Advances in Mineral Resources Management and 1 Environmental Geotechnology, Hania 2004, Greece

K.G. Tsakalakis School of Mining and Metallurgical Engineering, National Technical University of Athens, Greece ABSTRACT This work presents a new two-parameter empirical model, which can be used for the representation of the hydrocyclone corrected-efficiency curves. With the help of this model it is possible to represent the distribution of the various size fractions, in the overflow and underflow products during classification processes (corrected and actual efficiency). Experimental actual efficiencies for various size fractions from earlier classification tests were correlated with the new equation.

The interesting features of the new model are: (a) it is a simple physical model (b) the accuracy of its parameters, computed applying linear regression is comparable to that obtained from the already known (Lynch, Plitt-Reid and Harris) models, and (c) the proposed model can also be graphically represented, from which all of its parameters are easily predicted. Comparisons with the other models showed a good agreement, not only for the prediction of the hydrocyclone corrected and actual efficiencies, but for the characteristic separation sizes of the classification procedure as well.

1. INTRODUCTION For quite some time now, the use of mathematical models, depicting the performance of mineral processing operations, has become important.

Significant attempts have been made in the field of wet grinding to express mathematically the effectiveness of the classifier. The proposed models can be used later for the simulation of the grinding circuits.

A wet closed grinding circuit, in its simplest

form, consists of two units, the mill and the mill- product classifier (hydrocyclone or spiral classifier). The classifier controls through the maximum size of the product the recirculating load to the mill.

It is evident that the use of reliable models, expressing the classifier efficiency, is essential for the effective control and the simulation of the wet closed grinding circuits.

The classifier, fed with the mill product, divides it into two products; the coarse fraction, which represents the recirculating load fed back to the mill, and the fine product with particles of the appropriate size for the next stages of the circuit.

Due to short-circuiting of fines caused by the coarse product liquid, the actual classifier efficiency, instead of being Ec, becomes Ea. The relationship between the observed classification efficiency Ea and the corresponding Ec (corrected) is shown in Figure 1 and is given by:

1a f

cf

E RE

R−

=

−

(1)

where: Ec: is the mass fraction of particles of a

given size referred to the coarse product (underflow) of the classifier, due to the classifying action

Ea: is the mass fraction of the given size (same as above), which actually appears in the underflow and

Rf : is the fraction of the feed liquid, which is distributed in the underflow.

The actual separation size (50% of the size-fraction in the underflow and 50% in the overflow) is referred to as d50 (cut size or cut

Use of a new model to represent hydrocyclone corrected-efficiency curves

2 Advances in Mineral Resources Management and Environmental Geotechnology, Hania 2004, Greece

point) and the corresponding corrected one as d50c. From the relative position of the two curves in Fig. 1, it comes out that d50c is greater than d50. Their difference obviously depends on the value of Rf (Eq. 1).

Many equations, expressing the corrected classifier efficiency Ec as a function of the particle size d, were extensively presented in an earlier paper (Luckie and Austin, 1974).

From those the most widely used are: [ ]

[ ]

50

50

( / )

( / )

12

c

c

a d d

c a d d a

eEe e

−=

+ −

(2)

and:

( )500.6931 /1

mcd d

cE e −

= − (3)

The first model (Eq. 2) was proposed by Lynch (1965) and the second (Eq. 3), which followed a few years later, simultaneously but independently by Plitt (1971) and Reid (1971). The above equations have two parameters. The d50c gives the particle size in which the separation was expected to happen and the other (a or m) characterizes the sharpness of separation.

Another model has been also proposed (Harris, 1972), which is a three-parameter equation:

[ ]1 1 ( / )maxrm

E d dc = − − (4) where m, r are shape parameters and dmax the particle size at which Ec = 1.

Plitt's or Reid's model offers a number of considerable advantages: first, it can be represented as a straight line on a Rosin-Rammler graph paper from which the d50c and m values can be easily derived and secondly it is based on a physical model.

Equation (4), due to its greater number of parameters, presents a better goodness of fit to the data. It also provides corrected efficiency Ec = 1 at a finite size dmax, but the calculation of its parameters is much more tedious.

A “fish hook” phenomenon at the fine end of the hydroclone performance curve, with sufficient frequency to warrant attention, was reported from Finch (1983). This behavior of the hydroclone performance curve was modeled by introducing an entrainment component to recovery to the underflow which is a function of particle size. Roldan-Villasana et al. (1993) realized also this phenomenon, which is a dip of the “S” shape function in regions of the finer particle sizes (typically less than 20 µm) and it was thought to be of considerable industrial significance. They made a critical assessment of previous modeling work and a theory to account for the phenomenon and a new empirical model were proposed and tested.

Nageswararao (1999) examined the Plitt-Reid and Lynch (simple and that modified by Whiten) models and concluded that, both the Plitt-Reid and Whiten functions are satisfactory in describing most efficiency curves observed in practice. The latter seemed to be more accurate while the former is simpler to use.

Chen et al. (2000), using experimental data, compared 7 different (empirical, theoretical and semi-theoretical) models and they found that there has not been so far a single model, which can simulate most hydrocyclone operations. They stated that, it is impossible to have one model performing a good simulation of the pressure drop, cut size, grade efficiency and the flow split. However, these models can be realized as a good tool for estimating hydrocyclone performance when applied in the right domain.

Pasquier and Cilliers (2000) observed that

1.0

Cut size- d50

Sharpness of separation

Particle size0

0.5

d50C

Bypass

Rf

Frac

tion

to u

nder

flow

EC

Ea

Figure 1: Typical actual and corrected classification curves.

Advances in Mineral Resources Management and 3 Environmental Geotechnology, Hania 2004, Greece

small diameter (10 mm) hydrocyclones, used in sub-sieve classification, exhibit a fish-hook partition curve and a high bypass fraction, which results in high particle recovery to the underflow and a high concentration ratio. The fish-hook partition curve observed can be accurately described by a general classification model proposed.

"Nageswararao (2000) claims that the infrequently reported fish-hook effect, in hydrocyclone classifications, seems to be of partial acceptance. The fundamental models to date do not predict a "fish-hook" phenomenon. Current theories to explain it, based on the size dependent bypass mechanism, are mere mathematical transformations and it is still a long way before it could be universally accepted as a scientifically significant physical effect."

Kraipech et al. (2002), based on experimental data, attempted to model the fish-hook phenomenon by proposing two different approaches (the classification approach A and the bypass Approach B). For each of these approaches a different model was derived.

2. DERIVATION OF THE NEW MODEL It is known (Daniel and Wood, 1980) that the following model

blE ae

−

= (5)

where α, b are parameters > 0, has a plot shown in Fig. 2 (curve for n=1).

Equation (5) was extensively tested before (Tsakalakis, 1988) in order to be used as a screen size-separation model. Finally, the following model, incorporating one more parameter n, was proved to have a better fitting capability than Eq. (5) to screening data and was used as a screening function, i.e.

nblE ae

−

= (6)

where E was the cumulative undersize recovery of the screen, l was the screen length in cm and α, b parameters > 0.

It was noted that, when the value of n varies between 0.5 and 2.0 (0.5 < n < 2.0), the form of the curves changes slightly as is shown in Fig. 2 (curves for n=0.7 or 2.0). This fact, in some cases, leads to better goodness of fit to the data. It is evident that the curves in Fig. 2 resemble in shape the curve Ec shown in Fig. 1, which represents the corrected efficiency of a classifier. By reason of this observation, Eq. (6) was suitably recast and then subjected to mathematical analysis in order to verify its fitting capability when used as a classifier corrected-efficiency function.

From Eq. (6), substituting (e/2 = 1.359) for a, (d50c)n for b and d for l , we get

( )50 /1.359n

cd dcE e−

= (7)

where: Ec: assigned as the fraction of the corrected

efficiency d: is the particle size in µm (geometric

mean) d50c: as assigned before and n: is a power-law parameter. Equation (7) is a two-parameter (d50c, n) function and it presents the basic characteristics of a classifier corrected-efficiency function (e.g., when d = d50c, then Ec = 1.359 x 0.3679 = 0.5 or 50%). Applying simple linear regression to this equation, the two parameters can be easily computed.

3. CALCULATION OF THE PARAMETERS N, D50C AND DMAX Table 1 presents the experimental data referred in the classification test (Finch and Plitt, 1975)

Cum

ulat

ive

unde

rsiz

e re

cove

ry E

, %

n = 1

n = 2

n = 0.7

0

a

100

50

Screen length l, cm

Figure 2: Graphical representation of the cumulative undersize recovery as a function of the screen length.

4 Advances in Mineral Resources Management and Environmental Geotechnology, Hania 2004, Greece

and the hydrocyclone corrected-efficiency values Ec calculated from Eq. (1). For the test under consideration all the parameters were predicted, not only those of the proposed model, but also those for the already known models as well.

For the Lynch and Harris models, the various parameters were predicted using Mathcad and applying a non-linear regression procedure to the pairs (Ec, d) of Table 1, whereas for the other two models (Plitt-Reid and the new model) their parameters were computed using a simple linear regression to the linearized forms of the initial equations. Table 2 presents the values of the parameters calculated for the four models.

4. COMPARISON OF THE VARIOUS MODELS For purposes of comparison Table 3 was produced, using data from Table 2 and Eq. (1) for Rf = 47.9 %.

Table 3 contains the experimental data, the calculated actual-efficiency values and the

corresponding d50 and dmax sizes computed from the models and Eq. (1).

Table 3 shows that the already known models (Lynch, Plitt-Reid, Harris) give a better fitting capability for coarse particle sizes, while the proposed model for the fine particles.

The calculated d50 separation sizes differ slightly between the already known models (from 11.83 to 16.7 µm), but that predicted from the proposed model (28.3 µm) is closer to the experimental one, which in any case must be greater than 22 µm as shown in Table 1.

Another significant characteristic of the proposed model is that Ec = 1 at a finite particle size dmax = 436.2 µm, i.e. very close to that predicted from Harris’ model (dmax = 433.1 µm), overcoming the inherent disadvantage of the other two models (Lynch, Plitt-Reid).

From the data calculated from the various models and Eq. (1) (corrected efficiencies) and those given in Table 3 (actual efficiencies), Figures 3 and 4 were constructed. From these figures it is seen that the calculated values from

Table 1: Experimental data of the classification test (Finch and Plitt, 1975).

Particle size, Mesh Tyler

Geometric mean

size d, µm

Experimental actual

efficiency Ea, % to

underflow

Corrected efficiency Ec % from

Eq.1, (Rf=47.9%)

+ 10 - 100.0 100.0 -10+20 1173 100.0 100.0 -20+28 701 100.0 100.0 -28+35 496 99.6 99.2 -35+48 351 99.2 98.5 -48+65 247 92.7 86.0

-65+100 175 79.2 60.1 -100+150 124 75.9 53.8 -150+200 88 70.1 42.6 -200+325 57 57.2 17.9

-325 22* 48.9 0.2 *Arithmetic mean

Table 2. Values of the parameters for the various models and methods of predictionModel Lynch Plitt-Reid Harris New model

d50c = 123 µm d50c = 122 µm dmax = 433.1 µm d50c = 116 µm a = 1.602 m = 1.42 m = 1.263 n = 0.8922 Values of the

parameters r = 2.878

Method of prediction Non-linear regression

Simple linear regression & graphically

Non-linear regression & graphically (very

complicated)

Simple linear regression & graphically

0 10 20 30 40 50 60 70 80 90 100Corrected efficiencies predicted from various models (%)

0

10

20

30

40

50

60

70

80

90

100

Cor

rect

ed e

ffici

ency

(%) c

alcu

late

d fro

m s

ize

anal

ysis

New model (K.T.)LynchHarrisPlitt-Reid

Rf = 0.479 or 47.9%

Figure 3: Comparison of the corrected efficiency Ec, calculated from the experimental actual efficiency Ea, with the corrected efficiencies predicted from the various models. Solid line corresponds to y=x.

Advances in Mineral Resources Management and 5 Environmental Geotechnology, Hania 2004, Greece

the proposed model are relatively close to those predicted from the others.

5. GRAPHICAL REPRESENTATION OF THE NEW MODEL Equation (7) can be rearranged to give:

( )501.359n

cd dcE e−

= (8)

Substituting A for (Ec/1.359) gives: ( )50c

nd dA e−

= (9) Equation (9) is a modified Rosin-Rammler equation. From Eq. (9) it is clear that, when d=d50c then A=0.3679 or 36.79%, which

corresponds to Ec=0.5 or 50%. Similarly, when Ec=1 or 100%, then A=0.7358 or 73.58 %. Taking into account the above observations, the ordinate (y-axis) of a Rosin-Rammler graph was modified, putting in the points of 36.79% and 73.58% retained, the values 50% and 100% for Ec, respectively.

Equation (9) can be represented as a straight line on the new graph. The parameter n is positive, but the slope of the line is negative, as shown below:

( )( )50

log log 1.359 log log log(2.303)c c

E n d n d= − + −

(10) Thus d50c can be easily obtained from the point on the abscissa (x-axis) corresponding to the corrected efficiency Ec=0.5 or 50%. Applying the same procedure, dmax can be calculated from the graph as corresponding to Ec = 1.

Figure 5 shows the new graph, which represents the pairs (Ec, d) from Table 1.

6. CONCLUSIONS The present model is a powerful two-parameter model. All the parameters describing the performance of a classifier can be mathematically and graphically obtained with accuracy comparable to that presented by the already known models. It can be used as an alternative or in parallel with the other already applied models for the calculation of d50c, dmax and d50 (actual separation size).

It was pointed out that d50 is closer to the

50 60 70 80 90 100

Actual efficiencies predicted from various models (%)

50

60

70

80

90

100

Exp

erim

enta

l act

ual e

ffici

ency

Ea

(% to

und

erflo

w)

Actual Efficiency (Ea)LynchHarrisPlitt-ReidNew Model (K.T.)

Figure 4: Comparison of the actual efficiencies predicted from various models to the actual efficiency (Ea) from size analysis. Solid line corresponds to y=x.

Table 3: Comparison of the actual efficiencies Ea computed from the various modelsCalculated actual efficiency (% to underflow) from the

various models for Rf = 47.9% Geometric mean size d, µm

Experimental actual efficiency Ea, % to

underflow Lynch Plitt-Reid Harris New Model 1173 100.0 100.0 100.0 100.0 100.0 701 100.0 99.98 100.0 100.0 100.0 496 99.6 99.69 99.69 100.0 100.0 351 99.2 97.93 97.66 99.21 96.66 247 92.7 92.60 92.08 92.60 90.41 175 79.2 83.78 83.64 82.70 83.28 124 75.9 74.16 74.37 73.17 75.46 88 70.1 66.20 66.29 65.51 67.54 57 57.2 59.22 58.84 58.65 58.63

22* 48.9 51.92 50.97 51.27 48.76 Computed actual separation size d50, µm 11.83 16.7 15 28.3 dmax at Ec =1, in µm - - 433.1 436.2

6 Advances in Mineral Resources Management and Environmental Geotechnology, Hania 2004, Greece

experimental one than those predicted from the other models. This is due probably to the superior fitting capability of the proposed model for the fine size fractions.

It can also be thought as an advantage of the proposed model that Ec is predicted to be 1.359 or 135.9% at infinite particle size, whereas Ec = 1 or 100% at a finite particle size dmax, as it actually happens in wet classification.

Actually, the proposed model is in most cases reliable and adequate for the representation of the classifier efficiency (corrected and actual), but it needs further testing for its applicability to other classification tests. However, for the case examined here it was found to be valid.

ACKNOWLEDGEMENT The author is indebted to Professors A. Frangiskos and E. Mitsoulis for helpful discussions and suggestions.

REFERENCES Chen, W., N. Zydek and F. Parma, (2000). Evaluation of

hydrocyclone models for practical applications, Chemical Engineering Journal, 80, pp. 295-303.

Daniel, C. and F.S. Wood, (1980). Fitting Equations to Data, John Wiley & Sons, New York.

Finch, J.A. and R.R. Plitt, (1975). Introduction to Modelling the Comminution Circuit and Analysing the Concentrator Circuit, Ch. II Modelling the Classifier, In: Professional Development Seminars in Mineral Engineering, McGill Univ., Nov.-Dec. pp. II 1-II 24.

Finch, J.A., (1983). Modelling a fish-hook in Hydrocyclone Selectivity Curves, Powder Technology, 36, pp.127-129.

Harris, C.C., (1972). Graphical representation of classifier-corrected performance curves, Trans. Instn Min. Metall., Sect. C 81, pp. 243-245.

Kraipech, W., W. Chen, F.J. Parma and T. Dyakowski, (2002). Modelling the fish-hook effect of the flow within hydrocyclones, Int. J. Miner. Process. 66, pp. 49-65.

Lynch, A.J., (1965). The characteristics of hydrocyclones and their application as control units in comminution circuits, Prog. Rep. Dep. Min. Metall., Queensland Univ., No 6.

Luckie, P.T. and L.G. Austin, (1974). Technique for derivation of selectivity functions from experimental data, In: Proceedings of 10th Int. Miner. Process. Congress, IMM, London, pp. 773-790.

Nageswararao, K., (1999). Reduced efficiency curves of industrial hydrocyclones-An analysis for plant practice, Minerals Engineering, vol. 12, No. 5, pp.517-544.

Nageswararao, K., (2000). A critical analysis of the fish-hook effect in hydrocyclone classifiers, Chemical Engineering Journal, 80, pp. 251-256.

Pasquier, S. and J.J. Cilliers, (2000). Sub-micron particle dewatering using hydrocyclones, Chemical Engineering Journal, 80, pp. 283-288.

Plitt, L.R., (1971). The analysis of solid-solid separations in classifiers, CIM Bull. 64, pp. 42-48.

Roldan-Villasana, E.J., R.A. Williams and T. Dyakowski, (1993), The origin of fish-hook effect in hydrocyclone separators, Powder Technology 77, pp. 243-250.

Reid, K.J., (1971). Derivation of an equation for classifier-reduced performance curves, Can. Metall. Q., 10, pp. 253-4.

Tsakalakis. K.G., (1988). Some basic factors affecting the operation of horizontal vibrating screens, Ph.D. thesis, Nat. Tech. Univ. of Athens (in Greek, unpublished), 213 pp.

Figure 5: Hydrocyclone classification tests: modified Rosin-Rammler y-axis versus log plot.