The Application of Numerical Taxonomy Technique in the Iron Ore Flotation to Determine Appropriate pH and Particle Size Distribution

: The numerical taxonomic technique is one of the multivariate decision-making techniques for assessment and ranking. It is widely used for planning and development studies. In this research, flotation experiments were conducted for Band Narges mine iron ore using different pH and particle size ranges. Numerical taxonomy technique, as one of the most important multi-attribute decision-making techniques, was used to determine the best range of particle sizes for the input feed of the flotation cell as well as the pH of the pulp. For this purpose, two different particle size ranges were selected for six different pH values as options. Criteria for evaluating particle size ranges and pHs were determined. Subsequently, effective criteria were determined by performing various tests. Finally, the ranking of the tests was determined using numerical taxonomy. Based on the F i value (a parameter indicating the appropriate status or developmental rate of the option), the particle sizes <74 µm and pH = 9 were proposed as the optimum conditions to float Band Narges Iron Ore.


Introduction
Froth flotation is a complex physicochemical separation process based on the difference in the surface properties of minerals that is commonly used to separate valuable minerals and unwanted gangue minerals. Improving knowledge and understanding of the process leads to significant progress in achieving higher grade and recovery. However, the amount of energy used in the flotation process is not as much as the grinding process, but its optimization would reduce energy consumption. In recent years, the advent of gigantic float cells and energy-saving competition to reduce operating costs have led to efforts to optimize energy consumption. Energy is used to rotate agitators in mechanical flotation cells. The rotation of the agitator in the flotation cell gives the energy required to suspend the particles, distribute the bubbles and produces an interaction between the bubbles and the particles. Part of the energy consumes to rotate the pulp inside the cell, but a large part of the energy is wasted by micro-interactions between the three-phase include air, water and solids (Ahmed and Jameson, 1985;Amini et al., 2013;Chau, 2009;Deglon et al., 1998;Drzymala and Kowalczuk, 2018;Newell and Grano, 2006;Pyke et al., 2003;Reis et al., 2019;Schubert and Bischofberger, 1978;. Flotation performance depends on a variety of parameters, including distribution of particle size, solid percent pulp, reagent dosage, Eh and pH. In practice, selectivity in complex separation depends on the balance between reagent dosage and pH (Bahrami et al., 2019;Dehghani et al., 2013;Rezai et al., 2010;Tan et al., 2018;Zanin et al., 2019). The floatability of the sulfide minerals is strongly related to the potential difference of the mineral-solution interface (pulp potential). It is possible to float or dissipate minerals by oxidation or reduction. The measured potentials, therefore, determine whether or not the mineral will float. It is between the potential of the minerals and the potential of the solution. In the same solution, different materials have different Eh values. Controlling Eh-pH data in an operating process can help to reduce reagent additions and provide useful additional and useful information to solve the operational problem of flotation plants. Some studies related to Eh/pH have shown that Eh is an appropriate indicator of mineral floatability (Chimonyo et al., 2017;Göktepe, 2002;Liu et al., 1994).
Particle size is another important factor that plays a critical role in the flotation performance. It affects the probability of the particles colliding with the bubbles, the attachment of the particles to the bubbles after the collision and the stability of the attachment Ross, 1997). The rate of recovery of fine particles is low due to a lower probability of particle-bubble collisions, prone to entrainment and large specific areas, which may lead to excessive reagent adsorption and other effects associated with chemically active particles. On the other hand, there is an essential difficulty in the flotation of coarse particles in the high bubble-particle detachment efficiency (Pyke et al., 2003), as well as a decrease in the buoyancy of the particle-bubble aggregate relative to the pulp. Furthermore, increasing particle sizes may result in longer induction times and a commensurate deterioration in floatability (Eskanlou et al., 2019;Jameson et al., 1977).
Previous studies showed that some multi-attribute decision-making methods, such as AHP and TOPSIS, have been used to select appropriate methods and parameters in the mining, mineral processing industries. In one study, AHP and PROMETHEE methods were used to determine the type and concentration of coagulants for the physical and chemical purification of wastewater in a textile factory (Aragonés-Beltrán et al., 2009). Another study used the Fuzzy TOPSIS method to select the type of collector and optimize the dosage of it in the flotation of lead and zinc sulfide minerals (Kostović and Gligoric, 2015). Some researches utilized Fuzzy TOPSIS (Safari et al., 2012) and the AHP methods (Safari et al., 2010) to determine the suitable location for the construction of a processing plant for an iron ore mine. Nowever this method was used wildy in other fields to make a final decision, but the number of research that used the method in mineral processing, especially in flotation process is limited. Thus, in this research, it was applied to determine the appropriate ph and particle sizes.
Selecting the appropriate pH and particle size range can improve flotation performance by increasing grade and recovery. The applicable particle size ranges and pH pulp for the Band Narges iron ore flotation process were therefore investigated and determined using a mathematical method based on Multi-Attribute Decision-Making techniques (MADM). For this purpose, pH at six different levels and particle size distribution at two different levels have been considered. Effective parameters have been determined. Finally, the optimum distribution of particle sizes and PH values was proposed using the mathematical method of numerical taxonomy.

Research Methodology
Representative samples for the experiments were taken from the Band Narges iron ore mine located 60 km northeast of Badrud, Isfahan City, Iran. The degree of freedom of iron ore (d80) was determined as 150 µm by mineralogical studies. In the next step, the samples were ground and floated without pre-concentration. Samples were ground to a size of less than 150 µm. Then, six flotation series tests were carried out at different pHs for a particle size distribution of 0-150 µm. Each test series was repeated three times and the average amount was calculated.

Taxonomy Analysis Method
One of the multi-attribute decision-making methods is a taxonomic analysis method. It was first proposed by Adenson in 1763 and was extended to a group of mathematicians in 1950. Taxonomy analysis has been used in different sciences for different categories and its specific type is numerical taxonomy. Numerical taxonomy is used to assess the similarity and proximity between taxonomic units to classify these elements into taxonomic groups. This method is based on an analysis of existing alternatives using several criteria and provides a complete ranking for the evaluation of options. The method is widely used in various engineering fields (Feoli and Ganis, 1984;Soltanpanah et al., 2010;Rahman et al., 2013;Yari et al., 2015).
One of the advantages of using the numerical taxonomy method in comparison with other multiattribute decision-making methods is that it is not necessary to determine the relative importance of criteria based on expert judgments in this method. As a result, expert judgments have less interference with the analysis of the data so the results are more accurate than others. Taxonomy analysis includes eight steps as follows (Sneath and Freeman, 1975).

Step 1: Identify Options and Set Assessment Criteria
At first step, experts shall identify and determine the m alternative (A2, A1, ..., Am) and the n assessment criteria (C1, C2, ..., Cn) in accordance with the problem under consideration.

Step 2: Set up the Decision Matrix and Calculate the Mean and Standard Deviation of the Data
Subsequently, after evaluating the alternatives based on different criteria, the decision matrix is established, as shown in Table 1. In this table, rij represents the amount of the ith option in quantitative or qualitative terms based on the jth criterion. After the data matrix is formed, the mean and standard deviation of each column are calculated. .
Step 3: Formation of Standard Matrix (Normalized) In the decision matrix, the options are expressed in terms of criteria that have different measurement scales. At this stage, attempts are being made to eliminate their various units (measuring scales) or to make them dimensionless. The equation below is used to get this purpose: The mean of each matrix column j : The standard deviation of each matrix column It should be noted at this stage that criteria with a negative aspect (cost criteria) must be reversed or, in other ways, considered negative. Finally, in conventional methods, for each column of the standard matrix, the maximum positive or negative criteria will be chosen as ideal positive or negative. It's defined as the DOj.
Step 4: Determine the Compound Distance Between Options At this stage, based on the standard matrix, the distance of each option is determined from the following equation: where, a and b are two options to be assessed. This operation is a kind of paired comparison of the two options. The main features of the operation are as follows: 1) The distance of both options is zero (Dab = Dba = 0) 2) The distance between a and b is equal to the distance b of a. (Dab = Dba) According to the above items, the compound distance matrix is formed for options that the main diameter represents the distance of each option with itself is equal to zero.
Step 5: Determine the Shortest Distance The minimum distance in each row of the matrix is determined in this step. Then, their mean and standard deviations are obtained. For the shortest distance, the same method is used.
Step 6: Homogenizing Options There may be options that have much more or far fewer distances than other options. Therefore, heterogeneous options should be removed them from the set. To this end, the upper and lower limits are obtained using the following equations: In this case, the dr has been coordinated between the upper and lower limits. The options outside this specified range must be deleted. The decision matrix is again established without the deleted options. The steps are repeated.
Step 7: Determination of Options Patterns At this stage, the distance between each option is calculated from the ideal value (as specified in step 4). The short distance from the ideal indicates the appropriate condition of the option and a large distance indicates its incorrect status. Option patterns are also derived from the following equation: Step 8: Ranking the Developmental Rate of Options (Fi) The ranking of the development rate and the status of the options are discussed at this stage. If Fi is the developmental rate of an option (the appropriate status of an option), then: where, Cio is a pattern of each option and Co is the upper limit of the development. To calculate Co, the mean and deviation of Co should be determined. This was done at the end of step seven and calculated as follows: The Fi is between zero and one. If it is closer to zero, it indicates more development and better conditions for the option. If it is closer to one, it indicates its bad status and the lack of potential development of the option. In this case, the taxonomic technique is completed and the options ranking is specified.

Discussion
Flotation is one of the most important mineral arrangement methods established for particles in the fluid and airflow environments. Airflow causes bubbles in the pulp. Increasing the free surface of the particles (at constant mass), causes more contact and stronger bonding to the air bubble and vice versa. By increasing the mass of the particles when the free surface is constant, the force of gravity increases and the probability of separation increases (Rahimi et al., 2012;Van Deventer et al., 2004). A number of flotation tests were carried out on particle sizes of less than 150 µm with 500 g/ton Sodium Silicate as a dispersant, Dirol and Alke 724 with values of 200 and 600 g/ton as frother and collector, respectively, to investigate the appropriate flotation size ranges. Lime was used to adjust the pH. The results of the flotation are shown in Table 2. As it can be seen from the table at pH equal to 9 showed the best recovery and grade, 86.94 and 48.24%, respectively.
The next set of flotation experiments was performed for particle sizes below 74 µm with different pH values. For this purpose, ore samples were ground to 0 to 74 µm in size. The results are shown in Table 3.
The optimum particle size range, as well as pH, were determined using a mathematical method based on MADM called the numerical taxonomy mathematical method, after measurement and calculation of concentrate and tailing grades and recovery for all tests.
As can be seen from Table 2 and Fig. 1, at pH = 9 for particle size fraction 0-150 µm, we have the highest recovery and grade levels (86.94 and 48.24%, respectively). As pH increases, the grade and recovery of flotation decrease. Table 3 and Fig. 2 also show a similar trend for particle size fractions between 0-74 µm as the highest recovery and grade are at pH = 9, 86.74 and 49.52%, respectively. According to Table 2 and 3, 0-74 µm fraction has higher floatability; however, 0-150 µm fraction shows higher recovery.
We deal with a large number of data during the flotation process, so it is difficult to choose the data that can show optimum conditions. We, therefore, analyze and rank the data using a mathematical multi-attribute taxonomy decision-making method to overcome the difficulties. In the next step, the data with the highestranking using this method will be selected as an appropriate condition for the flotation process.

Determination of Appropriate Particles Sizes and pH using the Numerical Taxonomic Multi-Attribute Decision-Making Technique
In this study, the proper particle size of the input feed of the flotation cells and the pH of the pulp will be determined based on the following model Fig. 3.
Step 1: Select Options Initially, to study the effect of particle size ranges and pH, flotation tests were conducted for two particles sizes fractions (0-74 and 0-150 µm) and 6 different pH values (9, 9.5, 10, 10.5, 11, 11.5). They were used as alternative options according to the first step, Fig. 3.

Step 2: Determination of Appropriate Criteria to Evaluate Particles Sizes Fractions and pH
Criteria for assessing the particle size ranges and pH values have been determined. They were shown with the related symbols and their effects in Table 4. According to Table 4, both C1 and C3 have positive effects. The higher the criteria for an option, the higher the priority of that option. On the other hand, C2 has a negative impact as the number of criteria for one option increases, the priority level of the choice option decreases. Also, all criteria for small quantities have been determined based on the tests.

Step 3-8: Formation of Decision Matrix and Use of Numerical Taxonomy
At this stage, the measured values of the C1, C2, C3 for different pH values (1 to 12) are considered to be the decision matrix. In this case, a decision matrix with a dimension of 3*12 was obtained. Where 12 represents the number of options (alternative) and 3 represents the number of criteria that were presented in Table 5. The numerical taxonomic method was used to evaluate the parameters obtained from different experiments in order to select the best particle size ranges for the input feed of the flotation cell and the best pH value for the pulp. For the next steps, the taxonomic procedure for the decision matrix were performed. The results were shown in Tables 6-10.
The mean and standard deviation of the columns for each decision matrix were calculated Table 6. To normalize the matrix, Eq. 1 was used. As a result, standard matrixes have been obtained, see Table 7. In the next step, the largest positive number of each column for positive aspect criteria and the largest negative number of each column for negative aspect criteria were selected. They were identified as ideal (DOj). Equations 2 and 3 were used to determine the compound distance between the alternatives for each of the criteria. The results are shown in Table 8.           Equations 5 and 6 were used to homogenize the options and the upper and lower limits for each of the matrixes were obtained. To determine the pattern of options (Cio) in each matrix, at this stage, the distance of each option is calculated from the ideal values (DOj) as shown in Table 9. As noted, the low distance from the ideal value is an appropriate condition for this option. In the last step, the upper limit of development (Co) was calculated for data based on Eq. 9. To rank the development rate of the options, an index (Fi) was obtained for each of them by using Eq.
8. This index is between zero and one. Whenever the indicator is closer to zero, it indicates the better development of the option (better situation) and higher ranking. Whenever the indicator is closer to one, it indicates its poor status and lower ranking. The results of these calculations have been shown in Table  10. According to Table 10, the lowest rating for flotation is in the case of particle size of 0-74 µm and a pulp pH of 9, which is the first rank. This option is recommended as an appropriate condition for flotation using numerical taxonomy.

Conclusion
Consideration and evaluation of key factors, such as pH and particle size ranges, will enable us to have adequate knowledge of these parameters to optimize the flotation process. It is, therefore, necessary to classify and use the appropriate technique in order to understand the relationship between the parameters and finally their effects on flotation performance. Numerical taxonomy is one of the techniques most widely used to discover this relationship. Numerical taxonomy analysis is an excellent method of grading, classifying and comparing different parameters in engineering science, especially in mineral processing.
In this study, it was noted that factors such as pH and particle size ranges have a significant impact on the flotation process. The smaller the particles, the greater the degree of freedom of valuable minerals. An increased degree of particle freedom will, however, increase the flotation performance, but decreasing the particle size from certain ranges will have a negative effect on the flotation performance. Determining appropriate particle sizes with an appropriate degree of freedom is, therefore, an important issue for mineral processing plants. In addition, the surface of the particles should be properly charged in order to have an appropriate flotation performance. So pH is another important parameter.
In this research, numerical taxonomic mathematical methods clearly showed the effect of pH and particle size in the flotation process. This method has been shown experiments with similar pH value and smaller particles have a higher ranking. In other words, they have higher flotation performance. The results showed the particle sizes <74 µm and pH = 9 were proposed as the optimum conditions to float Band Narges Iron Ore based on the Fi value. In addition, according to the results, using a mathematical method such as numerical taxonomy is a strong tool to evaluate flotation performance.

Funding Information
Step 1: Select various options to evaluate (Different values of pH, different particles sizes fractions) pH1 to pH12 Step 2: Determine the appropriate criteria for evaluating different PH and particle sizes fractions (Concentrate grade, tailing grade, recovery) C1, C2, C3 Step 3: Formulate Decision Matrix and Use the Numerical