W

ELSEVIER Powder Technoh,-y 88 (1996) 155-163

Micromechanical analyses of the pressure-volume relationships for powders under confined uniaxial compression

M.J. A d a m s , R. M c K e o w n Unilever Research Port Sunlighl laboratory. Quarry Road Eg~t Bebington. Wirral. Mer,~eyside, 1~53 3JW, [JE

Received 22 March 1995; revised 2 January 1996

L.betract

Many pressure-volume relationships for powders under confined uniaxial compression have been proposed. However, they are essentially empirical in origin with the exception of the lumped parameter relationship developed in previous work. In the current study, it is shown that either the latter relationship or that due to Kawakita, Powder Technol., 74 ( 1970/71 ) 61, may be employed to determine the tensile sueagth of agglomerates provided that the influence oftbe friction at the wall oftbe compaction cell is taken into account. A micromechanical model of powder compaction is proposed which leads to an explicit form of the Kawakila relationship. It is shown that for particles which deferm plastically, rather than fracture, the compaction behaviour may be described by a reduced form of this explicit relationship. The mieromeebenical modelling is based upon the affine deformation of particles in load carrying chains or columns such that particles between the chains are redundant with respect to the load carrying capacity,

Eeywords: Contacl mechanics; Compac:,.in; Friction; St~agth

1. Introduction

Particulate materials in the form of agglomerates may be subject to attrition or fragmentation processes during han- dling, conveying or mixing operations. It has been established that the resistance to such damage correlates with the bulk strength of the agglomerates [ 1 ]. The determination of the strength of single agglomerates is time consuming since there is often a wide distribution of values and, consequently, a relatively large number have to be measured. A more con- venient technique is the confined uniaxial compression of agglomerate beds. This involves confining a bed of agglom- erares in a cylindrical cell and measuring the force applied to a flat punch, which is in contact with the top surface of the bed, as a function of the displacen~ent of the punch. Many empirical relationships have been pro;3osed [2] to describe the resulting data which may be expr~sed equivalently in terms of stress-strain, pressure-volume or pressu:¢-dansity since the natural strain, for example., is equ~l to the natural log of the ratio of the initial bed height or volume m the current height or volume respectively. In previous work [ 3], a relationship was derived that explicitly included tac single particle or agglomerate strength to describe such data, thus allowing this quantity to be determined indirectly. A lumped parameter approach was adopted in which the applied

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uniaxial stress was assumed to be propagated along columns of particles forming the bed. The validity of the model was examined using agglomerates, having a range of strengths, which were prepared from quartz sand and polyvinyl-pyrrol- idone (PVP) as a binder. It was found that the measured pressure-volume relationships were ec, as~stent with those predicted by the model. In addition, '.hare was a linear cue- relation between the strengths r:,easured directly by single agglomerate compression and those calculated from the measured pressuro--volame data using the model. The pr'o- portianality const,mt between t:'_3 meastLred and calculated values was !.J.

These measurements were made at a finite and fixed final bed aspect ratio (height/diameter) of 0.5. It is well esmb lisbed that wall friction significantly increases the resistance to deformation [4] so that the true proportionality constant would be smaller. In order to determine absolute values of strength, it is dearly important m establish the magnitude of the constant. This was one of the aims of the current work. The approach was to make measurements at different bed heights and extrapolate to a zero aspect ratio in order to eliminate the influence of wall friction.

In the derivation of the lumped parameter model [3], a number of simplifying assumptions were made about the

156 M.J. AdanLr. R. McKeown /Powder Technology 88 (1996) 155-163

evolution of the bed microstructure as indicated previously. The actual failure process was supposed to be similar to that for isolated particles except that the induced lateral stress field acting on the panicles would increase the uniaxially applied stress required to cause fragmentation. This phenom- enon is embedded in the Mohr-Coulomb failure criterion which was employed in the model. The sand agglomerates referred to previously were relatively brittle and exhibited un stable crack growth. Thus, it would only have been possible to compare the fragmentation debris of single and bed com- pressed agglomerates although this was not attempted. In the current work, more plastically deforming granules were investigated for which the extension of any cracks would be arrested following the removal of the applied load. In this way, it was possible to determine if the evolving damage in a compressed bed was similar in nature to that for single agglomerates subjected to a comparable diametral strain. The work was also an opportunity to examine further the validity of the model particularly using agglomerates with mechanical characteristics distinctly different from the sand agglomerates studied previously. In addition to being more plastically deforoting, the agglomerates studied here were about an order of magnitude weaker.

As indicated previously, there are many relationships available for describing powder compaction data. One of the most widely employed is due to Heckel [5] which inconpo- rates a parameter that is considered to be closely related to the yield stress of the individual panicles. Roberts and Rowe [6] carried out some elegant work involving the application of this relationship to a wide range of powders. They observed that there was reasonable agreement between the yield stresses obtained with the Heckel relationship and values measured independently using indentation hardness. How- ever, the most widely employed relationship is due to Kawak- ita and co-workers [2,7] which also includes a parameter that is considered to be related to the yield stress [7]. The terms strength and yield stress will be defined more precisely later but may be interpreted here as some measure of the failure stress associated with fracture or plastic deformation. In pre- vious work [ 3 ], it was found that the agglomerate strengths obtained from the lumped parameter model were linearly related to and of similar magnitude to the above Kawakita parameter. A further aim of the current work was to establish the most useful of these relatioaships for deriving the strengths of single agglomerates from bed compaction data. In practice, as described later, the Heekel relationship is a poor representation of the pressure--volume characteristics of agglomerate beds while the values of the Kawakita parameter are similar to the agglomerate strengths values derived from the lumped parameter model. Consequently, it was of interest to develop a theoretical basis for the Kawakita relationship in order to understand the origins for the simi- larity in agglomerate strengths derived from the two nomi- nally vet 7 different expressions for describing powder compaction.

2. Experimental

2.1. Materials

A number of types of experimental agglomerates contain- ing a fine inorganie particulate phase and a range of soft binder phases were investigated. These agglomerates were approximately spherical in shape. They were sieved but mechanical measurements were made on all sieve fractions for only one type. For the other types, measurements were restricted to a single size fraction corresponding to a mean diameter of 550/~m.

2.2. Confined uniaxial compression measurements

Granules were introduced into a cylindrical cell to give the required initial tapped aspect ratio. The internal diameter of the cylinder was 16 ram. A rod of the same diamater was placed in the cylinder and the assembly was compressed between parallel platens using an Instron model 6022 mechanical tester with a 10 kN load cell. Engineering strains of 40% were applied and the force--displacement data were monitored using a digital transient recorder (from Kontron). It was found that the data were relatively insensitive to velocity and an arbitrary slow compression velocity of 0.05 mms - ~ was selected which corresponded to nominal uniaxiai strain rates of 0.003 to 0.01 s -~ depending on the bed height. In some experiments, the compression was arrested at strains less than 40% and the compressed agglom- erates were subjected to scanning electronic microscopy.

The measured compaction data were fitted ~o Lhe following expression using the procedure described previously [ 3] :

In P = In(~o'/a') + c¢'~+ In[ 1 - exp( - a'E) ] (1)

where P is the applied pressure and • is the natural strain which is given by:

• = ln (ho lh ) ( 2 )

where h a and h are the initial and current heights of the bed respectively. The quantity ~'o' is the apparent single agglom- erate strength which is related to the actual strength, %, as follows:

~'o' = kll'o (3)

where k I is a constant. The quantity a ' is related to the pres- sure coefficient, a, of the agglomerate strength by the follow- ing expression:

a' = k e a (4)

where k z is a constant.

2.3. Single agglomerate measurements

These measurements were also performed using a univer- sal mechanical testing machine ( from Instron) but with a 0.1 N load cell (Maywood Instruments) mounted on the cross-

MJ. Adams, R McKeown I Powder Technology 88 (1996)/55-163 157

head. A glass convex lens with a radius of curvature of 50 mm was attached to the transducer in order to act as an upper platen. A lower planar platen was fixed to an XY micromotion stage so that the test agglomerate could be axially aligned with the lens. The displacement of the lens was measured using a linearly variable differential transducer (LVDT) (RDP Electronics) and the force-displacement data were monitored using a digital transient recorder. Test agglomer- ates were compressed at a velocity of 0.01 mms - * until they were observed to fracttae; for an agglomerate diameter of 550/,Lm this velocity corresponds to a nominal aniaxiai strain rate of about 0.02 s - ~ which is similar to the range employed in the bed compaction. Fifteen single agglomerates of each type were measured in this way. Some agglomerates were compressed to smaller strains and subsequently examined by scanning electron microscopy.

There was a distinct first maximum in the force--displace- ment date that corresponded approximately to the point at which the test agglomerate was observed to fracture. The nominal strength, to,, was calculated from the maximum force, Fr,~, using the following expression:

4 • ~, = ~ - / m , (5)

where d is the diameter of the agglomerate.

3. Results

Fig. 1 shows a typical plot of the measured force as a function of the natural strain for a single agglomerate which had a diameter of 570/.tm. It shows elastoplastic deformation and fracture characteristics. The initial region, for strains less than 0.05, corresponds to elastic Hertalan deformation. There is an intermediate linear plastic region with a slope propor- tional to the yield stress. At higher strains, the plastic defor- mation becomes non-linear. Finally, at the peak force, the agglomerate undergoes stable crack propagation; unstable growth is charecterised by a sharp reduction in the force following the maximum fracture value.

Representative data obtained from bed compaction are shown in Fig. 2. The curve is the best fit of Eq. ( 1 ), it can be seen that the theoretical relationship adequately describes the data. The effect of the final aspect ratio on the apparent strengths derived from fitting this relationship for one type of agglomerate is exemplified in Fig. 3. This demonsWates that wall friction has a significant influence on the magnitude of the bed compressibility for these materials. For the range of aspect ratios examined, the agglomerate strength appears to increase linearly with the aspect ratio. The number of nominal agglomerate layers corresponding to the different aspect ratios is also given in the figure. It may be seen that the agglomerate strength at a zuro aspect ratio may be obtained by linear extrapolat ion. Data corresponding to a very few particle layers, where other factors such as punch friction may become important, were not mensured in this study. We

O.O7

O.O6

O-O6

i 0.04 0.1}3

0.02

0.01

% o~. o., olo 0.6 Natural strain £

Fig. L The compressive force rcc~ured as a fuaction of the natural stntlrt for a single agglon'craze.

3.5

2.5

0.5

0 ~ . . . . . . . . 0.1 0,2 0.3 0,4 0.5 0.6

Natural strain f. Fig. 2. The pressure me~med during the confined cornlne~ion of a bed of agglomerates as a funcxion of the asmml stnfin; the curve is tl~ Ucst fit of F.q. (t) to me data.

No. ot parUc~ ~yam 10 20 30 40 :Et" o.n , ~

O.5

"0 o 3

~ 0.2

~ o.1 g

, t '~ 0.2 ' 0'.4 0.6 ' 0.6

Aspect refio Fig. 3. The agglommme sUength of singk =~,lomezat¢~ d=rivcd from bed compo~don, as ~ fuuctim= of the initi~ aspect r=io of d~e bed md aho the number of honied l ~ l e lay~s,

would emphasise that no theoretical justification for this extrapolation ptecedure may be offered at this time. The readeris referredto lsherwood [4] for an accountofthemost useful approaches to deconvoluting the intrinsic compactio*l relationship from experimental dam although tbesc approaches were not applicable here.

Scanning elecnon micrographs ( × 100) axe shown in Fig. 4 for an andeformed agglomerate and similar agglom-

158 M.J. Adams. K McKeown / Powdor Technology 88 (1996) 1~5-163

Fig. 4. Scanning electron micrographs ( × 100) of (a) an undefom~ed agglomerate; (b) and (¢) single agglomerates compz~ssed to natural strains of 0.35 and 0.50. respectively; (d) and (e) agglomerates compressed in a bed to natural strains of 0.35 and 0.~0, respectively.

grates compressed to natural strains of 0.35 and 0.50 both as single agglomerates and as beds. The damage is similar with evidence of partial through-thickness (stable) cracks and the formation of large fragments at the higher strain. There are two important points that are worth pointing out about the nature of the damage. The first is that the failures appear to be associated with crack opening or tensile fracture (mode

I) rather than with in-plane shear fmctare (mode II). Fur- thermore, it was observed, from the examination of a large number of the agglomerates, that the extent of crack opening was significantly less for those agglomerates compressed in a bed compared with those compressed individually. This is evidence of the constraint to crack opening imparted to bed compressed agglomerates due to neighbouring agglomerates.

M.J. Adams, R. Mcgeown / Powder Technology 88 (1996) 155-163 159

~_ 0.6

" 0.5

0.4

~ 0.2

~ 0,1

~ 0 , i , r i 0.5 ! 1.5

Particle diameter d, mm Fig. 5. Agglomerate st~ngth as a function of agglomerate diameter obtained by single agglomerate compmsslon (O) and by the compre~slon of agglom- erate [~eds and extrapolating to a zero aspect ratio (O).

0.7

'~ o.6

0,5

~ 0.4

"~ O.S

~ 0,2

~ 0.1

~ o 0,1 0,2 0.3 0,4 0~5 Agglomerate strength'Cos, MPa

Fig. 6. Agglomerate strength obtained by the compression of beds at an inidal aspect ratio of 0,5 (O) and after extrapolating to a zero ratio (O) as a function of the corresponding values obtained by single agglomerate compression.

Fig. 5 shows the apparent strength data for the agglomer- ates for which all the sieve fractions were measured. These data refer to values obtained after extrapolation to a zero aspect ratio. The standard deviations of the single agglom- erate strength values are also included in the figure. The strength values obtained from the bed compression are gen- erally close to these deviations: the mean of the ratios %'Pro, is 0.91. It will also be noted for this powder, that there were no systematic trends of the slmngth with agglomerate diameter.

Fig. 6 shows plots of the apparent strengths obtained from bed compression as a function of the corresponding single agglomerate strengths for a range of agglomerates with a mean diameter of 550/.tin. For values extrapolated to a zero aspect ratio, a linear correlation of slope equal to 0.91 is a reasonable approximation and consistent with the ratio obtained for the data in Fig. 5. Data are also shown for bed values measured at a final aspect ratio of 0.5; in this case, the ratio of the bed to the single agglomerate values is about 1.32.

As described in the introductory section, the bed compac- tion data were also analyses using the Heckel and Kawakita relationships. The Heckel [ 5 ] relationship may be written in the following form:

In 1 - - ~ = AP+A (6)

where D is the relative density and K and A are constants for a particular powder. Heckel [ 5] argued that the parameter K is equal to t /3 0.0 where 0.0 is the yield stress of the particles. Roberts and Rowe [6] pointed out that for n perfectly plastic material, 30" 0 is equal to the indentation hardness (see later). They obtained a reasonable agreement between yield stresses derived from Hecbel plots and from indentation measure- ments for a range of particle types. For the agglomerates studied in the current work, it was found that plots of I n [ l / ( l - D ) ] as a function of P deviated from linearity which is not consistent with Eq. (6). A typical plot is shown in Fig. 7. Similar problems were encountered for other agglomerate types examined by the authors and, conse- quently, it appears that the Heckel relationship is not suitable for such powders.

The Kawakita relationship [2,7] is given as follows:

P 1 P - = - - + - (7) e ab a

where e( = (h o - h) ~he)iS the engineering s~ain and a and b are constants. The parameter 1/b is thought to be rela~:l to the failure stress of the individual particles as discussed ear- lier. In the current and also previous work on sand agglom- erates [3] this relationship provided an excellent fit to the data. A typical plot in linear form for one of the agglomerates studied here is shown in Fig. 8.

Fig. 9 shows plots of 1/b as a function of the corresponding individually measured agglomerate strengths at a final aspect ratio of 0.5 and after linearly extrapolating to a zero aspe~ ratio. As with the lumped parameter results, there is a reason- able linear correlation between the strengths derived from bed compaction and single agglomerate measorements. A similar effect of wall friction on the single agglomerate strength, corresponding to the finite aspect ratio, is also evi- dent. These data may be expressed as:

1

1.4

9 1 on

o.e

0.4

0,2

J

012'014 01.6'018 ' ; '1~2' 114 Pressure, MPa

Hg. 7. A Hcche! plol for one of lhe agglomerates sn~ied here.

160 M.,L Adams. R. MeKeow,n/ Powder Technology 88 (1996) 155-163

12

1o

" N

o.

o 1 2 a

Pressure P, MPa Fig. S. A Kawakita plot for one of the agglomerates studied here.

i , i , i i , i , ~ 0 5 4 0.6 Agglomerate strength '~o=, MPa

m 0.6 Q. 0.5 ~ o.4

~ 0.3

~ 0 , 2

0.1

~ o~

Fig. 9. The Kawakita parameter, l/b, obtained by tile compression of beds at an initial aspect ratio of 0.5 (OI and after extrapolating to a zero aspect ratio (QI as a function of the corresponding values obtained by single agglomerate compression,

where k3 is a proportionality constant equal to 1.16 and 0.98 for the data obtained at an aspect ratio of 0.5 and after extrap- olating to a zero aspect ratio respectively, These values are comparable to those obtained with the lumped parameter model. However, the data appear to be less sensitive to wall friction. Moreover, on average, the values of the parameter 1/b obtained after extrapolating to a zero aspect ratio are almost exactly equal to the corresponding values of the meas- ured single agglomerate strengths.

4 . D i s c u s s i o n

It may be assumed that the inability of the Heckel relation- ship to describe compaction data for agglomerates, in contrast to the observations of Roberts and Rowe [ 6]. implies that this relationship does n at take account of the pressure depen d- euce of the failure of agglomerates. They studied single par- tides which are unlikely to exhibit a significant pressure sensitivity. Conversely, that the Kawakita relationship leads to a rather close approximation to the failure strength indi- cates that it does account for the pressure dependence. This is supported by the analysis carried out previously [ 3 ] which showed that the lumped parameter model, mathematically at

least, is related closely to the Kawakita relationship. Further- more, the Kawakita relationship has the form of the current pressure being the sum of a failure stress and a pressure term, This linear form suggests that it has a relatively simple basis,

The simplest assumption that may be made in compaction is that each agglomerate or particle under load undergoes a uniaxial affine deformation, that is the uniaxial strain expe- rienced by each particle is equal to that imposed on the bed, In the case of such a mean deformation field, the engineering strain for a particle in a column will be given by:

2~ e=~ (9)

where 23 is the total relative approach of the particle at the points of contact. We now assume that there are vertical load carrying chains of pazticles separated by a mean centre-te- centre distance of s d /2 where s is a spacing factor. For the ease of a simple close-packed cubic arrangement and equal load sharing on all particles, s will he equal to 2. However, computer simulation studies [8] have demonstrated that while the stress is transmitted along chains or columns of particles, there are significant numbers of particles between these chains that carry little or no load. Thus more generally. for equitable load sharing across all chains, the bed pressure will be given as follows:

F P = ( s d / 2 ) ~ (10)

where F is the force acting on a single chain and will be some function of the relative approach, ~, The Kawakita relation- ship follows if this functionality is given by that of a plasti- cally deforming sphere with a uniaxial yield stress, o'a, thus I91:

F=TwlflcOo ( l l )

where c is a constmim factor equal to about 3 for a rigid- plastic material. In the ease ofelasto-piastic materials its value will be in the range 1 to 3 depending on the ratio E/cr o where E is Young's modulus. It was pointed out earlier that the initial non-linear deformation behaviour shown in Fig. l is typical of an elastic Hertzian response (see later). This sug- gests that the constraint factor could be less than 3. However, estimates of Elcr o from the single agglomerate compaction measurements were found to be typically of order 160 when c takes a value of approximately 3 [9]. For a Mohr--Coulomb material, Eq. ( 11 ) would be re-written as:

F = ~rd&(~ro+ uP') (12)

where P' is the lateral pressure acting on the agglomerate due to the deformation of neighboming agglomerates and cro is the yield stress at a zero value of P'. As in the derivation of the lumped parameter model, we will assume that P' is pro portioanl to the applied bed pressure P; thus [3]:

P' = k4P (13)

M.J. Adams. R. McKeown / Po~vder Tedmology 88 (1996) 155-163

where k4 is a proportionality constant. Combining Eqs. (9)- (13) we obtain:

P 2~rc 2Trc -7,,o + - -~-~e (14) ; =

This is of the form of the Kawakita relationship with 11b given by:

]_=Zzo (15) b crk4

The present data suggest that the product (xk., is of order unity. In the development of the lumped parameter model, the agglomerate strength was equated to the shear failure stress, ¢o, rather than the uniaxiai value, O'o [3]. In the absence of any visual evidence about the mode of breakdown, it was assumed that the failure occurred in shear. The current work demonstrates that the agglomerates fail by a tensile or crack- opening mechanism. Actually, this is not important for esti- mating the failure stress of single agglomerates since this is taken as the failure force divided by the cross-sectional area. It will be appreciated that this is a ernde scaling argument but ore which is apparently a good approximation, that is, for either the single agglomerates or those in beds, only a nominal value of the strength is being defined. However, if ro is replaced by O'o in Eq. ( 5 ), then this e~pres~ion bet.nines equal to that used to calculate the tensile strength of an agglomerate in a Brazilian or diametrai crush test [ 10]. A more exact definition strength would involve the application of fracture mechanics. This has been carried out previously for agglom- erates [ I 1,12 ] and it has been shown that crack propagation is associated with a plastic or process ahead of the crack tip. Consequently, the failure stress in crack opening corresponds to the uniaxiai yield stress which will increase in value with the stresses acting normal to the direction of crack propaga- tion according to the Mohr-Coulomb criterion. Thus, the basis for deriving the Kawakita relationship is that the yield value controls both the gross deformation characteristics of the agglomerates (Eq. (12)) and also the stress required to propagate cracks which increases as the compaction process proceeds due to the increase in the local lateral stresses.

A similar exercise to the above may be carried nut for plastically deforming single particles having apressure insen- sitive yield stress. The compaction behaviour then may be described by El. (14) with the pressure coefficient, a , set to zero, thus:

p= 6=#% (16) 8 2

In writing this equation for a perfectly plastic material, the constraint factor e has been taken as 3. Spherical antinomy hardened lead shot ( IMI Ltd, UK) with a diameter of 3 mm were compacted in beds to examine the validity of this rela- tionship. Fig. 10 shows a plot of the pressth~ applied to a bed of such particles as a function of the engineering strain which is linear at low strains as expected from El. (16). Unlike the

2 5

2 0

m (n

i i i O.20

Engineering strain, e Fig. I 0. Bed compaction data for 3 mm diameter lead spheres.

161

Enginm)ring slrain, e 0.~0 0.01 O.Q2 O.Q~ 6O

5O

tiff40

8.~o o.o2 o.o, o.o~ o.= Displacement 2G, mm

Fig I I. The compressive force as a function of total disptseen-,~t c~ engi- neering slxain for a single lead sphexe.

agglomerates, the gradient of such plots was independent of the nspeet ratio. We are unable to explain this apparent insco- sitivity to the wail ~ :dan. In order to evaluate the parameter s, a single lead sphere was c o m p r ~ . The data are shown in Fig. 11 andwere analysed using Eq. (11) with the parmn- eter c taken as being equal to 3 which led to a value of 42 MPa for o" o. Substitution of this value into Eq. ( 16), together with the slope of the best linear fit to ,re data in Fig. lO, resulted in a value for s of 2.90. This suggests that the mean spacing between load bearing chains is of the order of the particle radius.

Since the yield slxess of lead is independent of pressure, the bed compaction date should also be well described by a Huekel plot as discussed earlier. In practice this is the case (Fig. 12) but the yield strees obtained from fitting the data to the Heckel relationship was 15.0 MPa which is not consis- tent with the value of 42 MPa measuzed above for a single lead sphere. Roberts and Rowe [6] also reported a similar diserepnaey for the yield sUess values obtained from a Heekel plot and by indentation for PTFE.

The expression (E t. (14)) of the form of the Kawakita relationship was derived previously on the basis that the par- ticles exhibited pressure dependent yielding. As a conse- quence, the Kawakita relationship should provide a poor

162

1.4

1.2

E3 1

~0.8 0.6

¢' 0.4

0.2

0

J J

M.J. Adams. R. McKeown /Powder Teetmology 88 (1996) 155-163

to a zero aspect ratio, as a function of the corresponding measured single agglomerate strengths (the data obtained at a final aspect ratio of O.5 is also shown). The single agglom- erate strength is denoted % in the plot in order to be consistent with earlier plots. As explained previously, it should strictly be denoted o'o~ to indicate a tensile value. If the slope of the best linear fit to the data is k5 then the following may he written

[ 2'rrc ~ = ks~u,=-7,~o (17)

20 and thus with ~b, = oo and c = 3

6 ~ ks=~- (18)

This leads to a value of 2.6 for s which is approaching that expected for a simple close-packed cubic arrangement with uniform stress transmission. The computer simulation work referred to earlier [8] would suggest that the spacing is sig- nificantly greater. However, more recent work [ 13 ] suggests that the stress transmission becomes more uniform for softer particles such as those studied in the current work.

5 10 15 Pressure, MPa

Fig. 12, Heckel plot for a bed of lead spheres

f

2o ,'o Be ' io ' leo Pressure P, MPa

Fig. 13. Kawakita plot for a bed of lead spheres

2oo

~:. 15o c_

too

2

5e n.

n

c

0 1 0~2 013 0~4 0.5 Agglomerate s~.':'ngth'Eos, MPa

~. 1 . 4

.~ 1.p

~ t ~ 0 . 8

0.4 ~0.2

Fig. 14. The Kawakita parameter, I/ab, obtaint.d by the compression or beds at an initial aspect ratio of 0.5 (O) and after extrapolating Io a zero ~.~pect ratio (lid) as a function of the corresponding values obtained by single agglomerate corapression.

description of the bed compaction characteristics of lead par- ticles. Fig. 13 shows that this is the case.

Having established the value of applying a simple affine deformation model, it is useful to re-examine the explicit form of the Kawakita relationship (Eq, ( 14 ) ). It may be seen that the intercept, I /ab, of a Kawakita plot should be given by 2 ~ c ~ % / s ~. If c is taken as being equal to 3 and o" o is the single agglomerate strength then it is possible to determine a value for s. Fig. 14 is a plot of l /ob , obtained after extrapolating

5. C o n c l u s l o n s

Agglomerates undergo tensile fracture when a bed is sub- jected to confined uniaxial compaction for the range of imposed strains studied here (40%). This mode of break- down is similar to that observed during the diamctral com- pression of single agglomerates for similar strains. However, it was evident from the examination of a large number of electron microgmphs of agglomerates fractured in the two loading configurations, that the extent of crack opening was significantly greater for individually compressed agglomer- ates compared with those in beds. Previous work [3] on the lumped parameter pressure-volume relationship should be modified to take account of the tensile rather than the shear failure mode that was proposed. In particular, the Mohr- Coulomb failure criterion used to derive this relationship should be written in terms of tensile stresses where the pres- sure coefficient represents the influence of local lateral stresses on the tensile strength. It is well established that the resistance to crack propagation is greater when the failure surfaces are not traction free [ 14]. The lumped parameter relationship then may be writtea in terms of the tensile rather than the shear strength.

The lumped parameter relationship provides a good description of bed compaction data for agglomerates with a wide range of mechanical properties. The tensile strengths derived from fitting this relationship increase linearly with the aspect ratio of the compressed bed due to the contribution of the wall friction to the compaction pressure. If the strength data are extrapolated to a zero aspect ratio then a value for the tensile strength is obtained which is on average about

M.J. Adcan¢, R McKeown /Powder Technology 88 H996) 155-163

90% of that obtained by diametral compression for the single agglomerates studied here. Measurements on other types of agglomerates would be required in order to establish if this proportionality factor is generally applicable. Such measure- ments would need to involve agglomerates showing a range of mechanical properties and sizes.

Similar comments to the above apply to the Kawakita rela- tionship although, in this case, the tensile strengths derived from bed compaction are on average about equal to the single agglomerate values. Moreover, the tensile strengths derived are rather less sensitive to the wall friction.

The Heckel relationship provides a poorer fit to the pres- sure-volume characteristics of agglomerate beds.

A modified lumped parameter relationship may be derived by assuming that the particles undergo an affine plastic defor- mation and fracture according to the Mohr--Coulomb crite- rion. This leads to an explicit form of the Kawakita relationship which may be simplified for the special case where the particles undergo only plastic deformation.

Greek letters

a pressure coefficient a ' k z a (Eq . ( 4 ) )

particle deformation at contact point ( m ) e natural strain o" o uniaxial failure or yield stress (N m - z) ¢o single agglomerate strength derived from bed

compaction ( N m - 2) "r o' kl'ro (E q. ( 3 ) ) t N m -2) T~ single agglomerate strength (N m - 2) v Poisson ratio

6. List of symbols

A parameter in Heekel 's equation a parameter in Kawakita 's equation b parameter in Kawakita 's equation (m - 2 N - t) c yielding constraint factor D relative density d agglomerate diameter (m) E Yoang 's modulus (N m -2) e engineering strain F applied force ( N ) F . m single agglomerate failure force (N) h current bed height (m) ha initial bed height (m) K parameter in Heckel ' s equation (m - 2 N - ' ) k~ proportionality constant ( i = 1 to 5) P pressure or uniaxial stress (N m - z) s particle spacing parameter

163

Acknowledgements

The work was carried out as part of a M A F F / D T I LINK Scheme on " 'The Optimisation of Solid Food Processing Operations".

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