Modelling of hydrocyclones at high feed-solids concentrations

The Chemical Engineering

Journal, 34 (1987)

81 - 88

81

Modelling of Hydrocyclones at High Feed-solids Concentrations M. A. DOHEIM,

G. A. IBRAHEIM

Faculty of Engineering, (Received

September

and A. A. AHMED

University of Assiut, Assiut (Egypt) 11,1985;

in final form

April 21,1986)

ABSTRACT

Hydrocyclones (HCs) are being extensively used and are finding widespread applications in many industries. The successful design and operation of HCs depend on the availability of sufficient data and proper quantitative correlations. Very few data and correlations exist in the literature for conditions of high feed-solids concentration and wide particle size distribution. In this work, classification tests were carried out over wide ranges of vortex finder diameter, apex diameter, feedsolids concentration and length of vortex finder. White sand and a cyclone of diameter 100 mm were used. An empirical model for the prediction of HC performance was developed, giving the corrected separation size, the flow rate of water in the overflow, the sharpness of separation and the capacity of the cyclone. Good agreement exists between the model and the experimental data. The findings for high levels of feed-solids concentration (up to 50% by weight) should be very useful in the successful design and operation of industrial HC classification units.

1. INTRODUCTION

Hydrocyclones (HCs) are a very efficient and attractive tool for many purposes in solid liquid handling and processing, e.g. classification, separation, de-watering, etc. [ 1, 21. HC design and operation are conventional for dilute slurries and narrow particle size distributions (PSDs) [3 - 51. In the case of slurries with high solids concentrations and wide PSDs, the literature data are however very limited [6, ‘I] which makes the design and operation of HCs very difficult. Mathematical models of HCs are very useful in the design and operation of such devices. 0300-9467/87/$3.50

This paper presents work on HCs at high feed-solids concentrations and wide PSDs, studying the effect of four important variables: the overflow diameter (vortex finder), the underflow diameter (apex), the solid content of the feed pulp and the length of the vortex finder extended inside the cyclone. The paper also presents a mathematical model to help in the design, to describe the operation and to predict the performance of HC units.

2. MODELLING

OF

HYDROCYCLONE

OPERA-

TION

A mathematical model of an HC, describing its design and operation, consists of a series of equations which relate the design and operating variables to the separation achieved. There are two approaches to the development of an HC model, namely the empirical model approach and the fundamental mathematical modelling approach. The first approach has been adopted by many investigators [ 1, 6 - 141, while the second approach was presented by Holland-Batt [ 151 and Boysan et al. [ 161 who analysed the performance of an HC in terms of bulk flow behaviour and average retention times. The HC model equations describe (a) separation size dsa, (b) HC capacity (pressurethroughput relationship), (c) water flow ratio (water split) and (d) classification efficiency. The separation size d 50, defined as the size which has an equal probability of being separated with either the underflow or the overflow, is of limited utility and a more useful quantity is the corrected size d5aC. The calculation to determine dsoC assumes that a fraction of feed solids completely by-passes the classification process and goes directly to the underflow by short@ Elsevier

Sequoia/Printed

in The Netherlands

82

circuiting. This solids fraction is in direct proportion to the fraction of feed water returning to the underflow [ 171. Many expressions [6 - 15,18 - 211 based upon theory and experiment have been given by various investigators to predict the separation size in terms of the different design and operating variables. Most of the empirical equations were based on data obtained from systems using dilute pulp, feed with narrow particle size distributions and small HCs. Relatively few equations were obtained for conditions of high concentrations of feed pulp. Lynch and Rao [6], Plitt [13], and Svarovsky and Marasinghe [7], presented equations corresponding to various degrees of pulp concentration. With respect to HC capacities, several correlations [l, 2, 4, 6, 9, 12 - 14, 18, 20, 22 - 241 have been developed to relate cyclone capacity to the more important system variables. The equations obtained for high feed pulp concentrations are few [6,13,17]. Regarding the water flow ratio (water split), the terms usually used to describe this aspect of HC performance are the volume split $ pulp flow ratio R, and water flow ratio Rf, which are related by the equations R,=

s

1+S

TABLE 1 Size distribution of feed material used in tests Size (pm)

Wt.% retained

500 400 315 200 160 125 90 63 45 30 20 15 Pan

0.00 0.75 2.97 35.58 9.36 8.75 10.38 9.81 5.97 6.33 5.65 2.61 1.84

3.2. Apparatus The experimental circuit used is shown in Fig. 1. The hydrocyclone dimensions are: cylindrical section diameter, 100 mm; cylindrical section length, 175 mm ; vortex finder diameter d,, variable; underflow orifice (apex) diameter d,, variable; inlet diameter di, 15 mm; vortex finder length 1, variable; cone angle 19,20”. During operation the product streams were discharged at atmospheric pressure (free discharge condition). The feed pressure was 24 psi.

and Rf =

R, - R,V

1-v

Many correlations exist [ 1, 21 and they have been summarized by Marlow [ 121. For the HC classification efficiency, the sharpness of separation depends on the slope of the central section of the classification curve (Tromp curve). The closer to vertical is the slope, the higher is the efficiency [25].

3. EXPERIMENTAL

PROCEDURE

3.1. Materials White sand with specific gravity 2.65 was used in the present work. It was ground in a dry ball mill to 500 pm. The particle size distribution is given in Table 1.

Fig. 1. Hydrocyclone test circuit: 1, cyclone; 2, pressure gauge; 3, feed tank; 4, agitator; 5, variable speed pump; 6, by-pass valve; 7, discharge boxes.

83

In the experimental circuit, the feed tank was baffled to eliminate vortex formation and an agitator was mounted in the tank to ensure satisfactory and uniform suspension of the solids. The cyclone overflow and underflow were returned to the feed tank via two conical receivers capable of taking simultaneous samples of the two products. F’recautions were taken to avoid any overflow siphon effect. 3.3. Procedure The cyclone dimensions were set as required. The desired cyclone feed pulp density was adjusted and the system was allowed to run for sufficient time to ensure thorough dispersion of the solids. Simultaneous samples of the overflow and underflow products of the cyclone were obtained. For each of the samples, the total volume, the total weight of pulp and the dry weight of solids were measured. The dried sample was then rolled on a rubber mat so that a representative sample of 100 g could be obtained. The size distribution for each sample was determined by dry sizing for particles of size greater than 90 pm and by elutriation for those smaller than 90 lrn.

4. EXPERIMENTAL

RESULTS

In studying the behaviour of minerals in HCs the effects of the following important variables were investigated: (a) percentage of solids by weight in HC feed; (b) vortex finder diameter (overflow diameter); (c) apex diameter (underflow diameter); (d) vortex finder length inside the HC. This work was characterized by the use of feed pulps of high solids concentrations and solids of wide particle size distribution. The literature data pertaining to such conditions are very limited. During the work, other HC dimensions were kept constant as follows: diameter of feed inlet, 15 mm; total cone angle, 20”; height of cylindrical section, 175 mm; and HC diameter 100 mm. The variable dimensions and other test conditions can be seen in Table 2 together with other data. During these experiments there prevailed a spray type of discharge in the underflow. The quantities directly measured during the experimental work were the volume of the pulp, the weight of the pulp, the weight of dry solids and the particle size distribution,

TABLE 2 Summary of dsoc, Q, WOF and m values for all variables investigated* Variable and unit

Variable values

dsoc (Pm)

WOF (1 min-‘)

m

z min-l)

Variable HC dimensions and test conditions

Sf (wt.%)

10 20 30 40 50

24 31 35 37 43

207 168 166 190 177

118 90 89 89 83

3.0 2.1 1.8 1.7 1.5

d, = 50 d, = 40 1=45

4 (mm)

40 50 60 70 80

25 37 42 50 60

237 223 230 267 263

98 122 144 175 170

1.4 2.0 2.2 2.3 2.8

Sf = 40 d, = 40 1=45

35 40 55 65

48 45 40 36

193 200 204 204

125 127 112 77

2.3 2.2 2.1 1.8

Sf = 40 d, =70 1=45

25 45 100 150 175

40 40 43 50 52

204 204 196 197 196

127 112 117 117 115

2.3 2.1 2.0 1.9 1.8

Sf = 40 d, = 70 d, = 55

aFeed pressure, 24 psi.

84 TABLE

3

Calculation Geometric mean size

of the corrected

Feed (kg min-l) (A)

efficiency

Overflow (kg min-‘)

curve

Underflow (kg min-‘) (B)

(Pm) 450 358 258 180 142 108 76 54 38 25 18

0.43 1.89 22.38 5.89 5.61 7.00 6.26 3.82 3.85 3.68 1.78 1.13

aShort-circuit

0.00 0.02 0.02 0.02 0.05 0.01 0.27 0.37 1.03 1.62 0.93 1.13 to underflow

0.43 1.87 22.36 5.87 5.56 6.99 5.99 3.45 2.82 2.06 0.85 0.00 Rf = 52.691141.96

Short-circuit to underflowa (C = RfA) (kg min-l)

Feed classified

0.16 0.70 8.28 2.18 2.07 2.59 2.32 1.41 1.42 1.36 0.66 0.42

0.27 1.19 14.10 3.71 3.54 4.41 3.94 2.41 2.43 2.32 1.12 0.71

(A-C) (kg min-l)

To underflow by true classification (B - C) (kg min-l)

Corrected efficiency (B - C)/ (A - C) (“ro)

0.27 1.17 14.08 3.69 3.49 4.40 3.67 2.04 1.40 0.70 0.19 0.00

100.00 98.30 99.80 99.40 98.60 99.80 93.10 84.60 57.60 30.90 17.00 0.00

Reduced efficiency

12.85 10.23 7.37 5.14 4.06 3.08 2.17 1.54 1.08 0.71 0.51

= 0.37.

each of these being measured separately for the two discharge streams of the HC. The determined and calculated quantities are the capacity of the HC, the water flow rate in the overflow and underflow products and the percentage of solids by weight in the two discharge streams. Material balance calculations were carried out for all runs to give the necessary details for the feed, overflow and underflow streams of the HC. Having performed the size analysis of the underflow, the classification curve (performance or efficiency curve) can be plotted. This curve is a representation of the cyclone efficiency. A corrected efficiency curve is then calculated, as shown in Table 3, by taking into consideration the feed solids which short-circuit to the underflow. The “corrected” curve gives the amount of solids that leaves in the underflow due to classification only. The uncorrected and corrected classification curves are shown in Fig. 2. From these curves, the classification size ds, (or dsoc) can be obtained, which is often at that point on the curve for which 50% of particles of that size in the feed return to the underflow. The reduced efficiency curve is obtained by plotting d/d,,c instead of d, as shown in Fig. 3. This curve can only be determined when the d,, value for the test is above the smallest size. In our representations, we used the corrected curve. The corrected classification curves calculated from the results were obtained for all the variables

60

I

I

120

180

2

4

6 d’dsoc

Fig. 3. Reduced

efficiency

300

I 360

lpm)

Size

and corrected Fig. 2. Uncorrected dso, 20 pm; dsOC, 35 pm.

0

I

240

curve.

efficiency

8

curves:

IO

I2

85

investigated and dsoc values were obtained as explained above.

5. MODEL DEVELOPMENT AND DISCUSSION

To develop the model, c&c values were obtained from the corrected classification curves, and the capacity and the water flow to overflow were obtained from the material balance calculations. The classification efficiency m was obtained using the equation ]I31 y = 1 - exp{-0.6931(d/d,,c}m

(3)

where m serves as a direct measure of the sharpness of separation [17, 25, 261. A condensed summary of the data obtained is given in Table 2. The data were processed, through a linear regression procedure, using different functional forms expressing the model parameters in terms of the design and operating variables. The computer programs were capable of fitting the data to three different functional forms. More details are given elsewhere [27]. The four following equations were thus obtained. For the separation size d 5,,c = 2.31d, 0.829du-0.63~~0.312~0.185 (4)

0

I

I

I

IO

20

30 d50C

40

Observed

I

I

50

60

(pm)

Fig. 4. Observed and calculated values for ds,,~.

For the water flow rate in the overflow WOF = 0.76WF - 1.78d, + 1.37d, - 19.1

I

(5) For the classification efficiency m = 0.844d,0.839d,-0.201~~-0.415E_0.0831

60

a0

I

100 WOF

I

I

I

120

140

160

Observed

(I

I

180

mfn-‘)

Fig. 5. Observed and calculated values for WOF.

(6)

And for HC capacity Q = exp(5.16 + O.O0477d, - O.O0402d, + 0.00217& - 0.0005121) (7) Equations (4) - (7) constitute the mathematical model of the HC which enables the performance of an HC to be calculated. The predicted and observed values of the model parameters are shown in Figs. 4 - 7. The effect of the different design and operating variables on the separation size dsoC and other performance parameters can now be examined through use of the model. The effects on dsoC of vortex finder diameter, apex diameter, solids concentration of feed pulp and vortex finder length are shown

in Table 2 and a sample plot is given in Fig. 8. These effects and behaviour are uniquely characterized by being under pulp conditions of high solids density and wide particle size distribution. However, the general behaviour agrees with that reported by Yoshioka and Hotta [9] and Bradley [ 191 for d,, with that from Plitt [ 131 for d,, with that from Lynch and Rao [6] and Plitt [13] for Sf and with that from Tarr [28] for 1. Similarly the effects of d, and d, on WOF are shown in Fig. 9. The effect of water content in feed pulp WF on WOF is shown in Fig. 10. The general trends are similar to those reported by Lynch and Rao [23] and Marlow [12]. The sharpness of separation increases with the ratio do/d, and decreases with the feed-

86

34-

I80 160-

30-

“,

26-

‘0 =, % 0 E

22-

I8I 50

I 40

I 60

I 70

Diameter

I

I

14

18

I

I

I

I

22

26

30

34

I

I 80

,

(mm)

Fig. 9. Overflow water split us. apex and vortex finder diameters: 0, d, ; 0, d,.

m Observed

Fig. 6. Observed and calculated values for m. 280

7

260

I 80

140 GO

I 120

I 100 WOF

ii

I 140

I 160

mln-‘)

Fig. 10. Overflow water split us. water in feed. 0

18C 260 I )

180

I

200

I

220

Q Observed

0

I

240

260

280

(I rnln-‘)

Fig. 7. Observed and calculated values for Q.

60

r u 5: -cl

40

-T

50 20 /

0

40

50

60 Dameter

30

I

A I

I

I

I

/

40

50

60

70

80

Vortex

finder

diameter

(mm)

Fig. 8. Effect of vortex finder diameter on the size of separation.

solids concentration and the vortex finder length. This is in good agreement with the work of Fahlstrom [ 111 and Dahlstrom [18].

70

80

(mm)

Fig. 11. Cyclone capacity us. spex and vortex finder diameters: 0, d,; 0, cl,.

Figure 11 shows the effect of vortex finder diameter on HC capacity, which was Jso indicated by Fahlstrom [ 111, Harry and Shiou [29] and Lynch [5]. It also shows that change in the apex diameter has a slight effect (an increase) on the capacity, as found by Lynch [5] and Lynch and Rao [23],

87

experimental results. The vortex-finder diameter, the apex diameter and the feed-solids concentration seem to be the most influential parameters.

280

REFERENCES

I

I

IO

I

I

I

I

20

30

40

50

Solids

in feed

I

pulp (%)

Fig. 12. Cyclone capacity us. percentage of solids in feed.

T

280

-

240

-

6

7

G E

0

200-

-~~L_c_

8 9

160-

0

I

I

25

45 Vortex

I

I

I

100

150

175

finder

length

10

(mm)

Fig. 13. Cyclone capacity vs. vortex finder length.

although Fahlstrom [ 111 observed a decrease in capacity with increase in the apex diameter. Figure 12 indicates that the throughput decreases slightly as the mass fraction of solids in the feed increases or as the water content of the feed decreases, which agrees well with the data of Lynch and Rao [23], although Tarr [28] reported the opposite effect. Figure 13 indicates the limited effect of the vortex finder length on the capacity, which agrees in general with the only observed relationship mentioned in the literature (Plitt [ 131).

11

12

13 14

15

16

6. CONCLUSIONS 17

Model equations for predicting the performance of a hydrocyclone were obtained for high pulp concentrations of feed and solids of wide particle size distribution. Good agreement exists between the model and the

18

19

D. Bradley, The Hydrocyclone, Pergamon, London, 1965. L. Svarovsky, Hydrocyclones, Holt, Rinehart and Winston, New York, 1984. K. Rietema and C. G. Verver (eds.), Cyclones in Industry, Elsevier, Amsterdam, 1961. L. Svarovsky (ed.), Solid-Liquid Separation, Butterworths, London, 1981, Ch. 6. A. J. Lynch, Mineral Crushing and Grinding Circuits - Their Simulation, Optimization and Control, Elsevier, Oxford, 1977, Chapters 5 and 6. A. J. Lynch and T. C. Rao, Modelling and scaleup of hydrocyclone classifiers, Proc. 11 th Znt. Miner. Proc. Congr., Cagliari, 1975, Paper 9. L. Svarovsky and B. S. Marasinghe, Performance of hydrocyclones at high feed-solids concentration, Znt. Conf. on HCs, 1 - 3 October, Churchill College, Cambridge, U.K., 1980. E. 0. Lilge, Hydrocyclone fundamentals, Trans. Inst. Min. Metall., 71 (1962) 285. N. Yoshioka and Y. Hotta, Liquid cyclone as a hydraulic classifier, Chem. Eng. Jpn., 9 (1955) 632. K. Rietema, The mechanism of the separation of finely dispersed solids in cyclones, in K. Rietema and C. G. Verver (eds.), Cyclones in Industry, Elsevier, Amsterdam, 1961, Chapter 4, p. 46. P. H. Fahlstrom, Studies of the hydrocyclone as a classifier, Proc. 6th Znt. Miner. Proc. Congr., Cannes, 1963, p. 87. D. R. Marlow, A mathematical analysis of hydrocyclone data, M.&z. Thesis, University of Queensland, 1973. L. R. Plitt, A mathematical model of the hydrocyclone classifier, CZM Bull. 69 (1976) 114. A. L. Mular and A. Jull, The selection of cyclone classifiers, . . . , for grinding circuits, in Mineral Processing Plant Design, 2nd edn., American Institute of Mechanical Engineers, New York, 1980, Chapter 17. A. B. Holland-Blatt, A bulk model for separation in hydrocyclones, Trans. Inst. Min. Metall., Sect. C, 91 (1982) 21. F. Boysan, W. H. Ayers and J. Swithenbank, A fundamental mathematical modelling approach to cyclone design, Trans. Inst. Chem. Eng., 60 (1982) 222. B. A. Wills, Mineral Processing Technology, 2nd edn., Pergamon, Oxford, 1981, p. 233. D. A. Dahlstrom, Fundamentals and applications of the liquid cyclone, Chem. Eng. Progr. Symp. Ser. 50 (15) (1954) 41. D. Bradley, A theoretical study of the hydraulic cyclone, Znd. Chem., 34 (1958) 473.

88

G. E. Agar and J. A. Herbst, The effect of fluid viscosity on cyclone classification, Trans. SOC. Min. Eng. (June 1966) 145. 21 D. F. Kelsall, A study of the motion of solid particles in a hydraulic cyclone, Trans. Inst. Chem. Eng., 30 (1952) 87. 22 I. R. M. Chaston, A simple formula for calculating the approximate capacity of a hydrocyclone,

20

Trans.

Inst. Min. Metall.,

67 (1985)

203.

A. J. Lynch and T. C. Rao, Studies on the operating characteristics of hydrocyclone classifiers, Znd. J. Tech., 6 (1968) 106. 24 K. Nageswararao, T. C. Rao and A. J. Lynch, A throughput expression for industrial hydrocyclones, Tech. Report for period July 1 - December 31, 1974, pp. 27 - 33 (Julius Kruttschnitt Mineral Research Centre). 25 S. P. Barber, G. W. Cutting and P. J. Orrock, Cyclones as classifiers in ore grinding circuits a case study: paper 2, Proc. Znt. Conf. on HCs,

23

Churchill 1980.

College,

Cambridge,

U.K.,

October

1

do du d 50 Got h

1

L m

-2

H. Trawinski, Theory, applications, and practical operation of HCs, Eng. Min. J., 177 (9) (1976) 155. 27 A. A. Ahmed, Ph.D. Thesis, University of Assiut, 1984. 28 D. T. Tarr, The influence of variables on the separation of solid particles in HCs, Proc. 45th Annu. Meet. AZME, Minnesota, 1972, p. 64. 29 G. P. Harry and Shiou Chuansun, Cyclone classification of artificial abrasive powders, Trans. Sot. Min. Eng., 226 (Dec. 1963) 461. 26

Sf s

V W

APPENDIX de

4

A: NOMENCLATURE

inside diameter (ID) of cylindrical section of an HC (mm) ID of HC inlet (mm)

WF WOF Y

ID of HC overflow (mm) ID of HC underflow (mm) uncorrected cut size (pm) corrected cut size (pm) free vortex height of a cyclone (length from spigot to bottom of the vortex finder) (mm) length of vortex finder inside the cyclone (mm) total length of HC (mm) parameter in the Rosin-Rammler expression for a classification curve which serves as a measure of the sharpness of separation volumetric flow rate of HC feed pulp (1 min- ‘) recovery of water to the underflow product (underflow water/feed water) recovery of feed solids to the underflow product recovery of feed pulp to the underflow product (underflow rate of pulp/ feed flow rate of pulp) percentage of solids by weight in feed Pulp to volumetric ratio of underflow overflow pulp volumetric fraction of solids in feed Pulp volumetric fraction of water in the feed flow rate of water in the feed (1 min- ’ ) flow rate of water in the overflow (1 min- ‘) classification efficiencv I