THE MECHANICS OF DRY, COHESIVE POWDERS1

Jürgen Tomas Mechanical Process Engineering, The Otto-von-Guericke-University Magdeburg Universitätsplatz 2, D – 39 106 Magdeburg, Germany Phone: ++49 391 67 18 783, Fax: ++49 391 67 11 160 e-mail: juergen.tomas@vst.uni-magdeburg.de 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 µm) 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

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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 ........................................................... 29 2.4.3 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 include bridging, 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 σ ≈ 0.1 - 100 kPa and slow frictional flow with shear rates vS < 1 m/s - and thus shear s s

kPa12/v2 ≈⋅ρ<τ - of fine, compressible and cohesive powders (particle size d < 10 µm)

should be described here. A powder bulk Reynolds number is less than unity for τ > 1 kPa (hS

tress contribution

z height of shear zone, ηb apparent bulk viscosity, ρb bulk density):

1kPa1s

m/kg1000m1vhvRe 2

32b

2S

b

bSzSb =

⋅⋅

=τρ⋅

≈η

ρ⋅⋅= (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 Tüzün [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) viscoelastic contact deformation (Yang [67], Rumpf [68] and Sadd [61]) which is reversible and 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 a spherical 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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( ) 20F

rsls,H2

0Fr

r2

0F

rsls,H0H a12

hCa/h12

h/d1a12

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 FH0 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

8

3K2,1N hr*E

32F ⋅⋅⋅= , (3)

with the averaged radius of particle 1 and 2

212,1 r/1r/1

1r+

= (4)

and the averaged modulus of elasticity E* of both particles 1and 2 (ν Poisson’s ratio): 1

2

22

1

21

E1

E12*E

−

ν−+

ν−⋅= (5)

Due to the parabolic curvature FN(hK), the particle contact becomes stiffer with increasing displacement hK or contact radius rK and particle radius r1,2 (kN is the contact stiffness in normal direction):

KK2,1K

NN r*Ehr*E

dhdFk ⋅=⋅⋅== (6)

When one applies an increasing load FN the contact starts at pmax = pf with plastic yielding at partial plate-plate contact. This elastic-plastic contact deformation response, see in Fig. 3 panel c), results in an additional contribution to adhesion force between these two particles, Krupp [62], Rumpf et al. [68] and Molerus [22, 23]. The total force can be obtained by the particle con-tact force equilibrium between attraction (-) and elastic as well as soft plastic repulsion (+) or force response ( coordinate of annular elastic contact area): *

Kr

∫∑ ⋅⋅π⋅+⋅π⋅+−⋅π⋅−−==K

pl,K

r

r

*K

*K

*Kel

2pl,KfN

2KVdW0H drr)r(p2rpFrpF0F (7)

Superposition provided, this leads to a very useful linear force displacement model (for κA ≈ const.) with the particle centre approach of both particles hK [43], shown in Fig. 4 as elastic-plastic yield boundary (or limit):

( ) KpAf2,10HN hprFF ⋅κ−κ⋅⋅⋅π=+ (8)

Thus, the contact stiffness decreases with smaller particle size 2,1r4d ⋅= (or micro-roughness

radius hr of non-deformed contact) of cohesive powders, predominant plastic yield behaviour provided [43]:

( pAf2,1K

Npl,N pr

dhdFk κ−κ⋅⋅⋅π== ) (9)

This size-dependent contact “softness” contributes essentially to a lot of adhesion effects of nanoparticles besides its large surface. Consequently, it makes sense to introduce here the model “stiff particles with soft contacts”. The particles may have a certain material stiffness so that the volume deformation is negligible. Any irreversible contact deformation should not have too large influence on the particle shape which is equivalent to a model of “healing contacts”.

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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 µm, 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 FH0 (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 pmax = 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. 3 panel d). This actual particle separation is considered for the calculation by KA,K0F hhaa −+= =

10

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(FN), see next Fig. 5.

repat F+

The plastic repulsion coefficient κp describes a dimensionless ratio of attractive van der Waals pressure pVdW (adhesion force per unit planar surface area) to repulsive particle micro-hardness pf for a plate-plate model (e.g. pVdW ≈ 3 – 600 MPa):

f0F

sls

f3

0F

sls,H

f

VdWp pa

4pa6

Cp

p⋅σ⋅

=⋅⋅π⋅

==κ==

(10)

This attraction term pVdW can also be expressed by surface tension, e.g. σsls ≈ 3.10-4 – 0.06 J/m²,

20F

sls,H

aVdWsls a24

Cda)d(p

21

0F =

∞

⋅π⋅=−=σ ∫

=

(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 mobile adsorption 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 κA 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

plA h

h311

AA

31

32

⋅−=⋅+=κ , (12)

with the centre approach hK,f for incipient yielding at point Y in Fig. 4, pel(rK = 0) = pmax = pf: 2

ff,K *E2

pdh

⋅⋅π

⋅= (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, κA = 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, κA = 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,H3

A,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*E32F +−⋅⋅⋅−= (15)

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

A,KVdW2,10HA,H hprFF ⋅⋅⋅π+= (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 ⋅κ+⋅−= − (17)

The actual particle separation a can be used by a long-range hyperbolic adhesion force curve with the displacement h3

Z,N aF −∝ K,A for incipient contact detachment by Eq. (17):

3

0F

K

0F

A,K

A,KVdW2,10HKZ,N

ah

ah

1

hprF)h(F

−+

⋅⋅⋅π+−=

==

(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

hKKunload,N

h

hKKreload,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 mass related 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 ≈ π/ε [22]): s

32,1P r3/4m ρ⋅⋅π⋅=

12

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

22,1

A,KU,Kf

2/5

2,1

A,KU,K

sdiss,m hhh

r32hhp3

rhh

20*EW −⋅κ−⋅κ

ρ⋅ε⋅⋅−⋅⋅π⋅

+

−ρ⋅ε⋅

−= ) (20)

This specific energy of 1.6 to 31 µJ/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 σM,st = 3.3 to 25 kPa (or major principal stress σ1 = 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,H22,1

0HNfp

22,10HN0H)1(,A,H FF

FF1

*Er2FF3prFFFF

+

−+⋅

⋅⋅+⋅

⋅⋅κ⋅⋅π−+⋅κ+= (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 κ 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 ⋅κ+⋅κ+=⋅

κ−κ

κ+⋅

κ−κκ

= (22)

The dimensionless elastic-plastic contact consolidation coefficient (strain characteristic) κ is given by the slope of adhesion force FH influenced by predominant plastic contact failure.

pA

p

κ−κ

κ=κ (23)

This elastic-plastic contact consolidation coefficient κ 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 κ⋅FH0. Thus, the total adhesion force consists of a stiff contribution FH0 and a con-tact strain influenced component ( N0H FF )+⋅κ , Fig. 5.

This Eq. (22) can be interpreted as a general linear particle contact constitutive model, i.e. linear in 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:

( ) fr0F20Fr

r

K

plfr0FZ,N pha

2a/h12h/d1

A3A

32pha

2F ⋅⋅⋅

π−≈

+⋅+⋅

⋅

+⋅⋅⋅⋅π

−= ==

= (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 ≈ 3⋅σf 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 σ1,Z of the constitutive consolidation function σc(σ1), 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 µm, surface moisture XW = 0.5 %, specific surface area AS,m = 9.2 m²/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 κ = 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 20°C 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 macroscopic stresses in powders [44, 47, 94]. Additionally, one can obtain a direct correlation between the

14

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 105 - 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 f

30F

sls,H

f

VdWp pa6

Cp

p⋅⋅π⋅

==κ=

tp

K

VdWt,p ⋅

η=κ

Constitutive models of combined contact deformation

elastic-plastic elastic-plastic and viscoplastic

Contact area ratio ( )elpl

plA AA3

A32

+⋅+=κ ( )elvispl

visplt,A AAA3

AA32

++⋅

++=κ

Contact consolidation coefficient

pA

p

κ−κ

κ=κ

t,ppt,A

t,ppvis κ−κ−κ

κ+κ=κ

Intersection with FN- axis (abscissa)

( )sls,Hf2,1r0FZ,N CfphaF ≠⋅⋅⋅π−≈ = ( )sls,HKf

f2,1r0Ftot,Z,N Cf

/tp1pha

F ≠η⋅+

⋅⋅⋅π−≈ =

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 κp + κp,t, 5th line in Table 1. These are characterized by the micro-yield strength pf, apparent contact viscosity and time ηK/t. Hence the repulsion effect of “cold” viscous flow of comparatively strongly-bonded adsorption layers on the particle surface is taken into consideration [45, 46, 104]. Hence with the

15

total viscoplastic contact consolidation coefficient κvis, 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,pp0H

t,ppt,A

t,Atot,H FF1FFF ⋅κ+⋅κ+=⋅

κ−κ−κ

κ+κ+⋅

κ−κ−κκ

= (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 ⋅κη< )

S,m = 9.2 m²/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 ∆⋅ϕ⋅ψ−±δ∆⋅⋅⋅⋅ψ⋅= (26)

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

the averaged shear modulus is given as: 1

2

2

1

1

G2

G22*G

−

ν−+

ν−⋅= (27)

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

KT

T r*G4ddFk ⋅⋅=δ

= (28)

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

( )ν−ν−⋅

=212

kk

N

T (29)

Hence this ratio ranges from unity, for ν = 0, to 2/3, for ν = 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].

16

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 reversible displacement 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.

17

Using increasing shear stress τ 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-called volumetric strain 0V h/)s(h∆=ε . 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 τ(σ) are taken into consideration, the negatives mean opposite directions. Mainly, compressive stresses (pressures) σ 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.

18

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 τ. During this shear the shear zone expands dV > 0, Fig. 6 bottom panel. This dilatancy h

r∆ 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 ν. If we connect all stress pairs τ(σ) we may obtain a straight line which is called as yield locus. The slope of this line is the angle of internal friction ϕi. The intersection with the ordinate σ = 0 represents the cohesion τc a shear resistance caused solely by particle adhesion effects without any external normal stress σ. 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 τc 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 σ1 and σ2, i.e. the largest (major) and the smallest (minor) normal stress without applying any shear stress τ = 0. The Mohr circle which intersects the origin, i.e. minor principle stress σ2 = 0, gives us the uniaxial compressive strength σc 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 σZ,1 for σ2 = 0, see left in Fig. 7 below. The intersection of the yield locus with abscissa, the isostatic tensile strength σZ represents the internal contribution of adhesion forces FH(FN) to the total stress, i.e. the sum of external normal stress σ plus σZ. Thus, this characteristic σZ 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 σ1 and σ2 and equivalent to these the radius and centre stresses σR,st and σM,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 ϕst. To extrapolate the stationary yield locus, the isostatic tensile strength σ0 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-

19

mental characteristic σ0 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 σ, e.g. by weight of powder layers, plus (by absolute value) an internal contribution by the pre-consolidation dependent adhesion, the isostatic tensile stress σZ.

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 plastic failure which leads to a consolidation of the particle packing dV < 0. This line may have the

20

same inclination as the slope of yield locus, the angle of internal friction ϕi, and intersects the abscissa at an isostatic normal stress σiso. This isostatic stress state means that all principal stresses have the same value in all three spatial directions σiso = σ1 = σ2 = σ3. 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 h

r∆ 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 +⋅κ+⋅κ+=+= (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

dFd

1d/hd

dE→→σ

⋅⋅εε−

=σ+σ

= (31)

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

( ) 3/1

Z23/1

22,1

V,N0H2

b *E61

4*E

r2FF)1(*E3

41E

σ⋅⋅

εε−

⋅=

⋅⋅κ+⋅κ+⋅⋅

⋅ε⋅ε−

= (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

614*E

r2F*E3

41E

σ⋅⋅

εε−

⋅=

⋅⋅⋅

⋅ε⋅ε−

= (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].

21

Fig. 9. Bulk shear modulus - centre stress diagram for load and unload of limestone powder (d50 = 1.2 µm). 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 to be G = 60 kN/mm² and Poisson ratio ν = 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 ∆δ/d ≈ ∆s/hSz (hSz characteristic height of the shear zone):

( ) 0F

T

0Szb

TddF

d1

h/sddG

→→τδ

⋅⋅εε−

=τ

= (34)

( ) 3/1

Z23/1

22,1

V,N0Hb *E

61*Gr*E2

FF)1(3*G1G

σ⋅⋅

εε−

⋅=

⋅⋅⋅κ+⋅κ+⋅

⋅⋅εε−

= (35)

3/1

0

2

0

0

3/1

22,1

0H

0

00,b *E

61*Gr*E2

F3*G1G

σ⋅⋅

εε−

⋅=

⋅⋅⋅

⋅⋅εε−

= (36)

This simple model, Eq. (35), overestimates the shear modulus Gb = 60 - 120 N/mm² for lime-stone (particle size d50 = 1.2 µm) compared to the shear modulus obtained from direct shear tests Gb,load = 180 - 270 kN/m² for loading and Gb,unload = 220 - 340 kN/m² for unloading [85]. Obvi-

22

ously, the shear modulus Gb depends on the pre-consolidation of the isostatic tensile strength σZ = f(σM,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):

( )ν−ν−⋅

===212

kk

EG

EG

N

T

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 ν = 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 σ+σ⋅ϕ=σ (38)

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

)(tan 0ststst σ+σ⋅ϕ=τ (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⋅

εε−

=σ . (40)

This isostatic tensile strength σ0 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 loose packing s0,b0 /1 ρρ−=ε and ρb,0 according to Eq. (59).

In some cases one may observe cohesionless steady-state flow, i.e. σ0 = 0 in Eq. (38), which is described by the effective yield locus according to Jenike [8] with the effective angle of internal friction ϕe as slope:

st,Mest,R sin σ⋅ϕ=σ (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 ϕe, the stationary angle of internal fric-tion ϕst and the centre stress σM,st:

σσ

+⋅ϕ=ϕst,M

0ste 1sinsin (42)

23

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 hopper design method, the latter is necessary for the calculation of flow factor ff )),(( W1e ϕσϕ [8, 24, 90] and

effective wall stress σ , 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 σM,st can be replaced by the major principle stress σ1 during steady-state flow

st

st01st,M sin1

sinϕ+ϕ⋅σ−σ

=σ , (43)

and one obtains

σ⋅ϕ−σ

σ+σ⋅ϕ=∆=ϕ

0st1

01ste sin

sintansin (44)

24

which is in accordance with the daily experimental experience in shear testing, Fig. 10 upper diagram [43]. If the major principal stress σ1 reaches the stationary uniaxial compressive strength σc,st, Fig. 10 diagram below,

st

0stst,c1 sin1

sin2ϕ−σ⋅ϕ⋅

=σ=σ (45)

the effective angle of internal friction amounts to ϕe = 90° and for σ1 → ∞ follows ϕe → ϕst. In soil mechanics [86] an effective angle of friction φ’ is used as slope of an auxiliary line which connects the preshear points of yield loci σpre, or approximately, the maxima of end Mohr circles at σM,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), consolidation by 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 σV (1) and pre-consolidated by twisting the cover. Than a smaller normal stress σpre < σV (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 σ < σpre (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 σ and unloaded to τ = 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.

25

2.3.3 Incipient yield

To combine the angle of internal friction ϕi for incipient contact failure (slope of yield locus) with the stationary angle of internal friction ϕst following relation is used [22, 47]:

( ) ist tan1tan ϕ⋅κ+=ϕ (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 (σM,st, σR,st centre and radius of Mohr circle for steady-state flow as parameter of powder pre-consolidation):

( )

σ−

ϕσ

+σ⋅ϕ=σ+σ⋅ϕ=τ st,Mi

st,RiZi sin

tantan (47)

It is worth to note here that only the isostatic tensile strength σZ for incipient yield depends di-rectly on the consolidation pre-history and is given by:

0i

stst,M

i

stst,M

i

st,RZ sin

sin1sinsin

sinσ⋅

ϕϕ

+σ⋅

−

ϕϕ

=σ−ϕ

σ=σ (48)

The smaller a radius stress for pre-consolidation σVR < σR,st, the larger is the centre stress σVM > σM,st right of largest Mohr circle for steady-state flow in Fig. 11, and the smaller can be the pow-der tensile strength σZ.

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 µm, solid density ρs = 2740 kg/m3, shear rate vS = 2 mm/min, surface moisture XW = 0.5 %.

26

2.3.4 Incipient consolidation

The consolidation locus represents the envelope of all Mohr circles for consolidation stresses, i.e. the radius σVR and centre σVM stresses between the Mohr circle for steady-state flow and the isostatic stress σiso, 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 locus with negative slope -sinϕi which is symmetrically with the linear yield locus, Eq. (52):

)(sin isoVMiVR σ+σ−⋅ϕ=σ (49)

Due to this symmetry between yield and consolidation locus, one can directly estimate the isostatic powder compression σ1 = σ2 = σVM = σiso from Fig. 8 for the radius stress σVR = 0:

0i

stst,M

i

stst,M

i

st,RZst,Miso sin

sin1sinsin

sin2 σ⋅

ϕϕ

+σ⋅

+

ϕϕ

=σ+ϕ

σ=σ+σ⋅=σ (50)

2.3.5 The three flow parameters

Generally, when we use these radius σR and centre stresses σM, 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 σ+σ+σ−⋅ϕ=σ , (51)

for incipient yield, the yield locus (YL),

( ) st,Rst,MMiR sin σ+σ−σ⋅ϕ=σ (52)

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

( )0st,Mstst,R sin σ+σ⋅ϕ=σ (38)

These yield functions are completely described only with three material parameters plus the characteristic pre-consolidation stress σM,st or average pressure influence, see Tomas [47]: (1) ϕi – incipient particle friction of failing contacts, i.e. Coulomb friction; (2) ϕst – steady-state particle friction of failing contacts, increasing adhesion by means of flat-

tening of contact expressed with the contact consolidation coefficient κ, or by friction an-gles ( ist sinsin )ϕ−ϕ 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) σ0 – extrapolated isostatic tensile strength of unconsolidated particle contacts without any contact deformation, equals a characteristic cohesion force in an unconsolidated powder;

(4) σM,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 tensile strength of yield loci via the cohesive steady-state flow as the stress history of the powder.

27

2.3.6 Consolidation functions

These physically based flow parameters are necessary to derive the uniaxial compressive strength σc which is simply found from the linear yield locus, Eq. (52) and Fig. 11, for σc = 2.σR (σ2 = 0 and σR = σM) as a linear function of the major principal stress σ1, Fig. 13, [43]:

( )( ) ( )

( )( ) ( ) 0,c110

ist

ist1

ist

istc a

sin1sin1sin1sin2

sin1sin1sinsin2

σ+σ⋅=σ⋅ϕ−⋅ϕ+ϕ+⋅ϕ⋅

+σ⋅ϕ−⋅ϕ+

ϕ−ϕ⋅=σ (53)

Equivalent to this linear function of the major principal stress σ1 and using again Eq. (52), the absolute value of the uniaxial tensile strength σZ,1 is also found for σZ,1 = 2.σR (σ1 = 0 and σR = -σM):

( )( ) ( ) 011, sin1

sin2sin1sin1

sinsin2σ⋅

ϕ+ϕ⋅

+σ⋅ϕ+⋅ϕ+

ϕ−ϕ⋅=σ

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 µm, 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 ϕi = 37°, stationary angle of internal friction ϕi = 43°, isostatic tensile strength of the unconsolidated powder σ0 = 0.65 kPa.

28

Both flow parameters σc and σZ,1 depend on the pre-consolidation level of the shear zone which is expressed by the applied consolidation stress for steady-state flow σ1. A considerable time consolidation under this major principal stress σ1 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 /σσ= is not constant and depends on the pre-consolidation level σ1. Ap-

proximately, one can write for a small intercept with the ordinate σc,0, Fig. 13, the stationary angle of internal friction is equivalent to the effective angle ϕst ≈ ϕe and Jenike’s [8] formula is obtained:

( ) (( )

)ie

iec sinsin2

sin1sin1ffϕ−ϕ⋅ϕ−⋅ϕ+

≈ (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 κ(ϕi = 30°)

flow function ffc κ-values ϕst in deg evaluation Examples 100 - 10 0.01006 – 0.107 30.3 - 33 free flowing dry fine sand

4 - 10 0.107 – 0.3 33 – 37 easy flowing moist fine sand 2 - 4 0.3 – 0.77 37 – 46 cohesive dry powder 1 - 2 0.77 - ∞ 46 – 90 very cohesive moist powder < 1 ∞ - non flowing 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 κ in the general linear particle contact constitutive model, Eq. (22). Thus one can directly correlate κ with flow function ffc [47]:

( ) 1

sin1ff2sin)1ff2(1

1

1sin1ff2tan

sin)1ff2(12

ic

icici

ic −

ϕ+−⋅ϕ⋅−⋅+

−

⋅ϕ+−⋅⋅ϕ

ϕ⋅−⋅+=κ (56)

A characteristic value κ = 0.3 for ϕi = 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

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]. Köhler [88] has experimentally confirmed this thesis for alumina powders (α-Al2O3) down to the sub-micron range (σc,0 ≈const.= 2 kPa, d50 median particle size in µm):

62.050c 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 =⋅ κ

0st,M

st,M

b

b dn

pdpnd

σ+σσ

⋅=⋅=ρρ

(58)

The total pressure including particle interaction p = σM,st + σ0 should be equivalent to a pressure term 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 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. (58) is solved:

n

0

st,M0

0,b

b

σσ+σ

=ρρ

(59)

Therefore, this physically based compressibility index n ≡ 1/κad 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 for σ1 = 1 – 100 kPa

index 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 (ρb,0 = 720 kg/m³). Both functions are shown in Fig. 14. Obviously, for the loose packing near the origin σM,st = - σ0, the compression rate (slope of bulk density) is maximum by particle rearrangement and incipient contact deformations, Fig. 14 dashed line.

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:

( )

−

σσ

+⋅ρσ

⋅−

=σρσ

⋅=−σ

∫ 11n1

ndnW

n1

0

st,M

0,b

0

0 st,Mb

st,Mb,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 µm, surface moisture XW = 0.5 %.

It describes the correlation between the external work (lower limit σM,st = 0) as the function of average pressure for steady-state flow σM,st only for compression. The specific compression work starts at the origin, Fig. 15, and comprises only the contribution of 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 flow for constant bulk density ρb of shear zone is obtained as ( Szprepre h/s=γ preshear distortion, spre

preshear displacement):

( )n1

0

st,M

0,b

0

Sz

stpre

0preprepre

bpre,b,m 1

h22sins

d1Wpre −τ

σσ

+⋅ρσ

⋅⋅

ϕ⋅≈γγτ⋅

ρ= ∫ (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)

31

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

m2.0s/m81.9

kg/J2g

WH 2

pre,b,mb ≈== (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

h22sinv

dtdW

P−

σσ

+⋅ρσ

⋅⋅

ϕ⋅== (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 µm, 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-cracking up 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 the powder consolidation function increases and the powder is more compressible, Fig. 16.

32

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.

33

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'

1σ

=σ (65)

The prime (´) denotes an effective stress. This effective wall stress σ1’ of the cohesive bridge has to be larger than the uniaxial compressive strength σc. Than the desired bridge failure condition is obtained by Eqs. (53) and (65):

ffa1 1

0,ccrit,c

'1 ⋅−

σ=σ≥σ , (66)

and finally for the outlet width b ≥ bmin:

( )( )ffa1g

2sin)1m(b

1b

W0,cmin ⋅−⋅⋅ρ

θ+ϕ⋅σ⋅+= (67)

Inserting the coefficients a1 and σc,0, the hopper opening width bmin is calculated (g gravitational acceleration, m = 1 conical hopper, m = 0 wedge hopper, ϕW angle of wall friction, θ hopper angle versus vertical, ρb bulk density):

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

sinsin12sin)1m(2bististb

0stiWmin −⋅⋅ϕ−ϕ−ϕ⋅ϕ−⋅⋅ρ

σ⋅ϕ⋅ϕ+⋅θ+ϕ⋅+⋅= (68)

The essential consolidation functions necessary for reliable design are provided in Fig. 17. For the cohesive steady-state flow σ1 = σc,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(2bϕ−⋅⋅ρ

σ⋅ϕ⋅θ+ϕ⋅+⋅= (69)

As starting value, bmin,st should be used for numerical calculation of both functions ϕe(σ1) and ρb(σ1) as well as the flow factor )),((ff W1e ϕσϕ [7, 8, 24, 90]. However, the hopper angle

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

ϕϕ

−ϕ−

ϕ⋅ϕ−

−°≤θe

Ww

e

econe sin

sinarcsinsin2sin1arccos180

21

(70)

( )

ϕ⋅⋅°+°

ϕ−⋅

°ϕ−°

°+°≤θ

e

Wewedge 06.0exp131.03.42

173.7

50arctan7.15

15.6021

(71)

34

Fig. 17. Consolidation functions of a cohesive powder for the purpose of hopper design for reliable flow. The compression function ρb(σ1) Eq. (59) with Eq. (43), the effective angle of internal friction ϕe(σ1) Eq. (44) and the uniaxial compressive strength σc(σ1) 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 µm, σ0 = 0.65 kPa, ρb ≈ 770 kg/m3, friction an-gles ϕi = 37°, ϕst = 43°, ϕW = 30°, maximum hopper angle θ ≤ 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

35

= 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 σc(σ1), 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 ρb(σ1) and ϕe(σ1) 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:

( )( )

( )

σ−σ+σ−⋅ϕ−σσ+σ⋅ϕ−σσ−σ−σ⋅ϕ−σ

==Φ)CL(locusionconsolidat

)SYL(locusyieldstationary)YL(locusyield

sinsin

sin0

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 ϕi, stationary angle of internal fric-tion ϕst, isostatic tensile strength of an unconsolidated powder σ0 plus the characteristic pre-con-solidation (average pressure) influence σM,st. The compressibility index n as an additional consti-tutive bulk powder parameter was introduced and the classification 0 ≤ n < 1 recommended. A direct correlation between flow function ffc and elastic-plastic contact consolidation coefficient κ 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

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. Gröger, 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).

37

6. SYMBOLS

a nm molecular separation A m² area AK nm² particle contact area b m hopper outlet width to avoid bridging CH,sls J Hamaker constant according to Lifshitz [71] d µm particle size E kN/mm² modulus of elasticity ff - flow factor according to Jenike [8] ffc - flow function according to Jenike [8] F N force G kN/mm² shear modulus h mm height of sample Hb 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 radius s mm shear displacement vS m/s preshear rate Wm J/kg mass related work γ - preshear distortion δ nm tangential contact displacement ε - porosity εV - volumetric strain ηK Pa.s apparent contact viscosity θ deg hopper angle κ - elastic-plastic contact consolidation coefficient κA - elastic-plastic contact area coefficient κp - plastic contact repulsion coefficient ν - Poisson ratio ϕe deg effective angle of internal friction according to Jenike [8] ϕi deg angle of internal friction ϕst deg stationary angle of internal friction ϕW deg kinematic angle of wall friction ρ kg/m³ density

38

σ kPa normal stress σiso kPa isostatic normal stress σM,st kPa centre stress for steady-state flow σR,st kPa radius stress for steady-state flow σ1 kPa major principal stress for steady-state flow σ2 kPa minor principal stress for steady-state flow σ0 kPa isostatic tensile strength τ 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 dependent V pre-consolidation VdW van der Waals vis viscoplastic W wall Z tensile 0 initial, zero point

39

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