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Solution Strategies to Stochastic Design of Mineral Flotation Plants
Mario R. Eden, John D. Siirola and Gavin P. Towler (Editors) Proceedings of the 8 th International Conference on Foundations of Computer-Aided Process Design – FOCAPD 2014 July 13-17, 2014, Cle Elum, Washington, USA © 2014 Elsevier B.V. All rights reserved.
Solution Strategies to Stochastic Design of Mineral Flotation Plants Nathalie Jamett,a Luis Cisternas,a,b,* Juan Pablo Vielmac a
Universidad de Antofagasta, Av. Universidad de Antofagasta s/n, Antofagasta, Chile CICITEM, Avda. José Miguel Carrera Nº170, Antofagasta, Chile c Sloan School of Management, Massachusetts Institute of Technology, USA [email protected] b
Abstract The aim of this paper is to analyze different strategies for the solution of a two-stage stochastic problem applied to a copper flotation circuit. In the optimization problem, we want to find the optimal configuration of a superstructure, equipment design and operational conditions such as residence time and stream flows, among others variables. Variability is considered in the copper price and ore grade. This variability is represented by scenarios with respective probabilities of occurrence. Two solution strategies are compared. As a first step, we solve the fully adaptable model, and then two partially adaptive strategies are analyzed. One of the partially adaptive models delivers the best profit, and it is closer to the reality of the operation. Keywords: Flotation, Uncertainty, Stochastic, Copper.
1. Introduction Flotation is a physicochemical process that allows the separation of minerals, such as copper sulfide and molybdenum, from the remaining minerals that form most of the parent rock substrate, including contaminants such as arsenic minerals (Bruckard et al., 2010). The methods employed in flotation circuit design can be generally classified as heuristic, rigorous or hybrid. Heuristic designs are based on the application of rules that permit feasible solutions to be attained and employ the experience of the designer (Chan and Prince, 1989). Rigorous designs involve models that describe the phenomenon or the relations among the problem variables and propose a superstructure that represents a universe of alternatives for circuits, which, after an optimization technique has been applied, allow the optimal alternative to be determined (Loveday and Brouckaert, 1995; Cisternas et al., 2006; Mendez et al., 2009). Hybrid designs are a combination of the two preceding methods in which graphic tools are employed in addition to models that describe the process, computational tools and the experience of the designer (e.g. Loveday and Brouckaert, 1995; Herrera et al., 2009). Most of the preceding methods for the design of flotation circuits do not consider that the design may contain parameters that cannot be completely defined or values that may be subject to degrees of uncertainty. These parameters and values usually involve external factors, such as product demand, economic factors, environmental factors or internal plant conditions, including kinetic and diffusion constants, which may lead to an inefficient process design (Kraslawski, 1989). Jamett et al. (2012) applied stochastic programming to flotation design using a simple flotation model. Large numbers of variables are handled in the design of flotation circuits, some of which may involve uncertainty, such as the feed grade, metal or product price, toxic-
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N. Jamett et al.
element composition, mineral distribution and particle size. Simonsen and Perry (1999) have indicated that the uncertainty in mining operations includes market characteristics (especially price), mineral reserves and their composition (feed grade), the functioning of the process (concentrate grades, recoveries), operational and capital costs and the lengths of the planning phases. This study applies two-stage stochastic programming with recourse (Birge and Louveaux, 1997) to design the copper flotation process, including stochastic parameters. In this study, different strategies for the resolution of the equivalent deterministic model of the stochastic model (expected value) are applied with the aim of finding the best resolution strategy. Furthermore, the design of the process is assessed by considering the mean values of the stochastic parameters to determine the importance of the previously used tools.
2. Model development 2.1. Superstructure The flotation process depends on several design and operational variables. We consider a superstructure that includes the following three flotation stages: the rougher, which processes the feed; the cleaner, which generates the final concentrate; and the scavenger, which generates the final tailing, as shown in Fig. (1). This is a simple superstructure but is used here as an example.
Figure 1. Superstructure of the flotation circuit.
The objective is to maximize the total income with respect to the operation conditions and process design. The decision variables to be optimized are divided into design and
Solution Strategies to Stochastic Design of Mineral Flotation Plants
341
operating variables. The design variables include equipment dimensions, such as the cell volume and total number of cells for each stage. The operating variables correspond to operating times for each cell at each stage and the directions of tails and concentrate streams. In stochastic problems, the operating variables (second level variables) are able to adapt to each scenario to increase the total income. Moreover, the design variables (first level) are the same for all scenarios. 2.2. Deterministic equivalent model of the stochastic problem The resolution of the stochastic two-stage model can be performed through an equivalent deterministic model (Birge and Louveaux, 1997). In the equivalent deterministic model, the uncertain parameters are represented through scenarios with their respective probabilities of occurrence. In addition, first- and second-level variables must be identified. The first-level variables are fixed for all scenarios, and the secondlevel variables change for each scenario. See Eq. (1) – Eq. (3). ் ή ݕ௪ ݔܽܯെܿ ் σ௪ ሺݓሻ ή ݀௪ s.t ܣή ݔൌ ܾ ܶ௪ ή ݔ ܹ௪ ݕ ڄ௪ ൌ ݄௪ ݓ ݔ Ͳǡ ݕ௪ Ͳ
(1) (2) (3)
Eq. (1) represents the profits of the process, where ܿ ் is the total cost of the flotation cells. The term ሺݓሻ corresponds to the probability of occurrence of each scenario. The ் ή ݕ௪ term corresponds to the income of the plant, primarily depending on ݀௪ concentrated flows and the concentrate grade obtained, which are variable in each scenario. Eq. (2) represents the first-level restrictions, such as volume restrictions. Finally, Eq. (3) represents the second-level constraints, such as mass balances per cell. The complete model corresponds to a MINLP model, which is difficult to solve due to its large number of binary and continuous variables.
3. Application example Consider the superstructure shown in Fig. (1), which involves the following three flotation stages: the rougher, the scavenger and the cleaner. The feed has four mineralogical species, fully liberated chalcopyrite (CuFeS2), partially liberated chalcopyrite, tennantite (Cu12As4S13), and gangue. Each of these species has a specific floatability. We can therefore identify the following three chemical species of interest to our problem: copper, arsenic and gangue. The treatment plant capacity is 1,200 t d-1. The stochastic parameters are the feed-copper grade and the copper price. The copper price varies between 1 - 4 kUS/t, and the total copper grade of the feed ranges from 0.45 to 2.4%. Because of the complexity of the problem, the numbers of cells by bank are fixed according to the notes made by Bourke (2002). To solve the optimization problem, the following two strategies are proposed: x First, solve the fully adaptive model, in which all variables become second-stage variables and change with each scenario. Each scenario is run separately. x After determining the optimum for each scenario, as was done in the fully adaptive model, the volume (first-level variable) is fixed to the maximum value of all of the volumes found in each of the scenarios. The reason for fixing to the maximum value is ensure the capacity of each flotation stage. Each scenario is optimized, finding new second-level variables. We denote this Solution Methodology A.
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x As a second option, after the solution of the fully adaptive model, the binary variable of the stream direction of the superstructure is fixed for each scenario. In this manner, all binary variables are became parameters, making the problem easier to solve. The optimal solution is found in a single model that is a function of w (scenarios). Thus, the first-level variables are optimized considering all scenarios at once. We denote this Solution Methodology B.
4. Results The problem is solved in a computer with an Intel core i7 processor (2.7 GHz) and 6 GB of RAM. The optimization software used was GAMS with BONMIN as the MINLP solver. At first, other MINLP solvers were used, such as BARON and COUENNE, but because of the amount of variables and equations, it was not possible to find a solution to the stochastic problem. Employing solution methodology B ensures that profits are maximized, by finding the optimal configuration and by minimizing costs through the cost of the cells. By employing the methodology of solution A, the variable volume is set to the maximum found, corresponding to scenario 9 (high price, high grade) in the fully adaptive model. These dimensions correspond to the optimum for that particular scenario but are oversized for others, increasing total costs. The difference between the two methodologies is approximately 30,000 US$/y. Although the difference between each methodology is not very large, it is noteworthy that through this methodology is possible to solve the problem and find an optimal solution. Table 1 shows the incomes and the differences in volume for both methodologies. Table1: Comparison of the results of solution methodologies A and B. Total Income MMUS$/y 3
Volume Rougher Cell m
3
Volume Scavenger Cell m 3
Volume Cleaner Cell m
Solution Methodology A
Solution Methodology B
2.7299
2.7599
75
27
41
17
6
5
Once the resolution of the stochastic model is possible, its effectiveness is evaluated as a tool for the design of the flotation process. The design and profits obtained by the stochastic and deterministic model are compared. In the deterministic model, the resolution is carried out through weighted averages of stochastic parameters. In this case, the feed grade corresponds to 0.56% chalcopyrite, 2.39% partially liberated chalcopyrite and 0.45% tennantite. The price of the metal corresponds to 5018 US$/ton. A single circuit and single cell design are obtained, which should be able to cushion any variability of the parameters. By running each scenario with this configuration to calculate the actual profit, we found that the first scenario, which considers a lower copper price and lower feed grade, is not capable of operating at the design conditions. It is for this reason that the use of mean values it is not the best option. However, it is not possible to determine the real value of total income; therefore, comparison with other alternatives is not possible. Fig. (2A) shows the optimal configuration for the deterministic model, and Fig. (2B) shows the optimal configuration for scenario 1 using the stochastic model. The optimal configuration for that scenario is to recirculate the scavenger concentrate to the rougher stage and not to the cleaner stage. This
Solution Strategies to Stochastic Design of Mineral Flotation Plants
343
configuration ensures a higher copper grade in the concentrate, cushioning the drop in the feed grade and copper price, lowering the costs for treatment, refining and penalization.
A
B
Figure 2: Optimal configuration of the deterministic model (A) and optimal configuration of stochastic model scenario 1 (B).
5. Conclusions A model for the design of flotation circuits under uncertainty has been presented. Uncertainty is represented by scenarios that include changes in the feed grade and in the metal price. The model allows the operating conditions (residence time and mass flows of each stream) and flow structure (tail and concentrate stream of cleaner and scavenger stage) to be changed for each scenario while the fixed design (size of cells in flotation stages) for all scenarios is maintained. The model can be modified to include other uncertainties and other adaptive variables. To solve the two-stage stochastic model, two solution strategies were proposed. The results show that the use of average values for the stochastic parameters leads to an inefficient design and hence a decrease in the profits made in the process. Finally, it can be concluded that the use of stochastic programming can be a beneficial tool in the design of a metallurgical process, specifically the copper flotation process. The optimal configuration is capable of adapting to uncertainty, leading to an increase in the company profits.
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6. Acknowledgments Financial support from CONICYT (Fondecyt 1120794), CICITEM (R10C1004) and the Antofagasta Regional Government is gratefully acknowledged.
7. References J. Birge, F. Louveaux, 1997, Introduction to stochastic programming. P. Bourke, 2002, Selecting flotation cells: How many and what size?, MEI Online, http://www.min-eng.com/frothflotation/4.html. W. Bruckard, K. Davey, F. Jorgensen, 2010, Development and evaluation of an early removal process for the beneficiation of arsenic-bearing copper ores, Minerals Engineering, 23 (15), 1167-1173. W. Chan, R. Prince, 1989, Heuristic Evolutionary Synthesis with Non – Sharp Separators, Computers Chem. Eng.13, 1207-1219. L. Cisternas, D. Méndez, E. Gálvez, R. Jorquera, 2006, A MILP model for design of flotation circuits with bank/column and regrind/no regrind selection, International Journal of Mineral Processing, 79 (4), 253-263. E. Gálvez, 1998, A shortcut procedure for the design of mineral separation circuits, Mineral Engineering, 31(2), 113-123. G.Herrera, E. Gálvez, L. Cisternas, 2009, Mineral Processing Flow Sheet Design Through A Group Contribution Method, Computer Aided Chemical Engineering, 26, 213-218. N. Jamett, J. Vielma, L. Cisternas, 2012, Design of Flotation Circuits Including Uncertainty and Water Efficiency, Computer Aided Chemical Engineering, 30, 1277-1281 A. Kraslawski, 1989, Review of Applications of Various Types of Uncertainty in Chemical Engineering, Chem, Eng. Process, 26, 185-191. B. Loveday, C. Brouckaert, 1995, An analysis of Flotation Circuit Design Principles. The Chemical Engineering Journal, 59, 15-21. D. Méndez, E. Gálvez, L. Cisternas, 2009, Modeling of grinding and classification circuits as applied to the design of flotation processes, Computers & Chemical Engineering, 33 (1), 97111. H. Simonsen, J. Perry, 1999, Risk identification, assessment and management in the mining and metallurgical industries, J. of the South African Institute of Mining and Metallurgy, 99 (6) 321- 329.
Solution Strategies to Stochastic Design of Mineral Flotation Plants Nathalie Jamett,a Luis Cisternas,a,b,* Juan Pablo Vielmac a
Universidad de Antofagasta, Av. Universidad de Antofagasta s/n, Antofagasta, Chile CICITEM, Avda. José Miguel Carrera Nº170, Antofagasta, Chile c Sloan School of Management, Massachusetts Institute of Technology, USA [email protected] b
Abstract The aim of this paper is to analyze different strategies for the solution of a two-stage stochastic problem applied to a copper flotation circuit. In the optimization problem, we want to find the optimal configuration of a superstructure, equipment design and operational conditions such as residence time and stream flows, among others variables. Variability is considered in the copper price and ore grade. This variability is represented by scenarios with respective probabilities of occurrence. Two solution strategies are compared. As a first step, we solve the fully adaptable model, and then two partially adaptive strategies are analyzed. One of the partially adaptive models delivers the best profit, and it is closer to the reality of the operation. Keywords: Flotation, Uncertainty, Stochastic, Copper.
1. Introduction Flotation is a physicochemical process that allows the separation of minerals, such as copper sulfide and molybdenum, from the remaining minerals that form most of the parent rock substrate, including contaminants such as arsenic minerals (Bruckard et al., 2010). The methods employed in flotation circuit design can be generally classified as heuristic, rigorous or hybrid. Heuristic designs are based on the application of rules that permit feasible solutions to be attained and employ the experience of the designer (Chan and Prince, 1989). Rigorous designs involve models that describe the phenomenon or the relations among the problem variables and propose a superstructure that represents a universe of alternatives for circuits, which, after an optimization technique has been applied, allow the optimal alternative to be determined (Loveday and Brouckaert, 1995; Cisternas et al., 2006; Mendez et al., 2009). Hybrid designs are a combination of the two preceding methods in which graphic tools are employed in addition to models that describe the process, computational tools and the experience of the designer (e.g. Loveday and Brouckaert, 1995; Herrera et al., 2009). Most of the preceding methods for the design of flotation circuits do not consider that the design may contain parameters that cannot be completely defined or values that may be subject to degrees of uncertainty. These parameters and values usually involve external factors, such as product demand, economic factors, environmental factors or internal plant conditions, including kinetic and diffusion constants, which may lead to an inefficient process design (Kraslawski, 1989). Jamett et al. (2012) applied stochastic programming to flotation design using a simple flotation model. Large numbers of variables are handled in the design of flotation circuits, some of which may involve uncertainty, such as the feed grade, metal or product price, toxic-
340
N. Jamett et al.
element composition, mineral distribution and particle size. Simonsen and Perry (1999) have indicated that the uncertainty in mining operations includes market characteristics (especially price), mineral reserves and their composition (feed grade), the functioning of the process (concentrate grades, recoveries), operational and capital costs and the lengths of the planning phases. This study applies two-stage stochastic programming with recourse (Birge and Louveaux, 1997) to design the copper flotation process, including stochastic parameters. In this study, different strategies for the resolution of the equivalent deterministic model of the stochastic model (expected value) are applied with the aim of finding the best resolution strategy. Furthermore, the design of the process is assessed by considering the mean values of the stochastic parameters to determine the importance of the previously used tools.
2. Model development 2.1. Superstructure The flotation process depends on several design and operational variables. We consider a superstructure that includes the following three flotation stages: the rougher, which processes the feed; the cleaner, which generates the final concentrate; and the scavenger, which generates the final tailing, as shown in Fig. (1). This is a simple superstructure but is used here as an example.
Figure 1. Superstructure of the flotation circuit.
The objective is to maximize the total income with respect to the operation conditions and process design. The decision variables to be optimized are divided into design and
Solution Strategies to Stochastic Design of Mineral Flotation Plants
341
operating variables. The design variables include equipment dimensions, such as the cell volume and total number of cells for each stage. The operating variables correspond to operating times for each cell at each stage and the directions of tails and concentrate streams. In stochastic problems, the operating variables (second level variables) are able to adapt to each scenario to increase the total income. Moreover, the design variables (first level) are the same for all scenarios. 2.2. Deterministic equivalent model of the stochastic problem The resolution of the stochastic two-stage model can be performed through an equivalent deterministic model (Birge and Louveaux, 1997). In the equivalent deterministic model, the uncertain parameters are represented through scenarios with their respective probabilities of occurrence. In addition, first- and second-level variables must be identified. The first-level variables are fixed for all scenarios, and the secondlevel variables change for each scenario. See Eq. (1) – Eq. (3). ் ή ݕ௪ ݔܽܯെܿ ் σ௪ ሺݓሻ ή ݀௪ s.t ܣή ݔൌ ܾ ܶ௪ ή ݔ ܹ௪ ݕ ڄ௪ ൌ ݄௪ ݓ ݔ Ͳǡ ݕ௪ Ͳ
(1) (2) (3)
Eq. (1) represents the profits of the process, where ܿ ் is the total cost of the flotation cells. The term ሺݓሻ corresponds to the probability of occurrence of each scenario. The ் ή ݕ௪ term corresponds to the income of the plant, primarily depending on ݀௪ concentrated flows and the concentrate grade obtained, which are variable in each scenario. Eq. (2) represents the first-level restrictions, such as volume restrictions. Finally, Eq. (3) represents the second-level constraints, such as mass balances per cell. The complete model corresponds to a MINLP model, which is difficult to solve due to its large number of binary and continuous variables.
3. Application example Consider the superstructure shown in Fig. (1), which involves the following three flotation stages: the rougher, the scavenger and the cleaner. The feed has four mineralogical species, fully liberated chalcopyrite (CuFeS2), partially liberated chalcopyrite, tennantite (Cu12As4S13), and gangue. Each of these species has a specific floatability. We can therefore identify the following three chemical species of interest to our problem: copper, arsenic and gangue. The treatment plant capacity is 1,200 t d-1. The stochastic parameters are the feed-copper grade and the copper price. The copper price varies between 1 - 4 kUS/t, and the total copper grade of the feed ranges from 0.45 to 2.4%. Because of the complexity of the problem, the numbers of cells by bank are fixed according to the notes made by Bourke (2002). To solve the optimization problem, the following two strategies are proposed: x First, solve the fully adaptive model, in which all variables become second-stage variables and change with each scenario. Each scenario is run separately. x After determining the optimum for each scenario, as was done in the fully adaptive model, the volume (first-level variable) is fixed to the maximum value of all of the volumes found in each of the scenarios. The reason for fixing to the maximum value is ensure the capacity of each flotation stage. Each scenario is optimized, finding new second-level variables. We denote this Solution Methodology A.
342
N. Jamett et al.
x As a second option, after the solution of the fully adaptive model, the binary variable of the stream direction of the superstructure is fixed for each scenario. In this manner, all binary variables are became parameters, making the problem easier to solve. The optimal solution is found in a single model that is a function of w (scenarios). Thus, the first-level variables are optimized considering all scenarios at once. We denote this Solution Methodology B.
4. Results The problem is solved in a computer with an Intel core i7 processor (2.7 GHz) and 6 GB of RAM. The optimization software used was GAMS with BONMIN as the MINLP solver. At first, other MINLP solvers were used, such as BARON and COUENNE, but because of the amount of variables and equations, it was not possible to find a solution to the stochastic problem. Employing solution methodology B ensures that profits are maximized, by finding the optimal configuration and by minimizing costs through the cost of the cells. By employing the methodology of solution A, the variable volume is set to the maximum found, corresponding to scenario 9 (high price, high grade) in the fully adaptive model. These dimensions correspond to the optimum for that particular scenario but are oversized for others, increasing total costs. The difference between the two methodologies is approximately 30,000 US$/y. Although the difference between each methodology is not very large, it is noteworthy that through this methodology is possible to solve the problem and find an optimal solution. Table 1 shows the incomes and the differences in volume for both methodologies. Table1: Comparison of the results of solution methodologies A and B. Total Income MMUS$/y 3
Volume Rougher Cell m
3
Volume Scavenger Cell m 3
Volume Cleaner Cell m
Solution Methodology A
Solution Methodology B
2.7299
2.7599
75
27
41
17
6
5
Once the resolution of the stochastic model is possible, its effectiveness is evaluated as a tool for the design of the flotation process. The design and profits obtained by the stochastic and deterministic model are compared. In the deterministic model, the resolution is carried out through weighted averages of stochastic parameters. In this case, the feed grade corresponds to 0.56% chalcopyrite, 2.39% partially liberated chalcopyrite and 0.45% tennantite. The price of the metal corresponds to 5018 US$/ton. A single circuit and single cell design are obtained, which should be able to cushion any variability of the parameters. By running each scenario with this configuration to calculate the actual profit, we found that the first scenario, which considers a lower copper price and lower feed grade, is not capable of operating at the design conditions. It is for this reason that the use of mean values it is not the best option. However, it is not possible to determine the real value of total income; therefore, comparison with other alternatives is not possible. Fig. (2A) shows the optimal configuration for the deterministic model, and Fig. (2B) shows the optimal configuration for scenario 1 using the stochastic model. The optimal configuration for that scenario is to recirculate the scavenger concentrate to the rougher stage and not to the cleaner stage. This
Solution Strategies to Stochastic Design of Mineral Flotation Plants
343
configuration ensures a higher copper grade in the concentrate, cushioning the drop in the feed grade and copper price, lowering the costs for treatment, refining and penalization.
A
B
Figure 2: Optimal configuration of the deterministic model (A) and optimal configuration of stochastic model scenario 1 (B).
5. Conclusions A model for the design of flotation circuits under uncertainty has been presented. Uncertainty is represented by scenarios that include changes in the feed grade and in the metal price. The model allows the operating conditions (residence time and mass flows of each stream) and flow structure (tail and concentrate stream of cleaner and scavenger stage) to be changed for each scenario while the fixed design (size of cells in flotation stages) for all scenarios is maintained. The model can be modified to include other uncertainties and other adaptive variables. To solve the two-stage stochastic model, two solution strategies were proposed. The results show that the use of average values for the stochastic parameters leads to an inefficient design and hence a decrease in the profits made in the process. Finally, it can be concluded that the use of stochastic programming can be a beneficial tool in the design of a metallurgical process, specifically the copper flotation process. The optimal configuration is capable of adapting to uncertainty, leading to an increase in the company profits.
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N. Jamett et al.
6. Acknowledgments Financial support from CONICYT (Fondecyt 1120794), CICITEM (R10C1004) and the Antofagasta Regional Government is gratefully acknowledged.
7. References J. Birge, F. Louveaux, 1997, Introduction to stochastic programming. P. Bourke, 2002, Selecting flotation cells: How many and what size?, MEI Online, http://www.min-eng.com/frothflotation/4.html. W. Bruckard, K. Davey, F. Jorgensen, 2010, Development and evaluation of an early removal process for the beneficiation of arsenic-bearing copper ores, Minerals Engineering, 23 (15), 1167-1173. W. Chan, R. Prince, 1989, Heuristic Evolutionary Synthesis with Non – Sharp Separators, Computers Chem. Eng.13, 1207-1219. L. Cisternas, D. Méndez, E. Gálvez, R. Jorquera, 2006, A MILP model for design of flotation circuits with bank/column and regrind/no regrind selection, International Journal of Mineral Processing, 79 (4), 253-263. E. Gálvez, 1998, A shortcut procedure for the design of mineral separation circuits, Mineral Engineering, 31(2), 113-123. G.Herrera, E. Gálvez, L. Cisternas, 2009, Mineral Processing Flow Sheet Design Through A Group Contribution Method, Computer Aided Chemical Engineering, 26, 213-218. N. Jamett, J. Vielma, L. Cisternas, 2012, Design of Flotation Circuits Including Uncertainty and Water Efficiency, Computer Aided Chemical Engineering, 30, 1277-1281 A. Kraslawski, 1989, Review of Applications of Various Types of Uncertainty in Chemical Engineering, Chem, Eng. Process, 26, 185-191. B. Loveday, C. Brouckaert, 1995, An analysis of Flotation Circuit Design Principles. The Chemical Engineering Journal, 59, 15-21. D. Méndez, E. Gálvez, L. Cisternas, 2009, Modeling of grinding and classification circuits as applied to the design of flotation processes, Computers & Chemical Engineering, 33 (1), 97111. H. Simonsen, J. Perry, 1999, Risk identification, assessment and management in the mining and metallurgical industries, J. of the South African Institute of Mining and Metallurgy, 99 (6) 321- 329.