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The packing density of `perfect' binary mixtures
Powder Technology 103 Ž1999. 145–150
The packing density of ‘perfect’ binary mixtures S.M.K. Rassouly
1
EÕaluation Section, Department of Technical Cooperation, International Atomic Energy Agency, 5 Wagramers-trasse, P.O. Box 200, A1400 Vienna, Austria Received 29 May 1998; received in revised form 9 October 1998
Abstract Packing density of perfect binary mixtures of uniformly sized large and small spheres has been developed from the structural analysis of the mixture. Analogy has been made to a homogeneous random mixture from which the packing structure of large spheres has been allowed to expand due to the trapping of small ones in their contact zones. On this basis, the packing density of the local structure of the mixture for different numbers of small spheres trapping at the contact points of large spheres and the corresponding probability of occurrence have been determined by the application of statistics and theories of stochastic processes. The resulting expression has shown to be valid for all proportions of the components but for size ratios less than 0.3. A simpler expression for the packing density of perfect binary mixtures has been developed for all values of size ratios but for proportions of large spheres less than 0.6; this equation giving results which are indistinguishable from those of the more complicated expressions. The validity of these equations has been verified by experiments. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Packing density; Binary mixtures; Spheres
1. Introduction Packing density and homogeneity of powder mixtures have been proposed, and widely reported, to have significant effects on most powder processing operations and quality of the resultant products. Both subjects have received considerable attention in theory and practice though almost in complete isolation from one another. Therefore very little is known on the effect of mixture homogeneity upon the packing density of the particulate beds. Packing density or the equivalent porosity of particulate mixtures has been experimentally determined in many reports w1–12x. A number of mathematical models have been proposed w13–23x for the prediction of packing density from known characteristics of particulate mixture. Despite such widespread attempts to characterize the packing of particulate mixtures, experimental and theoretical explanation to the complex phenomena involved are not yet conclusive and final. The validity of some of the proposed models w16x have been demonstrated with only limited amount of data. Some of the model predictions w15,17,19–21x were not always in
1
Former ENTC Director, Esfahan, Iran.
good agreement with the experimental results, in which cases a corrective factor was needed w15x or further improvement to the model was thought necessary w22x. These experimental and theoretical approaches often fail to identify the mechanisms involved in the distortion of the packing of individual components when they interact in the mixture and so they resort to simplified or semi-empirical models of the real packing situations. Despite suspected occurrence of segregation in the experimental packing w12,15x, they also fail to account for the effect of mixture uniformity on the results of packing density. Experimental measurements of packing density of mixtures without any assessment on their homogeneity, have been widely used as those of uniform mixtures and corresponded to model prediction for such particulate beds. The failure or agreement of these model predictions with such experimental results are therefore in question and so it is not known for what level of mixture homogeneity these models hold. In these models the curve of packing density vs. proportion of component sizes consists of two w19,21,23x or three w13x different curves. These curves suggest a single point for maximum packing density where packing density variation falls from one curve to another. However a region of maximum packing density with continuous variation of packing density seems more logical.
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 8 . 0 0 2 2 3 - X
146
S.M.K. Rassoulyr Powder Technology 103 (1999) 145–150
In the present, theoretical treatment of packing density of binary mixtures, not only a continuous variation of packing density Žon a single curve. with respect to the proportion of components has been achieved, a somewhat flat region of maximum packing density, for certain size ratios, has been obtained. Theoretical treatment of the packing density of uniform binary mixtures should first resort to identifying the structure of a uniform mixture. On this basis, the treatment should then establish: Ø the extent of dilation of large particles packing due to the presence of small particles in the uniform mixture. Ø the extent of distortion of small particles packing due to the presence of large particles in the uniform mixture. With such approach, the theoretical treatment of the packing density of a uniform binary mixture has been reported in the present paper. Expected values of packing density of uniform mixtures have been estimated from the results of many experiments on the packing density of mixtures and compared with theoretical predictions.
2. Theoretical treatment Theoretical treatment of the packing density of uniform mixtures has so far been based on the structure of a homogeneous ordered mixture in which the position of particles is uniformly arranged for the sake of easier modeling. However highest attainable uniform mixture in practice is a homogeneous random mixture in which the particles are assumed to uniformly take up random arrangements. Achievement of such uniform mixture is also hardly practical since different particle properties and unideal mixing actions, in practice, can rarely meet the requirement of randomized displacement of individual particles. Therefore an analogy of a perfect real mixture, as the highest attainable uniformity in practice, to a homogenous random mixture is inevitable and well adequate for the purpose of this treatment. In this analogy, local solid proportion of components, on a unit cell 2 scale, at any point in the mixture, is assumed to be randomly arranged around the mean local concentration; this in turn being equal to solid proportion of components in the whole mixture. Therefore the local packing density of a perfect real mixture of large and small particles is considered equal to the overall packing density of the mixture while the packing density of the unit cells are distributed around the overall packing density according to a frequency function, which is correlated to the random variation of small particles in the unit cells. 2 A disorderly packed bed of uniformly sized spheres may be considered to be built up of a large number of unit cells; a unit cell being a volume which contains one complete sphere. The corresponding coordination numbers and void volumes are distributed according to specified frequency distributions.
To facilitate easier realization of random arrangements of particles in the unit cell of a perfect real mixture, small particles, when mixing with the large ones to achieve a uniform mixture, are considered to approach the unit cell, one by one. Each small particle should then, theoretically, be allowed to take up a position at the contact points of large particles Žcausing dilation of the large particles. or in the corresponding void space; each with certain probabilities. On consideration of a unit cell of large spheres with Nc contact points, in which Np small spheres are to be packed due to the mixing actions, small spheres approach such contact points with certain probabilities. This on average results in the trapping of some small spheres between the large ones, which in turn results in the expansion of the unit cell and the corresponding void volume Žcausing reduction of packing density.. Consequently, new probabilities for new trapping arrangements of small spheres at the next transition are achieved. Considering a two dimensional presentation of the proposed situation ŽFig. 1., the center of a small sphere should theoretically be allowed to take up a position in the shaded volume, causing the dilation of large spheres packing. Hence, theoretically, the probability of dilating the large spheres packing in the unit cell, when small spheres are introduced, is the ratio of volume represented by the shaded area to the total void volume. Considering the geometry of the unit cell, the following calculations can easily be done. Ž1. Volume expansion due to trapping of k 1 small spheres at one contact point can be calculated w24x from the geometry of all possible arrangements and the corresponding probabilities. Ž2. Total contact volumes and the void volume expansion, VeŽ k ., due to all combination of k small spheres trapped at different contact points, considering the corre-
Fig. 1.
S.M.K. Rassoulyr Powder Technology 103 (1999) 145–150
sponding probabilities, have been calculated w24x in order to calculate the probability of Ž k q 1.th sphere getting trapped at a contact point at the next transition. Ž3. If each random approach of a small sphere to the unit cell is considered as a transition and resulting number of trapped spheres is considered as a state of the system, then system characteristics of the unit cell appear to conform with the behavior of a homogeneous Markov Chain process Žparticle displacements and resulting states are discrete-valued parameters.. In each transition l small spheres may leave the contact points thus Ž l q 1. small spheres are probable, according to Ž l q 1. term binomial distribution 3 of trapping probability calculated in step 2, to get trapped or fall in the void. On application of the theories of Markov Chain and using the above probability distribution, states transition matrix of the system can be determined. The number of ‘binomial expansion terms’ should be decided when comparing the calculated packing densities obtained through this treatment with those of experimental results. This parameter which controls the behavior of mathematical model as compared with the real system, will be called the l parameter, where l is the maximum number of small spheres which may leave contact zones in one transition. With such treatment the probability distribution of states Žunit cell containing Np small spheres, k of which are packed at contact points. can be calculated. Ž4. All the elements are now determined for the calculation of the packing density of the unit cell except for the density with which small spheres pack in the void and on the surfaces of large spheres. To this effect, the extent of distortion of the packing density of small spheres in the mixture was assumed to be directly proportional to the size of small spheres Ž r . and the total surface area of large spheres Ž S v . on which small spheres are packed and inversely proportional to the void volume between the large spheres, Ž Vv .; i.e., rS v Ž P2 y P2 m . A Ž 1. Vv where P2 and P2m are the packing densities of small spheres in a comparatively large container and in the mixture, respectively. With this assumption, the packing density of small spheres in the mixture, for two distinct cases, was shown w24x to be: P2 m s P2 y ASP1r Ž 1 y P1 . ,
Ž 2.
when small spheres are less than enough to fill the voids between the large spheres. X
P2 m s P2r 1 q A SFr Ž 1 y F . ,
In these equations, F is the proportion of large spheres in the mixtures, S is the size ratio of smallrlarge spheres and A and AX are the constants of proportionality to be obtained through experimental verification. P1 is the packing density of large spheres in a comparatively large container. Having determined the above values, the packing density of the unit cell was then derived w24x as Pm s Ž 1 q Np S 3 . r 1 q Np
q
Ý ksm c
ž
1 y P1
Np S 3
m cy1
P2 m
ms0
q
P1
Ý P Ž m.
Ve Ž k . 4r3p R 3
/
PŽ k. ,
Ž 4.
where P Ž m. and P Ž k . are the states probability distribution and the term VeŽ k .rŽ4r3p R 3 . is calculated as explained in step 2 above. m c is the critical number of small spheres, above which the number of trapping results in an expanded volume large enough for all small spheres packing in the void of the unit cell. Ž5. Now the calculated packing density of the unit cell should be averaged over the random variance of Np and distribution of Nc to yield the packing density of perfect binary mixture. It has been shown w24x that in practical cases, only an approximate estimation of random variance, with negligible values, is possible. Therefore, since calculation of Pm for one value of Nc and Np involves a large amount of data processing then Nc and Np are assumed to be constant and equal to 8 w25,26x Žfor close and near contacts. and Ž1 y F .rŽ S 3 F . w24x respectively. Hence Pm is assumed to predict the packing density of a perfect binary mixture. It has been shown w24x that application of Eq. Ž4. is limited to the region of size ratios below 0.3. Now, let us consider a perfect real mixture in which the proportion of components are such that no two large spheres are in contact. In such case the specific solid volume of the mixture Ž1rPm . is equal to the solid volume of large spheres plus the specific solid volume of small spheres in the mixture, i.e., 1 Pm
s F q Ž 1 y F . rP2 m
Ž 5.
in which P2m is obtained from Eq. Ž3..
3. Experimental investigations
Ž 3.
when small spheres are more than enough to fill the voids between the large spheres. 3
147
Position of a small sphere and the corresponding probabilities has dual states; either in a contact zone or in the void volume.
The objectives of the experimental investigations are to: Ø Verify the validity of Eqs. Ž2. and Ž3.; Ø Obtain experimental values for the packing density of perfect binary mixtures with different proportions and size ratios of components, in order to compare them with those calculated from Eqs. Ž4. and Ž5..
S.M.K. Rassoulyr Powder Technology 103 (1999) 145–150
148
Fig. 2.
Towards the objective of the first investigation, the cavity between the cubical array of eight steel balls, of 1.5 inches in diameter, was blocked from sides and bottom, to simulate a curved container Žas an analogy to a real mixture situation. for the determination of small spheres’ packing density in the mixture Ž P2m .. The results of six to ten times repeated tests for nearly 25 different glass beads are plotted in Figs. 2 and 3. Although there are considerable scatters in the experimental results, which is logical specially for large S, reasonable straight line correlations are evident. Such correlations confirm the assumption made for derivation of Eqs. Ž2. and Ž3. and the following least square lines are shown in Figs. 2 and 3. P2 m s 0.62 y 0.56SP1r Ž 1 y P1 . Ž 6. P2 m s 0.63r 1 q 0.6SFr Ž 1 y F . Ž 7. Although the validity of these equations were verified by the ‘cubical packing experiments’ and the resulting data did not very closely fit the lines, but the packing density of small particles so determined is a much better approximation than the customary assumption in the previous works that the packing density of small particles in the mixture is equal to the value when they are packed in a large container alone. Determination of the packing density of perfect real mixture in practice poses great difficulties both in attaining such highly homogeneous mixture and also defining its quality of mixedness, to which the measured packing density is to be attributed. To avoid the above difficulties and in line with the purpose of further investigation aimed at determination of packing density of non-uniform mixture, the following scheme was adopted. The packing density of several mixtures of large and small spheres with different uniformity was treated both theoretically and experimentally to obtain the trend of packing density variation with respect to degree of homogeneity of the mixture. 4 The extrapolated value of packing density which followed this trend to the maximum
degree of homogeneity was taken as the expected packing density of a perfect real mixture. Experiment on the packing density of mixtures of different homogeneity was carried out in a special apparatus with different combinations of glass bead powders. This apparatus consisted of 10 rectangular hollow plates of Perspex, which were clamped together with a bottom closing plate through two accurately fitting guide bolts, to form a cubical container. Specified proportion of two uniformly sized powders with different respective mean size ratios were mixed in the container. The bulk density of the bed was measured and then thin brass sheets were slipped between the loosened perspex plates to slice the mixtures into a number of equal layers. The material of each layer was then overtured into a special divider screen with equally exact geometry of the plates. With this procedure, the material of each layer was further divided into 36 samples. Consequently the whole mixture Ždivided in to layers. was divided into nearly 300 samples, from which 30 were selected at random. Experiments were carried out to verify the accuracy of sampling technique, specially for the extent of disturbances involved. Analysis of the results of these experiments revealed minimal disturbances and good accuracies of sampling technique w24x. Definition of some criteria to specify the degree of homogeneity of a mixture Ž M . has been the subject of a long standing debate. However, the following criterion was shown w24x to be most appropriate and adequate for the purpose of our treatment. M s 1 y s 2rso2
Ž 8.
in which s 2 is the samples variance and so2 in the unmixed state variance, i.e., F Ž1 y F .. Observed packing densities plotted against degree of homogeneity of the mixture Ždetermined as above. for one value of S and F is presented in Fig. 4. Solid and broken curves are the theoretical values of two different mixing models presented in the earlier mentioned paper.
4
Paper under preparation titled ‘Packing density of imperfect binary mixtures’.
Fig. 3.
S.M.K. Rassoulyr Powder Technology 103 (1999) 145–150
149
controlling parameter in reflecting the behavior of the hypothetical model, as compared to experimental results, which through minimization of the squared difference between experimental and calculated results, was obtained to be three. Calculated packing density of perfect real mixture using Eq. Ž4. for different proportion of components at size ratios identical to experimental cases, are plotted in Fig. 5. It is seen that the theoretically determined curve for all F but S - 0.3 fits the experimentally determined points extremely well and so the validity of the theoretical treatment underlying Eq. Ž4. is confirmed. It was argued earlier that for F - 0.6 Eq. Ž5. can also give the packing density of perfect real mixture. Minimizing the squares of deviation between the packing densities given by Eq. Ž5. and the experimentally expected values resulted in the value of AX as 0.604 in Eq. Ž3., which is extremely close to the earlier finding for this parameter in the ‘eight ball packing’ experiments ŽEq. Ž7.. Calculated packing densities from Eq. Ž5. for all S and for F - 0.6 are also plotted in Fig. 5 which falls exactly on the curve obtained from the calculated values of Eq. Ž4. for the same size ratio and proportions of components. This coincidence of values given by these equations is an interesting confirmation to the validity of complicated reasoning underlying Eq. Ž4.. However great simplicity of Eq. Ž5. favors its application to the treatment of packing density of perfect binary mixtures for F - 0.6. A point of particular interest which has so far not been reported, is the nearly flat region of theoretical packing density curve for values of F between about 0.6 and 0.8 and for S between 0.1 and 0.2. This phenomenon can be explained by the fact that as S increases, a smaller proportion of the voids, associated with the large particles,
Fig. 4.
As can be seen from the results, the variation trend of the packing densities with respect to degree of homogeneity, both on the theoretical and experimental cases, can lead us to the expected value of packing density at M s 1.0. Packing densities of perfect real mixtures estimated in this way for several sets of results, different in size ratio and proportion of components, have been plotted in Fig. 5. Such estimated results have been used to verify the theoretical treatment of the particulate system presented earlier. As explained earlier the l parameter is noticed to be the
Fig. 5.
150
S.M.K. Rassoulyr Powder Technology 103 (1999) 145–150
becomes available for small particles packing. In the region of maximum packing density, an increase in the amount of small particles Ždecrease in F . results in the loosening of the packing of large particles Žas explained in the theoretical treatment., as a result a larger volume between the large particles becomes available for small particle packing. Hence the packing density, in this region, depends upon the balance between the increased packing density due to the filling of the voids between the large particles by the small particles, and the decrease in the packing density due to the loosening of large particles’ packing. The former process, for small size ratios, occurs to a greater extent while the latter predominates for larger size ratios. Theoretical and experimental findings of the present work regarding the flat region of maximum packing density are also confirmed by the experimental findings of previous workers w13x.
4. Conclusion The packing density of small spheres, in a mixture of large and small spheres, has been deduced by consideration of the extent of packing distortion brought about by the presence of large spheres. This is found to depend upon the proportion, size ratio and packing densities of the components. Calculated packing densities from these equations are in agreement with experimental results. Theoretical treatment of a perfect real mixture of large and small spheres, applying theories of statistics and stochastic process, has led to the values of packing densities which are in close agreement with the experimentally expected values. Such theoretical values, however, are valid for size ratios less than 0.3. Simpler theoretical treatment for the packing density of perfect real mixtures for F - 0.6 has shown similar results to those of more complex treatment and closely fitted to the experimental results.
As the size ratio increases, the form of the theoretical curve of packing density of a perfect real mixture plotted against the proportion of large particles, shows a flat region and a shift of its maximum to lower value of this proportion. These observations are explained on theoretical grounds and are in accordance with the present and previous observational evidence.
References w1x R.K. McGeary, J. Am. Ceram. Soc. 44 Ž1961. 513. w2x K. Srinagesh, M.R. Seshadri, A. Ramachandran, The British Foundryman, July 1964, p. 307. w3x R. Ben Aim, P. le Goff, Powder Technol. 1 Ž1967. 281. w4x C.C. Furnas, Ind. Eng. Chem. 23 Ž1931. 1052. w5x H.J. Fraser, J. Geol. 43 Ž1935. 910. w6x S. Yerazunis, J.W. Bartlitt, A.H. Nisson, Nature 195 Ž1962. 33. w7x W.H. Visscher, H. Bolsterli, Nature 239 Ž1972. 504. w8x A.E.R. Westman, M.R. Hugil, J. Am. Ceram. Soc. 13 Ž1930. 767. w9x N. Epstein, M.J. Young, Nature 196 Ž1962. 885. w10x J.E. Ayer, F.E. Soppet, J. Am. Ceram. Soc. 48 Ž1965. 180. w11x K. Ridgway, K.J. Trabuck, Chem. Process Eng. 49 Ž1968. 103. w12x E.K.H. Li, P.D. Funkenbusch, Mat. Sci. Eng. A 157 Ž1992. 217. w13x A. Wieckowski, F. Strek, Chemia Stosowana B 1 Ž1966. 95. w14x S. Yerazunis, S.W. Cornell, B. Winter, Nature 207 Ž1965. 835. w15x M. Leitzelement, C.S. Lo, J. Dodds, Powder Technol. 41 Ž1985. 159. w16x M. Cross, W.H. Douglas, R.P. Fields, Powder Technol. 43 Ž1985. 27. w17x M. Suzuki, T. Oshima, Powder Technol. 43 Ž1985. 147. w18x A. Marmur, Powder Technol. 44 Ž1985. 249. w19x T. Stovall, F. De Larrard, M. Buil, Powder Technol. 48 Ž1986. 1. w20x N. Standish, A.B. Yu, Powder Technol. 49 Ž1987. 249. w21x A.B. Yu, N. Standish, Powder Technol. 52 Ž1987. 233. w22x N. Standish, A.B. Yu, Powder Technol. 53 Ž1987. 69. w23x R.M. German, Metallurgical Trans. A 23 Ž1992. 1455. w24x S.M.K. Rassouly, PhD Thesis, University of London, King’s College, 1982. w25x W.O. Smith, P.D. Foote, P.F. Busang, Phys. Rev. 34 Ž1929. 1271. w26x K. Ridgeway, K.J. Tarbuck, Br. Chem. Eng. 12 Ž1967. 384.
The packing density of ‘perfect’ binary mixtures S.M.K. Rassouly
1
EÕaluation Section, Department of Technical Cooperation, International Atomic Energy Agency, 5 Wagramers-trasse, P.O. Box 200, A1400 Vienna, Austria Received 29 May 1998; received in revised form 9 October 1998
Abstract Packing density of perfect binary mixtures of uniformly sized large and small spheres has been developed from the structural analysis of the mixture. Analogy has been made to a homogeneous random mixture from which the packing structure of large spheres has been allowed to expand due to the trapping of small ones in their contact zones. On this basis, the packing density of the local structure of the mixture for different numbers of small spheres trapping at the contact points of large spheres and the corresponding probability of occurrence have been determined by the application of statistics and theories of stochastic processes. The resulting expression has shown to be valid for all proportions of the components but for size ratios less than 0.3. A simpler expression for the packing density of perfect binary mixtures has been developed for all values of size ratios but for proportions of large spheres less than 0.6; this equation giving results which are indistinguishable from those of the more complicated expressions. The validity of these equations has been verified by experiments. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Packing density; Binary mixtures; Spheres
1. Introduction Packing density and homogeneity of powder mixtures have been proposed, and widely reported, to have significant effects on most powder processing operations and quality of the resultant products. Both subjects have received considerable attention in theory and practice though almost in complete isolation from one another. Therefore very little is known on the effect of mixture homogeneity upon the packing density of the particulate beds. Packing density or the equivalent porosity of particulate mixtures has been experimentally determined in many reports w1–12x. A number of mathematical models have been proposed w13–23x for the prediction of packing density from known characteristics of particulate mixture. Despite such widespread attempts to characterize the packing of particulate mixtures, experimental and theoretical explanation to the complex phenomena involved are not yet conclusive and final. The validity of some of the proposed models w16x have been demonstrated with only limited amount of data. Some of the model predictions w15,17,19–21x were not always in
1
Former ENTC Director, Esfahan, Iran.
good agreement with the experimental results, in which cases a corrective factor was needed w15x or further improvement to the model was thought necessary w22x. These experimental and theoretical approaches often fail to identify the mechanisms involved in the distortion of the packing of individual components when they interact in the mixture and so they resort to simplified or semi-empirical models of the real packing situations. Despite suspected occurrence of segregation in the experimental packing w12,15x, they also fail to account for the effect of mixture uniformity on the results of packing density. Experimental measurements of packing density of mixtures without any assessment on their homogeneity, have been widely used as those of uniform mixtures and corresponded to model prediction for such particulate beds. The failure or agreement of these model predictions with such experimental results are therefore in question and so it is not known for what level of mixture homogeneity these models hold. In these models the curve of packing density vs. proportion of component sizes consists of two w19,21,23x or three w13x different curves. These curves suggest a single point for maximum packing density where packing density variation falls from one curve to another. However a region of maximum packing density with continuous variation of packing density seems more logical.
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 8 . 0 0 2 2 3 - X
146
S.M.K. Rassoulyr Powder Technology 103 (1999) 145–150
In the present, theoretical treatment of packing density of binary mixtures, not only a continuous variation of packing density Žon a single curve. with respect to the proportion of components has been achieved, a somewhat flat region of maximum packing density, for certain size ratios, has been obtained. Theoretical treatment of the packing density of uniform binary mixtures should first resort to identifying the structure of a uniform mixture. On this basis, the treatment should then establish: Ø the extent of dilation of large particles packing due to the presence of small particles in the uniform mixture. Ø the extent of distortion of small particles packing due to the presence of large particles in the uniform mixture. With such approach, the theoretical treatment of the packing density of a uniform binary mixture has been reported in the present paper. Expected values of packing density of uniform mixtures have been estimated from the results of many experiments on the packing density of mixtures and compared with theoretical predictions.
2. Theoretical treatment Theoretical treatment of the packing density of uniform mixtures has so far been based on the structure of a homogeneous ordered mixture in which the position of particles is uniformly arranged for the sake of easier modeling. However highest attainable uniform mixture in practice is a homogeneous random mixture in which the particles are assumed to uniformly take up random arrangements. Achievement of such uniform mixture is also hardly practical since different particle properties and unideal mixing actions, in practice, can rarely meet the requirement of randomized displacement of individual particles. Therefore an analogy of a perfect real mixture, as the highest attainable uniformity in practice, to a homogenous random mixture is inevitable and well adequate for the purpose of this treatment. In this analogy, local solid proportion of components, on a unit cell 2 scale, at any point in the mixture, is assumed to be randomly arranged around the mean local concentration; this in turn being equal to solid proportion of components in the whole mixture. Therefore the local packing density of a perfect real mixture of large and small particles is considered equal to the overall packing density of the mixture while the packing density of the unit cells are distributed around the overall packing density according to a frequency function, which is correlated to the random variation of small particles in the unit cells. 2 A disorderly packed bed of uniformly sized spheres may be considered to be built up of a large number of unit cells; a unit cell being a volume which contains one complete sphere. The corresponding coordination numbers and void volumes are distributed according to specified frequency distributions.
To facilitate easier realization of random arrangements of particles in the unit cell of a perfect real mixture, small particles, when mixing with the large ones to achieve a uniform mixture, are considered to approach the unit cell, one by one. Each small particle should then, theoretically, be allowed to take up a position at the contact points of large particles Žcausing dilation of the large particles. or in the corresponding void space; each with certain probabilities. On consideration of a unit cell of large spheres with Nc contact points, in which Np small spheres are to be packed due to the mixing actions, small spheres approach such contact points with certain probabilities. This on average results in the trapping of some small spheres between the large ones, which in turn results in the expansion of the unit cell and the corresponding void volume Žcausing reduction of packing density.. Consequently, new probabilities for new trapping arrangements of small spheres at the next transition are achieved. Considering a two dimensional presentation of the proposed situation ŽFig. 1., the center of a small sphere should theoretically be allowed to take up a position in the shaded volume, causing the dilation of large spheres packing. Hence, theoretically, the probability of dilating the large spheres packing in the unit cell, when small spheres are introduced, is the ratio of volume represented by the shaded area to the total void volume. Considering the geometry of the unit cell, the following calculations can easily be done. Ž1. Volume expansion due to trapping of k 1 small spheres at one contact point can be calculated w24x from the geometry of all possible arrangements and the corresponding probabilities. Ž2. Total contact volumes and the void volume expansion, VeŽ k ., due to all combination of k small spheres trapped at different contact points, considering the corre-
Fig. 1.
S.M.K. Rassoulyr Powder Technology 103 (1999) 145–150
sponding probabilities, have been calculated w24x in order to calculate the probability of Ž k q 1.th sphere getting trapped at a contact point at the next transition. Ž3. If each random approach of a small sphere to the unit cell is considered as a transition and resulting number of trapped spheres is considered as a state of the system, then system characteristics of the unit cell appear to conform with the behavior of a homogeneous Markov Chain process Žparticle displacements and resulting states are discrete-valued parameters.. In each transition l small spheres may leave the contact points thus Ž l q 1. small spheres are probable, according to Ž l q 1. term binomial distribution 3 of trapping probability calculated in step 2, to get trapped or fall in the void. On application of the theories of Markov Chain and using the above probability distribution, states transition matrix of the system can be determined. The number of ‘binomial expansion terms’ should be decided when comparing the calculated packing densities obtained through this treatment with those of experimental results. This parameter which controls the behavior of mathematical model as compared with the real system, will be called the l parameter, where l is the maximum number of small spheres which may leave contact zones in one transition. With such treatment the probability distribution of states Žunit cell containing Np small spheres, k of which are packed at contact points. can be calculated. Ž4. All the elements are now determined for the calculation of the packing density of the unit cell except for the density with which small spheres pack in the void and on the surfaces of large spheres. To this effect, the extent of distortion of the packing density of small spheres in the mixture was assumed to be directly proportional to the size of small spheres Ž r . and the total surface area of large spheres Ž S v . on which small spheres are packed and inversely proportional to the void volume between the large spheres, Ž Vv .; i.e., rS v Ž P2 y P2 m . A Ž 1. Vv where P2 and P2m are the packing densities of small spheres in a comparatively large container and in the mixture, respectively. With this assumption, the packing density of small spheres in the mixture, for two distinct cases, was shown w24x to be: P2 m s P2 y ASP1r Ž 1 y P1 . ,
Ž 2.
when small spheres are less than enough to fill the voids between the large spheres. X
P2 m s P2r 1 q A SFr Ž 1 y F . ,
In these equations, F is the proportion of large spheres in the mixtures, S is the size ratio of smallrlarge spheres and A and AX are the constants of proportionality to be obtained through experimental verification. P1 is the packing density of large spheres in a comparatively large container. Having determined the above values, the packing density of the unit cell was then derived w24x as Pm s Ž 1 q Np S 3 . r 1 q Np
q
Ý ksm c
ž
1 y P1
Np S 3
m cy1
P2 m
ms0
q
P1
Ý P Ž m.
Ve Ž k . 4r3p R 3
/
PŽ k. ,
Ž 4.
where P Ž m. and P Ž k . are the states probability distribution and the term VeŽ k .rŽ4r3p R 3 . is calculated as explained in step 2 above. m c is the critical number of small spheres, above which the number of trapping results in an expanded volume large enough for all small spheres packing in the void of the unit cell. Ž5. Now the calculated packing density of the unit cell should be averaged over the random variance of Np and distribution of Nc to yield the packing density of perfect binary mixture. It has been shown w24x that in practical cases, only an approximate estimation of random variance, with negligible values, is possible. Therefore, since calculation of Pm for one value of Nc and Np involves a large amount of data processing then Nc and Np are assumed to be constant and equal to 8 w25,26x Žfor close and near contacts. and Ž1 y F .rŽ S 3 F . w24x respectively. Hence Pm is assumed to predict the packing density of a perfect binary mixture. It has been shown w24x that application of Eq. Ž4. is limited to the region of size ratios below 0.3. Now, let us consider a perfect real mixture in which the proportion of components are such that no two large spheres are in contact. In such case the specific solid volume of the mixture Ž1rPm . is equal to the solid volume of large spheres plus the specific solid volume of small spheres in the mixture, i.e., 1 Pm
s F q Ž 1 y F . rP2 m
Ž 5.
in which P2m is obtained from Eq. Ž3..
3. Experimental investigations
Ž 3.
when small spheres are more than enough to fill the voids between the large spheres. 3
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Position of a small sphere and the corresponding probabilities has dual states; either in a contact zone or in the void volume.
The objectives of the experimental investigations are to: Ø Verify the validity of Eqs. Ž2. and Ž3.; Ø Obtain experimental values for the packing density of perfect binary mixtures with different proportions and size ratios of components, in order to compare them with those calculated from Eqs. Ž4. and Ž5..
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148
Fig. 2.
Towards the objective of the first investigation, the cavity between the cubical array of eight steel balls, of 1.5 inches in diameter, was blocked from sides and bottom, to simulate a curved container Žas an analogy to a real mixture situation. for the determination of small spheres’ packing density in the mixture Ž P2m .. The results of six to ten times repeated tests for nearly 25 different glass beads are plotted in Figs. 2 and 3. Although there are considerable scatters in the experimental results, which is logical specially for large S, reasonable straight line correlations are evident. Such correlations confirm the assumption made for derivation of Eqs. Ž2. and Ž3. and the following least square lines are shown in Figs. 2 and 3. P2 m s 0.62 y 0.56SP1r Ž 1 y P1 . Ž 6. P2 m s 0.63r 1 q 0.6SFr Ž 1 y F . Ž 7. Although the validity of these equations were verified by the ‘cubical packing experiments’ and the resulting data did not very closely fit the lines, but the packing density of small particles so determined is a much better approximation than the customary assumption in the previous works that the packing density of small particles in the mixture is equal to the value when they are packed in a large container alone. Determination of the packing density of perfect real mixture in practice poses great difficulties both in attaining such highly homogeneous mixture and also defining its quality of mixedness, to which the measured packing density is to be attributed. To avoid the above difficulties and in line with the purpose of further investigation aimed at determination of packing density of non-uniform mixture, the following scheme was adopted. The packing density of several mixtures of large and small spheres with different uniformity was treated both theoretically and experimentally to obtain the trend of packing density variation with respect to degree of homogeneity of the mixture. 4 The extrapolated value of packing density which followed this trend to the maximum
degree of homogeneity was taken as the expected packing density of a perfect real mixture. Experiment on the packing density of mixtures of different homogeneity was carried out in a special apparatus with different combinations of glass bead powders. This apparatus consisted of 10 rectangular hollow plates of Perspex, which were clamped together with a bottom closing plate through two accurately fitting guide bolts, to form a cubical container. Specified proportion of two uniformly sized powders with different respective mean size ratios were mixed in the container. The bulk density of the bed was measured and then thin brass sheets were slipped between the loosened perspex plates to slice the mixtures into a number of equal layers. The material of each layer was then overtured into a special divider screen with equally exact geometry of the plates. With this procedure, the material of each layer was further divided into 36 samples. Consequently the whole mixture Ždivided in to layers. was divided into nearly 300 samples, from which 30 were selected at random. Experiments were carried out to verify the accuracy of sampling technique, specially for the extent of disturbances involved. Analysis of the results of these experiments revealed minimal disturbances and good accuracies of sampling technique w24x. Definition of some criteria to specify the degree of homogeneity of a mixture Ž M . has been the subject of a long standing debate. However, the following criterion was shown w24x to be most appropriate and adequate for the purpose of our treatment. M s 1 y s 2rso2
Ž 8.
in which s 2 is the samples variance and so2 in the unmixed state variance, i.e., F Ž1 y F .. Observed packing densities plotted against degree of homogeneity of the mixture Ždetermined as above. for one value of S and F is presented in Fig. 4. Solid and broken curves are the theoretical values of two different mixing models presented in the earlier mentioned paper.
4
Paper under preparation titled ‘Packing density of imperfect binary mixtures’.
Fig. 3.
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controlling parameter in reflecting the behavior of the hypothetical model, as compared to experimental results, which through minimization of the squared difference between experimental and calculated results, was obtained to be three. Calculated packing density of perfect real mixture using Eq. Ž4. for different proportion of components at size ratios identical to experimental cases, are plotted in Fig. 5. It is seen that the theoretically determined curve for all F but S - 0.3 fits the experimentally determined points extremely well and so the validity of the theoretical treatment underlying Eq. Ž4. is confirmed. It was argued earlier that for F - 0.6 Eq. Ž5. can also give the packing density of perfect real mixture. Minimizing the squares of deviation between the packing densities given by Eq. Ž5. and the experimentally expected values resulted in the value of AX as 0.604 in Eq. Ž3., which is extremely close to the earlier finding for this parameter in the ‘eight ball packing’ experiments ŽEq. Ž7.. Calculated packing densities from Eq. Ž5. for all S and for F - 0.6 are also plotted in Fig. 5 which falls exactly on the curve obtained from the calculated values of Eq. Ž4. for the same size ratio and proportions of components. This coincidence of values given by these equations is an interesting confirmation to the validity of complicated reasoning underlying Eq. Ž4.. However great simplicity of Eq. Ž5. favors its application to the treatment of packing density of perfect binary mixtures for F - 0.6. A point of particular interest which has so far not been reported, is the nearly flat region of theoretical packing density curve for values of F between about 0.6 and 0.8 and for S between 0.1 and 0.2. This phenomenon can be explained by the fact that as S increases, a smaller proportion of the voids, associated with the large particles,
Fig. 4.
As can be seen from the results, the variation trend of the packing densities with respect to degree of homogeneity, both on the theoretical and experimental cases, can lead us to the expected value of packing density at M s 1.0. Packing densities of perfect real mixtures estimated in this way for several sets of results, different in size ratio and proportion of components, have been plotted in Fig. 5. Such estimated results have been used to verify the theoretical treatment of the particulate system presented earlier. As explained earlier the l parameter is noticed to be the
Fig. 5.
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becomes available for small particles packing. In the region of maximum packing density, an increase in the amount of small particles Ždecrease in F . results in the loosening of the packing of large particles Žas explained in the theoretical treatment., as a result a larger volume between the large particles becomes available for small particle packing. Hence the packing density, in this region, depends upon the balance between the increased packing density due to the filling of the voids between the large particles by the small particles, and the decrease in the packing density due to the loosening of large particles’ packing. The former process, for small size ratios, occurs to a greater extent while the latter predominates for larger size ratios. Theoretical and experimental findings of the present work regarding the flat region of maximum packing density are also confirmed by the experimental findings of previous workers w13x.
4. Conclusion The packing density of small spheres, in a mixture of large and small spheres, has been deduced by consideration of the extent of packing distortion brought about by the presence of large spheres. This is found to depend upon the proportion, size ratio and packing densities of the components. Calculated packing densities from these equations are in agreement with experimental results. Theoretical treatment of a perfect real mixture of large and small spheres, applying theories of statistics and stochastic process, has led to the values of packing densities which are in close agreement with the experimentally expected values. Such theoretical values, however, are valid for size ratios less than 0.3. Simpler theoretical treatment for the packing density of perfect real mixtures for F - 0.6 has shown similar results to those of more complex treatment and closely fitted to the experimental results.
As the size ratio increases, the form of the theoretical curve of packing density of a perfect real mixture plotted against the proportion of large particles, shows a flat region and a shift of its maximum to lower value of this proportion. These observations are explained on theoretical grounds and are in accordance with the present and previous observational evidence.
References w1x R.K. McGeary, J. Am. Ceram. Soc. 44 Ž1961. 513. w2x K. Srinagesh, M.R. Seshadri, A. Ramachandran, The British Foundryman, July 1964, p. 307. w3x R. Ben Aim, P. le Goff, Powder Technol. 1 Ž1967. 281. w4x C.C. Furnas, Ind. Eng. Chem. 23 Ž1931. 1052. w5x H.J. Fraser, J. Geol. 43 Ž1935. 910. w6x S. Yerazunis, J.W. Bartlitt, A.H. Nisson, Nature 195 Ž1962. 33. w7x W.H. Visscher, H. Bolsterli, Nature 239 Ž1972. 504. w8x A.E.R. Westman, M.R. Hugil, J. Am. Ceram. Soc. 13 Ž1930. 767. w9x N. Epstein, M.J. Young, Nature 196 Ž1962. 885. w10x J.E. Ayer, F.E. Soppet, J. Am. Ceram. Soc. 48 Ž1965. 180. w11x K. Ridgway, K.J. Trabuck, Chem. Process Eng. 49 Ž1968. 103. w12x E.K.H. Li, P.D. Funkenbusch, Mat. Sci. Eng. A 157 Ž1992. 217. w13x A. Wieckowski, F. Strek, Chemia Stosowana B 1 Ž1966. 95. w14x S. Yerazunis, S.W. Cornell, B. Winter, Nature 207 Ž1965. 835. w15x M. Leitzelement, C.S. Lo, J. Dodds, Powder Technol. 41 Ž1985. 159. w16x M. Cross, W.H. Douglas, R.P. Fields, Powder Technol. 43 Ž1985. 27. w17x M. Suzuki, T. Oshima, Powder Technol. 43 Ž1985. 147. w18x A. Marmur, Powder Technol. 44 Ž1985. 249. w19x T. Stovall, F. De Larrard, M. Buil, Powder Technol. 48 Ž1986. 1. w20x N. Standish, A.B. Yu, Powder Technol. 49 Ž1987. 249. w21x A.B. Yu, N. Standish, Powder Technol. 52 Ž1987. 233. w22x N. Standish, A.B. Yu, Powder Technol. 53 Ž1987. 69. w23x R.M. German, Metallurgical Trans. A 23 Ž1992. 1455. w24x S.M.K. Rassouly, PhD Thesis, University of London, King’s College, 1982. w25x W.O. Smith, P.D. Foote, P.F. Busang, Phys. Rev. 34 Ž1929. 1271. w26x K. Ridgeway, K.J. Tarbuck, Br. Chem. Eng. 12 Ž1967. 384.