Minerals
Engineering,
0892-6875(01)00070-X
Vol. 14, NO.I, pp. 741-751.2001 0 2001 Ebevier Science Ltd All rights reserved 0892-6875/01/$ - see front matter
PREDICTION OF DENSE MEDIUM CYCLONE PERFORMANCE FROM LARGE SIZE DENSITY TRACER TEST S. HUB,B. FIRTH” , A. VINCEq and G. LEES’ $ CSIRO/Division of Energy Technology, P.O. Box 883, Kenmore, Qld. 4069, Australia E-mail:
[email protected] 1 BHP Coal Australia (Received 15 February 2001; accepted 19 April 2001)
ABSTRACT A modified Suspension-Partition Model has been developed to predict partition curves of the smaller particle size fractions from experimental data of density tracer test at large particle sizes in dense medium cyclones. In this model, the sedimentation flux caused by a centrifigal force is balanced by the turbulent dijfusionflux. In addition, medium segregation and the dependence of particle diffusivity on particle size are also taken into account in the model. Three model parameters are estimated by fitting experimental data for a large particle size fraction to the model, and the model can be then used to calculate partition curves of the smaller particle size fractions. The capacity to predict the partition coeficients offiner particle sizes is demonstrated in a number of examples. The benefit of using this improved mathematical model is a cost reduction in the experimental determination of the performance of a given dense medium cyclone. 0 2001 Elsevier Science Ltd. All rights reserved.
Keywords Modeling; dense medium separation; particle size
INTRODUCTION The dense medium cyclone is the most efficient separation process used for beneliciating coal in the size range of 0.5 mm to 50 mm. This process is a density concentration technique whereby particles of mixed sizes, shapes, and densities are separated from each other due to the differential settling in a dense medium fluid with controllable density under the influence of a centrifugal force. Although dense medium cyclones are simple and robust devices from both mechanical and metallurgical points of view, separation efficiency is rarely as high as expected in the plant design. This is because coal preparation plants are not always operated in accordance with the design parameters such as the original cyclone dimensions, medium density and viscosity, optimum feed rate and size distribution, and minimum washability variation. In many cases, significant inefficiencies have been identified in association with poor medium control, excess cyclone wear, abnormal operating conditions such as out of range operating pressures, and poor feed control with excessive coal to medium ratios. In order to identify problem areas and improve separation efficiencies, density tracer techniques have been utilized as a tool to examine effects of various factors on the performance of dense medium cyclones [Davis et al., 19851.
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Density tracers are synthetic particles prepared to known specific gravity levels, with each specific gravity fraction rendered identifiable through color coding. The tracers are introduced into the feed stream of a dense medium cyclone, and on recovery from the product and reject streams, are sorted into the appropriate specific gravity fractions, counted, and the points for the partition curve can then be determined. The size of tracer particles used in operating plant test-work is usually 32 mm, which is significantly larger than the average size of feed. The large tracers tend to report to the top of the coal bed due to the stratification on vibrating screens, and therefore they are relatively easily seen and recovered prior to discharge of the screen oversize. It was found [Davis et al., 19851 that tracers of 7 and 15 mm do not report to the top of the particle bed and therefore significant losses occur. If only one large particle size (i.e. 32 mm) of tracers is used, it is necessary to consider how the separation of a bulk coal feed is related to the partitioning behavior of the large tracers. The separation effected by a dense medium cyclone varies with particle size, and fines are separated at a higher cut point and less efficiently than coarse. Based on the analysis of published data relating separating efficiency to coal particle size for cyclones of 600 mm or larger diameter, Davis et al. [1985] found that separation E,, is not very sensitive to particle size larger than 5 mm, but the cut point for the separation of a given size fraction does change with particle size. It has been identified that the effect of particle size on E,, and cut point become pronounced for particles smaller than approximately 5 mm. Researchers from The Julius Kruttschnitt Mineral Research Centre (JKMRC) [Wood, 1990; Clarkson and Wood, 19931 have proposed an empirical correlation for partition number of any size-density class. The correlation is given by:
PN=
100 1+ EXP(l.O99(p,A,- pJlEJ
(1)
where PN is the partition number, pjO*is the separation density for coarse particles, pP is the particle density of a given size fraction, and Ep is Ecart probables for particles with size of dp, and given by
(2) This empirical model is valid only for the range of experimental conditions from which it was derived. It provides only approximate estimations due to its over-simplification. Sometimes significant errors in predictions have been observed [Wood et al., 19891. Therefore, this model may not be reliable for the prediction of partition curves of any particle size fractions from 32 mm tracer test, where the plant operating conditions could be poor. By extending Schubert and Neesse’s suspension-partition model for turbulent cross-flow wet classification [Schubert and Neesse, 19731, Clarkson [1989] developed a simple model of dense medium cyclone performance as follows:
PN _ EXP(-aQAp)-1 100-
EXP(-aAp) - 1
(3)
where PN is partition number, Q is the slurry split to overflow factor, Ap is density difference between particle and medium, and the IXis given by
kd; a=--_ Dt P
(4)
where k is a constant dependent on cyclone parameters, d,, is the mean particle size of a given size fraction, D, is the eddy diffusivity, and u is the medium viscosity.
prediction of dense medium cyclone performancefrom large size densit tracertest
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It has been demonstrated that Equation (3) performs well in terms of regressing with plant data on size-bysize partition curves [Clarkson, 19891. However, Equation (4) does not reflect the observed relationship between 01and dp [Clarkson and Wood, 19931. Therefore, the model is not capable of predicting partition curves for different particle sizes. The review of above mentioned literature has indicated that there have been no suitable models of dense medium cyclones for predicting partition curves of any particle size fractions from experimental dam of large density tracer test. This situation necessitates the development of improved mathematical models for dense medium cyclones.
MODEL DESCRIPTION The physics describing multi-phase flows in dense medium cyclones is complex due to the interaction between particles and fluid and between the particles themselves. Detailed modeling has to be based on a fundamental approach involving the solution of velocity and phase distributions for both the fluid and particle phases [Hu & Firth, 1996 a&b]. But the model equations have to be solved using sophisticated numerical methods which are very time consuming and not readily applicable to routine calculations. A simpler model with reasonable capability for the prediction of effects of particle size on partition curves would be more useful for practical applications. The objective of this work is to develop a modified Suspension-Partition Model with considerations given to particle settling velocity, medium density distribution and eddy diffusivity. The model permits the partition curves of any particle size fractions to be determined from density tracer test data at a large particle size. Having made extensive simplifying assumptions, Schubert and Neesse [1973] developed a suspensionpartition model of turbulent cross-flow wet classification as shown in Figure 1. In this model, the sedimentation flux caused by a centrifugal force is balanced by the turbulent diffusion flux, i.e.
where cr, = volume fraction of particles, y = distance from wall, v,= settling velocity of particles, D, = particle turbulent diffusivity.
vOF
VUF Fig. 1 Suspension-partition model. By neglecting the effects of particle size and concentration on the turbulence, and assuming a constant settling velocity, Schubert and Neesse [1973] obtained the solids concentration distribution of a given
S. Hu er al.
744
particle fraction. An expression similar to the form of Equation (3) was derived for the calculation of partition number of a given particle fraction. It has been considered that this model can reflect the nature of hydrocyclone separation better than other empirical models [Schubert, 19851. However, the density difference between liquid and solids phases in a dense medium cyclone is a function of radial distance from wall due to medium segregation under centrifugal force. Therefore, settling velocity, depending on the phase density difference is also a function of radial distance from wall. Hindered settling and turbulence dampening may also play a considerable role in industrial hydrocyclones under high solids concentrations. Moreover, particle turbulent diffusivity is dependent on particle size. The Schubert and Neesse model has been modified to describe the separation process in a dense medium cyclone by taking into account the effects of medium segregation, the dependence of settling velocity on particle size and the particle turbulent diffusivity. The settling velocity is , therefore, given by
(6)
where
signpm = + if pP - pm~4, otherwise = - , pp = particle density, p,,, = medium density, d,, = particle size, V = tangential velocity, R = radial position, CD= drag coefficient.
The medium density is assumed to be a linear function of y:
The parameter, G can be determined by using the relationship that the mean value of pm is equal to the medium density of inlet feed. flmis a model parameter to be determined from density tracer test. The drag coefficient is estimated by
where h is the medium viscosity. al and bl are parameters depending on particle size and density, and their values can be found in Shook and Roco’s book [1991] (page 13). The use of Equation (8) avoids the calculation of a particle Reynolds number before evaluating the CD. The particle turbulent diffusivity is modeled as [Walton, 19951
D, = Dfd,
(9)
where Dp is a constant, Substituting Equations (6) and (9) into Equation (5) and assuming a constant centrifugal force, i.e. V’/R=Cfz, we have
dac_ --arc
[I
4 Pp-Pm
signpm 3
where KJ= C,/op.
Pm
Prediction of dense medium cyclone performance from large size densit tracer test
145
the volume fraction distribution of a particle fraction with a size of d,, and a density of p,, can be calculated by the numerical integration of the above equation. The partition number of the particle fraction can then be calculated from Thus
(11) where &,G and CQ are the mean values of a, in the ranges of HG and H,,respectively. HG and H, are parameters in the suspension-partition model as shown in Figure 1. It has been found that a good estimate of Htis the radius of the dense medium cyclone. The three model parameters, HG.KJ and p,,, contained in the modified suspension-partition model can be estimated by fitting the experimental partition curve obtained from a large size density tracer test or a large particle size fraction to the model equations. The model can be then used to predict partition curves of different size fractions. RESULTS AND DISCUSSION A total of ll’data sets have been used to evaluate the quality of model predictions. 8 are from Wood [ 19901, and the other 3 from Deurbrouck and Hudy [ 19721. Figure 2 shows an example of the results. Three model parameters were obtained by fitting the experimental data for 32 mm tracer test, and then the model is used to predict the partition curves for
Fig.2 Comparison between model predictions and experimental data set #1302 [Wood, 19901. fractions of 315x16, 16x8, 8x4, 2x1 and 1x0.5 mm. It is clear that model predictions agree well with experimental data for large particle size fractions for all cases. Although predictions with finer particle size fractions (~2x1) are not as good as those for large particles, the prediction accuracy for the fines should be considered to be acceptable given the larger experimental errors involved with finer particle size fractions. Model parameters estimated for the 11 cases are summarized in Table 1.
S. Hu et al.
TABLE
1 Summary
of model parameters
estimated for 11 experimental
data setsr1’21
[l]: Cases #201, #322, #323, #407, #1015, #1301, #1302 and #1303 from [Wood [1990]; Cases #A, #B and #C from Deurbrouck and Hudy [ 19721. [Z]: D,= cyclone diameter, m; Qf = volumetric flowrate, LJsec; pin =relative medium density
of the feed; p,, -A = density difference between underflow and overflow; Q,/Qf =ratio of volumetric flowrate of the underflow to overflow; M.Grade = magnetite grade. [3]: SF=supertine; UF=ultrafine. System performance variables (Le. Separation density Dso, Ecart Brobables Ep and separation density offset Dso-Of) have been calculated from predicted partition curves at two selected size fractions. The comparison of the variables from predicted curves with those from experimental curves is shown in Table 2. For cases from Wood [1990], the predicted values of D 5(h and Dso-Df are close to those obtained from Wood’s experimental curves. Ep values from predicted curves are also in good correspondence with those from experimental curves for large particle size fractions. However for finer size fractions, Ep values from predicted curves are generally higher than those from experimental curves. As mentioned previously experimental errors with fines are higher and Ep values from experimental curves are highly dependent on the method of drawing the partition curves through experimental data points. Sensitivity analysis of variations in model parameters, Ha Kd and fl,,, has been carried out, and results for Case #323 for two size fractions of 16x8 and 2x1 are shown in Figure 3. System performance variables, Djo, Ep and Dso-Df calculated from the curves in Figure 3 are shown in Figures 4 to 6. Similar results for other cases were also obtained. For variations of &40% in any model parameter, the corresponding changes in Dso, Ep and Djo-Df are generally less than 10% relative to the base case, except for changes of Ep with larger variations in HG and Kd for finer size fractions. Results in Figures 4 to 6 also indicate that predictions are more sensitive to large negative variations (e.g. -80%) than to large positive variations in HG and Kd. This finding is useful for the improvement of estimations, of the model parameters. The model parameters determined by fitting an experimental partition curve for tracer or large particles always contain errors. Although D5& E,, and Dso-Df for large particle size fractions are not sensitive to the errors, even large negative ones, these performance variables for finer size fractions would change significantly with large negative errors in model parameters. Therefore increasing the estimated values of HG and Kd from tracer tests by +lO-20% could improve the predictions for very fine size fractions. It has also been found that the relationships between the variations in fi,,, and changes in DsLhE,, and D~J-Of are basically linear.
Prediction of dense medium cyclone performance from large size densit tracer test
TABLE 2 Comparison experimental curves
of system performance
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variables from predicted curves with those from
d,=10.9 1.569 1.569 1.519 1.529
0.045 0.039
0.044 0.036
0.189 0.199
0.189 0.209
d&84 1.654 1.636 0.051 1.616 1.571 0.058
0.073 0.081
0.274 0.296
0.256 0.251
1.559
0.036
0.033
0.139
0.139
1.674
0.071
0.254
0.214
1.567
1.634
0.064
i\_/--IT?ir’
,’
I
:
.w/.
:
,I’/
_I
Fig.3
Effects of variations in Case #323.
mm
d@3*
in model parameters on predictions
of partition curves for two size fractions
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S. Hu et al.
1.7
d,,=2xl
d,.,=18x8 1.6
1.6
1.5 a”
1.5 c?
1.4
1.4
1.3
1.3
1.2
1.2
,Jyyy -40%-m% 0
20%40%
80%
Variation of H,
Variation of HO
0.050 0.045
dpl6x8
t
L
0.040
w” 0.035 0.030 0.U2.5 0.020
Variation of )I, 0.5 ,
d,=2xl 0.4 -
0.004,,,,, 40%
,,,,,,,,,,,,,, ‘40%~aw40
i 20%40%
80%
Variation of H,
Fig.4
Variation of I-&
Effect of variations in Ho on the changes of D5& Ep and DjO-Dffor size fractions of 16x8 and 2x1 in Case #323.
CONCLUSIONS A modified Suspension-Partition Model has been developed to predict partition curves of the smaller particle size fractions from experimental data of large size tracer test. In this model, the sedimentation flux caused by a centrifugal force is balanced by the turbulent diffusion flux. In addition, medium segregation and the dependence of particle diffusivity on particle size are also taken into account in the model. The model contains three parameters, Ho, Kd and & l
. l
Ho is flow cut thickness to underflow. Kd is proportional to the ratio of centrifugal and diffusive forces. fi,,, is a parameter relating to the extent of medium segregation.
749
Prediction of dense medium cyclone performance from large size densit tracer test
.4O?b-20?4020%40%
BOK
Variation of G
_._
0.11 , d,,=l6x8
0.06 _
0.09
0.05 -
4
I
d,=2xl
0.10
0.08 0.04 -
w” 0.07
0.03 0.m;
0.06 ,\-----,
0.01 ao%
40%-20%020%40%
Box
L
0.05
-
0.04
~,,,,,....,...,.....I -80%
Variation of G
-40%~20%0
20x40%
80%
Variation of G
0.120 dp=lBxB
0.118
0.28 0.26
0.116
g
0.24
0.112
0.22
'g cl
0.20 0.18 0.16
0.110 0.108 I$- 0.114I
d
\.
-----I
0.14 0.12 0.10-I 1,,,,,,,,1,1 -90% 4o?hm%o
I,,,,,, 20x40%
60%
Variation of q
Fig.5
Effect of variations in Kd on the changes of D 5(hE,, and DSO-Df for size fractions of 16x8 and 2x1 in Case #323.
These parameters can be estimated by fitting experimental data for a large particle size fraction to the model, and the model can be then used to calculate partition curves of the smaller particle size fractions. Eleven case studies have been used to investigate the capability of the model. It has been found that predictions of partition curves of the lower particle size fractions are in good correspondence with experimental results. Predictions of D j(h Ep and offset (DJ,,-Df) were also found to be in good agreement with those determined from experimental data. Sensitivity analysis of the model parameters indicate that *40% variations in model parameters cause less than flO% changes in predictions of system performance variables (i.e. DT& Ep and DJo-Or>, except for changes of Ep with variations in Hc and KJ for small particles. Model predictions are more sensitive to large negative variations (e.g. - 80%) than to large positive variations in HG and Kd.
750
S. Hu et al. 1.48
1.48
1.46
1.46
1.44
1.44
I .42
1.42
1.40
1.40 a”
c?
:; 1.34
1.38 1.36
1.32
1.34
1.30
1.32
1.28
1.30
1.26 -40x-2O%Ow%40%
80%
Variation of pm
w” o.022 0.020 0.018 0.016 dp=16x6
0.014 %0%
aO%a%o
20x40%
m%
Variation of pm
Fig. 6
Effect of variations in Case #323.
40%-xl%0
x)x40%
60%
Variation of &
in h on the changes of D 56 E,, and DsO-Dffor size fractions of 16x8 and 2x1
REFERENCES Clarkson, C.J., A model of dense medium cyclones. Coal Preparation, 1989,7,159-174. Clarkson, C.J. and C.J. Wood, A model of dense medium cyclone performance. Coal Preparation, 1993, 12,101-l 15. Davis, J.J., C.J. Wood, and G.J. Lyman, The use of density tracers for the determination of dense medium cyclone partition characteristics. Coal Preparation, 1985,2, 107-125. Deurbrouck, A.W. and Hudy, J., Performance characteristics of coal-washing equipment: dense medium Cyclones. Report of Investigations 7673, US Bureau of Mines, 1972. Hu, S. and Firth, B., Numerical studies of phase redistribution phenomena in slurry flow Splitting. In Computational Techniques and Applications: CTAC95, World Scientific Publishing Corp, Singapore, 1996a, pp395-402, Hu, S. and Firth, B., Application of Multiphase Flow CFD to the Analysis of Slurry Flow Subdivision Devices. In Australian Engineering Mathematics Conference, ABMC’ 96, Sydney,l996b, pp293-300. Schubert, H. and T Neesse, The role of turbulence in wet classification. In Proceedings of the 10’h International Mineral processing Conference, London, 1973, pp213-239. Schubert, H., A hydrocyclone separation model in consideration of the turbulent multi-phase flow. Particulate Science and Technology, 1985,3, 1-13. Shook, C.A. and Roco, M.C., Slurry Flow: Principles and practice. Butterworth-Heinemann, Stoneham, USA, 1991, page 13.
Prediction of dense medium cyclone performance from large size densit tracer test
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Walton, I.C., Eddy diffusivity of solid particle in a turbulent liquid flow in a horizontal pipe, AZChE J., 1995,41(7), 18151820. Wood, C.J., A performance model for coal washing dense medium cyclones, PhD Thesis. University of Queensland , 1990. Wood, C.J., J.J. Davis and G.J. Lyman, Towards a medium behavior based performance model for coalwashing dense medium cyclones. Coal Preparation, 1989,7, 183-197.
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