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Porosity calculations of multi-component mixtures of spherical particles
Powder
Technology,
5.2 (1987)
233 - 241
233
Porosity Calculations of Multi-component
Mixtures of Spherical Particles
A. B. YU and N. STANDISH Department of Metallurgy and Materials Engineering, Wollongong, N.S. W. 2500 (Australia) (Received January 28,1987;
of Wollongong,
PO Box 1144,
in revised form March 17, 1987)
SUMMARY
It is shown that the relative maximum void contraction and the corresponding relative fractional solid volume are only the functions of size-ratio for binary mixtures, which can be determined experimentally. A linear analytical model, directly based on the experimental results of binary mixtures, is developed. The results show that the calculated porosities for ternary mixtures are within 8% relative error of measurements. Good agreement between the theoretical and experimental results was also obtained for Gaussian and log-normal size distributions. It is concluded that the porosity of multi-component mixtures of particles may be confidently predicted from the results of binary mixtures. INTRODUCTION
The behaviour and packing efficiency, or its equivalent porosity, of particulate mixtures is important in all industries processing granular materials. In the past, many experimental studies have been done on the packing of mixtures, especially on the packing of binary mixtures of spherical particles [l - 61. It is a well-known empirical fact that the packing fraction varies with the size distribution of the materials involved. After an extensive experimental study of the porosity and permeability of multi-component mixtures of uniform and irregular shape particles, one of the authors concluded that it should be possible to predict the porosity of multicomponent mixtures from the results of binary mixtures [7 - lo]. Recently, a number of mathematical models have been proposed to estimate the porosity of particulate 0032-5910/87/$3.50
The University
mixtures from the knowledge of particle sizes involved and their proportion in the mixture [ll - 131. Unfortunately the porosity predided from these generally somewhat complex models often did not satisfactorily match the experimentally measured porosities. In fact, considering the models proposed by Ouchiyama et al. [12] and Stovall et al. [ 131, the porosity calculation for a mixture is still based on the analysis of the mutual effect of two sizes of particles, though the former analyse a so-called simplified packing model and the latter use interdiary analysis. Unavoidably, there are always some discrepancies between the theory and experiment, even for binary mixtures, which may result in large error for multi-component mixtures. It was considerations of the above kind and the requirement to improve model predictability that have recently prompted us to include in a mathematical model some measurements as prescribed information [ 141. We used the simplex-centroid design, which gave the calculated porosities within 2% relative error of measurements for ternary mixtures [ 141. Moreover, it was shown there that the method could be used, not only for mixtures of spherical particles, but for any actual mixtures, e.g. irregular-shaped, or nonuniform mixtures. However, the prescribed data required for the method increases sharply for each additional component in a mixture, which may limit its general applicability. The purpose of this paper is to present a new approach to the porosity prediction for multi-component mixtures, based on the results of binary mixtures alone, that does not have the same degree of limitation as the above model has. 0 Elsevier Sequoia/Printed in The Netherlands
234 THEORETICAL
TREATMENT
Relative void contraction of binary mixtures Much work has been done experimentally on the packing of binary mixtures. Figure 1 illustrates typical results obtained by McGear-y [ 31 and Jeschar et al. [ 151. Similar results can be found elsewhere [l, 2, 4 - 61. It is evident from Fig. 1 that as the fractional solid volume of large particles increases, the fractional void volume in packing will contract to a minimum and then expand and, that the maximum void contraction increases with the decrease of size-ratio (small/large). It has been found that the packing fraction of a random packing of spheres ranges between two well-defined limits, i.e. the loose and dense packing, having packing fractions of about 0.60 and 0.64 respectively [16, 171. In fact, different experimental conditions may yield different voidages. However, different initial voidages, which reflect experimental conditions to a great degree, will result in the change of the maximum void contraction and the corresponding fractional solid volume. Therefore, the maximum void contraction AE and the corresponding fractional solid volume, XLmaX , should be a function of size-ratio r and initial voidage E’.
When the size-ratio equals zero, the maximum contraction AC(O), and the corresponding fractional solid volume XLmax(o) are given by Ae(o) = e”( 1 - e”)
(1)
Xr,max(0) = -&
(2)
In order to make the existing literature data comparable, we introduce the relative maximum void contraction, which is defined as the ratio of the maximum void contraction of size-ratio r, Ae(r), to the maximum void contraction, Ahe( Similarly, we have the corresponding relative fractional solid volume, i.e., XLmaX(r)/XLmaX(o). It may be expected that Ae(r)/Ae(o) and XLmax(r)/ XLmaX(o) are the only functions of sizeratio which can be determined experimentally. It was Furnas who first proposed a quadratic equation to represent the correlation [ 11. Subsequently, Ridgway and Tarbuck suggested a modified quadratic regression
161: Ac At(o)
_
1-
2.35r + 1.35r2
i0
r < 0.741 r > 0.741
(3)
The results of these and other workers are summarised in Fig. 2. As shown in Fig. 2, if r < 0.35, Ayer et al. [ 51 also gave an equation between packing fraction and size-ratio the results of which, after appropriate recalculation, are almost the same as those calculated using eqn. (3). Comparing the results obtained under different experimental conditions [4, 22, 231 with those calculated using
0.
0
x 20 ill
1
z
2
-‘-‘------
0
Ridgway et al [61 Furnas c 11 Ayer et al [51 Epstein et al [II
z 2; -3
0.
c
a ” 1
0.2
0.4
0.6
0.8
Fractlcnal SolId Volume of Large hrtlcles
Fig. 1. Comparison between measurements [3, 151 and model predictions. Size-ratio: O---, 0.0129; se-, O.O524;x---, O.O89;A--- -, 0.25; *--.-, 0.5.
Fig. 2. Relative maximum void contraction ratio.
us. size-
235
solid volumes of small and large particles be Xs and XL, respectively. The packing fraction pLT is then PLT = PL + PLT&
(8)
or PLT=
- PL 1-Y .I.
0.
I 0
0.4
cl.2
0
0.6
U.8
I.”
r
Fig. 3. Relative fractional solid volume us. size-ratio.
eqn. (3) shows that, as expected, there is not much difference between them. The correlation between XLmax(r)/ XLmax(o) and r can be discussed similarly. As shown in Fig. 3, there is a good correlation between them which is given by XLm”“(r)/XLmax(o) = 1 - r2 Therefore, have
combining
eqns.
(4) (1) - (4), we
e”(1 - e?( 1 - 2.35r + 1.35r2) r < 0.741 r > 0.741
(5)
(6) or X
S
max =
l-
1 -r2 1 - E0
Binary rnk tures The theory of the packing of binary mixtures with very small size-ratio is well known [ 2,131. The packing of binary mixtures with infinitely small size-ratio will now be examined and the possibility of extending the theory to the packing of binary mixtures with larger size-ratio will be discussed later. Consider a packing of large particles having a packing fraction pL. If small particles are introduced into the packing, and noting that the size-ratio is small enough, we may obtain a mixture such that the original voids among the large particles are filled with the small particles. The volume of the packing will not change. Let the fractional
A.9
Similarly, when large particles are considered to be introduced into a packing of small particles having a packing fraction ps and if the volume, except that occupied by large solid particles, is fully packed with small particles, the volume of the packing will not change either. The packing fraction psT is PST = PSTXL + Ps(l _PSTXL)
(10)
or Ps PsT = 1 - (1 -ps)xL
(11)
Equations (9) and (ll), in terms of the specific volume V, defined as the apparent volume occupied by unit solid volume of particles, i.e., the reciprocal of packing fraction, become VLT = v, - v,xs
(12)
vsr = vs - (Vs - 1)XL
(13)
Equations (12) and (13) describe the limitation of binary mixtures as discussed by Westman and Hugill [2]. However, the most important conclusion from eqns. (12) and (13) is that the relationship between the specific volume and the fractional solid volume of binary mixtures with small size-ratio can be truly described by simple linear equations. To extend the theory to the packing of binary mixtures, we assume that the interaction between two sized particles only results in the change of maximum void contraction and the corresponding fractional solid volume, and the specific volume will still vary linearly with the fractional solid volume for any size-ratio which, as shown later, does not result in great error in the porosity calculations of multi-component mixtures. Based on the above assumptions, the development of analytical equations is not too complicated. As shown in Fig. 4, the specific volumes of mixtures given by lines i and j are given by eqns. (14) and (15), respectively.
j r;
236
“i
The maximum packing fraction pij and the corresponding fractional solid volume Xii or Xii GUI be calculated using eqns. (5) - (7). The packing fraction of a mixture is the lower of the two values of PiT and piT. It may be of interest to note that Lee [18] assumed that it was the packing fraction and not the specific volume that varies linearly with the fractional solid volume. However, Lee’s conclusion is not supported by analytical considerations, and his model did not consider the change of the fractional solid volume XnmaX and the effect of packing modes. His model prediction may therefore be expected to result in large errors.
"ij
/’
. .
,R--
-1.
-xj
Fig. 4. Specific volume for binary mixtures.
VT= vixi+
vi-
us. fractional
vi-b
xj
xij
yT=
Xi
solid volume
(14)
i
+
&Xj
Multi-component mixtures It is assumed that mixtures are composed of n components of equal-density spherical particles. Here we do not consider mixtures composed of irregular-shaped particles. Component i particles have diameters di and initial packing fraction pi. The diameters are ordered so that
(15)
where Vij is the minimum specific volume of components i and j, and X, or Xii is the corresponding fractional solid volume. Equations (14) and (15), in terms of the packing fraction, become Pi
PiT =
d,>d2>d3>...>d,
(16)
and the fractional solid volume Xi should satisfy the equation xi+xz+x3+...+xn=1
Pi
Pi
T=
(17)
As the extension of the above packing theory of binary mixtures, we assume that the specific volumes of multi-component mixtures vary linearly with the fraction solid volumes, and there exist no co-interactions among the components. Therefore, the calculation of the interaction between components i and j in multi-component mixtures is identical to that of binary mixtures. The packing fractions of a multi-component mixture are thus given by
or, in general form, Pi
PiT =
(18)
(19)
Pij = Pji
Pi
PiT =
(20)
(i = 1, 2, . . . . n)
(21)
or Pi
piT = 1-c
i-l j=l
1-z i
(i = 1, 2, . . . . n)
$IJ
1
i zj
j=i+l
1-F i
2 rj 1
j
(22)
237
where the maximum can be, as mentioned
packing fraction pii and the corresponding fractional above, calculated using eqns. (5) - (7). That is,
pi + Pi(1 -Pi)(
solid volume,
Xij
rij G 0.741 rij > 0.741
1 - 2.35rij + 1.35rij2)
Pij = Pi
(23)
i
1 - rij2
j
(24)
z--Pi
Xij =
l-
1 - rij2 2
I
jai
-Pi
4
(25)
i>j
;i; rij =
&j i
Min. pT(d) =
0.741d 1 _ VI-
where, according
0
ps(4
+p(d)[l
-p(d)]
pL(d, x) =p(d)
+p(d)[l
-p(d)]
x) = l-
1 - (x/d)2 2 -P(d)
X,(d,
x) =
Xl
1
Xs(4
1, 2, . . . . n) values. For a distribution. The packing
1_ p(d) 1
(26)
f(x)
p,(d.
Xl
x1
Xr,(d,
& x)
to eqns. (23) - (25), we have
ps(d, x) =p(d)
Xs(d,
p(d)
=
- 2.35 ;
(27)
(23)
(29)
1 - (d/x)2 2 -p(d)
+ 1.35
Comparing the calculated results with the experimental results in Fig. 1, it is obvious that there is good agreement between the
(30)
The porosities of multi-component mixtures of discrete size particles and mixtures with continuous size distributions can be predicted from eqns. (22) - (25) and (26) (30) respectively. .I RESULTS AND DISCUSSION 0.1
I
1
0.5
0.0
The calculated results of binary mixtures under the experimental conditions in [ 3,151 are shown in Fig. 1.
Fractional
Solid
1.0
Volume of Large Particles
Fig. 5. Calculated results for loose (- - ) and dense packings for binary mixtures. ( -)
238
theory and the experiment, especially for small size-ratios. The porosities of the socalled dense and loose packings are also given in Fig. 5, which in fact shows the effect of initial porosities on the porosity distribution of binary mixtures. There are several ternary mixture porosities measurements in the literature [ 2,6 - 10, 151. Here we will only compare the calcu-
lated results with the experimental results for spherical systems. These are shown in Figs. 5 - 8. To account for different experimental conditions used, the initial porosities reported in [6, 10, 151 were chosen as prescribed information. By inspection of Figs. 5 - 8 it is obvious that the calculated porosities are in good agreement with the experimentally measured
lb) Fig. 6. Ternary
(a)
mixture
porosities.
(a), Measurements
from
Volume % 2~117mm -
[6];
(b), model predictions.
(b)
Fig. 7. Ternary
mixture
porosities.
(a), Measurements
from
[lo];
(b), model
predictions.
Fig. 8. Ternary
mixture
porosities.
(a), Measurements
from
[15];
(b), model predictions.
239
results over the entire range of compositions. In fact, the calculated results in Figs. 5 - 8 are within 8% relative error of the measurements. This value of the relative error may be compared with those of 5 - 13%, 3 - 11% and 5 - 15% of the Cross et al. [ 191, Ouchiyama and Tanaka [12] and Leitzelement et al. [11] model predictions, respectively. It may be of interest to remark that the model proposed by Cross et al. is essentially an application of the Ouchiyama and Tanaka model to non-spherical particles. It may also be noteworthy that the porosities measured by Ridgway and Tarbuck [6] were regarded as “unusual”. These authors concluded that the ternary results could not have been predicted from the binary behaviour. In the light of elementary considerations, that ternaries consist of three binaries, quaternaries consist of four ternaries, and so on, then given the same mechanism, this conclusion is difficult to understand. In fact, the results of Ridgway and Tarbuck can be predicted well by the present model and the only “unusual” aspect is simply that the size-ratio of small to medium particles in their experiment is greater than 0.741 which, as shown in Fig. 2, results in the almost same behaviour of small and medium particles. Finally, as an example of the use of eqn. (25) for continuous size distributions, a comparison has been made with some data of Sohn and Moreland [20], who have measured the packing fractions of sands obeying Gaussian and log-normal size distributions by weight. However, if the density of sand is assumed to be constant and independent of particle size, then the size distribution by volume and that by weight are identical. Sohn and Moreland [20] have found that the packing fraction is only dependent on a dimensionless standard deviation and particle shape. They also found that the packing fractions increase continuously as the dimensionless standard deviations c/x for Gaussian and ug for log-normal increase except when dimensionless standard deviations are small. Such relationship, it should be remarked, is also predicted by eqn. (26). Calculations were performed on a computer according to eqn. (26), combining with Gaussian or log-normal size distribution and using the complex method of M. J. Box [21]
to solve this optimisation problem and find the minimum packing fraction. Good agreement between the calculated results and the measured results was obtained as can be seen in Figs. 9 and 10. One of the reasons for the apparent difference between the results in Figs. 9 and 10 arises from the difference of the particle shapes of sands used - something that was specifically also noted by Sohn and Moreland [ 201.
x’ Sand (4 0.5 OS 0.8
OS
’ ----
Fig. 9. Calculated results () and experimental data (- - -) for Gaussian size distribution.
o*:,I. Qi and experimental Fig. 10. Calculated results ( -) data ( - - -) for log-normal size distribution.
240
For the above conditions,
eqn. (22) can be written
as
Pi
PiT =
(31)
i-l
1 - (1 -Pi)
x
g”(i,
i>Xj
j=l
-
i j=i+l
f%
dxj
where 1
g”(i, j) = -
1-E
l--Pi
f”(i,j)
=
li
:_
(
U
E
1
(32) ij
$ iJ
1
iJ
As may be expected from analytical considerations, eqn. (31) here, and eqn. (22) of Stovall et ~2. [13] have the same form. However, as shown in Fig. 11, the values calculated using their model are always greater than those calculated using eqns. (32) and (33). Therefore, it is not surprising that their model predicts optimistically large values of the packing density even for binary mixtures and these errors will be compounded for multi-component mixtures. Obviously, as the linear analytical model presented here is based on the experimental results of binary mixtures, its predictability would be expected to be superior to those of the geometrical/ mathematical models thus far developed.
0.0
0.5
1.0
Size- ratio
Fig. 11. Comparison between the model of Stovall et al. [ 131 (- - -) and this model (-).
CONCLUSIONS
We have shown that the relative maximum void contraction and the corresponding relative fractional solid volume are only the
functions of size-ratio for binary mixtures. The linear analytical model has been developed to calculate the porosity of multi-component mixtures, using the initial porosities and the size-ratios as prescribed information. The results show that the calculated porosities for ternary mixtures are within 8% relative error of measurements. Good agreement between the theoretical and experimental results has also been obtained for Gaussian and log-normal size distributions. However, the model may be used only for spherical systems as it is directly based on the experimental results of spherical binary mixtures. Nevertheless, it is confirmed that the porosity of multi-component mixtures of particles may be confidently predicted from the results of binary mixtures.
REFERENCES 1 C. C. Furnas, Bur. Mines, Rept. Investigations 2874, 7 (1978); Bur. Mines. Bull., 307 (1929) 74. 2 A. E. R. Westman and H. R. Hugill, J. Am. Ceram. Sot., 13 (1930) 767. 3 R. K. McGeary, J. Am. Ceram. Sot., 44 (1961) 513. 4 N. Epstein and M. J. Young, Nature, 196 (1962) 885. 5 J. E. Ayer and F. E. Soppet, J. Am. Ceram. Sot., 48 (1965) 180. 6 K. Ridgway and K. J. Tarbuck, Chem. Process. Eng., 49 (1968) 103. 7 N. Standish and D. E. Borger, Powder Technol., 22 (1979) 121. 8 N. Standish and D. G. Mellor, Powder Technol., 27 (1980) 61. 9 N. Standish and P. J. Leyshon, Powder Technol., 30 (1981) 119. 10 N. Standish and D. N. Collins, Powder Technol., 36 (1983) 55. 11 M. Leitzelement, Cho Seng Lo and J. A. Dodds, Powder Technol., 41 (1985) 159. 12 N. Ouchiyama and T. Tanaka, Z & EC Fundam., 23 (1984) 490. 13 T. Stovall, F. De Larrard and M. Buil, Powder Technol., 48 (1986) 1. 14 N. Standish and A. B. Yu, Powder Technol., 49 (1987) 248. 15 R. Jeschar, W. Potke, V. Petersen and K. Polthier,
241
16 17 18 19
in N. Standish (ed.), Blast Furnace Aerodynamics, Aust. I.M.M. Press, Wollongong, N.S.W., 1975, p. 136. C. D. Scott, Nature, 188 (1960) 908. R. Rutgers, Nature, 193 (1962) 465. D. I. Lee, J. Paint Technol., 42 (1970) 579. M. Cross, W. H. Douglas and R. P. Fields, Powder Technol., 43 (1985) 27.
20 H. Y. Sohn and C. Moreland, Can. J. Chem. Eng., 46 (1968) 162. 21 J. L. Kurster and J. H. Mize, Optimization Techniques with FORTRAN, McGraw-Hill, New York, 1973. 22 C. Lemaignan, Acta Met., 28 (1980) 1657. 23 M. Iannella, B. Met. Thesis, Univ. of Wollongong (1985).
Technology,
5.2 (1987)
233 - 241
233
Porosity Calculations of Multi-component
Mixtures of Spherical Particles
A. B. YU and N. STANDISH Department of Metallurgy and Materials Engineering, Wollongong, N.S. W. 2500 (Australia) (Received January 28,1987;
of Wollongong,
PO Box 1144,
in revised form March 17, 1987)
SUMMARY
It is shown that the relative maximum void contraction and the corresponding relative fractional solid volume are only the functions of size-ratio for binary mixtures, which can be determined experimentally. A linear analytical model, directly based on the experimental results of binary mixtures, is developed. The results show that the calculated porosities for ternary mixtures are within 8% relative error of measurements. Good agreement between the theoretical and experimental results was also obtained for Gaussian and log-normal size distributions. It is concluded that the porosity of multi-component mixtures of particles may be confidently predicted from the results of binary mixtures. INTRODUCTION
The behaviour and packing efficiency, or its equivalent porosity, of particulate mixtures is important in all industries processing granular materials. In the past, many experimental studies have been done on the packing of mixtures, especially on the packing of binary mixtures of spherical particles [l - 61. It is a well-known empirical fact that the packing fraction varies with the size distribution of the materials involved. After an extensive experimental study of the porosity and permeability of multi-component mixtures of uniform and irregular shape particles, one of the authors concluded that it should be possible to predict the porosity of multicomponent mixtures from the results of binary mixtures [7 - lo]. Recently, a number of mathematical models have been proposed to estimate the porosity of particulate 0032-5910/87/$3.50
The University
mixtures from the knowledge of particle sizes involved and their proportion in the mixture [ll - 131. Unfortunately the porosity predided from these generally somewhat complex models often did not satisfactorily match the experimentally measured porosities. In fact, considering the models proposed by Ouchiyama et al. [12] and Stovall et al. [ 131, the porosity calculation for a mixture is still based on the analysis of the mutual effect of two sizes of particles, though the former analyse a so-called simplified packing model and the latter use interdiary analysis. Unavoidably, there are always some discrepancies between the theory and experiment, even for binary mixtures, which may result in large error for multi-component mixtures. It was considerations of the above kind and the requirement to improve model predictability that have recently prompted us to include in a mathematical model some measurements as prescribed information [ 141. We used the simplex-centroid design, which gave the calculated porosities within 2% relative error of measurements for ternary mixtures [ 141. Moreover, it was shown there that the method could be used, not only for mixtures of spherical particles, but for any actual mixtures, e.g. irregular-shaped, or nonuniform mixtures. However, the prescribed data required for the method increases sharply for each additional component in a mixture, which may limit its general applicability. The purpose of this paper is to present a new approach to the porosity prediction for multi-component mixtures, based on the results of binary mixtures alone, that does not have the same degree of limitation as the above model has. 0 Elsevier Sequoia/Printed in The Netherlands
234 THEORETICAL
TREATMENT
Relative void contraction of binary mixtures Much work has been done experimentally on the packing of binary mixtures. Figure 1 illustrates typical results obtained by McGear-y [ 31 and Jeschar et al. [ 151. Similar results can be found elsewhere [l, 2, 4 - 61. It is evident from Fig. 1 that as the fractional solid volume of large particles increases, the fractional void volume in packing will contract to a minimum and then expand and, that the maximum void contraction increases with the decrease of size-ratio (small/large). It has been found that the packing fraction of a random packing of spheres ranges between two well-defined limits, i.e. the loose and dense packing, having packing fractions of about 0.60 and 0.64 respectively [16, 171. In fact, different experimental conditions may yield different voidages. However, different initial voidages, which reflect experimental conditions to a great degree, will result in the change of the maximum void contraction and the corresponding fractional solid volume. Therefore, the maximum void contraction AE and the corresponding fractional solid volume, XLmaX , should be a function of size-ratio r and initial voidage E’.
When the size-ratio equals zero, the maximum contraction AC(O), and the corresponding fractional solid volume XLmax(o) are given by Ae(o) = e”( 1 - e”)
(1)
Xr,max(0) = -&
(2)
In order to make the existing literature data comparable, we introduce the relative maximum void contraction, which is defined as the ratio of the maximum void contraction of size-ratio r, Ae(r), to the maximum void contraction, Ahe( Similarly, we have the corresponding relative fractional solid volume, i.e., XLmaX(r)/XLmaX(o). It may be expected that Ae(r)/Ae(o) and XLmax(r)/ XLmaX(o) are the only functions of sizeratio which can be determined experimentally. It was Furnas who first proposed a quadratic equation to represent the correlation [ 11. Subsequently, Ridgway and Tarbuck suggested a modified quadratic regression
161: Ac At(o)
_
1-
2.35r + 1.35r2
i0
r < 0.741 r > 0.741
(3)
The results of these and other workers are summarised in Fig. 2. As shown in Fig. 2, if r < 0.35, Ayer et al. [ 51 also gave an equation between packing fraction and size-ratio the results of which, after appropriate recalculation, are almost the same as those calculated using eqn. (3). Comparing the results obtained under different experimental conditions [4, 22, 231 with those calculated using
0.
0
x 20 ill
1
z
2
-‘-‘------
0
Ridgway et al [61 Furnas c 11 Ayer et al [51 Epstein et al [II
z 2; -3
0.
c
a ” 1
0.2
0.4
0.6
0.8
Fractlcnal SolId Volume of Large hrtlcles
Fig. 1. Comparison between measurements [3, 151 and model predictions. Size-ratio: O---, 0.0129; se-, O.O524;x---, O.O89;A--- -, 0.25; *--.-, 0.5.
Fig. 2. Relative maximum void contraction ratio.
us. size-
235
solid volumes of small and large particles be Xs and XL, respectively. The packing fraction pLT is then PLT = PL + PLT&
(8)
or PLT=
- PL 1-Y .I.
0.
I 0
0.4
cl.2
0
0.6
U.8
I.”
r
Fig. 3. Relative fractional solid volume us. size-ratio.
eqn. (3) shows that, as expected, there is not much difference between them. The correlation between XLmax(r)/ XLmax(o) and r can be discussed similarly. As shown in Fig. 3, there is a good correlation between them which is given by XLm”“(r)/XLmax(o) = 1 - r2 Therefore, have
combining
eqns.
(4) (1) - (4), we
e”(1 - e?( 1 - 2.35r + 1.35r2) r < 0.741 r > 0.741
(5)
(6) or X
S
max =
l-
1 -r2 1 - E0
Binary rnk tures The theory of the packing of binary mixtures with very small size-ratio is well known [ 2,131. The packing of binary mixtures with infinitely small size-ratio will now be examined and the possibility of extending the theory to the packing of binary mixtures with larger size-ratio will be discussed later. Consider a packing of large particles having a packing fraction pL. If small particles are introduced into the packing, and noting that the size-ratio is small enough, we may obtain a mixture such that the original voids among the large particles are filled with the small particles. The volume of the packing will not change. Let the fractional
A.9
Similarly, when large particles are considered to be introduced into a packing of small particles having a packing fraction ps and if the volume, except that occupied by large solid particles, is fully packed with small particles, the volume of the packing will not change either. The packing fraction psT is PST = PSTXL + Ps(l _PSTXL)
(10)
or Ps PsT = 1 - (1 -ps)xL
(11)
Equations (9) and (ll), in terms of the specific volume V, defined as the apparent volume occupied by unit solid volume of particles, i.e., the reciprocal of packing fraction, become VLT = v, - v,xs
(12)
vsr = vs - (Vs - 1)XL
(13)
Equations (12) and (13) describe the limitation of binary mixtures as discussed by Westman and Hugill [2]. However, the most important conclusion from eqns. (12) and (13) is that the relationship between the specific volume and the fractional solid volume of binary mixtures with small size-ratio can be truly described by simple linear equations. To extend the theory to the packing of binary mixtures, we assume that the interaction between two sized particles only results in the change of maximum void contraction and the corresponding fractional solid volume, and the specific volume will still vary linearly with the fractional solid volume for any size-ratio which, as shown later, does not result in great error in the porosity calculations of multi-component mixtures. Based on the above assumptions, the development of analytical equations is not too complicated. As shown in Fig. 4, the specific volumes of mixtures given by lines i and j are given by eqns. (14) and (15), respectively.
j r;
236
“i
The maximum packing fraction pij and the corresponding fractional solid volume Xii or Xii GUI be calculated using eqns. (5) - (7). The packing fraction of a mixture is the lower of the two values of PiT and piT. It may be of interest to note that Lee [18] assumed that it was the packing fraction and not the specific volume that varies linearly with the fractional solid volume. However, Lee’s conclusion is not supported by analytical considerations, and his model did not consider the change of the fractional solid volume XnmaX and the effect of packing modes. His model prediction may therefore be expected to result in large errors.
"ij
/’
. .
,R--
-1.
-xj
Fig. 4. Specific volume for binary mixtures.
VT= vixi+
vi-
us. fractional
vi-b
xj
xij
yT=
Xi
solid volume
(14)
i
+
&Xj
Multi-component mixtures It is assumed that mixtures are composed of n components of equal-density spherical particles. Here we do not consider mixtures composed of irregular-shaped particles. Component i particles have diameters di and initial packing fraction pi. The diameters are ordered so that
(15)
where Vij is the minimum specific volume of components i and j, and X, or Xii is the corresponding fractional solid volume. Equations (14) and (15), in terms of the packing fraction, become Pi
PiT =
d,>d2>d3>...>d,
(16)
and the fractional solid volume Xi should satisfy the equation xi+xz+x3+...+xn=1
Pi
Pi
T=
(17)
As the extension of the above packing theory of binary mixtures, we assume that the specific volumes of multi-component mixtures vary linearly with the fraction solid volumes, and there exist no co-interactions among the components. Therefore, the calculation of the interaction between components i and j in multi-component mixtures is identical to that of binary mixtures. The packing fractions of a multi-component mixture are thus given by
or, in general form, Pi
PiT =
(18)
(19)
Pij = Pji
Pi
PiT =
(20)
(i = 1, 2, . . . . n)
(21)
or Pi
piT = 1-c
i-l j=l
1-z i
(i = 1, 2, . . . . n)
$IJ
1
i zj
j=i+l
1-F i
2 rj 1
j
(22)
237
where the maximum can be, as mentioned
packing fraction pii and the corresponding fractional above, calculated using eqns. (5) - (7). That is,
pi + Pi(1 -Pi)(
solid volume,
Xij
rij G 0.741 rij > 0.741
1 - 2.35rij + 1.35rij2)
Pij = Pi
(23)
i
1 - rij2
j
(24)
z--Pi
Xij =
l-
1 - rij2 2
I
jai
-Pi
4
(25)
i>j
;i; rij =
&j i
Min. pT(d) =
0.741d 1 _ VI-
where, according
0
ps(4
+p(d)[l
-p(d)]
pL(d, x) =p(d)
+p(d)[l
-p(d)]
x) = l-
1 - (x/d)2 2 -P(d)
X,(d,
x) =
Xl
1
Xs(4
1, 2, . . . . n) values. For a distribution. The packing
1_ p(d) 1
(26)
f(x)
p,(d.
Xl
x1
Xr,(d,
& x)
to eqns. (23) - (25), we have
ps(d, x) =p(d)
Xs(d,
p(d)
=
- 2.35 ;
(27)
(23)
(29)
1 - (d/x)2 2 -p(d)
+ 1.35
Comparing the calculated results with the experimental results in Fig. 1, it is obvious that there is good agreement between the
(30)
The porosities of multi-component mixtures of discrete size particles and mixtures with continuous size distributions can be predicted from eqns. (22) - (25) and (26) (30) respectively. .I RESULTS AND DISCUSSION 0.1
I
1
0.5
0.0
The calculated results of binary mixtures under the experimental conditions in [ 3,151 are shown in Fig. 1.
Fractional
Solid
1.0
Volume of Large Particles
Fig. 5. Calculated results for loose (- - ) and dense packings for binary mixtures. ( -)
238
theory and the experiment, especially for small size-ratios. The porosities of the socalled dense and loose packings are also given in Fig. 5, which in fact shows the effect of initial porosities on the porosity distribution of binary mixtures. There are several ternary mixture porosities measurements in the literature [ 2,6 - 10, 151. Here we will only compare the calcu-
lated results with the experimental results for spherical systems. These are shown in Figs. 5 - 8. To account for different experimental conditions used, the initial porosities reported in [6, 10, 151 were chosen as prescribed information. By inspection of Figs. 5 - 8 it is obvious that the calculated porosities are in good agreement with the experimentally measured
lb) Fig. 6. Ternary
(a)
mixture
porosities.
(a), Measurements
from
Volume % 2~117mm -
[6];
(b), model predictions.
(b)
Fig. 7. Ternary
mixture
porosities.
(a), Measurements
from
[lo];
(b), model
predictions.
Fig. 8. Ternary
mixture
porosities.
(a), Measurements
from
[15];
(b), model predictions.
239
results over the entire range of compositions. In fact, the calculated results in Figs. 5 - 8 are within 8% relative error of the measurements. This value of the relative error may be compared with those of 5 - 13%, 3 - 11% and 5 - 15% of the Cross et al. [ 191, Ouchiyama and Tanaka [12] and Leitzelement et al. [11] model predictions, respectively. It may be of interest to remark that the model proposed by Cross et al. is essentially an application of the Ouchiyama and Tanaka model to non-spherical particles. It may also be noteworthy that the porosities measured by Ridgway and Tarbuck [6] were regarded as “unusual”. These authors concluded that the ternary results could not have been predicted from the binary behaviour. In the light of elementary considerations, that ternaries consist of three binaries, quaternaries consist of four ternaries, and so on, then given the same mechanism, this conclusion is difficult to understand. In fact, the results of Ridgway and Tarbuck can be predicted well by the present model and the only “unusual” aspect is simply that the size-ratio of small to medium particles in their experiment is greater than 0.741 which, as shown in Fig. 2, results in the almost same behaviour of small and medium particles. Finally, as an example of the use of eqn. (25) for continuous size distributions, a comparison has been made with some data of Sohn and Moreland [20], who have measured the packing fractions of sands obeying Gaussian and log-normal size distributions by weight. However, if the density of sand is assumed to be constant and independent of particle size, then the size distribution by volume and that by weight are identical. Sohn and Moreland [20] have found that the packing fraction is only dependent on a dimensionless standard deviation and particle shape. They also found that the packing fractions increase continuously as the dimensionless standard deviations c/x for Gaussian and ug for log-normal increase except when dimensionless standard deviations are small. Such relationship, it should be remarked, is also predicted by eqn. (26). Calculations were performed on a computer according to eqn. (26), combining with Gaussian or log-normal size distribution and using the complex method of M. J. Box [21]
to solve this optimisation problem and find the minimum packing fraction. Good agreement between the calculated results and the measured results was obtained as can be seen in Figs. 9 and 10. One of the reasons for the apparent difference between the results in Figs. 9 and 10 arises from the difference of the particle shapes of sands used - something that was specifically also noted by Sohn and Moreland [ 201.
x’ Sand (4 0.5 OS 0.8
OS
’ ----
Fig. 9. Calculated results () and experimental data (- - -) for Gaussian size distribution.
o*:,I. Qi and experimental Fig. 10. Calculated results ( -) data ( - - -) for log-normal size distribution.
240
For the above conditions,
eqn. (22) can be written
as
Pi
PiT =
(31)
i-l
1 - (1 -Pi)
x
g”(i,
i>Xj
j=l
-
i j=i+l
f%
dxj
where 1
g”(i, j) = -
1-E
l--Pi
f”(i,j)
=
li
:_
(
U
E
1
(32) ij
$ iJ
1
iJ
As may be expected from analytical considerations, eqn. (31) here, and eqn. (22) of Stovall et ~2. [13] have the same form. However, as shown in Fig. 11, the values calculated using their model are always greater than those calculated using eqns. (32) and (33). Therefore, it is not surprising that their model predicts optimistically large values of the packing density even for binary mixtures and these errors will be compounded for multi-component mixtures. Obviously, as the linear analytical model presented here is based on the experimental results of binary mixtures, its predictability would be expected to be superior to those of the geometrical/ mathematical models thus far developed.
0.0
0.5
1.0
Size- ratio
Fig. 11. Comparison between the model of Stovall et al. [ 131 (- - -) and this model (-).
CONCLUSIONS
We have shown that the relative maximum void contraction and the corresponding relative fractional solid volume are only the
functions of size-ratio for binary mixtures. The linear analytical model has been developed to calculate the porosity of multi-component mixtures, using the initial porosities and the size-ratios as prescribed information. The results show that the calculated porosities for ternary mixtures are within 8% relative error of measurements. Good agreement between the theoretical and experimental results has also been obtained for Gaussian and log-normal size distributions. However, the model may be used only for spherical systems as it is directly based on the experimental results of spherical binary mixtures. Nevertheless, it is confirmed that the porosity of multi-component mixtures of particles may be confidently predicted from the results of binary mixtures.
REFERENCES 1 C. C. Furnas, Bur. Mines, Rept. Investigations 2874, 7 (1978); Bur. Mines. Bull., 307 (1929) 74. 2 A. E. R. Westman and H. R. Hugill, J. Am. Ceram. Sot., 13 (1930) 767. 3 R. K. McGeary, J. Am. Ceram. Sot., 44 (1961) 513. 4 N. Epstein and M. J. Young, Nature, 196 (1962) 885. 5 J. E. Ayer and F. E. Soppet, J. Am. Ceram. Sot., 48 (1965) 180. 6 K. Ridgway and K. J. Tarbuck, Chem. Process. Eng., 49 (1968) 103. 7 N. Standish and D. E. Borger, Powder Technol., 22 (1979) 121. 8 N. Standish and D. G. Mellor, Powder Technol., 27 (1980) 61. 9 N. Standish and P. J. Leyshon, Powder Technol., 30 (1981) 119. 10 N. Standish and D. N. Collins, Powder Technol., 36 (1983) 55. 11 M. Leitzelement, Cho Seng Lo and J. A. Dodds, Powder Technol., 41 (1985) 159. 12 N. Ouchiyama and T. Tanaka, Z & EC Fundam., 23 (1984) 490. 13 T. Stovall, F. De Larrard and M. Buil, Powder Technol., 48 (1986) 1. 14 N. Standish and A. B. Yu, Powder Technol., 49 (1987) 248. 15 R. Jeschar, W. Potke, V. Petersen and K. Polthier,
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16 17 18 19
in N. Standish (ed.), Blast Furnace Aerodynamics, Aust. I.M.M. Press, Wollongong, N.S.W., 1975, p. 136. C. D. Scott, Nature, 188 (1960) 908. R. Rutgers, Nature, 193 (1962) 465. D. I. Lee, J. Paint Technol., 42 (1970) 579. M. Cross, W. H. Douglas and R. P. Fields, Powder Technol., 43 (1985) 27.
20 H. Y. Sohn and C. Moreland, Can. J. Chem. Eng., 46 (1968) 162. 21 J. L. Kurster and J. H. Mize, Optimization Techniques with FORTRAN, McGraw-Hill, New York, 1973. 22 C. Lemaignan, Acta Met., 28 (1980) 1657. 23 M. Iannella, B. Met. Thesis, Univ. of Wollongong (1985).