Void Fraction of Multi-Component Randomly Packed Beds with Size Distributions t

Michitaka Suzuki and Toshio Oshima Department of Chemical Engineering Himeji Institute of Technology* Hisaki lchiba Shin Nippon Kouki Co., Ltd.** Isamu Hasegawa Kokuyo Co., Ltd.*** Experimental measurements and computer simulation are reported on the void fraction of multi-component randomly packed beds with log-normal, log-uniform, Rosin-Rammler or Andreasen (Gaudin-Schuhmann) size distributions. Packing experiments were done by two methods, i.e. prodding and tapping. Glass beads (spherical particles) and crushed glass (irregularly shaped particles) were used for these experiments. The following results were obtained from the experiments and the simulations. A minimum void fraction of a packed bed with Andreasen distribution exists in the range of Fuller constant q = 0.50.8, and the void fractions of packed beds with the other three distributions decrease as the size distribution spreads. The experimental and the simulated results for void fraction were compared with values calculated by our model and agreement between them was fairly good. This model is useful for estimating the void fraction in a multi-component randomly packed bed of spherical and irregularly shaped particles with size distribution.


Introduction
for mathematical analysis.Real packing state, however, involves particles with a variety of particle-size distribution which has much effects on packing characteristics.
As well known, packing of particles is one of the fundamental characteristics of powder bed.In particular, packing of spheres with equal radii has been investigated by many workers because it is convenient to deal with t This report was originally printed in Kagaku Kogaku Ronbun-shu, 11, 438-443 (1985) in Japanese, before being translated into English with the permission of the editorial committee of the Soc.Chemical Engineers, Japan.
Void fraction of randomly packed particles of multi-components with particle-size distribution has been determined by Anderegg 1 ), Sohn and Moreland 8 ), Kawamura et al. 4 ), and Arakawa and Nishino 2 l.Each of them has been obtained by using a few kinds of material with a single particle-size distribution through one packing method.Unfortunately, there appears to be no report presenting how void fraction depends on the kind of material and its particlesize distribution or on the packing method.
The aim of the present work is to study experimentally the influence of particle-size distribution, particle shape and packing method on void fraction of randomly packed beds of multi-component system.For the packing experiments, materials used are glass beads (spherical shape) and crushed glass particles (irregular shape) with several different particle-size distributions: log-normal, log-uniform, Rosin-Rammler, and Andreasen (Gaudin-Schuhmann) distributions.Each experiment employs both prodding and tapping for packing particles.
Experimentally obtained data are compared with the calculated results of a mathematical model for estimating void fraction in a multicomponent system which was originally built for three-component (different in size) system 6 ), and also computer simulation.

Material
Two kinds of material with different shape were used as samples: spherical glass beads and irregularly shaped crushed glass particles-tr with a density of 2.5 x 10 3 kg• m-3 • After classified by sieve into 16 segments indicated in Table l, each of the materials were mixed at a given ratio to prepare multi-component packed bed with an arbitrary particle-size distribution.The mixed particles were dried at 120° C over 2 hours and then got cool naturally in desiccator, before being used for experiments which were made in the laboratory room conditioned at 20°C and a relative humidity of 50 percent.

Packing method
The sample particles were packed by prodding and tapping of which detail was stated in the previous paper 6 ).In each case of the present work, the particles arranged to have a given particle-size distribution were put into a container after they were mixed sufficiently in a vinyl bal-tr.After completion of packing, spare amount of particles were discarded and the mass of the whole sample was weighed to obtain the void fraction.Then, the particles were sieved for 1 2 minutes-tr-tr-tr and the mass of particles remained on each sieve was weighed to check the particle-size distribution after operation.
The computer simulation program for random packing of two-component spherical particles 5 ) was revised to extend the simulation to 4-to-20-component spherical particles.The total number of packed particles used for each simulation was 1000 to 2000.Table 1 lists the measured result of void fraction of packed beds involving uniform-size particles classified into 16 segments and the corresponding sieve openings that were averaged arithmetically.The void fractions € i were o btained for glass beads (tapping and prodding), computer simulation, and crushed glass beads (prodding and tapping) in order of quantity.The value of € i crushed glass particles increased with decreasing particle size, while that of glass bead was kept almost constant, independently of particle size.The estimate of e was carried out by using the values 'Ei and the mathematical model mentioned later.Computer simulation was executed by assuming 'Ei as a constant value of 0.423 that would be independent of particle size.

Mathematical model for predicting void fraction
Let us consider first a single particle in a packed bed consisting of m-component particles.When it contacts with surrounding particles, the number of combination is m 2 , as schematically indicated in Fig. 1.In this case, the void fractions of each core particle surrounded by several particles are defined as e( 1 , 1 ), e(l, 2 ), . .• , e(m,m) that can be calculated - -------(:m  on the basis of the geometrical data, for example, particle-size ratio and so forth (refer to Appendix 1 ).The mathematical model employed in the present work, similar to the model for three-component systems, permits combination of m 2 particle void fractions to express the value of e over the whole packed bed.
Let Sak be an fractional area of particle k and e u. k) be a partial void fraction around a particle j in direct contact with particles k.The partial void fraction ei around a particle j in mcomponent packed bed can be written as follows if ei is assumed to be the sum of the product of Sak and e(i,k) where {3i is the proportionality constant, which is derived from 'Ei measured in a packed bed that consists of only the assembly of particle j as follows.
The value of Sak> on the other hand, can be obtained by using a fractional volume of particle k, Svk: where Dpk and Dpi are the sizes of particle k and particle i, respectively.The values of Svk and Svi are calculated from cumulative undersize percentage, D-ti, of each size distribution.
The void fraction over the packed bed is finally obtained as follows, assuming that e is composed of the total summation of ei multiplied by Svi: The above expressions imply that e is obtainable from particle-size distributions and void fractions in a packed bed consisting of uniformly sized particles Ej.
* In the present work, it is assumed that the fractional mass is equal to fractional volume since the density of sample particles used has been ascertained to be constant.

Comparison of measured results with computer simulation and model equation
The present work dealt with packed beds consisting of multi-component system having typical four particle-size distributions: the distribution of log-normal, log-uniform, Rosin-Rammler, and Andreasen.The void fraction obtained from above each case is discussed in this section, in comparison with the results of computer simulation and model calculation.

1 Log-normal distribution
Log-normal distribution of particle size is defined as where D, Dp, Dp 50 , and In a g denote the cumulative undersize mass percentage; the particle diameter, the mass-based cumulative 50 percent particle diameter, and the standard deviation of the distribution, respectively.A packed bed consisting of multi-component system was prepared to satisfy the above di~~ribution.
Typical particle-size data of glass beads obtained by sieve classification after the experiments are found to nearly fall on the straight line in Fig. 2 that represents the calculated result of Eq. ( 5).This suggests the sufficient accordance of the experimental data with the log-normal distribution.
Based on this preparatory investigation, a lot of experiments and computer simulations were carried out in the wide range of the standard deviation In a g to gain the relationship between In a g and E, and to compare with calculated result derived from the mathematical model.Figure 3 shows the calculated results by curves and experimental or simulated ones by plotted circles (The same expressions are used in Figs. 5, 7 and 9).The void fraction E decreased in order of crushed glass particles (prodding and tapping), computer simulation, and glass beads (prodding and tapping), similar to ei referred to in Table 1.This implies that the void fraction would decrease in any case with increasing Fig. 3 Comparison between calculated, simulated and experimental void fractions for spheres and irregularly shaped particles with lognormal size distribution ln ag, that is, broadly spreading particle-size distributions.Moreover, the calculated values derived from the model are found to be in suitable agreement with experimental and simulated data.

2 Log-uniform distribution
This particle-size distribution is defined by where a and b are the slope and the constant, respectively, in a semi-logarithmic graph paper where the horizontal axis denotes the particle diameter Dp.Typical protted particle sizes of crushed glass particles used in the experiments are shown in Fig. 4 and found to be sufficiently consistent with the straight lines that represent the calculated results of Eq. ( 6).
Figure 5 shows the relationships of a and E obtained from experiments, computer simulations, and model calculations.The void fraction decreased with broadly spreading particlesize distribution, similar to the case of lognormal distribution.The effect of packing method was scarcely observed in the experiment for glass beads, whereas the effect of particle shape and packing method was similar to that of log-normal distribution.The calculated values derived from the model were in suitable agreement with experimental and simulated

3 Rosin-Rammler distribution
This distribution is expressed as where n and De are the distribution constant and the absolute size constant, respectively.Figure 6 shows an example of the measured results of glass beads.It is ascertained that these data would be fairly consistent with the straight lines representing the calculated result of Eq. (7).A large number of experiments and simulations were conducted by varying the n value to obtain the relationship between n and E, and compare with the calculated result of the model.As shown in Fig. 7, E decreased with decreasing n and the calculated result of the model was again in satisfactory agreement with the experimental and simulated data.As for the three particle-size distributions mentioned above, every distribution permitted the void fraction to be reduced with broadly spreading size distributions.This is probably because the void fraction would decrease with broadly spreading distribution due to the enlarged ratio of the largest particle size to the smallest one and the presence of smaller particles in the unoccupied space among larger particles.The minimum void fraction was at- Fig. 7 Comparison between calculated, simulated and experimental void fractions for spheres and irregularly shaped particles with Rosin-Rammler size distribution 20 tained in the case of the log-uniform distribution; yet the difference of it among the three distributions was insignificant, i.e., in the range of 0.005 to 0.01, which was insufficient to clarify the effect of the distributions.

4 Andreasen (Gaudin -Schuhmann) distribution
The maximum and the minimum particle sizes depend upon the shape of the particlesize distribution that was referred to in the previous sections.The following expression of particle-size distribution presented by Andreasen is based on the state where the maximum and the minimum particle sizes are fixed and the mixing ratio of particles varies.
Dp q where q is the Fuller constant, which has influence on the fractional volume of each size; the rise and drop of this value provide the increasing volume fraction of larger and smaller particles contained, respectively.As examples of measured results of crushed glass particles is shown in Fig. 8.The measured data are found to be in good agreement with curves which represent the calculated result of Eq. (8).A lot of experiments and simulations were conducted by varying the Fuller constant, q, to obtain the relationship between q and E, KONA No. 4 (1986) if'- .!!! and compare with the calculated result of the model.As shown in Fig. 9, the void fraction became minimum in any case within the Fuller constant range of 0.5 to 0.8.This is probably because unoccupied spaces among particles would decrease at q of 0.5 to 0.8 which means the furthest state from packing particles of equal size.It was also reported by Kawamura et al. 4 ) that the void fraction could become minimum at q = 0.6.The result of computer simulation is found to be less dependent on the Fuller constant, compared with the experimental data.This is due to the difference in the ratio of maximum particle size to minimum one, DPmax / DPmin; the ratio was 12.2 in the experiment and 4. 7 5 in the simulation.The calculated result of the model was in good agreement with the results of the experiments using glass beads and computer simulation.The measured values of void fraction in the case of crushed glass particles, however, were larger than that of the calculation by 0.02 to 0.04.This is because of the occurence of segregation at charge and the inapplicability of the assumption of uniform packing in the model.

Conclusion
In order to determine the void fraction, randomly packed beds of multi-component system were prepared which have several different particle-size distributions, that is, lognormal, log-uniform, Rosin-Rammler, and Andreasen distribution.Also two kinds of Fig. 9 Comparison between calculated, simulated and experimental void fractions for spheres and irregularly shaped particles with Andreasen (Gaudin-Schuhmann) size distribution packing method were employed: prodding and tapping.In addition to such a packing experiment, computer simulation for randomly packed spheres was carried out.The results obtained in the present work are as follows.
First, the smallest value of void fraction was obtainable at the Fuller constant of 0.5 to 0.8 in the case of Andreasen distribution.Any distribution except Andreasen distribution allowed the void fraction to decrease with increasing broadly spreading distribution; yet, the dependance on the kind of distribution was not obvious.
Second, the void fractions were determined for crushed glass particles (prodding and tapping), computer simulation, and glass beads (nearly independent of packing method) in order of quantity.
Finally, the comparison of the experimental and simulated data was carried out with the three-component model already presented by the authors by extending it to multi-component systems.This model was ascertained to hold in applications since the calculated values of it were in good agreement in the experimental and simulated data.proportionality constant in Eq.(A-5) proportionality constant in Eq. ( 1) void fraction in a multi-component mixture partial void fraction around particle j partial void fraction around particle j in direct contact with particle k void fraction in a uniform-sized particle bed geometric standard deviation in lognormal distribution function

Fig. 1
Fig. 1 Types of contact in a multi-component mixture

Fig. 2
Fig. 2 Examples of log-normal size distribution for glass beads

Fig. 5
Fig. 4 Examples of log-uniform size distribution for crushed glass

Fig. 6
Fig. 6 Examples of Rosin-Rammler size distribution for glass beads

u
Fig. 8 Examples of Andreasen (Gaudin-Schuhmann) size distribution for crushed glass

1 B 1 q 1
UndeterminatedconstantinEq.(A-7) [m1 b constant in log-uniform size dis-percent based on mass absolute size constant in Rosin-Rammler size distribution function diameter of particle, i, j or kin a multi-component mixture mass median diameter maximum particle diameter minimum particle diameter number of component in a mixture coordination number on particle j in direct contact with particle k cordination number in a uniformsized particle bed distribution constant in Rosin-Fuller constant in Andreasen size distribution function [ ~ Sak fractional area of particle k in a multiv €j component mixture fractional volume of particle k in a multi-component mixture volume of a hypothetical sphere in Fig. A-1 volume of a cut sphere in Fig. A-1

Table 1
Particle sizes and void fractions used for packing