Bulks Solids Behavior

7/31/2019 Bulks Solids Behavior

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The last set of tests includes solid indicizers pioneered by Johann-son. These include the Flow Rate and Hang-Up Indicizers [ef. Bellet al., Practical Evaluaton of the Johanson Hang-Up Indicizer, BulksSolids Handling, 14(1), 117 (Jan. 1994)]. They represent simplied ver-sions of permeability and shear cell tests. Assumptions are made withregard to typical pressures and wall frictions, and based on these, a

SOLIDS MIXING 21-33flow ranking is created. Their degree of success in an application willlargely rest on the validity of the property assumptions. For definedconditions, they can give similar ranking to shear cell and permeabil-ity tests. The choice of use is less warranted than in the past due to theprogress in automating shear cell and permeability tests, which hassimplified their ease of use.

SOLIDSMIXINGGENERAL REFERENCES: Fan, Chen, and Lai, Recent Developments in SolidsMixing, Powder Technology, 61, 255-287 (1990); N. Hamby, M. F. Edwards,A. W. Nienow (eds.), Mixing in the Process Industries, 2d ed., Butterworth-Heinemann, 1992; B. Kaye, Powder Mixing, 1997; Ralf Weinekotter and Her-man Gericke, Mixing of Solids, Particle Technology Series, Brian Scarlett (ed.),Kluwer Academic Publishers, Dordrecht 2000.

PRINCIPLES OF SOLIDS MIXINGIndustrial Relevance of SolidsMixing The mixing of powders,particles, flakes, and granules has gained substantial economic impor-tance in a broad range of industries, including, e.g., the mixing ofhuman and animal foodstuff, pharmaceutical products, detergents,chemicals, and plastics. As in most cases the mixing process adds sig-nificant value to the product, the process can be regarded as a key unitoperation to the overall process stream.By far the most important use of mixing is the production of ahomogeneous blend of several ingredients which neutralizes varia-tions in concentration. But if the volume of material consists of oneingredient or compound exhibiting fluctuating properties caused byan upstream production process, or inherent to the raw material itself,the term homogenization is used for the neutralization of these fluctu-ations. By mixing, a new product or intermediate is created for whichthe quality and price are very often dependent upon the efficiency ofthe mixing process. This efficiency is determined both by the materi-als to be mixed, e.g., particle size and particle-size distribution, den-sity, and surface roughness, and by the process and equipment usedfor performing the mixing. The design and operation of the mixingunit itself have a strong influence on the quality produced, butupstream material handling process steps such as feeding, sifting,weighing, and transport determine also both the quality and thecapacity of the mixing process. Downstream processing may alsodestroy the product quality due to segregation (demixing). Continu-ous mixing isone solution which limits segregation by avoiding storageequipment.The technical process of mixing is performed by a multitude ofequipment available on the market. However, mixing processes arenot always designed with the appropriate care. This causes a signifi-cant financial loss, which arises in two ways:1. The quality of the mix is poor: In cases where the mixing pro-duces the end product, this will be noticed immediately at the prod-uct's quality inspection. Frequently, however, mixing is only one in aseries of further processing stages. In this case, the effects of unsatis-factory blending are less apparent, and might possibly be overlookedto the detriment of final product quality.2. The homogeneity is satisfactory but the effort employed is toogreat (overmixing): Overmixing in batch blending is induced by an over-long mixing time or too long a residence time in the case of continuousblending. This leads to increased strain on the mixture, which can havean adverse effect on the quality of sensitive products. Furthermore,larger or more numerous pieces of equipment must be used than wouldbe necessary in the case of an optimally configured mixing process.Mixing Mechanisms: Dispersive and Convective Mixing

The mixing process can be observed in diagrammatic form as an over-

lap of dispersion and convection (Fig. 21-42). Movement of the par-ticulate materials isa prerequisite of both mechanisms. Dispersion isunderstood to mean the completely random change of place of theindividual particles. The frequency with which thelarticles of ingre-dient A change place with those of another is relate to the number ofparticles of the other ingredients in the direct vicinity of the particlesof ingredient A. Dispersion is therefore a local effect (micromixing)taking place in the case of premix systems where a number ofparticlesof different ingredients are in proximity, leading to a fine mix localizedto very small areas. If the ingredients are spatially separated at thebeginning of the process, long times will be required to mix themthrough dispersion alone, since there is a very low number of assortedneighbors. Dispersion corresponds to diffusion in liquid mixtures.However, in contrast to diffusion, mixing in the case of dispersion isnot caused by any concentration gradient. The particles have to be inmotion to get dispersed. Convection causes a movement of largegroups of particles relative to each other (macromixing). The wholevolume of material is continuously divided up and then mixed againafter the portions have changed places (Fig. 21-42). This forced con-vection can be achieved by rotating elements. The dimension of thegroups, which are composed of just one unmixed ingredient, is con-tinuously reduced splitting action of the rotating paddles. Convectionincreases the number of assorted neighbors and thereby promotes theexchange processes of dispersive mixing. A material mass isdivided up

IIFIG. 2142 The mixing process can be observed in diagrammatic form as anoverlap of dispersion and convection. Mixture consists of two components A andB; A is symbolized by the white block and B by the hatched block. Dispersionresults in a random arrangement of the part icles; convection results in a regularpattern.


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21-32 SOLID-SOLID OPERATIONS AND PROCESSINGf.2 -,,---------------~-------------,

"0~~o 0 ,8IIe .." 1 : : 1'-.Es'II,-,iii OA-M'-~. . r : : : :l; l 0.2-is ( l lli,w " , w l _ _ ,A ," d 2p g I' A " ,8 t- ... 1.48 x 10 - mp g

5 " 0 0o + , - - - - - - - - - - ~ - - - - - - - - ~ - - - - - - - - - - ~ - - - - - - - - ~o .20001 1 0 0

dp (11m)1500

FIG.21-41 The impact of decreasing particle size and bulk permeability on mass discharge rate.

tumbling avalanche methods. These methods should be used withcaution, as (1) they are often a strong function of the test methodand instrument itself, (2) engineering properties useful for eitherscale-up or a priori design are not measured, (3) they are only acrude characterization of flowability, and often suffer from lack ofreproducibility, (4) they lack a fundamental basis of use, and (5)they suffer from the absence of validation powders and methods.The first two points are particularly crucial, the end result of whichis that the ranking of powders determined by the apparatus cannotbe truly linked to process performance, as the states of stress in theprocess may differ from the apparatus, and further, the ranking ofpowders may very well change with scale-up. In contrast, shear cellsand permeability properties may be used directly for design, withno need for arbitrary scales of behavior, and the effect of changingstress state with scale-up can be predicted. Having said this, manyof these methods have found favor due to the misleading ease ofuse. In some defined cases they may be useful for quality control,but should not be viewed as a replacement for more rigorous flowtesting offered by shear cell and permeability testing.Various angles of repose may be measured, referring to the hori-

zontal angle formed along a powder surface. These include the angleof a heap, the angle of drain for material remaining in a flat-bottombin, the angle of sliding occurring when a dish of powder is inclined,rolling angles in cylinders, and dynamic and static discharge anglesonto vibrating feed chutes (Thompson, Storage of Particulate Solids,in Handbook of Powder Science and Technology, Fayed and Otten(eds.), VanNostrand Reinhold Co., 1984). From Eq. (21-37) describ-ing the impact of the angles of friction-as measured by shear cell-oncohesive strength, the angle ofrepose maybe demonstrated to lacka true connection to flowability. For cohesive powders, there will belarge differences between the internal and effective angles of friction,and the unconfined strength increases with an increase in the differ-ence in sine of the angles. When one ismeasuring the angle of reposein this case, wide variations in the angle of the heap willbe observed,and it likely varies between the angles of friction, making the mea-surement of little utility in a practical, measurement sense. However,when the difference in the angles of friction approaches zero, theangle of repose willbe equal toboth the internal and effective angle offriction. But at that point, the cohesive strength of the powder iszero[Eq. (21-37)], regardless of the angle of repose.In is likely the above has formed the basis for the use of rotatingavalanche testers, where the size and frequency of avalanches

formed on the sliding, rotating bed are analyzed as a deviation of thetime between avalanches, as well as strange attractor diagrams. This

approach ismore consistent with the variation in the angle of reposebeing related topowder strength [Eq. (21-37)]'The typical density ratios are the Carr and Hausner ratios, given

by

and

(21-57)where pb(tapped) is the equilibrium packed bulk density achieved undertapping. It could equally be replaced with a bulk density achievedunder a given pressure. The Carr index is a measure of compress-ibility, or the gain inbulk density under stress, and isdirectly relatedto gain in powder strength. Large gains in density are connected todifferences in the state of packing in the over- and critically consoli-dated state defined above (see "YieldBehavior ofPowders"), which inturn results in differences between the internal and effective angles offriction, leading to a gain in unconfined yield strength [Eq. (21-37)]'However, the results are a function of the method and may not be dis-criminating for free-flowing materials. Lastly, changes in density areonly one ofmany contributions to unconfined yield stress and powderflowability. Hence, Carr and Hausner indices may incorrectly rankflowability across ranges ofmaterial class that vary widely in particlemechanical and surface properties.Two methods of hopper flow characterization are used. The

first is the Flowdex tester, which consists of a cup with inter-changeable bottoms of varying orifice size. The cup is filled from afunnel, and the covering lid then drops from the opening. The min-imum orifice inmillimeters required for flow to occur isdeterminedas a ranking of flowability. This minimum orifice is analagous to theminimum orifice diameter determined from shear cell data for hop-per design. Alternatively, the mass discharge rate out of the cup orfrom a funnel may be determined. Various methods of vibrationboth before and after initiation of flow may be utilized. Mass dis-charge rates, as expected, rank with the correlations describedabove. The disadvantage of this characterization method is that it isa direct function of hopper/cup geometry and wall friction, and hasa low state of stress that may differ from the actual process. If aprocess hopper differs in vertical half-angle, wall friction, openingsize, solids pressure, filling method, or a range ofother process para-meters, the ranking of powder behavior in practice may differ fromthe lab characterization, since scalable engineering properties arenot measured.


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21-30 SOLID-SOLID OPERATIONS AND PROCESSING

- ; ; ; -c..:::-10If)If)eiii 8: uQ). . c :e n 6

4

2

o 15kPa_J_15kPa_PIII 9kPa_J

-9kPa_Po 6kPa_J

-o-6kPa P{:, 3kPa-J-.:.-3kPa::P

BC R

2 4 10 12 14 16Normal Stress (kPa)

6 8

(a)

( Ccmpa l1 s on . t o BCR -1 16 Ref.):140 -------- ---------------

: 1 0 0

5 t .l n d o r d { 6Ol I i' ,m' J Im l r \l l l>or \ l (G~ I II oni',o 5W l d or d ( '10 8/"""1 Im tr umor \l (\10 ,10m') Si.and"d 1150~/c""lo Im ll '\ .m lm l 1 1 >0 if"",ll

ao -

B I I -

2 1 1

- f 0 1 C ; ( l .~(I r Q ( lNorma l . s tre ss ts/cmZ)

(b)FIG. 21-39 Shear cell BCR-1l6limestone validation yield loci. (a) Comparison of Jenike translational to Peschl rotary shear cell data (DuPont,1994, used with permission). (b) Typical validation set performed on an iShear rotary shear cell as compared to BCR standard (2005). (CourtesyE&G Associates, Inc.)

Mass Discharge Rates for Coarse Solids The mass dis-charge rate from a flat-bottom bin with a circular opening of diame-ter B has been shown experimentally to be independent of bindiameter D and bed fill height H, for H > 2B. Dimensional analysisthen Indicates that the mass discharge rate W must be of the formW =cpYgB 5!2, where C is a constant function of powder friction.Such a form was verified by Beverloo [Beverloo et al., Chern. Eng.Sci., 15, 260 (1961)] and Hagen (1856), leading to the Beverlooequation of mass discharge, orWo =CPbVg(B - kdp J 2 5 ~ 0.52 PbA\ 12gB for B dp

(21-44)

z = o1

Here, p, is loose poured bulk density, C - 0.58 and is nearly indepen-dent of friction, k = 1.5 for spherical particles and is somewhat largerfor angular powders, d; is particle size, and A is the area of the open-ing. The correction term of particle size represents an excluded annu-lus effective lowering the opening diameter. See Nedderman (Staticsand Kinematics of Granular Materials, Cambridge University Press,1992) and Brown and Richards (Principles of Powder Mechanics,Pergamon Press, 1970) for reviews.The Beverloo relation for solids discharge may be contrasted withthe mass flow rate of an inviscid fluid from an opening of area A, orW= 0.64p1AV2gH (21-45)

Vertical stress < J zI

dz*stresstadial stressII Ia, =K,p(Jo

FIG. 21-40 Stresses in a vertical cylinder. [From Measuring Powder Flowability and Its Applications, E&G Associates,2006, with permission.)


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SOLIDS HANDLING: BULK SOLIDS FLOW CHARACTERIZATIONOther workers assume a linear form with a nonzero intercept f c o . Thisimplies a minimum powder strength in the absence of gravity or anyother applied consolidation stresses. Asdescribed above, the flow func-tion isoften curved, likely due to the angles of friction being a functionof applied stress, and various fitting relations are extrapolated to zero todetermine . f c o . While this is a typical practice, it has questionable basisas the flow function may have pronounced curvature at low stress.The flow function and powder strength have a large impact on min-imum discharge opening sizes of hoppers to prevent arching and ratholing, mass discharge rates, mixing and segregation, and compactstrength.One may compare the flowability of powders at similar pressures bycomparing their unconfined yield strcss j,at a single normal stress, orone point off a flow function. In this case one should clearly state thepressure of comparison. Flow indices have been defined to aid suchone-point comparisons, given by the ratio of normal stress to strength, or

(21-38)r

The first is due to Peschl (Peschl and Colijn, New Rotational ShearTesting Technique, Bulk Solids Handling and Processing Conference,Chicago, May 1976). For powders in the absence of caking ithas a min-imum value of 1 for a perfectly plastic, cohesive powder. The seconddefinition is due to Jenike (Jenike, Storage and Flow of Bulk Solids,Bull. 123, Utah Eng Expt. Stn., 1964). The reciprocal of these relativeflow indices represents a normalized yield strength of the powder,normalized by maximum consolidation shear in the case of Peschl andconsolidation stress in the case of Jenike. Flowability increases withdecreasing powder strength, or increasing flow index. Table 21-5 pro-vides typical ranges of behavior for varying flow index. For powders ofvarying bulk density, absolute flow indices should be used, or

(21-39)Therefore, for powders of equal powder strength, flowabilityincreases with increasing bulk density for gravity-driven flow.Shear Cell Standards and Validation While shear cells vary in

design, and may in some cases provide differing values of powderstrength, the testing does have an engineering basis in geotechnicalengineering, and engineering properties are measured, i.e., yieldstresses of a powder versus consolidation. As opposed to other phe-nomenological, or instrument-specific, characterizations of powderflowability, shear cells generally provide a common reliable ranking offlowability, and such data are directly used in design, as discussedbelow. (See also "Solids Handling: Storage, Feeding, and Weighing.")Rotary split cells (ASTM D6682-01), translation Jenike cells (ASTMD6128-97), and rotary annular ring cells (ASTM D6682-01) all haveASTM test methods. In addition, units may be validated against anindependent, international powder standard, namely, the BCR-1l6limestone validation powder for shear cell testing (Commission of theEuropean Communities: Community Bureau of Reference). Table21-6 provides an excerpt of shear values expected for the standard,and Fig. 21-39 provides a yield loci comparison between differing celldesigns and a comparison to the standard values.Stresses in Cylinders Bulk solids do not uniformly transmitstress. Consider the forces acting on a differential slice of material in,TABLE 21-5 Typical Ranges of Flowability for Varying FlowIndex, Modified after PeschlFlow index

Caked material, time consolidatedWet massMagstearate, starch (nongravity)Coarse organicsGranules inorganicsHard silica, sandIffine, floodable

Level of cohesion ExampleBonding, solidPlastic materialExtremely cohesiveVery cohesiveCohesiveSlightly cohesiveCohesionless

Relp: < 1=11-22- 44-1010-1515-25

From Measuring Powder Flowability and Its Applications, E&G Associates,2006, with permission.

21-29TABLE 21-6 BCR-116 Limestone Validation Powder for ShearCell Testing

COMMISION OF THE EUROPEAN COMMUNITIESCERTIFIED REFERENCE MATERIALCERTIFICATE OF MEASUREMENTCRM 116LIMESTONE POWDER FOR JENIKER SHEAR TESTING

CONSOLIDATIONNORMAL STRESSkPaSHEAR MEAN UNCERTMNTYNORMAL STRESS SHEAR STRESSkPa kPa kPa

3.03.03.03.03.03.0

3.02.01.751.51.251.0

2.141.751.641.541.411.27

0.31 0.19 0.17 0.14 0.13 0.10

say, a cylindrical bin (Fig. 21-40). Prior to failure or within the elasticlimit, the axial stresses 0', and radial stresses 0'" under the assumptionthey are principal stresses, are related byV(]r= --0,I- v (21-40)

where v is the Poisson ratio. Under active incipient failure, the axialand radial stresses are related by a lateral stress coefficient K, given byc, 1- sin < j J cK =- =__ ----'c_

a 0', 1 + sin < j J c (active) (21-41)In the case of wall friction, the axial and radial stresses differ some-what from the true principal stresses, and the stress coefficientbecomes

o, 1 - sin < j J c cos( 0) - < j J w )K; = a :- = 1+ sin < j J c cos(O)- < j J w )

where . sin < j J wsmO)=---sin < j J c(21-42)

This may be contrasted to, e.g., the isotropic pressure developed in afluid under pressure, with only nonnewtonian fluids able to developand sustain a nonisotropic distribution of normal stress. In addition,the radial normal stress acting at the wall develops a wall shear stressthat opposes gravity and helps support the weight of the powder. Asoriginally developed by Janssen [Zeits. D. Vereins Deutsch lng.,39(35), 1045 (1895)], from a balance offorces on a differential slice,the axial stress 0', as a function of depth z is given by0', = P b g D (1- e-(4f1wKjD),)4 J , ! w K a (21-43)

where D is the diameter of the column. Several comments may bemade of industrial practicality:1. Pressure initially scales with height as one would expect for afluid, which may be verified by expanding Eq. (21-35) for small z. Or0', ~ P b g Z .2. For sufficient depth (at least one diameter), the pressurereaches a maximum value given by 0', = P b g D / ( 4 J , ! w K a ) . Note that thispressure scales with cylinder diameter, and not height. This is a criti-cal property to keep in mind in processing; that diameter often con-trols pressure in a powder rather than depth. A commonplaceexample would be comparing the tall aspect ratio of a corn silo to thatof a liquid storage vessel. The maximum pressure in the base of sucha silo is controlled by diameter, which is kept small.3. The exact transition to constant pressure occurs at roughly 2zc,where z; = D/( 4 J , ! w K a ) .Stress transmission in powders controls flow out of hoppers, feed-ers, filling of tubes, and compaction problems such as tableting androll pressing. (See "Powder Compaction.")


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00,

~0 0. . . . t - >~ C I " " " ' ; " ' _ - - - ' ' - I -s:rJ)

Normal stressc rFIG. 2137 The yield loci of a powder, reflecting the increased shear stressrequired for flow as a function of applied normal load. YLl through YL3 repre-sent yield loci for increasing previous compaction stress. EYL and Wr'L are theeffective and wall yield loci, respectively.

(21-35)where u , is the effective coefficient of powder friction and


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SOLIDS HANDLING: BULK SOLIDS FLOW CHARACTERIZATIONShear stress .o r (D)

'fm' 1 ' , =' }1I(0')

-Yield point (elastic limit)

Strain r

21-27Density p _(I ' l) OverconsoUidated

CriticaL statestatic density

Underc,onsQl idiatedstate

FIG. 2135 Examples of yield behavior. (From Measuring Powder Flowability and Its Applications, E&G Associates, 2006, withpermission. )

Strain r

the powder as a function of normal load, in the overconsolidated orovercompacted state, each time reconsolidating the powder beforeperforming the next shear step.Powder Yield Loci For a given shear step, as the applied shearstress is increased, the powder will reach a maximum sustainableshear stress 't, at which point it yields or flows. The functional rela-tionship between this limit of shear stress r and applied normal load 0"is referred to as a yield locus, i.e., a locus of yield stresses that mayresult in powder failure beyond its elastic limit. This functional rela-tionship can be quite complex for powders, as illustrated in both prin-cipal stress space and shear versus normal stress in Fig. 21-36. SeeNadia (loc. cit.), Stanley-Wood (loc. cit.), and Nedderman (loc. cit.)for details. Only the most basic features for isotropic hardening ofthe yield surface are mentioned here.1. There exists a critical state line, also referred to as the effec-tive yield locus. The effective yield locus represents the relationshipbetween shear stress and applied normal stress for powders always ina critically consolidated state. That is, the powder is not over- orundercompacted but rather has obtained a steady-state density. Thisdensity increases along the line with increases in normal stress, andbed porosity decreases.2. A given yield locus generally has an envelope shape; the initialdensity for all points forming this locus prior to shear isconstant. Thatis, the locus represents a set ofpoints all beginning at the porosity; thiscritical state porosity is determined by the intersection with the effec-tive yield locus.3. Points to the left of the effective yield locus are in a state of over-consolidation, and they dilate upon shear. If sheared long enough, thedensity and shear stress will continue to drop until reaching the effec-tive yield locus. Points to the right are underconsolidated and compactwith shear.

r

4. For negative normal stresses, a state of tension exists in the sam-ple along the yield locus. This area is generally not measured by directshear cells, but can be measured by triaxial shear and tensile split cells.5. Multiple yield loci exist. As a powder isprogressively compactedalong the effective yield locus, it gains strength as density rises, reach-ing progressively higher yield loci. Yield loci of progressively largerenvelope size have higher critical density and lower critical voidage, asshown in Fig. 21-36. Therefore, the shear strength of a powder r is afunction of the current normal stress 0", as well as its consolidation his-tory or stress O"cowhich determined the starting density prior to shear.Currently in industrial practice, we are most concerned with theovercompacted state of the powder, and applications of the under-compacted and tensile data are less common, although they are find-ing applications in compaction processes of size enlargement (see"Powder Compaction").Although the yield locus in the overcompacted state may possesssignificant curvature, especially for fine materials, a common Mohr-Coulomb linear approximation to the yield locus as shown in Fig. 21-37is given by

't=c+J..lO"=c+O"tan


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(a) Normllllo'ado

Cell pressure,cr""O"oiil

Normal loadIT

(e) f

FIG. 2133 Examples of powder shear cells . Triaxial cel ls: (a ) Traditional triaxial cell; (b ) true triaxial. Directshear cells: (c) Translational split, Collin (1846), Jenike (1964); (d ) rotational annulus, Carr and Walker (1967),Schulze (2000); (e) rotational split, Peschl and Colijn (1976), iShear (2003). (From Measuring Powder Flowabil-ity and Its Applications, E&G Associates, 2006, with permission.)

Consolidation Stage1:2

Shear Stage. . a . a Ir 'r _, " " '

v . : - - . - lJ r- .--- --Iioomai-5Jiea~ ~~ " " " " ~~ - 1-Den~It!{ "..I~' ". L..-/1 II t I

0..400

O,:~5!li I .: M OQ . 2 : : > Q E~0. :


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SOLIDS HANDLING: BULK SOLIDS FLOW CHARACTERIZATION 21-25

FIG.2132 iShear rotary, full annulus split cell, illustrating normal load weight application, rotational base,and shear stress/torque measurement. Vertical displacement of lid is monitored by displacement transducer.(Courtesy E&G Associates, Inc.)

triaxial shear cells (Fig. 21-33). There are two such types of triaxialshear cells. In the traditional cylindrical triaxial cell, the axial andradial pressures acting on the sample contained within a rubber mem-brane are directly controlled through applied axial force and radialhydraulic oil pressure. Deviatoric stress, i.e., shear stress due to dif-ference in axial and radial pressure, is applied to the sample until fail-ure. In a true triaxial cell, all three principal stresses may be varied;whereas only the major and minor principal stresses are controlled inthe traditional cylindrical triaxial cell. Lastly, shear displacements aremeasured through a variety of strain gauges, and both the drained andundrained tests are possible. Such tests refer to simultaneous measure-ment ofpressure of any interstitial fluid or gas. Interstitial fluid can havepronounced effects on mitigating powder friction and changing flowproperties. While triaxial cells are not typically employed for powdercharacterization in industrial processing, they do provide the most com-prehensive information as well as a knowledge base of application insuch results for bulk solids flow, including detailed simulations of multi-phase flow of such systems. Their disadvantage is their difficulty of useand time required to perform measurements. Future advances inemploying these designs are likely.Direct shear cellswere introduced due to drastically reduced testingtimes, although the exact nature of stresses in the failure zone is not asprecisely defined as with triaxial cells. Direct cells have undergone sub-stantial automation in the last two decades. All have as a common featurethat only the applied axial force or axial stress is controlled (Fig. 21-33).The shear stress required to accomplish failure ismeasured as a functionof the applied axial stress, where translational or rotational motion isemployed. Both cup and split cell designs are available. Rotational cellsinclude both fullannulus and ring cells.For a properly designed directshear cell, failure occurs within a specific region, in which both the planeof failure and the acting stresses may be clearly defined. In addition,direct shear cells may be validated against an independent vendor stan-dard, or the BCR1l6 limestone powder (see "Shear Cell Standards andValidation"). Rotary split cell designs have two possible advantages: (1)Unlimited displacement ofthe sample ispossible, allowing ease of sampleconditioning and repeated sample shear on a single sample. (2) The shearplane isinduced in a defined region between the two cell halves, allowingunconfined expansion in the shear plane (Fig. 21-32).Yield Behavior of Powders The yield behavior of a powderdepends on the existing state of consolidation within the powderbed when it is caused to flow or yield under a given state of stress,defined by the acting normal and shear stresses. The consolidationstate controls the current bed voidage or porosity. Figure 21-34 illus-trates a times series of shears occurring for the BCR1l6 limestone

standard for a rotary shear cell. For each shear step, torque is applied

to the sample by cell rotation until sample failure; the cell is thenreversed until the shear force acting on the sample is removed. Twostages of a typical experiment may be noted. The first is a consolida-tion stage wherein repeated shears take place on the sample until theshear stress 1: reaches a steady state, defined by either the maximumvalue or the steady value occurring after an initial peak. This occurswith a constant normal consolidation stress 0=o, acting on the sam-pie. During this step, the sample reaches a characteristic or criticaldensity or critical porosity e, related to the consolidation normalstress. A set of shear steps is then performed during a shear stagewith progressively smaller normal loads. In all cases, each shear step ispreceded by a shear at the original consolidation normal stress.Three characteristic displacement profiles may be observed duringshear for shear stress and density (Fig. 21-35), which are unique to thestate of consolidation:1. Critically consolidated. Ifa powder is sheared sufficiently, itwill obtain a constant density or critical porosity s, for this consolida-tion normal stress c;This is defined as the critical state of the powder,discussed below. If a powder in such a state is sheared, initially thematerial will deform elastically, with shear forces increasing linearlywith displacement or strain. Beyond a certain shear stress, the mate-rial will fail or flow, after which the shear stress will remain approxi-mately constant as the bulk powder deforms plastically. Depending onthe type of material, a small peak may be displayed originating fromdifferences between static and dynamic density. Little change in den-sity is observed during shear, as the powder has already reached thedesired density for the given applied normal consolidation stress o;2. Overconsolidated. If the same sample is sheared, but at alower normal stress of0< 0, the shear stress will increase elasticallyto a peak and then fail, with this peak being less than that observed forthe critically consolidated state, as the applied normal stress is lower.After the failure peak, the shear stress will decrease as the powderexpands due to dilation and density decreases, eventually leveling offto a lower shear stress and lower density. Overconsolidated shears areobserved during the shear stage of a shear cell experiment.3. Underconsolidated. If the same sample is sheared, but at ahigher normal stress of 0> 0, the shear stress will progressivelyincrease to some value, while the material simultaneously densifies.Such underconsolidated responses are observed in the consolidationstage of an experiment.In practice, following the filling of a cell, the powder is in an under-consolidated state. A set of shear steps is performed under a chosenconsolidation stress in the consolidaton stage to increase its densityand bring it into a critical state. A set of shears is then performed atsmall normal stresses in the shear stage to determine the strength of


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21-24 SOLID-SOLID OPERATIONS AND PROCESSING100 .113

10

Plug or slug flow ~1.

\2 liD8 ' , MlnlmulIIl1uldlzalio

o 1 . . . .0.1

Plug or slug flow t--.Q1

1 - Mustard seed2 - P otye lhY rle ne powd er3 - S late dus t4 -1 , O OO - ! - l m send5 - Z irc on san d6 - Cem&rll7 - Pu lv e rized coal8 ~ P ulv erize d fue l,as ll (gri'ts)9 ~ Pol)'eth~ll In9 pellets1.0 - G ra!lu iat\ld sug ar11 ~ Pearlite12 - f lour13 - IPu llle nzed !ue l 8 sh

0.01 o 40 e o , 1200 160 200P re ss ur e d ro p p er u ni t l ength I A P I 1 . lcImbar/m)

o 160 200(a)

80 12.0

FIG. 21-30 Classification of pneumatic conveying based on (a) permeability factor and (b) deaeration factor. [From Mainwaring and Reed, Bulk SolidsHandling, 7,415 (1987) with permission.]

(b )P re s su re d ro p p er u Jl: itle ng tlil (APII. l

Bulks Solids Behavior

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