Estimation of breakage parameters in grinding operations using a direct search method

International Journal o[ Mineral Processing, 23 (1988) 137-150

137

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Estimation of B r e a k a g e Parameters in Grinding Operations Using a Direct Search Method V.R. K O K A

and O. T R A S S

Department o[ Chemical Engineering and Applied Chemistry, University of Toronto, Ont. M5S 1A4 (Canada) (Received February 19, 1986; accepted after revision April 28, 1987)

ABSTRACT

Koka, V.R. and Trass, 0., 1988. Estimation of breakage parameters in grinding operations using a direct search method. Int.J. Miner. Process.,23: 137-150. The breakage parameters, S and B, of individual particle size intervals can be obtained easily from size distributions at different grind times by using direct search optimization. The parameters for both batch and continuous grinding operations are considered, and the effectiveness of the search method is demonstrated. The direct search optimization method of Luus and Jaakola (1973) is easy to program, and exhibits good convergence of the optimal values even if the initial estimates are far from the optimal values.

INTRODUCTION

Since the introduction of population balance models (Reid, 1965; Kelsall and Reid, 1965; Austin, 1971; Mika, 1975) to describe material breakage in mills, considerable progress has been made in milling circuit design, operation and control. These models are based on size-mass balances on narrow size intervals of the particulate mass subjected to breakage in the mill, and are formulated in terms of two parameters: the selection function S and the cumulative breakage function B, which are defined below. The breakage of a x/~ size interval, i, of material in various mills (Austin et al., 1976) is found to follow a first-order rate law, where the breakage rate constant, Si, is called the selection function. On breakage, particles of size j, produce a set of fragments represented by the cumulative breakage function, Bij, i=j+ 1..... n, where n is the smallest size interval (sink interval). Bij is the weight fraction of the material produced by primary breakage of size j material, that reports below size i. Hence, each size interval has a breakage function, and very often it is found to be normalizable, i.e., Bij = Bh,~when i - j - - k - I. 0301-7516/88/$03.50

© 1988 Elsevier Science Publishers B.V.

138 When normalizable, it is sufficient if the breakage function of the top size interval of the particulate mass in the mill is known. Relationships of the above functions with mill operating conditions and mill dimensions are useful for scale-up, circuit design, operation and control. Normally, the breakage parameters for individual size intervals are determined separately by batch grinding tests (Austin et al., 1981a,b,c). These tests are tedious and time-consuming, requiring feeds which have to be accurately prepared. To overcome this problem, a number of workers (Klimpel and Austin, 1970, 1977; Gardner and Sukanjnatjee, 1972; Fruhwein and Schonert, 1975; Gupta et al., 1981; Rajamani and Herbst, 1983) have found an easier way to determine the parameters by calculating them using the size-mass balance models which apply to the breakage process, and experimental size distributions obtained by grinding a feed of known size distribution for several grind times. Most of them have used empirical functional forms for the relationship between the breakage parameters and particle size due to reasons stated elsewhere (Klimpel and Austin, 1977). The equations in the population balance models are nonlinear, hence, nonlinear optimization techniques are required for the estimation of the parameters. Klimpel and Austin (1970, 1977) have used the conjugate gradient technique of Goldfarb and Lapidus (1968), Gupta et al. (1981) used Powell's method (1965), and Rajamani and Herbst (1983) used a modified Gauss Newton method, Bard algorithm (Bard, 1970). The above methods require some amount of knowledge of optimization theory and programming skill. In addition, mathematical operations, such as partial derivatives of the objective function with respect to the parameters, linearization of the model equations, etc., are required for applying these methods. It would be useful if the breakage parameters could be estimated by a simpler technique that can be programmed with minimum effort, and yet possess a reasonable speed of convergence. This paper presents the application of an optimization method proposed by Luus and Jaakola (1973) which involves direct, random search and systematic reduction of the size of the search region. The authors (Koka and Trass, 1985a,b) have earlier used LJ optimization for simple parameter estimation problems in the area of size reduction. LJ optimization is well suited for a number of nonlinear programming problems (Luus and Jaakola, 1973; Luus, 1974a,b, 1980) and has a high reliability in obtaining the global optimum in nonunimodal systems (Wang and Luus, 1978). PROBLEM STATEMENT In the present study we shall be considering a batch grinding and a continuous grinding operation.

139

Batch grinding Mass balance over a size interval, i, of material subjected to batch grinding in a mill, assuming the breakage parameters to be time-invariant gives the familiar linear batch grinding equation (Reid, 1965): i--1

dwi_ - S i w i ( t ) + ~, ( B i - l j - B i j ) S j w j ( t ) dt 1=1

(1)

n>i>j>l where wi(t) and wi(t) are the weight fractions of size i and size j material, respectively. The solution of the above equation gives the size distribution of the material in the mill at various grind times, as shown below:

w(t) =DBw(0)

(2)

where w (t) and w ( 0 ) are the size distribution vectors of length n, and the elements wi, i = 1, 2, ..., n, of these vectors are the weight fractions of individual size intervals. DB is the batch grinding transfer matrix whose elements are given by:

I O

i
e --Sit

ds ( i j ) =

i-1

Ci,kCj,k (e -s~t - e -s")

(3)

i>j

k=j

where: j--1

- ~ Ci,k Cj,k

J
k=i

Cij =

1

i =j

(4 )

i--1

1 -

S)

~,Sk(Bi-l.h--Bi.k)Chj =i

i>j

If 6U(tl), ~(t2) ..... ~(tr) represent the experimentally obtained size distribution vectors at time tl, te ..... tr, respectively, then the problem is to estimate the breakage parameters, which will minimize the quadratic objective function: j=l

i=1

Multi-pass grinding Consider a multi-pass grinding operation where the steady-state product from the first pass is collected and passed through the mill. The steady-state,

140 second-pass product is collected and passed again through the mill, and so on for m passes. If the operating conditions are maintained constant throughout the operation, and assuming the residence time distribution of the material in the mill can be represented by the tanks-in-series model and is independent of the material size distribution in the mill, we have (Koka and Trass, 1985b) :

p(m) =D~ f

(6)

w h e r e p ( m ) and fare the size distribution vectors of the product after m passes and the initial feed, respectively. De is the steady-state, open-circuit transfer matrix, whose elements are given by: 0 (1 + S~t/N) - g ~-1 Ci,hCi.h[(l+SkE-/N ) --N_ (1+Sit'/N)-N]

de(i j ) =

i
(7)

i>j

k=j

here N and t-are the number of tanks in series and the mean residence time, respectively. The C values in the above equation are given by eq. 4. Let (1), ~ (2), ..., ~ ( m ) represent the experimentally obtained size distribution vectors after 1, 2 ..... m passes, respectively. The problem now is to estimate the values of the parameters which would minimize the quadratic objective function:

j=l

i=1

REDUCTION OF THE PARAMETERS If the selection and breakage functions for each size interval of the material in the mill are to be estimated, it would be a difficult task due to the large number of parameters involved. Fortunately, the S and B values for different size intervals are found to be related to particle size in various grinding mills for different materials (Austin et al., 1976, 1981a,b,c ), by the following empirical functional forms (Austin and Luckie, 1972):

Si.~.A(xi/xl) c~

Bij =O(xi/xj) ~ + (1-0) ( x ~ / x f

(9)

(10)

where xi is the mean size of size interval i. There have been instances when the breakage functions for the size intervals are not normalizable (Austin et al., 1976, 1981a; Jindal and Austin, 1976); finer breaking sizes tend to produce a greater fraction of reduced finer sizes. In such cases the non-normalized B values have been represented by the following form:

Bij =Oi(xi/xj) ~'+ (1--~)i)(xJxi) p

(11)

141 where:

~j =¢1 (x~/x~) - ~ It should be realized that other functional forms can be easily used for the breakage parameters, e.g., Gupta et al. (1981) have used polynomials of different orders for S and a modified form of eq. 11 involving a few additional parameters for B. If the breakage parameters are related to particle size by the above relationships the number of parameters to be estimated is reduced to five with the normalized breakage function and to six with non-normalized breakage function. I2 OPTIMIZATION

The following is a brief description of LJ optimization. Luus and Jaakola (1973) give a detailed methodology and description of the technique; a listing of the computer program in F O R T R A N is given by Jaakola and Luus (1974). Consider the problem of determining UI, [/2,...,Un which willminimize the objective function: P = f ( 81, U2 .... , U,,)

(12)

subject to the constraints:

gi (UI, U2,...,Un) <0

i = 1 , 2, .... m

(13)

hj (UI, U2 .... , U n ) > 0

j = l , 2,...,q

(14)

Let [ U °, U °, ..., U ° ] and [ r °, r °, ..., r ° ] be the initial estimates and the regions within which the optimum values of the variables are expected to be. A number of sets of random values, say 100, for [ U1, [/2, ...,/fin ] are taken about the initial point within the regions and tested with the constraints. With the sets which satisfy the constraints, the function P is calculated and the set [ U~, U~2..... U~ ] which gives the minimum value P* is retained. Next, the regions of search are decreased by, say, 5%, i.e., r=0.95 r °. Again 100 sets of random values of Ui are taken around the point [ U~, U~2,..., U~] within the new regions. The objective function is calculated with the sets which satisfy the constraints, and the set which gives the lowest value of P is picked out. If the value of P is less than P*, then it becomes the new P* and the corresponding set of values becomes [ U~, U~2,..., U~ ]. Thus, a number of iterations, say 100, is carried out and the final set [ U~, ..., U'n] gives the best values of the variables. For each of the variables the initial region of search should be chosen according to one's understanding of the problem. For example, if Ui lies between 0 and 10, it would be inappropriate to take an initial region < 5.0, unless the initial estimate is very close to the true value. Also, it should be realized that

142

by a region reduction rate of 5% after each iteration, the region of search is reduced to 5.9.10 -3 of its initial value after 100 iterations. Since the values of the variables are chosen around the best point determined in the previous iteration, there is a good likelihood of convergence to the optimum. OPTIMIZATION PROCEDURE

In order to apply the optimization technique to estimate the breakage parameters, the following are required: random numbers, constraints, initial estimates and regions of search, region reduction rate, and number of iterations. The standard modulus method was used to generate 2000 pseudo-random numbers in the range - 0 . 5 to 0.5. With these numbers and the regions of search, random values of the parameters were taken about their initial estimates. At each iteration 100 sets of random numbers were used. The constraints in the present study are essentially the limits within which the parameters are expected to be. These can be obtained from literature and one's understanding of the problem. For example, in ball milling the value of 0 is always found to be between 0 and 1.0. The initial estimate of a parameter should be chosen to be within the constraints, and if an experimentally determined value of the parameter for the same material obtained under very similar operating conditions is available, it should be used. However, when good initial estimates are not available, then the center of the region between the constraints for the parameter can be taken as an initial estimate. The initial regions of search will depend on the initial estimates. If the initial estimates are good and close to the true values, then the initial regions of search can be small. For example, if 0.6 is the initial estimate for ¢) whose true value is 0.5, then an initial region of search of 0.3 is more than sufficient. If a good initial estimate for a parameter is not available, it is advisable to take the region of search to be 1 to 1.3 times the region between the constraints for the parameter. The region reduction rate is an important factor. If the rate is too high (10-20%) then the chance of not converging to the global optimum is high. Experience has shown that a rate of 5% or less gives good results in most of the problems encountered. In the present study a rate of 3% was used in most of the examples. The number of iterations to be used depends on the desired accuracy of the final estimates of the parameters. Hence, it is a function of the initial regions of search and the region reduction rate. 150 iterations were used in most of the examples in this study. EXAMPLES

Batch grinding with normalized B In this case the functional forms for the breakage parameters are eqs. 9 and 10. With known values of the parameters, the selection and breakage functions

143

TABLE I Estimated parameters from simulated batch grindingdata (normalized breakage function) Parameters

True values

Estimated values

Constraints

Initial estimates

Initial regions

A

0.30 0.75 0.68 0.18 3.00

0.300 0.746 0.692 0.188 3.284

0 < A < 1.5 0 < ~ < 1.5 0 < ¢~< 1.0 0 < v < 1.5 2.0
0.75 0.75 0.50 0.75 3.50

1.5 1.5 1.0 1.5 3.0

¢~ v fl F B = 0 . 1 1 2 . 1 0 -4

for the size intervals were calculated, and using these values and eqs. 2-4, the size distributions were simulated for three grind times (3, 5, and 8 rain). The initial feed was assumed to consist of 40% in the top size interval, 55% in the second, and 5% in the third; the total number of size intervals in the material being ten. The size distribution data were truncated to two decimal places to get more realistic values. For example, a weight percent of 5.9567 was taken as 5.95. Applying LJ optimization on the simulated distributions, the five parameters were estimated (see Table I). Comparison of these values with the true values shows that they are similar. When the simulated data were not truncated the estimated values were still better. The CPU time on an IBM 3033 was 22 s. It should be noted that the constraints for the parameter values are wide. If we consider narrow constraints about the true values, the number of iterations can be reduced to 70 and the region reduction rate increased to 5% after each iteration. In Table II the estimated values using narrower conT A B L E II Estimated parameters using narrower constraints,from simulated batch grinding data (normalized breakage function) Parameters

True values

Estimated values

Constraints

Initial estimates

Initial regions

A

0.30 0.75 0.68 0.18 3.00

0.299 0.734 0.684 0.184 3.140

0.00 < A < 0.75 0.40 < ~ < 1.10 0.45 < ¢ < 0.95 0.00 < v < 0.70 2.25
0.30 0.75 0.70 0.35 3.00

0.75 0.75 0.50 0.75 1.50

v fl FB =0.359.10 -5

144 TABLE III Effect of the initial estimates on the convergence of the optimization procedure Parameters True 1 2 3 4 values Initial Estimated Initial Estimated Initial Estimated Initial Estimated estimates values estimates values estimates values estimates values A a p fl

0.30 0.75 0.68 0.18 3.00

0.75 0.75 0.50 0.75 3.50

0.300 0.746 0.692 0.188 3.284

Fs=0.112-10 -'~

1.00 1.00 0.30 0.75 3.30

0.300 0.720 0.690 0.190 3.338

FB=0.141"10

0.50 0.50 0.85 0.50 2.50

0.300 0.759 0.674 0.173 2.850

Fs=0.353-10 -~

0.30 0.90 0.30 0.80 3.00

0.300 0.749 0.681 0.180 3.063

FB=0.942" 10 -~

straints and a lower number of iterations are shown. The values are slightly better than the estimated values in Table I. The CPU time was only 10 s. Next, to find the effect of the starting estimates on the convergence to the true values, four different sets of initial estimates for the same problem were used, without changing the constraints and the initial regions of search which are given in Table I. Table III which gives the estimated values, shows that the convergence is very good for all different sets of initial estimates. To test the convergence of the gradient method ( Goldfarb and Lapidus, 1968) for the above example the same sets of initial estimates were used and the parameters were estimated with the maximum number of iterations being 10. The convergence was found to be equally good and the CPU time was about 17.5 s.

Batch grinding with non-normalized B In this example the functional forms for the selection and breakage functions of the various size intervals are given by eqs. 9 and 11. With parameter values shown in Table IV, size distributions were simulated at different grind times (1, 3, and 6 min) for (i) a feed used in the previous example, and (ii) a feed with a size distribution resulting from a previous breakage process, i.e., consisting of a wide distribution of material in different size intervals. The constraints, and initial estimates and regions of search are summarized in Table IV. Applying LJ optimization, the six parameters were estimated and are shown in Table IV. Although the FB values are low, the values of ~1 and 5 are not very close to the true values. This is due to the large regions between the constraints that were used. Hence, using the estimated values in Table IV as initial estimates, reducing the size of the regions between the constraints, and consequently the initial regions of search, the parameters were again estimated (see

145 TABLE IV Estimated parameters from simulated batch grinding data (non-normalized breakage function) ; first stage Parameters

A ~, u fl 5

True values

0.850 0.810 0.1245 0.682 3.530 0.390

FB

Estimated values coarse feed

wide feed distribution

0.849 0.813 0.104 0.645 3.400 0.531

0.853 0.817 0.154 0.643 3.590 0.225

0.709" 10-5

0.635"10-8

Constraints

Initial estimates

Initial regions

0 < A < 1.5 0<~<1.5 0<¢<1.0 0 < u < 1.5 2
0.75 0.75 0.50 0.75 3.50 0.35

2.0 2.0 1.5 2.0 5.0 1.0

T a b l e V ). T h e FB values have b e e n r e d u c e d f u r t h e r a n d t h e values o f O~ a n d 5 are c o m p a r a t i v e l y better. N e x t , t h e o p t i m i z a t i o n t e c h n i q u e was t e s t e d w i t h e x p e r i m e n t a l l y o b t a i n e d size d i s t r i b u t i o n s of A u s t i n et al. (1984) o n b a t c h g r i n d i n g o f c e m e n t clinker in a ball mill. W i t h a feed c o n s i s t i n g 99% of 40 × 50 m e s h material, size distrib u t i o n s were d e t e r m i n e d at 1, 3, 4.5, 7 a n d 10 rain o f grinding. I n addition, t h e r e was sufficient e x p e r i m e n t a l evidence t h a t t h e B values for t h e size intervals c o u l d n o t be n o r m a l i z e d . T h e p a r a m e t e r s were e s t i m a t e d f r o m t h e experi m e n t a l size d i s t r i b u t i o n s u s i n g t h e c o n j u g a t e g r a d i e n t t e c h n i q u e ( K l i m p e l a n d Austin, 1977 ) a n d are s h o w n in T a b l e VI. U s i n g this data, t h e p a r a m e t e r TABLE V

Estimated parameters from simulated batch grinding data (non-normalized breakage function); second stage Parameters

A ¢1 fl 5 FB

True values

0.850 0.810 0.1245 0.682 3.530 0.390

Estimated values coarse feed

wide feed distribution

0.850 0.811 0.117 0.671 3.480 0.447

0.850 0.811 0.140 0.659 3.580 0.300

0.945"10 -6

0.171"10-6

Constraints

Initial regions

0.6 < A < 1.0 0.6< ~ < 1.0 0.0 < ~ < 0.40 0.4 < v < 0.8 3.3
0.52 0.52 0.52 0.52 1.30 0.65

146 TABLE VI Estimated parameters from experimental batch grinding data of Austin et al. (1984) Parameters

A (~ 0~ fl /i

Experi- Austin mental et al. values values

0.29 0.91 0.60 0.90 4.01 0.20

0.29 0.89 0.61 0.90 3.99 0,18

F~

Estimated values

Constraints

Initial regions

Initial estimates

first stage

second stage

first

second

first

second

first

second

0,290 0,860 0.588 0.864 3,880 0,116

0,290 0.910 0.610 0.896 3,970 0.208

0-1.5 0-1.5 0-1.0 0-1.5 2.0-5.0 0-0.6

0.10-0.5 0.60-1.0 0.40-0.8 0.65-1.05 3.60-4.6 0.00-0.6

1.95 1.95 1.30 1.95 4.00 0.78

0.52 0.52 0.52 0.52 1.30 0.78

0.75 0.75 0.50 0.75 3.50 0.30

0.290 0.860 0.588 0,864 3.88 0.116

0.621.10 -~ 0.542.10 -4

values were estimated using two stages of LJ optimization as discussed before. The results are given in Table VI, and it can be observed that the estimated values are very close to the estimated values of Austin et al.

Multi-pass grinding Recently, Koka and Trass (1985b) determined the breakage parameters of individual size intervals of dry Pittsburgh coal in a laboratory Sze~o mill. The relationships with particle size for the selection function and the breakage functions which were found to be normalizable, could be well represented by eqs. 9 and 10. The residence time distributions obtained by a stimulus-response technique showed that material flow in the mill could be characterized by the tanks-in-series model. Since the assumptions involved in using the steady-state, multi-pass grinding model (eq. 6 ) were found to be satisfied, the above parameters and residence time distributions were introduced in the model and the size distributions after various numbers of passes through the mill were predicted. The predicted distributions (solid lines) are compared with experimental distributions at different mill rotational speeds in Fig. 1. The summed square difference between the predicted and experimental size distributions is given by the Frn
147

%

%

% ©.

960 RPM ~ / , / / .

/ ##'1~ ~

"e

FEED

e~

/i •~

e

FEED

/o°

u~ ~.

/

/

•

e

•

FMlexp) = 0 . 3 0 1 x 10 - 2 FM

%

~ o

= 0 . 1 9 1 x 10 - 2 2

~

= 0.582x10 -2

FM

~ ~ ~&~&~

•

FM(exp)= 0 . 8 6 3 x 1 0 - 2

~ o

~

PARTICLE 5 I Z E

~

FM(exp )= 0.864xi0-2 FM

= 0° 6 1 2 x 1 0 - 2

/

~ ~~

IN MICRON5

Fig. I. Comparison of sizedistributionsobtained (left)experimentally (• ), (middle) by corn), and (right) by computing with estimated puting with experimental parameter values ( parameter values ( - - - ). %

%

%

500 RPM

'-'~

790 RPM

31 G/s

~

31 G/S

,-, 960 RPM :r-

=,'4 (04

o~ //

./

i N

i , ,~

st

FI"I

~?

! o

ii

b- ~. b=" r~. % 02

~

~

~, ~ & ~

PABTICLE SlZE IN MICBONS Fig. 2. Comparison of experimental ( - [ 3 - ) and estimated ( - - - ) selection functions for dry Pittsburgh coal in the Szego mill, at three rotational speeds.

present problem it can not be concluded that the estimated selection and breakage functions are better than the experimentally determined functions because there was some evidence of briquetting of very fine particles forming large agglomerates during the breakage process ( K o k a and Trass, 1985b). Bri-

148

2-

%

%

%

5°0

!l

o~. co )'9

960

•

j/

RPH

z//t

'o

~

©-

>

~n-

e-c2:

~"

¢o

©'t ~4

~4 ='4

m-

2 % 0-2'

3o 0-2'

0~

,

, ,,i,,~0-1

,

,

, ,,t~,

0o

RELATIVE SIZE X i / Xj

Fig. 3. Comparisonof experimental ( ) and estimated ( - - - ) normalizedbreakage functions of dry Pittsburghcoal in the Szego mill, at three rotationalspeeds. quetting was found to be negligible at low mill rotational speeds. The multipass grinding model that has been used considers only material breakage, and hence, does not completely apply to the breakage of coal in the Szego mill. In order to take into account the briquetting process in the model, more information is required which is presently not available. The above example illustrates the care necessary when estimating breakage parameters from experimental size distributions. First, the breakage mechanisms at various operating conditions should be identified, verifying if there are any material effects and grinding environment effects present, and the appropriate model should be developed. Then, parameter estimation can be carried out, which will give parameter values which normally tend to provide a better fit to the experimental size distributions. SUMMARYAND CONCLUSIONS Application of the direct search LJ optimization technique to estimate breakage parameters from experimental size distributions at different grind times, has been demonstrated in several examples. When compared to other optimization methods, it is very simple to understand and to program the problem, thereby making it attractive for parameter estimation problems in size reduction. The technique was first tested with batch g r i ~ n g data simulatedwith known

149

parameter values considering both normalized and non-normalized breakage functions. The convergence to the true values was very good in spite of the wide regions between the constraints and consequently large initial regions of search used for the parameter values. Different sets of initial estimates had minimal effect on the convergence to the true values. The conjugate gradient method used by Klimpel and Austin {1970, 1977) showed equally good con, vergence when tested with different sets of initial estimates, but required considerably more background and skill in understanding the method and programming the problem. Next, the LJ method was tested on experimental batch grinding data of Austin et al. (1984) and the estimated parameter values were very close to the experimental values and those estimated by the conjugate gradient technique. Then, the method was applied to estimate parameters from multi-pass grinding data obtained by grinding dry Pittsburgh coal in the Szego mill. The estimated values were to some extent different from experimentally determined selection and breakage functions, yet they gave a better fit to the experimental size distributions. The major reason for this discrepancy is the evidence of some amount of briquetting of fine particles into large agglomerates, which was not considered in the model for multi-pass grinding. ACKNOWLEDGEMENTS

The authors are grateful to Professor R. Luus for his useful suggestions and comments during this study. The authors thank Professors R.R. Klimpel and L.G. Austin for providing the computer program of their parameter estimation technique. Financial assistance from Natural Sciences and Engineering Research Council of Canada under the Strategic Grants Program, is appreciated. Computations were carried out with the facilities of the University of Toronto Computer Center.

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