Pattern scaling in the axial segregation of granular materials in a rotating tube

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by  Christopher R. J. Charles

Pattern scaling in the axial segregation of granular materials in a rotating tube
Christopher R. J. Charles, Zeina S. Khan and Stephen W. Morris
Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7
(Dated: February 2, 2008)
Granular mixtures frequentlysegregate bygrain size along theaxis of partially-filled, horizontal,
rotating tubes. When segregation approaches saturation at the surface, a well-defined pattern of
5 bands with wavelength λ emerges. The long-term dynamics of the pattern involves a much slower
0 coarseningprocess. Wecharacterizedtheinitialsaturatedwavelengthλasafunctionofthediameter
0 of the tube D for a filling fraction of 30 %, for 1.9 cm≤D ≤11.5 cm. We also studied the initial
2 growth-rateofthebandsasDvaries. Wefindthatλ/Disnotconstant,butratherincreasesrapidly
for small D. The growth-rate of bands decreases with smaller D and segregation is suppressed
n
completely for sufficiently small D. These relatively simple features are not captured byany of the
a
J existing models of axial segregation.

5
PREPRINT will besubmitted to Phys. Rev. E. brief reports
2

PACSnumbers: 46.10.+z,64.75.+g
]
t
f
o
s A peculiar phenomenon occurs when two granular
. specieswithsimilardensitiesbutdifferentgrainsizesare
t
a
mixed together in long partially-filled, horizontal rotat-
m
ing tube. Rather than mixing together as one might ex-
- pect, the grains segregatepartially or completely by size
d
n [1, 2, 3]. Two distinct segregationphenomena have been
o observed, termed radial and axial. In radial segregation,
c thesmallercomponentmovestowardtheaxisofrotation
[
and forms a buried core. The core typically threads the

1 entire tube and, depending on the rotation speed, forms
v ratherquickly,oftenwithinthefirstfewtube revolutions
6 [4, 5]. The buried core is not usually visible from the FIG. 1: Three tubes of different diameters, showing segre-
0
surface. Radial segregation is followed by slower axial gated bands. Noclearsegregation was seen for D<2.35 cm.
6
1 segregation in which the mixture separates into bands
0 arranged along the axis of the tube as shown in figure
5 1. The array of axial bands usually forms after several φ, were held fixed[6, 11]. The scaling behavior we ob-
0 hundredtoseveralthousandtuberotations. Itexhibitsa serve, although very simple, is nevertheless a strong test
/
t reasonably well-defined and quasistable emergent wave- of models of this effect[13, 14, 15, 16, 17, 18].
a
m lengthλ. TheFourierspectrumofthebandpatternprior Nine glass tubes 1 m long with diameters 1.9 cm ≤
to saturation reveals a strong low-λ cutoff, a sharp peak D ≤ 11.5 cm were rotated at a constant rate of ω =
-
d with a long tail to higher wavelengths (see Fig. 5 of 2.5 rad/s. The grains consisted of glass spheres. The
n Ref. [6]). After a very long time, further rotation yields larger (black) grains were 750 µm in diameter while the
o either complete segregation [7] or a smaller number of smaller (transparent) grains had diameters in the range
c
bands produced by coarsening effects[8, 9, 10]. 297 - 420 µm. Thus, d = 550µm. The tubes were filled
:
v An obvious question of pattern scaling presents itself. withequalvolumesofeachsize,sothatφ=0.5,toafilled
i
X Howdoestheemergentwavelengthλscalewiththeother volume fraction η = 0.30. The grains were randomly
r simple length scales in the system, the tube diameter D pre-mixed and loaded into the tubes using a U-shaped
a andthe averagegrainsize d? Although the phenomenon channel [6, 11, 12]. Static charging effects were reduced
of axial segregation has been the subject of numerous by adding a small quantity of Larostat FPE-S antistatic
studies[1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, powder[20]. The tube was illuminated from above while
18, 19], its scaling behavior has not been systematically a CCD camera viewed the grains perpendicular to the
explored. A recent study[19] of the existence of axial flowing surface. Spacetime images like those shown in
segregation as a function of the dimensionless parame- Fig. 2 were constructed from horizontal strips 1 pixel
ter δ = D/d suggests that a scaling approach could be high and 640 pixels long, stacked at 2 s or 3 s intervals.
fruitful. In this paper, we investigate the scaling of λ The horizontal strip captured the central 79 cm of the
as a function of D, for fixed d. Other parameters that tube.
might be relevant, such as the rotation frequency ω, the For each D, the number of axial bands was counted
filledvolumefractionη andthesizecompositionfraction after 5 min and averaged over 6 or 7 runs. We thus ob-
2
(a)
)
m10
c n
( o
l gati
h e
gt gr
en 5 Se
el o
v N
a
W
0
0 5 10
Tube diameter D (cm)
1.5 (b)
n
o
ati
D eg
/ egr
l S
1.0 o
N
0.7
0 5 10
FIG. 2: Spacetime images of (a) D = 2.85 cm and (b) D = Tube diameter D (cm)
11.5cmshowingtheemergentwavelengthλ. Imagesarefalse-
coloured with red corresponding to thelight colored grains.
FIG. 3: Axial segregation results showing (a) λ vs. D and
(b) λ/D vs. D. No segregation is observed for tubes smaller
than D=2.35 cm
taining the averagewavelengthλ for each tube diameter
D. To investigate the growth rate of the axial pattern,
we examined the power in the largest Fourier peak as a rate of axially segregated bands is a rapidly increasing
function of time. These were found to increase approxi- function of D.
mately exponentially toward saturation[6]. We fit these Ourresultsfor1.9cm≤D ≤11.5cmarebroadlycon-
to extract the growth rate, which was a strong function sistent with the empirical survey presented in Ref. [19].
of D, as discussed below. That study concluded that if the ratio δ =d/D is larger
Axial segregation was observed for tube diameters in than a critical value δc ≈ 55, axial bands always form,
the range 2.85 cm ≤ D ≤ 11.5 cm. No segregation was whileifδ <40segregationisneverfound. Foroursmall-
observed for D = 1.9 cm even after several hours of ro- est tube, D = 1.9 cm, in which no segregation was ob-
tation. For D = 2.35 cm, segregation was irregular and served, δ = 35. For D = 2.35cm, we find δ = 43 and
incomplete for short times (less than 2 hours) but long- segregation was incomplete even after 2 hours of rota-
termsegregationwasobservedafter15hoursofrotation. tion. For D = 2.85cm, δ = 51 ∼ δc, but we found clear,
The averaged wavelength λ for each D is plotted in if rather slow, axial segregation. All other values of D
Fig. 3(a). A linear fit does not intercept the origin,indi- have δ >δc and clear axial segregationwas observed.
catingthatλisnotdirectlyproportionaltoD. Theslope Continuum models of axial segregation[13, 14, 15, 16,
is 0.73±0.03,while the intercept is 1.8±0.2 cm. Figure 17, 18] do not appear to make any useful predictions for
3(b)showsthe ratioλ/D vs. D fromthe same data. For this scaling. In general, the models are scaled so that
small D, λ/D increases toward the region where segre- all lengths are expressed in units of D, and hence for
gation is not observed. For large D, λ/D tends toward constant parameters they apparently predict a constant
a constant equal to the slope from Fig. 3(a). Turning to λ/D. In practice, the parameters of the various models
thegrowthrates,Fig.4showstheresultsforthesixtube probably contain some D dependence, which is however
diameters where axialsegregationoccurred. The growth unanalyzedwithin the models. Some clue about this de-
3
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In conclusion, we have examined the scaling of the
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less δ = D/d > 55. No model of which we are aware
[20] Manufactured by BASF Specialty Products, Mt. Olive
correctly predicts the scaling of the pattern with either
New Jersey, USA,07828-1234.
the tube diameter D, or the mean grain size d. Thus, [21] S.B. Savage, in Disorder and Granular Media, edited by
even the simplest scaling features of this striking effect D. Bideau and A. Hansen (North-Holland,Amsterdam,
remain unaccounted for. 1993), p.255.
We wish to thank Zahir Daya and Wayne Tokaruk.
This work was supported by the Natural Science and
Engineering Research Council of Canada.
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