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A 3-parameter packing density model for angular rock aggregate particles
Powder Technology 274 (2015) 154–162
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A 3-parameter packing density model for angular rock aggregate particles A.K.H. Kwan ⁎, V. Wong, W.W.S. Fung Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 25 September 2014 Received in revised form 20 December 2014 Accepted 22 December 2014 Available online 11 January 2015 Keywords: Packing density Packing models Particle mechanics Particle shape
a b s t r a c t The authors have in recent studies incorporated the wedging effect to develop a 3-parameter model for packing density prediction of binary and ternary mixes of spherical particles. This model has been restricted to only spherical particles because only the test results of spherical particles were used for derivation and validation. For more general applications to other types of particles, such as angular particles, the model needs to be further developed. In this study, an experimental program on the packing density of binary mixes of angular rock aggregate particles was carried out and the experimental results were used to derive the interaction functions of the three parameters (the loosening, wall and wedging effect parameters) for extending the 3-parameter model to binary mixes of angular particles. Apart from the experimental results obtained herein, the test results published by de Larrard and those obtained by the authors in an earlier study were also used to validate the extended 3-parameter model. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Particulate materials are widely used in many processing industries, such as the mineral, metallurgical, pharmaceutical and food industries [1–4]. Product design and manufacturing across this wide spectrum of industries are largely a practice of particle technology, which generally requires the knowledge of particle packing. For this reason, the importance of particle packing is well known in the processing industries, but somehow not in the concrete production industry. Actually, a typical concrete mix may be regarded as a mixture of aggregate particles and cement paste. The cement paste fills the voids between the aggregate particles and binds the aggregate particles upon hardening to give strength to the concrete. Among the various ingredients, cement is the most expensive and has the highest carbon footprint [5,6]. To slash the cost and carbon footprint of concrete production, a promising way is to maximize the packing density of the aggregate so as to reduce the volume of cement paste needed to fill the voids [6]. For this task, it would be helpful to have a particle packing model for predicting the packing density of rock aggregate particles, which are usually angular in shape. Numerous particle packing models have been developed but most of these models are limited to spherical particles. In the 1930s, Westman and Hugill [7] and Furnas [8] pioneered to derive analytical equations for predicting the packing density of binary mixes of spherical particles by considering two extreme cases: first, when the size ratio is close to ⁎ Corresponding author. Tel.: +852 2859 2647. E-mail address: [email protected] (A.K.H. Kwan).
http://dx.doi.org/10.1016/j.powtec.2014.12.054 0032-5910/© 2015 Elsevier B.V. All rights reserved.
zero, and second, when the size ratio is close to unity (the size ratio is the ratio of the size of smaller particles to the size of larger particles). Since then, substantial progress has been made in the packing density prediction of binary mixes of spherical particles with an intermediate size ratio, in which case, interactions between particles of different sizes occur [9–16]. Two interaction effects have been identified, namely: the loosening effect, which occurs when the larger particles are dominant and the wall effect, which occurs when the smaller particles are dominant. These two effects are taken into account using two parameters, called the loosening effect parameter and the wall effect parameter, respectively. Both these two parameters are functions of the size ratio (these functions are called interaction functions) derived by curve fitting against experimental results. The particle packing models, which employ two parameters to account for the loosening and wall effects, may be referred to as the 2-parameter models. In most of the 2-parameter models [10,12,13,16], the relationship between the specific volume of the particle system and the volumetric fractions of the various size classes of particles (each size class is a collection of mono-sized particles) is assumed to be linear. However, it has been observed from test results [9,17,18] that the relationship between the specific volume and the volumetric fractions is in fact nonlinear. To take into account such nonlinear relationship and improve the accuracy of the 2-parameter models, de Larrard [17] incorporated a compaction index, which is dependent on the packing process, to develop a more advanced compressible packing model, whereas Kwan et al. [18] added a wedging effect parameter, which accounts for a newly identified wedging effect, to develop a more sophisticated 3-parameter model. The 3-parameter model developed
A.K.H. Kwan et al. / Powder Technology 274 (2015) 154–162
by Kwan et al. [18] is only for binary mixes of spherical particles. Later, Wong and Kwan [19] refined this model for application to ternary mixes of spherical particles. Most existing particle packing models are applicable only to spherical or rounded particles because they were validated by comparing with experimental results of only spherical or rounded particles. For more general applications to non-spherical particles, further development of the particle packing models to allow for the effects of particle shape is needed. Overall, the particle shape may have two effects: first, on the packing density of each mono-sized fraction of the particles, and second, on the interactions between the different mono-sized fractions of particles. The effect of particle shape on the packing density of mono-sized particles has been studied extensively using regular particles such as ellipsoids, spheroids, disks and cylinders [20,21]. Moreover, the packing density of irregular rock aggregate particles has been measured and correlated to the various shape parameters [22]. Anyway, whatever the effect of particle shape is, the packing density of mono-sized particles can be measured directly, and provided the measured packing densities of the mono-sized fractions are used in the packing density prediction, the effect of particle shape on the packing density of mono-sized particles should already be allowed for. To characterize the size of a mono-sized fraction of non-spherical particles, Yu and Standish [23] and Yu et al. [24] introduced the concept of equivalent packing diameter and employed this concept to extend their 2-parameter model for application to non-spherical particles. Meanwhile, Goltermann et al. [25] modified the packing models of Aїm and Goff [10] and Toufar et al. [11] for application to rock aggregates and introduced the concept of position parameter for characterizing the size of aggregate particles having a certain particle size distribution ranging over several sieve sizes. On the other hand, for aggregate particles passing a certain sieve (the upper sieve) but retained on the next smaller sieve (the lower sieve), of which the size range is relatively narrow, de Larrard [17] assigned the geometric mean of the upper and lower sieve sizes to be the characteristic size. Herein, the 3-parameter model [18,19] is extended for application to binary mixes of angular rock aggregate particles. The measured packing densities of the mono-sized fractions are used in the packing density prediction and thus the effect of particle shape on the packing density of each mono-sized fraction should have been allowed for. An experimental program has been carried out to provide data for deriving the interaction functions. Two sets of interaction functions are derived, one for the uncompacted condition and the other for the compacted condition. Finally, the extended 3-parameter model is compared to the test results obtained in this study and the test results by de Larrard [17] and by Fung and Kwan [26] to verify the applicability and accuracy of the model. 2. The 3-parameter model for spherical particles The 3-parameter model was originally developed by Kwan et al. [18] for binary mixes of spherical particles. It requires pre-determination of the optimum volumetric fractions yielding the maximum packing density before applying the equations contained therein to predict the packing density. For easier application and dealing with ternary mixes, the 3-parameter model has been refined by Wong and Kwan [19] in two ways. First, the equations are modified so that they are not dependent on the optimum volumetric fractions and pre-determination of the optimum volumetric fractions is not needed any more. Second, the equations are expanded to cater for the packing density prediction of ternary mixes. The refined 3-parameter model by Wong and Kwan [19] is adopted herein as the basic framework for extension to apply to angular particles. For easy reference, an outline of this 3-parameter model is presented in the following. Consider a binary mix composed of two size classes of mono-sized spherical particles: size class i and size class j, in order of increasing particle size. Let the diameters of particles of size class i and size class j
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be di and dj, respectively (note that di ≤ dj), the volumetric fractions of size class i and size class j be ri and rj, respectively (note that ri + rj = 1), and the packing densities of size class i and size class j be ϕi and ϕj, respectively. The packing density of the binary mix when size class i is dominant (denoted by ϕi*) and the packing density of the binary mix when size class j is dominant (denoted by ϕj*) are given respectively by: 1 ¼ ϕi
! r h r i rj ri j þ − 1−bi j 1−ϕ j 1−ci j 2:6 j −1 ϕi ϕ j ϕj
ð1Þ
1 ¼ ϕ j
! r h r i rj ri þ − 1−ai j i 1−ci j 3:8 i −1 ϕi ϕ j ϕi
ð2Þ
in which aij, bij and cij are the loosening effect parameter, wall effect parameter and wedging effect parameter, respectively, and are given as functions of the size ratio. The final packing density of the binary mix of particles can be determined simply as min.{ϕi*, ϕj*}. The loosening effect occurs in the case when the larger particles are dominant due to the smaller particles squeezing into the voids between the larger particles causing loosening of the packing of the larger particles. The wall effect occurs in the case when the smaller particles are dominant due to the boundaries of the larger particles acting like walls causing disruption of the regular packing of the smaller particles. However, the wedging effect takes place in both cases. In the case when the larger particles are dominant, the wedging effect occurs when some isolated smaller particles are entrapped in the gaps between the larger particles instead of filling into the voids thereby wedging the larger particles apart. In the case when the smaller particles are dominant, the wedging effect takes place when some gaps between the larger particles are too narrow to accommodate even one complete layer of smaller particles leading to the presence of only isolated smaller particles at the gaps wedging the larger particles apart. Fig. 1 demonstrates the loosening effect, wall effect and wedging effect in binary mixes of particles. It should be noted that if the values of cij in Eqs. (1) and (2) are set to zero, Eqs. (1) and (2) would revert back to the packing density equations in the conventional 2-parameter models. With the additional terms accounting for the wedging effect, the relationship between the specific volume and the volumetric fractions is no longer linear because the wedging effect varies non-linearly with volumetric fractions. For spherical particles, Kwan et al. [18] and Wong and Kwan [19] have derived the following interaction functions for the evaluation of the loosening effect, wall effect and wedging effect parameters: 3:3
−2:6 s ð1−sÞ
1:9
−2:0 s ð1−sÞ
a ¼ 1−ð1−sÞ
b ¼ 1−ð1−sÞ
3:6
ð3aÞ
6:0
ð3bÞ
c ¼ 0:322 tanhð11:9 sÞ
ð3cÞ
When larger particles are dominant
When smaller particles are dominant
Loosening effect
Wedging effect
Wall effect
Wedging effect
Fig. 1. A schematic diagram showing the loosening, wall and wedging effects.
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in which s is the size ratio. The values of aij, bij and cij in Eqs. (1) and (2) may be obtained by substituting the value of sij = di/dj as s into the above equations. Note that when s = 0, aij = bij = cij = 0, and when s = 1, aij = bij = 1 and cij = 0.322. 3. Extension of the 3-parameter model for application to angular particles It is postulated herein that the 3-parameter model presented in the previous section can be applied also to binary mixes of angular aggregate particles provided that (1) the particle size of each size class (defined as the diameter of the particles in the original model for spherical particles) is re-defined as the characteristic size of the particles; (2) the measured packing density of each size class is used as input in the packing density prediction so as to allow for the effect of particle shape on the packing density of each size class of particles; and (3) the interaction functions for the loosening effect, wall effect and wedging effect parameters are re-derived from experimental packing density results of angular aggregate particles. In the field of concrete technology, the particle size distribution of aggregate particles is usually determined by mechanical sieving of the aggregate particles through a set of sieves starting from the sieve with the largest sieve size on top and then successively downwards through sieves with smaller and smaller sieve sizes [17,25,26]. After sieving, the aggregate particles are divided into different size classes by obtaining the aggregate particles retained on each sieve and taking the aggregate particles on the same sieve as belonging to the same size class. The sieve on which the particles are retained is called the lower sieve whereas the next larger sieve that the particles have passed through is called the upper sieve. Each size class of aggregate particles has a particle size ranging from the lower sieve size to the upper sieve size and is therefore not truly mono-sized. Nevertheless, since the size range of each size class is fairly narrow, the characteristic size is simply taken as the geometric mean of the upper and lower sieve sizes, as recommended by de Larrard [17]. 4. Experimental program An experimental program has been launched to measure the packing densities of binary mixes of fine and coarse aggregates. Crushed granite rock aggregate was used in the tests. As for other crushed rock aggregates, it was angular in shape. The crushed rock aggregate was divided into six size classes by mechanical sieving using steel plate sieves with square apertures. The sieve sizes were 0.6 mm, 1.18 mm, 2.36 mm, 5.0 mm, 10 mm, 14 mm and 20 mm. Each size class comprised of the aggregate particles retained on one of the sieves. The six size classes have particle size ranges of 0.6-1.18 mm, 1.18-2.36 mm, 2.36-5 mm, 5-10 mm, 10-14 mm, and 14-20 mm, respectively, as tabulated in Table 1. Their characteristic sizes, each taken as the geometric mean of the upper and lower sieve sizes, are 0.84 mm, 1.67 mm, 3.44 mm, 7.07 mm, 11.83 mm and 16.73 mm, respectively. Following usual practice, the size classes finer than 5.0 mm are regarded as fine aggregates, whereas those coarser than 5.0 mm are regarded as coarse aggregates. Hence, the six size classes are named as F1, F2, F3, C1, C2, Table 1 Sieve size ranges and characteristic sizes of the six size classes of aggregates. Size class
Sieve size range Lower sieve size (mm)
Upper sieve size (mm)
F1 F2 F3 C1 C2 C3
0.60 1.18 2.36 5.0 10.0 14.0
1.18 2.36 5.0 10.0 14.0 20.0
Characteristic size (mm)
0.84 1.67 3.44 7.07 11.83 16.73
and C3 in ascending order of their sizes, as listed in Table 1, where F denotes fine aggregate and C denotes coarse aggregate. By blending one size class of fine aggregate (F1, F2 or F3) and one size class of coarse aggregate (C1, C2 or C3) together at various volumetric fractions, a total of nine binary mix series were produced for packing density tests. An identification code in the form of Fx-Cy is assigned to each binary mix series, in which Fx represents the size class of the fine aggregate blended and Cy represents the size class of the coarse aggregate blended. The nine binary mix series produced are: F1-C1, F2-C1, F3C1, F1-C2, F2-C2, F3-C2, F1-C3, F2-C3 and F3-C3. In each binary mix series, the volumetric fraction of Fx was varied from 0 to 100% in steps of 10% and the volumetric fraction of Cy was set equal to 100% minus the volumetric fraction of Fx. Altogether, there were 6 packing density tests for the different size classes (F1 to F3 and C1 to C3), and 81 packing density tests for the binary mixes of aggregates. The test methods stipulated in the British Standard BS 812-2: 1995 [27] were employed to measure the uncompacted and compacted packing densities of the aggregates. Since the container specified in the British Standard has an internal diameter of about 10 times the size of the largest size class of coarse aggregate C3, the wall effect of the container should be small enough to be ignored [28]. The test method for measuring the uncompacted packing density is outlined as follows. First, a sample of the aggregate (a single size class of aggregate or a binary mix of fine and coarse aggregates) was prepared. In the case of preparing a sample of a binary mix of aggregates, the fine and coarse aggregates to be blended were accurately weighed to the required volumetric fractions and mixed thoroughly in a steel tray manually until uniform mixing was achieved. Then, the aggregate sample was placed into the container shovel by shovel instead of pouring into the container to avoid segregation until the container was over-filled. After then, the excess aggregate particles were removed and the surface was leveled off. Finally, the container fully-filled with aggregate particles was weighed. The foregoing procedures were carried out three times so as to obtain the average weight of aggregate inside the container. The test method for measuring the compacted packing density was almost the same except that the aggregate sample was placed into the container in three equal portions by means of a shovel and after each portion of aggregate was filled, the aggregate in the container was compacted by applying 30 compactive blows with a tamping rod evenly distributed over the surface. Each compactive blow was applied by releasing the tamping rod at a height of 50 mm above the surface of aggregate. The saturated solid density and oven-dried solid density of the aggregate were measured in accordance with BS812-2: 1995 [27] as 2609 kg/m3 and 2590 kg/m3 respectively. Since air-dried aggregate was used in the packing density tests, the air-dried solid density of the aggregate should be used for the determination of packing density. From the measured moisture content of the aggregate, the air-dried solid density of the aggregate was calculated as 2598 kg/m3. Hence, the solid volume of the aggregate in the container was calculated as the average weight of the aggregate in the container divided by the air-dried solid density of 2598 kg/m3. Finally, the packing density was determined as the ratio of the solid volume of aggregate in the container to the volume of the container. 5. Experimental results The packing density results of the six size classes tested are plotted against the respective characteristic sizes in Fig. 2, wherein two curves are presented, one for uncompacted condition and the other for compacted condition. As expected, for each and every size class, the compacted packing density is higher than the uncompacted packing density. This is because the compactive blows applied during compaction had enabled rearrangement of the aggregate particle to achieve a denser packing. Overall, under both the uncompacted and compacted conditions, there is a general trend that the packing densities of the
A.K.H. Kwan et al. / Powder Technology 274 (2015) 154–162
0.65
0.80
Compacted
0.70
Packing density
Packing density
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.60 0.55 0.50
Uncompacted
0.45
0.65 0.60 0.55 0.50
0.40
F1-C2 (s = 0.071) F2-C2 (s = 0.141) F3-C2 (s = 0.290)
0.45 0.40
0.35 0.0
2.0
4.0
0.0
6.0 8.0 10.0 12.0 14.0 16.0 18.0 Characteristic size (mm)
Fig. 2. Variation of packing density of a single size class with characteristic size.
three size classes of fine aggregate (F1, F2 and F3) are significantly lower than those of the three size classes of coarse aggregate (C1, C2 and C3). One most likely reason for the lower packing density of the finer particles is the presence of inter-particle cohesive forces, such as van der Waals and electrostatic attractive forces, which cause agglomeration of the particles and formation of voids inside the agglomerates [29–31]. Such agglomeration effect is generally larger in finer particles and that is why the packing density of finer particles is usually lower than that of coarser particles. The packing density results of the nine binary mixes tested are plotted against the respective volumetric fractions of the fine aggregate as data points in Figs. 3, 4 and 5 for the uncompacted binary mixes containing C1, C2 and C3, respectively, and in Figs. 6, 7 and 8 for the compacted binary mixes containing C1, C2 and C3, respectively. It can be seen from the data points plotted that in each binary mix series, as the volumetric fraction of the fine aggregate increases from zero, the packing density first increases and after reaching a certain maximum value, starts to decrease. The maximum packing density attained is generally higher when the size ratio is smaller (or in other words, when the size difference between the two size classes of particles blended together is larger). Overall, the general trends of the variation of the packing density with the volumetric fractions and the variation of the maximum packing density attained with the size ratio are the same as those for binary mixes of spherical particles.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Volumetric fraction of fine aggregate, r1 Fig. 4. Uncompacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C2, F2-C2 and F3-C2.
6. Interaction functions of the three parameters From the data points for each binary mix series under either uncompacted or compacted condition presented in Figs. 3 to 8, one set of best-fit values of the loosening effect parameter a, the wall effect parameter b, and the wedging effect parameter c has been derived by fitting the packing density equations, Eqs. (1) and (2), to the 11 packing density results in the series. The best-fit values of the three parameters, a, b and c, so derived are plotted against the size ratio as data points in Figs. 9, 10 and 11, respectively, using hollow symbols for uncompacted results and solid symbols for compacted results. Based on the best-fit values, the interaction functions of the three parameters were derived by regression analysis. For the uncompacted condition, the interaction functions of the three parameters so derived are as follows:
5:0
−1:9 s ð1−sÞ
1:9
−2:1 s ð1−sÞ
a ¼ 1−ð1−sÞ
b ¼ 1−ð1−sÞ
3:1
10:5
ð4aÞ
7:6
−0:2 ð1−sÞ
ð4bÞ
c ¼ 0:335 tanhð26:9 sÞ
0.80
ð4cÞ
0.80
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.70
0.70 Packing density
Packing density
157
0.65 0.60 0.55 0.50
0.60 0.55 0.50
F1-C1 (s = 0.119) F2-C1 (s = 0.236) F3-C1 (s = 0.486)
0.45
0.65
F1-C3 (s = 0.050) F2-C3 (s = 0.100) F3-C3 (s = 0.205)
0.45
0.40
0.40
0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 3. Uncompacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C1, F2-C1 and F3-C1.
0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 5. Uncompacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C3, F2-C3 and F3-C3.
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0.80
0.80
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.70 Packing density
0.70
Packing density
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.65 0.60 0.55 0.50
0.60 0.55 0.50
F1-C1 (s = 0.119) F2-C1 (s = 0.236) F3-C1 (s = 0.486)
0.45
0.65
F1-C3 (s = 0.050) F2-C3 (s = 0.100) F3-C3 (s = 0.205)
0.45
0.40
0.40 0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
0.0
1.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 6. Compacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C1, F2-C1 and F3-C1.
Fig. 8. Compacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C3, F2-C3 and F3-C3.
For the compacted condition, the interaction functions of the three parameters so derived are as follows:
size. Consequently, when squeezing into a void, the angular particle would push the larger particles a greater distance apart causing a larger loosening effect. From Fig. 10, it is evident that most of the time, the wall effect parameter is marginally larger under compacted condition than uncompacted condition. This is probably because under compacted condition, the smaller particles are more closely packed and thus apparently the wall effect of the larger particles disrupting the regular packing of the smaller particles is larger. Nevertheless, except at size ratio smaller than 0.2, the wall effect parameter is almost the same for both the angular particles and spherical particles. Hence, the wall effect parameter is not sensitive to the particle shape. More importantly, it appears that the wall effect parameter can become negative when the size ratio is smaller than 0.2, meaning that the wall effect can sometimes increase rather than decrease the packing density of the smaller particles. This is against our intuition and further study is needed to ascertain that such observed phenomenon is not caused by experimental errors or mistakes. In fact, the same phenomenon has been observed by de Larrard [17]. More discussions on this issue will be given in the next section. From Fig. 11, it is noted that the wedging effect parameter is similar under the uncompacted or compacted condition, indicating that the wedging effect parameter is not sensitive to the compaction applied.
7:1
−1:9 s ð1−sÞ
2:2
−0:7 s ð1−sÞ
a ¼ 1−ð1−sÞ
b ¼ 1−ð1−sÞ
3:1
9:3
ð5aÞ 10:6
−0:2 ð1−sÞ
ð5bÞ
c ¼ 0:335 tanhð26:9 sÞ
ð5cÞ
For comparison with the corresponding best-fit values, the above interaction functions of parameters a, b and c are plotted as curves in Figs. 9, 10 and 11, respectively. To study the effect of particle shape, the interaction functions of the three parameters derived from the test results of spherical particles under uncompacted condition, i.e. Eqs. (3a) to (3c), are also plotted in Figs. 9, 10 and 11. From Fig. 9, it can be seen that overall speaking, the loosening effect parameter is slightly larger under compacted condition than uncompacted condition. This is probably because under compacted condition, the larger particles are more closely packed and thus relatively the loosening effect of the smaller particles squeezing into the voids between the larger particles is larger. Moreover, in general, the loosening effect parameter is slightly larger for angular particles than for spherical particles. One possible reason is that each angular particle actually occupies a larger circumscribing volume (the convex volume circumscribing the particle) than a spherical particle with the same solid volume and thus would have a larger equivalent
1.0
Loosening effect parameter a
0.80
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
Packing density
0.70 0.65 0.60 0.55 0.50
F1-C2 (s = 0.071) F2-C2 (s = 0.141) F3-C2 (s = 0.290)
0.45
Angular particles
0.8
Spherical particles
0.6
0.4
Uncompacted Compacted Eqn. (3a) Eqn. (4a) Eqn. (5a)
0.2
0.40 0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 7. Compacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C2, F2-C2 and F3-C2.
0.0 0.0
0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 9. Best-fit values and interaction functions of the loosening effect parameter.
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1.0
Angular particles
Wall effect parameter b
0.8
Spherical particles
0.6
Uncompacted Compacted Eqn. (3b) Eqn. (4b) Eqn. (5b)
0.4 0.2 0.0 0.0
0.2
-0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 10. Best-fit values and interaction functions of the wall effect parameter.
In fact, the interaction function of the wedging effect parameter derived for the uncompacted condition, Eq. (4c), and that derived for the compacted condition, Eq. (5c), are the same. Hence, the wedging effect occurs under both the uncompacted and compacted conditions, and application of compaction would not mitigate the wedging effect. Moreover, the wedging effect parameter is marginally larger for angular particles than spherical particles. This may be attributed to the slightly larger equivalent size of angular particles than spherical particles having the same solid volume. 7. Verification of the extended 3-parameter model 7.1. Comparison with experimental results by de Larrard de Larrard [17] had carried out similar tests on the packing densities of rounded (uncrushed) and angular (crushed) rock aggregates. For each type of aggregate, five size classes were prepared by sieving. Each size class was obtained from the aggregate particles retained on a sieve and thus was of size within two successive sieve sizes following the French Standard. By blending at each time two different size classes
159
together, seven binary mix series were produced from each type of aggregate for testing and in each binary mix series, the volumetric fractions of the two size classes were varied from 0 to 100%. The packing density of each aggregate sample was measured by filling the aggregate into a container, applying compression on top of the aggregate and subjecting the container to vibration on a vibrating table. Hence, the results are the packing densities of the aggregate samples under compacted condition. From the variations of the measured packing density with the volumetric fractions in each binary mix series, the best-fit values of the loosening effect parameter and the wall effect parameter (there was no wedging effect parameter in de Larrard’s model) were derived by curve fitting. The loosening effect parameters from this study are compared to those by de Larrard in Fig. 12. It is seen that the loosening effect parameters by de Larrard are very scattered. Moreover, the loosening effect parameters for the rounded particles are sometimes larger and sometimes smaller than those for the angular particles. Hence, de Larrard’s results reveal no clear trend of whether the loosening effect parameters for the rounded particles are larger or smaller than those for the angular particles. Comparing the loosening effect parameters for angular particles under compacted condition from this study, as represented by Eq. (5a), to those obtained by de Larrard, it is found that whilst some of the loosening effect parameters obtained by de Larrard agree closely with Eq. (5a), several of them do not agree well with Eq. (5a). Overall, the agreement between the loosening effect parameters from this study and those obtained by de Larrard is not good. de Larrard’s results are too scattered to be used as benchmark for validation and the above comparison has not fulfilled the intended aim of verifying the accuracy of the extended 3-parameter model. The wall effect parameters from this study are compared to those by de Larrard in Fig. 13. It is seen that the wall effect parameters by de Larrard are also very scattered. Again, the wall effect parameters for the rounded particles are sometimes larger and sometimes smaller than those for the angular particles, thus revealing no clear difference between the wall effect parameters for the rounded and angular particles. Comparing the wall effect parameters for angular particles under compacted condition from this study, as represented by Eq. (5b), to those obtained by de Larrard, it is found that although some of the wall effect parameters obtained by de Larrard agree closely with Eq. (5b), some others do not agree well with Eq. (5b). Overall, the agreement between the wall effect parameters from this study and those
1.0
1.0
Wedging effect parameter c
0.8
Loosening effect parameter a
Uncompacted Compacted Eqn. (3c) Eqn. (4c) Eqn. (5c)
0.6
Angular particles 0.4
Spherical particles
0.2
Angular particles
0.8
Spherical particles
0.6
Rounded [17] Angular [17] Uncompacted Compacted Eqn. (3a) Eqn. (4a) Eqn. (5a)
0.4
0.2
0.0
0.0
0.0
0.0
0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 11. Best-fit values and interaction functions of the wedging effect parameter.
0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 12. Comparison of the loosening effect parameters obtained from this study with those by de Larrard [17].
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0.8
1.0
Spherical particles
0.6
Theoretical packing density
0.8
Wall effect parameter b
R2 = 0.980
Angular particles
Rounded [17] Angular [17] Uncompacted Compacted Eqn. (3b) Eqn. (4b) Eqn. (5b)
0.4 0.2
0.7
0.6
0.5
0.0 0.0 -0.2
0.2
0.4 0.6 Size ratio s
0.8
1.0
Line of equality
0.4 0.4
Fig. 13. Comparison of the wall effect parameters obtained from this study with those by de Larrard [17].
0.5
0.6
0.7
0.8
Experimental packing density Fig. 14. Theoretical uncompacted packing densities against experimental packing densities obtained from this study.
obtained by de Larrard is not satisfactory. The above comparison has not verified the accuracy of the extended 3-parameter model and further comparison with more experimental results is needed. More importantly, at a size ratio of 0.0625, negative best-fit values of the wall effect parameter were obtained by de Larrard for both the rounded and angular particles. de Larrard attributed such negative wall effect parameters to the effect of vibration or compaction. However, negative wall effect parameters have been obtained in this study under uncompacted condition. Therefore, the negative wall effect parameters were not caused by vibration or compaction. One possible cause of such negative wall effect parameter is the interaction between the agglomeration effect and the wall effect. When the size ratio is small, the packing density of the smaller particles tends to be smaller because of agglomeration and presence of voids inside the agglomerates. At the wall-like boundaries of the larger particles, the agglomerates could be broken and the voids originally inside the agglomerates could merge with the voids caused by the wall effect leading to an overall smaller volume of voids than having separately the agglomeration effect and the wall effect. As a result, the wall effect could appear to be smaller than when there is no such interaction and might even appear to be negative. This is nevertheless just a speculation and further study on this phenomenon is recommended.
Figs. 6, 7 and 8 for comparison with the experimental results plotted therein as data points. Again, the theoretical predictions agree closely with the experimental results. For more precise evaluation of the accuracy of the model, the theoretical packing densities are plotted against the experimental packing densities as data points in Fig. 15. As before, all the data points lie closely to the line of equality, indicating very good agreement between the theoretical predictions and the experimental results. From these results, it has been calculated that the percentage error in packing density ranges from − 1.15% to 2.22% and the overall mean percentage error is 0.08%. Overall, with the interaction functions modified to cater for angular rock aggregate particles under uncompacted or compacted condition, the 3-parameter model is accurate to within an absolute error of 3.6%. Actually, part of the errors might be due to experimental errors and intrinsic randomness of particle packing. The very small mean percentage errors indicate that the 3-parameter model is on average neither under-
0.8 R2 = 0.987
The theoretical uncompacted packing densities predicted by the 3-parameter model using the interactions functions given by Eqs. (4a) to (4c) for the nine binary mix series tested are plotted as curves in Figs. 3, 4 and 5 for comparison with the experimental results plotted therein as data points. The comparison reveals that in general, the theoretical predictions agree closely with the experimental results. For more precise evaluation of the accuracy of the model, the theoretical packing densities are plotted against the experimental packing densities as data points in Fig. 14. Generally, all the data points lie closely to the line of equality, indicating very good agreement between the theoretical predictions and the experimental results. From these results, it has been calculated that the percentage error in packing density ranges from − 2.51% to 3.55% and the overall mean percentage error is − 0.03% (negative error means under-estimation while positive error means over-estimation). The theoretical compacted packing densities predicted by the 3-parameter model using the interactions functions given by Eqs. (5a) to (5c) for the nine binary mix series tested are plotted as curves in
Theoretical packing density
7.2. Comparison with experimental results from this study
0.7
0.6
0.5 Line of equality
0.4 0.4
0.5 0.6 0.7 Experimental packing density
0.8
Fig. 15. Theoretical compacted packing densities against experimental packing densities obtained from this study.
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estimating nor over-estimating the uncompacted or compacted packing densities of angular rock aggregate particles. 7.3. Comparison with experimental results by Fung and Kwan Fung and Kwan [26] had in a previous study measured the packing densities of binary mixes of angular rock aggregate after flowing through a V-funnel into a container. The aggregate tested was crushed granite rock aggregate similar to the aggregate used in this study. Seven size classes of aggregate were prepared by sieving. The seven size classes were of size 0.3–0.6 mm, 0.6–1.18 mm, 1.18–2.36 mm, 2.36–5 mm, 5–10 mm, 10–14 mm and 14–20 mm, respectively. Their characteristic sizes were each taken as the geometric mean of the upper and lower sieve sizes. Fifteen binary mix series were prepared for packing density tests. The packing density measurement procedures adopted by Fung and Kwan were almost the same as those for measuring the uncompacted packing density of aggregate in this study except that the aggregate sample was discharged from a V-funnel after the V-funnel flow test into the container instead of being placed into the container shovel by shovel. It is noteworthy that while being discharged from the V-funnel to the container, the aggregate sample might have been slightly compacted due to fast flow rate into the container and might have become slightly non-uniform due to segregation. Nevertheless, these test results are still regarded as useful data for verifying whether the 3-parameter model can also be applied to packing density prediction in such condition. For comparison, the theoretical packing densities predicted by the 3-parameter model using the interactions functions given by Eqs. (4a) to (4c) are plotted against the experimental packing densities by Fung and Kwan in Fig. 16. Most data points in the figure are fairly close to the line of equality, indicating reasonably good agreement between the theoretical predictions and the experimental results. The percentage error ranges from − 5.76% to 5.92% and the mean percentage error is −2.00%. Overall, when applied to predict the packing densities of the binary mixes tested by Fung and Kwan, the 3-parameter model is accurate to within an absolute error of 5.9%. The error in this particular case is larger than the error when applied to the binary mixes tested in this study probably because there have been slight compaction and segregation of the aggregate particles during the packing density tests conducted by Fung and Kwan. Nevertheless, the maximum absolute error of 5.9% is still regarded as small enough to verify the applicability and accuracy of the 3-parameter model.
161
8. Conclusions The 3-parameter packing density model, originally developed for spherical particles, has been extended for application to angular rock aggregate particles. In the extended model, the measured packing density of each size class of particles is used in the packing density prediction so as to allow for the effect of particle shape on the packing density of each size class of particles and the interaction functions of the three parameters are re-derived from experimental packing density results of angular aggregate particles so as to allow for the influence of particle shape on the loosening, wall and wedging effects. An experimental program of measuring the uncompacted and compacted packing densities of binary mixes of angular rock aggregates has been carried out to provide data for the determination of the loosening, wall and wedging effect parameters. The three parameters so obtained are not quite the same as those for spherical particles, indicating that the particle shape does have certain influence. On the whole, the loosening and wall effect parameters are slightly larger under compacted condition than uncompacted condition. However, the wedging effect parameter is similar under the uncompacted or compacted condition. Negative values of wall effect parameter have been obtained. Since similar negative values have been obtained by de Larrard, it is postulated that these are not experimental errors. Further study on this observed phenomenon, which could be caused by the interaction between the agglomeration effect and the wall effect, is recommended. Two sets of interaction functions have been derived, one for the uncompacted condition and the other for the compacted condition. Using these interaction functions, the 3-parameter model has been extended for application to angular rock aggregate particles. The extended 3-parameter model has been compared to the experimental results by de Larrard [17], those from this study and those from Fung and Kwan [26]. Whilst the experimental results by de Larrard are too scattered to verify the accuracy of this model, comparison of the theoretical predictions by this model to the experimental results from this study and those from Fung and Kwan yielded very good agreement and reasonably good agreement, respectively. Acknowledgement The research work reported herein was fully funded by a GRF grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 17203514).
0.8
Theoretical packing density
R2 = 0.903
References
0.7
0.6
0.5 Line of equality
0.4 0.4
0.5 0.6 0.7 Experimental packing density
0.8
Fig. 16. Theoretical packing densities against experimental packing densities obtained from Fung and Kwan [26].
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
A 3-parameter packing density model for angular rock aggregate particles A.K.H. Kwan ⁎, V. Wong, W.W.S. Fung Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 25 September 2014 Received in revised form 20 December 2014 Accepted 22 December 2014 Available online 11 January 2015 Keywords: Packing density Packing models Particle mechanics Particle shape
a b s t r a c t The authors have in recent studies incorporated the wedging effect to develop a 3-parameter model for packing density prediction of binary and ternary mixes of spherical particles. This model has been restricted to only spherical particles because only the test results of spherical particles were used for derivation and validation. For more general applications to other types of particles, such as angular particles, the model needs to be further developed. In this study, an experimental program on the packing density of binary mixes of angular rock aggregate particles was carried out and the experimental results were used to derive the interaction functions of the three parameters (the loosening, wall and wedging effect parameters) for extending the 3-parameter model to binary mixes of angular particles. Apart from the experimental results obtained herein, the test results published by de Larrard and those obtained by the authors in an earlier study were also used to validate the extended 3-parameter model. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Particulate materials are widely used in many processing industries, such as the mineral, metallurgical, pharmaceutical and food industries [1–4]. Product design and manufacturing across this wide spectrum of industries are largely a practice of particle technology, which generally requires the knowledge of particle packing. For this reason, the importance of particle packing is well known in the processing industries, but somehow not in the concrete production industry. Actually, a typical concrete mix may be regarded as a mixture of aggregate particles and cement paste. The cement paste fills the voids between the aggregate particles and binds the aggregate particles upon hardening to give strength to the concrete. Among the various ingredients, cement is the most expensive and has the highest carbon footprint [5,6]. To slash the cost and carbon footprint of concrete production, a promising way is to maximize the packing density of the aggregate so as to reduce the volume of cement paste needed to fill the voids [6]. For this task, it would be helpful to have a particle packing model for predicting the packing density of rock aggregate particles, which are usually angular in shape. Numerous particle packing models have been developed but most of these models are limited to spherical particles. In the 1930s, Westman and Hugill [7] and Furnas [8] pioneered to derive analytical equations for predicting the packing density of binary mixes of spherical particles by considering two extreme cases: first, when the size ratio is close to ⁎ Corresponding author. Tel.: +852 2859 2647. E-mail address: [email protected] (A.K.H. Kwan).
http://dx.doi.org/10.1016/j.powtec.2014.12.054 0032-5910/© 2015 Elsevier B.V. All rights reserved.
zero, and second, when the size ratio is close to unity (the size ratio is the ratio of the size of smaller particles to the size of larger particles). Since then, substantial progress has been made in the packing density prediction of binary mixes of spherical particles with an intermediate size ratio, in which case, interactions between particles of different sizes occur [9–16]. Two interaction effects have been identified, namely: the loosening effect, which occurs when the larger particles are dominant and the wall effect, which occurs when the smaller particles are dominant. These two effects are taken into account using two parameters, called the loosening effect parameter and the wall effect parameter, respectively. Both these two parameters are functions of the size ratio (these functions are called interaction functions) derived by curve fitting against experimental results. The particle packing models, which employ two parameters to account for the loosening and wall effects, may be referred to as the 2-parameter models. In most of the 2-parameter models [10,12,13,16], the relationship between the specific volume of the particle system and the volumetric fractions of the various size classes of particles (each size class is a collection of mono-sized particles) is assumed to be linear. However, it has been observed from test results [9,17,18] that the relationship between the specific volume and the volumetric fractions is in fact nonlinear. To take into account such nonlinear relationship and improve the accuracy of the 2-parameter models, de Larrard [17] incorporated a compaction index, which is dependent on the packing process, to develop a more advanced compressible packing model, whereas Kwan et al. [18] added a wedging effect parameter, which accounts for a newly identified wedging effect, to develop a more sophisticated 3-parameter model. The 3-parameter model developed
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by Kwan et al. [18] is only for binary mixes of spherical particles. Later, Wong and Kwan [19] refined this model for application to ternary mixes of spherical particles. Most existing particle packing models are applicable only to spherical or rounded particles because they were validated by comparing with experimental results of only spherical or rounded particles. For more general applications to non-spherical particles, further development of the particle packing models to allow for the effects of particle shape is needed. Overall, the particle shape may have two effects: first, on the packing density of each mono-sized fraction of the particles, and second, on the interactions between the different mono-sized fractions of particles. The effect of particle shape on the packing density of mono-sized particles has been studied extensively using regular particles such as ellipsoids, spheroids, disks and cylinders [20,21]. Moreover, the packing density of irregular rock aggregate particles has been measured and correlated to the various shape parameters [22]. Anyway, whatever the effect of particle shape is, the packing density of mono-sized particles can be measured directly, and provided the measured packing densities of the mono-sized fractions are used in the packing density prediction, the effect of particle shape on the packing density of mono-sized particles should already be allowed for. To characterize the size of a mono-sized fraction of non-spherical particles, Yu and Standish [23] and Yu et al. [24] introduced the concept of equivalent packing diameter and employed this concept to extend their 2-parameter model for application to non-spherical particles. Meanwhile, Goltermann et al. [25] modified the packing models of Aїm and Goff [10] and Toufar et al. [11] for application to rock aggregates and introduced the concept of position parameter for characterizing the size of aggregate particles having a certain particle size distribution ranging over several sieve sizes. On the other hand, for aggregate particles passing a certain sieve (the upper sieve) but retained on the next smaller sieve (the lower sieve), of which the size range is relatively narrow, de Larrard [17] assigned the geometric mean of the upper and lower sieve sizes to be the characteristic size. Herein, the 3-parameter model [18,19] is extended for application to binary mixes of angular rock aggregate particles. The measured packing densities of the mono-sized fractions are used in the packing density prediction and thus the effect of particle shape on the packing density of each mono-sized fraction should have been allowed for. An experimental program has been carried out to provide data for deriving the interaction functions. Two sets of interaction functions are derived, one for the uncompacted condition and the other for the compacted condition. Finally, the extended 3-parameter model is compared to the test results obtained in this study and the test results by de Larrard [17] and by Fung and Kwan [26] to verify the applicability and accuracy of the model. 2. The 3-parameter model for spherical particles The 3-parameter model was originally developed by Kwan et al. [18] for binary mixes of spherical particles. It requires pre-determination of the optimum volumetric fractions yielding the maximum packing density before applying the equations contained therein to predict the packing density. For easier application and dealing with ternary mixes, the 3-parameter model has been refined by Wong and Kwan [19] in two ways. First, the equations are modified so that they are not dependent on the optimum volumetric fractions and pre-determination of the optimum volumetric fractions is not needed any more. Second, the equations are expanded to cater for the packing density prediction of ternary mixes. The refined 3-parameter model by Wong and Kwan [19] is adopted herein as the basic framework for extension to apply to angular particles. For easy reference, an outline of this 3-parameter model is presented in the following. Consider a binary mix composed of two size classes of mono-sized spherical particles: size class i and size class j, in order of increasing particle size. Let the diameters of particles of size class i and size class j
155
be di and dj, respectively (note that di ≤ dj), the volumetric fractions of size class i and size class j be ri and rj, respectively (note that ri + rj = 1), and the packing densities of size class i and size class j be ϕi and ϕj, respectively. The packing density of the binary mix when size class i is dominant (denoted by ϕi*) and the packing density of the binary mix when size class j is dominant (denoted by ϕj*) are given respectively by: 1 ¼ ϕi
! r h r i rj ri j þ − 1−bi j 1−ϕ j 1−ci j 2:6 j −1 ϕi ϕ j ϕj
ð1Þ
1 ¼ ϕ j
! r h r i rj ri þ − 1−ai j i 1−ci j 3:8 i −1 ϕi ϕ j ϕi
ð2Þ
in which aij, bij and cij are the loosening effect parameter, wall effect parameter and wedging effect parameter, respectively, and are given as functions of the size ratio. The final packing density of the binary mix of particles can be determined simply as min.{ϕi*, ϕj*}. The loosening effect occurs in the case when the larger particles are dominant due to the smaller particles squeezing into the voids between the larger particles causing loosening of the packing of the larger particles. The wall effect occurs in the case when the smaller particles are dominant due to the boundaries of the larger particles acting like walls causing disruption of the regular packing of the smaller particles. However, the wedging effect takes place in both cases. In the case when the larger particles are dominant, the wedging effect occurs when some isolated smaller particles are entrapped in the gaps between the larger particles instead of filling into the voids thereby wedging the larger particles apart. In the case when the smaller particles are dominant, the wedging effect takes place when some gaps between the larger particles are too narrow to accommodate even one complete layer of smaller particles leading to the presence of only isolated smaller particles at the gaps wedging the larger particles apart. Fig. 1 demonstrates the loosening effect, wall effect and wedging effect in binary mixes of particles. It should be noted that if the values of cij in Eqs. (1) and (2) are set to zero, Eqs. (1) and (2) would revert back to the packing density equations in the conventional 2-parameter models. With the additional terms accounting for the wedging effect, the relationship between the specific volume and the volumetric fractions is no longer linear because the wedging effect varies non-linearly with volumetric fractions. For spherical particles, Kwan et al. [18] and Wong and Kwan [19] have derived the following interaction functions for the evaluation of the loosening effect, wall effect and wedging effect parameters: 3:3
−2:6 s ð1−sÞ
1:9
−2:0 s ð1−sÞ
a ¼ 1−ð1−sÞ
b ¼ 1−ð1−sÞ
3:6
ð3aÞ
6:0
ð3bÞ
c ¼ 0:322 tanhð11:9 sÞ
ð3cÞ
When larger particles are dominant
When smaller particles are dominant
Loosening effect
Wedging effect
Wall effect
Wedging effect
Fig. 1. A schematic diagram showing the loosening, wall and wedging effects.
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in which s is the size ratio. The values of aij, bij and cij in Eqs. (1) and (2) may be obtained by substituting the value of sij = di/dj as s into the above equations. Note that when s = 0, aij = bij = cij = 0, and when s = 1, aij = bij = 1 and cij = 0.322. 3. Extension of the 3-parameter model for application to angular particles It is postulated herein that the 3-parameter model presented in the previous section can be applied also to binary mixes of angular aggregate particles provided that (1) the particle size of each size class (defined as the diameter of the particles in the original model for spherical particles) is re-defined as the characteristic size of the particles; (2) the measured packing density of each size class is used as input in the packing density prediction so as to allow for the effect of particle shape on the packing density of each size class of particles; and (3) the interaction functions for the loosening effect, wall effect and wedging effect parameters are re-derived from experimental packing density results of angular aggregate particles. In the field of concrete technology, the particle size distribution of aggregate particles is usually determined by mechanical sieving of the aggregate particles through a set of sieves starting from the sieve with the largest sieve size on top and then successively downwards through sieves with smaller and smaller sieve sizes [17,25,26]. After sieving, the aggregate particles are divided into different size classes by obtaining the aggregate particles retained on each sieve and taking the aggregate particles on the same sieve as belonging to the same size class. The sieve on which the particles are retained is called the lower sieve whereas the next larger sieve that the particles have passed through is called the upper sieve. Each size class of aggregate particles has a particle size ranging from the lower sieve size to the upper sieve size and is therefore not truly mono-sized. Nevertheless, since the size range of each size class is fairly narrow, the characteristic size is simply taken as the geometric mean of the upper and lower sieve sizes, as recommended by de Larrard [17]. 4. Experimental program An experimental program has been launched to measure the packing densities of binary mixes of fine and coarse aggregates. Crushed granite rock aggregate was used in the tests. As for other crushed rock aggregates, it was angular in shape. The crushed rock aggregate was divided into six size classes by mechanical sieving using steel plate sieves with square apertures. The sieve sizes were 0.6 mm, 1.18 mm, 2.36 mm, 5.0 mm, 10 mm, 14 mm and 20 mm. Each size class comprised of the aggregate particles retained on one of the sieves. The six size classes have particle size ranges of 0.6-1.18 mm, 1.18-2.36 mm, 2.36-5 mm, 5-10 mm, 10-14 mm, and 14-20 mm, respectively, as tabulated in Table 1. Their characteristic sizes, each taken as the geometric mean of the upper and lower sieve sizes, are 0.84 mm, 1.67 mm, 3.44 mm, 7.07 mm, 11.83 mm and 16.73 mm, respectively. Following usual practice, the size classes finer than 5.0 mm are regarded as fine aggregates, whereas those coarser than 5.0 mm are regarded as coarse aggregates. Hence, the six size classes are named as F1, F2, F3, C1, C2, Table 1 Sieve size ranges and characteristic sizes of the six size classes of aggregates. Size class
Sieve size range Lower sieve size (mm)
Upper sieve size (mm)
F1 F2 F3 C1 C2 C3
0.60 1.18 2.36 5.0 10.0 14.0
1.18 2.36 5.0 10.0 14.0 20.0
Characteristic size (mm)
0.84 1.67 3.44 7.07 11.83 16.73
and C3 in ascending order of their sizes, as listed in Table 1, where F denotes fine aggregate and C denotes coarse aggregate. By blending one size class of fine aggregate (F1, F2 or F3) and one size class of coarse aggregate (C1, C2 or C3) together at various volumetric fractions, a total of nine binary mix series were produced for packing density tests. An identification code in the form of Fx-Cy is assigned to each binary mix series, in which Fx represents the size class of the fine aggregate blended and Cy represents the size class of the coarse aggregate blended. The nine binary mix series produced are: F1-C1, F2-C1, F3C1, F1-C2, F2-C2, F3-C2, F1-C3, F2-C3 and F3-C3. In each binary mix series, the volumetric fraction of Fx was varied from 0 to 100% in steps of 10% and the volumetric fraction of Cy was set equal to 100% minus the volumetric fraction of Fx. Altogether, there were 6 packing density tests for the different size classes (F1 to F3 and C1 to C3), and 81 packing density tests for the binary mixes of aggregates. The test methods stipulated in the British Standard BS 812-2: 1995 [27] were employed to measure the uncompacted and compacted packing densities of the aggregates. Since the container specified in the British Standard has an internal diameter of about 10 times the size of the largest size class of coarse aggregate C3, the wall effect of the container should be small enough to be ignored [28]. The test method for measuring the uncompacted packing density is outlined as follows. First, a sample of the aggregate (a single size class of aggregate or a binary mix of fine and coarse aggregates) was prepared. In the case of preparing a sample of a binary mix of aggregates, the fine and coarse aggregates to be blended were accurately weighed to the required volumetric fractions and mixed thoroughly in a steel tray manually until uniform mixing was achieved. Then, the aggregate sample was placed into the container shovel by shovel instead of pouring into the container to avoid segregation until the container was over-filled. After then, the excess aggregate particles were removed and the surface was leveled off. Finally, the container fully-filled with aggregate particles was weighed. The foregoing procedures were carried out three times so as to obtain the average weight of aggregate inside the container. The test method for measuring the compacted packing density was almost the same except that the aggregate sample was placed into the container in three equal portions by means of a shovel and after each portion of aggregate was filled, the aggregate in the container was compacted by applying 30 compactive blows with a tamping rod evenly distributed over the surface. Each compactive blow was applied by releasing the tamping rod at a height of 50 mm above the surface of aggregate. The saturated solid density and oven-dried solid density of the aggregate were measured in accordance with BS812-2: 1995 [27] as 2609 kg/m3 and 2590 kg/m3 respectively. Since air-dried aggregate was used in the packing density tests, the air-dried solid density of the aggregate should be used for the determination of packing density. From the measured moisture content of the aggregate, the air-dried solid density of the aggregate was calculated as 2598 kg/m3. Hence, the solid volume of the aggregate in the container was calculated as the average weight of the aggregate in the container divided by the air-dried solid density of 2598 kg/m3. Finally, the packing density was determined as the ratio of the solid volume of aggregate in the container to the volume of the container. 5. Experimental results The packing density results of the six size classes tested are plotted against the respective characteristic sizes in Fig. 2, wherein two curves are presented, one for uncompacted condition and the other for compacted condition. As expected, for each and every size class, the compacted packing density is higher than the uncompacted packing density. This is because the compactive blows applied during compaction had enabled rearrangement of the aggregate particle to achieve a denser packing. Overall, under both the uncompacted and compacted conditions, there is a general trend that the packing densities of the
A.K.H. Kwan et al. / Powder Technology 274 (2015) 154–162
0.65
0.80
Compacted
0.70
Packing density
Packing density
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.60 0.55 0.50
Uncompacted
0.45
0.65 0.60 0.55 0.50
0.40
F1-C2 (s = 0.071) F2-C2 (s = 0.141) F3-C2 (s = 0.290)
0.45 0.40
0.35 0.0
2.0
4.0
0.0
6.0 8.0 10.0 12.0 14.0 16.0 18.0 Characteristic size (mm)
Fig. 2. Variation of packing density of a single size class with characteristic size.
three size classes of fine aggregate (F1, F2 and F3) are significantly lower than those of the three size classes of coarse aggregate (C1, C2 and C3). One most likely reason for the lower packing density of the finer particles is the presence of inter-particle cohesive forces, such as van der Waals and electrostatic attractive forces, which cause agglomeration of the particles and formation of voids inside the agglomerates [29–31]. Such agglomeration effect is generally larger in finer particles and that is why the packing density of finer particles is usually lower than that of coarser particles. The packing density results of the nine binary mixes tested are plotted against the respective volumetric fractions of the fine aggregate as data points in Figs. 3, 4 and 5 for the uncompacted binary mixes containing C1, C2 and C3, respectively, and in Figs. 6, 7 and 8 for the compacted binary mixes containing C1, C2 and C3, respectively. It can be seen from the data points plotted that in each binary mix series, as the volumetric fraction of the fine aggregate increases from zero, the packing density first increases and after reaching a certain maximum value, starts to decrease. The maximum packing density attained is generally higher when the size ratio is smaller (or in other words, when the size difference between the two size classes of particles blended together is larger). Overall, the general trends of the variation of the packing density with the volumetric fractions and the variation of the maximum packing density attained with the size ratio are the same as those for binary mixes of spherical particles.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Volumetric fraction of fine aggregate, r1 Fig. 4. Uncompacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C2, F2-C2 and F3-C2.
6. Interaction functions of the three parameters From the data points for each binary mix series under either uncompacted or compacted condition presented in Figs. 3 to 8, one set of best-fit values of the loosening effect parameter a, the wall effect parameter b, and the wedging effect parameter c has been derived by fitting the packing density equations, Eqs. (1) and (2), to the 11 packing density results in the series. The best-fit values of the three parameters, a, b and c, so derived are plotted against the size ratio as data points in Figs. 9, 10 and 11, respectively, using hollow symbols for uncompacted results and solid symbols for compacted results. Based on the best-fit values, the interaction functions of the three parameters were derived by regression analysis. For the uncompacted condition, the interaction functions of the three parameters so derived are as follows:
5:0
−1:9 s ð1−sÞ
1:9
−2:1 s ð1−sÞ
a ¼ 1−ð1−sÞ
b ¼ 1−ð1−sÞ
3:1
10:5
ð4aÞ
7:6
−0:2 ð1−sÞ
ð4bÞ
c ¼ 0:335 tanhð26:9 sÞ
0.80
ð4cÞ
0.80
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.70
0.70 Packing density
Packing density
157
0.65 0.60 0.55 0.50
0.60 0.55 0.50
F1-C1 (s = 0.119) F2-C1 (s = 0.236) F3-C1 (s = 0.486)
0.45
0.65
F1-C3 (s = 0.050) F2-C3 (s = 0.100) F3-C3 (s = 0.205)
0.45
0.40
0.40
0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 3. Uncompacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C1, F2-C1 and F3-C1.
0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 5. Uncompacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C3, F2-C3 and F3-C3.
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0.80
0.80
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.70 Packing density
0.70
Packing density
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
0.65 0.60 0.55 0.50
0.60 0.55 0.50
F1-C1 (s = 0.119) F2-C1 (s = 0.236) F3-C1 (s = 0.486)
0.45
0.65
F1-C3 (s = 0.050) F2-C3 (s = 0.100) F3-C3 (s = 0.205)
0.45
0.40
0.40 0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
0.0
1.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 6. Compacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C1, F2-C1 and F3-C1.
Fig. 8. Compacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C3, F2-C3 and F3-C3.
For the compacted condition, the interaction functions of the three parameters so derived are as follows:
size. Consequently, when squeezing into a void, the angular particle would push the larger particles a greater distance apart causing a larger loosening effect. From Fig. 10, it is evident that most of the time, the wall effect parameter is marginally larger under compacted condition than uncompacted condition. This is probably because under compacted condition, the smaller particles are more closely packed and thus apparently the wall effect of the larger particles disrupting the regular packing of the smaller particles is larger. Nevertheless, except at size ratio smaller than 0.2, the wall effect parameter is almost the same for both the angular particles and spherical particles. Hence, the wall effect parameter is not sensitive to the particle shape. More importantly, it appears that the wall effect parameter can become negative when the size ratio is smaller than 0.2, meaning that the wall effect can sometimes increase rather than decrease the packing density of the smaller particles. This is against our intuition and further study is needed to ascertain that such observed phenomenon is not caused by experimental errors or mistakes. In fact, the same phenomenon has been observed by de Larrard [17]. More discussions on this issue will be given in the next section. From Fig. 11, it is noted that the wedging effect parameter is similar under the uncompacted or compacted condition, indicating that the wedging effect parameter is not sensitive to the compaction applied.
7:1
−1:9 s ð1−sÞ
2:2
−0:7 s ð1−sÞ
a ¼ 1−ð1−sÞ
b ¼ 1−ð1−sÞ
3:1
9:3
ð5aÞ 10:6
−0:2 ð1−sÞ
ð5bÞ
c ¼ 0:335 tanhð26:9 sÞ
ð5cÞ
For comparison with the corresponding best-fit values, the above interaction functions of parameters a, b and c are plotted as curves in Figs. 9, 10 and 11, respectively. To study the effect of particle shape, the interaction functions of the three parameters derived from the test results of spherical particles under uncompacted condition, i.e. Eqs. (3a) to (3c), are also plotted in Figs. 9, 10 and 11. From Fig. 9, it can be seen that overall speaking, the loosening effect parameter is slightly larger under compacted condition than uncompacted condition. This is probably because under compacted condition, the larger particles are more closely packed and thus relatively the loosening effect of the smaller particles squeezing into the voids between the larger particles is larger. Moreover, in general, the loosening effect parameter is slightly larger for angular particles than for spherical particles. One possible reason is that each angular particle actually occupies a larger circumscribing volume (the convex volume circumscribing the particle) than a spherical particle with the same solid volume and thus would have a larger equivalent
1.0
Loosening effect parameter a
0.80
Lines: predictions by the extended 3-parameter model Points: experimental results
0.75
Packing density
0.70 0.65 0.60 0.55 0.50
F1-C2 (s = 0.071) F2-C2 (s = 0.141) F3-C2 (s = 0.290)
0.45
Angular particles
0.8
Spherical particles
0.6
0.4
Uncompacted Compacted Eqn. (3a) Eqn. (4a) Eqn. (5a)
0.2
0.40 0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volumetric fraction of fine aggregate, r1
0.9
1.0
Fig. 7. Compacted packing density against volumetric fraction of fine aggregate for binary mix series F1-C2, F2-C2 and F3-C2.
0.0 0.0
0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 9. Best-fit values and interaction functions of the loosening effect parameter.
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1.0
Angular particles
Wall effect parameter b
0.8
Spherical particles
0.6
Uncompacted Compacted Eqn. (3b) Eqn. (4b) Eqn. (5b)
0.4 0.2 0.0 0.0
0.2
-0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 10. Best-fit values and interaction functions of the wall effect parameter.
In fact, the interaction function of the wedging effect parameter derived for the uncompacted condition, Eq. (4c), and that derived for the compacted condition, Eq. (5c), are the same. Hence, the wedging effect occurs under both the uncompacted and compacted conditions, and application of compaction would not mitigate the wedging effect. Moreover, the wedging effect parameter is marginally larger for angular particles than spherical particles. This may be attributed to the slightly larger equivalent size of angular particles than spherical particles having the same solid volume. 7. Verification of the extended 3-parameter model 7.1. Comparison with experimental results by de Larrard de Larrard [17] had carried out similar tests on the packing densities of rounded (uncrushed) and angular (crushed) rock aggregates. For each type of aggregate, five size classes were prepared by sieving. Each size class was obtained from the aggregate particles retained on a sieve and thus was of size within two successive sieve sizes following the French Standard. By blending at each time two different size classes
159
together, seven binary mix series were produced from each type of aggregate for testing and in each binary mix series, the volumetric fractions of the two size classes were varied from 0 to 100%. The packing density of each aggregate sample was measured by filling the aggregate into a container, applying compression on top of the aggregate and subjecting the container to vibration on a vibrating table. Hence, the results are the packing densities of the aggregate samples under compacted condition. From the variations of the measured packing density with the volumetric fractions in each binary mix series, the best-fit values of the loosening effect parameter and the wall effect parameter (there was no wedging effect parameter in de Larrard’s model) were derived by curve fitting. The loosening effect parameters from this study are compared to those by de Larrard in Fig. 12. It is seen that the loosening effect parameters by de Larrard are very scattered. Moreover, the loosening effect parameters for the rounded particles are sometimes larger and sometimes smaller than those for the angular particles. Hence, de Larrard’s results reveal no clear trend of whether the loosening effect parameters for the rounded particles are larger or smaller than those for the angular particles. Comparing the loosening effect parameters for angular particles under compacted condition from this study, as represented by Eq. (5a), to those obtained by de Larrard, it is found that whilst some of the loosening effect parameters obtained by de Larrard agree closely with Eq. (5a), several of them do not agree well with Eq. (5a). Overall, the agreement between the loosening effect parameters from this study and those obtained by de Larrard is not good. de Larrard’s results are too scattered to be used as benchmark for validation and the above comparison has not fulfilled the intended aim of verifying the accuracy of the extended 3-parameter model. The wall effect parameters from this study are compared to those by de Larrard in Fig. 13. It is seen that the wall effect parameters by de Larrard are also very scattered. Again, the wall effect parameters for the rounded particles are sometimes larger and sometimes smaller than those for the angular particles, thus revealing no clear difference between the wall effect parameters for the rounded and angular particles. Comparing the wall effect parameters for angular particles under compacted condition from this study, as represented by Eq. (5b), to those obtained by de Larrard, it is found that although some of the wall effect parameters obtained by de Larrard agree closely with Eq. (5b), some others do not agree well with Eq. (5b). Overall, the agreement between the wall effect parameters from this study and those
1.0
1.0
Wedging effect parameter c
0.8
Loosening effect parameter a
Uncompacted Compacted Eqn. (3c) Eqn. (4c) Eqn. (5c)
0.6
Angular particles 0.4
Spherical particles
0.2
Angular particles
0.8
Spherical particles
0.6
Rounded [17] Angular [17] Uncompacted Compacted Eqn. (3a) Eqn. (4a) Eqn. (5a)
0.4
0.2
0.0
0.0
0.0
0.0
0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 11. Best-fit values and interaction functions of the wedging effect parameter.
0.2
0.4 0.6 Size ratio s
0.8
1.0
Fig. 12. Comparison of the loosening effect parameters obtained from this study with those by de Larrard [17].
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0.8
1.0
Spherical particles
0.6
Theoretical packing density
0.8
Wall effect parameter b
R2 = 0.980
Angular particles
Rounded [17] Angular [17] Uncompacted Compacted Eqn. (3b) Eqn. (4b) Eqn. (5b)
0.4 0.2
0.7
0.6
0.5
0.0 0.0 -0.2
0.2
0.4 0.6 Size ratio s
0.8
1.0
Line of equality
0.4 0.4
Fig. 13. Comparison of the wall effect parameters obtained from this study with those by de Larrard [17].
0.5
0.6
0.7
0.8
Experimental packing density Fig. 14. Theoretical uncompacted packing densities against experimental packing densities obtained from this study.
obtained by de Larrard is not satisfactory. The above comparison has not verified the accuracy of the extended 3-parameter model and further comparison with more experimental results is needed. More importantly, at a size ratio of 0.0625, negative best-fit values of the wall effect parameter were obtained by de Larrard for both the rounded and angular particles. de Larrard attributed such negative wall effect parameters to the effect of vibration or compaction. However, negative wall effect parameters have been obtained in this study under uncompacted condition. Therefore, the negative wall effect parameters were not caused by vibration or compaction. One possible cause of such negative wall effect parameter is the interaction between the agglomeration effect and the wall effect. When the size ratio is small, the packing density of the smaller particles tends to be smaller because of agglomeration and presence of voids inside the agglomerates. At the wall-like boundaries of the larger particles, the agglomerates could be broken and the voids originally inside the agglomerates could merge with the voids caused by the wall effect leading to an overall smaller volume of voids than having separately the agglomeration effect and the wall effect. As a result, the wall effect could appear to be smaller than when there is no such interaction and might even appear to be negative. This is nevertheless just a speculation and further study on this phenomenon is recommended.
Figs. 6, 7 and 8 for comparison with the experimental results plotted therein as data points. Again, the theoretical predictions agree closely with the experimental results. For more precise evaluation of the accuracy of the model, the theoretical packing densities are plotted against the experimental packing densities as data points in Fig. 15. As before, all the data points lie closely to the line of equality, indicating very good agreement between the theoretical predictions and the experimental results. From these results, it has been calculated that the percentage error in packing density ranges from − 1.15% to 2.22% and the overall mean percentage error is 0.08%. Overall, with the interaction functions modified to cater for angular rock aggregate particles under uncompacted or compacted condition, the 3-parameter model is accurate to within an absolute error of 3.6%. Actually, part of the errors might be due to experimental errors and intrinsic randomness of particle packing. The very small mean percentage errors indicate that the 3-parameter model is on average neither under-
0.8 R2 = 0.987
The theoretical uncompacted packing densities predicted by the 3-parameter model using the interactions functions given by Eqs. (4a) to (4c) for the nine binary mix series tested are plotted as curves in Figs. 3, 4 and 5 for comparison with the experimental results plotted therein as data points. The comparison reveals that in general, the theoretical predictions agree closely with the experimental results. For more precise evaluation of the accuracy of the model, the theoretical packing densities are plotted against the experimental packing densities as data points in Fig. 14. Generally, all the data points lie closely to the line of equality, indicating very good agreement between the theoretical predictions and the experimental results. From these results, it has been calculated that the percentage error in packing density ranges from − 2.51% to 3.55% and the overall mean percentage error is − 0.03% (negative error means under-estimation while positive error means over-estimation). The theoretical compacted packing densities predicted by the 3-parameter model using the interactions functions given by Eqs. (5a) to (5c) for the nine binary mix series tested are plotted as curves in
Theoretical packing density
7.2. Comparison with experimental results from this study
0.7
0.6
0.5 Line of equality
0.4 0.4
0.5 0.6 0.7 Experimental packing density
0.8
Fig. 15. Theoretical compacted packing densities against experimental packing densities obtained from this study.
A.K.H. Kwan et al. / Powder Technology 274 (2015) 154–162
estimating nor over-estimating the uncompacted or compacted packing densities of angular rock aggregate particles. 7.3. Comparison with experimental results by Fung and Kwan Fung and Kwan [26] had in a previous study measured the packing densities of binary mixes of angular rock aggregate after flowing through a V-funnel into a container. The aggregate tested was crushed granite rock aggregate similar to the aggregate used in this study. Seven size classes of aggregate were prepared by sieving. The seven size classes were of size 0.3–0.6 mm, 0.6–1.18 mm, 1.18–2.36 mm, 2.36–5 mm, 5–10 mm, 10–14 mm and 14–20 mm, respectively. Their characteristic sizes were each taken as the geometric mean of the upper and lower sieve sizes. Fifteen binary mix series were prepared for packing density tests. The packing density measurement procedures adopted by Fung and Kwan were almost the same as those for measuring the uncompacted packing density of aggregate in this study except that the aggregate sample was discharged from a V-funnel after the V-funnel flow test into the container instead of being placed into the container shovel by shovel. It is noteworthy that while being discharged from the V-funnel to the container, the aggregate sample might have been slightly compacted due to fast flow rate into the container and might have become slightly non-uniform due to segregation. Nevertheless, these test results are still regarded as useful data for verifying whether the 3-parameter model can also be applied to packing density prediction in such condition. For comparison, the theoretical packing densities predicted by the 3-parameter model using the interactions functions given by Eqs. (4a) to (4c) are plotted against the experimental packing densities by Fung and Kwan in Fig. 16. Most data points in the figure are fairly close to the line of equality, indicating reasonably good agreement between the theoretical predictions and the experimental results. The percentage error ranges from − 5.76% to 5.92% and the mean percentage error is −2.00%. Overall, when applied to predict the packing densities of the binary mixes tested by Fung and Kwan, the 3-parameter model is accurate to within an absolute error of 5.9%. The error in this particular case is larger than the error when applied to the binary mixes tested in this study probably because there have been slight compaction and segregation of the aggregate particles during the packing density tests conducted by Fung and Kwan. Nevertheless, the maximum absolute error of 5.9% is still regarded as small enough to verify the applicability and accuracy of the 3-parameter model.
161
8. Conclusions The 3-parameter packing density model, originally developed for spherical particles, has been extended for application to angular rock aggregate particles. In the extended model, the measured packing density of each size class of particles is used in the packing density prediction so as to allow for the effect of particle shape on the packing density of each size class of particles and the interaction functions of the three parameters are re-derived from experimental packing density results of angular aggregate particles so as to allow for the influence of particle shape on the loosening, wall and wedging effects. An experimental program of measuring the uncompacted and compacted packing densities of binary mixes of angular rock aggregates has been carried out to provide data for the determination of the loosening, wall and wedging effect parameters. The three parameters so obtained are not quite the same as those for spherical particles, indicating that the particle shape does have certain influence. On the whole, the loosening and wall effect parameters are slightly larger under compacted condition than uncompacted condition. However, the wedging effect parameter is similar under the uncompacted or compacted condition. Negative values of wall effect parameter have been obtained. Since similar negative values have been obtained by de Larrard, it is postulated that these are not experimental errors. Further study on this observed phenomenon, which could be caused by the interaction between the agglomeration effect and the wall effect, is recommended. Two sets of interaction functions have been derived, one for the uncompacted condition and the other for the compacted condition. Using these interaction functions, the 3-parameter model has been extended for application to angular rock aggregate particles. The extended 3-parameter model has been compared to the experimental results by de Larrard [17], those from this study and those from Fung and Kwan [26]. Whilst the experimental results by de Larrard are too scattered to verify the accuracy of this model, comparison of the theoretical predictions by this model to the experimental results from this study and those from Fung and Kwan yielded very good agreement and reasonably good agreement, respectively. Acknowledgement The research work reported herein was fully funded by a GRF grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 17203514).
0.8
Theoretical packing density
R2 = 0.903
References
0.7
0.6
0.5 Line of equality
0.4 0.4
0.5 0.6 0.7 Experimental packing density
0.8
Fig. 16. Theoretical packing densities against experimental packing densities obtained from Fung and Kwan [26].
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