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The cylindrically ordered packing of equal spheres
The Cylindrically Ordered Packing of Equal Spheres F_ A . ROCKE Australian Atomic Energy Commission Research Establishment, Private Mail Rag, Sutherland N_S_II' . 2232 (Australia) (Received May 13, 1970)
SUMMARY A homogeneous ordered packing of equal spheres can be formed by vibration in vertical cylindrical vessels with certain base shapes. The shapes are described. The packing consists of a concentric array of cylindrical "shells" of spheres which fills the vessel. The spheres in a shell are arrayed hexagonally, and their fitting against those in an adjacent shell is cyclic and undulatory with respect to a circle concentric with the shells. A theory based on the observed structure yields infinite-bed valuesforsome parameters of the packing which are consistent with extrapolated experimental values- In particular, the infinite-bed void fraction is about 0.29- The number of shells in a bed with given vessel-to-sphere diameter ratio is predictable over certain ranges of the ratio . INTRODUCTION
When a bed of equal spheres is packed randomly in a vertical cylindrical vessel, those lying against the wall tend to form a peripheral "shell" in which there is some ordering and in which the packing is closer than that in the interior of the bed_ During an investigation of this effect and its relation to pebblebed reactors, it was found that an ordered packing, consisting entirely of such shells& in a concentric array (Fig. 1), may be formed by vibration in vertical cylindrical vessels with certain base shapes_ This array was called "the cylindrically ordered packing of equal spheres". The structure of the packing was determined by external and internal examination of beds . Experimental beds were prepared, with various values of the vessel-to-sphere diameter ratio, and the measured values of some parameters were studied theoretically in relation to the structure and this ratio. After the work had been completed, that of McGearyt came to the author's attention . McGeary Ponder Technology - Elsevier
Sequoia S_A., Lausanne
Fig . 1_ Concave conical "free surface at top of bed illuminated from the right. with vessel-to-sphere diameter ratio D,1d =253 and d=4.77 mm. showing A spheres of Fir . 4 spotted in two rings
used "nearly spherical' particles in vibrated cylindrical vessels with unstated base shape. and observed an "arrangement" which seems to resemble the present packing. But there are crucial differences between the respective findings on structure and void fraction, from which it is concluded that he did not have the same packing.
STRUCTURE OF TH`' PACKING
Apart from small-scale irregularities and a small proportion of vacancies, each shell is a "rolled up" hexagonal array of spheres, as well as a vertical
CYLINDRICALLY ORDERED PACKING OF EQUAL SPHERES
stack of horizontal rings of spheres (Fig 2)_ The spotted spheres in the axially halved bed (Fig 3) show that the packing is a vertical stack of downward and upward pointing "cones" of rings. as well as horizontal sets of alternately raised and depressed rings, here termed layers . Corresponding concave conical (Fig . I), convex conical, or horizontal "free surfaces" may be formed by removing spheres . The detailed structure is like that in Fig 4. The spheres of a ring in one shell fit cyclically into the cusps of an adjacent shell, with a corresponding vertical and radial undulation of the ring . (The vertical undulation may be seen by sighting obliquely along the rings of Fig. 2_) The spheres are generally closely packed in a ring in contact or with small gaps between them- (For convenience of drawing, Fig. 4 shows the shells "unrolled", with spurious gaps between some spheres, and the rings of one shell without undulation .) The fitting cycles can be identified in Fig 1 from the pairs of radially aligned "A spheres" (Fig. 4) which demarcate them. as in the case of the spotted spheres of the two outermost rings_ Except for a central region of the bed where the cycles are hard to define- the number of cycles in a ring with respect to an adjacent shell is
-`__ _
JIF
-,-,% tU • ri~, -a r : r S
equal to the difference in number of spheres per ring between the shells& because of the progressive vernier-like variation of fit (Figs . I and 4 (Plan)). On the average this difference is -27, spheres. corresponding to a difference between ring circumferences" which is sphere diameters, because the shells are packed at radial intervals -- I sphere diameter . Thus the number of cycles per ring is restricted, while the number of spheres per cycle increases on the average with increasing ring circumference. In general the number of spheres per cycle varies from cycle to cycle in a given ring .
.Y
t t
181
Fig. 3 . "Free surface" of axialh hared bed with %esscl-to-sphere diameter ratio D d=253 and d=4_77 mm_ showing ring-grout cd bare plate and spo .ted spheres indicating cones and :a%cr Or : :SS ; Cycle
0- ...C al
I
~i
Fig. 2 Peripheral shell of bed Kith vessel-to-sphere diameter ratio DJd=253 and d=4_77 mm.
Fig-4_ "unrolled" portions of adjacent shells tietved from abose bed (Plan). horizontally from axis of bed (Elevation) and as in Fig. 3 (Profile) . showing A spheres corresponding to spotted spheres of Fig_ f . Note spurious gaps between spheres and spuriously straight horizontal lines_
Poet den Teehnol . . 4 (1970 71) 1 80-186
182
F_ A_ ROCKS METHODS OF FORMING BEDS
The beds studied were formed of smooth steel spheres, for which the maximum relative variation of diameter from-sphere to sphere in a given bed was --1 part in 1000. The possible effects of surface roughness or of using materials other _than steel are not known. Perspex vessels were used and the spheres were added slowly by pouring thetas onto the bed without regard to position, while the vessel was being vibrated . The vessel, which stood without attachment on the vibrator table, was held down firmly by hand. The vibration of the table had a frequency of 50 Hz and a somewhat distorted sinusoidal form . It was found that beds could be formed with peak values of the vertical acceleration of the table ranging from about 20 to about 50 m s -2, with an optimum around 40 m s - =_ The peak horizontal acceleration of the table was - 10% of the vertical . The acceleration of the vessel approximated that of the table. The following vessel base shapes were used successfully (i) a smooth convex (upward pointing) cone with a slope of 27° to the horizontal, (ii) a convex cone with ring steps cut to fit the shells, and 27 ° slope, (iii) a concave (downward pointing) form of (ii). (iv) a horizontal plate with ring grooves cut to fit alternate shells (Fig . 3). Vibration on a smooth concave cone with 27° slope, and an ungrooved horizontal plate, and hand-stacking one sphere at a time on (i) without vibration, were unsuccessful. Tne base cone slope angle of 27 ° was found by choosing that angle out of several which produced, at any height in the bed, a layer which was "flat", that is, not dished or peaked . This was assumed to be the undistorted state of the packing because the dishing or peaking of layers above base cones with other angles decreased with increasing bed height, and eventually disap feared_ The axially halved bed (Fig 3) was obtained by forming a full bed on a ring-grooved plate in a vessel consisting oftwo semi-cylinders. The top of this bed was restrained with a second ring-grooved plate, and then the vessel was laid on its side with the edges of the semi-cylinders in the same horizontal plane. The top semi-cylinder was lifted off and surplus spheres were removed with a suction tube to obtain the "free surface", which was photographed from above.
Complete cylindrical ordering was obtained in all cases up to a vessel-to-sphere diameter ratio of 40.4, which was the highest value used. The practical upper limit of this ratio for forming the packing is not known. SHELL SPACING, LAYER SPACING, ANGULAR PARAMETER AND VOID FRACTION
Definitions of parameters
The radius of a shell is defined as the mean radial position of sphere centres in the shell, a central column of spheres (Figs. 1 and 3) being treated as a shell with zero radius. The (radial) shell spacing Ar (Fig. 4) is the difference between radii of adjacent shells. The (vertical) laver spacing Ah (Fig. 4) is defined in a similar wad,. The means of Ar and Ni over the bed are Ar and Nt. Because of departures from a strict configurationa definite slope or gradient cannot be assigned to a cone of rings (Fig_ 3)_ However, the above definitions and the Profile in Fig_ 4 show that on the average the conical structure rises or falls a distance Ah/2, and advances a distance Ar between successive rings of a cone. In this sense a gradient parameter Ah/2Ar can be assigned to the packing as a whole. A corresponding angular parameter 8 is defined by B - tan'
Alt
(1)
2Ar
If Vb and V are respectively the volume of a bed and the total volume of spheres in the bed, the fraction of Vb which is empty, or the void fraction*, is e-
Vb -
Vb
V -
V 2 -Vb
(2)
If Vb were defined by the vessel wall and plane end boundaries in contact with the top and bottom layers of the bed, e would include end effects which would change in importance as the height of the bed changed_ A void fraction e 'corrected for end effects is defined by replacing V in eqn . (2) with the product of the volume of a sphere and the mean number m of spheres per layer, and Vb with the mean increment in Vb per layer away from end layers . This increment is the volume of a cylinder with • The corresponding parameter with local (point) values is sometimes termed the porosity or voidage_ Its space-average of er the bed is equal to the void fraction as defined in eqn_ (2).
Powder Technol_,d (1970/71) 180-186
183
CYLINDRICALLY ORDERED PACKING OF EQUAL SPHERES
diameter that of the vessel, and height Oh . Thus
d 3r (2 e'-1-
3
2
n(D)
_ xrn
2d3 xin =1- 3D 2 xAh' xt~h
where d is the sphere diameter and D is the vessel inside diameter. As defined here, e' corresponds to the void fraction for a bed with infinite height .
central and peripheral shells and the number of shells, Ah being determined in a similar way from widely separated layers . Spheres were removed and counted to determine in, so that the effect of vacancies on e' was includedThe results are shown in Fig. 5 (upper points), Figs. 6 and 7 and Fig 8 (lower points)*, those for Ar and eh being made relative to the sphere dia-
000 0
Experiments Measurements were made on eight experimental beds with values of the vessel-to-sphere diameter ratio D/d ranging from 7.5 to 40.4, formed on smooth convex base cones with 27° slope- The value of Ar was determined in each case from the radii of the
0
0
0
N 0 L W zo
a
m
1_0
pR
000 0 a a a a
d
t r 100 OD5 o/a OiO 0-15
0
0
0
0
. essel Fig. 7. Angular parameter 0 tan -1 (Ah,?Ar) rs sphcr--to-diameter ratio d D.
a
to
a
a
la
a 0 Observed
(for bed with one shell more than observed bed)
O.8
A Hypothetical
1-E Q4
1 Q05
d/p
0.70
0.15
Fig. 5_ Observed and hypothetical rtlatitc mean shell spacing Mid rs. sphere-to-sessel diameter ratio d,D.
C 0
.6 0
v0 LL. 0.4 0 00 Q.2
0.4
0
0.05
0
0
0
0
o Void fraction e • Solid fraction 1-e'
0.05
I
015 0-10 d/f) Fig- 8_ Void fraction a and solid fraction I -e- both corrected for end c`r cts, rs sphere-to-vessel diameter ratio d D. 0.10
0)5
d/D Fig. 6. Relative mean la}er spacing Ah/d rs sphere-to-vessel diameter ratio WD.
• Fitting of smooth monotonic tunes could be misleading .
especially in Fig . 5, because the deviation of the data from such curves is not entirely random . Hence no curves ar : drawn_ Powder Tech ol w 4 (1970=71) 180-186
F. A . ROCKE
184
meter d- Since the solid fraction 1-e is sometimes considered, 1-e' is plotted in Fig . 8 as well as e . In
_The lower points in Fig . 5 represent a hypothetical Ar obtained by calculating from the data the values
each figure d/D is used as abscissa since an extrapo-
that Ar would have taken if v had been one shell greater in each bed. It was postulated on the basis
lation to d/D=0 may show the asymptotic behaviour of the packing as D/d-->oo, and indicate an infinite-bed value for the parameter plotted
of Fig 5 that, for given D/d, v takes the largest value compatible with Ar A: AR_ This can be shown to imply that, if
Theory A theory based on the structure in Fig . 4 was
N=D_
+1,
developed to account for the extrapolated data in Figs . 5-8 . This theory, which excludes vacancies and gaps, entails a convergence assumption and a hypothetical geometry at infinite D/d defined by letting the curvature of the shells vanish and the number of spheres per fitting cycle (Fig . 4) go to infinity- The resulting theoretical infinite-be parameters corresponding to Ar, Ali, 0 and e' are AR, AH, (9 and E'. (Note that E' corresponds to both infinite bed height and infinite D/d-) AR is determined by an integration over a path corresponding to the heavy line in Fig . 4 (Plan)_ AH is the spacing between rows of spheres in a hexagonal array, corresponding to the horizontal rows in Fig. 4 (Elevation) . (9 is tan-'(AH/2AR) . E' is determined by replacing V in eqn . (2) with the volume of one sphere and Vb with the mean bed volume per sphere . which is d x AR x AH_ The theoretical values of AR/d, AH/d, Q and E'
d
1 3,/ = + /66 4
-, (1 sin
_ 0.8497 . . .
`3
AH `/3 = 0 .8660 . . . a=
v=n
(3)
(Equation (3) is consistent with Figs . I and 3 where D/d=25.3, N=15.3, n=15 and a count of shells yields v=15 .) Experiments The general validity of eqn- (3) was tested with two sets of experiments, in which the inside diameter D of a vessel was progressively enlarged, the packing being formed at each value of D on a smooth convex base cone with 27° slope. In the first set, about 200 approximately uniformly distributed values of D/d from 7 .8 to 24-8 were used, the corresponding range of N being from 5-0 to 15-0. In these experiments, whenever a new shell appeared it did so as a central column_ As Did increased, this column collapsed, then formed a
are given by : AR
and n is the whole-number part of N, then
satisfied eqn . (3) except over a relatively narrow range of N around each point at which N was an
2 AH
0- = tan-'
shell with 3, and then 4, spheres per ring- As it reached 5 spheres per ring a new central column appeared inside it, and so on . The observed values of v
integer, that is, at which eqn- (3) predicts the appea-
_ ?7 .003 __
2AR a d s 3n2 d x AR x AH
=028846 . . .
rance of a new shell- Within these ranges the new shell could be made to appear at a lower or higher N than the integer value, according as the vibration was sufficiently light or heavy, so that v could have either of two values throughout a range- For
6 AR x AH T d These values are shown on the appropriate ordinates in Figs. 5-8 for comparison with the extrapolated data. NUMBER OF SHELLS IN BEDS
instance, the 10th shell could be made to appear at values of N slightly lower or higher than 10, so that a bed with any N between these values could have either 9 or 10 shells, depending on the vibration used to form it. The second set of experiments was designed to detect, with higher resolution than the first, the
Theory The number of shells in a bed is denoted by v, with a central column of spheres (Figs . 1 and 3) being counted as a shell-
extent of the above ranges associated with the integer values 6, 10, 13 and 17 of N_ With vibration which appeared to be the lightest and heaviest for which an orderly packing would form, each range
Powder Technol., 4 (1970171) 180-186
CYLINDRICALLY ORDERED PACKING OF EQUAL SPHERES
extended from -0_1 below to associated integer value .
-0_1 above the
DISCUSSION AND CONCLUSIONS
The consistency of the theoretical infinite-bed parameters with the extrapolated experimental data (Figs . 5-8) indicates that the parameters of the packing can be related to its structure (Fig . 4). If the extrapolated value of the void fraction e' is significantly higher than E' (Fig. 8), this might be due to a small non-vanishing asymptotic effect of vacancies and gaps, which are included in the experiments but not in the theory . As well as being consistent with the data for (Fig. 7), O is close to the value 27' chosen for the base cone slope angle . These results support the choice of © (eqn. 1) as a parameter of the packing and the criterion followed in choosing the base cone slope angle (see Methods of Forming Beds) . The results for the number of shells r in the bed show that. eqn . (3)_ and the postulate on which it was based, are not universally valid . Nevertheless, for the cases examined, there are known wide ranges of N, and hence of D/d, where eqn. (3) gives r. and known narrow ranges where r takes either of two predictable values_ The value of about 029 obtained for the infinite bed void fraction may be compared with those for other packings of equal spheres . A theorem of Rogers2 implies that no packing of equal spheres can have a fraction less than 1- [cos-'(N1)-2-7r] x 3,12 = 022036. . . but it does not imply that this bound is necessarily attainable. The lowest fraction known to be attained is 1 - 3
185
excludes the close fitting attained by the A spheres and those midway between them (Fig 4) and by all spheres in the rhombohedral packings . obtained experimental results which are consistent with his calculated fraction 0.375. In view of these comparisons it is not surprising that the promotion of ordering and fitting ofspheres can produce packings Kith relatively low void fractions. In fact the present packing is a special case of such compaction of a vibrated single-size granular materiaL The packing is interesting both as a structure of spherical elements and a vibratory product of the interaction of boundary shape, fitting and, presumably, a tendency to minimum potential energy_ The hexagonal packing of spheres in a shell (Fig 2) and the cyclic and undulatory fitting (Fig_ 4) is perhaps an optimum combination, compatible with the cylindrical shape, which minimizes the height of a shell and maximizes the number of shells in a bed. If this were so it would minimize the bed height for a given number of spheres. and hence also the potential energy and the void fraction_ The failure of hand-stacking. one sphere at a time without vibration. suggests that cooperative behaviour of the spheres is necessary for forming the bed . The work discussed should relate to the general effects of boundary shape and vibration on packings of equal spheres. ACKNOWLEDGENIEtiTS
I am grateful to Messrs . G_ W_ K_ Ford_ E_ Szomanski and G. A_ Tingate for encouragement and helpful discussions, and to Messrs E . W_ Clarke and H_ N . Harvey for technical advice and .assistance_ Miss B_ A. Beard checked some of the ca'.cu'_atiolts. Mr. N. H_ Clark of the National Standards Laboratory . CS.I.RO, measured the -vibration accelerations-
= 0-25952---
."/2 which is that of the rhombohedral ordered packings known as hexagonal close-packed and face-centered cubic. With random packings the fraction typically lies in a region surrounding 0 .38, and Scott' concludes from his experiments that it is unlikely that it can lie outside the limits 0 .36(3) and 0.39(9). Clancy' studied computer-generated random configurations, hypothetically packed in cylinders according to rules which were intended to imitate physical stacking of spheres . With two different models he obtained the calculated fractions 0.40 and 0.41. McGeary' (see Introduction above), who
LIST OF SYMBOLS
D d
vessel inside diameter sphere diameter e - 1- VJVb. the void fraction of a bed E' theoretical infinite-bed void fraction e' void fraction corrected for end effect corresponding to that for a bed with infinite height AH theoretical infinite-bed mean (vertical) layer - spacing Ah mean (vertical) layer spacing m mean number of spheres per layer Fowder TechnoL, 4 (1970,x71) 180-186
186 N=
F_ A_ ROCKE
D/d-1 } 1 2AR/d
n whole-number part of N AR theoretical infinite-bed mean (radial) shell _ spacing Ar mean (radial) shell spacing 1b volume of a bed Y total volume of spheres in a bed e=tan - ' (AH/2AR), a theoretical infinite-bed angular parameter
B = tan - ' (Ah/2Ar), an angular parameter v number of shells in a bed REFERENCES 1 R. K. McGEARY, Mechanical packing of spherical particles, J. Am . Ceram. Soc_, 44 (1961) 513 . 2 C A RoGExs, The packing of equal spheres, Proc. London Math_ Soc- 3 (8) (1958) 6093 G. D. Scorn, Packing of spheres, Nature, 188 (1960)908_ 4 B. E_ CtarcY, Calculations of hard sphere packdngs in large cylinders. Brookharen NarL Lab . Rept. BNL 997 (T-424) (Physics-TID-4500), 1966_
Powder Techno!_, 4 (1970111) 180-186
SUMMARY A homogeneous ordered packing of equal spheres can be formed by vibration in vertical cylindrical vessels with certain base shapes. The shapes are described. The packing consists of a concentric array of cylindrical "shells" of spheres which fills the vessel. The spheres in a shell are arrayed hexagonally, and their fitting against those in an adjacent shell is cyclic and undulatory with respect to a circle concentric with the shells. A theory based on the observed structure yields infinite-bed valuesforsome parameters of the packing which are consistent with extrapolated experimental values- In particular, the infinite-bed void fraction is about 0.29- The number of shells in a bed with given vessel-to-sphere diameter ratio is predictable over certain ranges of the ratio . INTRODUCTION
When a bed of equal spheres is packed randomly in a vertical cylindrical vessel, those lying against the wall tend to form a peripheral "shell" in which there is some ordering and in which the packing is closer than that in the interior of the bed_ During an investigation of this effect and its relation to pebblebed reactors, it was found that an ordered packing, consisting entirely of such shells& in a concentric array (Fig. 1), may be formed by vibration in vertical cylindrical vessels with certain base shapes_ This array was called "the cylindrically ordered packing of equal spheres". The structure of the packing was determined by external and internal examination of beds . Experimental beds were prepared, with various values of the vessel-to-sphere diameter ratio, and the measured values of some parameters were studied theoretically in relation to the structure and this ratio. After the work had been completed, that of McGearyt came to the author's attention . McGeary Ponder Technology - Elsevier
Sequoia S_A., Lausanne
Fig . 1_ Concave conical "free surface at top of bed illuminated from the right. with vessel-to-sphere diameter ratio D,1d =253 and d=4.77 mm. showing A spheres of Fir . 4 spotted in two rings
used "nearly spherical' particles in vibrated cylindrical vessels with unstated base shape. and observed an "arrangement" which seems to resemble the present packing. But there are crucial differences between the respective findings on structure and void fraction, from which it is concluded that he did not have the same packing.
STRUCTURE OF TH`' PACKING
Apart from small-scale irregularities and a small proportion of vacancies, each shell is a "rolled up" hexagonal array of spheres, as well as a vertical
CYLINDRICALLY ORDERED PACKING OF EQUAL SPHERES
stack of horizontal rings of spheres (Fig 2)_ The spotted spheres in the axially halved bed (Fig 3) show that the packing is a vertical stack of downward and upward pointing "cones" of rings. as well as horizontal sets of alternately raised and depressed rings, here termed layers . Corresponding concave conical (Fig . I), convex conical, or horizontal "free surfaces" may be formed by removing spheres . The detailed structure is like that in Fig 4. The spheres of a ring in one shell fit cyclically into the cusps of an adjacent shell, with a corresponding vertical and radial undulation of the ring . (The vertical undulation may be seen by sighting obliquely along the rings of Fig. 2_) The spheres are generally closely packed in a ring in contact or with small gaps between them- (For convenience of drawing, Fig. 4 shows the shells "unrolled", with spurious gaps between some spheres, and the rings of one shell without undulation .) The fitting cycles can be identified in Fig 1 from the pairs of radially aligned "A spheres" (Fig. 4) which demarcate them. as in the case of the spotted spheres of the two outermost rings_ Except for a central region of the bed where the cycles are hard to define- the number of cycles in a ring with respect to an adjacent shell is
-`__ _
JIF
-,-,% tU • ri~, -a r : r S
equal to the difference in number of spheres per ring between the shells& because of the progressive vernier-like variation of fit (Figs . I and 4 (Plan)). On the average this difference is -27, spheres. corresponding to a difference between ring circumferences" which is sphere diameters, because the shells are packed at radial intervals -- I sphere diameter . Thus the number of cycles per ring is restricted, while the number of spheres per cycle increases on the average with increasing ring circumference. In general the number of spheres per cycle varies from cycle to cycle in a given ring .
.Y
t t
181
Fig. 3 . "Free surface" of axialh hared bed with %esscl-to-sphere diameter ratio D d=253 and d=4_77 mm_ showing ring-grout cd bare plate and spo .ted spheres indicating cones and :a%cr Or : :SS ; Cycle
0- ...C al
I
~i
Fig. 2 Peripheral shell of bed Kith vessel-to-sphere diameter ratio DJd=253 and d=4_77 mm.
Fig-4_ "unrolled" portions of adjacent shells tietved from abose bed (Plan). horizontally from axis of bed (Elevation) and as in Fig. 3 (Profile) . showing A spheres corresponding to spotted spheres of Fig_ f . Note spurious gaps between spheres and spuriously straight horizontal lines_
Poet den Teehnol . . 4 (1970 71) 1 80-186
182
F_ A_ ROCKS METHODS OF FORMING BEDS
The beds studied were formed of smooth steel spheres, for which the maximum relative variation of diameter from-sphere to sphere in a given bed was --1 part in 1000. The possible effects of surface roughness or of using materials other _than steel are not known. Perspex vessels were used and the spheres were added slowly by pouring thetas onto the bed without regard to position, while the vessel was being vibrated . The vessel, which stood without attachment on the vibrator table, was held down firmly by hand. The vibration of the table had a frequency of 50 Hz and a somewhat distorted sinusoidal form . It was found that beds could be formed with peak values of the vertical acceleration of the table ranging from about 20 to about 50 m s -2, with an optimum around 40 m s - =_ The peak horizontal acceleration of the table was - 10% of the vertical . The acceleration of the vessel approximated that of the table. The following vessel base shapes were used successfully (i) a smooth convex (upward pointing) cone with a slope of 27° to the horizontal, (ii) a convex cone with ring steps cut to fit the shells, and 27 ° slope, (iii) a concave (downward pointing) form of (ii). (iv) a horizontal plate with ring grooves cut to fit alternate shells (Fig . 3). Vibration on a smooth concave cone with 27° slope, and an ungrooved horizontal plate, and hand-stacking one sphere at a time on (i) without vibration, were unsuccessful. Tne base cone slope angle of 27 ° was found by choosing that angle out of several which produced, at any height in the bed, a layer which was "flat", that is, not dished or peaked . This was assumed to be the undistorted state of the packing because the dishing or peaking of layers above base cones with other angles decreased with increasing bed height, and eventually disap feared_ The axially halved bed (Fig 3) was obtained by forming a full bed on a ring-grooved plate in a vessel consisting oftwo semi-cylinders. The top of this bed was restrained with a second ring-grooved plate, and then the vessel was laid on its side with the edges of the semi-cylinders in the same horizontal plane. The top semi-cylinder was lifted off and surplus spheres were removed with a suction tube to obtain the "free surface", which was photographed from above.
Complete cylindrical ordering was obtained in all cases up to a vessel-to-sphere diameter ratio of 40.4, which was the highest value used. The practical upper limit of this ratio for forming the packing is not known. SHELL SPACING, LAYER SPACING, ANGULAR PARAMETER AND VOID FRACTION
Definitions of parameters
The radius of a shell is defined as the mean radial position of sphere centres in the shell, a central column of spheres (Figs. 1 and 3) being treated as a shell with zero radius. The (radial) shell spacing Ar (Fig. 4) is the difference between radii of adjacent shells. The (vertical) laver spacing Ah (Fig. 4) is defined in a similar wad,. The means of Ar and Ni over the bed are Ar and Nt. Because of departures from a strict configurationa definite slope or gradient cannot be assigned to a cone of rings (Fig_ 3)_ However, the above definitions and the Profile in Fig_ 4 show that on the average the conical structure rises or falls a distance Ah/2, and advances a distance Ar between successive rings of a cone. In this sense a gradient parameter Ah/2Ar can be assigned to the packing as a whole. A corresponding angular parameter 8 is defined by B - tan'
Alt
(1)
2Ar
If Vb and V are respectively the volume of a bed and the total volume of spheres in the bed, the fraction of Vb which is empty, or the void fraction*, is e-
Vb -
Vb
V -
V 2 -Vb
(2)
If Vb were defined by the vessel wall and plane end boundaries in contact with the top and bottom layers of the bed, e would include end effects which would change in importance as the height of the bed changed_ A void fraction e 'corrected for end effects is defined by replacing V in eqn . (2) with the product of the volume of a sphere and the mean number m of spheres per layer, and Vb with the mean increment in Vb per layer away from end layers . This increment is the volume of a cylinder with • The corresponding parameter with local (point) values is sometimes termed the porosity or voidage_ Its space-average of er the bed is equal to the void fraction as defined in eqn_ (2).
Powder Technol_,d (1970/71) 180-186
183
CYLINDRICALLY ORDERED PACKING OF EQUAL SPHERES
diameter that of the vessel, and height Oh . Thus
d 3r (2 e'-1-
3
2
n(D)
_ xrn
2d3 xin =1- 3D 2 xAh' xt~h
where d is the sphere diameter and D is the vessel inside diameter. As defined here, e' corresponds to the void fraction for a bed with infinite height .
central and peripheral shells and the number of shells, Ah being determined in a similar way from widely separated layers . Spheres were removed and counted to determine in, so that the effect of vacancies on e' was includedThe results are shown in Fig. 5 (upper points), Figs. 6 and 7 and Fig 8 (lower points)*, those for Ar and eh being made relative to the sphere dia-
000 0
Experiments Measurements were made on eight experimental beds with values of the vessel-to-sphere diameter ratio D/d ranging from 7.5 to 40.4, formed on smooth convex base cones with 27° slope- The value of Ar was determined in each case from the radii of the
0
0
0
N 0 L W zo
a
m
1_0
pR
000 0 a a a a
d
t r 100 OD5 o/a OiO 0-15
0
0
0
0
. essel Fig. 7. Angular parameter 0 tan -1 (Ah,?Ar) rs sphcr--to-diameter ratio d D.
a
to
a
a
la
a 0 Observed
(for bed with one shell more than observed bed)
O.8
A Hypothetical
1-E Q4
1 Q05
d/p
0.70
0.15
Fig. 5_ Observed and hypothetical rtlatitc mean shell spacing Mid rs. sphere-to-sessel diameter ratio d,D.
C 0
.6 0
v0 LL. 0.4 0 00 Q.2
0.4
0
0.05
0
0
0
0
o Void fraction e • Solid fraction 1-e'
0.05
I
015 0-10 d/f) Fig- 8_ Void fraction a and solid fraction I -e- both corrected for end c`r cts, rs sphere-to-vessel diameter ratio d D. 0.10
0)5
d/D Fig. 6. Relative mean la}er spacing Ah/d rs sphere-to-vessel diameter ratio WD.
• Fitting of smooth monotonic tunes could be misleading .
especially in Fig . 5, because the deviation of the data from such curves is not entirely random . Hence no curves ar : drawn_ Powder Tech ol w 4 (1970=71) 180-186
F. A . ROCKE
184
meter d- Since the solid fraction 1-e is sometimes considered, 1-e' is plotted in Fig . 8 as well as e . In
_The lower points in Fig . 5 represent a hypothetical Ar obtained by calculating from the data the values
each figure d/D is used as abscissa since an extrapo-
that Ar would have taken if v had been one shell greater in each bed. It was postulated on the basis
lation to d/D=0 may show the asymptotic behaviour of the packing as D/d-->oo, and indicate an infinite-bed value for the parameter plotted
of Fig 5 that, for given D/d, v takes the largest value compatible with Ar A: AR_ This can be shown to imply that, if
Theory A theory based on the structure in Fig . 4 was
N=D_
+1,
developed to account for the extrapolated data in Figs . 5-8 . This theory, which excludes vacancies and gaps, entails a convergence assumption and a hypothetical geometry at infinite D/d defined by letting the curvature of the shells vanish and the number of spheres per fitting cycle (Fig . 4) go to infinity- The resulting theoretical infinite-be parameters corresponding to Ar, Ali, 0 and e' are AR, AH, (9 and E'. (Note that E' corresponds to both infinite bed height and infinite D/d-) AR is determined by an integration over a path corresponding to the heavy line in Fig . 4 (Plan)_ AH is the spacing between rows of spheres in a hexagonal array, corresponding to the horizontal rows in Fig. 4 (Elevation) . (9 is tan-'(AH/2AR) . E' is determined by replacing V in eqn . (2) with the volume of one sphere and Vb with the mean bed volume per sphere . which is d x AR x AH_ The theoretical values of AR/d, AH/d, Q and E'
d
1 3,/ = + /66 4
-, (1 sin
_ 0.8497 . . .
`3
AH `/3 = 0 .8660 . . . a=
v=n
(3)
(Equation (3) is consistent with Figs . I and 3 where D/d=25.3, N=15.3, n=15 and a count of shells yields v=15 .) Experiments The general validity of eqn- (3) was tested with two sets of experiments, in which the inside diameter D of a vessel was progressively enlarged, the packing being formed at each value of D on a smooth convex base cone with 27° slope. In the first set, about 200 approximately uniformly distributed values of D/d from 7 .8 to 24-8 were used, the corresponding range of N being from 5-0 to 15-0. In these experiments, whenever a new shell appeared it did so as a central column_ As Did increased, this column collapsed, then formed a
are given by : AR
and n is the whole-number part of N, then
satisfied eqn . (3) except over a relatively narrow range of N around each point at which N was an
2 AH
0- = tan-'
shell with 3, and then 4, spheres per ring- As it reached 5 spheres per ring a new central column appeared inside it, and so on . The observed values of v
integer, that is, at which eqn- (3) predicts the appea-
_ ?7 .003 __
2AR a d s 3n2 d x AR x AH
=028846 . . .
rance of a new shell- Within these ranges the new shell could be made to appear at a lower or higher N than the integer value, according as the vibration was sufficiently light or heavy, so that v could have either of two values throughout a range- For
6 AR x AH T d These values are shown on the appropriate ordinates in Figs. 5-8 for comparison with the extrapolated data. NUMBER OF SHELLS IN BEDS
instance, the 10th shell could be made to appear at values of N slightly lower or higher than 10, so that a bed with any N between these values could have either 9 or 10 shells, depending on the vibration used to form it. The second set of experiments was designed to detect, with higher resolution than the first, the
Theory The number of shells in a bed is denoted by v, with a central column of spheres (Figs . 1 and 3) being counted as a shell-
extent of the above ranges associated with the integer values 6, 10, 13 and 17 of N_ With vibration which appeared to be the lightest and heaviest for which an orderly packing would form, each range
Powder Technol., 4 (1970171) 180-186
CYLINDRICALLY ORDERED PACKING OF EQUAL SPHERES
extended from -0_1 below to associated integer value .
-0_1 above the
DISCUSSION AND CONCLUSIONS
The consistency of the theoretical infinite-bed parameters with the extrapolated experimental data (Figs . 5-8) indicates that the parameters of the packing can be related to its structure (Fig . 4). If the extrapolated value of the void fraction e' is significantly higher than E' (Fig. 8), this might be due to a small non-vanishing asymptotic effect of vacancies and gaps, which are included in the experiments but not in the theory . As well as being consistent with the data for (Fig. 7), O is close to the value 27' chosen for the base cone slope angle . These results support the choice of © (eqn. 1) as a parameter of the packing and the criterion followed in choosing the base cone slope angle (see Methods of Forming Beds) . The results for the number of shells r in the bed show that. eqn . (3)_ and the postulate on which it was based, are not universally valid . Nevertheless, for the cases examined, there are known wide ranges of N, and hence of D/d, where eqn. (3) gives r. and known narrow ranges where r takes either of two predictable values_ The value of about 029 obtained for the infinite bed void fraction may be compared with those for other packings of equal spheres . A theorem of Rogers2 implies that no packing of equal spheres can have a fraction less than 1- [cos-'(N1)-2-7r] x 3,12 = 022036. . . but it does not imply that this bound is necessarily attainable. The lowest fraction known to be attained is 1 - 3
185
excludes the close fitting attained by the A spheres and those midway between them (Fig 4) and by all spheres in the rhombohedral packings . obtained experimental results which are consistent with his calculated fraction 0.375. In view of these comparisons it is not surprising that the promotion of ordering and fitting ofspheres can produce packings Kith relatively low void fractions. In fact the present packing is a special case of such compaction of a vibrated single-size granular materiaL The packing is interesting both as a structure of spherical elements and a vibratory product of the interaction of boundary shape, fitting and, presumably, a tendency to minimum potential energy_ The hexagonal packing of spheres in a shell (Fig 2) and the cyclic and undulatory fitting (Fig_ 4) is perhaps an optimum combination, compatible with the cylindrical shape, which minimizes the height of a shell and maximizes the number of shells in a bed. If this were so it would minimize the bed height for a given number of spheres. and hence also the potential energy and the void fraction_ The failure of hand-stacking. one sphere at a time without vibration. suggests that cooperative behaviour of the spheres is necessary for forming the bed . The work discussed should relate to the general effects of boundary shape and vibration on packings of equal spheres. ACKNOWLEDGENIEtiTS
I am grateful to Messrs . G_ W_ K_ Ford_ E_ Szomanski and G. A_ Tingate for encouragement and helpful discussions, and to Messrs E . W_ Clarke and H_ N . Harvey for technical advice and .assistance_ Miss B_ A. Beard checked some of the ca'.cu'_atiolts. Mr. N. H_ Clark of the National Standards Laboratory . CS.I.RO, measured the -vibration accelerations-
= 0-25952---
."/2 which is that of the rhombohedral ordered packings known as hexagonal close-packed and face-centered cubic. With random packings the fraction typically lies in a region surrounding 0 .38, and Scott' concludes from his experiments that it is unlikely that it can lie outside the limits 0 .36(3) and 0.39(9). Clancy' studied computer-generated random configurations, hypothetically packed in cylinders according to rules which were intended to imitate physical stacking of spheres . With two different models he obtained the calculated fractions 0.40 and 0.41. McGeary' (see Introduction above), who
LIST OF SYMBOLS
D d
vessel inside diameter sphere diameter e - 1- VJVb. the void fraction of a bed E' theoretical infinite-bed void fraction e' void fraction corrected for end effect corresponding to that for a bed with infinite height AH theoretical infinite-bed mean (vertical) layer - spacing Ah mean (vertical) layer spacing m mean number of spheres per layer Fowder TechnoL, 4 (1970,x71) 180-186
186 N=
F_ A_ ROCKE
D/d-1 } 1 2AR/d
n whole-number part of N AR theoretical infinite-bed mean (radial) shell _ spacing Ar mean (radial) shell spacing 1b volume of a bed Y total volume of spheres in a bed e=tan - ' (AH/2AR), a theoretical infinite-bed angular parameter
B = tan - ' (Ah/2Ar), an angular parameter v number of shells in a bed REFERENCES 1 R. K. McGEARY, Mechanical packing of spherical particles, J. Am . Ceram. Soc_, 44 (1961) 513 . 2 C A RoGExs, The packing of equal spheres, Proc. London Math_ Soc- 3 (8) (1958) 6093 G. D. Scorn, Packing of spheres, Nature, 188 (1960)908_ 4 B. E_ CtarcY, Calculations of hard sphere packdngs in large cylinders. Brookharen NarL Lab . Rept. BNL 997 (T-424) (Physics-TID-4500), 1966_
Powder Techno!_, 4 (1970111) 180-186