THE MECHANICS OF DRY, COHESIVE POWDERS 1

Transcription

1 THE MECHANICS OF DRY, COHESIVE POWDERS Jürgen Tomas Mechanical Process Engineering, The Otto-von-Guericke-University Magdeburg Universitätsplatz, D 39 6 Magdeburg, Germany Phone: , Fax: 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 (d 5. 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. eywords: 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. Paper at Bulk India 3, 9 Dec. 3 Mumbai, review version

2 . Introduction...3. Slow Frictional Flow of Cohesive Powder...4. Particle contact constitutive models Normal force - displacement functions of particle contact Energy absorption in a contact with dissipative behaviour Adhesion force - normal force model Viscoplastic contact behaviour and time dependency Tangential contact force Biaxial stress states in a sheared particle packing Shear force - displacement relation Shear stress normal stress diagram Cohesive powder flow criteria Elasticity of pre-consolidated powder Cohesive steady-state flow Incipient yield Incipient consolidation The three flow parameters Consolidation functions Powder consolidation and compression functions Powder Flowability Powder Compressibility Powder compression and preshear work Design Consequences for Reliable Flow Conclusions Acknowledgements Symbols Indices References...39

3 . INTRODUCTION 3 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 market. Fig.. Storage in containers - mechanical behaviours of solid, liquid, gas and bulk solid according to alman [4]. Solids or parcels and fluid products are comparatively easy to handle. But the mechanical behaviours of powders or granulates [ - 3] depend directly on pre-stressing history. This can be demonstrated by a simple tilting test of storage containers [4]. Depending on how to fill the container, tilt and bring it back different shapes of the bulk surface will be generated, Fig.. A cohesive 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 problems in powder processing and handling equipment. These well-known flow problems of cohesive 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

4 4 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 consolidation 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. shows also that the powder has a memory concerning its physical-chemical product properties. In terms of mineral genesis it can be of global historic periods. These peculiarities of cohesive powders connected with its strong individualism to flow or not to flow, we try now to understand it from a fundamental point of view:. SLOW FRICTIONAL FLOW OF COHESIVE POWDER A comparatively low consolidation in a pressure range of about σ. - kpa and slow frictional flow with shear rates v S < m/s - and thus shear stress contributions τ < ρ v / kpa - of fine, compressible and cohesive powders (particle size d < µm) should be described here. A powder bulk Reynolds number is less than unity for τ > kpa (h S height of shear zone, η b apparent bulk viscosity, ρ b bulk density): 3 vs hsz ρb vs ρb m kg / m Reb () η τ s kpa b 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 - ], and the Warren-Spring-Equations [3-6], Birks [8 - ], and the approach by Tüzün [] etc., were supplemented by Molerus [3-5] 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 [6] and Horne [7]. Parallel to it, Nedderman [8, 9], Jenkins [3], Savage [3] and others discussed the rapid and non-rapid particle flow as well as Tardos [3, 33] the slow and the so-called intermediate, frictional flow of compressible powders without any cohesion from the fluid mechanics point of view. z

5 5 Fig. : 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 [] 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. [6] 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.

6 6 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 adhesion forces at particle contacts, Molerus [, 3]. 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 [4 to 47]:. 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: () elastic contact deformation (Hertz [49], Huber [5], Cattaneo [5], Mindlin [5, 53], Greenwood [54], Dahneke [55], Derjaguin (DMT theory) [57], Johnson (JR theory) [58], Thornton [59] and Sadd [6]) which is reversible, independent of deformation rate and consolidation time effects and valid for all particulate solids; () plastic contact deformation with adhesion (Derjaguin [56], rupp [6], Schubert [63], Molerus [, 3], 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 [6]) 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... Normal force - displacement functions of particle contact These normal force - displacement models are shown as characteristic constitutive functions in Fig.. 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 a F to form a direct contact, see Fig. 3 panel a). Consequently, a long-range adhesion force is created because of van der Waals interactions of both surfaces. This adhesion force F H can be modelled as a single rough sphere-sphere-contact [54], additionally, with a characteristic hemispherical micro-roughness height or radius h r < d instead of particle size d [73, 74]:

7 F H C h d / h C 7 h H,sls r r H,sls r + () a F ( + h r / a F ) a F Fig. 3: Particle contact approach, elastic, elastic-plastic deformation and detachment. After approaching a a F, 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 H and an increasing normal load F N, the contact starts at the yield point p max p f 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 p max and an annular elastic pressure distribution dependent on radius r,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 F N,Z then the contact plates fail and detach with the increasing distance a > a F, panel d). After loading with an external compressive normal force F N the previous contact point is deformed to a small contact area. As the contact deformation response, a non-linear function between this elastic force and the centre approach (indentation height or overlap) h is obtained according to Hertz [49], demonstrated in Fig. 3 and Fig. 4

8 F N 8 3 E* r, h, (3) 3 with the averaged radius of particle and r, (4) / r + / r and the averaged modulus of elasticity E* of both particles and (ν Poisson s ratio): ν ν E* + E E (5) Due to the parabolic curvature F N (h ), the particle contact becomes stiffer with increasing displacement h or contact radius r and particle radius r, (k N is the contact stiffness in normal direction): k df N N E* r, h E* r (6) dh When one applies an increasing load F N the contact starts at p max p f 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, rupp [6], Rumpf et al. [68] and Molerus [, 3]. The total force can be obtained by the particle contact force equilibrium between attraction (-) and elastic as well as soft plastic repulsion (+) or * force response ( coordinate of annular elastic contact area): r * * * F F p π r F + p π r + π p (r ) r dr (7) H VdW N f,pl Superposition provided, this leads to a very useful linear force displacement model (for κ A const.) with the particle centre approach of both particles h [43], shown in Fig. 4 as elasticplastic yield boundary (or limit): F N H, f ( κa κp ) h + F π r p (8) Thus, the contact stiffness decreases with smaller particle size d 4 r, (or micro-roughness radius h r of non-deformed contact) of cohesive powders, predominant plastic yield behaviour provided [43]: k dfn N,pl π r, pf ( κa κp ) (9) dh 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. r r,pl el

9 9 Fig. 4. Force - displacement diagram of recalculated characteristic contact deformation of cohesive limestone particles as spheres, median diameter d 5. µm, surface moisture X W.5 %. Pressure and compression are defined as positive but tension and extension are negative, above panel. The origin of this diagram h is equivalent to the characteristic adhesion separation for direct contact a F. After approaching from an infinite distance - to this minimum separation a F the sphere-sphere-contact without any contact deformation is formed by the attractive adhesion force F H (the so-called jump in). As the response, from 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 p f with plastic yielding, Fig. 3 panel c). This yield point Y is located below the abscissa, i.e. contact force equilibrium F N includes a certain elasticplastic deformation as response of adhesion force F H. 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 A el (thickness r,el ) and circular plastic A pl (radius r,pl ) contact area, Fig. 3 panel c). After unloading between the points U A the contact recovers elastically according to Eq. (4) to a displacement h,a. The reloading curve runs from point A to U to the displacement h,u, Eq. (5). If one applies a certain pull-off force F N,Z - F H,A as given in Eq. (6) but here negative, the adhesion boundary line at failure point A is reached and the contact plates fail and detach with the increasing distance a a + h h, Fig. 3 panel d). This actual particle separation is considered for the calculation by F,A

10 a hyperbolic adhesion force curve F N,Z -,A F H a 3 of the plate-plate model Eq. (8). 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 FN,U. Thus, the variation in adhesion forces F H,A between particles depend directly on this frozen irreversible deformation, the so-called contact pre-consolidation history F H (F N ), see next Fig. 5. The plastic repulsion coefficient κ p describes a dimensionless ratio of attractive van der Waals pressure p VdW (adhesion force per unit planar surface area) to repulsive particle micro-hardness p f for a plate-plate model (e.g. p VdW 3 6 MPa): κ p p p VdW f C 6 π a H,sls 3 F p f 4 σsls a p F f This attraction term p VdW can also be expressed by surface tension, e.g. σ sls J/m², σ sls C H,sls p VdW (d) da 4 π a a F F fist introduced by Bradley [69] and Derjaguin [56]. The characteristic adhesion distance in Eqs. () and () lies in a molecular scale a a F nm. It depends mainly on the properties of liquid-equivalent packed adsorbed layers and can be estimated for a molecular interaction potential minimum du / da F F + or force equilibrium [7, 87]. Provided that these at F rep molecular contacts are stiff enough compared with the soft particle contact behaviour, this separation a F is assumed to be constant. The particle surface behaviours are influenced by mobile adsorption layers due to molecular rearrangement. The Hamaker constant C H,sls [7] includes these solid-liquid-solid interactions of continuous media. Thus C H,sls can be calculated due to Lifshitz theory and depends on dielectric constants and refractive indices [7, 7]. The elastic-plastic contact area coefficient κ A represents the ratio of plastic particle contact deformation area A pl to total contact deformation area A A + A which includes a certain elastic displacement [43] A h pl,f κ 3 A +, () 3 3 A 3 h with the centre approach h,f for incipient yielding at point Y in Fig. 4, p el (r ) p max p f : pl el () () h,f π pf d (3) E* Constant mechanical bulk properties provided, the finer the particles the smaller is again the yield point h,f which is shifted towards zero centre approach. Thus, an initial pure elastic contact deformation A pl, κ A /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-

11 ers elastically in the compression mode and remains with a perfect plastic displacement h,e. For this pure plastic contact deformation A el and A A pl, κ A 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 h,a : F N,unload 3 E* r, ( h h,a ) FH, A (4) 3 Consequently, the reloading runs along the symmetric curve from point A to point U: 3 F N,reload E* r, ( h,u h ) + FN, U (5) 3 If one applies a certain pull-off force F N,Z - F H,A, here negative, F H,A F + π r p h (6) H, VdW,A the adhesion (failure) boundary at point A is reached and the contact plates are failing and detaching with the increasing distance a a + h h. The displacement h,a at point A of F contact detachment is calculated from Eqs. (8), (4) and (6) as an implied function (index () for the beginning of iterations) of the displacement history point h,u : h ( h + κ h ) 3,A,() h,u h,f,el pl,u,a,() (7) The actual particle separation a can be used by a long-range hyperbolic adhesion force curve 3 F N a with the displacement h,a for incipient contact detachment by Eq. (7):,Z,A F N,Z (h F ) H + π r h + a,,a F p VdW h a F h,a 3 (8) This hyperbolic force - separation curve is shown in Fig. 4 bottom panel d)... Energy absorption in a contact with dissipative behaviour Additionally, if one considers a single elastic-plastic particle contact as a conservative mechanical system without heat dissipation, the energy absorption equals the lens-shaped area between both unloading and reloading curves A - U in Fig. 4: h,u h,u W F (h ) dh F (h ) dh (9) diss N,reload h,a N,unload h,a With Eqs. (4) and (6) for F H,A and (5), (8) for F N,U, one obtains finally the specific or mass related energy absorption W k W / m, which includes the averaged particle mass m 4 / 3 π 3 P r, ρ s m,diss diss P. In addition, the resultant Eq. () includes a characteristic contact number in the bulk powder (coordination number k π/ε []):

12 W m,diss E* h h 5/ 3 π p ( h,u h,a )[ κa h,u κp ( h,u h,a ],U,A f s r + ) () ε ρ, 3 r, ε ρs This specific energy of.6 to 3 µj/g for the limestone powder example mentioned was dissipated during one unloading - reloading - cycle in the bulk powder with an average pressure of only σ M,st 3.3 to 5 kpa (or major principal stress σ 5.9 to 4 kpa)...3 Adhesion force - normal force model The slopes of elastic-plastic yield and adhesion boundaries in Fig. 4 are characteristics of irreversible particle contact stiffness or compliance. Consequently, if one eliminates the centre approach h of the loading and unloading functions, Eqs. (8) and (4), an implied non-linear function between the contact pull-off force F H,A - F N,Z at the detachment point A is obtained for the normal force at the unloading point F N F N,U : F ( F + F ) 3 ( F + F ) N H H,A,() H H,A,() FH + κ N H π r, κp pf r, E* + FN + F () H 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, elasticplastic contact consolidation coefficient κ and, additionally, the influence of adhesion, stiffness, average particle radius r,, 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 r,. Practically, a linear function F H f(f N ) is used to evaluate the correlation between adhesion and normal force [43] which is more complex than the ideal plastic model of Molerus [3], Fig. 5: F κ κ ( + κ) FH + κ FN A p H FH + FN () κa κp κa κp The dimensionless elastic-plastic contact consolidation coefficient (strain characteristic) κ is given by the slope of adhesion force F H influenced by predominant plastic contact failure. κ p κ (3) κ κ A p This elastic-plastic contact consolidation coefficient κ is a measure of irreversible particle contact stiffness or softness as well. A shallow slope implies low adhesion level F H F H because of stiff particle contacts, but a large slope means soft contacts, or i.e., a cohesive powder flow behaviour. This model considers, additionally, the flattening of soft particle contacts caused by the adhesion force κ F H. Thus, the total adhesion force consists of a stiff contribution F H and a contact strain influenced component κ ( F H + FN ), Fig. 5. This Eq. () 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 () F F / 3

13 3 with abscissa (F H ) in the negative extension range of consolidation force F N is surprisingly independent of the Hamaker constant C H,sls, Fig. 5: F π A d / h pl r N,Z a F h r pf a F h r pf 3 3 A + + (4) ( + h r /a F ) Considering the model prerequisites for cohesive powders, this minimum normal (tensile) force limit F N,Z combines the opposite influences of a particle stiffness, micro-yield strength p f 3 σ f or resistance against plastic deformation and particle distance distribution. The last-mentioned is characterised by roughness height h r as well as molecular centre distance a F. It corresponds to an abscissa intersection σ,z of the constitutive consolidation function σ c (σ ), which is shown by Eq. (53) and Fig. 3 in section.3. π Fig. 5. Adhesion force normal force diagram of recalculated particle contact forces of limestone (median diameter d 5. µm, surface moisture X W.5 %, specific surface area A S,m 9. m²/g) according to the linear model Eq. () and non-linear model Eq. (6) for instantaneous consolidation t as well as a linear function for time consolidation t 4 h [45, 46, 4] using data of Fig. 3. The points characterise the pressure levels of YL to YL 4 according to Fig. 3. A characteristic line with the slope κ.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 X W.5 % is accurately analysed with arl-fischer titration. This is equivalent to idealised mono- to bimolecular adsorption layers being in equilibrium with ambient air temperature of C and 5% humidity. Generally, the linearised adhesion force equation () 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 4 micromechanical elastic-plastic particle contact consolidation and the macro-mechanical powder flowability expressed by the semi-empirical flow function ff c according to Jenike [8]. It should be pointed out here that the adhesion force level in Fig. 5 is approximately 5-6 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...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 [7, 4, 6, 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 [4, 4, 4, 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 [4, 4, 4]. Table : Material parameters for characteristic adhesion force functions F H (F N ) in Fig. 5 Constitutive models of contact deformation Repulsion coefficient Constitutive models of combined contact deformation Contact area ratio Contact consolidation coefficient Intersection with F N - axis (abscissa) Instantaneous contact consolidation Time dependent consolidation plastic κ p p p VdW elastic-plastic κ A 3 A f + 3 κp κ κ κ F C 6 π a p N,Z π a F A H,sls 3 F ( A + A ) pl pl h r, el p p f f f ( C ) H,sls viscoplastic κ p,t p η VdW t elastic-plastic and viscoplastic κ κ F A,t vis N,Z,tot 3 κ + 3 A,t κ p κ A ( A + A + A ) + κ p pl p,t pl κ f + A p,t vis vis π a F h r, pf + p t / η el f ( C ) H,sls 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 κ p + κ p,t, 5 th line in Table. These are characterized by the micro-yield strength p f, apparent contact viscosity and time η /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, 4]. Hence with the

15 5 total viscoplastic contact consolidation coefficient κ vis, which includes both the elastic-plastic and the viscoplastic repulsion, the linear correlation between adhesion and normal force F H (F N ) from Eq. () can be written as: F κ κ + κ ( + κ vis ) FH + κ vis FN A,t p p,t H,tot FH + FN (5) κa,t κp κp,t κ A,t κp κp,t This rheological model is only valid for a short term indentation of ( f t < η / κ p ), here approximately t < 6 h for the high-disperse (ultra-fine), cohesive limestone powder with a certain water adsorption capacity (specific surface area AS,m 9. m²/g). All the essential material parameters are collected in Table and the total adhesion force F H,tot is demonstrated in Fig. 5 above line...5 Tangential contact force The influence of a tangential force in a normal loaded spherical contact was considered by Cattaneo [5] and Mindlin [5, 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, 6]: F T, ( ψ) tan ϕi FN 4 ψ G * r h δ ± (6) Here δ is the tangential contact displacement, ψ the loading parameter dependent on loading, G E + ν the shear modulus, and unloading and reloading, ϕ i the angle of internal friction, ( ) the averaged shear modulus is given as: G* ν G ν G + (7) Thus, with ψ the ratio of the initial tangential stiffness k df dδ T T 4 G * r (8) to the initial normal stiffness according to Eq. (6) is: k k T N ( ν) ν (9) Hence this ratio ranges from unity, for ν, to /3, for ν.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, 8, 8].

16 6. Biaxial stress states in a sheared particle packing After this introduction into the fundamentals we have to look at what a volume element of particles 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... Shear force - displacement relation At a shear test after a certain elastic shear displacement, we can distinguish between () incipient consolidation, () incipient yield and (3) steady-state flow of a particle packing. This is demonstrated in a shear force - displacement diagram F S (s), Fig. 6. If we apply a certain shear force F S then the powder shows an elastic distortion with reversible displacement s elastic 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) h.

17 7 Using increasing shear stress τ beyond the elastic displacement s elastic the packing generates a shear zone, which can be ellipsoidal for a Jenike-type shear cell [9, ]. The powder flow or irreversible 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 ε V h(s) / h. This dependent variable is measured by the cover height h(s) and related to the initial height h... 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 () shear and dilatancy (expansion) of the shear zone, cohesion, uniaxial pressure and tension, isostatic tension.

18 8 First we turn to incipient yield. This state can be measured point by point with an overconsolidated sample which reduces the shear resistance after obtaining a peak stress τ. During this shear the shear zone expands dv >, 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 directions 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 ϕ i. The intersection with the ordinate σ represents the cohesion τ c 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 F N. To avoid too much confusion we have cut out to draw the tangential force components F T for every contact. The black colour at all contacts shall demonstrate the irreversible deformation. The shear resistance τ c is directly caused by this internal contribution of adhesion forces F H (F N ) and depends on the stressing pre-history as discussed in section.. The intersections of Mohr (semi-)circle with abscissa are the so-called principal stresses σ and σ, i.e. the largest (major) and the smallest (minor) normal stress without applying any shear stress τ. The Mohr circle which intersects the origin, i.e. minor principle stress σ, gives us the uniaxial compressive strength σ 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 σ Z, for σ, see left in Fig. 7 below. The intersection of the yield locus with abscissa, the isostatic tensile strength σ Z represents the internal contribution of adhesion forces F H (F N ) to the total stress, i.e. the sum of external normal stress σ plus σ Z. Thus, this characteristic σ Z depends directly on the stressing or pre-consolidation history, see section.. 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 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 characterised by an endpoint and the largest Mohr circle with the major and minor principle stresses σ and σ and equivalent to these the radius and centre stresses σ R,st and σ M,st, Fig. 8 (and Fig. ). 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 ϕ st. To extrapolate the stationary yield locus, the isostatic tensile strength σ of very loose packing density is obtained, Fig. 8. This is typically for an unconsolidated powder, i.e. direct particle contacts but without any contact deformation. Thus, the funda-

19 9 mental characteristic σ does not depend on pre-consolidation. This is equivalent to the adhesion F H 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 () stationary 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 σ Z. The incipient consolidation is described by the so-called consolidation locus which lies at the right hand side of Fig. 8 (and Fig. ) represents the envelope of all stress states with plastic failure which leads to a consolidation of the particle packing dv <. This line may have the

20 same inclination as the slope of yield locus, the angle of internal friction ϕ i, and intersects the abscissa at an isostatic normal stress σ iso. This isostatic stress state means that all principal stresses have the same value in all three spatial directions σ iso σ σ σ 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 h r is here negative and can be microscopically explained by both contact loading, particle rearrangement and structural compression. 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..3 Cohesive powder flow criteria.3. 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 h /d is assumed [8-84]. For that purpose, the micro/macro-transition [47] with the normal stress - force relation Eq. (4) and the contact stiffness due to Eq. (6) are applied to a packing of smooth spheres. We have to consider the total normal force F N,tot of a characteristic particle contact which includes the contribution of pre-consolidation dependent adhesion F H (F N,V ): N,tot H N,V N ( + κ) FH + κ FN,V FN F F (F ) + F + (3) By the first derivative near F N we can write for the bulk modulus of elasticity (contact radius according to r r h and E* per Eq. (5)) E b, ( σ + σz ) ε dfn,tot ( h / d) ε d dh σ F d (3) d Using a micro/macro-transition Eq. (4) we obtain finally: E (( + κ) F + κ F ) N / 3 / 3 ε 3 E * H N,V E * ε 6 σ Z b (3) 4 ε r, 4 ε E * For a unconsolidated loose packing of a cohesive powder E b, follows [47, 45]: E b, / 3 / 3 ε 3 E * F H E * ε 6 σ 4 r, 4 (33) ε ε E * A free-flowing powder is unable to sustain a tensile stress F H and E b refers solely to compression in a mould with stiff walls [83].

21 Fig. 9. Bulk shear modulus - centre stress diagram for load and unload of limestone powder (d 5. µm). The physical model Eq. (35) was multiplied by a fit factor of 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 G b is about 3 orders of magnitude smaller than the shear modulus of particle material assumed to be G 6 kn/mm² and Poisson ratio ν.8. Consequently, the initial shear stiffness (shear modulus) for elastic shear displacement can be derived from Eq. (6), provided that the shear displacements at a characteristic particle contact and in the bulk are equivalent δ/d s/h Sz (h Sz characteristic height of the shear zone): G G b b T (34) d dτ ( s / h Sz ) ε d dδ F τ ε df (( + κ) F + κ F ) T / 3 / 3 ε 3 H N,V ε 6 σ Z G * G * (35) ε E * r, ε E * G b, ε G * 3 F / 3 ε G * 6 σ H E * r (36) ε, ε E * / 3 This simple model, Eq. (35), overestimates the shear modulus G b 6 - N/mm² for limestone (particle size d 5. µm) compared to the shear modulus obtained from direct shear tests G b,load 8-7 kn/m² for loading and G b,unload - 34 kn/m² for unloading [85]. Obvi-

22 ously, the shear modulus G b depends on the pre-consolidation of the isostatic tensile strength σ Z f(σ 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 stiffness Eq. (3) of the bulk equals the contact stiffness ratio k T /k N as in Eq. (9): G E b b G E b, b, k k T N ( ν) ν Approximately for cohesive powders, the shear stiffness is equivalent to the normal stiffness, e.g. G b /E b.8 for a common Poisson ratio ν.3 of the particle material. (37).3. Cohesive steady-state flow Using the elastic-plastic particle contact constitutive model Eq. () 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 stationary yield locus in radius stress f(centre stress) - coordinates ( σ + ) σ (38) R,st sin ϕst M,st σ or in the τ(σ)-diagram of Fig. [43]: τ st tanϕst ( σst + σ ) (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 ε F H σ. (4) ε d This isostatic tensile strength σ of an unconsolidated powder without any particle contact deformation is obtained from the adhesion force F H, Eq. (), with the initial porosity of very loose packing ε ρ ρ and ρ b, according to Eq. (59). b, / s In some cases one may observe cohesionless steady-state flow, i.e. σ in Eq. (38), which is described by the effective yield locus according to Jenike [8] with the effective angle of internal friction ϕ e as slope: σ (4) R,st sin ϕe σm,st Replacing the radius stresses in Eqs. (38) and (4) and we obtain a simple correlation between the stress-dependent effective angle of internal friction ϕ e, the stationary angle of internal friction ϕ st and the centre stress σ M,st : σ sin ϕ e sin ϕst + (4) σm,st

23 3 Fig. : Friction angle consolidation stress diagram and shear stress - normal stress diagram to show the correlation between the cohesive stationary yield locus [4] as envelope of all Mohr circles for steadystate 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 ϕ ( σ ), ϕ ) [8, 4, 9] and ' ( e W effective wall stress σ, Eq. (65), with respect to the radial stress field during discharging, see chapter 3. The termination locus [, 9] 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 termination loci are directly dependent on stress history, Eq. (44). The centre stress σ M,st can be replaced by the major principle stress σ during steady-state flow σ σ sin ϕ st σ M,st, (43) + sin ϕst and one obtains σ + σ sin ϕ e tan sin ϕst (44) σ sin ϕst σ

24 4 which is in accordance with the daily experimental experience in shear testing, Fig. upper diagram [43]. If the major principal stress σ reaches the stationary uniaxial compressive strength σ c,st, Fig. diagram below, sin ϕ σ st σ σc,st (45) sin ϕst the effective angle of internal friction amounts to ϕ e 9 and for σ follows ϕ e ϕ st. 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 σ pre, or approximately, the maxima of end Mohr circles at σ M,st. This so-called termination locus [, 9] is directly dependent on stress history and describes only the cohesionless steady-state flow. Fig. : 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 σ V () and pre-consolidated by twisting the cover. Than a smaller normal stress σ pre < σ V () for preshear is applied. The cell is presheared as far as the steady-state flow (3) is obtained for a constant volume dv of the shear zone. Than the cell is unloaded and loaded by a smaller normal stress σ < σ 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 > of the shear zone. The shear zone relaxes to steady-state flow at the given small normal stress level σ and unloaded to τ (4). The cell is weighed, opened and the shear zone is observed to evaluate it as a suitable good test. All these steps () (5) are repeated n-times (generally x 4) for fresh and identically prepared powder samples.

25 5.3.3 Incipient yield To combine the angle of internal friction ϕ i for incipient contact failure (slope of yield locus) with the stationary angle of internal friction ϕ st following relation is used [, 47]: st ( + κ) tan ϕi tan 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., is obtained from resolution of a general square function [47], is simply to use (σ M,st, σ R,st centre and radius of Mohr circle for steady-state flow as parameter of powder pre-consolidation): τ σr,st ( ) tan ϕ i σ + σz tanϕi σ + σm, st (47) sinϕi It is worth to note here that only the isostatic tensile strength σ Z for incipient yield depends directly on the consolidation pre-history and is given by: σ sinϕ σ sinϕ + σ R,st st st σ Z σm,st M,st sin (48) ϕi sinϕi sinϕi The smaller a radius stress for pre-consolidation σ VR < σ R,st, the larger is the centre stress σ VM > σ M,st right of largest Mohr circle for steady-state flow in Fig., and the smaller can be the powder tensile strength σ Z. Fig. : Shear stress - normal stress diagram of yield loci (YL) and stationary yield locus (SYL) of limestone powder, straight line regression fit.98, d 5. µm, solid density ρ s 74 kg/m 3, shear rate v S mm/min, surface moisture X W.5 %.

26 6.3.4 Incipient consolidation The consolidation locus represents the envelope of all Mohr circles for consolidation stresses, i.e. the radius σ VR and centre σ VM stresses between the Mohr circle for steady-state flow and the isostatic stress σ iso, Fig.. Provided that the particle contact failure is equivalent to that between incipient powder flow and consolidation, one can write for a linear consolidation locus with negative slope -sinϕ i which is symmetrically with the linear yield locus, Eq. (5): σ sin ϕ ( σ + σ ) (49) VR i VM iso Due to this symmetry between yield and consolidation locus, one can directly estimate the isostatic powder compression σ σ σ VM σ iso from Fig. 8 for the radius stress σ VR : σr,st sin ϕ st sin ϕst σ iso σm,st + σz + σm,st M,st + sin + i sin ϕ σ i sin ϕ σ (5) ϕ i.3.5 The three flow parameters Generally, when we use these radius σ R and centre stresses σ M, the essential flow parameters are compiled as one set of linear constitutive equations, i.e. for instantaneous consolidation, the consolidation locus (CL), ( σm + σm,st ) + R, st σ, (5) R sin ϕi σ for incipient yield, the yield locus (YL), ( σm σm,st ) + R, st σ (5) R sin ϕi σ and for steady-state flow, the stationary yield locus (SYL): ( σ + ) σ (38) R,st sin ϕst M,st σ These yield functions are completely described only with three material parameters plus the characteristic pre-consolidation stress σ M,st or average pressure influence, see Tomas [47]: () ϕ i incipient particle friction of failing contacts, i.e. Coulomb friction; () ϕ st steady-state particle friction of failing contacts, increasing adhesion by means of flattening of contact expressed with the contact consolidation coefficient κ, or by friction angles ( sin ϕ sin ϕ ) as shown in the next Eqs. (53) and (54). The softer the particle con- st i tacts, the larger are the difference between these friction angles the more cohesive is the powder; (3) σ extrapolated isostatic tensile strength of unconsolidated particle contacts without any contact deformation, equals a characteristic cohesion force in an unconsolidated powder; (4) σ 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 tensile strength of yield loci via the cohesive steady-state flow as the stress history of the powder.

27 7.3.6 Consolidation functions These physically based flow parameters are necessary to derive the uniaxial compressive strength σ c which is simply found from the linear yield locus, Eq. (5) and Fig., for σ c. σ R (σ and σ R σ M ) as a linear function of the major principal stress σ, Fig. 3, [43]: c ( sin ϕst sin ϕi ) ( + sin ϕ ) ( sin ϕ ) st i sin ϕst ( + sin ϕi ) σ a σ + σc, ( + sin ϕ ) ( sin ϕ ) σ σ + (53) st Equivalent to this linear function of the major principal stress σ and using again Eq. (5), the absolute value of the uniaxial tensile strength σ Z, is also found for σ Z,. σ R (σ and σ R - σ M ): σ ( sin ϕst sin ϕi ) sin ϕst σ + σ ( + sin ϕ ) ( + sin ϕ ) + sin ϕ Z, (54) st i st i Fig. 3. Powder strength - consolidation stress diagram of constitutive consolidation functions of limestone, straight line regression fit.98, median particle size d 5. µm, surface moisture X W.5 % accurately analysed by arl Fischer titration. Additional flow properties according to the basic Eqs. (53) and (54) are the averaged angle of internal friction ϕ i 37, stationary angle of internal friction ϕ i 43, isostatic tensile strength of the unconsolidated powder σ.65 kpa.

28 8 Both flow parameters σ c and σ Z, depend on the pre-consolidation level of the shear zone which is expressed by the applied consolidation stress for steady-state flow σ. A considerable time consolidation under this major principal stress σ after one day storage at rest is also shown in Fig. 3. Equivalent linear functions are also used to describe these time consolidation effects [4-47]..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, 4]..4. Powder Flowability c c In order to assess the flow behaviour of a powder, Eq. (53) shows that the flow function due to Jenike [7, 8] ff σ / σ is not constant and depends on the pre-consolidation level σ. Approximately, one can write for a small intercept with the ordinate σ c,, Fig. 3, the stationary angle of internal friction is equivalent to the effective angle ϕ st ϕ e and Jenike s [8] formula is obtained: ff c ( + sin ϕe ) ( sin ϕi ) ( sin ϕ sin ϕ ) (55) e i 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 : Table : Flowability assessment and elastic-plastic contact consolidation coefficient κ(ϕ i 3 ) flow function ff c κ-values ϕ st in deg evaluation Examples free flowing dry fine sand easy flowing moist fine sand cohesive dry powder very cohesive moist powder < - non flowing moist powder Obviously, the flow behaviour is mainly influenced by the difference between the friction angles, Eq. (55), as a measure for the adhesion force slope κ in the general linear particle contact constitutive model, Eq. (). Thus one can directly correlate κ with flow function ff c [47]: + ( ff c ) sin ϕi κ (56) tan ϕi ( ff c + sin ϕi ) + ( ff c ) sin ϕi ff c sin + ϕi A characteristic value κ.3 for ϕ i 3 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. Due to the consolidation function, a small slope designates a free

29 9 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. 3. Obviously, the finer the particles the softer are the contacts and the more cohesive is the powder [4, 43]. öhler [88] has experimentally confirmed this thesis for alumina powders (α- Al O 3 ) down to the sub-micron range (σ c, const. kpa, d 5 median particle size in µm): ff. (57).6 c d 5.4. Powder Compressibility A survey of uniaxial compression equations was given by awakita [89]. Thus in terms of a moderate cohesive powder compression, to draw an analogy to the adiabatic gas law κad p V const., a differential equation for isentropic compressibility of a powder ds, i.e. remaining stochastic homogeneous (random) packing without a regular order in the continuum, is derived, beginning with: dρ ρ b b n dp p dσm,st n σ + σ M,st The total pressure including particle interaction p σ M,st + σ should be equivalent to a pressure p + a / V V b R in van der Waals equation of term with molecular interaction ( ) ( ) T VdW m state to be valid near gas condensation point. A condensed loose powder packing is obtained ρ b ρ b,, if only particles are interacting without an external consolidation stress σ M,st, e.g. particle weight compensation by a fluid drag, and Eq. (58) is solved: m (58) ρ ρ b b, σ + σ σ M,st n (59) Therefore, this physically based compressibility index n /κ ad lies between n, i.e. incompressible stiff bulk material and n, i.e. ideal gas compressibility. Considering the predominant 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 σ kpa index n evaluation examples flowability. incompressible gravel free flowing..5 low compressibility fine sand.5 -. compressible dry powder cohesive. - very compressible moist powder very cohesive Our limestone powder shows a compressible behaviour with the index n.5 (ρ b, 7 kg/m³). Both functions are shown in Fig. 4. Obviously, for the loose packing near the origin σ M,st - σ, the compression rate (slope of bulk density) is maximum by particle rearrangement and incipient contact deformations, Fig. 4 dashed line.

30 3.4.3 Powder compression and preshear work The mass related or specific compression work W m,b of a cohesive powder is obtained by an additional integration of the reciprocal compression function Eq. (59) for n : W m,b σ n M,st dσ σ σ M,st n M,st n + ρ ( σ ) M, n ρb, σ b st (6) Fig. 4. Bulk density centre stress diagram of compression function and compression rate of limestone powder according to Eqs. (58) and (59), curve regression fit.94, median particle size d 5. µm, surface moisture X W.5 %. It describes the correlation between the external work (lower limit σ M,st ) as the function of average pressure for steady-state flow σ M,st only for compression. The specific compression work starts at the origin, Fig. 5, 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 h Sz. Additionally, the energy input during this steady-state flow for constant bulk density ρ b of shear zone is obtained as ( γ s / h preshear distortion, s pre preshear displacement): W τ pre n spre sin ϕst σ σ M,st m,b,pre τpre ( γ pre ) dγ pre b h + ρ Sz ρb, σ (6) To compare this energy consumption in handling practice, e.g. W mb,pre J/kg is equivalent to the specific kinetic energy of a shear rate of v S,eq m/s, see Fig. 5, v W J / kg m / s (6) S,eq m,b, pre pre pre Sz

31 3 and to a lift height of bulk powder H b. m of the equivalent potential energy: Wm,b,pre J / kg. m (63) g 9.8 m / s H b From the specific preshear work, Eq. (6), we can derive the mass related power consumption of steady-state flow (v S ds pre /dt preshear rate): P dw v sin ϕ σ σ m,b,pre S st M,st m,b,pre dt h + (64) Sz ρb, σ n Fig. 5: Specific work centre stress diagram of mass related preshear and compression work and mass related power consumption of limestone powder according to Eqs. (6), (6) and (64), curve fit.97, median particle size d 5. µm, surface moisture X W.5 %. The mass related preshear work is essentially 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 fugitive 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. 6. 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. 6.

32 3 Fig. 6: Characteristic constitutive functions of stiff and compliant particle contact behaviours, free flowing 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., d) Powder strength - consolidation stress diagram of consolidation functions acc. to Fig. 3, e) Radius stress centre stress diagram of yield and consolidation loci (YL, CL) and stationary yield locus (SYL) according to Eqs. (38), (5) and (5), f) Bulk density - consolidation stress diagram of compression function according to the following Fig. 7 above panel.