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The influence of the fine particle content of the flow behaviour of bulk materials
Powder Technology. 37 (7984)
145
745 - 154
THE INFLUENCE OF THE FINE PARTICLE CONTENT ON THE FLOW BEHAVIOUR OF BULK MATERIALS 0. Molerus and M_ Nywlt Lehrstuhl fi_ir Mechanische Verfahrenstechnik, University of Erlangen-Ntirnberg, tdestGermany
Abstract Experimental results obtained from shear tests with well defined mixtures of coarse grained (=cohesionless)and fine grained (=cohesive) materials are presented. Evaluation of the experiments with respect to parameters based on considerations on particle interactions results in a meaningful interpretation of the observed material behaviour.
Introduction Some years ago, Kurz and Miinz 111 oresented experimental results of shear tests on limestone Fowders with different widths of particles size distribution. Even small changes in the fine particle content gave measurable changes in cohesiveness.
Pore recently, Fiirll 121 investigated the maximum permissible fine
particle content which does not affect the flow behaviour of coarse, cohesionless materials. are discussed, which were In the present uaper, experimental results obtained with different mixtures of fine grained, cohesive and coarse grained, cohesionless powders. The aim of this paper is to interpret the experimental results in terms of parameters, which define particle interactions_ Some simple geometrical considerations To estimate critical size and mass ratios of fine grained and coarse grained particles, binary mixtures of spheres are considered_ In a random packing of particles, local changes of the coarse particle content have to be admitted. The maximum diameter d of a sphere passing through the most dense packing of spheres of diameter d, is ~I;f;Wi;;~gtF/dg = 0,155 E31. r ndom packings of coarse grained irregular particles are E = O-4 - O-5. A corresponding regular arrangement of spheres is thus a cubic packing_ A fine particle content which defines the flow properties of the mixture is reached when the coarse particles are completely embedded in the fine particles. The appropriate ratio of the mass mf of the 00325910/84/s3.00
0 Elsevier Sequoia/Printed
in The Netherlands
146
fine
particles
mass mf + mc of the mixture
to the ,total
(I - Ef)(I - $dc3 “f = 1 + mf + m, = (1 - Ef)(I - gJdc3 i- 2 dc3 Taking the void coarse particles mf mf •l-mc
fraction Ef of the fine as Ef = 0 -5 yields
= 0.31
1
1
(I - Ef)(,
is thus .
6
- I)
parti cl es Surrounding
the
= 30%_
Preparation of the powder samples Keeping in mind the estimates derived in the preceding paragraph the preparation of the powder samples investigated is easily understood. A limestone powder with initial particles size distribution I (compare figure 1) was separated into two fine fractions Fl= F2 ‘and one coarse fraction C. From figure 1 one reads for the coarse fraction C and the fine fraction F2: d
= 45 pm; c50 df50 = 3.4 urn;
2% < 20 urn, 2% > 7 Urn
hence df50’dc50 Only mixtures parti cl es of ox,
= 0,076
< 0,155,
of fractions
F2 and C with mass contents
of fine
5%, lo%, i5%, 30%, 25%, 35%, 130%
were used for
the shear tests,
Shear tests The shear testswereperformedwith a translational shear tester similar to the Jenike tester according to that procedure which has been meanwhile standardised 141 (preconsolidation including twisting, shear unti 1 failure under critical 1 oad, shear under reduced normal load) _ For each mixture three different yield loci have been measured corresponding to three different load levels of consolidation, i.e. maximum principle stresses o1 of (J
1=0.4Ncm
-2
;
0.8 NcmW2;
l-2 NcmW2.
Evaluation of the shear tests with respect to hopper design As described in the 1 i terature C41, with respect to hopper design, shear tests usually are evaluated according to figure 2.
dipm1-o
Figvre
Czrmv__Zative mass distributions
I
fraction
C and
of izitiai: muteriaZ
-two _fine pa_zticZe
_fzwctions
f13
I, eoczxse
f2-
f
T
T&__
-_--
T_-_-
Gl
Figure
2
Evakation
of shear tests with respect
to hopper
CT--
design.
148
Primar_y measurement data are: The consolidating stresses (J , T and the stresses o, T measured under reduced normal loads. The’pargmeters which are relevant for hopper design are obtained from these primary data by the following graphical or computational procedures : (i> (ii)
Maximumprincipal consolidating stress o1 defined by the Mohr circle touching the yield locus and passing through the point (o ‘3 -QUnconfined yield stress f defined by the Mohr circle touchins the yield ‘locus and passifig through the origin. Ds indicated in figure 2, in some cases f can on7y be evaluated by extra701 ati on of measurement dsta.
(iii)
Effective angle of friction d according to Jenike 141 as the tangent of the end Mohr circle passing through the ori gin _ Usua77y. the parameters defined by procedures (i ) to (iii ) are represented as f = f (0 ) and 6 = 6(0 )_ Without do&t) tile Fe1 ations f (ol) and d (o ) reflect materi al nroperties. However, since they are geffned sole13 in the context of a dekign a reasonable interpretation in terms of particle interactions procedure, cannot be expected. This weakness of the two relations in question is confirmed by figures 3 and 4, in which f resoectively 6, are depicted as functions of the fine partic7e conten$‘for ‘the three load leve7s of consol i dati on appl i ed i n experiments.
Q6-
t f,[Ncm? QS-
-G,=
0-L
N
Figure
3
fine
--
1-r. -_v
cm-’
60
1
LO -
-
particle
80
I
I
content
100 [%I
lJnconf5ned yieZd stress f as a ~mction of -f&e particile content and vzaxim ptin&Ze stress during consoZidation, O1'
149
In figure 3, the results of two different evaluation schemes are represented (computational determination of f usi no 1 i near regressi of measured yield loci and direct graphical e6aluation according to figure
on
2).
50 t u-----
&I0
--
6,~ 0.4 N cme2 l.__
---e
/
40
I
20
1
1
40
60
80
particle
content I”/01
-
fine
I
100
Figure 3 as we1 1 as figure 4 reveal strong dependence on the fine parti cl e content, whi ch however, is not accessible to more detailed ana7ysis _ In particular, data which really deserve the designation “material
properti es ” should be - at least within some limits - invariants, definitely not functions of the consolidating stress aplied. Eva1 uation of the parti cl e i nteracti
experiments ons
with
respect
to
parameters
characterizing
In two previously published oapers 15.61 one of the present developed a theory for the yield of cohesive powders, whi ch on consi derati ons concerning parti cl e i nteracti ons _ The main of these theoretical considerations can be summarized as foil pare figure 5) _ (i)
The angle of inclination p of the individual mately considered as parallel straight lines) al friction of the bulk materi al _
authors
is based resul ts
ows
(corn-
yield loci (approxidefines the intern-
r’igurz
5
EvaZvttion descr-ibtng
of shear particZe
tests with respect interactions.
to parameters
of the envel ape of the IfQed esi gnates the angle of inclination end Mohr circles (approximately considered as a straight line), the difference Q - P represents the increase in adhesion forces between particleg with increasins consolidation_ This imolies, that @ itself has no ohysical significance. In narti cul&, @ - p”= 0 means adhesion forces independent of consolidation, wf?ere as comparably high values 4 - P designate strong dependence of adhesion forces on previous co%olidation. (iii ) 1f o,tanp defines the intersection of the effective yield locus with the -r-axis, the factor o designates the three-axial tensile strength of the unconsoli date8 bulk material. The evaluation of all shear tests according to points (i) to (iii) is represented in figure 6, in which p, @ - p and o are depicted as functions of fine particle content. Inethe 1 i oht 8f the considerations described above, these results are accessible&to a reasonable interpretati on _ With coarse particles a a nominal slip plane will be built up by a rather irregular and therefore rough inner surface in the bulk material _ Increasing the fine particle content will gradually smooth out such an inner rupture surface. Therefore it is not surprising, that an increase in the fine part-i cle content gradually decreases the angle of internal fri cti on P of the powder bul k _ With a fine particle content > 301, the coarse particles are comoletely embedded by the fine ones. Therefore it is not surprising that for a fine parti tie- content > 25% the. angle of internal friction remains constant _ (ii)
9 PI % x lO$N cm-*1 *e -9 PI \o t
o~0-40-o-
30
P
25
I---
u
5.
4
6
10
15
_ Figure
6
20
25
30
. i‘& 35
iO0
fine particle content Ioh]
Parameters which are based on co-nsiderat
the increase in adhesion forces with The difference $e- p represents increasing consolidation. For the coarse fraction, as expected , p = 0 is observed. With increasing fine particle content the Q up to a fine particle content of d?fference Q - p rises linearly are about 30%. Fgr fine oarti cle contents > 30% the coarse particles completely embedded in the fine ones. Assuming a same surface roughness for the fine grained particles as confor the coarse grai ned particles ) mixtures with a fine particle tent > 30% should show the same numerical value of & - p as for the fine particle fraction. This consideration coincidekewi th the experimental observation that within the experimental accuracy for a fine particle content > .25% no further change in 0 - p is observed. As described in detail in t63 the three-axial te&ile strength o. of an unconsolidated powder i related to the average tensile force per diameter). !Iith df/dc = 0.08 and contact H by o. s Ho/d 3 (d: particle assumingt ?i esamesurface roughness of fine qrained as for coarse grained particles, for the coarse fraction, the numerical value of u should be -z 1% of that for the fine particle fraction. For the coar!!e particles completely embedded by the fines, i.e. for a fine particle content > 30% the numerical value of (T should be that for the fine particle fraction. Again, this expecta?ion is confirmed by figure 6. From the interpretation of the experimental results depicted in figure 6, it can be concluded that the materi al data p, $e - p and u. therefore, which are derived from considerations on particle interacti on reasonably define the behaviour of cohesive powders.
152
Calculation
of
data
for hopper design from measured
material
properties
particle Obviously, the materi al proporties P, Cp - p and o define The practical %opli cabilify of these parai nteracti ons reasonably. meters can be demonstrated by cal cul ati on of the functions fc(ol) and a(ol> from measured values P, ee - P and oo_ As described
f,
in
detail
in
[6l
2(siWe - sinp) = (1 i- slnee)(l - slnp)
As can be read from relation between d, sind
=
comparison p. Qe - P,
it
holds 2cos@e(l
+ sinp)tano
ol
+ (1 + sln@,)(l
of
figures and ol:
a0
- sinp)
2 and 5 it
holds
(1)
aO-
for
the
1
(2)
1 + sirMe sin&.e + tanp
COS~~(~~/O~)
-
1
The values of f and d calculated from equations (1) and (2) using in figure 6 are reoresented measured data pt Q - p and o as depicted in fi aures 7 and Seresoecti ve?y as functions of the fine particle content for the three consolidating load levels investigated, i.e. in these figures renresent values 0.8, 1.2 Ncm-2. The points “1 = 0.4, calculated from measurements of p, Q - p and o . Full lines and dotted 1 ines renresent directly measured da%!a as depic?ed in fi cures 3 and 4 respecti vely _ one observes compl ete equivalence From figures 7 and 8 respectively, of the procedure proposed here in comparison to direct experimental determination of the functions f (al) and 8(01), respectively _ The advantage of this apparentlyCci rcunstanti al procedure is obvi ous from fioure 6, Since p, Q - p and o are comparably simple, singlevalued Functions of fine Farti cl e cogtent, even the behaviour of experimentally would be described mixtures , which were not investigated with sufficient accuracy. Concl usi ons The results
presented
here
suggest
the
foll
owing
conclusions
:
(i>
For better understanding of cohesive powders, it makes sense to investi oate well defined mixtures of cohesionless and cohesive materials because the results obtained in this manner can be interpreted rather simply _
(ii)
Parameters defined from consi derati ons of parti cl e interactions are accessible to reasonable exol anati on with changes thereof account for different fine particle contents_
(iii)
Since the parameters defined from considerations on parti cl e interactions show a simple and single-valued dependence on fine particle content, the behaviour of not investigated mixtures can be estimated safely.
-
to
153 Q6t f, [Ncni21
c-y_
//rn
/ J-
--
---_
---_
G, =1_2Ncm-2
-1_
--
Q5/----_
m
---__ Gr = 0.8N cm7
O---
--0
---
I-
---_
-Nr~-n-~
/--o--
-
~igupe
7
fme
particle
content
[%i
Uizeon_fined yieZd stress f as _ficr~ctionof ~%xe partZcZe eoxmcteria2 c7ata P, Q, - p, ~~ tent, ca2cirZated -from mea&-zeE conso2id&ion, CT-_ and maxZmxm pri-nciple stress &?-ng L
6, = O-4 NcmT2
ii
i0
2'0 -
Figure
8
$0 fine
particle
ab content
100 ID/o1
to Jenike [d] as Q Ef_fective ang2e oj=-_friciion 6, aceopdfng fwzetion of r-ine partie2e coz-kezt, eaZeu2cted _fz-ommeasuz*ed mater-La2 data p, Qe - p, o. and ma&mum ptincipo2 stress dxring eonso2ic&ztzon, 03-
154
Acknowledgement The experimental results presented here have been gained during a visit of one of us (M. Nywlt) at Loughborough University, United Kingdom. The authors thank R. Akers and B_ Scarlett for supervision of that work. References Kurz H-P, and M&z G.: Powder Technology, 11 (1975) 37-40. FiirllC.: Freiberger Forschungshefte, A 634 (1980) 127-141. Schmidt P,: Aufbereitungstechnik. 5 (1964) 335-365, Jenike A.W.: Storage and flow of solids, Utah Univ.Eng.Exp.Stn. Bull. 123 (1964). 5. Molerus 0.: Powder Technol., 12 (1975) 259-275. 6- Molerus 0,: Powder Technol., 20 (1978) 161-175, 1. 2. 34.
145
745 - 154
THE INFLUENCE OF THE FINE PARTICLE CONTENT ON THE FLOW BEHAVIOUR OF BULK MATERIALS 0. Molerus and M_ Nywlt Lehrstuhl fi_ir Mechanische Verfahrenstechnik, University of Erlangen-Ntirnberg, tdestGermany
Abstract Experimental results obtained from shear tests with well defined mixtures of coarse grained (=cohesionless)and fine grained (=cohesive) materials are presented. Evaluation of the experiments with respect to parameters based on considerations on particle interactions results in a meaningful interpretation of the observed material behaviour.
Introduction Some years ago, Kurz and Miinz 111 oresented experimental results of shear tests on limestone Fowders with different widths of particles size distribution. Even small changes in the fine particle content gave measurable changes in cohesiveness.
Pore recently, Fiirll 121 investigated the maximum permissible fine
particle content which does not affect the flow behaviour of coarse, cohesionless materials. are discussed, which were In the present uaper, experimental results obtained with different mixtures of fine grained, cohesive and coarse grained, cohesionless powders. The aim of this paper is to interpret the experimental results in terms of parameters, which define particle interactions_ Some simple geometrical considerations To estimate critical size and mass ratios of fine grained and coarse grained particles, binary mixtures of spheres are considered_ In a random packing of particles, local changes of the coarse particle content have to be admitted. The maximum diameter d of a sphere passing through the most dense packing of spheres of diameter d, is ~I;f;Wi;;~gtF/dg = 0,155 E31. r ndom packings of coarse grained irregular particles are E = O-4 - O-5. A corresponding regular arrangement of spheres is thus a cubic packing_ A fine particle content which defines the flow properties of the mixture is reached when the coarse particles are completely embedded in the fine particles. The appropriate ratio of the mass mf of the 00325910/84/s3.00
0 Elsevier Sequoia/Printed
in The Netherlands
146
fine
particles
mass mf + mc of the mixture
to the ,total
(I - Ef)(I - $dc3 “f = 1 + mf + m, = (1 - Ef)(I - gJdc3 i- 2 dc3 Taking the void coarse particles mf mf •l-mc
fraction Ef of the fine as Ef = 0 -5 yields
= 0.31
1
1
(I - Ef)(,
is thus .
6
- I)
parti cl es Surrounding
the
= 30%_
Preparation of the powder samples Keeping in mind the estimates derived in the preceding paragraph the preparation of the powder samples investigated is easily understood. A limestone powder with initial particles size distribution I (compare figure 1) was separated into two fine fractions Fl= F2 ‘and one coarse fraction C. From figure 1 one reads for the coarse fraction C and the fine fraction F2: d
= 45 pm; c50 df50 = 3.4 urn;
2% < 20 urn, 2% > 7 Urn
hence df50’dc50 Only mixtures parti cl es of ox,
= 0,076
< 0,155,
of fractions
F2 and C with mass contents
of fine
5%, lo%, i5%, 30%, 25%, 35%, 130%
were used for
the shear tests,
Shear tests The shear testswereperformedwith a translational shear tester similar to the Jenike tester according to that procedure which has been meanwhile standardised 141 (preconsolidation including twisting, shear unti 1 failure under critical 1 oad, shear under reduced normal load) _ For each mixture three different yield loci have been measured corresponding to three different load levels of consolidation, i.e. maximum principle stresses o1 of (J
1=0.4Ncm
-2
;
0.8 NcmW2;
l-2 NcmW2.
Evaluation of the shear tests with respect to hopper design As described in the 1 i terature C41, with respect to hopper design, shear tests usually are evaluated according to figure 2.
dipm1-o
Figvre
Czrmv__Zative mass distributions
I
fraction
C and
of izitiai: muteriaZ
-two _fine pa_zticZe
_fzwctions
f13
I, eoczxse
f2-
f
T
T&__
-_--
T_-_-
Gl
Figure
2
Evakation
of shear tests with respect
to hopper
CT--
design.
148
Primar_y measurement data are: The consolidating stresses (J , T and the stresses o, T measured under reduced normal loads. The’pargmeters which are relevant for hopper design are obtained from these primary data by the following graphical or computational procedures : (i> (ii)
Maximumprincipal consolidating stress o1 defined by the Mohr circle touching the yield locus and passing through the point (o ‘3 -QUnconfined yield stress f defined by the Mohr circle touchins the yield ‘locus and passifig through the origin. Ds indicated in figure 2, in some cases f can on7y be evaluated by extra701 ati on of measurement dsta.
(iii)
Effective angle of friction d according to Jenike 141 as the tangent of the end Mohr circle passing through the ori gin _ Usua77y. the parameters defined by procedures (i ) to (iii ) are represented as f = f (0 ) and 6 = 6(0 )_ Without do&t) tile Fe1 ations f (ol) and d (o ) reflect materi al nroperties. However, since they are geffned sole13 in the context of a dekign a reasonable interpretation in terms of particle interactions procedure, cannot be expected. This weakness of the two relations in question is confirmed by figures 3 and 4, in which f resoectively 6, are depicted as functions of the fine partic7e conten$‘for ‘the three load leve7s of consol i dati on appl i ed i n experiments.
Q6-
t f,[Ncm? QS-
-G,=
0-L
N
Figure
3
fine
--
1-r. -_v
cm-’
60
1
LO -
-
particle
80
I
I
content
100 [%I
lJnconf5ned yieZd stress f as a ~mction of -f&e particile content and vzaxim ptin&Ze stress during consoZidation, O1'
149
In figure 3, the results of two different evaluation schemes are represented (computational determination of f usi no 1 i near regressi of measured yield loci and direct graphical e6aluation according to figure
on
2).
50 t u-----
&I0
--
6,~ 0.4 N cme2 l.__
---e
/
40
I
20
1
1
40
60
80
particle
content I”/01
-
fine
I
100
Figure 3 as we1 1 as figure 4 reveal strong dependence on the fine parti cl e content, whi ch however, is not accessible to more detailed ana7ysis _ In particular, data which really deserve the designation “material
properti es ” should be - at least within some limits - invariants, definitely not functions of the consolidating stress aplied. Eva1 uation of the parti cl e i nteracti
experiments ons
with
respect
to
parameters
characterizing
In two previously published oapers 15.61 one of the present developed a theory for the yield of cohesive powders, whi ch on consi derati ons concerning parti cl e i nteracti ons _ The main of these theoretical considerations can be summarized as foil pare figure 5) _ (i)
The angle of inclination p of the individual mately considered as parallel straight lines) al friction of the bulk materi al _
authors
is based resul ts
ows
(corn-
yield loci (approxidefines the intern-
r’igurz
5
EvaZvttion descr-ibtng
of shear particZe
tests with respect interactions.
to parameters
of the envel ape of the IfQed esi gnates the angle of inclination end Mohr circles (approximately considered as a straight line), the difference Q - P represents the increase in adhesion forces between particleg with increasins consolidation_ This imolies, that @ itself has no ohysical significance. In narti cul&, @ - p”= 0 means adhesion forces independent of consolidation, wf?ere as comparably high values 4 - P designate strong dependence of adhesion forces on previous co%olidation. (iii ) 1f o,tanp defines the intersection of the effective yield locus with the -r-axis, the factor o designates the three-axial tensile strength of the unconsoli date8 bulk material. The evaluation of all shear tests according to points (i) to (iii) is represented in figure 6, in which p, @ - p and o are depicted as functions of fine particle content. Inethe 1 i oht 8f the considerations described above, these results are accessible&to a reasonable interpretati on _ With coarse particles a a nominal slip plane will be built up by a rather irregular and therefore rough inner surface in the bulk material _ Increasing the fine particle content will gradually smooth out such an inner rupture surface. Therefore it is not surprising, that an increase in the fine part-i cle content gradually decreases the angle of internal fri cti on P of the powder bul k _ With a fine particle content > 301, the coarse particles are comoletely embedded by the fine ones. Therefore it is not surprising that for a fine parti tie- content > 25% the. angle of internal friction remains constant _ (ii)
9 PI % x lO$N cm-*1 *e -9 PI \o t
o~0-40-o-
30
P
25
I---
u
5.
4
6
10
15
_ Figure
6
20
25
30
. i‘& 35
iO0
fine particle content Ioh]
Parameters which are based on co-nsiderat
the increase in adhesion forces with The difference $e- p represents increasing consolidation. For the coarse fraction, as expected , p = 0 is observed. With increasing fine particle content the Q up to a fine particle content of d?fference Q - p rises linearly are about 30%. Fgr fine oarti cle contents > 30% the coarse particles completely embedded in the fine ones. Assuming a same surface roughness for the fine grained particles as confor the coarse grai ned particles ) mixtures with a fine particle tent > 30% should show the same numerical value of & - p as for the fine particle fraction. This consideration coincidekewi th the experimental observation that within the experimental accuracy for a fine particle content > .25% no further change in 0 - p is observed. As described in detail in t63 the three-axial te&ile strength o. of an unconsolidated powder i related to the average tensile force per diameter). !Iith df/dc = 0.08 and contact H by o. s Ho/d 3 (d: particle assumingt ?i esamesurface roughness of fine qrained as for coarse grained particles, for the coarse fraction, the numerical value of u should be -z 1% of that for the fine particle fraction. For the coar!!e particles completely embedded by the fines, i.e. for a fine particle content > 30% the numerical value of (T should be that for the fine particle fraction. Again, this expecta?ion is confirmed by figure 6. From the interpretation of the experimental results depicted in figure 6, it can be concluded that the materi al data p, $e - p and u. therefore, which are derived from considerations on particle interacti on reasonably define the behaviour of cohesive powders.
152
Calculation
of
data
for hopper design from measured
material
properties
particle Obviously, the materi al proporties P, Cp - p and o define The practical %opli cabilify of these parai nteracti ons reasonably. meters can be demonstrated by cal cul ati on of the functions fc(ol) and a(ol> from measured values P, ee - P and oo_ As described
f,
in
detail
in
[6l
2(siWe - sinp) = (1 i- slnee)(l - slnp)
As can be read from relation between d, sind
=
comparison p. Qe - P,
it
holds 2cos@e(l
+ sinp)tano
ol
+ (1 + sln@,)(l
of
figures and ol:
a0
- sinp)
2 and 5 it
holds
(1)
aO-
for
the
1
(2)
1 + sirMe sin&.e + tanp
COS~~(~~/O~)
-
1
The values of f and d calculated from equations (1) and (2) using in figure 6 are reoresented measured data pt Q - p and o as depicted in fi aures 7 and Seresoecti ve?y as functions of the fine particle content for the three consolidating load levels investigated, i.e. in these figures renresent values 0.8, 1.2 Ncm-2. The points “1 = 0.4, calculated from measurements of p, Q - p and o . Full lines and dotted 1 ines renresent directly measured da%!a as depic?ed in fi cures 3 and 4 respecti vely _ one observes compl ete equivalence From figures 7 and 8 respectively, of the procedure proposed here in comparison to direct experimental determination of the functions f (al) and 8(01), respectively _ The advantage of this apparentlyCci rcunstanti al procedure is obvi ous from fioure 6, Since p, Q - p and o are comparably simple, singlevalued Functions of fine Farti cl e cogtent, even the behaviour of experimentally would be described mixtures , which were not investigated with sufficient accuracy. Concl usi ons The results
presented
here
suggest
the
foll
owing
conclusions
:
(i>
For better understanding of cohesive powders, it makes sense to investi oate well defined mixtures of cohesionless and cohesive materials because the results obtained in this manner can be interpreted rather simply _
(ii)
Parameters defined from consi derati ons of parti cl e interactions are accessible to reasonable exol anati on with changes thereof account for different fine particle contents_
(iii)
Since the parameters defined from considerations on parti cl e interactions show a simple and single-valued dependence on fine particle content, the behaviour of not investigated mixtures can be estimated safely.
-
to
153 Q6t f, [Ncni21
c-y_
//rn
/ J-
--
---_
---_
G, =1_2Ncm-2
-1_
--
Q5/----_
m
---__ Gr = 0.8N cm7
O---
--0
---
I-
---_
-Nr~-n-~
/--o--
-
~igupe
7
fme
particle
content
[%i
Uizeon_fined yieZd stress f as _ficr~ctionof ~%xe partZcZe eoxmcteria2 c7ata P, Q, - p, ~~ tent, ca2cirZated -from mea&-zeE conso2id&ion, CT-_ and maxZmxm pri-nciple stress &?-ng L
6, = O-4 NcmT2
ii
i0
2'0 -
Figure
8
$0 fine
particle
ab content
100 ID/o1
to Jenike [d] as Q Ef_fective ang2e oj=-_friciion 6, aceopdfng fwzetion of r-ine partie2e coz-kezt, eaZeu2cted _fz-ommeasuz*ed mater-La2 data p, Qe - p, o. and ma&mum ptincipo2 stress dxring eonso2ic&ztzon, 03-
154
Acknowledgement The experimental results presented here have been gained during a visit of one of us (M. Nywlt) at Loughborough University, United Kingdom. The authors thank R. Akers and B_ Scarlett for supervision of that work. References Kurz H-P, and M&z G.: Powder Technology, 11 (1975) 37-40. FiirllC.: Freiberger Forschungshefte, A 634 (1980) 127-141. Schmidt P,: Aufbereitungstechnik. 5 (1964) 335-365, Jenike A.W.: Storage and flow of solids, Utah Univ.Eng.Exp.Stn. Bull. 123 (1964). 5. Molerus 0.: Powder Technol., 12 (1975) 259-275. 6- Molerus 0,: Powder Technol., 20 (1978) 161-175, 1. 2. 34.