Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979 25
Interparticle Percolation: a Statistical Mechanical Interpretation Michael H. Cooke and John Bridgwater" Department of Engineering Science, Oxford University, Oxford OX 1 3PJ. England
For mixtures of free-flowing particles, the quality is controlled in part by interparticle percolation, the drainage of a smaller species through a larger under the action of strain. Employing data from a test cell, it has been found previously that the dimensionless percolation rate of isolated small spheres is controlled by the diameter ratio of small to large particles. Here it is shown that this relationship may be explained using statistical mechanics by supposing that a large particle has a fluctuating amount of free space associated with it and that percolation occurs if the length scale of this space exceeds a value sufficiently great for the small particle to pass.
Introduction When particulate materials of different types are mixed together, a random distribution of the components is rarely produced and deterioration of the mixture can occur on subsequent handling. During deformation it has been shown (Roscoe, 1970) that failure zones, 10-15 particle diameters thick, occur between blocks of material that are sliding over each other. In such zones interparticle percolation, the drainage of particles through the interstices between larger ones, takes place. Such percolation depends on the effects of gravity and strain on particles of different size, shape, or density and is thus important for individual particles in cohesionless materials or aggregates in cohesive materials. Fundamental studies of this process have been few. Willemse (1961) performed some poorly defined experiments in a stirred vessel. Later Campbell and Bridgwater (1973) studied percolation in a vertical failure zone adjacent to a roughened wall in a thin rectangular hopper. The only extensive studies have used simple shear apparatus developed from those for testing soils (Scott and Bridgwater, 1975; Bridgwater et al., 1978). A bed of particles was sheared so that its shape changed from a rectangle to a parallelogram and was thus analogous to a failure zone. The base plate was made from 1.5-in. phenolic resin spheres fixed to nine 0.5-in. diameter steel rods held in ball races in an undercarriage. These spheres rotate as the cell moves and allow the free passage of percolating particles up to about 15 mm diameter while retaining 18.6-mm bulk particles in the cell. Percolation rates were established by measuring the mean time taken for a single particle fed to the top of the cell to traverse the bulk and fall out the bottom. Transit times were generally measured using a light cell at the point at which the particle falls into the cell to start an electronic timer and a microphone mounted on a tray under the cell to stop the timer. A typical estimation of the residence time was the mean of about 500 readings. Percolation rates were measured from the slope of the linear relationship found between residence time vs. bed height, thus eliminating end effects. The percolation of particles depends mainly on the size of the percolating and bulk particles. Experiments with bulk particles of various sizes confirmed experimentally a dimensionless presentation of results. Theory A percolating particle progresses through the bulk particles which are undergoing random motion. Eventually this motion produces a hole under the percolating particle into which it can then move. Each bulk particle is assumed to lie in a cage, the limits of which are the other bulk 0019-7874/79/1018-0025$01 .OO/O
particles. A characteristic of each cage is the time-dependent quantity R, the longest distance between the surface of the particle and the extremity of the cage. If this distance is greater than some critical value R*, the percolating particle is able to squeeze past the bulk particle in the cage and thus fall a distance related to the diameter of the bulk particles. A similar suggestion has been followed in studies of molecular transport in liquids and glasses (Cohen and Turnbull, 1959). The frequency distribution of R may be divided into regions q, each region having average value R,. If, at some instant, Nq is the number of separations in region q , then
C NqRq = N&
(1)
4
where N b is the number of bulk particles and R is the average value of R. If the bulk particle diameter db is increased, R will increase and these are taken to be proportional, i.e. k,db = R
(2)
where k , is a constant. Also
EN, 4 The number of ways distributed is
= Nb
(3)
WRin which the lengths can be Nb! w, = IIN,!
(4)
9
whence In W , = In Nb! -
9
In Nq!
(5)
Using Stirling's approximation and differentiating d In W R =
C In Nq dN,
(6)
4
From the constraints on the system, eq 1 and 3 XdN, = 0
(7)
ERqdNq= 0
(8)
4
9
From eq 6-8, introducing Lagrangian multipliers a1and cy2, at equilibrium In Nq + a1 + a2Rq= 0 or 0 1979 American Chemical Society
26
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
\
Now
\
\
I\\ X '
\
and hence
\ x\
\ \ \
Now, from eq 1 and 10, passing to the continuum limit CN,R, 4 = LmNba2e-a2RqdRq = N$ -31
whence a2 = 1/R. The probability p(R = R,) that a value of R has value R, is
0.1
I
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
I 9
dP/db
Figure 1. Plot of h u/;ldb vs. d / d b ; d b = 18.6 mm, ;l= 0.4 s-l, normal stress on top surface = 1.4 kd/m2; bulk particles, phenolic resin; percolating particles, acrylic resin.
N, 1 exp(-R,/R) dR, _-Nb R and thus the probability that R exceeds R* is
X
p(R
I
> R*) = L I p ( R = R,) dR, = exp(-R*/R)
I
10 -
(12)
I
I
I I
Assume that each time the percolating particle finds a gap greater than R* it falls a distance k& where kf is a constant. The number of times this happens in a given interval is proportional to p(R > R*) and the rate of strain ?. Thus, if k, is another constant
I I /
I
8-
I
Vdb -
I
/
Y
U
6-
- = k, exp(-R*/R)
I
I
I
I
/x
?db where u is the percolating particle velocity. If R* is proportional to the percolating particle diameter d,, Le., R* = k,d, where k, is a constant, then (13)
or In (U/?db) =
kh
- (k,d,)/(k,db)
(14) I
where k h is a constant. A graph of In (u/?d$ vs. d,/db should therefore be linear with slope -k,db/R. Experimental data (Bridgwater et al., 1978) are plotted in this way in Figure 1 and agreement between the theory and experimental results is excellent. The slope of the line is about -8, which if k, is about 1, implies that the mean separation R is about one-eighth of the bulk particle diameter, a physically reasonable value. A restriction of this model is that it does not take into account the possibility of a percolating particle rising due to its being forced upward by the movement of the bulk particles, as would occur at high diameter ratios. If d, = db the percolation velocity should be zero, whereas the theory predicts a positive value. The theory would also not apply to smaller particles where the motion is controlled by spontaneous percolation (Bridgwater et al., 1969). The dependence of percolation velocity on particle size ratio has also been modeled by Scott and Bridgwater (1975). They argued that for a percolating particle to drop through the bulk material, one of the bulk balls below it must move approximately (d - d,) in a specified direction where d, is the diameter of tke largest particle which can percolate spontaneously. The time taken for this to occur is approximately db/u. Then, as the mean square lateral
0.3
Figure 2. Plot of ydb/u vs. [(d,/db) - 0.155]'. Conditions as for Figure 1.
displacement in the x direction is 2E,t, where E, is the bulk particle diffusion coefficient in the x direction 2E,db/u = ( d p -
(15)
Since for spheres d,/db = 0.155
Using the more recent, reliable and extensive data (Bridgwater et al. 1978), a plot of dbj//U vs. ( ( d /db) 0.155)2 is curved (Figure 2) and it is deduced t i a t the earlier analysis is inadequate. In conclusion, the primary factor in determining dimensionless percolation rates has been shown experimentally to be the diameter ratio of percolating to bulk particles. This relationship can be explained by employing statistical mechanics. Nomenclature db, diameter of bulk particles ( L )
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
d,, diameter of percolating particles ( L ) d,, diameter .~ of largest particle that can percolate spontaneously ( L )
a1,cy2,Lagrangian
27
multipliers
i.,rate of shear strain
(Tl)
Literature Cited
E,, bulk particle diffusion coefficient ( L 2 T 1 ) kf,k,,kh,k$,, constants Nb, number of bulk particles in system N,, number of bulk particles having separation in region q p , probability q, region R , largest distance between surface of bulk particle and extremity of cage containing bulk particle (L) R*, critical value of R which, if exceeded, will lead to percolation ( L ) 8, mean value of R ( L ) R value of R in region g ( L ) L f i , number of ways of distributing the lengths R, Greek Letters
Bridgwater, J., Sharpe, N. W., Stocker, D.C., Trans. Inst. Chem. Eng., 47, T114 (1969).
Bridgwater, J., Cooke, M. H., Scott, A. M., Trans. Inst. Chem. Eng., 56, 157 (1978).
Campbell, A. P., Bridgwater, J., Trans. Inst. Chem. Eng., 51, 72 (1973). Cohen, M. H., Turnbull, D., J. Chem. Phys., 31, 1164 (1959).
Roscoe, K. H., Geotechnique, 20, 129 (1970). Scott, A. M., Bridgwater, J., Ind. Eng. Chem. Fundam., 14, 22 (1975). Willemse, Th. W.. Chem. Weekbl., 57, 377 (1961).
Received for review February 8, 1978 Accepted September 6, 1978
The work was made possible by the provision of an S.R.C.research grant and by a studentship from the S.R.C. and Thorn Lighting Ltd.
Experimental Behavior of Falling Liquid Films at High Surface Tension Numbers Myron A. Hoffman* and Winston W. Potts Department of Mechanical Engineering, University of California, Davis, Davis, California 956 76
The application of thin, falling liquid films of lithium to cool the inner walls of future laser fusion reactors has been investigated. In order to extend the existing experimentalresults on falling films to the higher surface tension numbers characberistic of alkali metals such as lithium, experiments using hot water on a vertical plate were run, primarily in the wavy laminar flow regime. Results show that very low minimum flow rates can fully wet the surface provided that the surface is carefully cleaned and preconditioned. Both the wave inception distance and the equilibrium wave arnplitude decrease as the surface tension number increases for a given flow Reynolds number. Based on these results, the scaling laws for the minimum wetting rate, the wave inception distance, and the equilibrium wave amplitude have been extended up to surface tension numbers of about 10 000.
Introduction This research was motivated by the possible application of falling liquid lithium films to protect the inner walls of a future laser fusion reactor from the micro-explosion products. Detailed studies of the effects of short pulses of photons and ions impinging on solid first walls made by Hovingh (1976) revealed some severe limitations. The energy deposition depth in first walls for the fusion micro-explosion products of a small pellet of deuterium and tritium is typically on the order of a few hundred micrometers for the CY particles, energetic deuterium and tritium ions, and the X-rays. Furthermore, the particle energies are deposited on a time scale of the order of a microsecond, while the X-rays are deposited over an even shorter time. This implies that very high pressure and temperature peaks may be reached in the front layers of the first wall material, which can severely limit the permissible micro-explosion energy yield if spalling is to be avoided. The use of thin films of liquid lithium to protect the first wall was proposed in a pioneering study by Booth (1973) at LASL (Los Alamos Scientific Laboratory). The present research program grew out of a modified version of the LASL wetted-first-wall concept proposed a t LLL (Lawrence Livermore Laboratory) and referred to as the suppressed-ablation coincept (Hovingh et al., 1974). Uniformity of the liquid coating is very important for these possible fusion reactor applications. If the coating 0019-7874/79/1018-0027$01.00/0
is locally too thin, too large a fraction of the ions and radiation will penetrate the liquid film and possibly cause spalling and/or sputtering of the solid first wall material. If the liquid coating is locally too thick, then excessive peak liquid temperatures can be reached in these regions, resulting in excessive ablation of the liquid lithium and larger pressure pulses to the substructure. (In addition, it is well known that if the liquid film is nonisothermal, dry spots can develop due to gradients in the surface tension coefficient.) In the first phase of this research reported here, we have concentrated on the hydrodynamics of the falling liquid films in the wavy laminar flow regime under near-isothermal conditions (Hoffman and Munir, 1976; Hoffman, 1977). Review of Thin Falling Liquid Flows Characteristics of Various Liquids. The basic properties of the more common liquids used in falling film research a t typical experiment temperatures are summarized in the upper part of Table I. The properties of some liquid metals (for which no experimental falling film data have been found) are given in the bottom part of the table. The key nondimensional parameters governing the isothermal falling film flow of pure liquids when heat and mass transfer effects are absent are the Reynolds number, N R E , the surface tension number No, defined as follows 4iib N Re = -4riz = - =4-r (1) -
-
PW
0
1979
American Chemical Society
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