CRUSHING STRENGTH OF ZINC
OXIDE AGGLOMERATES H.
P. M E I S S N E R , A .
S. M I C H A E L S , A N D
R O B E R T
K A I S E R ’
Massachusetts Institute of TeLhnoloty, Cambridge, Masr.
By action o f cohesive forces o f the van der Waals type, fine particles of zinc oxide spontaneously form agglomerates. Assuming the constituent particles all to b e spherical and of the same diameter d, then as a first approximation the average coordination number of these particles can b e calculated from + r the volume fraction of solids in the agglomerate. The force which an agglomerate of diameter D can support i s shown to be a unique function of D, d, and 4. This relation i s shown to be in reasonable agreement with experimental data for two different zinc oxide powders. VALENCE FORCES of attraction exist between particles in a powder bed. These forces are great enough so that as particle size is reduced, the number of cross-links between particles per unit volume of powder bed increases to the point where agglomerates of measurable size, density, and crushing strength appear. When the constituent particles are relatively large (as with ZnO particles of approximately 1 micron in diameter). then the physical properties of these agglomerates are not much altered by agitation. \\’hen the particles are relatively small ( Z n O of 0.2 micron and less), then tumbling in a rotating drum progressively increases the density and strength of these agglomerates (6, 9 ) . This process, which is known industrially as pelletization, results in 110 change in prime particle size. The purpose here is to consider the factors affecting the strength of these agglomerates. Consideration is limited to Z n O powders. The attractive forces between two spherical particles of equal diameter was shown by Lifschitz (7) to be: ECONDARY
where F is the interparticle force. d is diameter. and a is the distance of closest approach between neighboring surfaces. The constant B depends on particle composition and the nature of the surrounding medium, and has a value of about 10-19 erg-cm. for glass and similar polar materials (5) in air. For a rigid assembly of such cohesive spheres, randomly packed. the tensile strength in the z direction has been shown by Rumpf ( 7 7 ) to b e .
wherr 9 is the volume fraction solids and F is the mean bonding force between spheres. The coordination number k is the number of contact points between a single sphere and its neighbors. In a regular array. k is the same for all spheres, while in an irregular array k varies from sphere to sphere. For irregular arrays. the average coordination number k is drfined as : (3’)
.4 reasonable assumption is that the bonding force of attIaction between any two spherical particles is unaffected by the presence of neighboring particles. The tensile strength of an a ~ s e m b l yof spheres of equal size, in which the packing arrangement is macroscopically homogeneous. can then be obtained by combining Equations 1 and 2 : (4) I n developing Equation 4: all constants (including a , which is assumed constant for a given particle system) are combined into the single constant M . This relation Lvould. of cour:e. apply to an agglomerate made u p of equal-sized spheres. Average Coordination Number. T h e term “open-packed systems” is used here to denote those relatively loose arrangements in which at least some of the spheres would be free to move with respect to each other were they not constrained from doing so by a bonding force operating at each point of contact between spheres. I n “close-packed systems.” spheres would not be free to move in relation to each other. even in the absence of bonding forces. Regular and irregular closepacked systems of equal-sized spheres have been extensively studied ( 2 ) .with the finding that. as a good first approximation. k is a unique function of @. Manegold et ai. (8)calculated values of @ for regular close-packed arrays having G values of 6, 8, 10. and 12, as presented in Table I: and proposed the following empirical equation to fit these points :
Both the points and the equation are shown graphically in Figure 1. Normally. when smooth noninteracting spheres are thrown together in a container. irregular systems rather than such regular arrays are obtained. Thus. Smith et ai. (72) poured lead shot into a large vessel and filled the voids with acetic acid. Since lead acetate formed over the sphere surfaces except a t the points of contact. the cobrdination numbrr of the spheres in the system could be counted directly. ‘l‘he observed values of k lis. 9 for theFe irregular close-packed arrays were found to follo\v the following equation:
k where ,VIis the number of spheres in the array with a coordination numbrr of k ,
’ Prrsrnt addrrss, The A.I. 202
\V, Krllogg Go.. .Jersey City, N. J.
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
=
26.49 - (10.73/@)
(5)
Open-packed system? have been studied less extensively than close-packed systems. especially a t low values of 9 . The evidence available, however, indicates that there is much 9 than for close-packed greater scatter for data points of
x
LJJ.
systems. Y'hus. Heesch and Laves ( 3 )studied various regular packings having k values of 3 and 4. For a coordination number of 4, the simplest is the diamond packing Lvherein each sphere touches four other spheres. whose centers form the verticrc of a regular tetrahedron. For this arrangement, 0 is 0.338. .l'hey alqo noted that, ~vhilemaintaining a i,value of 4. theae spherer could be displaced to form tetrahedrons \vith a11 equilatrral triangle a t the base and isosceles triangles along the face,. re~ulringin I;) values as loiv as 0.123. Similarly, for i, valur. of 3. various 1)acking.i are described in ivhich Q ranges from 0.0.56to 0.1 8.5. hlelrnore (70) extended these results by developing othrr complex constructions for a k value of 3. for uhich Q is a, lo\\. a \ 0.042. These results are again shotvn in Figure 1 . Open-Packed Systems. To extend the d a m available for open-packed systems. the authors propose three sets of packing arrangements of equal-sized spheres. here named "open-beam cubic packing." "open-beam hexagonal packing:" and "openbeam dodecahedral packing." I n open-beam cubic packing. the splieres are arrangrd to form the edges of a cube. Lvith n spheres along each edge betiveen any t i i o corner spheres. Each cubical cell is therefore bounded b) eight corner spheres [each shared by eight neighboring cells) and 12n edge spheres (each shared by four neighboring cells) ; hence. the spheres per cubical cell are (1 f 3 n ) . The corner spheres have a k value of 6: and the orher spheres along the edges have a X value of 2 ; hence: by Equation 3: the average coordination number is : /.
=
6 f 6n 1 3n ~~~~~
+
(7)
From the geometry of the basic cell: which obviously has (1 7 1 ) spheres per edge :
+
I n open-beam hexagonal packing, \vhich can be vielved as a n expansion of close-packed tetragonal packing, the spheres form the edges of equal-sided parallelepipeds having vertex angle. of 60' and 120'. Each unit cell again is bounded by eight corner cells (each shared by eight neighboring cells) and 1212 edge sphere: (each shared by four neighboring cells) ; hence. the spheres per cell are again (1. f 3 n ) . The corner spheres again have a k value of 6. each of the six neighbors of each corner sphere have a k value of 4 , and the remaining spherea again have a X value of 2. 'I'hus. the average coordination number is :
Lvhile the volume fraction is:
(1
Q =
3
+ 3.)
+
1)3
Equations 9 and 10 are valid only when 72 is a 2 or greater. \$'hen the structure involves t\vo spheres on a side (a situation for lvhich n is zero, but to which these t\vo equations d o not apply). this arrangement reduces to close-packed tetragonal packing for ivhich Manegold aho\ved k to be 12 and Q to be 0.74. TVhen 'i is unity. then all spheres have a coordination number of 6 and Q is 0.35:agreeing \vith the calculations of Heesch and Laves. \$'hen n become? infinite. then k is again 2 and Q is zero. In open-beam dodecahedral packing, the spheres are arranged to form the edges of a regular dodecahedron. This polyhedron has 12 pentagonal faces, 30 edge, and 20 corners, There are again TI spheres along each edge betLveen each t\vo corner spheres: hence there are 20 corner spheres (each shared by four neighbors) and 30n edge spheres (each .hared by three neighbors). The number of spheres per unit cell therefore equal, ( 5 10n). Each corner sphere has a coordination number of 4: lvhile the edge spheres have a coordination number of 2 ; hence :
+
\t-hen ri is zero. thi> arrangement reduces to simple cubic packing, and Q is 0.524. \$'hen n is infinite: thiq arrangement become5 a chain of spheres through empty space. for \tzhich k i \ 2 and Q is zero.
'The volume of a regular dodecahedron ( 7 ) is 8 O X 3 . where L is the lengi h of a n edge; hence for a unit cell.
0 325 ( 2 n 9 =
E -1
i'
1 01
02
03
04
(n
+
+ 1) ~~
Coordination Number Calculations. Different values of of k and 9 \ v e x calculated from Equations through 12 for various values of n. Lvith resuirs as in Table I . Figure 1. on Ivhich a r r precrnred graphically all the results of Tables I and 11. shoivs thar i,is a function not only of 9 but of the packing arrangemenr. Particularly in this open-packed range, therefore. 2. is onl) approximately fixed \vhen 0 is fixed. .A11 thrse scattered points nevertheless fall reasonabl! close to line 3 of I:igure 1. Jvhich can be represented by the expression :
T 3:e- beom *etrogorcl, Eqs. 9 and IC 0;en beam dodecchedro1,Eqs. I a n d I2
I
5
~~
0 5
05
9 Figure 1 . Values of coordination number fraction of solids 4
07
i I 0 8
-
k
vs. volume
Inspection shoivs this line to fit the data in the close-packed range about as \vel1 as curves 1 and 2> Lvhich are graphical reprerentations of the Lfanegold and Smith Equations 5 and 6: respFctively. Similarly, in the open-packed range. there a p pears to be little to choose benveen thr Xlanegold Equation 5 and Equation 13 proposed here dowm to 0 values of about 0 . 2 . .4t lo\ver Q values. Equation 1 3 indicates that as Q approaches VOL. 3
NO. 3
JULY
1964
203
Table 1.
Calculated and Experimental Values of Close-Packed Arrays
Coordznatzon yo. (x)
Tllppe of Packing
Rrgular arrays" Simple cubic Cubical tetrahedral Tetragonal spheroidal Pyi amidal retrahfdrdl Irregular arrays5
and p for
Volume firaction of Solzds ( @ )
?r/6 = 0 5236 ?r/3 4\/3= 0 6046 2n/9 = 0 6981 7r/3 42=0 7405 ?r/3 4 2 = 0 7405
6
8 10
12 12 6 92 7 34 8 06 9 51 9 14
Vomenclature of I)rreszewzez ( 2 )
a
k
0 0 0 0 0
553 560 574 628 641
D a t a of Y m t h et al ( 72)
Average Coordination Number vs. Volume Fraction
Table II.
of Solids for Different Packing Arrangements
Oben-Bran2 Cubic
Oben-Beam Hevrieonal
n 524
n 1 2 3 4
3 2 2 2 2 2
8 m
57 4 31 24 0
. . .
6 4 3 3 2 2
262 136 082 055 018
0 0 0 0 0 0
0
0 524
28 6 23 64 0
0 0 0 0 0
272 164 109
036 0
____Open-Beam Dodecahedral ~~~~
~
11
0 1
2 3 4 8
k (Eq. 7 7 ) 4 2 67 2 40 2.29 2 22
4 ( E q . 7.3) 0 325 0 122 0 061
2 12
0,0076
2.0
0.0
m
n 036 0.023
zero, k goes to 2 as in the open-beam packings discussed here. 'l'he Manegold relation, on the other hand, indicates that k goes to 3.1 as 9 goes to zero; hence, i t does not apply to openbeam packing and so is niore limited than Equation 13. The Smith Equation 6 obviously was not intended to apply to open-packed systems. Figure 1 shows that line 3 representing Equation 13 fits the points for open-beam dodecahedronal packing especially well. This type of packing is perhaps a more probable open-beam configuration than the other two! since it requires the crossing of only two rather than three beams at a single point. Agglomerate Crushing Strength
A sphere will fail when subjected to a critical load by compression between two flat parallel plates. Hertz (I) found that such failure occurred in tension a t the circumference of the circular surface of contact betlveen the slightly flattened spheres and the flat press platens. H e related the tensile strength u to P,the load causing failure, as follolvs: u =
0 6 1) '5: :I(
(
E2P D 2 )l
3
where D is sphere diameter. E is the modulus of elasticity. and r is Poisson's ratio. The modulus of elasticity of a n agglomerate can be expected to vary in some complex fashion with particle packing geometry. solids volume fraction, and the character of the interparticle bonds. If it is assumed that all strain is confined to the particles comprising the agglomerate and that the normalized stress vector in pure tension is parallel to the externally applied stress. thrn the true stress on the particle network will be inversely proportional to d, and, hence, the modulus of elasticity will be approximately proportional to $, By substitution of Equations 4 and 13 into Equation 14, setting E proportional to @, and combining constants :
Equations 9 and 10 not applicable-see text
a
According to this equation, the crushing strength of an agglomerate is a function solely of d, D, and 6.
I IO1
025
'
'
03
'
035
, 04/
,
045
,
~
05
i
IO
055
YJ Figure 2. Experimental values of crushing strength vs. volume fraction of solids Each point is a n a v e r a g e of 15 determinations
204
I&EC PROCESS DESIGN A N D DEVELOPMENT
Experimental Procedure. The value of 9 , namely the volume fraction of solids in the agglomerate. was determined from the measured average diameter and weight of the pellet. .4gglomerate strength was measured by crushing between two flat parallel plates. T h e apparatus used was a slight modification of a balance-buret arrangement previously described (73). .4n agglomerate was placed between two parallel microscope slides, the upper one held fixed in space, the lower one resting on one pan of an equilibrated overhead beam laboratory balance. The agglomerate was loaded by allo\ving water to run from a graduated buret into a beaker placed on the other balance pan. T h e load needed to cause failure was determined from the buret reading. T h e end point was easy to determine as there was very little agglomerate deformation prior to failure. Agglomerates of different values of diameter and volume fraction of solids were obtained by screening from beds of Z n O which had been tumbled for various periods of time in rotating glass bottles. T h e materials used (Kadox 72 and St. Joe G.L. 42) and the tumbling procedure have been described ( 9 ) . The particle diameter measurements (Kadox, 0.13 p ; St. Joe. 0.26 p ) were based upon the manufacturer's B.E.T. surface area measurements. T h e agglomerates studied ranged in size from -16 to +20 mesh (about 1 mm.) to -50 to +70 mesh (about 0.25 m m . ) , while values of ranged from 0.54 to 0.25. It was not possible to study agglomerates with smaller values of @ by the experimental technique used. Results. T h e experimental points (each a n average of 15 determinations) are presented in Figure 2, plotted as log
Ap@orne-s‘e
X -IS ~
L,
0 -30
Size
+ 2 0 mesh + 40 mesh
I
distribution, which would result in a greater strength than predicted for particles all of the same size. I n consequence, the values of K for these systems cannot be predicted a t this time.
1
A - 4 0 t 5 0 mesh
I I
Conclusions
T h e strengths of the agglomerates of the Z n O studied here were found to be a unique function of the particle diameter. agglomerate diameter, and volume fraction of solids within the agglomerate. T h e relationship developed can presumably be extended to other powder systems. Nomenclature a
=
B =
n = d
=
E =
F = K = k
=
L = .hf =
-v, = n
=
(P,.’D2)us. 9. T h e use of semilog coordinates here is merely for convenience, but it is interesting that both lines are straight a n d show the same slope. These data points are replotted in Figure 3 on the coordinates (P;’D2) us. the function ( 9 e 7 , 2 $ ) ) , \vhich by Equation 1 5 should yield a straight line. T h e fact that the points do, in fact, fall reasonably on a straight line in Figure 3 lends support to Equation 1 5 , and the assumptions on which it is based-namely, that a n agglomerate is made u p of a random chain network; k is related to q~ by Equation 13; the Rumpf Equation 2 applies to agglomerature systems; E is proportional to @; and (I and r are independent of 9. ’The slope of the line for the Kadox material in Figure 3 is twice that of the St. Joe material. T h e derivation of Equation 15 indicates that the value of K-;. namely, the slope of these lines-is a sensitive function of both d and u . No method for determining u is available in these powder systems. Moreover. the particles studied were not spheres, a n d there is uncertainty as to the proper value of the diameter to use in this equation. V a n der Waals forces between particles are a function of their shape ( , 5 ) ; hence K may be a weaker function of particle dimension for irregularshaped particles than for spheres. Again, the number of contaci points is higher in systems having a wider paiticle size ~
P = 7
=
u
=
m =
distance of closest approach between particle surfaces, cm. consiant in Et, 1 . erg-cm. diameter of a sphere or a spherical agglomerate, c‘m. or mm . diameter of particle, c m . modulus of elasticit)-, d p e s , sq. cm. interparticle attractive force. dynes constant in Eq. 13. grams force. sq. m m . coordination iiumber. dimensionless length of edge in dodecahedron, crn. constant in E,q. 4, dynes, cm. number of spheres of species j, dimensionless number of spheres bet\ieen two corner spheres along an edge in an array. dimensionless total compressive force causing failure, grams (force) Poisson‘s ratio. dimensionless tensile strength: dynes sq. c m . volume fraction solids: dimensionless
literature Cited
(1) Coxeter, H. S. M.. “Rrgular Polytopes,” p. 295. Methurn Br Co., London, 1948. (2) Deresiewiez. H., “Advancrs in Applied Mechanics,’‘ Vol. V, pp. 233-306, Academic Press, New York, 1958. (3) Heesch. H.: Laves, F., Z. Krzst. 85, 443 (1333). (4) Hertz, H., J . Reine Angew. M a t h . , p. 92 (1881). (5) Jongh, J. G. V. de, doctoral dissertation, Utrecht Cniv., The Netherlands. 1958. (6) Kaisrr, R., Sc.D. thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1962. (7) Lifschitz, E. M.. Soviet Phys. J E T P 2 9 , 94 (1954). (8) Manegold, E., Hofmann. R., Soff. K.. Koliozd-Z. 56, 142 (1931). (9)’ Meissner, H . P., Michaels. A. S.,Kaiser. R.. IXD.ENG.CHEM., PROCESS DESIGN DEVELOP. 3.197 (1964). (10) Melmore, S., Suture 149, 212%669 (1942). (11) Rumpf. H., Chem.-1ngr.-Tech. 30, 144 (1958). (12) Smith, W. 0..Foote, P. D., Busang. P. F.. Phys. Rer. 36, 524 iis?n\ \*’-”/’ (13) Sweitzer, C. D.: Columbian Carbon Co.. Princeton, 9. J., private communications. 1964; Chem. Eng. A V ~ w32, s 4287 (1954).
RECEIVED for review July 10. 1963 ACCEPTED December 9, 1963
VOL. 3
NO. 3
JULY
1964
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