International Journal of Mineral Processing, 28 (1990) 289-300 289 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Mode l l ing S c r e e n i n g as a Conjugate Rate P r o c e s s
G.K.N.S. SUBASINGHE, W. SCHAAP and E.G. KELLY
University of Auckland, Private Bag, Auckland (New Zealand)
(Received October 7, 1988; accepted revision July 20, 1989)
ABSTRACT
Subasinghe, G.K.N.S., Schaap, W. and Kelly, E.G., 1990. Modelling screening as a conjugate rate process. Int. J. Miner. Process., 28: 289-300.
It is shown that the screening process can be described as a combination of two first-order processes occurring simultaneously. The processes are those of segregation of undersize material on to the screen surface, and the passage of segregated material through the screen apertures. Also, it has been shown that by adopting such an approach, a unified model could be formulated to predict the screening results of different systems provided the rate constants are evaluated for each size fraction.
INTRODUCTION
Attempts in the past to quantitatively describe the performance of screening processes mathematically have adopted either a "probabilistic" or a "kinetic" approach (Kelly and Spottiswood, 1982). Most of the studies that used the rate approach assumed a single first-order process. Ferrara and Preti (1975, 1988) were amongst the first to show that two distinct processes needed to be considered. In their work, they found that initially a zero-order process oc- curred, and then, as the quantity of material on the screen declined, a first- order process took over. Given that it is widely accepted that three zones are typically found in screening (Fig. 1 ), Kelly and Spottiswood (1982) suggested that for a complete analysis, a third process needed to be considered: that of a first-order process to describe the stratification that occurred before the "high- rate" zero-order process.
This paper describes an alternative approach which uses two first-order rate processes to describe the three-region behaviour of Fig. 1.
SCREENING AS A RATE PROCESS
The shape of the curve representing the screening results shown in Fig. lc may be considered as resulting from two simultaneously occurring processes.
I W / / / / / / / / Y / ~ x x, Separated 0 W///////~////~ Screening
1.0-
~ @
.~_
o i J Screen Length
Fig. 1. The three major regions occurring along a screen (a), and their relationship to the rate of passage (b), and the fraction remaining above the screen (c).
One is the segregation of undersize particles through the particulate bed to reach the screen surface, and the second is the process of material present in the contact layer passing through the screen apertures. The former process has been shown by Olsen and Rippie (1964) to be first-order, with the rate con- stant depending on the relative size of the particles being segregated and the size of the surrounding particles, the concentration of a given size of particle, and the nature of the vibratory motion. It has also been shown (Rosato and Prinz, 1986) that when there are sufficiently more smaller particles than larger ones, as a result of vibration, the larger particles tend to be pushed upwards, while the smaller ones descend. This indicates that the migration of larger particles towards the screen surface is slower when smaller particles are pres- ent, but increases as the smaller particles are depleted.
The second process, that of particles passing through the screen apertures, has also been observed to be first-order for uniformly sized feed material (i.e.,
MODELLING SCREENING AS A CONJUGATE RATE PROCESS 291
in the absence of an associated segregation process). Kaye (1962) reported tha t for sieving, first-order kinetic behaviour was observed under conditions which gave rise to constant probability of passage. The condition under which he observed the first-order behaviour was when the sieving time was suffi- ciently large, so tha t only a small amount of material able to pass the sieve remained on the sieve in the presence of a relatively large quanti ty of oversize material. This is essentially the condition that prevails in screening when the particles of a given size have almost completed the process of segregation be- neath the remaining oversize particles.
It is possible, therefore, that the "non-first order" behaviour often found with the screening of material with a wide size distribution could be the result of the interaction of two first-order processes as described above.
DEVELOPMENT OF THE RATE MODEL
Let MH denote the mass of (sub-aperture) size d in a cross-sectional "thin- slice" of material tha t travels along the screen, momentari ly at a distance L from the feed end; Sd be the mass of material of tha t size present in the slice that is on and in contact with the screen surface; and Pd the mass of material of this size tha t passes through the screen up to the location L of the screen.
Assuming a first-order rate process for both segregation and screen passage, using length along the screen as a surrogate for screening time, with "rate" constants k~.d and kp,d (of dimension reciprocal length), respectively, then the overall screening process becomes one with two first-order processes in series, thus:
k~,d k,,d Md , S ' Pd
Hence, the amount of material of size d, not in contact with the screen sur- face is given by:
dMd dL - -k~,a Md (1)
which, on integration, leads to:
Md = Md,o exp ( -- k~,dL) (2)
where Md,o is the initial mass of material on the screen. If the rate of passage of material of size d through the screen is given by:
dPd dL --hP'd Sd (3)
then the rate at which material of size d is accumulating against the screen surface is given by:
292 G.K.N.S. SUBASINGHE ET AL.
a s d - ~ --~- k s ,dM d - - kp,d S d ( 4 )
Substitution of eq. 2 in eq. 4 results in:
d S d ---- ks,dMd,o exp ( - ks ,dL) - kp,dS d (5 )
On integration, and using the starting conditions of Sd = 0 at L = 0, and Pd = 0 at L = 0 (Denbigh and Turner, 1971 ):
Sd Md,o-- [ (ks .d / (kp, d - - k s , d ) ] [exp ( - ks ,dL) - exp ( - kp,dL) ]
and:
Pd Fexp(hp,dL) exp(ks,dL)1 Md,o-- 1 + [ (kp,a ks,a~ (kp,d-- ks,d) l L ~ ks,d J
But, because Ya, the fraction retained on the screen, is (1 -Pd/Md,o), it follows that
A number of limiting cases are worth noting. Firstly, with uni-sized material, the segregation process can be considered to occur instantly: thus ks,a = ~ , and eq. 6 reduces to:
Yd=exp( -- kp,dL) (7)
which represents a first-order process. Similarly, when ks,d >> kp,d, eq. 6 reduces to eq. 7. However, when kp,d >> ks,d,
eq. 6 reduces to:
Yd=exp (-ks,dL) (8)
which also represents a first order process. Thus, eqs. 7 and 8 show that when one rate constant is appreciably larger than the other, the screening process reduces to a first-order process.
On the other hand, as the two rate constants become equal in magnitude (i.e., ks,d-~kp,d), it follows from a symmetric application of l'HSspital's rule to eq. 6 that Yd can be expressed by the limiting conditions:
Yd-- (l+kp,dL) e x p ( - kp,dL)= (1 +ks,dL) e x p ( - ks,dL) (9)
MODELLING SCREENING AS A CONJUGATE RATE PROCESS 293
When the difference between ks,d and kp, d is very small:
e x p [ - (ks,d--kp,d)L] = 1 - (ks,d--kp,d)L (11)
and substitution of eq. 11 in eq. 10 yields eq. 9. However, from this second derivation it follows tha t eq. 9 is valid when (ks,d-- kp,d)L << 1, tha t is, not only when ks.d ~ kp,d, but also when L is small.
DISCUSSION
To evaluate the above analysis, the experimental data (Keen, 1985 ) in Table I was used. In this experimental work, undersize from a vibrating screen was collected in a sequence of trays (beneath the screen surface), then sieved into size fractions. The screen was 803 mm long, 130 mm wide, inclined at an angle of 20 °, and had 2.2 mm apertures. For the data in Table I, the feed rate of the crushed calcite was 7.68 g s - 1 c m - 1.
The difficulty in reliably evaluating parameters in multi-component rate equations such as eq. 6, using linear algorithms, is known (Lanczos, 1957). However, the problem with three negative exponentials given in this reference was re-analysed using the non-linear regression procedure NLIN in the SAS statistics package (SAS Institute, 1985). This procedure employs an algorithm proposed by Marquadt tha t involves regressing the residuals on the partial derivatives of the model with respect to the parameters, until the iterations converge. The method was successful in separating the three exponentials when the number of parameters to be evaluated was four or less, and tended to ov- ershoot the solution when all six parameters were used.
Thus, the non-linear method was considered preferable to linear algorithms
TABLE I
Experimental data: fraction passing as function of d/a and length
for evaluating the two parameters of eq. 6 from the experimental data. Al- though the procedure met the required convergence criteria, the resulting re- sidual sum of squares (and thus the values of ks,d and kp,d) were found to de- pend on the starting estimates of the two rate constants. Therefore, the procedure was repeated with different starting estimates of the two rate con- stants covering a grid of values ranging from 0.0001 to 1000. The values ob- tained for one value of d/a are tabulated in Table II. It can be seen that al- though there can be a large variation in the estimated values of kp,d and ks,d, there is a region where the values are relatively constant, and the sum of squares of residuals are at a minimum. The values of kp,d and ks,d where the residual sum of squares was a minimum were chosen as the best estimates of the param- eters, and the results are presented in Table III. (This procedure is justified, in that the search procedure in NLIN determines convergence by minimisation of the residual sum of squares, and the SAS Applications Manual (Freund and Littell, 1986) recommends a grid search procedure for lack of convergence. )
Table II also shows that the relative size of the starting estimates also de- termined the relative size of the evaluated parameters. However, the correct relative size can be decided from a consideration of the appropriate limits of kp.d: as d/a--. 1.0, kp.s-*0 (i.e., near mesh particles have a negligible rate of pas- sage), and as d/a -,0, kp,d---~GO (i.e., very small particles have a very high rate of passage).
Fig. 2a is a semi-log plot of k~,d and kp, d v e r s u s d/a determined on this basis. The values of kp,d show the correct trend, in tha t they increase as the particle size falls to zero, and tend to zero as the particle size approaches the aperture size. While k~,d shows less variation, it is more complex, as it is dependent on
TABLE II
Results of grid search for k8 and kp (d /a = 0.701 )
Star t ing est imate Regression result Residual sos
Fig. 2. Experimental rate constants: (a) from eq. 6, and (b) from eq. 9.
the nature of the surrounding environment, which changes along the length of the screen. It can be seen that for small and large d/a ratios, the difference between ks,d and kp,d is large. Under these conditions, the lower value k should approach the limiting first-order behaviour given by eq. 7 or 8. Table IV pre- sents values of k determined by NLIN assuming eq. 7 or 8 is valid (i.e., single first-order behaviour). It can be seen that for the largest d/a value, the values of k in Table IV tends to that of kp,d in Table III, while those for small values of d/a tend to ks,d. As would be expected, the residual sum of the squares in
Table IV gets progressively larger as the d/a values differ from the range of applicability of eqs. 7 and 8.
The results in Table III also show that for intermediate values of d/a, ks,d approaches kp,d, as would be expected if a dynamic equilibrium exists between the two processes.
To test the validity of the assumption that the two rate constants become equal, eq. 9 was fitted to the data of Table I, again using the non-linear regres- sion method NLIN. The results are tabulated in Table V, and a semi-log plot of kp,d versus d/a is shown in Fig. 2b. In the range 0.25 < d/a < 0.6, the values of k and the residual sum of squares are the same as those obtained from eq. 6, substantiating the reasoning above. However, the poorer fit (larger residual
MODELLING SCREENING AS A CONJUGATE RATE PROCESS 297
sum of squares) for eq. 9 outside this range is not unexpected, since the equa- tion is no longer a realistic approximation of eq. 6 as d/a tends to zero or one.
While eq. 9 cannot be used outside the range 0.25 < d/a < 0.6, there are two advantages in using this equation for this intermediate range. Firstly, it re- quires only one rate constant to be determined from experimental data, and thus it eliminates the convergence difficulties associated with non-linear regression techniques when trying to estimate two parameters of the same or- der, as occurs with eq. 6. Secondly, eq. 9 not only provides a convenient check on the values of kp,d and ks.d determined from eq. 6: it also substantiates the use of the minimum residual sum of squares as a criterion for selecting these values.
To demonstrate the goodness of fit of the two rate model, calculated versus observed values of the fraction of material retained at a given size are shown plotted in Fig. 3. The calculated results of Fig. 3 are based on eq. 6 with the following empirical expression fit ted to the data in Table I:
and, over the range 0.25 < d / a < 0.6, essentially predicts the same curves on Fig. 3 as eqs. 12 and 13.
It can be seen tha t the model gives a satisfactory description of the experi- mental data, even though this data shows abnormal behaviour as d/a tends to zero (a feature tha t is due to the small proportion of fines in the feed (Keen, 1985) ).
The most noticeable error occurs with the short screen length. However, recalculation of the data with all screen lengths reduced by the same amount indicates tha t much of this discrepancy can be at tr ibuted to the error in align- ing the first sample t ray with the true start of the screen surface.
The realistic evaluation of separation efficiency requires the separate eval- uation of equipment and particle characteristics (Kelly and Spottiswood, 1982 ). This approach has been widely applied to classifiers "sizing" particles, but, as yet, has not been so widely applied to screens. Much of this difficulty can be at tr ibuted to the fact tha t classifiers have performance (partit ion) curves that, when reduced (i.e., made dimensionless), are essentially independent of op- erating conditions, whereas this is not the case with screens. There are two main reasons for this difference in performance curve characteristics: one is due to the performance curve of a screen being "pinned" on the size axis by the
298 G.K.N.S. SUBASINGHE ET AL.
" o o
Q re
u .
1.0
0.5
I I I I I
L = /
1 4 6
r i
~ 0 3 m m
o I 1- ~ ~] I o d ~ 1 I I o o .5 1.o
d/a Fig. 3. C.leulated (lines) versus observed (data points) performance curves for various screen lengths, derived with k.,d and kp,d calculated from eqs. 10 and 11, and substituted in eq. 6.
aperture, and the other is due to the particles on a screen being more con- strained by their surroundings (i.e., the bed of particles). If, as is proposed above, kp.d does represent the passage process, and k~,d does represent the seg- regation process, then this model offers the possibility of separating out the effects of equipment and particle characteristics, and should provide a better understanding of screening. For example, it has already been noted above that the low proportion of fines in the feed material used in the experimental work contributed to the "fishhooking" of the performance curves. Given that the fraction retained rises at the lowest values of d/a, it can be argued that the effective ks,d is low in this region, and that the analysis has detected this. It must be appreciated, however, that a large amount of data will be necessary for a complete evaluation that is capable of distinguishing equipment from particle characteristics.
MODELLING SCREENING AS A CONJUGATE RATE PROCESS 299
CONCLUSIONS
In this work, a model has been developed to describe the screening process by treating screening as two first-order processes occurring simultaneously. The two processes are considered to be segregation of particles in the bed and rate of passage of particles through the aperture: processes that have generally been regarded to be first-order. This model contrasts with that of Ferrara and Preti (1975, 1988) which involves zero- and first-order processes occurring sequentially. However, the concept of two simultaneous first-order processes offers the advantage that it leads to the three sequential screening zones tra- ditionally considered to exist, and should allow equipment characteristics to be distinguished from particle characteristics.
From the best-fit estimates of ks,a and kv, d obtained from observed screening data it was seen that for small particles (d/a < 0.25 ) the rate of passage through the apertures is faster than the segregation of those particles from upper layers towards the screen surface. For larger particles (0.6 < d/a < 1.0) the trend was reversed. However, in the intermediate region (0.25 < d/a < 0.6), the rate of segregation approaches that of passage through the apertures, implying a dy- namic equilibrium between the two processes.
For very small and near mesh particles the two rate constants differ appre- ciably, and the screening process approaches, in the limiting cases, a single first-order process, described by eq. 7 or 8: a fact observed by many workers in the past. Significantly, in the intermediate region the behaviour differs from that of a first-order process, but complies with that depicted by eq. 9. Thus, considering the screening process to be comprised of two processes complies with the observed behaviour for all (undersize) particle sizes.
Eq. 6 can be used to describe screening results provided the rate constants ks.d and kp,d are evaluated by experimentation. Alternative single-parameter (kn,d) models can be used for extreme and intermediate d/a ratios, and these can also be valuable for obtaining estimates for the two-parameter model, thus avoiding the need for the grid search sometimes required with the non-linear regression method.
A P P E N D I X - - LIST OF SYMBOLS
a aperture size d particle size ( taken as the geometric mean of the sieve apertures) kn,a rate cons tan t for general first order process, of particles of size d kp,a rate cons tan t for the passage of particles of size d through the screen apertures k~,a rate cons tan t for the segregation, above the screen surface, of particles of size d L screen length Ma mass of material of a given size d present in the " t h i n slice" at a posi t ion L along the
screen Md,o Ma at feed poin t
300 G.K.N.S. SUBASINGHE ET AL.
Sd mass of material of a given size d present in the "thin slice" and in contact with the screen surface at a position L along the screen
Pd mass of material of a given size d that has passed through the screen up to the position L along the screen
Yd fraction of material of given size d retained on the screen at a position L along the screen
ACKNOWLEDGEMENTS
The authors wish to express their appreciation to the New Zealand Ministry of Energy and the Auckland University Research Committee for their support of this work.
REFERENCES
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Kelly, E.G. and Spottiswood, D.J., 1982. Introduction to Mineral Processing. Wiley Interscience, New York, N.Y., 474 pp.
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