BREAKTHROUGH PRESSURE FOR RANDOM SPHERE PACKINGS R A Y M O N D P. I C Z K O W S K I
Research Division, Allis-Chalmers, Milwaukee, Wis.
The pressure required to force mercury into random sphere packings is evaluated from the distribution function representing probable positions of neighbor particles and the penetration pressure of mercury between four spheres in contact.
pressures have been derived for openings between four spheres in contact by Mayer and Stowe (7) and Frevel and Kressley ( 3 ) . I n applying these results to determining the breakthrough pressure into a randomly packed aggregate of spheres, arrangements of cells with six (four-sided) faces were assumed which correspond to the distributions of neighboring spheres shown in Figure 1 A for the Mayer-Stowe model and Figure 1 B for the Frevel-Kressley model. The separations shown are between sphere surfaces and are expressed in units of sphere radii. The number of spheres in contact is given by the strength of the delta function a t zero. The distances in the range 0 5 s _< 2(21/2-1) represent separations on the smaller diagonals of the faces, while distances greater than 2(21j2-1) represent the corresponding larger diagonals of the same faces or the separations of spheres which are not nearest neighbors. I n the Frevel-Kressley model two of the six faces have zero separation while the other four have a separation greater than the analogous six faces of the Mayer-Stowe model. An experimentally determined ( 6 ) radial distribution function of dense randomly packed spheres was converted to units of radii and number per unit radius interval, multiplied by 4r(2 s ) ~ ,and the smallest interval was divided into discrete and continuous portions. The resulting distribution of spheres, shown in Figure lC, consists of a delta function a t zero, representing the number of spheres in contact, and a continuous distribution of separations of the smaller diagonals. T h e behavior of the continuous portion of the distribution function was extrapolated (4) into the interval of the bar representing the closest separation on the histogram of Mason and Clark (6) and the number of spheres represented by the area under the curve in this region was subtracted from the number represented by the bar to yield the number of spheres in contact with a given sphere, 5.85. Bernal (7) gives 6.4 as the number of spheres in contact with a given sphere. The total number of spheres around a central sphere within distances from zero to 2(2112-1) radii is 10.6 from the work of Mason and Clark and 13.5 from the work of Scott ( 8 ) . The cell model is simple enough to analyze mathematically and yet it yields reasonable results for the penetration pressure as a function of porosity (3, 7). Observations of the structure of packed spheres show that many cells of the kind assumed in the model actually occur ( 3 ) . The model accounts for triangular as well as quadrilateral faces, except that the triangular faces are constrained to occur in pairs sharing one side, and lying in the same plane. The number of spheres in contact with a given sphere and the number of spheres in the interval from zero to 2(21/2-1) radii which are expected for the model are 6 and 12, respectively. These values agree with the corresponding values obtained from the experimentally REAKTHROUGH
A
(4)
0
I
t; z
0 3
m
+
2
C
+
SEPARATION BETWEEN S P H E R E S U R F A C E S , s (UNITS OF RADIUS)
Figure 1.
Distribution of sphere separations
A. 8.
Mayer-Stowe model Frevel-Kressley model C. Dense random packing Strengths of delta functions;hown in parentheses. same for A, B, and C
Ordinates and abscissas
determined distribution functions, to the precision with which the latter are known. T h e distribution function of interparticle distances was used previously to determine the void volume of annular rings around points of contact and near contact in an aggregate of packed spheres containing mercury a t high pressure (4). I t is of interest to take into account the effect of the distribution function on the penetration pressure and the much larger volume changes which accompany filling of the cells in the cell model. Calculation
Penetration Pressure. If mercury enters a cell through one face, it will pass out through one of the other five faces a t the same pressure, provided the sphere separation on one of those faces is greater than or equal to the separation on the face through which it entered. This situation corresponds to VOL.
6 NO. 2
MAY 1967
263
a branching process ( 2 ) in which the number of branches is equal to five. The average number of new cavities which are filled from a typical cavity is p = 5p, where p is the probability that one of the exit faces will be larger than the entrance face. Mercury in cavities filled in the first generation may subsequently pass through any of five more faces to fill cavities in the second generation, etc. The breakthrough in mercury penetration occurs when the filling of the cavities becomes “chain reacting,” which corresponds (2) to fi = 1 or p = l / 5 . Characteristic of such processes ( Z ) , the amount of penetration will be very small until p = 1, after which it becomes very large. If the sizes of the faces are randomly distributed, the probability, p , that the exit size will be greater than any given value, s, is equal to the ratio of the number of separations greater than s, to the total number of separations of the smaller diagonals, which is six:
T h e area under the curve was evaluated by summing the discrete values from the histogram of Mason and Clark in the region s < 0.7413 and by integrating the curve dN/ds = 2.775 29.5(0.8284 - s ) ~ which , was fitted by the method of least squares to the last 11 points of the data of Mason and Clark and for the region 0.7413 5 s 5 2(21/2-1). Setting p = solving for s yields a separation s = 0.50, corresponding to breakthrough. The pressure a t which entrances of this size or greater will be filled can be obtained from the penetration pressure as a function of the relation derived by Mayer and Stowe shown in Figure 2. At s = 0.5, the reduced pressure P R l y , where X is the radius of the spheres and y is the surface tension, is 4.3 if the contact angle is 140’ and is 4.7 if the constant angle is 150’. The observed values (3, 5) are 4.06 and 4.0 for mercury penetration. Volume Penetrated. A face of a cell is said to be open if the separation of the spheres on that face is large enough to permit mercury penetration a t a given pressure. Otherwise, the face is closed. A cell having all closed faces will be empty. A cell having some open faces will also be empty if it is part of a closed configuration of cells-Le., a configuration of cells around which a boundary of closed faces can be drawn. An open face is said to be capped on a given side if the requirements for a closed configuration are satisfied on that side of the face. A face is capped on a given side if the adjacent cell on that side has all its other faces either closed or capped. T h e probability that a given face is capped is
+
w = ro
+
rlw
+ rzw2 + r3w3 + rpw4 + rgw5
(2)
where r k is the probability that the adjacent cell has exactly k other open faces (3) where
(;)
n!
k!(n - k)!
and w xis the probability that k open faces are capped. fore, w is the solution of the quintic equation rgw5
+ r4w4 + 73w3 + r2w2 + (rl - 1)w + ro = 0 5
(4) There-
(5)
From the identityxk=O r k = 1, w = 1 is aroot, and Equation 5 264
I&EC FUNDAMENTALS
2t OO
.I .2 .3 .4 .5 .e SEPARATION B E T W E E N SPHERE SURFACES,S (UNITS
I OF
.8 RADIUS)
.9
Figure 2. Breakthrough pressure as a function of separation between sphere surfaces
can be reduced to a quartic equation of which w is the positive real root. r5w4
+ + (76
r4)w3
+ + +d w 2+ (rs + r4 + + rz)w - ro (75
74
r3
=
o
(6)
A given cell will belong to a closed configuration if all of its faces are either closed or capped. T h e probability that a given cell belongs to a closed configuration is (7) where tk is the probability that the cell has exactly k open faces Q =
(3
p”1 - p ) 6 - k
W can be calculated as a function of p. Since p is a function of s, and s is a function of P R / y , W can be calculated as a composite function of P R / y . W represents the fraction of the cells remaining empty. If the volumes of the cells are randomly distributed, then W also represents the fraction of the total void volume remaining empty. W is plotted in Figure 3 for assumed contact angles of 140’ and 150’, along with the experimentally determined (3, 5) mercury penetration curves. Surface Effect. T h e calculation of the volume penetrated is valid in the limit that the ratio of the area over which the mercury originally is in contact with the sample to the volume of the sample approaches zero when the pressure is zero. I n most experiments this ratio is appreciable, but small, as evidenced by the small amount of penetration which occurs a t pressures below the breakthrough pressure. Mercury penetrates only a few cells lying near the surface. T h e need to consider the surface effect can be circumvented by repeating the experiment a t several values of the ratio of the contact area of the mercury with the surface of the sample
Or
surface, u is the average volume of the cells, and V 2is the total = Nu, where N is the total number of void volume. Since spheres (or cells),
vk
v=--
I
-1n(5p)
N,
N
Calculated upper limits for the surface penetration are shown for contact angles of 140’ and 150’ in Figure 3 taking N , / N = 0.01, which corresponds to the surface to volume ratio for Frevel and Kressley’s curve. Discussion
Figure 3. tion
T h e calculated penetration curves are in moderately good agreement with the experimentally determined curves, except that the calculated penetration pressure is 17% larger than the experimentally determined value, if the contact angle is 150°, and the calculated curve decreases more rapidly to zero void volume than the experimental curve. The difference in the penetration pressures may be due to the distribution function data which do not cover the entire range of integration and which appear to underestimate the distribution function when compared with the data of Scott ( 8 ) ,or to the neglect of nonplanarity of the cell faces and the lack of five- and higher sided cell faces in the theory, or it may be due to mechanical vibration of the sample (5) which causes the mercury meniscus to pass through openings which would otherwise be too small. I n deriving the theoretical curves, it was assumed that all of the branches from a given cell lead to other unfilled cells. Taking into account the decrease in the number of open faces due to intersections of the branches with each other may result in theoretical curves which decrease more slowly to zero.
Theoretical and experimental mercury penetra-
to the volume of the sample and extrapolating the results to a zero ratio. An upper limit to the surface penetration can be obtained by considering the penetration as a branching process neglecting the closed configurations. T h e validity of this approximation decreases as the volume penetrated increases. The expected size ( 2 ) of the nth generation of branches is ( 5 ~ ) ~For . a large sample, the total number of cells pene-
literature Cited
(9)
(1) Bernal, J. D., Mason, J., Nature 188, 910 (1960). (2) Feller, W., “Introduction to Probability Theory and Its Applications,” Vol. I, p. 223, Wiley, New York, 1950. (3) Frevel, L. K., Kressley, L. J., Anal. Chem. 35, 1492 (1963). (4) Iczkowski, R. P., IND. ENG. CHEM.FUNDAMENTALS 5 , 516 (1966). ( 5 ) Kruyer, S., Trans. Faraday SOC. 54, 1758 (1958). (6) Mason, G., Clark, W., Nature 207, 512 (1965). (7) Mayer, R. P., Stowe, R. A , , J . ColloidSci. 20,893 (1965). (8) Scott, G. D., Nature 194,956 (1962).
where NBis the number of spheres (or faces of the cells) a t the
RECEIVED for review June 27, 1966 ACCEPTED Jmuary 11, 1967
trated from an opening a t a given pressure is -l/ln(5p).
so-
(5~)~dnor
T h e fraction of the total volume penetrated is
ANALYTICAL ESTIMATES OF THE PERFORMANCE OF CHEMICAL OSCILLATORS J. M. DOUGLAS AND N. Y . G A I T O N D E Department of Chemical Engineering, University of Rochester, Rochester, N . Y .
processes which have constant inputs but generate periodic outputs have long been of interest to chemists and chemical engineers. Most of the early work in this field was concerned with chemical reactions in batch systems, so that the amplitude of the fluctuations decreased with time as the reactants became exhausted. Interesting reviews of much of this material were published by Frank-Kamenetrkii (4)and Hedges and Myers (7). Alternatively, when a system having continuous inputs is HEMICAL
designed to produce a stable periodic output, the amplitude of the fluctuations will not be damped and the oscillations will continue indefinitely. Aris and Amundson ( 7 ) observed this type of behavior when they were studying the feedback control of a nonisothermal continuous stirred tank reactor (CSTR). They found that the size of the limit cycles [closed trajectories in the ( X I ? X ~ )phase plane] decreased as the controller gain was increased and that oscillations disappeared when controller gain was high enough to make the equilibrium point stable. VOL. 6
NO. 2
M A Y 1967
265
