- Email: [email protected]
Financial implications of comminution system optimization
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Int. J. Miner. Process. 44-45 (1996) 209-221
Financial implications of comminution system optimization J.A. Herbst, Y.C. Lo Control Internationul, Inc., Suit Lake City, UT, USA
Abstract Systems optimization is an exciting new tool for improving the performance of existing comminution plants while incurring minimal capital expenditures and producing high rates of return. The goal of this approach is to achieve significant performance gains without installing new equipment or replacing older, less efficient equipment. Thus, rates of return can be much higher than the historical rates of return in the process industries. This paper describes a methodology for systems optimization that has produced significant rates of return for several comminution plants. The approach seeks to determine important cost-sensl,tive, synergistic interactions in a plant, find the optimum conditions that improve the performance of the whole plant and accurately identify the tradeoffs between incremental costs and additional revenue. The interactions are described by mathematical models of the processes occurring in the system which can then be programmed into plant simulators. Two case studies of comminution plant optimization which illustrate the concepts of systems optimization are presented in this paper. The first case study concerns a system optimization project for the SAG mill circuit at a western US gold mine. The financial rate of return for that project was 145%. The second case study is a systems optimization project with a rate of return of 115% for a large iron ore concentrator in Europe.
1. Introduction The area of plant performance improvement which has been most important for the mineral industry in the last decade has been optimization of unit operations (Mular and Klimpel, 1991; Herbst and Lo, 1991). The success of unit optimization, together with the knowledge, the experience, and the tools that have been gained, have brought the next important stage in this technology: the optimization of systems. The basic definition of a System is a group of interacting and interdependent items that form a whole. Systems in the process industries have the characteristic that they 0301-7516/96/$15.00 SSDI 0301-75
0 1996 Elsevier Science B.V. All rights reserved
16(95)00037-2
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210
+
J. Miner. Process. 44-45 (1996) 209-221
Crusher
4
Flotation Ball Mill
-
Induration
System 1
-
-l-.-Y
System 2 System 3 Fig. 1. System boundaries
defmed by process equipment.
yield a product possessing value (along with revenue derived from that value), and there are costs associated with producing that product. Systems optimization is the coordinated manipulation of system parts to yield better performance of the whole than would be achieved through independent changes in each part. The difference between unit optimization and systems optimization is that the latter emphasizes and takes advantage of the interactions and interdependence among operating variables and pieces of equipment in the plant. These interactions cause tradeoffs between incremental costs and additional revenues. For example, reducing costs in one area may be more than offset by fall of productivity caused in another area. Unless these relationships are fully known and employed, false conclusions about the financial feasibility of a project can be drawn, and there may be missed opportunities to exploit the interactions that improve process performance. Fig. 1 shows how a flowsheet (iron ore processing in this example) can be divided into three systems (an arbitrary number) with each system being encompassed by a larger system. The optimization of System 3 is enormously complex since there are a tremendous number of interacting variables. The difficulty, or time to find a solution, grows exponentially with the number of interacting variables. With the current state-ofthe-art, it is possible to optimize Systems 1 and 2; but, for now, optimization of System 3 might be out of reach. It must be noted that the optimization discussed here is much different than the optimization of, say, airline schedules or telephone networks. The later problems may involve thousands of variables, but the problem itself is direct, clear, and simple. The solution for many thousands of variables is exactly the same as for a few variables, only needing a faster computer to arrive at the solution. In the process industries, however, one must deal with processes that may be poorly understood, difficult to quantify, or difficult to measure. These are the difficulties that the methodology presented in this paper was designed to address.
J.A. Herbst, Y.C. Lo/h. Table 1 An examole
.I. Miner. Process. 44-45 (1996) 209-221
211
of false economy bv cost cuttine Media No.
Media cost, US$/year Revenue, US$/year Savings, US$/year
I
5,770,OOO 135,000,000
Media No. 2 5,290,OOO
119,500,OOo - 15,020,000
In order to accomplish systems optimization, an objective function is needed. The financially relevant objective function is the “profit” from a system expressed as: “PROFIT” = REVENUE - OPERATINGCOSTS The revenue from the product of a system may be real, such as from System 3, but in other systems, such as 1 and 2, the revenue represents the potential value of the product. Whether real or potential, the revenue should properly reflect the influence of the optimization variables, as should the amount for operating costs. Since the revenue amount in Eq. 1 may be potential, so will the profit be a potential amount. It will be, nonetheless, a good measure, since it is affected by the interactions between revenue and costs. An exciting prospect for systems optimization is the reduction of plant operating costs, which in the current economic climate is a top priority for many companies. Eq. 1 determines which cost cutting measures are effective and which are not. Consider first a case where cutting costs would not be attractive, i.e., a case in which reduction in cost is less than the loss of revenue (or productivity) resulting from the cost cutting change. Suppose, for example, that a 50,000 tpd iron ore regrinding plant is considering a switch from its current grinding media (No. 1) to a cheaper grinding media (No. 2). The cheaper media is known to have a higher wear rate and is irregularly shaped so that it will produce lower grinding rates. The cost of media No. 2 is 34% of the cost of media No. 1, and the wear rate per ton of ore will likely be 225% higher. The mill capacity using media No. 2 will probably drop by 11% so the total cost of media No. 2 is 8% less than media No. 1. Table 1 shows a considerable decrease in “profit” (Eq. I), i.e., a negative savings, so that the switch to grinding media No. 2 is economically unattractive. Next, consider the opposite case in which a reduction in cost is larger than the loss of revenue or productivity. Suppose that at a 100,000 tpd copper ore concentrator a coarser grind is being proposed to reduce mill operating costs. As a result, energy consumption
Table 2 An example of true economy by cost cutting
Energy costs, US$/year Revenue, lJS$/year Savings, US$/year
Base case
1% less liberation
15,840,000 253,011,OOo
14,520,OOO 252,702,450 1,011,450
212
/.A. Herbst, Y.C. Lo/h.
J. Miner. Process. 44-45 (1996) 209-221
Classifiertype Classifieroperation
Fig. 2. Variables for optimization
in a ball mill grinding circuit.
is expected to be reduced from 12 kWh to 11.O kWh. If the grind-recovery curve is fairly flat, the recovery may drop by 0.1 %, producing only a small reduction in revenue at the same feed tonnage. Table 2 shows a positive savings so that coarsening the grind is an economically attractive move. The authors and their co-workers have developed a systematic approach using mathematical models to analyze situations like these with a view toward determining promising alternatives for systems optimization in a financial sense. The first step of this methodology is concerned with steady-state operating conditions involving variables such as those shown in Fig. 2 for a ball mill grinding circuit. Even within a single-unit operation, there are many variables that can be manipulated. Thus, there exists considerable potential for optimization. When a larger system is considered, such as system 2 in Fig. 1, the number of manipulated variables goes up accordingly. The principle of coordination of manipulated variables still holds for the larger system. Steady-state optimization yields specific recommendations for steady-state target variables and predictions of the expected improvement after implementing such changes. In practical terms, plants never operate at steady-state, but rather are in a perpetual transient state due to a variety of disturbances (Herbst et al., 1988). The benefit predicted by steady-state simulations is for average or normal conditions. Process disturbances cause minute-by-minute deviations from that average either for the worse, harder ore/coarser feed, for example, or for the better, softer ore/finer feed. In this methodology, dynamic optimization must be performed to compensate for the process disturbances and to react to these disturbances in such a manner that plant productivity is continually maximized. Dynamic models in the form of plant simulators can be used for off-line control strategy development, and on-line dynamic models can be used to help make well-informed
responses
to each disturbance.
J.A. Herbst, Y.C. Lo/Int.
J. Miner. Process. 44-45 (1996) 209-221
213
2. Methodology The methodology which is described in this paper is summarized in Fig. 3. Steady-state optimization comprises batch and plant testing, modeling, computer simulation, and plant verification. A detailed description of plant and lab tests and the associated optimization procedure has been outlined in previous publications (Herbst et al., 1985; Lo et al., 1988). The effects of critical operating variables, such as ball size, mill percent solids, etc., are investigated during the lab experiments. The data obtained in laboratory or pilot-scale grinding experiments are used as input to the estimator/simulator GRINDS1M.S’“. Once the breakage kinetic parameters are determined, computer simulations are performed to determine energy requirements at a desired product size for different operating conditions and circuit configurations. Computer simulations yield specific recommendations for changing critical operating variables, classification and circuit configuration along with a prediction of the expected improvement. The recommended conditions are then implemented in the plant, where comparative tests are usually carried out on two mills in parallel. Plant tests provide verification of the methodology developed for optimization. The methodology for dynamic optimization involves plant disturbance testing, laboratory evaluation, model building and control strategy evaluation. The model for ore hardness disturbances (magnitude and frequency) is developed from plant samples taken at varying time intervals. These ore samples are subjected to laboratory grinding tests which yield hardness variations as a function of sampling time. From these tests it is possible to obtain a profile of the variations in ore hardness for the grinding circuit feed. The evaluation of alternative control strategies for a given application involves the use of a dynamic simulator (Rajamani and Herbst, 1980; Herbst and Pate, 1986).
D&e Objectiie
PlantAudit steadystate Dynamic
Receive Benefits
t Conduct Lab or
Evaluate In Plant
PilotScaleTests
lmptementBest Alternative
Evaluate lnteractbns Fig. 3. Methodology for optimization.
214
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(1996) 209-221
Dynamic simulations help eliminate costly experimental plant testing during preliminary screening of control strategies which may include manual control, stabilizing control and model-based control. For model-based control, the model is a reduced form of the one used in a dynamic simulator. The model must be simple enough for rapid on-line calculations, but detailed enough to faithfully reproduce the essential dynamic characteristic of the grinding process. In addition, an estimator and an optimizer are needed for model-based control. The estimator combines measurements within the process with model information to determine the state of the system at any instant in time. The optimizer uses the current state of the system and the model to select the path for manipulated variables that will achieve the process objectives in an optimal way. Dynamic optimization requires either parallel or serial on-off testing of the new control strategy until enough data has been collected for a statistically significant comparison to be made with the old control strategy. Then, Eq. l is used to compute measures such as the internal rate of return and pay back period for the systems optimization project. 3. Models for optimization System models are built or assembled from subsystem models connected to reflect the configuration or flowsheet of the system (Himmelblau and Bischoff, 1968; Guillaneau et al., 1992). A subsystem model itself may be composed of subprocess models that account for influential phenomena occurring in that process. When built in this manner the system model gives an accurate description of the interactions occurring within the system and of the influence of manipulated variables, provided that the subsystem models are an accurate description of subprocess interactions and the influence of manipulated variables. At the foundation of today’s comminution systems models are the breakage kinetics and the transport of material through the comminution device. The breakage kinetics for an assemblage of particles in a comminution device are quantified by the selection and breakage functions. The selection function si is the fractional rate of breakage for particles of size di. The breakage function 6,, is the size distribution of the fragments from the breakage of particles of size dj. The selection and breakage functions can be determined for almost any comminution device, but this determination must be done by experiment with device in the plant and, ideally, in the laboratory as well. The dependence of selection and breakage functions on particle size and operating conditions is unique for each type of device and also must be determined by experiment. The transport of material through a device can be quantified by its residence-time-distribution (or RID). The RTD of tumbling mills is typically well described by the iv-mixers-in-series model, which describes anything from a single perfect mixer to plug flow. The transport of material through a crusher is harder to envision, but the N-mixers-in-series model is adequate because of its flexibility. A rate expression that is applicable to the internal size distribution of crushers and tumbling mills (AG, SAG and ball) is: i-
qqP4
dr
= M,m,i
I
- M,m, - siHpmi + c bijsjHpmj + K, H,b” j= I
(2)
JA. Herbs?, Y.C. Lo/h.
J. Miner. Process. 44-4.5 (1996) 209-221
215
where mi is the mass fraction in the ith size interval, H is the total mass of material, the feed ram is M, and the discharge rate is M,. Self-breakage phenomena occurs primarily in AG and SAG mills where large pieces of ore act as the grinding media. The self-breakage rate is K, and the self-breakage function is b,!. The discharge rate depends on the type of comminution device. The optimization case studies that follow show how the rate expressions are developed for SAG, pebble, and ball mills.
4. Industrial applications 4.1. SAG mill optimization project The first example is concerned with the optimization of a SAG mill grinding circuit at a gold operation in the western US. Over 4,000 tpd of ore is ground in a SAG mill-ball mill circuit and then leached by CIL. Profit is very ore tonnage sensitive at this plant. An optimization project was initiated to increase the capacity of the SAG mill by at least 4%, since it was the bottleneck in the plant. The total cost of the project was $100,000 for steady state and dynamic optimization. Steady-state optimization was required. to investigate a number of options for increasing capacity. These options were ball load, mill percent solids and grate design. Dynamic optimization required the purchase of a PC-based control system. The financial break-even point for the project was a 2’% increase in feed rate. The typical increase in feed rate for projects of this type varies from 5 to 20%, depending on the number of plant variables available for manipulation.
Fig. 4. Flowsheet of SAC milling grinding circuit for gold ore.
216
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J. Miner. Process. 44-45 (1996) 209-221
The SAG mill circuit and instrumentation are shown schematically in Fig. 4. The feed to the circuit has been crushed to minus 15 mm by the primary crusher. The SAG mill is 6 m by 2 m and is driven by a single 1250 kW motor. The mill discharge is sent to a double deck vibrating screen with 12.7 and 7.9 mm openings. The 12.7 mm oversize is crushed in a 1.3 m short-head cone crusher and recycled to the SAG mill, while the 12.7 by 7.9 mm fraction is sent back to the SAG mill. The screen undersize constitutes the ball mill feed. SAG mill sampling was performed to evaluate the mill response to process variable changes, to measure the mean residence time of various ore types, and to determine the frequency and magnitude of the ore hardness disturbances. The process response tests were done on four ore types and based on a 2-by-2 factorial design with one center point. The manipulated variables for each test were the mill feed tonnage and the mill feed percent solids. The model for the SAG mill can be derived from Eq. 2 by first recognizing that within a SAG mill the larger rocks act as media to break smaller rocks and, eventually, themselves; balls in a SAG mill can break rocks no larger than some critical size. The different breakage mechanisms for big rocks and smaller rocks (which will be referred hereafter as particles) must therefore lead to grinding kinetic models appropriate for each size. In the context of this model, rocks are defined to be ore pieces larger than the grate opening and are assumed to break completely into particles at a rate proportional to the mass holdup of rocks. This assumption leads to the following expression for the kinetics of rock breakage:
(3) where H, is total mass of rocks in the mill, Ma is the mass flowrate of rocks in the mill feed and K, is the grindability parameter for rocks. K, depends upon ore characteristics such as hardness and feed size, and upon mill conditions such as percent solids, mill power and holdup. A useful relationship between K, and mill specific energy similar to one proven valid for ball mill grinding is: K,=K$-
R
where P is mill power and K, E is the upon ore type, mill percent solids, etc. laboratory testing. Eq. 2 is used to describe the particles. to the holdup of particles in the mill by M,=A,K,H,
(4) specific grindability parameter which depends This dependence must be found by plant and The mill discharge rate, M,, in Eq. 2 is related the following: (5)
where A, is the grate open area factor and K, is the discharge parameter for particles. A, describes the effect of the number and area of grate openings on the slurry discharge flowrate and how it changes with filling level. K, depends upon factors such as size
J.A. Herbst, Y.C. La/h.
J. Miner. Process. 44-45 (1996) 209-221
217
Table 3 Financial summary of SAG optimization project Original plant capacity increase in capacity due to Steady state optimization Dynamic optimization Project investment (principal and interest) Incremental revenue Marginal costs Project profit, 1st year Internal rate of return Simple pay back period
4,200 tpd 4.7% 4.4% $108,OQO $2,494,000 $909,000 $1,477,000 145% < 1 month
and mill conditions. The relationship between K, equation: K,
= Kg
and specific energy is given by an
$ P
where Kz is the specific discharge constant for particles. The data from the factorial design test were used to estimate model parameters. Both a steady state and a dynamic simulator of the SAG mill/crushing circuit were built. The steady state simulator was used to investigate the following interactions: percent solids/power ball load/power grate open area/power feed rate/power
percent solids/filling ball load/filling grate open area/filling feed rate/filling
Increasing the grate opening was found to provide the biggest boost to capacity; therefore, the grate was redesigned to increase the open area by 8%. The average feed rate after this modification was 4.7% higher than before. The dynamic simulator was used to test the effectiveness of control strategies in the presence of feed size and ore hardness variations. The frequency and magnitude of ore hardness variations determined from plant testing were used in the dynamic simulator. For dynamic optimization, an expert system was designed to work in conjunction with the process models to maximize feed rate to the SAG mill. The model-based expert system was found by simulation to be the best control strategy and so was implemented in the plant. To document the performance of the model-based expert control system, a series of on/off test were conducted in which the mill was operated alternatively for set time periods under model-based control and PI control of mill power. The model-based expert system produced an average of 4.4% more throughput than the PI controller. During the testing period energy consumption under PI control was 5.2 kWh/t, while under model-based expert control it was 4.9 kWb/‘t. Together the grate modification and control system increased the average tonnage from 175 tph to 191 tph. Increasing the grate open area also improved the control system’s quickness of response.
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J. Miner. Process. 44-45 (1996) 209-221
The financial benefits obtained from the grinding optimization are summarized in Table 3. The capital investment is the cost of the project plus 8% interest. This amount is paid by the end of the first year. The incremental revenue is the annual increase in revenue from the 16 tph increase in tonnage. The marginal costs are the total mining and processing costs (minus milling costs) for the additional 16 tph. For this analysis, typical mining and processing costs were used. Milling costs are not included in the incremental marginal costs for two reasons. The first is that most of the additional milling costs are for the control system, which is reported as capital investment. The second reason is that the reduction in energy per ton resulted in no increase in total milling costs at the higher tonnage. The profit is the incremental revenue minus the marginal costs minus the capital investment for the project. The internal rate of return for this project is: “Profit” IRR = 100% x
Capital Investment + Marginal Costs 1,477,OOo
= 100% x
108,000 + 909,000
= 145%
(7)
Clearly, this project generated a high rate of return and an extremely short pay back period. Even considering some uncertainty in marginal costs, this project is fully justified. The best estimates possible of incremental revenue and marginal costs are certainly desirable when justifying a systems optimization before it is started. 4.2. Grinding plant optimization project The second example is concerned with the grinding circuits in an iron ore concentrator in Europe. The concentrator and the pelletizing plant were built in the mid-1960s. The annual capacity was 1.5 Mt. Over the years, the capacity has been increased to 3.7 Mt annually by lengthening the pebble mills, increasing the mill speed and replacing the induration system. In 1988, it became apparent that the grinding circuits had become the
ol Fbpoae E Ebsize v”T v mafilhg VS Vafi&bmpad D !zt? W Wdghtometor : Lwd
Fig. 5. Grinding circuit and instrumentation.
PSM
J.A. Herbs& Y.C. Lo /ht.
J. Miner. Process. 44-4.5 (1996) 209-221
219
major limitation to further tonnage expansion. Before undertaking a major capital investment for expansion, an intermediate project was initiated with the objective of reaching 4.5 Mt pellets per year through systems optimization of the current facilities. The objective was to be achieved by decreasing the magnetite feed to the concentrator from minus 25 mm to minus 10 mm, converting from rod mill to ball mill grinding and by adding a supervisory control system. The grinding circuit and instrumentation are illustrated in Fig. 5. Ball mill tests were performed to evaluate the alternative of converting the 3.6 X 4.8 m rod mill to a ball mill. Grinding rates for different particle sizes are a strong function of the b’all size distribution in the mill (Lo and Herbst, 1986). So, lab grinding tests were conducted with top ball sizes of 38, 50, 63, 76 and 88 mm. In each case, the ball size distribution used approximated that of an “equilibrium charge distribution” often used in laboratory tests for scale-up design (Lo and Herbs& 1986). Computer simulations of the grinding process were conducted to predict continuous plant performance of the rod mill after conversion to a ball mill. Parameters for the simulation were estimated from plant data and from laboratory batch grinding tests. Mill throughput capacity in open and closed circuit configurations was predicted for different ball sizes and was compared to the base line feed rate of the rod mill. The expected product size was 83% minus 295 microns. Simulation results indicated that a 30% increase in capacity could be expected from a ball mill operating in closed circuit with 76 mm balls. Computer simulations were also used to determine the effect of percent solids on the throughput capacity of the pebble mills. Model parameters were estimated from grinding data. A. comparison of continuous mill capacity is provided in Table 4 for a pebble mill operating at various percent solids (base case 70% solids) and for a ball mill at 70% solids. The expected product size was 65% minus 45 microns. Simulation results indicatled that percent solids in the pebble mill should be maintained at 70%. The grinding efficiency (t/kWh) for grinding with steel balls is only about 2% higher than that for pebble milling, with the difference likely due to the non-spherical character of the pebbles (Herbst and Lo, 1989). Consistent with the recommendations of this study, one rod mill was converted to a ball mill. Computer simulation predicted a capacity increase of 50 tph for these conditions. Based on the major findings of steady-state and dynamic optimization, a supervisory controli system was designed and installed. Measurements of particle size and charge volume in the mill were combined with an on-line grinding model to provide optimal
Table 4 Simulation
results for different percent solids
Percent solids
Grinding media
Energy, kWh/t
Feed rate, tph
Increase in capacity,
65 70 75 70
Pebbles Pebbles Pebbles Balls
11.9 10.7 11.7 10.5
120 133 122 136
-9.5 base case - 8.4 t2.1
%
J.A. Herbst, Y.C. Lo/lnt.
220 Table 5 Financial
summary
of the grinding
J. Miner. Process. 44-45
plant optimization
(1996) 209-221
project
Original plant capacity Increase in capacity due to Steady state optimization Dynamic optimization Incremental revenue (net present amount) Marginal costs (net present amount) Project investment (principal and interest at net present value) Project profit Internal rate of return Simple pay back period
3.7 Mt/yr 18% 10% $123,660,000 $57,074,000 $590,000 $65,996,000 114% < 1 month
reactions to hardness disturbances. This model-based control system resulted in an average feed rate which is 10% higher than the conservatively operated manual control. The benefits obtained from steady-state and dynamic optimization of the grinding plant are summarized in Table 5. Processing costs were estimated from the company’s annual report. Here we note that the internal rate of return is greater than 100% and the pay back period is less than one month. After this project was completed, the demand for low phosphorus pellets increased so that a flotation circuit was brought on-line to reduce phosphorus content. An expert system manipulates reagent additions to keep phosphorus at the desired level without overdosing (which causes problems in the balling circuits). The next logical step in the systems approach will be to coordinate grinding and flotation control.
5. Conclusions Systems optimization is an exciting new tool for improving the performance of existing comminution plants with minimal capital expenditures and high rates of return. Steady state optimization typically results in 5-15% increases in throughput and/or efficiency. Dynamic optimization typically results in 3-10% increases in throughput and/or efficiency with increased product consistency. Systems optimization projects typically produce very high rates of return, up to 150%, and short pay back times ( I- 12 months). The next step for system optimization is to bigger systems such as coordinating grinding and flotation at steady state and dynamically. As systems optimization projects are applied to bigger systems, the benefits, both technically and economically, are expected to be greater and greater.
References Guillaneau et al., 1992. A new industrial tool for design and optimization PAC 2.0, APCOM ‘92. SME, Littleton, CO.
of mineral processing
plants: USIM
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Herbst, J.A., Lo, Y.C. and Rajamani, K., 1985. Population balance model predictions of the performance of large-diameter mills. Miner. Metallurg. Process.: 114 120. Herbst, J.fi. and Pate, W.T., 1986. The power of model-based control. In: IFAC Symposium on Automation in Mining, Mineral and Metal Processing, pp. 24-29. Herbst, J.A., Alba J., F., Pate, W.T. and Oblad, A.E., 1988. Optimal control of comminution operations. Int. J. Miner. Process., 22: 275-296. Herbst, J.A. and Lo, Y.C., 1989. Grinding efficiency with balls or cones as media. Int. J. Miner. Process., 26: 141-151. Herbst, J.A. and Lo, Y.C., 1991. Optimization of mineral processing plants around the world. In: 4th Asian Symposium on Mineral Processing. Armco-Marsteel Corp., Manila, pp. 144- 157. Himmelblau, D.M. and Bischoff, K.B., 1968. Process Analysis and Simulation: Deterministic Systems. McGmw-Hill, New York. Lo, Y.C. and Herbst, J.A., 1986. Considerations of ball size effect in population balance approach to mill scale-up. In: P. Somasundaran (Editor), Advances in Mineral Processing. SME, Littleton, CO, pp. 33-47. Lo, Y.C., Herbst, J.A., Rajamani, K. and Arbiter, N., 1988. Design considerations for large diameter ball mills. Int. J. Miner. Process., 22: 75-93. Mular, A.L. and Klimpel, R.R., 1991. Optimization procedures in mineral processing plants. In: D. Malhotra, R.R. Klimpel and A.L. Mular (Editors), Evaluation and Optimization of Metallurgical Performance. SME, Littleton, CO, pp. 91- 118. Rajamani, K., Herbst, J.A., 1980. A dynamic simulator for the evaluation of grinding circuit control strategies. In: K. Schiinert (Editor), Preprints European Symposium on Particle Technology, Vol. A.I. Dechema, Amsterdam, pp. 64-81.
JamnnL of
mInERA
PROtEmlG
Int. J. Miner. Process. 44-45 (1996) 209-221
Financial implications of comminution system optimization J.A. Herbst, Y.C. Lo Control Internationul, Inc., Suit Lake City, UT, USA
Abstract Systems optimization is an exciting new tool for improving the performance of existing comminution plants while incurring minimal capital expenditures and producing high rates of return. The goal of this approach is to achieve significant performance gains without installing new equipment or replacing older, less efficient equipment. Thus, rates of return can be much higher than the historical rates of return in the process industries. This paper describes a methodology for systems optimization that has produced significant rates of return for several comminution plants. The approach seeks to determine important cost-sensl,tive, synergistic interactions in a plant, find the optimum conditions that improve the performance of the whole plant and accurately identify the tradeoffs between incremental costs and additional revenue. The interactions are described by mathematical models of the processes occurring in the system which can then be programmed into plant simulators. Two case studies of comminution plant optimization which illustrate the concepts of systems optimization are presented in this paper. The first case study concerns a system optimization project for the SAG mill circuit at a western US gold mine. The financial rate of return for that project was 145%. The second case study is a systems optimization project with a rate of return of 115% for a large iron ore concentrator in Europe.
1. Introduction The area of plant performance improvement which has been most important for the mineral industry in the last decade has been optimization of unit operations (Mular and Klimpel, 1991; Herbst and Lo, 1991). The success of unit optimization, together with the knowledge, the experience, and the tools that have been gained, have brought the next important stage in this technology: the optimization of systems. The basic definition of a System is a group of interacting and interdependent items that form a whole. Systems in the process industries have the characteristic that they 0301-7516/96/$15.00 SSDI 0301-75
0 1996 Elsevier Science B.V. All rights reserved
16(95)00037-2
J.A. Herbst, Y.C. Lo/Int.
210
+
J. Miner. Process. 44-45 (1996) 209-221
Crusher
4
Flotation Ball Mill
-
Induration
System 1
-
-l-.-Y
System 2 System 3 Fig. 1. System boundaries
defmed by process equipment.
yield a product possessing value (along with revenue derived from that value), and there are costs associated with producing that product. Systems optimization is the coordinated manipulation of system parts to yield better performance of the whole than would be achieved through independent changes in each part. The difference between unit optimization and systems optimization is that the latter emphasizes and takes advantage of the interactions and interdependence among operating variables and pieces of equipment in the plant. These interactions cause tradeoffs between incremental costs and additional revenues. For example, reducing costs in one area may be more than offset by fall of productivity caused in another area. Unless these relationships are fully known and employed, false conclusions about the financial feasibility of a project can be drawn, and there may be missed opportunities to exploit the interactions that improve process performance. Fig. 1 shows how a flowsheet (iron ore processing in this example) can be divided into three systems (an arbitrary number) with each system being encompassed by a larger system. The optimization of System 3 is enormously complex since there are a tremendous number of interacting variables. The difficulty, or time to find a solution, grows exponentially with the number of interacting variables. With the current state-ofthe-art, it is possible to optimize Systems 1 and 2; but, for now, optimization of System 3 might be out of reach. It must be noted that the optimization discussed here is much different than the optimization of, say, airline schedules or telephone networks. The later problems may involve thousands of variables, but the problem itself is direct, clear, and simple. The solution for many thousands of variables is exactly the same as for a few variables, only needing a faster computer to arrive at the solution. In the process industries, however, one must deal with processes that may be poorly understood, difficult to quantify, or difficult to measure. These are the difficulties that the methodology presented in this paper was designed to address.
J.A. Herbst, Y.C. Lo/h. Table 1 An examole
.I. Miner. Process. 44-45 (1996) 209-221
211
of false economy bv cost cuttine Media No.
Media cost, US$/year Revenue, US$/year Savings, US$/year
I
5,770,OOO 135,000,000
Media No. 2 5,290,OOO
119,500,OOo - 15,020,000
In order to accomplish systems optimization, an objective function is needed. The financially relevant objective function is the “profit” from a system expressed as: “PROFIT” = REVENUE - OPERATINGCOSTS The revenue from the product of a system may be real, such as from System 3, but in other systems, such as 1 and 2, the revenue represents the potential value of the product. Whether real or potential, the revenue should properly reflect the influence of the optimization variables, as should the amount for operating costs. Since the revenue amount in Eq. 1 may be potential, so will the profit be a potential amount. It will be, nonetheless, a good measure, since it is affected by the interactions between revenue and costs. An exciting prospect for systems optimization is the reduction of plant operating costs, which in the current economic climate is a top priority for many companies. Eq. 1 determines which cost cutting measures are effective and which are not. Consider first a case where cutting costs would not be attractive, i.e., a case in which reduction in cost is less than the loss of revenue (or productivity) resulting from the cost cutting change. Suppose, for example, that a 50,000 tpd iron ore regrinding plant is considering a switch from its current grinding media (No. 1) to a cheaper grinding media (No. 2). The cheaper media is known to have a higher wear rate and is irregularly shaped so that it will produce lower grinding rates. The cost of media No. 2 is 34% of the cost of media No. 1, and the wear rate per ton of ore will likely be 225% higher. The mill capacity using media No. 2 will probably drop by 11% so the total cost of media No. 2 is 8% less than media No. 1. Table 1 shows a considerable decrease in “profit” (Eq. I), i.e., a negative savings, so that the switch to grinding media No. 2 is economically unattractive. Next, consider the opposite case in which a reduction in cost is larger than the loss of revenue or productivity. Suppose that at a 100,000 tpd copper ore concentrator a coarser grind is being proposed to reduce mill operating costs. As a result, energy consumption
Table 2 An example of true economy by cost cutting
Energy costs, US$/year Revenue, lJS$/year Savings, US$/year
Base case
1% less liberation
15,840,000 253,011,OOo
14,520,OOO 252,702,450 1,011,450
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Classifiertype Classifieroperation
Fig. 2. Variables for optimization
in a ball mill grinding circuit.
is expected to be reduced from 12 kWh to 11.O kWh. If the grind-recovery curve is fairly flat, the recovery may drop by 0.1 %, producing only a small reduction in revenue at the same feed tonnage. Table 2 shows a positive savings so that coarsening the grind is an economically attractive move. The authors and their co-workers have developed a systematic approach using mathematical models to analyze situations like these with a view toward determining promising alternatives for systems optimization in a financial sense. The first step of this methodology is concerned with steady-state operating conditions involving variables such as those shown in Fig. 2 for a ball mill grinding circuit. Even within a single-unit operation, there are many variables that can be manipulated. Thus, there exists considerable potential for optimization. When a larger system is considered, such as system 2 in Fig. 1, the number of manipulated variables goes up accordingly. The principle of coordination of manipulated variables still holds for the larger system. Steady-state optimization yields specific recommendations for steady-state target variables and predictions of the expected improvement after implementing such changes. In practical terms, plants never operate at steady-state, but rather are in a perpetual transient state due to a variety of disturbances (Herbst et al., 1988). The benefit predicted by steady-state simulations is for average or normal conditions. Process disturbances cause minute-by-minute deviations from that average either for the worse, harder ore/coarser feed, for example, or for the better, softer ore/finer feed. In this methodology, dynamic optimization must be performed to compensate for the process disturbances and to react to these disturbances in such a manner that plant productivity is continually maximized. Dynamic models in the form of plant simulators can be used for off-line control strategy development, and on-line dynamic models can be used to help make well-informed
responses
to each disturbance.
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2. Methodology The methodology which is described in this paper is summarized in Fig. 3. Steady-state optimization comprises batch and plant testing, modeling, computer simulation, and plant verification. A detailed description of plant and lab tests and the associated optimization procedure has been outlined in previous publications (Herbst et al., 1985; Lo et al., 1988). The effects of critical operating variables, such as ball size, mill percent solids, etc., are investigated during the lab experiments. The data obtained in laboratory or pilot-scale grinding experiments are used as input to the estimator/simulator GRINDS1M.S’“. Once the breakage kinetic parameters are determined, computer simulations are performed to determine energy requirements at a desired product size for different operating conditions and circuit configurations. Computer simulations yield specific recommendations for changing critical operating variables, classification and circuit configuration along with a prediction of the expected improvement. The recommended conditions are then implemented in the plant, where comparative tests are usually carried out on two mills in parallel. Plant tests provide verification of the methodology developed for optimization. The methodology for dynamic optimization involves plant disturbance testing, laboratory evaluation, model building and control strategy evaluation. The model for ore hardness disturbances (magnitude and frequency) is developed from plant samples taken at varying time intervals. These ore samples are subjected to laboratory grinding tests which yield hardness variations as a function of sampling time. From these tests it is possible to obtain a profile of the variations in ore hardness for the grinding circuit feed. The evaluation of alternative control strategies for a given application involves the use of a dynamic simulator (Rajamani and Herbst, 1980; Herbst and Pate, 1986).
D&e Objectiie
PlantAudit steadystate Dynamic
Receive Benefits
t Conduct Lab or
Evaluate In Plant
PilotScaleTests
lmptementBest Alternative
Evaluate lnteractbns Fig. 3. Methodology for optimization.
214
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Dynamic simulations help eliminate costly experimental plant testing during preliminary screening of control strategies which may include manual control, stabilizing control and model-based control. For model-based control, the model is a reduced form of the one used in a dynamic simulator. The model must be simple enough for rapid on-line calculations, but detailed enough to faithfully reproduce the essential dynamic characteristic of the grinding process. In addition, an estimator and an optimizer are needed for model-based control. The estimator combines measurements within the process with model information to determine the state of the system at any instant in time. The optimizer uses the current state of the system and the model to select the path for manipulated variables that will achieve the process objectives in an optimal way. Dynamic optimization requires either parallel or serial on-off testing of the new control strategy until enough data has been collected for a statistically significant comparison to be made with the old control strategy. Then, Eq. l is used to compute measures such as the internal rate of return and pay back period for the systems optimization project. 3. Models for optimization System models are built or assembled from subsystem models connected to reflect the configuration or flowsheet of the system (Himmelblau and Bischoff, 1968; Guillaneau et al., 1992). A subsystem model itself may be composed of subprocess models that account for influential phenomena occurring in that process. When built in this manner the system model gives an accurate description of the interactions occurring within the system and of the influence of manipulated variables, provided that the subsystem models are an accurate description of subprocess interactions and the influence of manipulated variables. At the foundation of today’s comminution systems models are the breakage kinetics and the transport of material through the comminution device. The breakage kinetics for an assemblage of particles in a comminution device are quantified by the selection and breakage functions. The selection function si is the fractional rate of breakage for particles of size di. The breakage function 6,, is the size distribution of the fragments from the breakage of particles of size dj. The selection and breakage functions can be determined for almost any comminution device, but this determination must be done by experiment with device in the plant and, ideally, in the laboratory as well. The dependence of selection and breakage functions on particle size and operating conditions is unique for each type of device and also must be determined by experiment. The transport of material through a device can be quantified by its residence-time-distribution (or RID). The RTD of tumbling mills is typically well described by the iv-mixers-in-series model, which describes anything from a single perfect mixer to plug flow. The transport of material through a crusher is harder to envision, but the N-mixers-in-series model is adequate because of its flexibility. A rate expression that is applicable to the internal size distribution of crushers and tumbling mills (AG, SAG and ball) is: i-
qqP4
dr
= M,m,i
I
- M,m, - siHpmi + c bijsjHpmj + K, H,b” j= I
(2)
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where mi is the mass fraction in the ith size interval, H is the total mass of material, the feed ram is M, and the discharge rate is M,. Self-breakage phenomena occurs primarily in AG and SAG mills where large pieces of ore act as the grinding media. The self-breakage rate is K, and the self-breakage function is b,!. The discharge rate depends on the type of comminution device. The optimization case studies that follow show how the rate expressions are developed for SAG, pebble, and ball mills.
4. Industrial applications 4.1. SAG mill optimization project The first example is concerned with the optimization of a SAG mill grinding circuit at a gold operation in the western US. Over 4,000 tpd of ore is ground in a SAG mill-ball mill circuit and then leached by CIL. Profit is very ore tonnage sensitive at this plant. An optimization project was initiated to increase the capacity of the SAG mill by at least 4%, since it was the bottleneck in the plant. The total cost of the project was $100,000 for steady state and dynamic optimization. Steady-state optimization was required. to investigate a number of options for increasing capacity. These options were ball load, mill percent solids and grate design. Dynamic optimization required the purchase of a PC-based control system. The financial break-even point for the project was a 2’% increase in feed rate. The typical increase in feed rate for projects of this type varies from 5 to 20%, depending on the number of plant variables available for manipulation.
Fig. 4. Flowsheet of SAC milling grinding circuit for gold ore.
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The SAG mill circuit and instrumentation are shown schematically in Fig. 4. The feed to the circuit has been crushed to minus 15 mm by the primary crusher. The SAG mill is 6 m by 2 m and is driven by a single 1250 kW motor. The mill discharge is sent to a double deck vibrating screen with 12.7 and 7.9 mm openings. The 12.7 mm oversize is crushed in a 1.3 m short-head cone crusher and recycled to the SAG mill, while the 12.7 by 7.9 mm fraction is sent back to the SAG mill. The screen undersize constitutes the ball mill feed. SAG mill sampling was performed to evaluate the mill response to process variable changes, to measure the mean residence time of various ore types, and to determine the frequency and magnitude of the ore hardness disturbances. The process response tests were done on four ore types and based on a 2-by-2 factorial design with one center point. The manipulated variables for each test were the mill feed tonnage and the mill feed percent solids. The model for the SAG mill can be derived from Eq. 2 by first recognizing that within a SAG mill the larger rocks act as media to break smaller rocks and, eventually, themselves; balls in a SAG mill can break rocks no larger than some critical size. The different breakage mechanisms for big rocks and smaller rocks (which will be referred hereafter as particles) must therefore lead to grinding kinetic models appropriate for each size. In the context of this model, rocks are defined to be ore pieces larger than the grate opening and are assumed to break completely into particles at a rate proportional to the mass holdup of rocks. This assumption leads to the following expression for the kinetics of rock breakage:
(3) where H, is total mass of rocks in the mill, Ma is the mass flowrate of rocks in the mill feed and K, is the grindability parameter for rocks. K, depends upon ore characteristics such as hardness and feed size, and upon mill conditions such as percent solids, mill power and holdup. A useful relationship between K, and mill specific energy similar to one proven valid for ball mill grinding is: K,=K$-
R
where P is mill power and K, E is the upon ore type, mill percent solids, etc. laboratory testing. Eq. 2 is used to describe the particles. to the holdup of particles in the mill by M,=A,K,H,
(4) specific grindability parameter which depends This dependence must be found by plant and The mill discharge rate, M,, in Eq. 2 is related the following: (5)
where A, is the grate open area factor and K, is the discharge parameter for particles. A, describes the effect of the number and area of grate openings on the slurry discharge flowrate and how it changes with filling level. K, depends upon factors such as size
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Table 3 Financial summary of SAG optimization project Original plant capacity increase in capacity due to Steady state optimization Dynamic optimization Project investment (principal and interest) Incremental revenue Marginal costs Project profit, 1st year Internal rate of return Simple pay back period
4,200 tpd 4.7% 4.4% $108,OQO $2,494,000 $909,000 $1,477,000 145% < 1 month
and mill conditions. The relationship between K, equation: K,
= Kg
and specific energy is given by an
$ P
where Kz is the specific discharge constant for particles. The data from the factorial design test were used to estimate model parameters. Both a steady state and a dynamic simulator of the SAG mill/crushing circuit were built. The steady state simulator was used to investigate the following interactions: percent solids/power ball load/power grate open area/power feed rate/power
percent solids/filling ball load/filling grate open area/filling feed rate/filling
Increasing the grate opening was found to provide the biggest boost to capacity; therefore, the grate was redesigned to increase the open area by 8%. The average feed rate after this modification was 4.7% higher than before. The dynamic simulator was used to test the effectiveness of control strategies in the presence of feed size and ore hardness variations. The frequency and magnitude of ore hardness variations determined from plant testing were used in the dynamic simulator. For dynamic optimization, an expert system was designed to work in conjunction with the process models to maximize feed rate to the SAG mill. The model-based expert system was found by simulation to be the best control strategy and so was implemented in the plant. To document the performance of the model-based expert control system, a series of on/off test were conducted in which the mill was operated alternatively for set time periods under model-based control and PI control of mill power. The model-based expert system produced an average of 4.4% more throughput than the PI controller. During the testing period energy consumption under PI control was 5.2 kWh/t, while under model-based expert control it was 4.9 kWb/‘t. Together the grate modification and control system increased the average tonnage from 175 tph to 191 tph. Increasing the grate open area also improved the control system’s quickness of response.
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The financial benefits obtained from the grinding optimization are summarized in Table 3. The capital investment is the cost of the project plus 8% interest. This amount is paid by the end of the first year. The incremental revenue is the annual increase in revenue from the 16 tph increase in tonnage. The marginal costs are the total mining and processing costs (minus milling costs) for the additional 16 tph. For this analysis, typical mining and processing costs were used. Milling costs are not included in the incremental marginal costs for two reasons. The first is that most of the additional milling costs are for the control system, which is reported as capital investment. The second reason is that the reduction in energy per ton resulted in no increase in total milling costs at the higher tonnage. The profit is the incremental revenue minus the marginal costs minus the capital investment for the project. The internal rate of return for this project is: “Profit” IRR = 100% x
Capital Investment + Marginal Costs 1,477,OOo
= 100% x
108,000 + 909,000
= 145%
(7)
Clearly, this project generated a high rate of return and an extremely short pay back period. Even considering some uncertainty in marginal costs, this project is fully justified. The best estimates possible of incremental revenue and marginal costs are certainly desirable when justifying a systems optimization before it is started. 4.2. Grinding plant optimization project The second example is concerned with the grinding circuits in an iron ore concentrator in Europe. The concentrator and the pelletizing plant were built in the mid-1960s. The annual capacity was 1.5 Mt. Over the years, the capacity has been increased to 3.7 Mt annually by lengthening the pebble mills, increasing the mill speed and replacing the induration system. In 1988, it became apparent that the grinding circuits had become the
ol Fbpoae E Ebsize v”T v mafilhg VS Vafi&bmpad D !zt? W Wdghtometor : Lwd
Fig. 5. Grinding circuit and instrumentation.
PSM
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major limitation to further tonnage expansion. Before undertaking a major capital investment for expansion, an intermediate project was initiated with the objective of reaching 4.5 Mt pellets per year through systems optimization of the current facilities. The objective was to be achieved by decreasing the magnetite feed to the concentrator from minus 25 mm to minus 10 mm, converting from rod mill to ball mill grinding and by adding a supervisory control system. The grinding circuit and instrumentation are illustrated in Fig. 5. Ball mill tests were performed to evaluate the alternative of converting the 3.6 X 4.8 m rod mill to a ball mill. Grinding rates for different particle sizes are a strong function of the b’all size distribution in the mill (Lo and Herbst, 1986). So, lab grinding tests were conducted with top ball sizes of 38, 50, 63, 76 and 88 mm. In each case, the ball size distribution used approximated that of an “equilibrium charge distribution” often used in laboratory tests for scale-up design (Lo and Herbs& 1986). Computer simulations of the grinding process were conducted to predict continuous plant performance of the rod mill after conversion to a ball mill. Parameters for the simulation were estimated from plant data and from laboratory batch grinding tests. Mill throughput capacity in open and closed circuit configurations was predicted for different ball sizes and was compared to the base line feed rate of the rod mill. The expected product size was 83% minus 295 microns. Simulation results indicated that a 30% increase in capacity could be expected from a ball mill operating in closed circuit with 76 mm balls. Computer simulations were also used to determine the effect of percent solids on the throughput capacity of the pebble mills. Model parameters were estimated from grinding data. A. comparison of continuous mill capacity is provided in Table 4 for a pebble mill operating at various percent solids (base case 70% solids) and for a ball mill at 70% solids. The expected product size was 65% minus 45 microns. Simulation results indicatled that percent solids in the pebble mill should be maintained at 70%. The grinding efficiency (t/kWh) for grinding with steel balls is only about 2% higher than that for pebble milling, with the difference likely due to the non-spherical character of the pebbles (Herbst and Lo, 1989). Consistent with the recommendations of this study, one rod mill was converted to a ball mill. Computer simulation predicted a capacity increase of 50 tph for these conditions. Based on the major findings of steady-state and dynamic optimization, a supervisory controli system was designed and installed. Measurements of particle size and charge volume in the mill were combined with an on-line grinding model to provide optimal
Table 4 Simulation
results for different percent solids
Percent solids
Grinding media
Energy, kWh/t
Feed rate, tph
Increase in capacity,
65 70 75 70
Pebbles Pebbles Pebbles Balls
11.9 10.7 11.7 10.5
120 133 122 136
-9.5 base case - 8.4 t2.1
%
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220 Table 5 Financial
summary
of the grinding
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project
Original plant capacity Increase in capacity due to Steady state optimization Dynamic optimization Incremental revenue (net present amount) Marginal costs (net present amount) Project investment (principal and interest at net present value) Project profit Internal rate of return Simple pay back period
3.7 Mt/yr 18% 10% $123,660,000 $57,074,000 $590,000 $65,996,000 114% < 1 month
reactions to hardness disturbances. This model-based control system resulted in an average feed rate which is 10% higher than the conservatively operated manual control. The benefits obtained from steady-state and dynamic optimization of the grinding plant are summarized in Table 5. Processing costs were estimated from the company’s annual report. Here we note that the internal rate of return is greater than 100% and the pay back period is less than one month. After this project was completed, the demand for low phosphorus pellets increased so that a flotation circuit was brought on-line to reduce phosphorus content. An expert system manipulates reagent additions to keep phosphorus at the desired level without overdosing (which causes problems in the balling circuits). The next logical step in the systems approach will be to coordinate grinding and flotation control.
5. Conclusions Systems optimization is an exciting new tool for improving the performance of existing comminution plants with minimal capital expenditures and high rates of return. Steady state optimization typically results in 5-15% increases in throughput and/or efficiency. Dynamic optimization typically results in 3-10% increases in throughput and/or efficiency with increased product consistency. Systems optimization projects typically produce very high rates of return, up to 150%, and short pay back times ( I- 12 months). The next step for system optimization is to bigger systems such as coordinating grinding and flotation at steady state and dynamically. As systems optimization projects are applied to bigger systems, the benefits, both technically and economically, are expected to be greater and greater.
References Guillaneau et al., 1992. A new industrial tool for design and optimization PAC 2.0, APCOM ‘92. SME, Littleton, CO.
of mineral processing
plants: USIM
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Herbst, J.A., Lo, Y.C. and Rajamani, K., 1985. Population balance model predictions of the performance of large-diameter mills. Miner. Metallurg. Process.: 114 120. Herbst, J.fi. and Pate, W.T., 1986. The power of model-based control. In: IFAC Symposium on Automation in Mining, Mineral and Metal Processing, pp. 24-29. Herbst, J.A., Alba J., F., Pate, W.T. and Oblad, A.E., 1988. Optimal control of comminution operations. Int. J. Miner. Process., 22: 275-296. Herbst, J.A. and Lo, Y.C., 1989. Grinding efficiency with balls or cones as media. Int. J. Miner. Process., 26: 141-151. Herbst, J.A. and Lo, Y.C., 1991. Optimization of mineral processing plants around the world. In: 4th Asian Symposium on Mineral Processing. Armco-Marsteel Corp., Manila, pp. 144- 157. Himmelblau, D.M. and Bischoff, K.B., 1968. Process Analysis and Simulation: Deterministic Systems. McGmw-Hill, New York. Lo, Y.C. and Herbst, J.A., 1986. Considerations of ball size effect in population balance approach to mill scale-up. In: P. Somasundaran (Editor), Advances in Mineral Processing. SME, Littleton, CO, pp. 33-47. Lo, Y.C., Herbst, J.A., Rajamani, K. and Arbiter, N., 1988. Design considerations for large diameter ball mills. Int. J. Miner. Process., 22: 75-93. Mular, A.L. and Klimpel, R.R., 1991. Optimization procedures in mineral processing plants. In: D. Malhotra, R.R. Klimpel and A.L. Mular (Editors), Evaluation and Optimization of Metallurgical Performance. SME, Littleton, CO, pp. 91- 118. Rajamani, K., Herbst, J.A., 1980. A dynamic simulator for the evaluation of grinding circuit control strategies. In: K. Schiinert (Editor), Preprints European Symposium on Particle Technology, Vol. A.I. Dechema, Amsterdam, pp. 64-81.