during the collaboration of the project

shear dynamics of cohesive, fine-disperse particle systems of the joint research program Be-

haviour of Granular Media of German Research Association (DFG).


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6. SYMBOLS

a nm molecular separation

A m area

AK nm particle contact area

b m hopper outlet width to avoid bridging

CH,sls J Hamaker constant according to Lifshitz [71]

d m particle size

E kN/mm modulus of elasticity

ff - flow factor according to Jenike [8]

ffc - flow function according to Jenike [8]

F N force

G kN/mm shear modulus

h mm height of sampleHb m lift height

hK nm total contact or indentation height (overlapp of spheres)

Kh& nm/h indentation rate

hSz mm characteristic height of shear zone

p kPa pressure

pf MPa plastic yield strength of particle contact

Pm W/kg mass related power input

r nm radiuss mm shear displacement

vS m/s preshear rate

Wm J/kg mass related work

- preshear distortion

nm tangential contact displacement

- porosity

V - volumetric strain

K Pa.s apparent contact viscosity

deg hopper angle

- elastic-plastic contact consolidation coefficient

A - elastic-plastic contact area coefficient

p - plastic contact repulsion coefficient

- Poisson ratio

e deg effective angle of internal friction according to Jenike [8]

i deg angle of internal friction

st deg stationary angle of internal friction

W deg kinematic angle of wall friction kg/m density


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kPa normal stress

iso kPa isostatic normal stress

M,st kPa centre stress for steady-state flow

R,st kPa radius stress for steady-state flow

1 kPa major principal stress for steady-state flow

2 kPa minor principal stress for steady-state flow

0 kPa isostatic tensile strength

kPa shear stress

7. INDICES

b bulk

c compressive

K total contact

e effective

el elastic

eq equivalent

H adhesion

i internal

l liquid

M centre of Mohr circle

N normal pl plastic

pre preshear

R radius of Mohr circle

s solid

sls solid-liquid-solid

st stationary

Sz shear zone

t time dependentV pre-consolidation

VdW van der Waals

vis viscoplastic

W wall

Z tensile

0 initial, zero point


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39

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