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The pivot phenomenon and difference-similarity of classifier particle distributions
Powder Technology 168 (2006) 152 – 155 www.elsevier.com/locate/powtec
Short communication
The pivot phenomenon and difference-similarity of classifier particle distributions B. Venkoba Rao Tata Research Development and Design Centre (TRDDC), 54 B, Hadapsar Industrial Estate, Pune 411 013, India Received 10 June 2005; received in revised form 15 February 2006; accepted 25 July 2006 Available online 1 September 2006
Abstract Truncated analytical expressions for product size distributions of mechanical classifiers have been proposed recently by considering classifier feed distribution in terms of Gates–Gaudin–Schumann (GGS) function and classifier efficiency in terms of Plitt function. In this work it is shown from theoretical considerations that the distributions when expressed in density form pivot at a common size referred to as pivot-size. Under the specified operating conditions of the unit, the physical interpretation of this pivot phenomenon along with the mass balance considerations reveal that the size distributions are difference-similar and collapse onto a single curve when the mathematical difference between any two distributions is scaled with corresponding maximum value of the differences and thus make them invariant of shape effects of feed and product distributions. The procedure is explained in simple terms using an illustrative example. © 2006 Elsevier B.V. All rights reserved. Keywords: Sizing; Classification; Difference-similarity; Pivot-phenomenon
1. Introduction Truncated analytical expressions for classifier size distributions have been proposed recently [1] in terms of the parameters of Gates–Gaudin–Schumann (GGS) function [2] and Plitt function [3] over the particle size range [0,dmax] using a practical transformation of the feed distribution. The distributions are expressed explicitly in continuous size form. The mathematical representations of the classifier feed, overflow and underflow distributions expressed in cumulative percent finer form are respectively represented as 8 d n < 100 for 0Vd Vdmax FðdÞ ¼ ð1Þ dmax : 100 for d Ndmax n g ;K 100 nm OðdÞ ¼ g ; K > max > m : 100 8 > > <
8 > > > <
n n mK ðmÞ −nð1−Rf Þg ; K m 100 n U ðdÞ ¼ ðmn Þ mK ; K −nð1−R Þg > max f max > > m : 100
for d Ndmax
E-mail address: [email protected]. 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.07.015
ð2Þ
for d Ndmax ð3Þ
where the values of K and Kmax are given by K ¼ lnð2Þ
d
m
d50c
Kmax ¼ lnð2Þ for 0VdVdmax
for 0VdVdmax
dmax m d50c
ð4Þ
The meaning of the symbols is explained in the nomenclature section. An expression for the yield of solid particles that report to the coarse stream i.e., underflow has been calculated by integrating all the feed particles weighted with respect to their
B.V. Rao / Powder Technology 168 (2006) 152–155
8 m d > > − > > < mlnð2Þðn=mÞ 2 d50c d n−1 n oðdÞ ¼ > n > d50c g ; Kmax > > m : 0
uðdÞ ¼
153
for 0Vd Vdmax
ð8Þ
for d Ndmax
m 1 8 0 d > > − > > C ðn=mÞ n−1 B > d @1−ð1−Rf Þ2 d50c A > < mnlnð2Þ > > > > > > :
0
n n=m n mKmax −nð1−Rf Þg ; Kmax d50c m
for 0Vd Vdmax for d Ndmax
ð9Þ Because the feed particles get separated in the classifier, the feed distribution intersects the overflow and underflow distributions when expressed in density form. The size at which the feed distribution intersects the overflow and the underflow distributions can be obtained by equating Eq. (7) with Eqs. (8) and (9) respectively and solving for the intersecting particle size. Equating Eqs. (7) and (8) and solving for the intersecting size d1 yields !!m1 n d50c n gðn=m; Kmax Þ d1 ¼ d50c −1:4427ln ð10Þ n m dmax lnð2ÞðmÞ
Fig. 1. Classifier feed, overflow and underflow distributions expressed in cumulative finer as well as density forms for the specific set of parameter values: n = 0.85; dmax = 450 μm; d50c = 160 μm; m = 1.5; Rf = 0.15 (Note: d* = 176.16 μm).
corresponding actual efficiency probabilities of report to coarse stream, and is given by Ssplit ¼ 1−
n
ð1−Rf Þ
m
ðlnð2ÞÞðn=mÞ
!
d50c dmax
n n g ; Kmax m
ð5Þ
Clearly, the feed and product distributions are related to each other by the following relation. FðdÞ ¼ Ssplit U ðdÞ þ ð1−Ssplit ÞOðdÞ
ð6Þ
Similarly equating Eqs. (7) and (9) yields the intersecting size d2 as d2 ¼ d50c
!!!m1 n=m 1 d50c n ðmKmax −nð1−Rf Þgðn=m; Kmax ÞÞ 1− −1:4427ln n ð1−Rf Þ dmax mlnð2Þm
ð11Þ Substituting the value of Kmax from Eq. (4) into Eq. (11) and simplification gives 0 11m1 0 n B B n d50c gðn=m; Kmax ÞCC CC B d2 ¼ d50c B n AA ¼ d1 @−1:4427ln@m dmax lnð2Þm ð12Þ Eqs. (10) and (12) show that both overflow and underflow density distributions intersect the feed density distribution at the
In this paper we extend the theoretical implications of the classifier distributions and explore a generic commonality that these distributions possess. 2. Present investigation Cumulative finer distributions of feed, overflow and underflow when expressed in density form, by differentiating the Eqs. (1), (2) and (3) with respect to particle size yield 8 < n d n−1 f ðdÞ ¼ dmax : dmax 0
for 0Vd Vdmax for d Ndmax
ð7Þ Fig. 2. Absolute density difference distributions for the parameters in Fig. 1.
154
B.V. Rao / Powder Technology 168 (2006) 152–155
butions of the classifier are proportional to one another over the entire particle size range [0,dmax] as given below. Abs½U ðdÞ−OðdÞ / Abs½FðdÞ−OðdÞ / Abs½U ðdÞ−FðdÞ
ð15Þ
Fig. 3. Absolute cumulative finer difference distributions for the parameters in Fig. 1.
same size referred hereafter as pivot size, d*. Fig. 1 shows feed, overflow and underflow distributions in cumulative finer as well as density forms with parameter values defined therein. The measured particle size distributions in density form for a hydrocyclone separator by Kraipech et al. [4] depict the pivot phenomenon in the cases that lack agglomeration during classification of particles. The measurements were made using a laser diffraction technique. 3. Discussions When the absolute differences between the density forms of feed and product distributions are considered, they indicate a minimum value of zero at the pivot size as shown in Fig. 2. This implies the existence of maximum value for the absolute cumulative difference distributions at the pivot size. Further, the maximum values of cumulative difference distributions at the pivot size imply individual cumulative distributions being separated by a maximum percentile at the pivot size. Fig. 3 shows the absolute difference of the cumulative finer distributions as a function of particle size for the parameters specified in Fig. 1. The existence of difference similarity of the classifier distributions over the entire particle size range can be shown by the following mass balance condition. Rearrangement of Eq. (6) gives Abs½FðdÞ−OðdÞ ¼ Ssplit Abs½U ðdÞ−OðdÞ
Thus the absolute difference distributions when scaled with corresponding reciprocals of their maximum distribution values collapse them on to a single curve, which indicates the difference-similarity among classifier distributions. Fig. 4 expresses the difference-similarity for the data presented in Fig. 1. Table 1 presents a simpler method of portraying difference-similarity of cumulative finer distributions for the published data of Lynch et al. [7] in the absence of any functional fit. As the proportionality between any two classifier distributions (refer Eqs. (13) and (14)) is independent of the functional forms of feed and product streams, the difference similarity is a generic phenomenon in separators/classifiers. Moreover, the yield of the product streams and therefore Eqs. (13) and (14) should be independent of the particle attribute under study. From a reverse analogy, a maxima on difference-similar cumulative distributions indicates presence of pivot phenomenon for any particle attribute that is under study. Thus under specified operating conditions of the separator a one-input two-output stream separator always shows coexistence of pivot point phenomenon and difference similarity among its distributions regardless of the particle attribute in terms of which the distributions are measured, say particle size or particle density or any other attribute. 4. Conclusions It has been shown that the classifier feed and product distributions when expressed in density form pivot at a common size called the pivot size. The pivot size corresponds to the size at which the cumulative distributions are separated by a maximum percentile. It follows from the pivot phenomenon and mass balance equations that the classifier distributions are difference-similar and can be collapsed on to a single curve and thus making them invariant of the shape effects of feed and product distributions. The coexistence of pivot phenomenon and difference similarity of separator distributions is ubiquitous
ð13Þ
Similarly from Eq. (6) it can be shown that Abs½U ðdÞ−FðdÞ ¼ 1−Ssplit Abs½U ðdÞ−OðdÞ
ð14Þ
In fact, Eq. (13) or (14) are applied to reconcile error prone classifier measurements [5,6]. Since Ssplit, which represents the flow split of feed particles to the underflow is a constant fraction regardless of the particle size considered, Eqs. (13) and (14) suggest that the absolute differences of cumulative finer distri-
Fig. 4. Cumulative finer difference-similarity curve for the parameters in Fig. 1.
B.V. Rao / Powder Technology 168 (2006) 152–155
155
Table 1 Difference-similarity of cumulative finer distributions for the Lynch et al. [7] data Passing size, μm
Cumulative percent finer distributions
Absolute cumulative finer difference distributions
F(d )
Difference-similarity distributions
U(d )
O(d )
Abs[O(d ) − U(d )]
Abs[O(d ) − F(d )]
Abs[F(d ) − U(d )]
Abs[O(d ) − U(d )]/m1
Abs[O(d ) − F(d )]/m2
Abs[F(d ) − U(d )]/m3
1200 99.9 99.7 850 98.4 96.1 600 90.6 75.4 420 82.0 54.1 300 75.9 41.0 210 70.3 34.0 150 65.3 30.3 75 54.6 24.2 53 51.2 22.5 Maximum values
100.0 100.0 99.9 99.3 97.1 92.4 86.5 72.8 68.1
0.3 3.9 24.5 45.2 56.1 58.4 56.2 48.6 45.6 58.4 m1
0.1 1.6 9.3 17.3 21.2 22.1 21.2 18.2 16.9 22.1 m2
0.2 2.3 15.2 27.9 34.9 36.3 35.0 30.4 28.7 36.3 m3
0.0051 0.0668 0.4195 0.7740 0.9606 1.0000 0.9623 0.8322 0.7808
0.0045 0.0724 0.4208 0.7828 0.9593 1.0000 0.9593 0.8235 0.7647
0.0055 0.0634 0.4187 0.7686 0.9614 1.0000 0.9642 0.8375 0.7906
Ssplit = m2 / m1.
regardless of particle attribute in terms of which distribution measurements are made. Nomenclature Abs Absolute value of a mathematical function d Passing particle size d1, d2 Intersecting particle sizes of feed density distribution with overflow and underflow density distributions respectively d50c Cut size of the Plitt efficiency curve d* Pivot particle size dmax Size modulus of GGS function f(d ) Feed distribution in density form F(d ) Feed distribution in cumulative percent passing form K A function of particle size defined by Eq. (8) Kmax A constant defined by Eq. (8) m Sharpness index of the Plitt efficiency curve n Distribution modulus of GGS function o(d ) Fine stream size distribution in density form O(d ) Fine stream size distribution in cumulative percent passing form Rf By-pass fraction of the Plitt efficiency curve Ssplit Solid flow-split to coarse stream u(d ) Coarse stream size distribution in density form U(d ) Coarse stream size distribution in cumulative percent passing form Greek symbols γ Gamma function
Acknowledgement The author acknowledges Prof. Mathai Joseph, Executive Director, TRDDC for his encouragement and for providing management support. The author is grateful to one of the reviewers for the helpful criticism of this investigation. References [1] B. Venkoba Rao, Analytical expressions for classifier product size distributions, Minerals Engineering 18 (2005) 557–560. [2] A.M. Gaudin, Principles of Mineral Dressing, McGraw-Hill, New York, 1939. [3] L.R. Plitt, The analysis of solid–solid separations in classifier, CIM Bulletin 64 (708) (1971) 42–47. [4] W. Kraipech, W. Chen, F.J. Parma, T. Dyakowski, Modelling the fish-hook effect of the flow within hydrocyclones, International Journal of Mineral Processing 66 (2002) 49–65. [5] K. Heiskanen, Particle Classification, Chapman and Hall, London, 1993. [6] B.A. Wills, Mineral Processing Technology, Butterworth-Heimemann, Oxford, 1997. [7] A.J. Lynch, T.C. Rao, K.A. Prisbrey, The influence of hydrocyclone diameter on reduced efficiency curves, International Journal of Mineral Processing 1 (2) (1974) 173–181.
Short communication
The pivot phenomenon and difference-similarity of classifier particle distributions B. Venkoba Rao Tata Research Development and Design Centre (TRDDC), 54 B, Hadapsar Industrial Estate, Pune 411 013, India Received 10 June 2005; received in revised form 15 February 2006; accepted 25 July 2006 Available online 1 September 2006
Abstract Truncated analytical expressions for product size distributions of mechanical classifiers have been proposed recently by considering classifier feed distribution in terms of Gates–Gaudin–Schumann (GGS) function and classifier efficiency in terms of Plitt function. In this work it is shown from theoretical considerations that the distributions when expressed in density form pivot at a common size referred to as pivot-size. Under the specified operating conditions of the unit, the physical interpretation of this pivot phenomenon along with the mass balance considerations reveal that the size distributions are difference-similar and collapse onto a single curve when the mathematical difference between any two distributions is scaled with corresponding maximum value of the differences and thus make them invariant of shape effects of feed and product distributions. The procedure is explained in simple terms using an illustrative example. © 2006 Elsevier B.V. All rights reserved. Keywords: Sizing; Classification; Difference-similarity; Pivot-phenomenon
1. Introduction Truncated analytical expressions for classifier size distributions have been proposed recently [1] in terms of the parameters of Gates–Gaudin–Schumann (GGS) function [2] and Plitt function [3] over the particle size range [0,dmax] using a practical transformation of the feed distribution. The distributions are expressed explicitly in continuous size form. The mathematical representations of the classifier feed, overflow and underflow distributions expressed in cumulative percent finer form are respectively represented as 8 d n < 100 for 0Vd Vdmax FðdÞ ¼ ð1Þ dmax : 100 for d Ndmax n g ;K 100 nm OðdÞ ¼ g ; K > max > m : 100 8 > > <
8 > > > <
n n mK ðmÞ −nð1−Rf Þg ; K m 100 n U ðdÞ ¼ ðmn Þ mK ; K −nð1−R Þg > max f max > > m : 100
for d Ndmax
E-mail address: [email protected]. 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.07.015
ð2Þ
for d Ndmax ð3Þ
where the values of K and Kmax are given by K ¼ lnð2Þ
d
m
d50c
Kmax ¼ lnð2Þ for 0VdVdmax
for 0VdVdmax
dmax m d50c
ð4Þ
The meaning of the symbols is explained in the nomenclature section. An expression for the yield of solid particles that report to the coarse stream i.e., underflow has been calculated by integrating all the feed particles weighted with respect to their
B.V. Rao / Powder Technology 168 (2006) 152–155
8 m d > > − > > < mlnð2Þðn=mÞ 2 d50c d n−1 n oðdÞ ¼ > n > d50c g ; Kmax > > m : 0
uðdÞ ¼
153
for 0Vd Vdmax
ð8Þ
for d Ndmax
m 1 8 0 d > > − > > C ðn=mÞ n−1 B > d @1−ð1−Rf Þ2 d50c A > < mnlnð2Þ > > > > > > :
0
n n=m n mKmax −nð1−Rf Þg ; Kmax d50c m
for 0Vd Vdmax for d Ndmax
ð9Þ Because the feed particles get separated in the classifier, the feed distribution intersects the overflow and underflow distributions when expressed in density form. The size at which the feed distribution intersects the overflow and the underflow distributions can be obtained by equating Eq. (7) with Eqs. (8) and (9) respectively and solving for the intersecting particle size. Equating Eqs. (7) and (8) and solving for the intersecting size d1 yields !!m1 n d50c n gðn=m; Kmax Þ d1 ¼ d50c −1:4427ln ð10Þ n m dmax lnð2ÞðmÞ
Fig. 1. Classifier feed, overflow and underflow distributions expressed in cumulative finer as well as density forms for the specific set of parameter values: n = 0.85; dmax = 450 μm; d50c = 160 μm; m = 1.5; Rf = 0.15 (Note: d* = 176.16 μm).
corresponding actual efficiency probabilities of report to coarse stream, and is given by Ssplit ¼ 1−
n
ð1−Rf Þ
m
ðlnð2ÞÞðn=mÞ
!
d50c dmax
n n g ; Kmax m
ð5Þ
Clearly, the feed and product distributions are related to each other by the following relation. FðdÞ ¼ Ssplit U ðdÞ þ ð1−Ssplit ÞOðdÞ
ð6Þ
Similarly equating Eqs. (7) and (9) yields the intersecting size d2 as d2 ¼ d50c
!!!m1 n=m 1 d50c n ðmKmax −nð1−Rf Þgðn=m; Kmax ÞÞ 1− −1:4427ln n ð1−Rf Þ dmax mlnð2Þm
ð11Þ Substituting the value of Kmax from Eq. (4) into Eq. (11) and simplification gives 0 11m1 0 n B B n d50c gðn=m; Kmax ÞCC CC B d2 ¼ d50c B n AA ¼ d1 @−1:4427ln@m dmax lnð2Þm ð12Þ Eqs. (10) and (12) show that both overflow and underflow density distributions intersect the feed density distribution at the
In this paper we extend the theoretical implications of the classifier distributions and explore a generic commonality that these distributions possess. 2. Present investigation Cumulative finer distributions of feed, overflow and underflow when expressed in density form, by differentiating the Eqs. (1), (2) and (3) with respect to particle size yield 8 < n d n−1 f ðdÞ ¼ dmax : dmax 0
for 0Vd Vdmax for d Ndmax
ð7Þ Fig. 2. Absolute density difference distributions for the parameters in Fig. 1.
154
B.V. Rao / Powder Technology 168 (2006) 152–155
butions of the classifier are proportional to one another over the entire particle size range [0,dmax] as given below. Abs½U ðdÞ−OðdÞ / Abs½FðdÞ−OðdÞ / Abs½U ðdÞ−FðdÞ
ð15Þ
Fig. 3. Absolute cumulative finer difference distributions for the parameters in Fig. 1.
same size referred hereafter as pivot size, d*. Fig. 1 shows feed, overflow and underflow distributions in cumulative finer as well as density forms with parameter values defined therein. The measured particle size distributions in density form for a hydrocyclone separator by Kraipech et al. [4] depict the pivot phenomenon in the cases that lack agglomeration during classification of particles. The measurements were made using a laser diffraction technique. 3. Discussions When the absolute differences between the density forms of feed and product distributions are considered, they indicate a minimum value of zero at the pivot size as shown in Fig. 2. This implies the existence of maximum value for the absolute cumulative difference distributions at the pivot size. Further, the maximum values of cumulative difference distributions at the pivot size imply individual cumulative distributions being separated by a maximum percentile at the pivot size. Fig. 3 shows the absolute difference of the cumulative finer distributions as a function of particle size for the parameters specified in Fig. 1. The existence of difference similarity of the classifier distributions over the entire particle size range can be shown by the following mass balance condition. Rearrangement of Eq. (6) gives Abs½FðdÞ−OðdÞ ¼ Ssplit Abs½U ðdÞ−OðdÞ
Thus the absolute difference distributions when scaled with corresponding reciprocals of their maximum distribution values collapse them on to a single curve, which indicates the difference-similarity among classifier distributions. Fig. 4 expresses the difference-similarity for the data presented in Fig. 1. Table 1 presents a simpler method of portraying difference-similarity of cumulative finer distributions for the published data of Lynch et al. [7] in the absence of any functional fit. As the proportionality between any two classifier distributions (refer Eqs. (13) and (14)) is independent of the functional forms of feed and product streams, the difference similarity is a generic phenomenon in separators/classifiers. Moreover, the yield of the product streams and therefore Eqs. (13) and (14) should be independent of the particle attribute under study. From a reverse analogy, a maxima on difference-similar cumulative distributions indicates presence of pivot phenomenon for any particle attribute that is under study. Thus under specified operating conditions of the separator a one-input two-output stream separator always shows coexistence of pivot point phenomenon and difference similarity among its distributions regardless of the particle attribute in terms of which the distributions are measured, say particle size or particle density or any other attribute. 4. Conclusions It has been shown that the classifier feed and product distributions when expressed in density form pivot at a common size called the pivot size. The pivot size corresponds to the size at which the cumulative distributions are separated by a maximum percentile. It follows from the pivot phenomenon and mass balance equations that the classifier distributions are difference-similar and can be collapsed on to a single curve and thus making them invariant of the shape effects of feed and product distributions. The coexistence of pivot phenomenon and difference similarity of separator distributions is ubiquitous
ð13Þ
Similarly from Eq. (6) it can be shown that Abs½U ðdÞ−FðdÞ ¼ 1−Ssplit Abs½U ðdÞ−OðdÞ
ð14Þ
In fact, Eq. (13) or (14) are applied to reconcile error prone classifier measurements [5,6]. Since Ssplit, which represents the flow split of feed particles to the underflow is a constant fraction regardless of the particle size considered, Eqs. (13) and (14) suggest that the absolute differences of cumulative finer distri-
Fig. 4. Cumulative finer difference-similarity curve for the parameters in Fig. 1.
B.V. Rao / Powder Technology 168 (2006) 152–155
155
Table 1 Difference-similarity of cumulative finer distributions for the Lynch et al. [7] data Passing size, μm
Cumulative percent finer distributions
Absolute cumulative finer difference distributions
F(d )
Difference-similarity distributions
U(d )
O(d )
Abs[O(d ) − U(d )]
Abs[O(d ) − F(d )]
Abs[F(d ) − U(d )]
Abs[O(d ) − U(d )]/m1
Abs[O(d ) − F(d )]/m2
Abs[F(d ) − U(d )]/m3
1200 99.9 99.7 850 98.4 96.1 600 90.6 75.4 420 82.0 54.1 300 75.9 41.0 210 70.3 34.0 150 65.3 30.3 75 54.6 24.2 53 51.2 22.5 Maximum values
100.0 100.0 99.9 99.3 97.1 92.4 86.5 72.8 68.1
0.3 3.9 24.5 45.2 56.1 58.4 56.2 48.6 45.6 58.4 m1
0.1 1.6 9.3 17.3 21.2 22.1 21.2 18.2 16.9 22.1 m2
0.2 2.3 15.2 27.9 34.9 36.3 35.0 30.4 28.7 36.3 m3
0.0051 0.0668 0.4195 0.7740 0.9606 1.0000 0.9623 0.8322 0.7808
0.0045 0.0724 0.4208 0.7828 0.9593 1.0000 0.9593 0.8235 0.7647
0.0055 0.0634 0.4187 0.7686 0.9614 1.0000 0.9642 0.8375 0.7906
Ssplit = m2 / m1.
regardless of particle attribute in terms of which distribution measurements are made. Nomenclature Abs Absolute value of a mathematical function d Passing particle size d1, d2 Intersecting particle sizes of feed density distribution with overflow and underflow density distributions respectively d50c Cut size of the Plitt efficiency curve d* Pivot particle size dmax Size modulus of GGS function f(d ) Feed distribution in density form F(d ) Feed distribution in cumulative percent passing form K A function of particle size defined by Eq. (8) Kmax A constant defined by Eq. (8) m Sharpness index of the Plitt efficiency curve n Distribution modulus of GGS function o(d ) Fine stream size distribution in density form O(d ) Fine stream size distribution in cumulative percent passing form Rf By-pass fraction of the Plitt efficiency curve Ssplit Solid flow-split to coarse stream u(d ) Coarse stream size distribution in density form U(d ) Coarse stream size distribution in cumulative percent passing form Greek symbols γ Gamma function
Acknowledgement The author acknowledges Prof. Mathai Joseph, Executive Director, TRDDC for his encouragement and for providing management support. The author is grateful to one of the reviewers for the helpful criticism of this investigation. References [1] B. Venkoba Rao, Analytical expressions for classifier product size distributions, Minerals Engineering 18 (2005) 557–560. [2] A.M. Gaudin, Principles of Mineral Dressing, McGraw-Hill, New York, 1939. [3] L.R. Plitt, The analysis of solid–solid separations in classifier, CIM Bulletin 64 (708) (1971) 42–47. [4] W. Kraipech, W. Chen, F.J. Parma, T. Dyakowski, Modelling the fish-hook effect of the flow within hydrocyclones, International Journal of Mineral Processing 66 (2002) 49–65. [5] K. Heiskanen, Particle Classification, Chapman and Hall, London, 1993. [6] B.A. Wills, Mineral Processing Technology, Butterworth-Heimemann, Oxford, 1997. [7] A.J. Lynch, T.C. Rao, K.A. Prisbrey, The influence of hydrocyclone diameter on reduced efficiency curves, International Journal of Mineral Processing 1 (2) (1974) 173–181.