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Calibrating computer models of mineral processing equipment using genetic algorithms
Minerals Engineering, Vol. 8, No. 9, pp. 989-998, 1995
Pergamon 0892-6875(95)00062-3
Elsevier Science Ltd Printed in Great Britain 0892-6875/95 $9.50+0.00
CALIBRATING COMPUTER MODELS OF MINERAL PROCESSING EQUIPMENT USING GENETIC ALGORITHMS
C. L. KARR§ and D. YEAGERt Engineering Science and Mechanics Department, University of Alabama, Box 870278, Tuscaloosa, AL 35487-0278, USA t U.S. Bureau of Mines, Tuscaloosa Research Center, P.O. Box L, University of AL Campus, Tuscaloosa, AL 35486-9777, USA (Received 5 April 1995; accepted 26 May 1995)
ABSTRACT Researchers at the U.S. Bureau of Mines have developed an approach to tuning empirical computer models that is substantially more efficient than some traditional approaches. The mineral processing industry relies heavily on empirical computer models for both design and process control applications. Unfortunately, the empirical constants associated with the computer models have traditionally been selected using either statistical methods which are quite limited in some domains or trial-and-error procedures which are quite time consuming. A new and more efficient approach to selecting empirical constants for computer models has been developed and implemented. This approach employs a genetic algorithm for selecting the empirical constants for computer models of mineral processing equipment. Genetic algorithms are search algorithms based on the mechanics of natural genetics. They rapidly locate near optimum solutions in difficult search spaces and are shown to be effective in the search for the constants associated with empirical computer models. Bureau researchers have used these innovative search algorithms to both dramatically reduce the time needed to select empirical constants for computer models and to substantially improve the accuracy of the resulting models. The effectiveness of the approach is demonstrated with several examples from the field of mineral processing.
Keywords Modelling, mineral processing, genetic algorithm, simulation
INTRODUCTION The increased speed and decreased cost of today's personal computers have allowed researchers to focus on improving computer modelling capabilities, and to utilize the resulting models to produce more efficiently designed equipment and better process control systems. A number of industries have taken full advantage of improved technology in developing computer models of complex systems. The aerospace industry, for instance, employs a number of high level, first-principle models of aircraft in both the design of vehicles, and in the development of flight simulators [1]. The result of these improved computer models has been aircraft that are more maneuverable, safer, and less expensive. In fact, effective first-principle computer models occasionally provide engineers with insights into the physics of systems, insights that simply were unavailable prior to the development of a first-principle model.
989
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C.L. KARRand D. YEAGER
Unfortunately, all industries have not been as receptive as the aerospace industry to the development of high level, first-principle computer models. The mineral processing industry, for example, still relies heavily on the use of empirical models. For empirical models to accurately predict the response of a physical system, correct values of the model parameters are required. The models often have no physical association to the system being modeled, thereby making the calibration of the empirical constants a difficult task. Although the resulting models may not provide any insight into the physics of the systems being modelled, they still can be used just as effectively as first-principle models to improve equipment designs and process control systems. However, the procedure used to select the values of the empirical constants remains inefficient. The empirical models used in the mineral processing industry rely, for the most part, on a trial-and-error process for selecting the empirical constants that allow the computer model to accurately predict the response of a physical system. The problem of selecting appropriate empirical constants is simply a search problem. Thus, researchers at the U.S. Bureau of Mines have employed a search procedure from the field of artificial intelligence, a genetic algorithm, to locate empirical constants for computer models of mineral processing systems. Genetic algorithms are search algorithms based on the mechanics of natural genetics [2]. They combine a Darwinian survival-of-the-fittest approach with a structured information exchange procedure, producing a scheme that is quite robust. Genetic algorithms are able to locate near-optimum solutions to a wide range of difficult search problems after having viewed but a small portion of the search space [3]. Genetic algorithms have been used to solve a number of difficult search problems [4], and prove to be quite effective in tuning empirical models of mineral processing systems. In this paper, an approach to tuning empirical models using genetic algorithms is presented. The effectiveness of the approach is demonstrated with several examples from the field of mineral processing. The approach proves to both dramatically reduce the time needed to select empirical constants for computer models and to substantially improve the accuracy of the resulting models.
HYDROCYCLONESEPARATOR Hydrocyclones have traditionally been modeled using empirical relationships [5]. Plitt [6] identified a model to predict the d5o or split size that is still used extensively today. The split size is that size particle (given by diameter of the particle) that has an equal chance of exiting the hydrocyclone either through the underflow or the overflow, and is often used to quantify a separation process. Plitt's model implements the following functional relationship: ds0 = f (Dc, D~, D O, D u, h, Q, dd, p)
where D c is the diameter of the hydrocyclone, D i is the diameter of the slurry input, D o is the diameter of the overflow, D u is the diameter of the underflow, h is the height of the hydrocyclone, Q is the volumetric flow rate into the hydrocyclone, ~ is the percent solids in the slurry input, and p is the density of the solids. More specifically, the model has the form: C2
C3
C4
CxDc Di Do e x p [ C s ~ ] d5 ° =
D uc~h CTQCspC9
(1)
where the empirical constants, C i, are selected so that the model accurately matches data that has been collected from actual hydrocyclone separators. A number of approaches have been developed for determining the empirical constants associated with this particular model of a hydrocyclone. Most of the approaches involve the use of statistical routines, each of which has some potential drawbacks [7]. The following section describes the use of a genetic algorithm for determining the empirical constants for a hydrocyclone separator.
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G E N E T I C A L G O R I T H M SELECTS E M P I R I C A L CONSTANTS Genetic algorithms are broadly applicable, efficient search algorithms based on the mechanics of natural genetics. They speculate on new points in the search space with expected improved performance by exploiting historical information. Genetic algorithms demonstrate some fundamental differences with more conventional search techniques. Because of these fundamental differences they are able to work effectively in search spaces for which no derivative information is available, and they do not require that the search space be continuous [3]. Although some genetic algorithms have become quite complex, good results can be achieved with relatively simple genetic algorithms. The simple genetic algorithm used in this study consists solely of reproduction, crossover, and mutation - - operators basic to most genetic algorithms [2]. Once the genetic algorithm operators to be used have been determined, there are only two decisions to be made when applying a genetic algorithm to a particular search problem: (1) how to code the parameters of the problem as a finite string, and (2) how to evaluate the merit of each string (each parameter set). The first decision involves coding the parameter set. Genetic algorithms require the natural parameter set of the problem be coded as a finite string of characters. The parameter set in this study, the empirical constants C 1 through C 9, is coded as strings of zeros and ones. This is accomplished by representing each constant as a binary string, and evaluating the bits as a binary number. Ten bits are allotted for defining each constant (although fewer or more bits can be used). The ten bits are interpreted as a binary number (1001010111 is the binary number 599). This value is mapped linearly between some user determined minimum (Cmin) and maximum (Cmax) values according to the following:
b C 1 = Cmin + - - ( C m a
2M-1
x - Cmin)
(2)
where b is the integer value represented by an M bit string. The values of Cmin and Cmax in a given problem are selected by the user based on personal knowledge of the problem. If necessary, a rapidly converging, coarse optimization method may be used for selecting the limiting values. This same form is used to represent each of the remaining constants, C 2 through C 9, and the nine ten-bit strings are concatenated to form a single ninety bit string representing the entire parameter set. The second decision involves establishing a fitness function which indicates the merit of each potential solution to the search problem. Tuning the empirical constants associated with the hydrocycione model is actually a curve fitting problem. Thus, several criteria can be used to establish the quality of the fit. For example, the most common curve fit measure is the least squares criteria. Least squares curve fit techniques attempt to minimize the distance between the measured data points and the estimated values as defined by the empirical model. However, traditional least squares curve fitting techniques are limited by their use of partial derivatives. The fitness function used in this study is: N f = K - ~(y/ i=1
- yi) 2
(3)
where f is the fitness function value, K is a large constant defined so that the fitness function value is always non-negative, N is the number of data points, y is an actual data value, and y' is an estimated data value using the parameters defined by a particular genetic algorithm string. This fitness function is minimized by a genetic algorithm. A genetic algorithm has been used to select the values of C! through C 9 used in the empirical model of a hydrocyclone as shown in equation (1). Figure 1 demonstrates the effectiveness of using a genetic algorithm for tuning the empirical constants. In this plot, the actual d50 size is plotted against the d50 size predicted using the model with the empirical constants as selected using a genetic algorithm. Note that the model
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C.L. KARR and D. YEAGER
would exactly reproduce the data if all of the points shown in Figure 1 fell on a line of 45o. Although the genetic algorithm tuned model does not exactly predict the ds0 size, it fits the data very well. 100 90 80 70 .-$ 3D O "Di
• ,~5 "5/
60
.5/
50 40 30 20 10 0
0
10
20
30
40
50
60
70
80
90
100
d~,Data
Fig. 1 Performance of genetic algorithm tuning of hydrocyclone model. It is important to note that the genetic algorithm determined the empirical constants in roughly 10 minutes, whereas a statistical package took roughly 9 hours to locate constants that provided the same quality fit to the data. This time of computation becomes critical when the tuning of an empirical model is being done as a part of a real-time, adaptive process control system such as the one developed by Karr [8] for the minerals industry.
REE-EYRING EQUATION One of the major concerns of the mineral processing industry is the dewatering of mineral slurries. Researchers at the U.S. Bureau of Mines have developed innovative techniques for dewatering numerous mineral slurries, and a number of these techniques are based on knowledge of the non-Newtonian components of viscosity of the slurry. The non-Newtonian components of viscosity can be found by curve fitting data obtained in the laboratory. The Ree-Eyring equation allows for the estimation of the non-Newtonian components of viscosity and is of particular interest to the minerals industry in the area of dewatering phosphatic clay wastes. The Ree-Eyring equation may be written as
y=C
1
+
C2sillh-l(C3x)
(4)
C3x
where y is the viscosity (the dependent variable), x is the shear rate (the independent variable), and C l, C e, and C 3 are empirical constants to be determined. In most instances, experimental viscosity and shear rate data is readily available through laboratory tests. However, curve fitting the data is not always easy because of the highly nonlinear form of the equation. Attempts to curve fit data to the Ree-Eyring equation using traditional least squares techniques were unsuccessful. The difficulty arises in solving for the constant C 3. A symbolic math package was used in this attempt, but the package failed to produce a solution. Thus, in this problem domain, traditional statistical techniques were not useful. Therefore, the genetic algorithm approach to selecting the empirical constants was not only convenient as in the previous examples, but it was necessary.
Calibrating computermodels of mineral processing
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Fortunately, a genetic algorithm was able to effectively tune the Ree-Eyring model. The process of tuning the model using a genetic algorithm is the same as in the previous example. The parameter set (C l, C 2, and C3) is represented as a thirty-three bit string of zeroes and ones, and are evaluated using a version of the previously discussed fitness function. For this example, however, the sum of the squares of the residuals was replaced with the median residual value. This approach is known as least median squares (LMS) curve fitting, and is an effective approach to negating the appearance of outliers in the data set [7]. The ease with which a LMS method can be substituted for a least squares approach demonstrates a further advantage of using a genetic algorithm for tuning empirical models. As in the previous example, the fitness function allows for the minimization of the distance between the data points and the resulting curve. Stanley, Webb, and Scheiner [9] used a transformation method to solve the Ree-Eyring equation for a sodium ion-exchanged clay. The data used in that report were also used here. Once again the genetic algorithm quickly converged to a quality solution; the smallest error produced by the genetic algorithm calculated curve (Error = 0.008) is less than that of the curve calculated by the transformation method used by Stanley, Webb, and Scheiner (Error = 1.57). Figure 2 shows how well the curve calculated by the genetic algorithm fits the data, and also how well the LMS approach overcomes the occurrence of the three outliers (inconsistent data points). C l, C 2, and C 3 were allowed to range between 0.0 and 200.0 for the coding necessary in the genetic algorithm application.
Genetic algorithm LMS (error = 0.008)
Sheer Rote
(10-4s -1)
Fig.2 Performance of genetic algorithm tuning of Ree-Eyring model. In the previous example involving tuning a model of a hydrocyclone separator, using a genetic algorithm to do the model tuning made sense because the genetic algorithm approach took less time and was more accurate than the traditional statistical algorithms. However, in the example of the Ree-Eyring equation presented in this section, a genetic algorithm was used because the traditional techniques were simply not applicable. It is encouraging to note that extending the genetic algorithm tuning approach used is easily altered to consider various criteria for judging the accuracy of the model; switching from a least squares to a LMS criteria merely involves altering the fitness function. The successful application of a genetic algorithm to the problem of tuning a model using the Ree-Eyring equation has enabled researchers at the U.S. Bureau of Mines to improve dewatering techniques. Results obtained using the genetic algorithm LMS technique represent potential cost savings for the minerals industry. Furthermore, these results are helping Bureau researchers to develop dewatering techniques that help reduce waste imparted to the environment.
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C.L. KARR and D. YEAGER GRINDING
Grinding is a necessary component in the processing of a number of minerals, and improvements in this area could provide substantial cost savings for the minerals industry. Optimizing the process of grinding, however, is a difficult task requiring innovative techniques and accurate computer models. The U.S. Bureau of Mines is quite interested in the grinding process because it plays a role in the processing of numerous minerals, and the substantial energy requirements associated with grinding provide substantial room for economic savings. Thus, because of the high energy costs combined with the considerable popularity of the process, improvements in the efficiency of the grinding process could have a dramatic economic impact on the minerals industry. A number of innovative optimization techniques have been developed that are applicable to the improvement of the grinding process [10]. Unfortunately, the success of these optimization techniques requires an effective computer model of the grinding process. Although empirical computer models of the grinding process have been developed [11 ], the tuning of these computer models is time-consuming and often quite difficult. Thus, using a genetic algorithm for tuning the grinding models is inviting. The grinding process is characterized by several performance measures, all of which are important in various circumstances. Generally, there are four measures that are especially important indicators of the efficiency of a particular grinding process: (1)fineness of the ground product, (2) e n e r g y costs associated with the process, (3) a v i s c o s i t y c o e f f i c i e n t , and (4) a v i s c o s i t y e x p o n e n t . Mehta, Kumar, and Schultz [11] have developed an empirical model of grinding that accurately mirrors the response of a coal grinding process. In that work, they studied the ability of a simple genetic algorithm to optimize the performance of their computer model of a particular grinding circuit. In the current study, the focus is shifted from using a genetic algorithm to optimize the performance of a tuned model, to employing a genetic algorithm to actually tune the computer model of a grinding process. Only two of the four indicators of grinding efficiency are considered: fineness and energy consumption. However, the steps used to tune the models that predict fineness and energy can be used to tune the models that predict the amount of dispersant addition and the viscosity characteristics. The pertinent modelling equations are: F
= C 1 + C 2 xs -
C6 x2
C3 xB + C 4 xO + C 5 x M
C7 X2
-
+
C8
+ Clo XsB + Cl1
XsD
+ CI3
XBM
XBD
C14
-
X2 - C9 X2 C12
-
(5)
XsM
-
CI5
XDM
+
C19
xD +
and
E
= C16 + C17 -
x:
C18
xs -
xB
-
-
+ C25
xss
+
C26
XsD -
+ C28
XBD
-
C29
XB M -
C20
xM
(6)
C27 X~M
C30
Xo M
where F is the fineness (or the weight percent less than 38 millimeters), E is the energy in kilowatt hours per ton, x s is the percent solids by weight, x B is the maximum ball size, x M is mill speed, x D is dispersant addition, xij represents the product of x i and xj, and C l through C30 are the empirical constants selected so that the model equations accurately reproduce data obtained from a physical system. It is important to note that C l through Cl5 can be determined entirely independent of Cl6 through C30. Thus, there are two separate fifteen parameter search problems to be solved. The development of an efficient empirical computer model of a grinding circuit actually depends on tuning fifteen parameters for each of four separate model equations. The very same approach used in the previous
Calibrating computer models of mineral processing
995
sections of employing a genetic algorithm for tuning computer models can be used here. The coding scheme outlined earlier can easily be used, as can the fitness function definition. Thus, all that is left is to present some results. Figures 3 and 4 demonstrate the effectiveness of using a genetic algorithm for tuning the empirical constants associated with the grinding models. In these plots, the actual fineness and energy consumption as measured in the physical system are plotted against the computer predicted values. These figures demonstrate the ability of a genetic algorithm to determine empirical constants for the grinding models. 90 80 70 6O '10
=~ 5o ~ 4o C
3O 20
/
10 /*
./
,/ 10
20
30
40
50
50
70
80
90
Finess Actual
Fig.3 Performance of the genetic algorithm in tuning the fineness model.
2O
O "10 O
iii
J
0 0
5
10
15
20
25
Energy Actual Fig.4 Performance of the genetic algorithm in tuning the energy model.
COLUMN FLOTATION CIRCUIT In recent years, numerous researchers have worked to better understand the principles that govern column flotation operation. As a result of these efforts, several computer models have been produced that attempt to predict the performance of column flotation units. Some of the more noteworthy research has been performed by Finch and Doby [12], Herbst and Rajamani [13], Luttrell, Adel, and Yoon [14], and Ynchausti, Herbst, and Hales [15]. In the current effort, the Finch and Doby model has been used because it has been well documented, is relatively straightforward to program, and provides accurate results when the effects of the froth zone are not too pronounced. HE 8:9-D
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c.L. KARRand D. YEAGER
The Finch and Doby model of column flotation relies heavily on a property of hydrophobic minerals called induction time. Induction time is the time required, once a hydrophobic particle has collided with a bubble, for the particle to attach to the bubble. Once the attachment has occurred, the particle will usually remain attached to the bubble until the bubble/particle aggregate rises to the pulp/froth interface level. Also, it is in practice often a good assumption that a particle which reaches the interface level will remain attached to its bubble and be carried out with the concentrate. Unfortunately, induction time is extremely difficult to measure in any flotation equipment. However, the induction time can be treated as an empirical constant, and thus can be computed using a genetic algorithm in the manner described in previous sections. The column circuit considered in this effort is shown in Figure 5. This circuit has been used because data is available in the thesis by Alford [16]. Alford's data is taken from a circuit used in an industrial application at the Mount Isa Mines Limited installation in Australia.
Concentrate flows New Feed
Fee0 5
>
Fee
J
Interfac/ee
/ / )
Tailings 1
Tailings 2
Final Tailing
Fig.5 Schematic of the column flotation circuit. A genetic algorithm has been used to tune selected parameters of the computer model in order to make the values predicted by the model match as nearly as possible the behavior of the actual circuit as recorded by the data in Alford's thesis. Besides the induction time for each hydrophobic mineral species (in this case sphalerite and galena) other parameters used in the tuning of the circuit were the vessel dispersion number and a first-order rate constant for the water and for each non-hydrophobic mineral species. Since vessel dispersion number is largely a function of the size and shape of the column and since the physical dimensions of the three columns were identical, the three were assumed to have the same vessel dispersion number. Since induction time is a chemical property, the sphalerite and galena induction times were also assumed to be the same for each of the three columns. The parameters used and the values assigned to those parameters by the genetic algorithm can be seen in Table 1. The genetic algorithm was set up to minimize the mean squared error between the balanced data values based on observed flows in the Mount Isa Mines circuit and the corresponding values obtained from the computer model. There were thirty values used in computing the mean squared error, namely the percent solids, the percentage of lead, the percentage of zinc, the percentage of iron, and the flow volume in each of the six flows produced by the circuit. The final mean squared error attained by the genetic algorithm was 0.0003. The precise values achieved by the model are listed in Table 2 along with the balanced values based on observed data for comparison. (The percent solids column is omitted.) Each table entry is of the form Vrn/Vb, where V m is the value produced by the computer model and V b is the balanced value from Alford's data.
Calibrating computermodels of mineral processing
997
T A B L E 1 Parameters used and values obtained by a genetic algorithm for the column flotation circuit. Parameter
Value
Vessel dispersion number Nd Galena induction time Tig Sphalerite induction time Ti s Water rate constant stage 1 Water rate constant stage 2 Water rate constant stage 3 Sulfides rate constant stage 1 Sulfides rate constant stage 2 Sulfides rate constant stage 3 Gangue rate constant stage 1 Gangue rate constant stage 2 Gangue rate constant stage 3
154.88 0.04665 sec 0.04415 sec 3.08770e-04 1.08730e-04 6.23246e-05 6.35543e-05 5.23178e-05 5.21631e-05 1.60884e-05 4.01268e-05 3.99883e-06
TABLE 2 Parameters used and values obtained by the genetic algorithm. Flow Name I
% Lead
% Zinc
% Iron
Flow Volume
Stage 1 Concentrate
11.7/11.8
41.0/40.8
12.4/12.5
12517.8/12632.1
5.7/5.7
14.6/14.5
26.6/26.7
31942.1/31828.1
11.1/10.7
36.5/37.4
14.0/14.2
4597.7/4457.6
5.0/5.0
11.6/11.7
28.3/28.2
39344.5/39370.4
11.1/11.1
34.3/32.5
17.5/16.9
2684.4/2710.0
4.5/4.5
9.7/9.8
29.2/29.2
48660.1/48660.5
Stage 1 Tails Stage 2 Concentrate Stage 2 Tails Stage 3 Concentrate Stage 3 Tails
SUMMARY Empirical computer models play a key role in the mineral processing industry for tackling both design and control problems. Generally, the calibration or tuning of these models, the selection of values for the empirical constants, is a time-consuming task. Fortunately, the search capabilities of genetic algorithms can be used to help solve the problem of selecting these constants. Genetic algorithms are search algorithms based on the mechanics of natural genetics. They are able to locate near-optimal solutions after having sampled only small portions of the search space, and they are flexible enough to be effective in a wide range of problems. Their novel approach to solving search problems is fundamentally different from more conventional search techniques. They have been used in this paper to tune the empirical constants associated with several different computer models used in the minerals industry.
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C.L. KARR and D. YEAGER REFERENCES
1.
2. 3. 4. 5. 6. 7. 8.
9.
10.
11.
12. 13.
14. 15. 16.
Hoffman, K.A., Computational Fluid Dynamics for Engineers. Englewood Cliffs, N J, Prentice-Hall, (1991). Holland, J.H., Adaptation in Natural and Artificial Systems. Ann Arbor, MI, The University of Michigan Press, (1975). Goldberg, D.E., Genetic Algorithms in Optimization, Search, and Machine Learning. Reading, MA, Addison-Wesley, (1989). Davis, L.D., The genetic algorithms handbook. New York, NY, Van Nostrand Reinhold Company, (1991). Wills, B.A., Mineral Processing Technology. Toronto, Pergamon Press, (1979). Plitt, L.R., A mathematical model of the hydrocyclone classifier. CIM Bulletin, 69:114--123 (1976). Karr, C.L., Least median squares curve fitting using a genetic algorithm. Engineering Applications of Artificial Intelligence, (1995), accepted. Karr, C.L., Strategy for adaptive process control for a column flotation unit. Proceedings of the Fifth Workshop on Neural Networks, SPIE2204:95-100 (1993). Stanley, D.A., Webb, S.W. & Scheiner, B.J., Rheology of ion-exchanged motmorillonite clays. Report of Investigations Number 8895. U.S. Department of the Interior, Bureau of Mines, Washington, DC, (1986). Karr, C.L., Optimization of a computer model of a grinding process using genetic algorithms. In H. El-Shall, B. Moudgil, and R. Weigel (Eds.), Beneficiation of Phosphate: Theory and Practice (pp. 339-345). Littleton, CO: Society for Mining, Metallurgy, and Exploration, Inc., (1993). Mehta, R.K., Kumar, K.K. & Schultz, C.W., Multiple objective optimization of a coal grinding process via simple genetic algorithm, Society for Mining, Metallurgy, and Exploration, Inc., Littleton, CO, preprint number 92-108, (1982). Finch, J.A. & Doby, G.S., Column flotation. Toronto: Pergamon Press, (1990). Herbst, J.A. & Rajamani, K., Models for the dynamic optimization of mineral processing plant performance. In K. V. S. Sastry and M. C. Fuerstenau (Eds.) Challenges in mineral processing. SME, 709-739, (1989). Luttrell, G.H., Adel, G.T. & Yoon, R.H., Modeling of column flotation. SME Annual Meeting, Preprint number 87-130, (1987). Ynchausti, R.A., Herbst, J.A. & Hales, L.B., Unique problems and opportunities associated with automation of column flotation cells. Column Flotation '88, SME, 27-33 (1988). Alford, R.A., Modelling and design of flotation column circuits. Doctoral dissertation, Queensland, Australia: The University of Queensland, (1991).
Pergamon 0892-6875(95)00062-3
Elsevier Science Ltd Printed in Great Britain 0892-6875/95 $9.50+0.00
CALIBRATING COMPUTER MODELS OF MINERAL PROCESSING EQUIPMENT USING GENETIC ALGORITHMS
C. L. KARR§ and D. YEAGERt Engineering Science and Mechanics Department, University of Alabama, Box 870278, Tuscaloosa, AL 35487-0278, USA t U.S. Bureau of Mines, Tuscaloosa Research Center, P.O. Box L, University of AL Campus, Tuscaloosa, AL 35486-9777, USA (Received 5 April 1995; accepted 26 May 1995)
ABSTRACT Researchers at the U.S. Bureau of Mines have developed an approach to tuning empirical computer models that is substantially more efficient than some traditional approaches. The mineral processing industry relies heavily on empirical computer models for both design and process control applications. Unfortunately, the empirical constants associated with the computer models have traditionally been selected using either statistical methods which are quite limited in some domains or trial-and-error procedures which are quite time consuming. A new and more efficient approach to selecting empirical constants for computer models has been developed and implemented. This approach employs a genetic algorithm for selecting the empirical constants for computer models of mineral processing equipment. Genetic algorithms are search algorithms based on the mechanics of natural genetics. They rapidly locate near optimum solutions in difficult search spaces and are shown to be effective in the search for the constants associated with empirical computer models. Bureau researchers have used these innovative search algorithms to both dramatically reduce the time needed to select empirical constants for computer models and to substantially improve the accuracy of the resulting models. The effectiveness of the approach is demonstrated with several examples from the field of mineral processing.
Keywords Modelling, mineral processing, genetic algorithm, simulation
INTRODUCTION The increased speed and decreased cost of today's personal computers have allowed researchers to focus on improving computer modelling capabilities, and to utilize the resulting models to produce more efficiently designed equipment and better process control systems. A number of industries have taken full advantage of improved technology in developing computer models of complex systems. The aerospace industry, for instance, employs a number of high level, first-principle models of aircraft in both the design of vehicles, and in the development of flight simulators [1]. The result of these improved computer models has been aircraft that are more maneuverable, safer, and less expensive. In fact, effective first-principle computer models occasionally provide engineers with insights into the physics of systems, insights that simply were unavailable prior to the development of a first-principle model.
989
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C.L. KARRand D. YEAGER
Unfortunately, all industries have not been as receptive as the aerospace industry to the development of high level, first-principle computer models. The mineral processing industry, for example, still relies heavily on the use of empirical models. For empirical models to accurately predict the response of a physical system, correct values of the model parameters are required. The models often have no physical association to the system being modeled, thereby making the calibration of the empirical constants a difficult task. Although the resulting models may not provide any insight into the physics of the systems being modelled, they still can be used just as effectively as first-principle models to improve equipment designs and process control systems. However, the procedure used to select the values of the empirical constants remains inefficient. The empirical models used in the mineral processing industry rely, for the most part, on a trial-and-error process for selecting the empirical constants that allow the computer model to accurately predict the response of a physical system. The problem of selecting appropriate empirical constants is simply a search problem. Thus, researchers at the U.S. Bureau of Mines have employed a search procedure from the field of artificial intelligence, a genetic algorithm, to locate empirical constants for computer models of mineral processing systems. Genetic algorithms are search algorithms based on the mechanics of natural genetics [2]. They combine a Darwinian survival-of-the-fittest approach with a structured information exchange procedure, producing a scheme that is quite robust. Genetic algorithms are able to locate near-optimum solutions to a wide range of difficult search problems after having viewed but a small portion of the search space [3]. Genetic algorithms have been used to solve a number of difficult search problems [4], and prove to be quite effective in tuning empirical models of mineral processing systems. In this paper, an approach to tuning empirical models using genetic algorithms is presented. The effectiveness of the approach is demonstrated with several examples from the field of mineral processing. The approach proves to both dramatically reduce the time needed to select empirical constants for computer models and to substantially improve the accuracy of the resulting models.
HYDROCYCLONESEPARATOR Hydrocyclones have traditionally been modeled using empirical relationships [5]. Plitt [6] identified a model to predict the d5o or split size that is still used extensively today. The split size is that size particle (given by diameter of the particle) that has an equal chance of exiting the hydrocyclone either through the underflow or the overflow, and is often used to quantify a separation process. Plitt's model implements the following functional relationship: ds0 = f (Dc, D~, D O, D u, h, Q, dd, p)
where D c is the diameter of the hydrocyclone, D i is the diameter of the slurry input, D o is the diameter of the overflow, D u is the diameter of the underflow, h is the height of the hydrocyclone, Q is the volumetric flow rate into the hydrocyclone, ~ is the percent solids in the slurry input, and p is the density of the solids. More specifically, the model has the form: C2
C3
C4
CxDc Di Do e x p [ C s ~ ] d5 ° =
D uc~h CTQCspC9
(1)
where the empirical constants, C i, are selected so that the model accurately matches data that has been collected from actual hydrocyclone separators. A number of approaches have been developed for determining the empirical constants associated with this particular model of a hydrocyclone. Most of the approaches involve the use of statistical routines, each of which has some potential drawbacks [7]. The following section describes the use of a genetic algorithm for determining the empirical constants for a hydrocyclone separator.
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G E N E T I C A L G O R I T H M SELECTS E M P I R I C A L CONSTANTS Genetic algorithms are broadly applicable, efficient search algorithms based on the mechanics of natural genetics. They speculate on new points in the search space with expected improved performance by exploiting historical information. Genetic algorithms demonstrate some fundamental differences with more conventional search techniques. Because of these fundamental differences they are able to work effectively in search spaces for which no derivative information is available, and they do not require that the search space be continuous [3]. Although some genetic algorithms have become quite complex, good results can be achieved with relatively simple genetic algorithms. The simple genetic algorithm used in this study consists solely of reproduction, crossover, and mutation - - operators basic to most genetic algorithms [2]. Once the genetic algorithm operators to be used have been determined, there are only two decisions to be made when applying a genetic algorithm to a particular search problem: (1) how to code the parameters of the problem as a finite string, and (2) how to evaluate the merit of each string (each parameter set). The first decision involves coding the parameter set. Genetic algorithms require the natural parameter set of the problem be coded as a finite string of characters. The parameter set in this study, the empirical constants C 1 through C 9, is coded as strings of zeros and ones. This is accomplished by representing each constant as a binary string, and evaluating the bits as a binary number. Ten bits are allotted for defining each constant (although fewer or more bits can be used). The ten bits are interpreted as a binary number (1001010111 is the binary number 599). This value is mapped linearly between some user determined minimum (Cmin) and maximum (Cmax) values according to the following:
b C 1 = Cmin + - - ( C m a
2M-1
x - Cmin)
(2)
where b is the integer value represented by an M bit string. The values of Cmin and Cmax in a given problem are selected by the user based on personal knowledge of the problem. If necessary, a rapidly converging, coarse optimization method may be used for selecting the limiting values. This same form is used to represent each of the remaining constants, C 2 through C 9, and the nine ten-bit strings are concatenated to form a single ninety bit string representing the entire parameter set. The second decision involves establishing a fitness function which indicates the merit of each potential solution to the search problem. Tuning the empirical constants associated with the hydrocycione model is actually a curve fitting problem. Thus, several criteria can be used to establish the quality of the fit. For example, the most common curve fit measure is the least squares criteria. Least squares curve fit techniques attempt to minimize the distance between the measured data points and the estimated values as defined by the empirical model. However, traditional least squares curve fitting techniques are limited by their use of partial derivatives. The fitness function used in this study is: N f = K - ~(y/ i=1
- yi) 2
(3)
where f is the fitness function value, K is a large constant defined so that the fitness function value is always non-negative, N is the number of data points, y is an actual data value, and y' is an estimated data value using the parameters defined by a particular genetic algorithm string. This fitness function is minimized by a genetic algorithm. A genetic algorithm has been used to select the values of C! through C 9 used in the empirical model of a hydrocyclone as shown in equation (1). Figure 1 demonstrates the effectiveness of using a genetic algorithm for tuning the empirical constants. In this plot, the actual d50 size is plotted against the d50 size predicted using the model with the empirical constants as selected using a genetic algorithm. Note that the model
992
C.L. KARR and D. YEAGER
would exactly reproduce the data if all of the points shown in Figure 1 fell on a line of 45o. Although the genetic algorithm tuned model does not exactly predict the ds0 size, it fits the data very well. 100 90 80 70 .-$ 3D O "Di
• ,~5 "5/
60
.5/
50 40 30 20 10 0
0
10
20
30
40
50
60
70
80
90
100
d~,Data
Fig. 1 Performance of genetic algorithm tuning of hydrocyclone model. It is important to note that the genetic algorithm determined the empirical constants in roughly 10 minutes, whereas a statistical package took roughly 9 hours to locate constants that provided the same quality fit to the data. This time of computation becomes critical when the tuning of an empirical model is being done as a part of a real-time, adaptive process control system such as the one developed by Karr [8] for the minerals industry.
REE-EYRING EQUATION One of the major concerns of the mineral processing industry is the dewatering of mineral slurries. Researchers at the U.S. Bureau of Mines have developed innovative techniques for dewatering numerous mineral slurries, and a number of these techniques are based on knowledge of the non-Newtonian components of viscosity of the slurry. The non-Newtonian components of viscosity can be found by curve fitting data obtained in the laboratory. The Ree-Eyring equation allows for the estimation of the non-Newtonian components of viscosity and is of particular interest to the minerals industry in the area of dewatering phosphatic clay wastes. The Ree-Eyring equation may be written as
y=C
1
+
C2sillh-l(C3x)
(4)
C3x
where y is the viscosity (the dependent variable), x is the shear rate (the independent variable), and C l, C e, and C 3 are empirical constants to be determined. In most instances, experimental viscosity and shear rate data is readily available through laboratory tests. However, curve fitting the data is not always easy because of the highly nonlinear form of the equation. Attempts to curve fit data to the Ree-Eyring equation using traditional least squares techniques were unsuccessful. The difficulty arises in solving for the constant C 3. A symbolic math package was used in this attempt, but the package failed to produce a solution. Thus, in this problem domain, traditional statistical techniques were not useful. Therefore, the genetic algorithm approach to selecting the empirical constants was not only convenient as in the previous examples, but it was necessary.
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993
Fortunately, a genetic algorithm was able to effectively tune the Ree-Eyring model. The process of tuning the model using a genetic algorithm is the same as in the previous example. The parameter set (C l, C 2, and C3) is represented as a thirty-three bit string of zeroes and ones, and are evaluated using a version of the previously discussed fitness function. For this example, however, the sum of the squares of the residuals was replaced with the median residual value. This approach is known as least median squares (LMS) curve fitting, and is an effective approach to negating the appearance of outliers in the data set [7]. The ease with which a LMS method can be substituted for a least squares approach demonstrates a further advantage of using a genetic algorithm for tuning empirical models. As in the previous example, the fitness function allows for the minimization of the distance between the data points and the resulting curve. Stanley, Webb, and Scheiner [9] used a transformation method to solve the Ree-Eyring equation for a sodium ion-exchanged clay. The data used in that report were also used here. Once again the genetic algorithm quickly converged to a quality solution; the smallest error produced by the genetic algorithm calculated curve (Error = 0.008) is less than that of the curve calculated by the transformation method used by Stanley, Webb, and Scheiner (Error = 1.57). Figure 2 shows how well the curve calculated by the genetic algorithm fits the data, and also how well the LMS approach overcomes the occurrence of the three outliers (inconsistent data points). C l, C 2, and C 3 were allowed to range between 0.0 and 200.0 for the coding necessary in the genetic algorithm application.
Genetic algorithm LMS (error = 0.008)
Sheer Rote
(10-4s -1)
Fig.2 Performance of genetic algorithm tuning of Ree-Eyring model. In the previous example involving tuning a model of a hydrocyclone separator, using a genetic algorithm to do the model tuning made sense because the genetic algorithm approach took less time and was more accurate than the traditional statistical algorithms. However, in the example of the Ree-Eyring equation presented in this section, a genetic algorithm was used because the traditional techniques were simply not applicable. It is encouraging to note that extending the genetic algorithm tuning approach used is easily altered to consider various criteria for judging the accuracy of the model; switching from a least squares to a LMS criteria merely involves altering the fitness function. The successful application of a genetic algorithm to the problem of tuning a model using the Ree-Eyring equation has enabled researchers at the U.S. Bureau of Mines to improve dewatering techniques. Results obtained using the genetic algorithm LMS technique represent potential cost savings for the minerals industry. Furthermore, these results are helping Bureau researchers to develop dewatering techniques that help reduce waste imparted to the environment.
994
C.L. KARR and D. YEAGER GRINDING
Grinding is a necessary component in the processing of a number of minerals, and improvements in this area could provide substantial cost savings for the minerals industry. Optimizing the process of grinding, however, is a difficult task requiring innovative techniques and accurate computer models. The U.S. Bureau of Mines is quite interested in the grinding process because it plays a role in the processing of numerous minerals, and the substantial energy requirements associated with grinding provide substantial room for economic savings. Thus, because of the high energy costs combined with the considerable popularity of the process, improvements in the efficiency of the grinding process could have a dramatic economic impact on the minerals industry. A number of innovative optimization techniques have been developed that are applicable to the improvement of the grinding process [10]. Unfortunately, the success of these optimization techniques requires an effective computer model of the grinding process. Although empirical computer models of the grinding process have been developed [11 ], the tuning of these computer models is time-consuming and often quite difficult. Thus, using a genetic algorithm for tuning the grinding models is inviting. The grinding process is characterized by several performance measures, all of which are important in various circumstances. Generally, there are four measures that are especially important indicators of the efficiency of a particular grinding process: (1)fineness of the ground product, (2) e n e r g y costs associated with the process, (3) a v i s c o s i t y c o e f f i c i e n t , and (4) a v i s c o s i t y e x p o n e n t . Mehta, Kumar, and Schultz [11] have developed an empirical model of grinding that accurately mirrors the response of a coal grinding process. In that work, they studied the ability of a simple genetic algorithm to optimize the performance of their computer model of a particular grinding circuit. In the current study, the focus is shifted from using a genetic algorithm to optimize the performance of a tuned model, to employing a genetic algorithm to actually tune the computer model of a grinding process. Only two of the four indicators of grinding efficiency are considered: fineness and energy consumption. However, the steps used to tune the models that predict fineness and energy can be used to tune the models that predict the amount of dispersant addition and the viscosity characteristics. The pertinent modelling equations are: F
= C 1 + C 2 xs -
C6 x2
C3 xB + C 4 xO + C 5 x M
C7 X2
-
+
C8
+ Clo XsB + Cl1
XsD
+ CI3
XBM
XBD
C14
-
X2 - C9 X2 C12
-
(5)
XsM
-
CI5
XDM
+
C19
xD +
and
E
= C16 + C17 -
x:
C18
xs -
xB
-
-
+ C25
xss
+
C26
XsD -
+ C28
XBD
-
C29
XB M -
C20
xM
(6)
C27 X~M
C30
Xo M
where F is the fineness (or the weight percent less than 38 millimeters), E is the energy in kilowatt hours per ton, x s is the percent solids by weight, x B is the maximum ball size, x M is mill speed, x D is dispersant addition, xij represents the product of x i and xj, and C l through C30 are the empirical constants selected so that the model equations accurately reproduce data obtained from a physical system. It is important to note that C l through Cl5 can be determined entirely independent of Cl6 through C30. Thus, there are two separate fifteen parameter search problems to be solved. The development of an efficient empirical computer model of a grinding circuit actually depends on tuning fifteen parameters for each of four separate model equations. The very same approach used in the previous
Calibrating computer models of mineral processing
995
sections of employing a genetic algorithm for tuning computer models can be used here. The coding scheme outlined earlier can easily be used, as can the fitness function definition. Thus, all that is left is to present some results. Figures 3 and 4 demonstrate the effectiveness of using a genetic algorithm for tuning the empirical constants associated with the grinding models. In these plots, the actual fineness and energy consumption as measured in the physical system are plotted against the computer predicted values. These figures demonstrate the ability of a genetic algorithm to determine empirical constants for the grinding models. 90 80 70 6O '10
=~ 5o ~ 4o C
3O 20
/
10 /*
./
,/ 10
20
30
40
50
50
70
80
90
Finess Actual
Fig.3 Performance of the genetic algorithm in tuning the fineness model.
2O
O "10 O
iii
J
0 0
5
10
15
20
25
Energy Actual Fig.4 Performance of the genetic algorithm in tuning the energy model.
COLUMN FLOTATION CIRCUIT In recent years, numerous researchers have worked to better understand the principles that govern column flotation operation. As a result of these efforts, several computer models have been produced that attempt to predict the performance of column flotation units. Some of the more noteworthy research has been performed by Finch and Doby [12], Herbst and Rajamani [13], Luttrell, Adel, and Yoon [14], and Ynchausti, Herbst, and Hales [15]. In the current effort, the Finch and Doby model has been used because it has been well documented, is relatively straightforward to program, and provides accurate results when the effects of the froth zone are not too pronounced. HE 8:9-D
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c.L. KARRand D. YEAGER
The Finch and Doby model of column flotation relies heavily on a property of hydrophobic minerals called induction time. Induction time is the time required, once a hydrophobic particle has collided with a bubble, for the particle to attach to the bubble. Once the attachment has occurred, the particle will usually remain attached to the bubble until the bubble/particle aggregate rises to the pulp/froth interface level. Also, it is in practice often a good assumption that a particle which reaches the interface level will remain attached to its bubble and be carried out with the concentrate. Unfortunately, induction time is extremely difficult to measure in any flotation equipment. However, the induction time can be treated as an empirical constant, and thus can be computed using a genetic algorithm in the manner described in previous sections. The column circuit considered in this effort is shown in Figure 5. This circuit has been used because data is available in the thesis by Alford [16]. Alford's data is taken from a circuit used in an industrial application at the Mount Isa Mines Limited installation in Australia.
Concentrate flows New Feed
Fee0 5
>
Fee
J
Interfac/ee
/ / )
Tailings 1
Tailings 2
Final Tailing
Fig.5 Schematic of the column flotation circuit. A genetic algorithm has been used to tune selected parameters of the computer model in order to make the values predicted by the model match as nearly as possible the behavior of the actual circuit as recorded by the data in Alford's thesis. Besides the induction time for each hydrophobic mineral species (in this case sphalerite and galena) other parameters used in the tuning of the circuit were the vessel dispersion number and a first-order rate constant for the water and for each non-hydrophobic mineral species. Since vessel dispersion number is largely a function of the size and shape of the column and since the physical dimensions of the three columns were identical, the three were assumed to have the same vessel dispersion number. Since induction time is a chemical property, the sphalerite and galena induction times were also assumed to be the same for each of the three columns. The parameters used and the values assigned to those parameters by the genetic algorithm can be seen in Table 1. The genetic algorithm was set up to minimize the mean squared error between the balanced data values based on observed flows in the Mount Isa Mines circuit and the corresponding values obtained from the computer model. There were thirty values used in computing the mean squared error, namely the percent solids, the percentage of lead, the percentage of zinc, the percentage of iron, and the flow volume in each of the six flows produced by the circuit. The final mean squared error attained by the genetic algorithm was 0.0003. The precise values achieved by the model are listed in Table 2 along with the balanced values based on observed data for comparison. (The percent solids column is omitted.) Each table entry is of the form Vrn/Vb, where V m is the value produced by the computer model and V b is the balanced value from Alford's data.
Calibrating computermodels of mineral processing
997
T A B L E 1 Parameters used and values obtained by a genetic algorithm for the column flotation circuit. Parameter
Value
Vessel dispersion number Nd Galena induction time Tig Sphalerite induction time Ti s Water rate constant stage 1 Water rate constant stage 2 Water rate constant stage 3 Sulfides rate constant stage 1 Sulfides rate constant stage 2 Sulfides rate constant stage 3 Gangue rate constant stage 1 Gangue rate constant stage 2 Gangue rate constant stage 3
154.88 0.04665 sec 0.04415 sec 3.08770e-04 1.08730e-04 6.23246e-05 6.35543e-05 5.23178e-05 5.21631e-05 1.60884e-05 4.01268e-05 3.99883e-06
TABLE 2 Parameters used and values obtained by the genetic algorithm. Flow Name I
% Lead
% Zinc
% Iron
Flow Volume
Stage 1 Concentrate
11.7/11.8
41.0/40.8
12.4/12.5
12517.8/12632.1
5.7/5.7
14.6/14.5
26.6/26.7
31942.1/31828.1
11.1/10.7
36.5/37.4
14.0/14.2
4597.7/4457.6
5.0/5.0
11.6/11.7
28.3/28.2
39344.5/39370.4
11.1/11.1
34.3/32.5
17.5/16.9
2684.4/2710.0
4.5/4.5
9.7/9.8
29.2/29.2
48660.1/48660.5
Stage 1 Tails Stage 2 Concentrate Stage 2 Tails Stage 3 Concentrate Stage 3 Tails
SUMMARY Empirical computer models play a key role in the mineral processing industry for tackling both design and control problems. Generally, the calibration or tuning of these models, the selection of values for the empirical constants, is a time-consuming task. Fortunately, the search capabilities of genetic algorithms can be used to help solve the problem of selecting these constants. Genetic algorithms are search algorithms based on the mechanics of natural genetics. They are able to locate near-optimal solutions after having sampled only small portions of the search space, and they are flexible enough to be effective in a wide range of problems. Their novel approach to solving search problems is fundamentally different from more conventional search techniques. They have been used in this paper to tune the empirical constants associated with several different computer models used in the minerals industry.
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C.L. KARR and D. YEAGER REFERENCES
1.
2. 3. 4. 5. 6. 7. 8.
9.
10.
11.
12. 13.
14. 15. 16.
Hoffman, K.A., Computational Fluid Dynamics for Engineers. Englewood Cliffs, N J, Prentice-Hall, (1991). Holland, J.H., Adaptation in Natural and Artificial Systems. Ann Arbor, MI, The University of Michigan Press, (1975). Goldberg, D.E., Genetic Algorithms in Optimization, Search, and Machine Learning. Reading, MA, Addison-Wesley, (1989). Davis, L.D., The genetic algorithms handbook. New York, NY, Van Nostrand Reinhold Company, (1991). Wills, B.A., Mineral Processing Technology. Toronto, Pergamon Press, (1979). Plitt, L.R., A mathematical model of the hydrocyclone classifier. CIM Bulletin, 69:114--123 (1976). Karr, C.L., Least median squares curve fitting using a genetic algorithm. Engineering Applications of Artificial Intelligence, (1995), accepted. Karr, C.L., Strategy for adaptive process control for a column flotation unit. Proceedings of the Fifth Workshop on Neural Networks, SPIE2204:95-100 (1993). Stanley, D.A., Webb, S.W. & Scheiner, B.J., Rheology of ion-exchanged motmorillonite clays. Report of Investigations Number 8895. U.S. Department of the Interior, Bureau of Mines, Washington, DC, (1986). Karr, C.L., Optimization of a computer model of a grinding process using genetic algorithms. In H. El-Shall, B. Moudgil, and R. Weigel (Eds.), Beneficiation of Phosphate: Theory and Practice (pp. 339-345). Littleton, CO: Society for Mining, Metallurgy, and Exploration, Inc., (1993). Mehta, R.K., Kumar, K.K. & Schultz, C.W., Multiple objective optimization of a coal grinding process via simple genetic algorithm, Society for Mining, Metallurgy, and Exploration, Inc., Littleton, CO, preprint number 92-108, (1982). Finch, J.A. & Doby, G.S., Column flotation. Toronto: Pergamon Press, (1990). Herbst, J.A. & Rajamani, K., Models for the dynamic optimization of mineral processing plant performance. In K. V. S. Sastry and M. C. Fuerstenau (Eds.) Challenges in mineral processing. SME, 709-739, (1989). Luttrell, G.H., Adel, G.T. & Yoon, R.H., Modeling of column flotation. SME Annual Meeting, Preprint number 87-130, (1987). Ynchausti, R.A., Herbst, J.A. & Hales, L.B., Unique problems and opportunities associated with automation of column flotation cells. Column Flotation '88, SME, 27-33 (1988). Alford, R.A., Modelling and design of flotation column circuits. Doctoral dissertation, Queensland, Australia: The University of Queensland, (1991).