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Ind. Eng. Chem. Res. 1996, 35, 3730-3741

Modifying the Linear Packing Model for Predicting the Porosity of Nonspherical Particle Mixtures A. B. Yu* and R. P. Zou School of Materials Science and Engineering, The University of New South Wales, Sydney, New South Wales 2052, Australia

N. Standish Department of Materials Engineering, The University of Wollongong, P.O. Box 1144, Wollongong, New South Wales 2500, Australia

Based on the similarity analysis between the spherical and nonspherical particle packings, a mathematical model, modified from the previous linear packing model for spherical particles, is proposed for predicting the porosity of nonspherical particle mixtures. The background for this development is discussed in detail. The applicability of the proposed model is validated by the good agreement between the measured and calculated results for various packing systems including binary, ternary, and multicomponent packing of spherical and/or nonspherical particles. Introduction Of the many variables affecting the packing of particles, the particle size distribution is probably the most significant. The question “Is there an optimum particle size distribution that would maximize the packing density of particles” has attracted the interest of mathematicians, scientists, physicists, and engineers for a few centuries (Cumberland and Crawford, 1987). The answer to this question is directly related to the understanding of the relationship between porosity or packing density and particle size distribution. Since the turn of this century, namely, after the classical work of Fuller (1907), many theoretical and experimental attempts have been made to understand and hence model this relationship as summarized by Gray (1968), Cumberland and Crawford (1987), and German (1989). Theoretical models are always highly desirable, but their development is extremely difficult, if not impossible. To date, mathematicians are still struggling for an answer to the question “what is the maximum packing density for uniform spheres”sKepler’s sphere packing problem (Stewart, 1991, 1992). On the other hand, physicists are looking for a better understanding of the physics of particle packing, and this investigation is to a great degree limited to the packing of uniform spheres [for example, see Meakin and Skjeltorp (1993) and Bideau and Hansen (1993)]. With the application of computers, our understanding of particle packing has been advanced significantly. However, as pointed out by Dodds (1980), the computer simulation is in essence just one type of experimental measurements and cannot give any insight into the structure of a packing without the aid of some interpretative model. Therefore, the mathematical modeling, although it may be empirical to some degree, is most useful, particularly from the point of view of engineering application. In the past decade or so, significant progress has been made in the modeling of the relationship between porosity and particle size distribution for spherical particles (Dodds, 1980; Ouchiyama and Tanaka, 1981, 1986; Suzuki and Oshima, 1985; Stovall et al., 1986; Yu and Standish, 1987, 1988, 1991). The developed models have been found to be useful in the solving of optimum packing * Phone: + 61 2 9385 4429. Fax: + 61 2 9385 5956. E-mail: [email protected].

S0888-5885(95)00616-6 CCC: $12.00

problems (Cross et al., 1985; Ouchiyama and Tanaka, 1989; Standish et al., 1991; Yu and Standish, 1993a), the theoretical evaluation of porosity-related properties, such as permeability (Leitzelement et al. 1985; Yu and Zulli, 1994) and viscosity (Gupta and Sashadri, 1986; Patlazhan, 1993a,b), and the mineral process modeling (Bailey et al., 1987). It is known that the particles involved in engineering practice are usually not spherical, and particle shape has a strong effect on porosity. For uniformly sized particles, it has been well-established that porosity generally increases with the decrease of sphericity, which is defined as the ratio of the surface areas between a sphere and a particle of the same volume (Brown et al., 1950), and a recent study shows that this relationship is dependent on particle shape and packing method (Zou and Yu, 1996). For multisized or multicomponent mixtures of particles, the effect of particle shape on porosity has also been recognized for many years. However, compared with the almost exhaustive work on the packing of spherical or nearly spherical particles, the studies of the packing of nonspherical particle mixtures appear to be very limited. For example, examination of the book by German (1989) indicates that our present understanding of nonspherical particle packing is to a large degree limited to the sphere-fiber (or sphere-cylinder) binary packing, which was intensively investigated by Milewski (1973, 1978, 1987). For this binary packing system, Starr (1986) attempted to formulate a predictive equation with very limited success in terms of accuracy and generality. On the other hand, some investigators have demonstrated that the packing models developed for spherical particles may be applied to predicting the porosity of nonspherical particle mixtures [for example, see Cross et al. (1985) and Standish et al. (1991)]. However, as demonstrated by Yu and Standish recently (1993b), such a direct application can only be made to nonspherical particles with their shape not much different from spherical. Obviously, a more fundamental and systematic study is necessary in order to model the packing of nonspherical particles. In the past few years, some useful steps have been made in this direction that may be summarized as follows: (1) the similarity between spherical and nonspherical particle packings has been properly © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3731

established, which provides a solid basis for applying the well-developed understanding of spherical particle packing to the investigation of nonspherical particle packing (Yu et al., 1992; Zou et al., 1996); (2) a new concept of equivalent packing diameter has been introduced in the particle characterization to quantify this similarity and hence facilitate its application (Yu and Standish, 1992, 1993c; Zou and Yu, 1996); and (3) by use of this newly developed concept, the porosity of the binary mixtures of nonspherical particles can satisfactorily be predicted from the packing of binary mixtures of spherical particles, which confirms the applicability of the model framework from spherical to nonspherical particle packing (Yu et al., 1993). It is believed that a sound basis has been established for the modeling of the relationship between porosity and particle size distribution for nonspherical particles. The purpose of this paper is to present a modified linear packing model that can predict the porosity of nonspherical particle mixtures. The background and mathematical formulation of this model will be discussed first. The validity of this model will then be assessed by an extensive comparison between predictions and measurements for different packing systems. Theoretical Treatments System Considered. We will consider the system composed of n components of equal-density particles. Component i (i ) 1, 2, ..., n) has equivalent volume diameter dvi, sphericity ψi, and volume fraction Xi. Under a given packing condition that may be reflected by the specific volume of uniform spheres Vs, the specific volume of the considered system V can be generally expressed as

V ) f(Vs, X1, X2, ..., Xn, dv1, dv2, ..., dvn, ψ1, ψ2, ..., ψn) (1) The mixing interaction between two components is dependent on their relative size, which should be quantified in terms of equivalent packing diameter (Yu and Standish, 1993c). The equivalent packing diameter of a particle can be determined by measuring and then relating its size-dependent packing property to the diameter of a sphere as a result of the similarity between spherical and nonspherical particle packings. Component i should therefore have an equivalent packing diameter dpi which is a function of dvi and ψi, given by Zou and Yu (1996):

dvi ) ψi2.785 exp[2.946(1 - ψi)] dpi

(2)

For spherical particles, the porosity of monosized particles, i.e., initial porosity s related to Vs, should be constant, generally taking the value of 0.4 for the loose random packing or 0.36 for the dense random packing (German, 1989). For nonspherical particles, initial porosity varies with particle shape. In general, we may have i ) f(s, ψi). For example, for cylindrical particles, the following equation has been formulated (Zou and Yu, 1996):

ln lc ) ψ5.58 exp[5.89(1 - ψ)] ln 0.40 for the loose random packing, and

(3a)

ln dc ) ψ6.74 exp[8.00(1 - ψ)] ln 0.36

(3b)

for the dense random packing. However, different shapes give different relationships between initial porosity and sphericity, which may complicate the problem. On the other hand, it should be noted that for a given packing system, the initial porosity i or specific volume Vi of all the components involved can be readily measured. In this case, Vi, either estimated from the shape measurement or directly measured, can be incorporated in a model calculation. Therefore, eq 1 can be alternatively written as

V ) f(X1, X2, ..., Xn, dp1, dp2, ..., dpn, V1, V2, ..., Vn) (4) The use of eq 4 implies that particle shape affects packing through its effect on equivalent packing diameter and initial specific volume. Technically speaking, once the particle sizes and initial porosities are known, the porosity of a nonspherical particle mixture can be predicted by means of a mathematical model proposed for spherical particles, e.g., the simplified packing model (Ouchiyama and Tanaka, 1989) or the linear-mixture packing model (Yu and Standish, 1991), as the porosity prediction by such a model is based on the same information. However, this application cannot generally provide satisfactory predictions as demonstrated by Yu and Standish (1993b). This is not surprising because the packing of spherical particles should be understood as a special case of the packing of nonspherical particles. Therefore, there is a need to refine the existing models for predicting the porosity of both spherical and nonspherical particles. Model Framework. Understanding the binary packing is important in the elucidation of packing mechanisms. Following the work of Graton and Fraser (1935), the role of each component has been analytically studied by use of the concept of partial specific volume (Yu and Standish, 1988). It is postulated that two components will either “mix” or “unmix” with each other depending on their size ratio, so that the specific volume decreases either due to the “joint action” and the “additive” or “inert” effects. This has been extended to the packing of multicomponent mixtures by use of the concept of controlling mixture to establish a general model framework for the packing of spherical particles (Yu and Standish, 1991). This model framework should also be applicable to the packing of nonspherical particles because of the similarity between the two systems. In this connection, the concept of specific volume variation, which is the difference in specific volume between the unmixing (different types of particles are put layer by layer) and the mixing states, should be used (Yu et al., 1992; Yu and Standish, 1993c; Zou et al., 1996). A large specific volume variation means a large interparticle interaction and hence a large porosity reduction due to the mixing of particles of different sizes. If the sizes of all the components in a considered system are represented by their corresponding equivalent packing diameters, then we will have an equivalent packing system composed of spherical particles only. As shown in Figure 1, it is considered that the specific volume variation increases mainly due to the joint effect of medium particles, i.e., the controlling mixture, and the additive or inert effect of large or small particles.

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Figure 1. Two-dimensional diagram illustrating the random packing structure of nonspherical particles with their sizes represented by equivalent packing diameters.

As discussed by Yu and Standish (1991), the partial specific volume of the medium particles corresponding to component i, V h Mi, is not affected by the addition of small or large particles, so that this controlling mixture can be treated as a pseudo-component and the partial specific volumes of small or large particles, V h Sj or V h Lj, are only dependent on their individual sizes relative to the representative size of this pseudo-component. Hence, the specific volume, as the weighted mean of the partial specific volumes of the three grouped particles, can be written as M-1

ViT )

∑ j)1

N

V h LjXj +V h Mi

∑

j)M

n

Xj +

∑

V h SjXj

(5)

volumes. Of particular interest here is the so-called linear packing model proposed by Stovall et al. (1986) and Yu and Standish (1987), respectively. In the following, it will be shown that a slightly modified version of this model can be used for predicting the porosity of nonspherical particle mixtures. Modified Linear Packing Model. The linear package model, as a special case of the above model framework, assumes that there is only one component in the controlling mixture, so that M ) N ) i and V h Mi ) Vi. According to Stovall et al. (1986), the partial specific volume of a small component V h Sj (i < j e n) is given by

j)N+1

V h Sj ) Vi[1 - f(r)]

(7)

ViT,

In the calculation of the medium particles of mixing effect is assumed to be composed of the Mth to Nth (1 e M e i e N e n) components, and hence the large and small particles of the unmixing effect should be composed of the 1st to (M - 1)th and (N + 1)th to nth components, respectively. Depending on the particle sizes and volume fractions involved, a component may be categorized into any of the above three groups of particles so that usually it is not known a priori whether a particle has the mixing or unmixing effect. However, this problem can be readily solved by the physical consideration that the volume occupied by a unit solid volume of particles should not be less than the volume that can accommodate all the particles. Therefore, the specific volume of a particle mixture V should be the maximum of the calculated ViT, i.e.

V ) max{V1T, V2T, ..., VnT}

(6)

The above treatments can be used as a general model framework for developing a porosity prediction model for both spherical and nonspherical particle packings. Different models, as they are developed based on different assumptions, may give different equations for evaluating the values of M and N and partial specific

and the partial specific volume of a large component V h Lj (1 e j < i) is

V h Lj ) [Vi - (Vi - 1)g(r)]

(8)

where f(r) and g(r) are referred to as the interaction functions between components i and j, and r is the (packing) size ratio between components i and j (small to large). The two interaction functions should meet the boundary conditions: f(0) ) g(0) ) 1 and f(1) ) g(1) ) 0. Therefore, when r ) 0, eqs 5 and 6 can be reduced to the linear equations proposed by Westman and Hugill (1930) to calculate the specific volume many years ago. Stovall et al. (1986) attempted to evaluate f(r) and g(r) based on the analytical assumptions. However, note that eq 5, when applied to binary packing, can be reduced to

ViT ) ViXi + Vi[1 - f(r)]Xj or

(9a)

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3733

VjT ) ViXi + [Vi - (Vi - 1)g(r)]Xj

(9b)

where Xi + Xj ) 1. Obviously, the evaluation of f(r) and g(r) can be made by directly fitting eq 9 to the wellestablished binary packing results of spherical particles. This approach should improve the predictability of the model and was therefore employed by Yu and Standish (1987) and then De Larrard (1989). However, these authors used different sources of literature data in their analysis, resulting in different equations for calculating f(r) and g(r). As part of the present modeling, f(r) and g(r) were re-analyzed for the literature data collected by Ben Aim and Le Goff (1967) and Yu et al. (1993), respectively. Notably, the f(r) and g(r) values for the data of Ben Aim and Le Goff could be obtained from the work of De Larrard (1989). As shown in Figures 2 and 3, the resulting data were found to be satisfactorily fitted by the following equations:

f(r) ) (1 - r)3.3 + 2.8r(1 - r)2.7

(10)

g(r) ) (1 - r)2.0 + 0.4r(1 - r)3.7

(11)

and

Equations 10 and 11 were formulated based on the packing results of spherical particles only. However, their dependence on particle shape can be ignored provided that the size ratio r is evaluated in terms of equivalent packing diameter. Equations 7 and 8 need to be modified in order that this linear model can be applied to the packing of nonspherical particles. This can be seen when examining the binary packing with r ) 0 and 1. The specific volumes under these two extreme conditions can be theoretically determined, as shown in Figure 4. For the packing of spherical particles, since Vi ) Vs (i ) 1, 2, ..., n), the predicted specific volumes by the use of eqs 7 and 8 are the same as those given in Figure 4. However, this is not the case for the packing of nonspherical particles where Vi is usually not equal to Vj. It is almost intuitive that to match the theoretical prediction eqs 7 and 8 should be respectively modified as

V h Sj ) Vj[1 - f(r)]

(12)

V h Lj ) [Vj - (Vj - 1)g(r)]

(13)

Figure 2. Variation of interaction function f(r) with size ratio r: O, data obtained by fitting eq 9 to measurements; s, calculated by eq 10.

and

Therefore, in this modified linear packing model, ViT is given by i-1

ViT )

∑ j)1

n

[Vj - (Vj - 1)g(r)]Xj + ViXi +

∑

Vj[1 -

j)i+1

f(r)]Xj (14a) or noting that i-1

Xi ) 1 i-1

ViT ) Vi +

∑ j)1

n

Xj -

∑

Xj

j)i+1 n

[Vj - (Vj - 1)g(r) - Vi]Xj + ∑ [Vj ∑ j)1 j)i+1

Figure 3. Variation of interaction function g(r) with size ratio r: O, data obtained by fitting eq 9 to measurements; s, calculated by eq 11.

diameter dpi are so ordered that dp1 g dp2 g ... g dpn. So r should be equal to the ratio of dpi to dpj when j < i and equal to dpj to dpi when j > i. As mentioned above, the specific volume of a particle mixture V should be the maximum of the calculated ViT (i ) 1, 2, ..., n). The above equations can be readily extended to the packing of particles with a continuous size distribution. For example, if the volume frequency distribution and initial specific volume, as a function of particle size dp (0 e dp < +∞) are respectively represented by fv(dp) and V0(dp), then according to eqs 6 and 14, we have

V ) max{VT(dp)}

Vjf(r) - Vi]Xj (14b) For convenience, in this paper the equivalent packing

and

(15a)

3734 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

Figure 4. Specific volumes for binary packing when r ) 0 and r ) 1.

Figure 6. Comparison between the measured (a) and calculated (b) porosities for spherical ternary system [size ratios: 28-14-7, s ) 0.37, Jeschar et al. (1975)].

Figure 5. Packing of binary mixtures of spheres for different size ratio: points, measurements; solid lines, predictions [s ) 0.375 for the measurements of McGeary (1961) or 0.40 for the measurements of Ridgway and Tarbuck (1968)].

∫ [V (x) - V (x)f(x/d ) V (d )]f (x) dx + ∫ [V (x) - (V (x) - 1)g(d /x) -

VT(dp) ) V0(dp) + 0

p

v

dp

0

0 +∞

dp

0

0

p

0

p

V0(dp)]fv(x) dx (15b) Results and Discussion As mentioned above, the present model should be able to predict the porosity of spherical and/or nonspherical particle mixtures. Consequently, in this section, its applicability will be examined for both spherical and nonspherical particle packings, although emphasis will be given to the latter. Application to the Packing of Spherical Particles. We will first consider the packing of binary mixtures and then extend our discussion to ternary and multicomponent packings. Figure 5 shows the comparison between the measured and the calculated results for binary packing. In this plot, specific volume

rather than porosity was used for the convenience of discussion. Following the work of Yu and Standish (1988), the results of McGeary (1961) and Ridgway and Tarbuck (1968) were used to represent the so many measurements in the literature. It is evident that the prediction is in good agreement with the measurement. This is expected because the interaction functions f(r) and g(r) are obtained from the fitting of eq 14 to the measurement. On the other hand, it should be noted that the linear relationship between specific volume and volume fraction is really not valid when the size ratio is large, implying a deficiency of this model that may result in a more significant problem for ternary packing as discussed below. Good agreement between measured and predicted porosities could be found for all the reported ternary systems mentioned by Yu and Standish (1991). Figure 6 gives one typical example to demonstrate this. The change of particle sizes involved in a ternary system may result in different porosity patterns in such a triangular diagram. As discussed by Yu and Standish (1991), a general understanding of ternary packing can be obtained by a single plot that combines the six ternary systems of representative sizes. Such a diagram can also be constructed by the present linear model, as shown in Figure 7. Inspection of the results suggests

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3735

Figure 7. Calculated porosities of six ternary systems with the dimensionless diameter of the central component equal to unity (s ) 0.4).

that the ternary packing is featured by the following (1) there is a minimum porosity for each ternary system; (2) this minimum porosity and its position in a triangle vary with size ratios; (3) starting with this minimum porosity, a triangular diagram may be considered to be composed of three regions dominated, respectively, by the large, medium, and small components. In general, these features are in good agreement with those obtained from the linear-mixture packing model (Yu and Standish, 1991). However, it can also be observed that the unsmooth changeover of the controlling/dominant component in the linear packing model may give discontinuous porosity. For example, for the ternary system of size ratios 2-1-0.7 in Figure 7, when X1:X3 ) 0.7:0.3, a small addition of the medium component will make this component be a controlling component, which may imply a sudden change in porosity. This is unlikely to be true in reality. This phenomenon could also be found in the linear packing model of Stovall et al. (1986) or Yu and Standish (1987), although it was not fully appreciated before. The above unrealistic prediction results from the deficiency of the linear approach. As discussed by Yu and Standish (1988, 1991), there are two packing mechanisms, namely, the mixing and unmixing mechanisms that correspond respectively to large and small size ratios. The linear model is developed based on the unmixing mechanism by assuming that the controlling mixture is composed of one and only one component. Consequently, it can be accurately applied to packing systems in which the size of the controlling component is quite different from those of other components, but not so for systems where other components may join this

controlling component to form a controlling mixture. Therefore, as a direction for future development, the present model should be further modified in order to take the mixing mechanism into account. On the other hand, as discussed below, in spite of the above-noted deficiency, the present linear packing model can provide reasonably accurate estimates of porosity for many packing systems, including multicomponent packing of spherical and/or nonspherical particles. To demonstrate the predictability of the present model for continuous size distribution systems, a comparison has been made with the measurement for the log-normal size distribution which is given by

fv(d) )

[ (

)]

1 1 1 1 ln d - ln d0.5 exp d ln σ 2 ln σg x2π g

2

(0 e d < +∞) (16) where d0.5 and σg are, respectively, the median size and standard geometric deviation of a distribution. For spherical particles, ψ ) 1, so that dp ) dv. That is, the particle size d in eq 16 can be dp or dv. The packing of particles with the log-normal distribution is dependent on σg only (Sohn and Moreland, 1968; Dexter, 1972). Figure 8 shows the results in terms of specific volume variation, indicating that the prediction matches the measurement well. It can also been seen that as σg increases, the difference between the measured and predicted porosities increases. This is mainly due to the inaccuracy in approximating eq 16 by a practically limited size range for large σg but not the model itself, as noted by Standish et al. (1991).

3736 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

Figure 8. Specific volume variation vs σg for particles with the size/length distribution given by eqs 16 and 20, respectively, for the loose random packing: O, spheres [data from Sohn and Moreland (1968) and Dexter (1972)]; 4, cylinders (refer to Figure 15); s, predictions.

The features and application of the linear packing model to the packing of spherical particles have been studied in detail by a number of investigators (Stovall et al., 1986; Yu and Standish, 1987, 1991; De Larrard, 1989). The present model should have the same features as the previous model. Since its interaction functions are formulated based on a more extensive data source, it should be able to estimate the porosity better than the previous versions as seen from the above examples. However, as discussed below, the use of eqs 12 and 13 rather than eqs 7 and 8 will allow this modified linear packing model to predict the porosity of nonspherical particle mixtures. Application to the Packing of Nonspherical Particles. The porosity prediction of nonspherical particle mixtures should be made by use of equivalent packing diameter. For a given component, its equivalent packing diameter dpi can be determined from its size and shape analysis according to eq 2. As illustrated elsewhere (Zou and Yu, 1996), the shape information can also be used to evaluate the initial porosity and hence specific volume Vi, as, for example, given by eq 3 for cylindrical particles. However, to improve the prediction accuracy, the measured Vi can be used directly in a model calculation. Unless otherwise specified, this treatment will be used in all the numerical examples below. Again our discussion will be started with the packing of binary mixtures and then extended to the packing of ternary and multicomponent mixtures. As demonstrated by Yu and Standish (1993), the previous packing models, e.g., the simplified packing model of Ouchiyama and Tanaka (1989) or the linearmixture packing model of Yu and Standish (1991), fail to predict the porosity when there is a large difference in initial porosity. For the example they considered, as shown in Figure 9, the present model can predict the porosity pattern well. It may be of interest to note that this figure also highlights the difference between spherical and nonspherical particle packings. For the packing of spherical particles, mixing particles of different sizes

Figure 9. Comparison between the measured (a) and calculated (b) porosities for binary mixtures of different size ratio r: 1 ) 0.4, 2 ) 0.8.

can always result in a decrease in porosity so that a concave porosity curve for a given size ratio should be observed. However, for the packing of nonspherical particles, both concave and convex porosity curves can be found and their appearance is dependent on size ratio. This complex feature has been reported by Milewski (1973, 1978) for the binary packing of cylindrical and spherical particles. Figures 10 and 11, respectively, show his measurements for cylinder-sphere or cylinder-cylinder binary packing together with the predictions by the present model, the results indicating that the predictions are in good agreement with the measurements. The packing of ternary mixtures of nonspherical particles was recently studied by Yu et al. (1992), mainly using well-defined particles such as disk, cylinder, and sphere. This modified linear packing model can predict their ternary systems satisfactorily. The data used in the model calculation for their two ternary systems were given in Figures 12 and 13, respectively, while detailed experimental conditions can be found in their publication. It is obvious from the two figures that the predictions are comparable with the measurements.

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3737

Figure 10. Porosities of binary mixtures of sphere and cylinders of different dimensionless length L when the diameter ratio of sphere to cylinder is 17.40: points, measurements of Milewski (1973); s, predictions.

Figure 12. Comparison between the measured (a) and calculated (b) specific volume variations for a disk-cylinder-cylinder ternary system: sizes: (DL)1 ) 19.374 × 0.071, (DL)2 ) 4.0 × 7.5, (DL)3 ) 6.0 × 2.0 mm; initial specific volumes: V1 ) 2.653, V2 ) 2.155, V3 ) 1.616 (Yu et al., 1992). Figure 11. Porosities of binary mixtures of short cylinder (L ) 3.91) and (long) cylinders of different L values, D ) 2.09 mm for all cylinders: points, measurements of Milewski (1973); s, predictions.

Note that only a limited number of experiments (13 points) were carried out to construct Figures 12a and 13a (Yu et al., 1992), which might give an uncertainty in evaluating the porosity patterns. To overcome this problem, direct comparison between the measured and calculated porosities has also been made, and Table 1 lists the results. Inspection of the results in Table 1 clearly indicates that the predictions are in reasonably good agreement with the measurements. If the shape of particles does not vary with particle size, then dvi can be directly used in the calculation because, as implied by eq 2, the ratio between dpi and dpj (i * j) in the evaluation of the interaction functions is the same as that between dvi and dvj. This is also the case if other equivalent spherical diameters are used, although the evaluation of equivalent packing diameter is dependent on the equivalent spherical diameters used (Yu and Standish, 1993c). This is of

practical interest because the variation of particle shape with particle size is usually not so significant (Allen, 1981). In this case, one can simply adjust the initial porosity to best fit the measured porosity and then use this for predictive purpose for a given packing system. This treatment was recommended by a number of investigators (Yu and Standish, 1988, 1991; Stovall et al., 1987; Ouchiyama and Tanaka, 1986). However, the present consideration of nonspherical particle packing provides a good fundamental reason for this seemingly arbitrary treatment. In practice, there are also many packing systems in which particle shape does vary with particle size. A typical example is the packing of fibers (cylindrical particles) with a length distribution, which is widely used in ceramic or plastic industries [see, for example, Milewski (1987)]. To demonstrate the applicability of the present model to such a system, we will consider the packing of cylinders of constant diameter D, with their dimensionless lengths L (defined as the ratio of actual length to diameter) distributed according to the

3738 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

Figure 14. Porosity of cylinders with the modified powder law length distribution for various length ranges for the loose random packing: points, measurements of Zou et al. (1996); - - -, predictions by eq 15; s, predictions with the experimental conditions considered.

Figure 13. Comparison between the measured (a) and calculated (b) specific volume variations for a disk-cylinder-sphere ternary system: disk, dp1 ) 20.59 mm, V1 ) 2.653; cylinder, dp2 ) 10.95 mm, V2 ) 2.155; sphere, dp3 ) 3.00 mm, V3 ) 1.639 (Yu et al., 1992).

modified power law length distribution given by

fv(L) )

mLm-1 Lmaxm - Lminm

(Lmin e L e Lmax)

(17)

This equation is the same as the modified GaudinSchuhman distribution where particle diameter rather than length is used (Allen, 1981; Yu and Standish, 1991). By definition, the equivalent volume diameter and sphericity of a cylinder of dimensionless length L are respectively

dv ) 1.145L1/3D

(18)

and

ψ)

2.621L2/3 1 + 2L

(19)

For cylinders (L g 1) of constant D, each L corresponds to one sized component, with its initial porosity calculated by eq 3. fv(L) can therefore be transformed to fv(dp). In this case, eq 15 can be used in the porosity

calculation for this continuous size system. However, if disks (L < 1) are also involved, then there is no oneto-one transformation between L and dp, and the use of eq 15 may be difficult. This problem can be overcome by simply increasing the number of length intervals in the discrete approach using eq 14. Theoretically, the result should be the same as that obtained by the use of eq 15 if the number of length intervals is large enough. For simplicity, this discrete approach was employed in the present calculation. The packing of cylinders with eq 17 was recently studied by Zou et al. (1995) for different length ranges under either loose or dense random packing conditions. Figure 14 shows their loose packing results together with the prediction. Obviously the predicted and measured porosities are comparable. The difference between them may be mainly due to the errors in formulating eq 3 and/or in approximating eq 17. In the experiment, the number of length intervals was very limited. As shown in Figure 14 also, better quantitative results could be obtained if the way used to discretize eq 17 and the measured initial porosities were taken into account in the calculation. This treatment becomes necessary if one considers the packing of cylinders with the log-normal length distribution given by

fv(L) )

[ (

)]

1 1 1 1 ln L - ln L0.5 exp L ln σ 2 ln σg x2π g

2

(0 e L < +∞) (20) In practice, the range of lengths is limited, and hence eq 20 can never be exactly obtained. In general, the difference between the theoretically expected and actually obtained distributions increases with the increase of σg. Therefore, the direct use of eq 20 may result in large errors, particularly for large σg. This remark is also applied to the packing of spherical particles as demonstrated by Standish et al. (1991) for coal mixtures. As mentioned earlier, the increased discrepance between the predicted and the measured specific volume with large σg can be well-explained from this consideration. For the packing of cylinders with the above log-

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3739 Table 1. Comparison between Measured and Calculated Porosities for Ternary Mixtures in Figures 12 and 13 volume fraction

Figure 12

Figure 13

point

X1

X2

X3

measured

calculated

measured

calculated

1 2 3 4 5 6 7 8 9 10 11 12 13

1 0 0 0.28 0.28 0.72 0.72 0 0 1/ 3 2/ 3 1/ 6 1/ 6

0 1 0 0.72 0 0.28 0 0.28 0.72 1/ 3 1/ 6 2/ 3 1/ 6

0 0 1 0 0.72 0 0.28 0.72 0.28 1/ 3 1/ 6 1/ 6 2/ 3

0.623 0.536 0.381 0.530 0.392 0.550 0.475 0.430 0.500 0.460 0.513 0.483 0.414

0.623 0.536 0.383 0.543 0.434 0.576 0.541 0.430 0.498 0.499 0.544 0.519 0.440

0.623 0.536 0.390 0.530 0.380 0.550 0.410 0.385 0.410 0.354 0.453 0.460 0.348

0.623 0.530 0.390 0.543 0.363 0.576 0.494 0.376 0.414 0.394 0.518 0.476 0.365

Figure 15. Variation of the porosity of cylinders with the lognormal length distribution with σg when L0.5 ) 8.0: points, measurements of Zou et al. (1996); s, predictions.

normal length distribution, Zou et al. (1996) found that porosity is dependent on both median length L0.5 and standard deviation σg. Figures 15 and 16 show their results together with the predictions obtained consistant with their experimental conditions. Again, it can be seen that the predictions are in reasonably good agreement with the measurements. It would be of interest to note that both eqs 16 and 20 are log-normal, with one in terms of particle size and the other in terms of cylinder length. For spherical particles, porosity decreases with increasing σg (Sohn and, 1968; Dexter, 1972; Yu and Standish, 1991). For nonspherical particles, however, porosity increases with increasing σg. From this comparison, one might find it difficult to accept the similarity between the spherical and the nonspherical particle packings. However, as discussed elsewhere (Yu et al., 1992; Yu and Standish, 1993c; Zou et al., 1996), this is because porosity or its related parameters such as packing density and specific volume give packing results compounded by both size and shape effects. The size effect, i.e., the decrease of porosity due to the mixing of particles of different sizes, should be depicted by the concept of specific volume variation. In doing so, one would immediately find out that the size effects for the two packing systems are similar. For example, the results in Figure 15 for

Figure 16. Variation of the porosity of cylinders with the lognormal length distribution with L0.5 when σg ) 3.5: points, measurements of Zou et al. (1996); s, predictions.

cylinders, when replotted in terms of specific volume variation, are qualitatively comparable with those for spherical particles as shown in Figure 8. This is also the situation for other packing systems (Yu et al., 1992; Zou et al., 1996). In fact, it is on the basis of this finding that the similarity between spherical and nonspherical particle packings can properly be estab-lished, and then the present linear packing model results. Conclusions In the past decade or so, significant progress has been made in the modeling of the relationship between porosity and particle size distribution for spherical particles. The developed mathematical models have significantly enhanced our understanding of the size effect and provided an effective means for solving various packing problems. However, particles involved in practice are usually not spherical, and particle shape strongly affects porosity. In the past, no systematic effort has been made to model the packing of nonspherical particles mainly because of the complexity of the problem. On the basis of the similarity analysis between the spherical and the nonspherical particle packings, a mathematical model, modified from the previous linear packing model for spherical particles (Stovall, 1986; Yu and Standish, 1987), has been developed for

3740 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

predicting porosity as a function of both particle size and shape distributions. The results in this paper clearly indicate that this packing model can generally be used for predicting the porosity of (convex) nonspherical particle mixtures. Acknowledgment The authors are grateful to Australian Research Council (ARC) and Energy and Research Development Corporation (ERDC) for financial support. Nomenclature d ) particle size, m D ) diameter of a cylindrical particle, m dp ) equivalent packing diameter, m dpi ) equivalent packing diameter of ith component, m dvi ) equivalent volume diameter of ith component, m f(r), g(r) ) interaction functions between two components of size ratio r, dimensionless fv(dp) ) volume-frequency distribution in terms of packing diameter dp, m-1 fv(L) ) volume-frequency distribution in terms of dimensionless length L, dimensionless i, j, M, N ) integer, dimensionless L ) ratio of length to diameter of a cylindrical particle, dimensionless L0.5 ) median length in eq 20, dimensionless Lmin, Lmax ) minimum and maximum dimensionless lengths in fv(L), respectively, dimensionless m ) exponent in eq 17, dimensionless n ) number of components in a packing system, dimensionless r ) size ratio between two components, dimensionless V ) overall specific volume of a packing system, dimensionless V0(dp) ) initial specific volume as a function of dp, dimensionless Vi ) initial specific volume of ith component, dimensionless ViT or VT(dp) ) calculated specific volume under the assumption that the controlling mixture corresponding to the component of particle size dpi or dp is the controlling component, dimensionless V h Lj ) partial specific volume of jth (large) component in calculating ViT, dimensionless V h Mi ) partial specific volume of the controlling mixture in calculating ViT, dimensionless Vs ) initial specific volume of spheres, dimensionless V h Sj ) partial specific volume of jth (small) component in calculating ViT, dimensionless x ) packing diameter used in eq 15, m Xi ) volume fraction of ith component, dimensionless Greek Letters dc ) initial porosity of cylinders of sphericity ψ for the dense random packing, dimensionless i ) initial porosity of ith component, dimensionless lc ) initial porosity of cylinders of sphericity ψ for the loose random packing, dimensionless s ) initial porosity of spheres, dimensionless σg ) standard deviation of the log-normal distribution, dimensionless ψ ) sphericity, dimensionless ψi ) sphericity of ith component, dimensionless

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Received for review October 11, 1995 Revised manuscript received June 5, 1996 Accepted June 6, 1996X IE950616A X Abstract published in Advance ACS Abstracts, August 15, 1996.