Ind. Eng. Chem. Res. 1993,32, 2179-2182
2179
RESEARCH NOTES Limitation of Proposed Mathematical Models for the Porosity Estimation of Nonspherical Particle Mixtures A. B. Yu' School of Materials Science and Engineering, The University of New South Wales, P.O. Box 1, Kensington, N.S. W . 2033, Australia
N. Standish Department of Materials Engineering, The University of Wollongong, P.O. Box 114.4, Wollongong, N.S. W . 2500,Australia
The porosity of nonspherical particles may technically be estimated by a mathematical model for estimating the porosity of spherical particles. The applicability of the simplified packing model and the linear-mixture packing model recently proposed in the literature is investigated. It is found that, in spite of the reported successful application to some situations, the two models cannot generally be used to estimate the porosity of nonspherical particle mixtures.
Introduction In the past few years a number of mathematical models have been proposed for estimating the porosity of a particle mixture (Stoval et al., 1986;Yu and Standish, 1988,1991; Ouchiyama and Tanaka, 1989). In general these models predict the porosity of a particle mixture from the knowledge of (1)the particle sizes, (2)the initial porosity (defined as the porosity of uniformly or single sized particles), and (3)the volume fractions of the components involved. Strictly speaking, such amodel is only applicable to spherical particles packed under either loose or dense random packing condition, where the particle size and initial porosity of a component is clearlydefined. However, the particles involved in engineering practice are usually not spherical and particle shape has a strong effect on porosity. Therefore, it is very useful to develop a mathematical model for estimating the porosity of nonspherical particle mixtures. The similarity between the packing systems of spherical and nonspherical particles suggests that nonspherical particle packing may be related to spherical particle packing (German, 1989;Yu et al., 1992). It is therefore likely that a model for predicting the porosity of spherical particle mixtures can be used to predict the porosity of nonspherical particle mixtures. Such application has been attempted, and the satisfactory agreement between model predictions and experimental measurements has been reported for particles the shape of which is not much different from spherical (Cross et al., 1985;Ouchiyama and Tanaka, 1988;Standish et al., 1991). Moreover, to overcome the uncertainty in determining the "size" of a nonspherical particle and hence facilitate the application of this approach, the concept of equivalent packing diameter has been introduced in the particle characterization (Yu and Standish, 1992, 1993). It has recently been reported that, by use of this newly developed concept, the porosity of binary mixtures of nonspherical particles can satisfactorily be predicted from the packing result of spherical particles (Yu et al., 1992). It is considered that the above approach may not be limited to binary mixtures and should be applicable to multicomponentmixtures. Technically speaking, once the particle sizes (packing diameters) and initial porosity of
the components involved have been determined, the porosity of a nonspherical particle mixture can always be predicted by means of the models available in the literature (Stoval et al., 1986;Yu and Standish, 1988,1991;Ouchiyama and Tanaka, 1989). However, because all these models are based on the understanding of the packing of spherical particles alone,their application to nonspherical particle packing should be carefully investigated. In this paper it will be shown that the proposed models cannot generally be used to estimate the porosity of nonspherical particle mixtures in spite of the successful applications to some situations as reported in the literature (Crosset al., 1985;Ouchiyama and Tanaka, 1988;Standish et al., 1991).
System and Models Considered For convenience,we will only consider the latest version of the simplified packing model developed by Ouchiyama and Tanaka (1989)and the linear-mixture packing model by Yu and Standish (1991)which for convenience are, respectively, referred to as the OT and YS models. Furthermore, our discussion will be limited to binary mixtures. In this case, the porosity of a particle mixture t should be a function of initial porosity BL and €5, size ratio r (small/large), and volume fraction X L or XS (=1X L ) ,i.e., = f(fL,t$,r,XL)
(1)
where subscripts L and S represent, respectively, lwge and small components. Explicit forms of eq 1 in the OT and YS models have been detailed elsewhere (Ouchiyama and Tanaka, 1988,1989;Yu and Standish, 1991;Standish et al., 1991), and for brevity they are not repeated here, though a slight modification of the previous YS model has been made as given in the Appendix.
Results and Discussion For spherical particles, BL = es = to, and to is usually equal to either 0.40 for loose random packing or 0.36 for dense random packing (German, 1989). However, in order to obtain estimates which are not only qualitatively but also quantitatively comparable with the measurements, the value of this initial porosity may be purposely assumed
0888-5885/93/2632-2179$04.~0J0 0 1993 American Chemical Society
2180 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
as used by a number of investigators (Cross et al., 1985; Stoval et al., 1986; Ouchiyama and Tanaka, 1988, 1989; Yu and Standish, 1987,1988,1991; Standish et al., 1991). Figure l b and Figure ICshow the results when t o is equal to 0.5, calculated by the OT model and YS model, respectively. Because the two models are based on different assumptions, their predicted results may differ to some degree. However, it is quite obvious that the predictions are comparable to each other and match reasonably well the experimental measurements shown in Figure la. The experimental measurements were generalized from an extensive experimental data by means of Westman's empirical treatment (Westman, 1936; Yu et al., 1992). It can be seen from Figure 1 that the packing of binary mixtures of spherical particles is characterized by the following: (1) for a constant r, porosity decreases to a minimum and then increases with volume fraction X L ;(2) for a constant XL, porosity with a small size ratio is always lower than that with a large size ratio unless the size ratios involved are nearly equal to unity. This result is clearly illustrated in Figure 2. The reasonably good agreement between the two model predictions and the experimental measurement is also observed in this figure. It may be worth noting that the above two features for spherical particles are independent of the initial porosity to. For nonspherical particles, ELand es are usually different from each other due to the effect of particle shape. This difference may strongly affect the porosity values of particle mixtures. However, as shown by Yu et al. (19921, the Westman equation can still apply. By use of the equivalent packing diameter, the size ratio between any two components can always be determined. In this case, the porosity of binary mixtures of nonspherical particles can be predicted from the generalized experimental results of binary mixtures of spherical particles. Details of this work can be found elsewhere (Yu et al., 1992). Figure 3a shows the porosity results when e~ = 0.4and es = 0.8 for different size ratios. It can be seen that, due to the effect of the large difference in initial porosity, depending on the size ratio involved the minimum porosity does not always occur at binary mixtures. In fact it may be obtained when a pure component with relatively low porosity is used. However, for a constant volume fraction the maximum porosity should always be obtained when size ratio is equal to unity, as clearly shown in Figure 4. It is evident that Figure 4 is qualitatively comparable with Figure 2 if we only examine the experimental measurements. This result is expected because of the similarity between the packing systems of spherical and nonspherical particles. Unfortunately, this feature cannot be observed in both the OT and YS model predictions. In fact, as shown in Figure 4, the maximum porosity predicted by the OT model is obtained when r is equal to 0.35. This implies that mixing particles of different sizes may increase porosity. This is obviously contrary to the present understanding of particle packing. As a result, the model predictions given in Figure 3b are quite different from the experimental measurements in Figure 3a. This disagreement, as discussed by Yu and Standish (1991,1993), is mainly due to the assumption that the porosity of a particle mixture of the same (packing) size but different initial porosity is
rather than (3)
0.5
0.4
I
0.0
0.3
0.2
I
0.0
1
0.2
I
0.4
V I
I
0.6
0.8
1 .o
Fractional Solid Volume of Large Component
0.2 0.0
0.2
0.4
0.6
0.8
1 .o
Fractional Solid Volume of Large Component
0.5
0
.e
8 a
0.4
0.3
0.2 u.0
I
0.2
I
0.4
I
I
0.6
0.8
1 .o
Fractional Solid Volume of Large Component Figure 1. Comparison betweenthe calculatedand measuredporosity for spherical particle packing when EO = 0.5: (a) experimental measurement; (b) OT model prediction; (c) YS model prediction.
where Xi and ti are, respectively, volumefraction and initial porosity of component i and Vi is the initial specific volume corresponding to ti. Analysis of the literature measurements confirms that eq 3 is a more reasonable assumption
Ind,.Eng. Chem. Res., Vol. 32,No.9,1993 2181
a
0.8
0.7 0.50
0.6
i3
0.45
.e
P
B
E
P
0.5
0.4
0.40
0.2 -
tL
0.0
0.30 0.0
0.2
0.4
0.6
0.8
.
I
0.2
0.4
0.6
1 .o
0.8
Fractional Solid Volume of Large Component
1 .o
b Size Ratio Figure 2. Porosity of binary mixtures of spherical particles as a function of size ratio when EO = 0.5 and XL = 0.5 (refer to Figure 1).
(Yu and Standish, 1992, 1993). In other words, the treatment proposed by Ouchiyama and Tanaka (1988, 1989)to incorporate the initial porosity information into their previous model may not be adequate-at least when the equivalent packing size is applied. On the other hand, though the linear-mixture model has taken eq 3 into account, its predicted result does not match the experimental result well. In fact, as shown in Figure 4 also, there is an evident disagreement between the model prediction and measurement. This model is developed by incorporating the linear and the mixture packing models, and the connection between the two models as arbitrarily determined at r = 0.154 is not so smooth (Yu and Standish, 1991). This unsmooth connection does not result in much error for spherical particles where there is almost no difference in initial porosity. However, it can be exaggerated and result in a very large error if there is a large difference in initial porosity as clearly seen in Figures 3 and 4.
i 0.0
I
I
I
I
0.2
0.4
0.8
0.8
1 .o
Fractional Solid Volume of Large Component C
Conclusion Both the simplified and the linear-mixture packing models cannot generally be used for porosity estimhtion of nonspherical particle mixtures. It is considered that they can only be used for packing systems in which the maximum difference between initial porosity is relatively small, say less than 0.1. The two models should therefore be refiied to predict the porosity of a nonspherical particle mixture. This consideration is perhaps not only applicable to nonspherical particles but ale0 to fine particles because both the particle shape and the absolute size of fine particles (
