338
Ind. Eng. Chem. Fundam.
1980, 19, 338-340
Estimation of the Average Number of Contacts between Randomly Mixed Solid Particles Norio Ouchiyama National Industrial Research Institute of Kyushu, Tow, Saga-ken, Japan
Tatsuo Tanaka’ Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan
A simple expression for the number of contacts between solid particles existing in a completely mixed packing of solid spheres is discussed theoretically based upon a simple packing model. Theoretical estimates of contact numbers show good correspondence not only with known facts concerning the packing of uniform spheres but also with past experimental data for random mixtures. For contacts between spheres of different size, computations from the new theory show a trend similar to computer simulation results. These results suggest a general application of the present theory.
In the course of developing our kinetic equation (Ouchiyama and Tanaka, 1974,1975) for the size distribution of solid particles due to granulation, it became necessary to have some knowledge of the number of contacts (pairs) existing in a completely mixed packing of various sized spheres. A simple expression for the number of contacts between spheres of arbitrarily selected sizes is of particular interest in the present paper and of primary importance for studies of the properties of powdered solids. Prior to the precise modeling of a completely mixed packing, some aspects of the coordination number will be referred to. Figure 1 illustrates the geometrical states of contact around a specified sphere surrounded by uniformly sized spheres. For a completely mixed packing of various sized spheres, the uniform diameter of the neighboring spheres would be replaced by the average diameter of the mixture. By comparing the coordination number around the specified sphere in Figure 1A with that in Figure lB, one expects, on the one hand, that the coordination number should be proportional to the spherical surface “area” of the central sphere. By comparing the coordination number around the sphere in Figure 1B with that in Figure lC, one expects, on the other hand, that the coordination number should be inversely proportional to the crosssectional area of a neighboring sphere. These considerations in reference to the area lead to a concept of twodimensional porosity as a characteristic parameter of the packing, that is, the “surface porosity” around a particle. Suppose a completely mixed packing system is characterized by the total number of spheres, N, the numberfrequency distribution of size, f ( x ) , and the surface porosity around a particle, tA. Here, the surface porosity is the void area fraction on a spherical surface of diameter x + a , where X is the average diameter of particles defined as R =
J m x f ( x ) dx
(1)
To describe the geometrical states of contact precisely, a simple packing model is introduced: every sphere in the system is in direct contact with neighbors having the average diameter (see Figure 2). The coordination number, C ( x ) , around a particle of size x is exactly expressed as
coordination” with the coordination number characteristic of the size of sphere, the total number of “points of coordination” in the system, CT,can be found.
CT = X m C ( x ) N f ( x dx )
The term “point of coordination” refers to a point at which two adjacent spheres have a possibility of making contact. In contrast with a segregated packing system, no selectivities in size are present in a randomly mixed packing, so that every point of coordination in the system should have an equal chance of Occurrence irrespective of the sizes of the spheres involved. Therefore, the probability that a specified point of coordination will be in contact with one of the points of coordination belonging to the size fraction between d and d + 6d should be equal to C ( d ) N f ( d ) 6d/CT. The number of contacts between the particles of size fraction D D + 6D and those of size fraction between d and d + 6d, denoted by n(D,d) 6D6d, is equal to the product of this probability and the number of the points of coordination around the spheres of size fraction between D and D 6D. It is expressed as
-
+
n(D,d) 6D6d = C ( D ) C ( d )W f ( D ) f ( d )6D6d (4) CT When the surface porosity can be assumed constant, eq 2, 3, and 4 can be rewritten as
CT = 16(1 - tA)N
3
+ [?/(a)2] 4
and n ( D , d ) 6D6d =
4 where
C ( X )= 16(1 - t By allotting to each sphere an equal number of “points of 0 196-4313/80/ 1019-0338$01.OO/O
(3)
0 1980 American Chemical Society
~ )
(6)
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980 338 B BY THE PREYNT THEORY
12 V
EXPERIMENTAL
0
0 :CALCULATED
8 -
4 Figure 1. Some aspects of the coordination number. Hatching calls attention to a central sphere considered in the theory. 0
0.2
0.4
6
0.6
0.8
1.0
Figure 3. Relationship between coordination number and volume porosity in the packing of uniform spheres. Experimental data for random packing by Smith et al. (1929) and Arakawa and Nishino (1973);calculated data for regular packing by Manegold et al. (1931). 2.0 I CO
=(
xlYt1 2
f
Figure 2. A simplified packing model.
By making use of the values of packing characteristics, it is now possible to evaluate the number of contacts between spheres of arbitrarily specified sizes. When the surface porosity can be regarded as constant, irrespective of the sizes of the spheres, the surface porosity tA is related to the volume porosity t of the packing of uniform sized spheres. For the packing of uniformly sized spheres, every sphere in the packing makes 16(1- tA) direct contacts with neighbors, each size x (see Figure 2). We approximate the volume porosity of the packing t by the fractional free volume within the spherical space of diameter 2x t = [ ~ ( 2 ~ )-~(7x3/6 /6 + 16(1 - tA)(13/16) X (7a3/Wll/ ~ ( 2 x ) ~ /(9) 6 or EA = (16t - 1)/13. The quantity (13/16)(ax3/12) is the exactly calculated volume contribution from a neighboring sphere. Past investigations have generally dealth with assemblages of uniform spheres by examining the relationship between the coordination number, C, and the volume porosity of the packing, t. According to the present theory, the relationship should be given by eq 9 as 32 C = 16(1 - EA) = - (7 - 8t) (10) 13 This relationship is shown in Figure 3, together with previous experimental data for random packing and calculated values for regular packings. Good agreement can be seen. Recently, Arakawa and Nishino (1973) reported experimental results from their studies of randomly packed mixtures composed of spheres having various sizes. By using their tabulated experimental data, the coordination number of each specified size divided by that of the average diameter, C ( x ) / C ( f ) is , plotted in Figure 4 against the dimensionless size ratio, x/R. In the figure, the theoretical relationship given by eq 5 is also depicted. Figure 4 also shows good agreement between the theory and experiments. A little higher estimate of the coordination number for larger particles suggests a possible existence of macropores around the bigger particles. In the present theory, however, this was omitted. When the surface porosity can be assumed to be constant, C ( f ) in eq 5 should be connected, as already mentioned, through the surface porosity to the volume porosity for the packing of uni-
Y 0
I
:
sample I1 I
05
l.Q 1.5 xlx
1
2.0
Figure 4. Relationship between coordination number of a sphere of specified size divided by that of a sphere of the average diameter and the dimensionless size ratio. Plotted from experiment of Arakawa and Nishino (1973, Table I, 11, and 111).
formly sized spheres. At present, however, reported data are not available for testing the theory in this respect. Investigations into the number of contacts between particles of arbitrarily selected sizes have never been carried out except for computer simulations. Kuno (1972) performed computer simulations on randomly packed circles of two different sizes in a two-dimensional container. He also examined the number of circles of different size around circles both of a larger size D (= 0.5) and of a smaller size d (= 0.2), respectively. According to the present theory, each of these numbers should be given as the quotient of n(D,d) 6D6d in eq 7 divided by either N f ( D ) 6D or N f ( d ) 6d. The variables f ( d ) 6d and f ( D ) 6 0 in the quotients are replaced by (1 - rD) and rD, respectively, where r D is the number fraction of the larger particles. Therefore, both the numbers ND and Nd can be calculated from
(D + f 1 2 (d + f12
4
[(I - r ~ or) r ~ (11) l where = L m x m f ( x )dx = DmrD + d" (1 - rD)
(rn = 1,2) (1la) Changes of each number with rD are illustrated in Figure 5, together with the simulation results of Kuno. Since the latter were obtained for a two-dimensional packing of circles, direct comparison with the present estimates has no direct meaning. However, the theoretical estimate that
340
Ind. Eng. Chem. Fundam. 1980, 79, 340-344
Nomenclature
C = coordination number in packing of uniform spheres,
dimensionless C ( x ) = coordination number in completely mixed packing,
dimensionless total number of points of coordination, dimensionless D , d = diameters of particles, m f ( x ) = size frequency distribution of particles, m-l n(D,d) = density function of number of contacts between particles, me2 N = total number of particles, dimensionless ND,.Nd = numbers of contacts between particles of different sizes, D and d , per central particle, dimensionless rD = number fraction of larger particles, dimensionless x = dummy variable for particle size, m f = average diameter of particles, m 2 = average square diameter of particles, m2 Greek Letters t = volume porosity of packing, dimensionless t A = surface porosity around a particle, dimensionless CT =
Figure 5. Number of contacts between particles of different sizes in a binary mixture (D= 0.5; d = 0.2) vs. number fraction of larger particles. Simulation data are replotted from the computer experiment of Kuno (1972, Figure 6).
values of No and Nd should be equal at rD = 0.5 can also be observed in the simulation data; additionally, the corresponding curves show similar trends. These results suggest general application of the present theory to various technological problems which may arise in connection with beds of solid particles.
Literature Cited Arakawa, M., Nishino, M., Zairyo, 22, 658 (1973). Kuno, H., Funtai Oyobi Funmatsuyakin, I S , 85 (1972). Manegold, E., Hofman, R.,Soif, K., KolloM Z.,56, 143 (1931). Ouchiyarna, N., Tanaka, T., Ind. Eng. Chem. Process D e s . Dev., 13, 383 (1974): 14, 286 (1975). Smith, W. O., Fwte, P. D.,Busang, P. F.. Phys. Rev., 34, 1271 (1929).
Acknowledgment
Miss Chieko Wakamatsu is gratefully acknowledged for typing the manuscript.
Received for review July 5 , 1979 Accepted June 19, 1980
Kinetic Study on Cycloaddition of Allyl Radical to Acetylene Daisuke Nohara' and Tomoya Sakal DepaHment of Chemical Reaction Engineering, Faculty of Pharmaceutical Sciences, Nagoya City University, Mizuho-ku, Nagoya 467, Japan
Formation of C5 hydrocarbons by addition of the allyl radical to acetylene was surveyed by use of diallyl oxalate as an allyl source at 430-510 O C . Cyclopentadiene amounted to more than 90% of total C5products, in distinct contrast to the addition of allyl radical to ethylene, which produced cyclopentene and 1-pentene in almost a 2 to 1 molar ratio.
c=cc.
+
C E C
- c=ccc=c. - 'Q -
c=cc*
+
c=c
- c=cccc*
-r-
*c, 0+ u
ti.
The overall second-order rate constant of cyclopentadiene formation with respect to allyl and acetylene, k = 1014.6 exp(-104000/RT) cm3 mol-' s-', was almost 10 times as large as that of cyclopentene formation in the ethylene case, k = exp(-48 000/RT).The rate-limiting step was concluded to be either a cyclization or an H. elimination step for the present reaction.
Introduction
It has been confirmed that the allyl radical plays a role of diene in cycloaddition with olefins analogous to butadiene which thermally combines with olefins to produce C6 cyclic compounds, as reported by Sakai et al. (1970).
Thus, Nohara and Sakai (1973) found the formation of C5 cyclic compounds such as cyclopentadiene and cyclopentene to be primary products in the pyrolysis of 1,5hexadiene (diallyl). Furthermore, Sakai and Nohara (1975) reported that cyclopentene and 1-pentene were produced
0196-4313/80/1019-0340$01.00/00 1980 American Chemical Society
