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Behavior of powder under weak perturbations: the concept of the effective surface. II.
E. Gurovich, S. Maruk
To cite this version:
E. Gurovich, S. Maruk. Behavior of powder under weak perturbations: the concept of the effective surface. II.. Journal de Physique II, EDP Sciences, 1991, 1 (10), pp.1167-1177. �10.1051/jp2:1991126�.
�jpa-00247582�
J. Phys. II France1 (1991) l167.l177 OCTOBRE1991, PAGE l167
Classification Physics Abstracts
05.40 05.60
Proofs not corrected by the authors
Behavior of powder under weak perturbations : the concept of
the effective surface. U.
E. V. Gurovich (*) and S. V. Maruk
Department of Theoretical Physics, Institute for High Pressure Physics, Moscow Region, Troisk, 142092, U-S-S-R-
(Received J3 May J99J, accepted in final farm J7 June J99J)
AbswacL- The dense and size separation curve is constructed. The phase rule for powder
mixture is derived. Dependence of segregation velocity on the shares of large and small grains is estimated. The density of free-poured powder is calculated.
1. Introduction.
In the first part [I] of our work we proposed a new way to describe the behavior of granular
materials and investigated the swelling and packing down processes and the origin of the instability of powders under vibration. The present part deals with two other problems which
were already briefly mentioned in [I], namely, with the dense and size separation (see Refs. [3, 7, 10, 14-16] in [I]), and vith the notion of the free-poured powder (see references
[17-24] in [I]). As will be shown, the concept of the effective surface makes it possible to obtain both qualitative and quantitative results in these questions too. Let us remind the
fundamental tenets of our approach (for all the details and exact definition see [I]).
Because of any state of powder is metastable it is not its initial state but only the behavior of
powder with a certain initial conditions under perturbations of certain intensity obeys physical law-governed nature. To describe stable in gravity field states of powder we introduced the notion of the effective surface u(x) which is connected with the local geometry of grain packing on the one hand and characterizes the average value of momentum transferred to
grains under shaking on the other hand. The value u determines by what magnitude the thickness of a horizontal layer is more in a loose random packing than the thickness is in the closed random one. The value 2 ix is equal approximately to a typical accidental distance on
which grains are shifted in the direction of the perpendicular to the gravity force at a single perturbation : the less the density and the more intensity of perturbation are the more the shift 2 ix is. So, under a k-th disturbance (such as a weak stirring, jolting or shaking) a grain
on average shifts in respect with its neighbour grains on the distance u~ u~
j along gravity
field and on the distance x~ xk-
i in the perpendicular direction. The repacking of powder grains at a single weak perturbation (the turning of powder from a metastable state with
density p~
j into another one with density p~) at the k-th shaking corresponds to ajump from
one point x~_
j, u~_ j into another one u~, x~. Walls of the effective surface are formed by
JOURNAL DE PHYSIQUE II T i, V 10. OCTOBRE 1991 51
l168 JOURNAL DE PHYSIQUE II N 10
surrounding impenetrable grains making it bnpossible for « average grain » to come out of it.
Note that the effective surface is defined in the configurational space and does not demand the localization of grain in the real space after numerous stirrings. The direction of the initial
momentum obtained by a grain free to move at the moment from underlaying ne1gllbors
depends on the density of powder. The more density the more vertical component of the
momentum transferred from the bottom to the top of the container (inside which powder is)
and tile more the tendency » for powder to swell. On the contrary the less density the less vertical component of « the average momentum » obtained by grains under perturbations and the more the «tendency» for powder to settle (see Fig. I here and Fig. 2-4 in [I] and
comments to them).
The concept of the effective surface developed in [I] allows to treat a powder under chaotic
perturbation as a new kind of self-organbing system and to use corresponding theoretical
physics and mathematical methods to describe it.
2. Density of free-poured powder.
The following simple models were shown in [I] paragraph 3 to approximate the effective surfaces for the plane powder of discs and tree-dimensional powder of spheres respectively
~
=a
)lj, (2.I)
D D
) ~f £ ~ (~~)
D 2 (D(
'
Here D is the grain diameter, the dimensionless coefficient a is of order of I and the
physically allowable values of the position x of the « average grain on the effective surface range vithin interval 0 0,I D.
Let us calculate density p of free poured powder. A dropped grain falls onto can arbitrary point of underlaying layer with an equal probability of hitting. If friction coefficient
@ between grains equals infinity, the fatling grain will stay at the point of first hitting with
underlaying grains. It means that particles of powder are uniformly distributed along the axis
x of the effective surface. So, in the one-particle approach, the free poured density in tile case of infinite friction forces is
ij)j~
o
~ ~~~ ~'~ j) ~ ~( ~~ ~ ~~
~~ ~~
~ ~ ~
~~~'~ dx
j~ ~~ (p))'~ ~~ dp
x(p) = o p> ax BP
Here p and p' are densities of the closed and the lossest (but stable) random packings respectively. The loose random packing was experimentally investigated in [2] recently.
The dependence p upon the ordinate u of the point on the effective surface has the
following universal form ([1] (2.9))
pip = I + u/d. (2.4)
Here I is the layer thickness of random closed packing, p is the density of the closed packing (0.637 for the powder of spheres and 0.821 for the plane powder of discs). Integrating (2.3)
one obtains that the free poured densities for the effective surfaces (2.1-2) are respectively
N 10 BEHAVIOR OF POWDER UNDER PERTURBATIONS II l169
#P' ~~#
~
~ " f-p' P'
p' 1/2 p' 1/2
p
~ = w =
# arc cos (2.5)
f-p' #
Note that the model parameter a determining the form of the effective surface does not appear in these expressions.
It follows from formula (2.3) that in the limit of the infinite friction forces the distribution function w~ of packing configurations of free-poured powder on densities is defined as follows
~ ~~ ~
~~
(2.6)
~°~ ~~"°'
j~ j~ (p))' it dP
p, x P
If the hypothesis of Edwards-Oakeshott [3, 4] about «microcanonical» distribution of configurations (according to which the probability of configuration with density p is
~ exp (- «p/p ), where « is unknown parameter) is correct formula (2.6) makes it possible to calculate the number of stable in gravity field configurations of the given density
p. For example, for ef§ective surface (2.I) it equals
r(p )
= w~ exp ~
x
)~ ~ ~~~~~~ ~'~
~ ~
p»p, p<p ,
In the case of friction forces absence the particle should be at the bottom of the effective surface and the density of powder should correspond to the closed random packing. The
presence of a friction holds up the particle on the wall of the effective surface. Having denoted by @ the effective coefficient of the friction between the particle and the walls of the effective surface the equilibrium position x~ of the former will be determined from the trivial condition
Grains having hit in a point u ~u(x~) are shifted by gravity force to the equilibrium position u(x~) in respect with particles of underlaying layer. The density of poured powder
will be
x~lw) xjp') x.(p) -i
p (~ = p (x) dx + p (x~) dx dx
=
xl#)
o
x,jp) xlp)
o
p.ia) >
= P, w~
~ ~ ~
dp + pm
~ j~
~
dp (2.7)
p~ p.ia)
This expression has the simple physical meaning at the finite friction local geometric configurations, which are characteristic and typical for a friable packing with p < p
~,
subside to the equilibrium density p~.
For the effective surface of form (2.2) the equilibrium position is given by the following expression
(x~, U~)
= (D~/2 a, D~ 2/4 a)
170 JOURNAL DE PHYSIQUE II N lo
and according to formula (2.4) the minimum density p~ of a local configuration stable in gravity field is equal to
P. (R )
= P (x~)
= (P~ ' + (4 aP?)~ ' DR ~)~' (2.8)
From these expressions it follows that the equilibriuln position u~ on the effective surface decreases (and density increases in diminishing friction @ between particles.
Having substitute these expressions in (2.7) we fmd the free poured density
P'~=P'~=m-Pl ~'pl"~l'lnlllllll-iilll~-
((P P') (P P~))'/~ + (PP.)"~
~"~°~ ~ ~
P
Here the dependence p~ on @ is given by (2.8), p, p' are densities of closed and loose random
packings respectively. Formula (2.8) determines in the explicit form the dependence of density of free poured powder on the friction coefficient @ between the grains.
For the surface (2.I) the equilibrium position and minimum density p~ of stable
configurations and, hence, the free poured density p (~ don't depend on the coefficient
@.
Friable powders are really [to be] very sensitive to value of the friction coefficient
@. An addition of a lubricant results in the increase of free-poured density. However, in
dhdnishing @ after some packing down the further increase of density does not occur. It
implies that the effective surface u(x) is given in its upper part by (2.2) and near the very
narrow vicinity of the bottom (in the vicinity of the closed random packing) by (2.I). Mean-
field theory leads to the same conclusions (see [I] Sect. 3).
3. The size and dense segregation~
The separation of a mixture of different grains in the size and mass under the action of stirring has already been referred (see [3, 7, 10, 14-16] in [I]). As yet the nature of this phenomenon,
on the microscopic level at any rate, is assumed to be unknown [10]. We propose the simplest an additional to ii0] mechanism of segregation basing on the concept of the effective
surface.
Let us consider a test particle of a larger or smaller size as regards to other particles of monodisperse powder. The local geometric configurations inside which a shift of the test grain
could occur are shown in figure I. The vertical component of the momentum vector
transferred to the test particle is seen to depend on the grain size. The more the diameter D' of the test particle is compared to the diameter D of all the other grains, the more is the
vertical component of a momentum transferred to the former, other factors being equal, and the more is « a tendency » to emerge for test grain under shaking.
In terms of the effective surface it means that in a polydispersive powder for a given grain
its form u(x) depends on relative size of the grain : u
= u(x, D'). The larger (smaller) the diameter D' of the grain, the smaller ~larger) the sloping of the walls » of the effective surface
M lo BEHAVIOR OF POWDER UNDER PERTURBATIONS : II l171
qualitatively shown in figure I. Note for the sake of completeness that the direction of the
passed to the test grain initial momentum depends on the distance i~ between the grains in layers of powder in which a test grain is (see [I] Figs, 1-3) as well. In a monodisperse powder,
the average distance i~ between grains in random horizontal layers depends on powder density and, hence, on the intensity of perturbation. In accordance with the results sections 3, 4 of [I] in the powder obtained after prolonged vibration with amplitude A and frequency
w this distance I, can be estimated (see [1] 4.1-5, 6.4) as follows
I,(A,
w )
=
1+ 2 ix,(A, w ) (3.2)
~ ~) ~ ~j~
~ ~
~
, '~
~ _-_
"
i~=j
~ l'
~, i
,
~
i ~
j /
i j
Fig, I. Local configurations larger of packing inside which a shift of the shaded grain of a) a larger, b)
a smaller size can occur ; c) typical configurations of packing in the mixture of large and small grains, if their concentration is commensurable. The position a) of a large grain (or of a grain in a closed packing of equal ones) and b) of a small particle (or of a grain in a loose packing of equal ones) on the effective
surface u(x). Due to the only size effect large grain has « tendency » to emerge and small on has
« tendency» to sink down under shaking of a given intensity, as well as the more density (the less distance f between grains in a layer), the more « tendency » to swell for powder of equal ones (see ill)-
Here I is the typical distance between neighbors in a layer of the closed random packing of
equal grains, the distance x~ is the equilibrium for the given perturbation A, w coordinate of
« the average grain on the effective surface, which for the two-dimensional powder of discs and three-dimensional powder of spheres is equal to
x~ = Da pm ~A/g(I +
a ~)
x~ = D (a g/#rw ~A)'/~/2 a respectively. (3.3)
Here g is the gravity acceleration, D is the powder grain diameter and
a ~l and
#r ~ l are the dimensionless parameters characteridng the forrn of the effective surface ii]
sections 4-6.
172 JOURNAL DE PHYSIQUE II N 10
In the real situation the behavior of a test grain depends not only on the local geometry of packing but also on its mass m'. As well as the geometry of test grain packing determines the
direction of a passed momentum, its mass m' does the module of the latter. To summarize, whether a test particle sinks or comes to the surface under stirring depends on its size, first, on
its mass, second, and, finally, on the quilibrium for the given perturbation distance (3.2-3)
between grains of medium I, (or, put in another way, on the characteristics of perturbations).
To obtain quantitative results, the simplest case will be dealt with. Sizes of large and small
particles are supposed to be commensurable quantities. All the other characteristics except the size and mass of a test grain and grains of powder are assumed to be equal. The correlation between the presence of the test grain and the geometry of packing of other grains
is neglected. particularly, the average distance between the grains of the medium near the test
grains is equal to the average distance I, far from the test grain. Cohesion forces between any two grains moving apart (if an instantaneous configuration of packing permits to do it) are
considered to turn to zero. In this case the vertical component VI of the velocity
V' obtained in an elastic collision by the test grain with the size D' and mass
m' from a grain of powder with the size D and mass m is equal to
(i~~~'
) (D
~j'
)~
~~ ~~'~~
Here V is the average value of the velocity passed during a repacking under weak
perturbation from one powder grain to another grain. The latter was introduced in [I]
sections 2, 3 in the description of the concept of the effective surface.
If the vertical component of the initial velocity (3.4) is more than the corresponding component of the powder grain, it is natural to expect that the test grain will come up to the surface under the shaking. On the opposite case the test particle Mill sink. The folloving
condition of the equilibrium
(i fi(A,
w )/D2)"2
=
~~ ~
(,
~~ (i 4 ii(A,
w )/(D + D~ )2)"2 (3.5)
determines the separation curve in the coordinates (D'/D, m'/m, I,(A,w)/D). The
separation curve and the section of the most interest are shown in figure 2. A small though
less dense, grain is seen to sink, but a large, even though denser, grain is seen to arise to the top « in spite » of Archimed's law. Near the coordinate origin, where the discrepancy in sizes of large and small grains is great, small ones simply filter down through the voids whatever
O.O O. 4 O. 8
j
't
, ~
Z~$
~ ~'
f
~ ~ ~~ ~'
/o)2
*
Fig. 2. The phase diagranl of separation and its toposight : the particles having parameters above the suTface shall sink, in the opposite case they will emerge.
N 10 BEHAVIOR OF POWDER UNDER PERTURBATIONS : II l173
their mass and shaking intensity. Vfhen large and small grains are of comparable size, the
segregation mechanism has a more complicated nature and demands a disturbance of a certain intensity.
Let us consider a polycomponent powder consisting of the n sorts of grains with masses m~ sizes D,, the number grains of the r-th sort being Nr where r
= 1, 2,
..., n. Under shaking
in gravity field the mixture can be separated into f homogeneous phases as follows
mj, m2;.
,
m
~
dj, d~,..
,
d,, NIP),NjP~,...,N,~P), where r
= 1, 2,
,
n, p = 1, 2,..
,
f and
Nli)~~~r(2)~_ ~~~ru)_~~r
The following question arises : how many phases can be obtained after an infinitely long shaking of a certain intensity? In the framework of effective surface concept it can be
answered, provided that
Nj»N~». N,.
Let grains of sorts jj,
~ dissolve in grains Nj of the first sort under perturbations of the given intensity in accordanie with condition (3.5). Some part of another sorts ii,
~,__ ~
N;~ ~ N;~ » N; (3.6)
let sink and second part of the rest sorts rj,~,_,
N, » N,~ » » N, (3.7)
let go to the top under the same perturbation. On the next step one investigates the sinking (or emerging) mixture of grains (3.6). In accordance with formula (3.5) the sinking mixture
shall be segregated into other three mixtures too : the first mixture will consist of those sorts
i~,__ ~ which dissolve in the grains of the ij-th sort, the second is formed by those sorts
i~,,,_,~ which sink in respect with the grains of the ij-th sort and third one is a mixture consisting of the remaining sorts i~
_,_,~
which emerge in respect with the grains of the
ij-th sort. The above reasons can be iontinued for all the
new mixtures once more and so on.
One makes sure that after all we obtain the following phase rule for powders
f « n (3.8)
Here f is number of separating phases which can be attained by prolonged stirring of a n- component powder (n is number of different sorts in initial mixture). Let us emphasize that the number f of phases depends not only on the grain sorts but also on the intensity of the
disturbance (for example, on the value of
w
~A for vibration). Compared with the well-known Gibbs phase rule
f«n+2
forrnula (3.8) means that degrees of freedom associated with temperature and pressure do not exist in granular materials (1).
(~) « A hot bucket of sand is mush the same as a cold one », Mechta, Edwards (1989).
