Advanced Powder Technol., Vol. 16, No. 2, pp. 123–135 (2005)© VSP and Society of Powder Technology, Japan 2005.Also available online - www.vsppub.com

Original paper

Flow characterization of fine powders using materialcharacteristic parameters

MOHAN MEDHE 1, B. PITCHUMANI 1,∗ and J. TOMAS 2

1 Particle Science and Technology Laboratory, Department of Chemical Engineering,Indian Institute of Technology, New Delhi 110016, India

2 Department of Process and Systems Engineering, Institute of Mechnical Process Engineering,Otto-von-Guericke University, Magdeburg 39106, Germany

Received 14 October 2003; accepted 29 March 2004

Abstract—Instantaneous flow properties of fine cohesive powders, i.e. limestone, glass-ballotini,alumina and talc, are evaluated using Jenike shear tester. The results are analysed in the light ofelastic–plastic interparticle contact deformation theory incorporating fundamentals of fine powderadhesion and consolidation in flow behavior characterization. Thus, unconfined yield strengths (σc)of the given samples at various consolidation levels are evaluated solely on the basis of materialcharacteristic parameters, i.e. angle of internal friction (φi), stationary angle of friction (φst),isostatic tensile strength of an unconsolidated powder (σ0) along with characteristic pre-consolidationinfluence (σM,st) and subsequently predicted flow functions (ffc) are compared with those evaluatedexperimentally. The correlation coefficient values for the comparison are found to be in the range of0.96–0.99, indicating the satisfactory nature of the above parameters in explaining the flow behavior.Further, compressibility indices and elastic–plastic contact consolidation coefficient (κ) characterizingpowder cohesiveness are calculated using experimental data.

Keywords: Powder flow behaviour; powder mechanics; yield locus; inter-particle adhesion force;contact deformation.

NOMENCLATURE

Apl plastic particle deformation contact area (nm2)Ael elastic particle deformation contact area (nm2)AK total particle deformation contact area (nm2)CH,sls Hamakar constant (J)d particle diameter (m)d0 separation distance (nm)FN normal force (nN)

∗To whom correspondence should be addressed. E-mail: bpmani@chemical.iitd.ernet.in

124 M. Medhe et al.

FH adhesion force (nN)FHO adhesion force for FN = 0 (nN)ffc flow function (—)H ∗ material hardness (N/m2)h̄W̄ Lifshitz–van der Waals constant (eV)kN,pl contact stiffness (N/m)n compressibility index (—)Pf plastic yield strength of particle contact (N/mm2)PVdW van der Waals force (kPa)R∗ characteristic dimensions of asperities (m)

Greek

ε porosity (—)ε0 loose packing porosity of powder (—)κA elastic–plastic contact area coefficient (—)κP plastic repulsion coefficient (—)κ elastic–plastic contact consolidation coefficient (—)μ coefficient of friction (—)σ normal stress (kPa)σ1 major consolidation stress (kPa)σc unconfined yield strength (kPa)σ0 isostatic tensile strength (kPa)σM,st center of Mohr circle for steady-state flow function (kPa)σR,st radius of Mohr circle for steady-state flow function (kPa)ρ particle density (kg/m3)ρb bulk density of powder (kg/m3)ρb,0 powder bulk density at negligible consolidation (kg/m3)τ shear stress (kPa)φe effective angle of internal friction (deg)φi slope of yield loci (deg)φw angle of wall friction (deg)φst steady-state angle of internal friction (deg)

1. INTRODUCTION

Handling and storage of fine particles are known to produce well-known problemslike bridging, channeling, fluctuating flow rates, etc., in process equipment and stor-age. After the introduction of various shear testers for powder flowability character-ization techniques by various researchers, a sound basis is available for designingvarious bulk solids handling equipments [1, 2]. However, in spite of these advancesin the field, the fundamentals of particulate solids, consolidation and flow behaviorat microscopic level are not yet fully understood. It is thus important to understand

Flow characterization of fine powders 125

various interparticle adhesion force effects in terms of their force–response behav-ior from a particle mechanics point of view. In the present case study an attempt ismade to analyze the flowability of various mineral powders in terms of basic mate-rial characteristic parameters using particle mechanics theories explaining elastic–plastic contact deformation effects. The current study is restricted to the flowabilityof powders exhibiting irreversible contact deformations at lower deformation rateand which are consolidation time invariant.

2. EXPERIMENTAL

Various fine mineral powders like talc, limestone, glass ballotini and alumina(α-alumina) were analyzed for instantaneous flow properties using a Jenike sheartester interfaced with a computer in the consolidation level range of 7–20 kPa as perstandard EUR14022 [3]. Flow properties characterization thus involves evaluationof yield loci at various consolidation levels to evaluate the parameters of majorprincipal stress σ1, unconfined yield strength σc, effective angle of internal frictionφe and angle of wall friction φw. Measured yield loci were observed to be linear forall the consolidation levels. Flowability of powders is further characterized in termsof flow function ffc, defined as the ratio of σ1/σc as suggested by Jenike [1], apartfrom their physical properties like particle size distribution, surface area, particledensity and hardness using standard methods. Table 1 shows hardness parametersof the samples along with other relevant physical properties.

3. RESULTS AND DISCUSSION

3.1. Effect of consolidation stress on yield locus

To understand the effect of consolidation stress on the yield locus, experiments werecarried out at various consolidation levels as mentioned in the previous section.Figure 1 shows a typical set of yield loci of fine limestone powder used in thestudies. It is seen that at a given consolidation level, the shear stresses (τ ) requiredto cause yield of the powder increase almost linearly with normal stresses (σ )according to contact failure criterion (Mohr–Coulomb criterion) as given by (1)

Table 1.Physical properties of samples used in the studies

Sample Median diameter Density Hardness [9](μm) (kg/m3)

Moh Knoop Vickers

Talc 3.5 2600 1 16–35 40–56Limestone 1.2 2670 3 — 120Glass ballotini 9.2 2500 5 373–500 500Alumina 3.5 3790 9 1802–2251 1800–2100

126 M. Medhe et al.

Figure 1. Effect of consolidation level on yield loci of fine limestone powder.

defining the angle of internal friction of powder φi. It assumes frictional behavior ofinterparticle contacts such that breakdown of adhesion at a contact will occur only ifthe ratio of the magnitude of shear stress to compressive stresses exceeds a criticalvalue of coefficient of friction (μ):

τ

σ� μ = tan(φi). (1)

As the consolidation forces on the powder increase, the yield locus of the resultingpowder bed shifts upward in an almost parallel manner indicating an increase inadhesive forces in interparticle contacts. These increased adhesive forces cause anincrease in the unconfined yield strength (σc) of the sample, leading to deteriorationof flow properties of the powder. Thus, it can be realized that interparticle adhesiveforces (FH) must depend on the normal force applied (FN) on the interparticlecontacts. Generally, in the case of dry powders, these adhesive forces are attributedto disperse attractive forces like van der Waals forces. The present case studyattempts to analyze the increase in cohesiveness of a sample as a function of appliednormal forces in the light of elastic–plastic contact deformation theory of fineparticles [4–6] complementing perfect plastic contact deformation theory suggestedby Molerus [7, 8]. Thus, in the case of dry fine particles under normal environmentalconditions, van der Waals forces are assumed to play the dominant role. Hence,direct solid–solid interaction is modified by adsorption layers at the solid materialsurface resulting in irreversible inter particle contact deformation, which determinesthe magnitude of the contact force between particles.

Rumph et al. [10] first developed a constitutive model describing the linearincrease of adhesion force (FH) with applied normal force (FN) caused by plastic

Flow characterization of fine powders 127

deformation as given by (2). Molerus [8], while developing particle contact plasticdeformation theory, obtained a similar formula as given by (3):

FH =(

1 + PVdW

Pf

)FHO +

(PVdW

Pf

)FN, (2)

FH = FHO + κPFN, (3)

κP is the plastic repulsion coefficient of particle contact and is defined using (4)describing a dimensionless ratio of attractive van der Waals pressure (PVdW) for aplate–plate model to repulsive particle micro-hardness (Pf) and FHO is an adhesionforce without any additional consolidation (FN = 0). This adhesion force inthe particles without any consolidation (FHO), which is present inherently in finepowders in varying degrees depending on their cohesive nature, can be approachedas a single rigid sphere–sphere contact:

κP = PVdW

Pf= CH,sls

6πd3oPf

, (4)

CH,sls is the Hamakar constant and d0 is the distance resulting in maximum attractiveforce based on the force balance of repulsive and attractive forces on the distancebetween particle surfaces. Additionally, Krupp [11] initially proposed an estimationof inherent cohesive forces (FHO) in fine powders dominated mainly by van derWaals forces with the help of material parameters as given by:

FHO = h̄W̄

8πd20

R∗[

1 + h̄W̄

8π2d30H ∗

], (5)

where h̄W̄ is the Lifshitz–van der Waals constant (eV), d0 ≈ 4 × 10−10 m,the distance at which maximum adhesion force is observed, H ∗ is the hardnessof material (N/m2) and R∗ is a characteristic measure of surface asperities (m).As aconsequence of inevitable surface asperities observed with real particles, aparameter R∗ characterizing the surface geometry has to be taken into account.According to Krupp [11], the evaluation of experiments yields a numerical valueof R∗ ∼= 0.1 μm for fine-grained particles. With these values he had predictedadhesive forces in the range of 7×10−7 to 8×10−8 N for various materials rangingfrom plastics to glass beads.

The characteristic irreversible plastic contact deformation in the case of twoisotropic stiff, mono-disperse mineral particles having soft contacts in the presenceof adhesive forces can thus be seen as a sequence of stages comprising of particleapproach, particle loading followed by elastic–plastic deformation of particle con-tacts resulting in increased interparticle forces and, finally, unloading of particleswith permanent plastic contact flattening. Tomas [4–6], based on the above analy-sis, developed a relationship between particle contact deformation and normal forceby taking into account the particle contact force equilibrium between attraction andelastic as well as plastic repulsion. This adhesion force model incorporates Hertziancontact theory while developing the extent of elastic and plastic deformation that

128 M. Medhe et al.

particle contacts are undergoing during yield to obtain a very useful formula foranalyzing the increase in interparticle adhesive forces as given by:

FH = (1 + κ)FHO + κFN, (6)

where κ is the elastic–plastic contact consolidation coefficient taking into accountthe extent of elastic–plastic deformation and ratio of resulting van der Waals forcesto plastic yield strength as given by (7) and (8), respectively. This elastic–plasticconsolidation coefficient describing the influence of predominant plastic contactdeformation characterizing particle contact stiffness or softness is therefore givenby the slope of the linear relationship between the adhesion force FH and normalforce FN:

κ = κP

κA − κP, (7)

κA is the elastic–plastic contact area coefficient representing the ratio of plasticparticle contact deformation area Apl to total contact deformation area AK =Apl + Ael including certain elastic displacement and is given by:

κA = 2

3+ 1

3

Apl

AK. (8)

Further, the contact stiffness (kN,pl) as given by (9) is shown to be decreasing pro-portionally due to its dependence on the particle size (d) and micro-hardness (Pf),especially for fine cohesive powders, resulting in plastic yielding tendencies [5]:

kN,pl = π

4d(κA − κP)Pf. (9)

Thus, theories developed by Molerus [8] and Tomas [4] advocate estimation andcharacterization of flow properties of fine powders using plastic repulsion coef-ficient (κP) and elastic–plastic contact consolidation coefficient (κ), both essen-tially being particle material characteristics, respectively, for elucidating the micro-processes occurring at the particulate surface level. Thus, for a softer material andfine particles contact consolidation coefficient will be higher in magnitude (κP → 1)as compared to those for a harder material and coarser particles (κP ≈ 0). Simi-larly, the elastic–plastic contact consolidation coefficient (κ), being a dimensionlessstrain characteristic is a measure of the irreversible particle contact stiffness or soft-ness. Thus, the κ value is expected to increase with increasing interparticle adhesiveforces. Hence, values of these coefficients evaluated based on shear testing of pow-ders can be used for understanding flow difficulties of powders. Tomas [4, 6], basedon elastic–plastic contact deformation theory, has also proposed a relationship be-tween κ and flow function ffc (due to Jenike [1]) given by:

κ = 1 + (2ffc − 1) sin φi

tan φi(2ffc − 1 + sin φi)

√√√√√1

1 −(

1 + (2ffc − 1) sin φi

(2ffc − 1 + sin φi)

)2 − 1. (10)

Flow characterization of fine powders 129

Figure 1, depicting fine limestone yield loci, shows that the angle of internalfriction for given sample varies in the range of 28.4–30.7◦ for the consolidationrange employed, indicating almost parallel shift in yield loci. These increaseddifficulties encountered by the limestone particles for instantaneous flow as reflectedby increasing σc with consolidation indicate an increase in van der Waals forces inthe interparticle contacts. The increase in interparticle cohesive forces can thereforebe attributed to the plastic contact deformation due to increased consolidation forcesresulting in increased interparticle adhesive forces. The increase in adhesive forces,in turn, causes an increase in shear forces required to achieve flow of the powderleading to an upward shift in the yield locus. The theoretically calculated value ofthe plastic repulsion coefficient (κP) using (4) for limestone particles is observedto be 0.249 for characteristic adhesion distances (do) of 0.4 nm, Hamakar constantCH,sls = 15×10−20 J and particle micro-hardness Pf = 1×108 Pa as compared to theexperimentally evaluated value of elastic–plastic contact consolidation coefficient of0.39, indicating agreement in order of magnitude of accuracy in understanding thesecontact plastic deformation effects. The theoretical values are generally observedto underestimate the forces due to the fact that they do not include any effect ofcharacteristic particle dimensional parameters of the concerned particles in the forcecalculation, unlike (5). Further, it is to be understood that the extent of elasticloading effects in the case of fine mineral powders such as limestone used in thepresent case studies might be considerably lower and hence may not be of muchconsequence in bulk solids handling of fine powders. Additionally, Tomas [4]has extended the use of the elastic–plastic contact consolidation coefficient forpowder flowability characterization as reported in Table 2. From Table 2, it isseen that for coarser particles the values of κ are considerably lower, indicatingthe stiffness of the contact. For softer interparticle contacts, on the other hand, thatare characteristic of fine particles, κ values are quite high.

It can also be observed from the above analysis that κ values are dependent onthe stress history during the consolidation, which the sample has undergone beforeyield. Hence, it needs to be taken into consideration for characterization of a givensample using κ as explained in the following section.

Table 2.Flowability assessment and elastic–plastic contact consolidation coefficient (for internal friction angleφi = 30◦)

κ (—) φst (deg) Evaluation Examples

0.01006–0.107 30.3–33.0 free flowing dry fine sand0.107–0.3 33.0–37.0 easy flowing moist fine sand0.3–0.77 37.0–46.0 cohesive dry powder0.77–∞ 46.0–90.0 very cohesive moist powder∞ — non-flowing, hardened hydrated powders

130 M. Medhe et al.

Figure 2. Evaluation of steady-state yield locus and associated parameters (φi = angle of internalfriction, φst = stationary angle of internal friction, σ0 = isostatic tensile strength of unconsolidatedpowder, σM,st = center of the Mohr circle for steady-state flow function, σR,st = radius of the Mohrcircle for steady-state flow function).

3.2. Estimation of material characteristic parameters for powder flowabilityanalysis

To arrive at the material characteristic parameters, the flowability results of thesamples employed in the present case study are analyzed in terms of parametersdefined from considerations of particle interactions depicted Fig. 2. It showsvarious particle interaction parameters, i.e. angle of internal friction (φi), increasein adhesive forces because of elastic–plastic contact deformation of fine particles(φst − φi), isostatic tensile strength of unconsolidated powder (σo) and center of theMohr circle for the steady-state flow function (σM,st), respectively, using the conceptof a cohesive steady-state yield locus.

The center of the Mohr circle for steady-state flow function (σM,st) and radius ofthe Mohr circle for steady-state flow function (σR,st), which are invariant stressesdue to the theory of plasticity, are thus calculated with (11) using major principalstress σ1 and minor principal stress σ2 from experimental yield loci data at givenconsolidation levels:

σM,st = σ1 + σ2

2, σR,st = σ1 − σ2

2. (11)

Hence, it is useful to evaluate the stationary yield locus in terms of invariant stressesσM,st and σR,st as given by (12), and these can be subsequently used to determinethe representative steady-state angle of internal friction (φst) and isostatic tensilestrength of unconsolidated powder (σo) at given consolidation levels. The contactconsolidation coefficient (κ), which is a material characteristic taking into accountthe extent of elastic–plastic deformation and ratio of resulting van der Waals forcesto plastic yield strength of given samples as a function of fines, is then evaluatedusing (13) for all the samples:

σR,st = sin φst(σM,st + σ0), (12)

Flow characterization of fine powders 131

Figure 3. Recalculated interparticle adhesion force between limestone particles.

κ = tan φst

tan φi− 1. (13)

Since all yield loci at given consolidation levels are observed to be almost parallel,the angle of internal friction (φi) to be used in (13) is taken to be an average of thevalues determined experimentally for all consolidation levels. This methodology ofevaluating the steady-state angle from invariant stresses σM,st and σR,st is necessaryinstead of calculating κ from ffc and φi directly using (10) in order to avoid anydependence of κ on consolidation levels. Thus, with the help of the above analysis,all the samples are characterized in terms of κ having a definite effect on the flowproperties of powders.

Figure 3 shows recalculated particle contact forces of limestone using (6). Thepoints characterize the experimental normal stress levels used for evaluation of yieldloci. From Fig. 3, it is seen that the experimentally observed increase in interparticleadhesive forces can be modeled quite well using (6). It demonstrates the effective-ness of the elastic–plastic contact consolidation coefficient κ in understanding therole of interparticle forces on flow behavior of powders. Due to their relativelylower hardness, limestone particles are prone to such contact flattening during nor-mal loading, leading to cohesive flow tendencies. The isostatic tensile strength σo ofpowders having characteristic diameter d without any particle contact deformation

132 M. Medhe et al.

effects for initial loose packing porosity ε0 can be calculated using (14) proposedby Molerus [7, 8]:

σ0 = 1 − ε0

ε0

FHO

d2. (14)

The required loose packing porosity of given powders εo can be estimated byknowing the compressibility characteristics of powder based on measured bulkdensity values at various consolidations as explained in next section.

3.3. Evaluation of powder compressibility chracteristics

Considering the deformability of interparticle contacts in fine powders, it is usefulto characterize the powders in terms of their compressibility index to facilitateunderstanding of the relationship between powder bulk density and normal stressesapplied. Tomas [4, 6] has proposed such a relationship for powders by extendinganalogies to the adiabatic gas law for isentropic compression as given by (15). Thefeatures of the resulting relationship include physically based compressibility index,n, in the range 0–1 indicating incompressible stiff bulk material for n = 0 and idealgas compressibility index for n = 1. These compressibility indices of powderscan thus be useful in categorizing the powders for their flowability as suggestedby Tomas [4]. In bulk solids handling practice these compressibility indicesrelating bulk densities to isostatic tensile strength of powders without additionalconsolidation and center Mohr circle stress can be used to arrive at the requiredbulk densities for estimation of characteristic storage silo dimensions:

ρb

ρb,0=

[σ0 + σM,st

σ0

]n

. (15)

For given experimental consolidation levels, bulk densities of corresponding pow-ders are measured and are then used to estimate the bulk density of powders havingnegligible consolidation using (15). Table 3 indicates the estimated values of com-pressibility indices, bulk densities at the unconsolidated state and correspondingisostatic tensile strength of powders used in the studies.

Table 3.Powder characterization parameters

Sample Elastic–plastic Compressibility Isostatic tensile Loose bulkcontact consolidation index, n (—) strength of density,coefficient, loose powder, ρb,0 (kg/m3)κ (—) σ0

a (kPa)

Talc 0.48 0.118 0.28 456.1Limestone 0.39 0.117 0.54 548.2Glass ballotini 0.24 0.042 0 836.6Alumina 0.81 0.137 1.21 1028.7

a With negligible consolidation.

Flow characterization of fine powders 133

3.4. Prediction of flow functions of powders

Further, linearized yield locus and unconfined yield strength of given powders canbe predicted, with the help of center and radius Mohr circle stresses as given by (16)and (17), respectively [4–6]. As mentioned earlier, the stationary yield locus is acharacteristic of a given material enveloping all the Mohr circles for steady-stateflow:

τ = tan φi

(σ + σR,st

sin φi− σM,st

), (16)

σc = 2(sin φst − sin φi)

(1 + sin φst)(1 − sin φi)σ1 + 2 sin φst(1 + sin φi)

(1 + sin φst)(1 − sin φi)σ0. (17)

Figure 4 compares the flow function of the limestone sample evaluated experi-mentally with that estimated using elastic–plastic contact consolidation theory. Thecorresponding correlation coefficient for the limestone sample is observed to be0.98, indicating the satisfactory nature of the method in understanding interparticleeffects in bulk solids handling.

A similar analysis was carried out for other samples, i.e. talc, glass ballotiniand alumina, respectively, for characterizing their flow properties in terms ofthe corresponding characteristic elastic-plastic contact consolidation coefficient asreported in Table 3. It can be seen from the table that for powders in a given particlesize and consolidation range, the elastic–plastic contact consolidation coefficient[and contact stiffness as defined by (9)] varies on the basis of the micro-hardness

Figure 4. Comparison of experimental and predicted flow functions of samples used in the studies.

134 M. Medhe et al.

of the particles. This observation confirms the effect of hardness as postulatedby Krupp [11] for estimation of interparticle adhesive forces. Therefore, it isreasonable to assume that the effect of hardness of particles will be relatively moreimportant as compared to particle size in understanding the effect of fine particleson flowability of powders. However, it is to be noted that these interparticle contactplastic deformation effects are likely to be confined to the so-called asperitiesor irregularities on the particle surfaces, which are not yet fully characterizable.Thus, the effect of micro processes occurring on the particle surface directly tendsto govern the macro-aspects of powder flow behavior in the presence of fines.However, direct measurement of these asperity height parameters affecting theextent of plastic deformation during particle loading and subsequent estimation ofadhesive forces is in the developmental stage with certain specialized techniqueslike atomic force microscopy [12]. Presently, as a part of further experimentation, aprogram for evaluating the effect of fines on powder flowability of mineralogicallysofter and harder materials is being carried out. The objective of the study is to findout the effect of incremental addition of very fine particles (below 10 μm havingcohesive flow character) to relatively coarser particles of talc and alumina, and toevaluate the critical amount of fines governing the flow categorization of powders,if any.

Figure 4 shows the comparison of experimental flow functions of talc, limestone,glass ballotini and alumina samples with those predicted using the elastic–plasticcontact deformation theory. The corresponding correlation coefficients are observedto be in the range of 0.96–0.99, indicating the usefulness of material characteristicparameters in understanding powder flowability characterization.

4. CONCLUSIONS

The present case study illustrates the successful use of the theory of plasticityin explaining the consolidation, instantaneous yield and stationary yield loci ofpowders. Powder flow parameters are analyzed completely using the basis ofmaterial characteristic parameters, i.e. angle of internal friction (φi), stationaryangle of friction (φst), isostatic tensile strength of an unconsolidated powder (σ0)along with characteristic pre-consolidation influence (σM,st). The micro-hardness ofthe particle plays an important role in flow characterization of powders and must betaken into consideration for understanding micro–macro interactions taking placeduring powder flow.

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