Ind. Eng. Chem. Res. 1988, 27, 723-126
Literature Cited
723
Union Carbide Corporation.
Bridgman, P. W. Phys. Rev. 1914, 3(4), 273. Bridgman, P. W. A Condensed Collection of Thermodynamic Formulas; Harvard University Press: Boston, 1925. Erben, T. M. Istanbul Technical University, personal communication, 1973. Lerman, F. J. Chem. Phys. 1937,5, 792. Shaw, A. N. Phil. Trans. R. SOC.1935,2344, 299. Sandler, S . I. Chemical and Engineering Thermodynamics; Wiley: New York, 1977. Tobolsky, A. J . Chem. Phys. 1942, 10, 644. Wylie, C. R. Advanced Engineering Mathematics; McGraw-Hill: New York, 1966.
* West Virginia University.
Manuk Colakyan,*' Richard Turtonj Union Carbide Corporation Technical Center S o u t h Charleston, W e s t Virginia 25314 and W e s t Virginia University Department of Chemical Engineering Morgantown, W e s t Virginia 26506 Received f o r review March 30, 1987 Accepted November 30, 1987
Modified Back-Calculation Method To Predict Particle Size Distributions for Batch Grinding in a Ball Mill This paper describes a back-calculation procedure based on the quasi-Newton method of optimization. T h e general solution of the integro-differential equation was undertaken, and with a careful study of the literature, various forms of breakage distribution function and selection function were assumed. An error function was defined as the root mean square error between the calculated and the experimental product size distribution$. The values of the parameters a t the lowest error were calculated, using the quasi-Newton optimization technique. This method was experimentally verified for different materials and for grinding processes following different forms of selection and breakage functions. T h e calculated size distribution compared well with the experimental size distribution. I t is concluded t h a t the present method is more useful than other back-calculation methods and can be used for both normalized and nonnormalized breakage distribution functions. It has a potential in industrial application, as i t does not demand grinding data with narrow size feed materials. Grinding of solid materials is the most energy-consuming unit operation in mineral processing and cement and allied industries. As expenditure on energy is a major portion of the total processing cost of the products, it is absolutely necessary to run size reduction processes at optimum operating conditions so that the maximum size reduction can be achieved (Devaswithin et al., 1985) a t the lowest operating cost. To evaluate the optimum operating conditions, it is essential (Devaswithin et al., 1987) to predict the complete product size distribution. There are many models (Reid, 1965; Luckie and Austin, 1971; Herbst and Mika, 1970; Kapur, 1970) for the prediction of the particle size distribution for batch ball mill grinding. These models relate the product size distribution with breakage and selection functions. There are many methods available in the literature (Reid and Stewart, 1970; Austin and Luckie, 1971/1972; Herbst and Fuerstenau, 1968; Kapur, 1982; Gupta et al., 1981) for the experimental determination of the breakage distribution and selection functions. Most of these methods demand elaborate experimentation. In this paper, an optimization technique based on the quasi-Newton algorithm is presented.
Equations for the Prediction of Particle Size Distribution For a finite size interval, the batch grinding equation in continuous time form is conveniently expressed as dWi(t)/dt = -SiWi(t) +
i-1
SjbijWj(t)
j=l,i>l
(1)
where Wi(t)is the weight fraction of the total material present in the size interval i at any time t;Si is the selection for breakage or specific rate of breakage of the material in the ith size interval with units of inverse time; and bij is the breakage distribution of particles in the size interval 0888-5885/88/2627-0723$01.5Q JQ
j broken into smaller fragments falling in the size interval i, Le., b, = B, - Bi+ljwhere B, is the cumulative breakage distribution function. For the first size interval, the solution of the batch grinding eq 1 is given (Reid, 1965) as
Wl(t) = Wl(0)e-slt
(2)
The second size interval is given as
and the ith size interval is given as
where
It can be seen that the solution for all the n size intervals will contain n selection function constants and n(n - 1)/2 breakage function constants. These constants are evaluated by a back-calculation method.
Experimental Section Batch grinding experiments were conducted with a known feed size distribution in a laboratory ball mill that was 0.2 m i.d. and 0.3 m long with a provision for changing the speed of revolution of the mill. The product size distributions are measured a t different times of grinding 0 1988 American Chemical Society
724 Ind. Eng. Chem. Res., Vol. 27, No. 4,1988 Table I. Variables and Their Ranges Studied no. variables range feed size of material, mm less than 4.0 mill speed 0.55-0.81 0.21-0.52 ball loading particle loading 0.61-1.21 time, min 0.5-3.2 0.045-4.0 sieve sizes, mm ball size, cm 0.95-4.4 0.1-1.1 variance of ball size, cm2 Table 11. Comparison of the Estimated Constants with the Data data from Austin et al. (1976) est. using direct optimization data constants est. technique assumed calcd ri
4 P I
K a
0.36 0.36 1.17 4.00 0.46 1.04
0.30 0.30 1.01 3.20 0.43 1.04
0.0 0.5 1.0 6.0
0.5 0.5
0.03 0.47 0.97 5.90 0.50 0.48
Method of Estimation of Breakage a n d Selection Functions As n(n + 1112 constants are involved in the prediction of the product size distribution, certain functional forms for the breakage distribution and selection functions are assumed. It can be seen from the direct estimation studies (Austin et al., 1976) that eq 5 and 6 can represent the breakage distribution and selection functions of various materials. = J/(xj/xJ'(xj/xi)'
+ (1 - J / ) ( x j / x 1 ) ' ( x j / x i I v
sj = K(Xj/Xl)0l
(5)
(6)
Assuming certain initial values for the constants I/, 6, p, v, k, and ct involved in eq 5 and 6, the entire breakage distribution function and selection function constants are calculated. With these constants, the product size distributions are predicted by using eq 4. The error in the prediction is estimated (Devaswithin et al. 1987) by eq 7. n
E = [C(Mip - M i E ) 2 / n ] ' / 2 i=l
20
30 LO
100
200
400
1000
, ( p m ) __c Figure 1. Comparison of the simulated data with the computed product size distributions. Particle s i z e
Table 111. Range of Comminution Parameters no. parameter range 1 K , min-' 0.0-1.5
by sieve analysis. Calcite was used for the study, and the ranges of experimental variables studied are given in Table I.
Bij
10
(7)
The calculated error is minimized by choosing suitable values for the constants J/,6, P, v, k,and a. A quasi-Newton algorithm (Gill and Murray, 1972) is used for the nonlinear optimization. The optimization technique requires the differentiated values of the error function with respect to the constants. This is obtained by numerical techniques. Initial values of the breakage distribution function and selection function constants are taken as IC, = 0.5, 6 = 0.0; B = 1.0; v = 5.0; k = 0.5, and a = 1.0. Results a n d Discussion A known feed size distribution has been taken as in Figure 1. By use of the comminution parameters given in Table 11, the product size distributions for different times of grinding were calculated accurately and are given in Figure 1. These feed and product size distributions were taken as data, and the back-calculation method was applied to calculate the comminution parameters. The
2 3 4 5 6
a
4 P Y
ri
0.0-1.5 0.0-1.0 0.0-2.0 3.5-6.0 0.0-0.5
back-calculated parameters thus calculated are given in Table I1 for comparison. Table I11 gives the ranges of values of the comminution parameters observed for different materials using the direct determination techniques (Austin et al., 1976). The calculations of the comminution parameters were repeated with different initial values of the comminution parameters taken in the ranges given in Table 111. The same values for the parameters given in Table I1 were obtained for all the repetitions. The calculated product size distributions using the thus obtained parameters are given in Figure 1. It can be seen from Figure 1 that the calculated comminution parameters and the calculated product size distributions tally with that of the experimental size distribution. This proves that the error function defined by eq 7 has a unique minimum which leads to the accurate estimation of the comminution parameters. The experimental product size distribution for silicon carbide of known feed size distribution in a ball mill of 19.5-cm diameter with balls of size 2.51 cm with a mill speed of 0.85 of the critical speed has been taken from Austin et al. (1976). The iterative procedure was used to back-calculate the breakage distribution and selection function constants. The constants estimated by Austin et al. (1976) by direct determination techniques and the constants estimated with the present optimization technique are compared in Table 11. Pilot plant data were also taken from Herbst et al. (1981) for testing the above method. They have given the product size distribution for batch dry grinding of limestone in a 76-cm-diameter ball mill. The back-calculation procedure using the optimization technique was adopted to calculate the breakage distribution and selection function constants of eq 5 and 6. With these values, by use of eq 5 and 6, the values of Bij and S . for all the sizes are calculated. By use of eq 4,the compiete product size distribution was predicted. The comparison of the experimental and the predicted product size distribution is shown in Figure 2. Batch grinding experiments have been conducted with calcite for different operating conditions as given in Table
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 725 10
[ 1:
-2
Material : Lime stone
0
07
L
Z 06 P
3 05
y.
E
2
04
03 ?
+
;0 2 f
u 01 0 001
002
004
01
02
Particle size ( m m )
04
10
16
Figure 2. Comparison of predicted product size distribution with experimental data (Herbst et al., 1981).
01 0
005
01
02
05
12
20
L
3
Particle size,( mm)J
Figure 4. Comparison of predicted and experimental size distributions.
:045
cessfully. The tests carried out using the simulated particle size distribution data showed that the comminution parameters calculated by using this method are independent of the initial conditions chosen. I t is found that the method of back-calculation predicts the particle size distribution of the ground products in pilot plant and laboratory mills accurately. The advantages of this method of back-calculation over the other direct determination techniques are as follows. (1) Minimum experiments are required to be done for the estimation of breakage and selection function parameters. (2) The preparation of single narrow size fraction feed is not necessary. (3) Data for short grinding times are not required.
-
0
_ _ _ _ r----r---’, 0
005
01
02
/
05
10
20
40
Particle size (mm-
Figure 3. Comparison of predicted and experimental size distributions.
I. For all the conditions, the constants of eq 5 and 6 were back-calculated from the experimental data by the iterative optimization technique. The entire Bjjand Sj values were evaluated from eq 5 and 6, and the product size distribution was predicted by eq 4. Figures 3 and 4 show the experimental and the predicted product size distributions for different operating conditions of the ball mill. It can be seen from Figures 2-4 that the predicted product size distributions match well with the experimental results. From Table I1 it is also seen that the calculated values of the constants are very close to the values obtained by direct determination techniques (Austin and Luckie, 1971/1972). This shows that the optimization procedure can be conveniently used for the prediction of the product size distribution in batch ball milling systems.
Conclusion A back-calculation method for the prediction of breakage and selection functions was developed and tested suc-
Nomenclature a = term defined by eq 4 B = cumulative breakage function b = differential breakage function d = diameter, cm E = error defined by eq 7 J = ball loading K = selection rate constant M = cumulative undersize weight fraction n = number of size intervals S = selection function constant, min-l t = time, min U = particle loading W = differential weight fraction x = particle size, mm Greek Symbols a = selection function parameter p, Y, 6, $ = breakage function parameters 2 = variance, cm2 4 = mill speed (fraction of critical speed) Subscripts B = balls E = experimental i, j , k = size intervals P = predicted M = ball mill
726 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988
Literature Cited Austin, L. G.; Luckie, P. T. “Methods of Determination of Breakage Distribution Parameters”. Powder Technol. 1971/1972, 5, 215-222. Austin, L. G.; Shoji, K.; Bhatia, V.; Jindal, V.; Savage, K. “Some Results on the Description of Size Reduction as a Rate Process in Various Mills”. Ind. Eng. Chem. Process Des. Dev. 1976, 15, 187-196. Devaswithin, A.; Krishnan, B.; Pitchumani, B.; Agarwal, V. K. “Prediction of Particle Degradation During Impact on a Flat Surface”. Wear 1987, 118, 281. Devaswithin, A.; Mani, B. P.; Viswanathan, K. “Increase Efficiency of Grinding Circuit Plant”. Int. Conf. on Hydraulic Tramp., Ranchi, March, 1985. Gill, P. E.; Murray, W. J. Inst. Math. Appl. 1972, 9, 91. Gupta, V. K.; Howein, D.; Berube, M. A.; Everell, M. D. “The Estimation of Rate and Breakage Distribution Parameters from Batch Grinding Data for a Complex Pyrite Ore using a BackCalculation Method”. Powder Technol. 1981,28, 97-106. Herbst, J. A.; Fuerstenau, “The Zero Order Production of Fine Sizes in Comminution and its Implications in Simulation”. Trans. SME 1968, Dec, 538-548. Herbst, J. A.; Mika T. Rudy 1970, 18 (3-4), 70-75. Herbst, J. A.; Siddique, M.; Rajamani, K.; Sanchez, E. “Population Balance Approach to Ball Mill Scale-up: Bench and Pilot Scale Investigations”. Trans. S M E AIME 1981, 272, 1945-1954. Kapur, P. C. “Kinetics of Batch Grinding-Part A: Reduction of Grinding Equation”. Trans. AIME 1970,247,299-303; “Part B.
An Approximate Solution to the Grinding Equation”. Trans. AIME 1970,247, 309-313. Kapur, P. C. “An Improved Method for Estimating the Feed-Size Breakage Distribution Functions”. Powder Technol. 1982, 33, 269-275. Luckie, P. T.; Austin, L. G. “A Review Introduction to the Solution of the Grinding Equation by Digital Computation”. Miner. Sci. Eng. 1971, 4 , 24-51. Reid, K. J. “A Solution to Batch Grinding Equation”. Chem. Eng. Sci. 1965, 20, 953-963. Reid, K. J.; Stewart, P. S. B. “An Analogue Model of Batch Grinding and its Application to the Analysis of Grinding Results”. Chemica 1970, 87-106. Indian Institute of Technology.
* Chr. Michelsen Institute.
A. Devaswithin,’ B. Pitchumani,*t S. R. de Silva* Chemical Engineering Department I n d i a n Institute of Technology N e w Delhi 110 016, India and Chr. Michelsen Institute Bergen, Norway Received f o r review June 16, 1987 Revised manuscript received November 23, 1987 Accepted December 21, 1987
