Modeling the Recovery of Froth Flotation Using Game Theory

Journal of Mining World Express (MWE) Volume 5, 2016 www.mwe‐journal.org doi: 10.14355/mwe.2016.05.001

Modeling the Recovery of Froth Flotation Using Game Theory Mohammed‐Noor N. Al‐Maghrabi Mining Engineering Department, College of Engineering, King Abdulaziz University, Jeddah 21589, P.O. Box 80204, KSA almagrabii@yahoo.com

Abstract For a better understanding of froth flotation principle and to explain the conflict in flotation a recovery model is formulated.

From the theoretical formulation, it transpires that based on the gain values of the various strategies of the three players in a game theory, it is possible to predict pure mineral recovery using flotation column by summing up the concentrate of the two players at a time. The present work is an attempt to apply the game theory to a mineral extraction problem where the aim of flotation optimization is to find the operating conditions which gives the highest grade and recovery using game theory with three players.

Keywords Column Flotation; Mineral Separation; Concentrate; Froth Flotation; Flotation; Recovery of Pure Mineral; Game Theory

Introduction Mineral separation involves physical separation of mineral particles from mineral/water slurry. Some particles have the ability to attach themselves to air bubbles which float to the surface and are removed in the form of concentrate. These particles are known as hydrophobic or non‐polar type (Wills 1988; Wills 2006). Other particles that remain wetted but not attached are called hydrophilic or polar type. However, they can attain hydrophobic property by chemical treatment. The process of mineral separation is achieved by the process of froth flotation. It is useful for finely‐grained ores which cannot be removed by conventional gravity method. Froth flotation is a selective process to separate minerals, suspended in water. It is accomplished by mechanical or non‐mechanical cells (Crozier 1992).

Among the non‐mechanical cells in flotation is the column flotation as shown in Figure 1. It provides a means for improving the effectiveness of froth flotation using countercurrent (Rubinstein 1995). The main features of column flotation are the use of countercurrent flow of air bubbles and solid particles. Air is introduced at the bottom of the column, and feeds above the midpoint of the column. The particles then dropped through a rising swarm of air bubbles. In comparison with conventional cells, washwater is added to the top of the column which accentuates the countercurrent flow, which forces all of the water which entered with the feed downward, to the tailings outlet. As a result, preventing entrained contaminants from reaching the froth. Therefore, columns produce a better product grade, similar or higher recovery, and lower power consumption. This is due to higher bubble/particle contact efficiency exhibited by columns than that exhibited by conventional cells. Yoon and Luttrell (1986) mentioned that, the flotation rate of both the coarser and finer particles is improved, in certain types of flotation columns, due to the reduction of bubble diameter.

No earlier work exists where game theory has been used. Game theory is applied to a thinking process where two players contest (are in conflict) by playing different strategies for maximizing their gains and minimizing their losses. Ostensibly, the present problem pertains to a non‐thinking process since the two or more players are not humans. However, it attains ‘thinking process’ when the two players, for instance, mineral particle and air bubble, are made to use their separate strategies to maximize their gains. Mineral Particle may apply strategies like different sizes of particles, while air bubble strategies may include the addition of chemical agents. In this way, the application of various conflicting strategies leads to a greater recovery, which can then be vectorially summed up to maximize the percentage recovery of concentrate.

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Washwate

Froth Outlet

Froth Phase‐ Cleaning Zone

Feed Inlet

Pulp Phase‐ Recovery Zone Collection Zone

Air or Air/Water Injection Tailings Outlet

FIGURE 1: SCHEMATIC OF A FLOTATION COLUMN

In carrying out a specific operation involving ratio of concentration and recovery, it is imperative to know the factors that impinge on the flotation system and their inter‐relationship. The flotation system given below in Figure 1 (Klimpel 1995) shows the inter‐relation of components of the total system which comprises several factors. It is therefore important to take all of these factors into account and not to study the effects of any single factor in isolation. This makes it difficult to develop predictive models for froth flotation, although work is being done to develop simple models that can predict the performance of the circuit from easily‐measurable parameters such as solids recovery and tailings solid content (Rao et al. 1995). A simple model using game theory sets the stage for predicting the performance of the circuit from easily measurable parameters such as solids recovery.

Chemistry Components Collectors Frothers Activators Depressants pH

Equipment Components Cell Design Agitation Air Flow Cell Bank Configuration Cell Bank Control

Operation Components Feed Rate Mineralogy Particle Size Pulp Density Temperature

FIG. 2 INTER‐RELATED COMPONENTS OF A FLOTATION SYSTEM (KLIMPEL 1995).

Problem Statement/Objective The objective of the present theoretical study is to develop an innovative sustainable recovery model using game theory technique for conflict in froth flotation operations of pure mineral using columns that predicts process changes and produces expected results by optimizing settings of various interrelated parameters.

Methodology Game Theory and Its Application Game theory is a mathematical model that is designed on the relationship between two independent intelligent

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competing entities associated with a thinking process, that is, with a strategic decision‐making. It is also known as “interactive decision theory” that engages models of conflict and cooperation between decision makers such that one person’s choice must allow for conflict and for possibilities for cooperation. The theory considers cooperative games of several players, or more than two players. It is applicable to a range of behaviourist relations. It tends to be a mixed strategy in a two‐person zero‐sum game with the two players which have “n” strategies, and each player uses the strategies to try to outdo their opponent. Associated with each pair of strategies is a payoff that one player receives from the other such that one person’s gains exactly equal the other participant’s loss (Wikipedia, February 26, 2015).

An important solution to the game theory is the minimax solution, which minimizes the maximum possible loss (or maximizes the minimum possible gain). Two‐person zero‐sum games provide a unique minimax solution because in this situation it is the best choice of the decision maker if the other player has chosen a minimax solution. This unique solution provides an optimal solution since it selects one or more strategies for each player such that any change in the chosen strategies does not improve the payoff to either player (this is equilibrium point). These solutions can be in the form of a single pure strategy (known as pure strategy) or several strategies mixed to specific probabilities (called mixed strategies). However, not all situations under pure strategy have a minimax equilibrium solution; in such situations, some probability mixture of the strategies (known as mixed strategy) is applied which will improve the position of both parties with respect to the available pure strategies.

Player 2

Player 3

Feed Water Strategy

Player 1

Recovery of Concentrate %

Feed Water Strategy

Air Bubble Strategy

(1.0 , 0

Air Bubble Strategy

(1.0 , 0

Possible Strategies of Particle

)

Possible strategies of Feed (0 , 1.0

)

FIGURE 3: CUBE WITH THREE PLAYERS

In the present paper, game theory has been adapted conceptually to solve the flotation problem assuming that there are three players instead of two players. Hence, there exists a 3‐D relationship between the three players as illustrated by Figure 3. Also, Table 1 and Table 2 clearly delineate two players 1 and 2, and players 1 and 3 with their respective strategies. The problem is one of the mixed strategy and the solution applied is minimax solution.

The matrix provides the payoff values. Since such an analysis having three players has not been attempted

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previously, it is not possible to compare differences. Furthermore, the present analysis is a theoretical conceptual study which can be verified only if an experiment is undertaken in a laboratory.

TABLE 1 STRATEGIES OF TWO PLAYERS 1 AND 2

Particle Size Strategies Air Bubble Strategies

p

p

a

+y

+y

a

+y

+y

TABLE 2 STRATEGIES OF TWO PLAYERS 1 AND 3

Feed Water Strategies Air Bubble Strategies

f

f

a

+y

+y

a

+y

+y

Problem Operationalization With the above in mind, the present study explores the development of a model based on game theory that predicts changes in other parts of the system if the settings of one factor changes. Herein use is made of game theory in a 3‐D version. Conventionally, there are two players having ‘n’ strategies (or options) competing for maximum gain (Luce and Raiffa, 1989; Taha, 2011); the proposed model introduces a third player with ‘n’ strategies (or options). The three players and their strategies (or options) are operationalized thus: Player 1 is:

Air Bubble having strategies and

, which are different chemical treatments

Player 2 is:

Particle Size having strategies and which are different particle sizes

Player 3 is:

Feed Water having strategies and , which are different feed rates

‘n’ is the number of strategies for each player. For this paper, ‘n’ is taken as equal to two for each player.

Assumptions In the illustration to follow (Table 1 and Table 2), it is assumed that each of the three players has two strategies and that variable values are assigned for all gains and losses in the 3‐D matrix, either following a linear path or a curvilinear path. This is represented by a cube (Figure 3) that details all the players and their strategies, and the matrices given in Table 1 and Table 2. The tables give the values of concentrate percentage viz. to Procedure and Implementation Strategies of Player 1 (Air Bubble) and Player 2 (Particle Size) Table 1 lists the strategies of two players Air Bubble and Particle Size. Based on Table 1, a graph is plotted and explained in Figure 4. The figure is drawn from values of concentrate % given in Table 1. and are allocated on the y‐axis keeping air bubble strategy constant. Any point on the path to is considered as a gain to the air bubble strategy . If we select for , we would choose mixed strategy (0.5 , 0.5 ).

The relationship between the variables on x‐axis and y‐axis may follow a linear or a curvilinear path with the usual interpretations, viz. in Figure 4. If the air bubble strategy follows a straight path, intermediate values of particle strategies and percentage recovery of concentrate will be as given in the graph. However, if the air bubble strategy moves on a curvilinear path, and assuming that intermediate values of particle strategies remain the same, it can be observed that concentrate recovery percentage is higher if the air bubble strategy path is convex; on the contrary, concentrate recovery percentage is lower if the air bubble strategy path is concave. The same explanation is true for Figure 5 with air bubble strategy ; and for particle strategy and with .

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air bubble strategy,

Recovery or Concentrate % Conc.%

(1.0 , 0.0 )

(0.5 , 0.5 ) Possible Strategies of Particle

(0.0 , 1.0 )

FIG. 4 CORRESPONDING STRATEGIES p AND p OF PARTICLE SIZE WITH AIR BUBBLE STRATEGY a

In Figure 5 given below, the relationship between the variables on the x‐axis and y‐axis is assumed, for the sake of simplicity, to follow a linear path. Thus, the lines of concentrates connecting and indicate the gain to the air bubble strategy . Because of similarity, the explanation is the same as given for Figure 4. By looking at thick dotted lines of Figure 5, if the particle follows the rule of minimizing the maximum concentrate, it will adopt the mixed strategy (0.4 , 0.6 ) and give the air bubble a concentrate of 0.4( ) + 0.6( ) = for strategy ; or 0.4( ) + 0.6( ) = for strategy . This amount cannot be exceeded regardless of any air bubble action.

Further, Figure 5 shows the line to , which represents the concentrate to air bubble for various mixed strategies for particle strategy . This line would indicate the minimum concentrate for strategies between (1.0 , 0.0 ) and (0.8 , 0.2 ). The line joining and of strategy indicates the increase in concentrate corresponding to various air bubble strategies. This particle strategy provides the minimum concentrate for air bubble action in the range (0.8 , 0.2 ) to (0.0 , 1.0 ).

In case the air bubble action is to maximize the minimum concentrate, it will adopt strategies (0.8 , 0.2 ). Thus we get a value 0.8( ) + 0.2( ) = when the particle follows strategy ; in addition, we also get a value 0.8( ) + 0.2( ) in case of particle following strategy . air bubble strategy,

particle strategy,

air bubble strategy,

particle strategy,

Recovery or Concentrate %

Conc. %

(0.4 , 0.6 )

(1.0 , 0.0 )

(0.5 , 0.5 )

(0.0 , 1.0 )

Possible Strategies of Particle (1.0 , 0.0

)

(0.8 , 0.2

)

(0.0 , 1.0

Possible Strategies of Air Bubble

)

FIG. 5 STRATEGIES p AND p OF PARTICLE SIZE WITH AIR BUBBLE STRATEGY a AND a STRATEGIES a AND a OF AIR BUBBLE WITH PARTICLE SIZE STRATEGY p AND p

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Strategies of Player 1 (Air Bubble) and Player 3 (Feed Water) Table 2 above gives the matrix of strategies of two players Air Bubble (strategies and ) and Feed Water (strategies and ) respectively. The assumption is that the player 3 (Feed Water) is following strategy .

However, if the strategy of the player 3 (Feed Water) changes from to , then the results will be different as shown in Table 2. All values assumed in Table 2 are variable values. It is further assumed that the player 2 (Particle Size) is following strategy . The schematic 3D figure 3 shows the behavior.

Going by the same reasoning, we have similar diagram and explanation as above except that the players have changed. Given below is Figure 6 and explanation based on the data of Table 2 for player 1 (Air Bubble) and player 3 (Feed Water), and their strategies and of Air Bubble and and of Feed Water, respectively.

By looking at the thick dotted lines given by to and by to , in case the feed water follows the rule of minimizing the maximum concentrate, it would have to give to air bubble, then it would require to follow the mixed strategies (0.4 , 0.6 ) and give the air bubble a concentrate % of 0.4( ) + 0.6( ) = on air bubble strategy ; or 0.4( ) + 0.6( ) = if air bubble strategy is . However, the maximum concentrate would remain as .

Furthermore, by looking at the thick part of the line to , as also the thick part of ‐ , if the feed water adjusts its strategy to , the lines would indicate the minimum concentrate for strategies between (1.0 , 0.0 ) and (0.8 , 0.2 ). The line joining and determines the concentrate enhancement as a result of various air bubble mixed strategies for the feed water strategy . Consequently, the above strategy provides the minimum concentrate for air bubble action in the range (0.8 , 0.2 ) to (0.0 , 1.0 ).

On the contrary, if the air bubble action is to maximize the minimum concentrate, it will adopt strategies (0.8 , 0.2 ) to give a value of 0.8( ) + 0.2( ) = if feed water strategy is ; and a value of 0.8( ) + 0.2( ) = for feed water strategy .

feed water strategy,

air bubble strategy, air bubble strategy,

Recovery or Concentrate %

feed water strategy,

Conc.%

(1.0 , 0.0 )

(0.5 , 0.5 ) (0.4 , 0.6 )

(0.0 , 1.0 )

Possible Strategies of Feed Water

(1.0 , 0.0

) (0.8 , 0.2 ) Possible Strategies of Air Bubble

(0.0 , 1.0

)

FIG. 6 STRATEGIES f1 AND f2 OF FEED WATER WITH AIR BUBBLE STRATEGY a AND a STRATEGIES a AND a OF AIR BUBBLE WITH FEED WATER STRATEGY f1 AND f2

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Results From the above, it can be found that combining strategies of players 1 and 2, i.e. summing up vectorially, of Table 1 and strategies of players 1 and 3, i.e. summing up vectorially, of Table 2 yields (for Air Bubble and Particle Size strategies, are seen in Figure 5; and for Air Bubble and Feed Water strategies, i.e. summing up vectorially, can be seen in Figure 6): (a) With Air Bubble strategy and Particle Size strategy (0.5 , 0.5 ), Recovery or Concentrate % is and Air Bubble strategy and Feed Water strategy (0.5 , 0.5 ), Recovery or Concentrate % is . Therefore, Total Recovery % = + (b) With Air Bubble strategy and Particle Size strategy (0.5 , 0.5 ), Recovery or Concentrate % is and Air Bubble strategy and Feed Water strategy (0.5 , 0.5 ), Recovery or Concentrate % is . Therefore, Total Recovery % = + (c) With Air Bubble strategy and Particle Size strategy (0.4 , 0.6 ), Recovery or Concentrate % is and Air Bubble strategy and Feed Water strategy (0.4 , 0.6 ), Recovery or Concentrate % is . Therefore, Total Recovery % = + (d) With Air Bubble strategy and Particle Size strategy (0.4 , 0.6 ), Recovery or Concentrate % is and Air Bubble strategy and Feed Water strategy (0.4 , 0.6 ), Recovery or Concentrate % is . Therefore, Total Recovery % = + (e) With Particle Size strategy and Air Bubble strategy (0.8 , 0.2 ), Recovery % is , and with Feed Water strategy and Air Bubble strategy (0.8 , 0.2 ), Recovery % is Therefore, Total Recovery % = + (f) With Particle Size strategy and Air Bubble strategy (0.8 , 0.2 ), Recovery % is , and with Feed Water strategy and Air Bubble strategy (0.8 , 0.2 ), Recovery % is Therefore, Total Recovery % = + .

Explanation of Results and Discussion Given that there are three players in the flotation problem, each has two strategies to counter the strategies of their opponents to optimize their individual gain. Strategy of the player (variable) Air Bubble signifies that it can change form from to if required, and in different proportions by the addition of reagents. Strategy Particle Size can have size changed from to ; and the strategy of the third player Feed Water addresses the amount that can be fed from to . Different proportions of the strategy indicate the mixed strategies of the players. A mix of strategies of any two players gives a payoff, meaning a gain to one player with an equivalent loss to the other. This means that the various proportions of strategies of two players yield a different payoff, or recovery percentage in flotation problem.

The line joining (maximum concentrate percentage value) to (minimum concentrate percentage value) is considered to be a gain to air bubble strategy . When it moves by the addition of chemical reagents, to becomes to , which can be seen in Fig. 5.

Similar is the explanation for the particle size becomes to , which can be seen in Fig. 5.

to

. When it moves by changing the particle size,

to

In the same way, the line joining to is a gain to the air bubble strategy. When the feed water volume changes, to becomes to , which can be seen in Fig. 6.

Application of the minimax solution, which minimizes maximum loss or maximizes minimum gain, obtains an optimal solution (or optimal recovery percentage). This point is also called the equilibrium point in a mixed strategy problem. For example, the explanation of part (a) of the Results tells that with Air Bubble strategy and Particle Size strategy (0.5 , 0.5 ), Recovery or Concentrate % is an amount equal to as seen in Fig. 5; and with Air Bubble strategy remaining same ( ) and Feed Water strategy (0.5 , 0.5 ), Recovery or Concentrate % is an amount , as seen in Fig. 6. Therefore, Total Recovery % with Air Bubble strategy remains the same equals an amount = + . Thus, Air Bubble strategy remains the same, and the proportion of Particle Size mixed strategies

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and Feed Water mixed strategies was given, this is the maximum recovery percentages that can be predicted at this level. However, (as given in part (b) of the Results) if the Air Bubble strategy changes to another value ( ), but the other two players have the same mixed strategies, the maximum predicted total recovery percentage = + .

Indeed, this is true for all the parts of the Results.

Interestingly, knowing the various strategies of the three players and how they are interacting; and also affecting changes in their strategy values can be used to predict the recovery percentages to within small tolerances. In practice, all three players have the capability to change; for instance reagents, catalysts can influence Air Bubble size, Particle Size can be varied, and volume of Feed Water can be changed to give a particular predicted percentage recovery.

Conclusions To understand flotation experiments that can explain the conflict in flotation, a recovery model has been formulated. From the above, it can be visualized that yields (concentrates of pure mineral resulting from conflict of various factors, viz. particle size, air bubbles, and feed water) of the various strategies of the three players in a game theory can be summed up vectorially (possibly using vector algebra) and it is possible to predict mineral recovery using froth flotation columns. It must be remembered that the game theory matrix of two players having three strategies is different from the present dispensation where the gains of two strategies of two players (player 1 and 2, and player 1 and 3) have been summed up to give the total yield.

Recommendation Experiments can be conducted in a lab to verify results. This study is a precursor to further research.

ACKNOWLEDGEMENT

No paper reaches the publication stage without the contribution of many people. In this context, my thanks are due to the various authors and sources that made my task easier and focused in the making of this paper. My appreciation also goes out to my dear friends and colleagues, Professor Dr H.E.M. Sallam, and Professor Dr Asif Haleem who made useful contributions in the approach to the problem and has been supportive of my efforts.

Their invaluable advice and timely encouragement have helped me in the preparation of the manuscript.

REFERENCES

[1]

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[2]

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[3]

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[4]

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[5]

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[6]

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[7]

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[8]

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[9]

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[10] En.wikipedia.org/wiki/Game theory

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