Abstract
The sustainable beneficiation of low-grade phosphate ores, especially those containing apatite, is vital to meet the increasing global demand for phosphorus in agricultural and industrial applications. This study proposes a novel, data-driven framework for optimizing collector combinations for direct apatite flotation, through the integration of experimental evaluation and multi-criteria decision-making analysis. A series of micro- and bench-scale flotation tests were conducted on samples from the Chadormalu deposit (Iran), utilizing a wide spectrum of surfactants, including anionic, nonionic, and ethoxylated reagents. Among the tested formulations, a combined collector system consisting of Tall Oil Fatty Acid (TOFA) and oleic acid polyethylene glycol ester (OAPEGE6), designated as TO6, exhibited superior performance, achieving 91.1% recovery and 22.2% P2O5 grade in alkaline conditions (pH = 9.5). To resolve the inherent trade-offs between metallurgical performance and economic feasibility, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was employed as a multi-criteria decision-making tool. Performance metrics including flotation efficiency, reagent cost, and availability were normalized and objectively weighted using Shannon entropy. The integrated analysis ranked eight collector formulations, identifying the TO6–diesel system as the optimal configuration, delivering enhanced flotation performance with competitive cost efficiency. This research establishes a reproducible and quantitative framework for collector selection in apatite flotation, bridging experimental mineral processing and decision-science methodologies. The proposed approach provides a scalable strategy for improving reagent optimization in industrial phosphate beneficiation circuits.
Keywords: Phosphate resources, Apatite, Flotation, Collector, TOPSIS decision-making method
Subject terms: Chemistry, Energy science and technology, Engineering, Environmental sciences, Materials science
Introduction
Phosphate-group minerals, including apatite, are prevalent in igneous, metamorphic, and sedimentary rocks1,2. Apatite is generally represented by the formula Ca5(PO4)3(F, Cl, OH), where Ca may be partially substituted by elements such as Na, Sr, and Mn3. The composition of apatite can vary depending on its type and extraction location. The survival of all living organisms requires access to this essential substance, prompting the extraction, exploitation, and processing of phosphate rock to render it a useful product4. Global demand for phosphate resources has increased steadily in recent years5, driven primarily by agricultural intensification and fertilizer production, as well as expanding industrial and medical applications. Population growth and the consequent need for increased food production have heightened the demand for fertilizers, thereby increasing the demand for apatite.
Furthermore, the rising demand for apatite is largely driven by the expansion of industrial sectors, notably the chemical and dental industries. However, addressing the global demand for apatite presents significant challenges. Apatite reserves are unevenly distributed worldwide, with the majority concentrated in a limited number of countries, including Morocco, China, Russia, and the United States6. This concentration creates potential supply chain vulnerabilities and strategic constraints. Additionally, the complex mineralogy and prevalence of low-grade reserves make apatite extraction particularly challenging. These conditions necessitate advanced beneficiation strategies to achieve acceptable concentrate grades and recoveries7,8. Flotation has proven to be an effective method for separating apatite from other minerals to address these challenges9. However, the efficiency of direct flotation is highly sensitive to reagent chemistry, particularly the type and formulation of collectors employed. Conventional separation techniques have consistently posed challenges for apatite processing facilities. A contributing factor is the widespread use of traditional and single collectors, often in the form of fatty acids10,11. This issue is particularly pertinent for ores with high levels of impurities, such as silicates, carbonates, and iron oxides. In such cases, the use of a single collector is insufficient for effectively separating apatite from gangue minerals, leading to reduced separation efficiency and compromised metallurgical performance12.
A range of collectors has been utilized in apatite flotation, each possessing distinct advantages and limitations. Fatty acids, such as oleic acid and stearic acid, are carboxylic acids characterized by long hydrocarbon chains and are extensively employed as apatite collectors due to their strong affinity for calcium ions on the mineral surface13–16. Salts of fatty acids, such as sodium oleate and potassium oleate, are effective collectors for apatite in alkaline media17–21. Additionally, sulfonate-based collectors, which are organic compounds containing a sulfonic acid group (-SO3H), are frequently used as co-collectors with fatty acids to enhance flotation performance22,23. In contrast, sulfates, which are organic compounds containing a sulfate group (-SO4H), are less commonly employed than fatty acids and sulfonates but can be effective under specific conditions24,25. Recent research has concentrated on developing novel collectors with improved selectivity, environmental sustainability, and cost-effectiveness, including mixed and bio-based collectors derived from renewable sources such as plants and microorganisms, offering environmentally friendly alternatives to traditional collectors. While individual collectors can enhance apatite flotation, their combination can often yield synergistic effects. The mechanisms underlying the action of combined collectors are complex. Different collectors may adsorb onto distinct sites on the mineral surface, enhancing overall hydrophobicity and improving flotation recovery26–32. In this context, collectors may interact with each other, forming a more stable adsorption layer on the mineral surface. Furthermore, the combination of collectors can alter the surface properties of the mineral, making it more amenable to flotation.
Numerous reagents have been identified as essential collectors for apatite flotation in various studies. Furthermore, when selecting the most suitable collector for mineral processing operations, economic considerations are as significant as technical criteria. Accordingly, a structured multi-criteria evaluation framework is required to assess alternative collector systems by simultaneously considering technical performance indicators and economic constraints. Therefore, an appropriate approach is required to design a stable collector and examine its behavior from both technical and economic perspectives. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria decision-making (MCDM) method that facilitates the ranking and selection of alternatives based on their proximity to an ideal solution33,34. By considering multiple criteria simultaneously, TOPSIS enables more balanced and informed decisions, potentially leading to improved process optimization outcomes. This straightforward yet powerful methodology has been applied in fields such as management and engineering35–38, but has found limited application in mineral processing39 conducted a study on a novel approach for flotation machine selection, where the optimal solution was identified based on the evaluation performed40 investigated the application of the TOPSIS technique for selecting the optimal collector from lead–zinc sulfide ores, focusing on the integration of technological and economic factors to evaluate collector performance and optimize flotation outcomes. The model successfully identified a specific collector dose value that outperformed all other alternatives among the 18 options, thereby enabling the selection of the most optimal mode.
Previous studies on apatite flotation have largely focused on the development and laboratory-scale evaluation of single or mixed collectors, with primary emphasis on flotation performance indicators such as recovery and concentrate grade. In parallel, multi-criteria decision-making (MCDM) tools, including the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), have been sporadically applied in mineral processing, predominantly for equipment selection or limited decision problems, and only rarely in the context of phosphate flotation under conditions relevant to industrial practice.
In the present study, an integrated experimental–analytical framework is proposed for the systematic optimization of collector systems in direct apatite flotation. Rather than introducing a novel reagent, the contribution of this work lies in coupling bench-scale flotation experiments with a TOPSIS-based multi-criteria model that simultaneously incorporates flotation efficiency, reagent unit cost, and market availability. This research not only identifies the most suitable collector but also provides a replicable framework for decision-making in similar contexts.
Materials and methods
Materials
Samples
The samples employed in this study were sourced from the tailings of magnetite-hematite ores at the Chadormalu Mine in Yazd, Iran, which are characterized by low phosphorus content. These tailings, exhibiting low phosphorus levels, are processed at the apatite processing facility operated by Miras-E-Kavir Mines Development Co., serving as feed for the processing plant, with an annual production capacity of 70,000 tons of concentrate. The tailings were utilized as reference samples for conducting micro-flotation and bench-flotation tests to identify the constituent minerals qualitatively and quantitatively. The X-ray diffraction analysis was conducted by scanning within the 4-60 degree (2θ) range. The analysis identified hematite, quartz, fluorapatite, calcite, dolomite, and albite as the major phase minerals, whereas talc, orthoclase, and muscovite were identified as minor phases. As indicated in Table 1, X-ray fluorescence analysis revealed significant concentrations of SiO2 (29.66%), Fe2O3 (22.94%), and CaO (15.52%) in the sample. Furthermore, the mineral grade of P2O5 (target mineral) was determined to be 8.49%. The particle size distribution of the feed revealed (d80 = 105 
), with approximately 65.12% of apatite liberated from gangue minerals at this fraction. These characteristics confirm that the samples employed in the experiments are representative of the industrial operation, providing realistic and relevant conditions for the assessment of collector performance.
Table 1.
Chemical composition of tailing sample and pure apatite.
| Component (Wt%) | SiO2 | Fe2O3 | CaO | P2O5 | Al2O3 | MgO | Na2O | K2O | SO3 | TIO2 | MnO | L.O.I | Other |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Tailing sample | 29.66 | 22.94 | 15.52 | 8.49 | 6.57 | 5.50 | 1.13 | 1.13 | 0.81 | 0.43 | 0.27 | 6.31 | 1.24 |
| Pure apatite | 1.02 | 0.23 | 53.95 | 39.82 | 0.29 | 0.15 | 0.01 | 0.03 | 0.11 | 0.02 | 0.03 | 3.55 | 0.79 |
A sample of apatite-magnetite was collected from the southeastern pit of the Chadormalu mine to prepare a pure sample of apatite minerals. To maximize the purity of the desired mineral, the sample was meticulously crushed and prepared in the size range of −125+25 
. Magnetite was completely separated using a high-intensity magnetic separator with a strength exceeding 7000 Gauss. The resultant apatite-rich fraction was characterized using X-ray fluorescence, which revealed a P2O5 mineral grade of 39.82%, as detailed in Table 1.
Reagents
Table 2 lists the reagents used in the study. These include pH modifiers, various surfactants for collector formulations (such as anionics, nonionics, and polymers), and depressants. Apatite flotation was performed using various chemical reagents to determine the optimal conditions for this case study.
Table 2.
Used reagents list.
| N.o | Name | Function | Supplier |
|---|---|---|---|
| 1 | Sodium Carbonate | pH Adjuster | Kave Sodium Carbonate |
| 2 | Sodium Hydroxide | pH Adjuster | Petro Gohar |
| 3 | Sodium Silicates | Depressant | Petro Kavir Sadr |
| 4 | Pine Oil | Foam Modifier | Merck |
| 5 | Fatty Acid Mixture | Surfactant | Acid Charban Shimy |
| 6 | Tall Oil Fatty Acid (TOFA) | Surfactant | Shandong Up Chemical Technology |
| 7 | Oleic Acid Polyethylene Glycol Ester 6-mol (OAPEGE6) | Surfactant | Kimyagaran |
| 8 | Oleic Acid Polyethylene Glycol Ester 9-mol (OAPEGE9) | Surfactant | Kimyagaran |
| 9 | Alkyl Hydroxamic Acid | Surfactant | Merck |
| 10 | Benzohydroxamic Acid | Surfactant | Merck |
| 11 | Polysorbate (Tween 80) | Surfactant | Merck |
| 12 | Sorbitan Monooleate (Span 80) | Surfactant | Merck |
| 13 | Ethoxylated Nonylphenol 6-mol (ENP6) | Surfactant | Merck |
| 14 | Ethoxylated Nonylphenol 10-mol (ENP10) | Surfactant | Merck |
| 15 | Ethoxylated Isotridecanol (EIT) | Surfactant | Merck |
| 16 | N-Oleoylsarcosine (NOS) | Surfactant | Merck |
| 17 | Kerosene | Surfactant | Merck |
| 18 | Diesel | Surfactant | IRANOL |
Methodology
Micro-flotation
Micro-flotation experiments were conducted in a 200-milliliter Hallimond tube. A 1:10 solution of sodium hydroxide and sodium carbonate was used to prepare pulp pH. For all the tests, the collector dosage was 20 mg/L. Each test was conducted with conditioning times of 5 min for pH adjustment, 2 min for collector adsorption, and 5 min for froth flotation. The micro-flotation test was performed using 1 g of the purified apatite sample. The airflow rate was maintained at 15 mL/min, and the pulp temperature was controlled at 30 °C to represent typical plant operating conditions in the studied region and to ensure stable reagent behavior throughout the experiments. The resulting concentrate was dried and its weight was measured. The results were averaged from duplicate experiments, and the average was taken as the outcome.
Bench flotation experiments
To increase the concentrate grade and recovery, bench-flotation experiments were conducted using Denver D-12 containing a 1-liter tank and an impeller with a diameter of 3.88 inches. For the experiment, a feed sample entering the plant under study was used to prepare pulp. The pulp was regulated to 32°C, with an average density of 1.30 and a solid content of 35%. For each test, 500 g of the dry feed sample obtained from the plant feed of the studied processing facility was accurately weighed and transferred into the flotation cell before pulp preparation. The rotor impeller speed was set at 800 rpm. A solution composed of a 1:10 ratio of sodium hydroxide to sodium carbonate was used to adjust the pH of the pulp. A 300 g/t of sodium silicate (depressant and dispersant) were added to each cell volume and conditioned for 4 min. Various surfactants were added to render apatite mineral hydrophobic, as indicated in Table 2. The pulp was conditioned for an additional 4 minutes. After aeration, froth products were collected for 5 min. The concentrate and tailings were thoroughly dried in an oven at 80°C for a day, the concentrate and tailings were thoroughly dried. When the weight and control samples were dried, they were analyzed for their P2O5 content using a spectrophotometer. Finally, equations 1 and 2 were used to calculate the recovery and separation efficiency (S. E.) of each test. Each bench-scale flotation test was conducted in duplicate under identical operating conditions, and the reported values represent the average of independent experiments.
 |
1 |
 |
2 |
Where C represents the weight of the concentrate, c denotes the grade of the P2O5 concentrate, F is the weight of the feed, and f is the feed grade. 
is the fraction of the total feed weight transferred to the concentrate and m represents the maximum attainable grade of the targeted mineral41.
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)
A decision-making process involves identifying available solutions to a problem, analyzing and evaluating the available options, and comparing their outcomes to achieve the goal. As part of the decision-making process, the decision-maker decides which solution will be optimal for achieving the objective from among several available alternative solutions42 introduced the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) as a multi-criteria decision-making method for ranking options based on their similarity to ideal solutions. In the TOPSIS technique, in addition to ranking, comparing, and selecting the best option, this method can also determine the distance and categorization among options. The TOPSIS technique selects options with the highest similarity to the ideal solution as the most suitable choice. The ideal solution is the option that offers the greatest benefits at the least cost. Conversely, the non-ideal solution has negligible benefits and the highest costs. The steps of the method are as follows:
Formation of the decision matrix
For the evaluation of the selected options and criteria, a matrix is needed, which is determined according to Equation 3.
 |
3 |
In the context of the above equation, if n criteria and m options exist in a MCDM problem, 
represents the performance of the 
option with respect to the 
criterion.
Normalization of the decision matrix
Following the creation of the decision matrix, the data were normalized according to Equation 4.
 |
4 |
Calculation of index weights
In the following stage, the Shannon entropy method was applied to determine the relative priority of each index as compared to another based on equations 5, 6, 7, and 8.
 |
5 |
For incommensuring the entries, 
is used.
 |
6 |
Where 
is used to determine the index entropy.
 |
7 |
In this example, 
represents the degree of uncertainty or deviation of each index.
 |
8 |
Where 
was used to determine the weight of each index. Subsequently, the weight vector of the index is applied to the decision matrix and Equation 9 is established (form the weighted normalized matrix).
 |
9 |
Determining the positive and negative ideal solutions
The ideal solution is symbolized by 
, the non-ideal solution by 
, the positive indices by 
, and the negative indices by 
. Therefore, the following equation is determined by Equations 10 and 11.
 |
10 |
 |
11 |
Determining the distance of options
Equations 12 and 13 were applied to determine the distance between each option and the positive and negative ideals.
 |
12 |
 |
13 |
The distance between the positive and negative ideals is determined by 
and 
, respectively.
Determining similarity index
Finally, the Relative Closeness (RC), which is used to rank selected options, is calculated in accordance with Equation 14.
 |
14 |
This indicates that RC is the similarity index, which is a variable with a value ranging from zero to one. When the similarity index is closer to one, the option is closer to the ideal solution. Conversely, if the similarity index is close to zero, then the option is closer to the non-ideal solution. As a result, the higher the value of this index, the higher the option's ranking will be compared to other options.
Results and discussion
Micro-flotation
Hallimond tube cell flotation experiments were conducted using a pure apatite sample and anionic surfactants from the carboxylate and hydroxamate families. These tests were aimed at determining the effectiveness of each surfactant at pH 4-11. Figure 1 presents data demonstrating that the TOFA-OAPEGE6 (TO6) compound achieved the best recovery rate among all surfactants studied, with a 92% recovery rate. Given that the adsorption of fatty acid-based collectors on apatite is chemically specific, its neutralization leads to the formation of carboxylate ions (RCOO-) at the solid-liquid interface. Ultimately, this ion reacts with the calcium on the apatite surface and is recognized as the adsorption mechanism for apatite43,44.
Fig. 1.
Micro-flotation analysis results with different surfactants on pure apatite (Collector dosage 20 mg/L).
Upon closer inspection, in the pH:4-6, only benzo-hydroxamic acid and, to a lesser extent, Alkyl Hydroxamic Acid demonstrated appreciable flotation performance. Benzo-hydroxamic acid achieved the highest recovery (25–45%) under acidic conditions, indicating its strong interaction with the apatite surface even at low pH. This behavior may be attributed to its aromatic structure and capacity to form stable complexes with surface calcium ions.
Natural fatty acid-based collectors, such as TOFA and TO6, are almost inactive in this pH range. This can be explained by the poor ionization of carboxylic acid groups under acidic conditions, limiting their adsorption into the apatite surface.
As the pH increased from 6 onwards, all collectors exhibited a noticeable improvement in flotation performance. This enhancement is primarily due to the gradual ionization of carboxylic acid groups (-COOH to -COO-), which promotes electrostatic interactions with positively charged calcium sites on the apatite surface.
Among the tested reagents, TO6 exhibited a pronounced increase in recovery near pH 8, indicating enhanced collector effectiveness as the system approached alkaline conditions. The favorable performance of the TO6 combined collector has been previously reported32, and the present results further confirm its strong response in alkaline media. In this region, maximum flotation recoveries were observed for all collectors, with TO6 achieving over 90% recovery at approximately pH 9.5. Similar improvements in apatite floatability under alkaline conditions have been reported for other anionic collector systems, where enhanced adsorption and surface hydrophobicity contribute to improved recovery45. In closely related results, floatability was 4% lower for TOFA-OAPEGE9 (TO9). An increase of 3 mol of ethylene oxide units on the other end of the chain indicated that it negatively affected the adsorption rate of the apatite mineral. Because oleic acid polyethylene glycol ester is produced by the reaction of ethylene oxide with fatty acids, it has a two-part structure. One part is hydrophobic, consisting of the alkyl chain of the fatty acid, and the other part is the hydrophilic polar head, which is the ethylene oxide chain. This property enables it to be soluble in both aqueous and lipid phases, thereby reducing the surface tension at the liquid-liquid interface. This became evident when fatty acids were used individually. Under these conditions, the flotation efficiency reached 73%.
Alkyl Hydroxamic Acid and Benzo Hydroxamic Acid also showed strong performance in this pH range, although to a slightly lesser extent than TO6 and TO9. Interestingly, benzo-hydroxamic acid displays a more gradual performance increase across the entire pH spectrum, indicating a more uniform and stable flotation response across a wide pH range.
Bench flotation experiments
Figure 2 presents a comparative evaluation of different collector formulations in bench-scale flotation experiments, emphasizing their impact on three key performance indicators: P2O5 grade, recovery, and separation efficiency (SE). Each formulation consisted of a base collector TO6 (a mixture of TOFA and OAPEGE6, 70 wt%) combined with various secondary reagents (30 wt%) including hydrocarbon oils (diesel, kerosene), ethoxylated nonylphenols (ENP6, ENP10), sorbitan-based surfactants (Tween 80, Span 80), and other ethoxylated compounds. Surfactants commonly used in both previous studies and industrial applications for phosphate rock flotation were selected.
Fig. 2.
The results of bench-flotation analysis with different surfactants on apatite (pH: 9.5).
Flotation of TO6 combined with diesel fuel demonstrated relatively high selectivity and overall performance. With a recovery of 91.1%, a concentrate grade of 22.2%, and a separation efficiency of 69.9%, this mixture significantly enhanced the hydrophobicity of the apatite surface under alkaline conditions. In addition to recovery and grade, selectivity against major carbonate gangue minerals such as calcite and dolomite is a critical factor in phosphate flotation systems. Since these carbonate minerals also contain calcium in their structure, competitive adsorption between apatite and calcite/dolomite surfaces is expected during flotation. The relatively higher concentrate grade and separation efficiency obtained for the TO6–diesel system suggest that collector–mineral interactions preferentially favored apatite surfaces under the selected alkaline conditions. Although detailed surface characterization techniques were beyond the scope of this study, the flotation trends provide indirect evidence of improved selectivity toward apatite over carbonate gangue minerals.
A similar but slightly lower performance was observed when kerosene was used instead of diesel, yielding a separation efficiency of 62.2% and a concentrate grade of 21.6%. This reduction in performance may indicate subtle differences in hydrophobic interaction strength and froth behavior between the two hydrocarbon modifiers, while still maintaining acceptable selectivity toward apatite under the tested conditions.
Formulations using ENP6 and ENP10 achieved moderate to high recovery (81.9% and 87.6%, respectively) and separation efficiency (52.3% and 53.5%, respectively), but their grade levels were relatively lower than the hydrocarbon-based combinations. These results suggest that ethoxylated nonylphenols enhance froth stability but may suffer from weaker selectivity. Evidence has shown that ethoxylated nonylphenols have detrimental environmental effects46,47. Therefore, alternatives for these compounds should be considered.
Upon further ranking, N-oleoyl sarcosine and ethoxylated isotridecanol exhibited separation efficiencies of 40.4% and 43%, respectively. It is evident from these results that there was a noticeable decline compared with the family of ethoxylated nonylphenols. However, these surfactants displayed a higher degree of frothing than the other surfactants evaluated in this study. Because of this elevated frothing, industrial applications may face control challenges. The following surfactant examined was Tween 80, which achieved a separation efficiency of 37.7%, closely resembling that of N-oleoyl sarcosine. This substance can be used alone or in combination with other hydrophobic emulsifiers. It can form various types of water-in-oil and oil-in-water emulsions. However, because of its hydrophilic-hydrophobic balance value of 15, it failed to provide relative satisfaction when selected as a component of the collector for apatite flotation. Compared to Tween 80, Span 80 produced a lower separation efficiency of 1.8%, but exhibited a similar grade. Compared to Tween 80, this product has a hydrophilic/hydrophobic balance value of 14.9, and its longer stearic acid chain makes it more lipophilic. It was, however, unable to surpass Tween 80 in terms of results.
TOPSIS Decision-making model
Due to various influential technical criteria and their nonlinear relationship with economic criteria, it is necessary to develop decision-making methods for selecting the best solution from the available options.
The range of chemical reagents available for apatite flotation presents challenges in ensuring a constant supply of each and considering the overall cost of these reagents. It is essential to consider this factor in conjunction with the importance of grade and recovery when selecting an optimal collector. Therefore, the results obtained from bench-flotation analysis were selected based on crucial impact criteria for selecting the most effective combination based on the results obtained from bench-flotation analysis. Several factors contribute to this assessment, including grade, recovery, separation efficiency, overall cost, and access factor. Separation efficiency was included as a decision criterion because it is widely used in mineral processing practice as an integrated performance indicator that reflects the combined effect of recovery and concentrate grade. In flotation systems where trade-offs between these two parameters are common, separation efficiency provides a concise measure of overall process effectiveness that is not always evident from individual performance metrics alone. It should be noted that all collector systems were evaluated under identical dosage conditions during bench-scale experiments. Therefore, the reagent unit price (USD/kg) was considered a proportional indicator of relative reagent cost per ton of processed ore.
In addition to flotation performance and cost-related criteria, an implementation-oriented parameter was incorporated into the decision-making framework to reflect practical constraints associated with reagent availability, supply chain reliability, and ease of procurement under the studied industrial conditions. Access factors 1 to 5, respectively, represent very poor, poor, average, good, and excellent accessibility. For the evaluation of the best combination among the chosen criteria, the TOPSIS decision-making method was employed due to the quantitative nature of the selected criteria. It was chosen because of its strong mathematical foundation. The results of this analysis are presented in Tables 3, 4, 5, and 6. This technique has also been proven to be an efficient mathematical optimization method for determining the most advantageous alternative among the available alternatives due to the proximity of the solution space for the alternatives, the multitude of evaluation criteria, the inability to exclude any of these metrics, and the presence of empirically validated data that eliminates the need for expert judgment.
Table 3.
Creating a decision-making matrix.
| Option | Primary reagent (70% Wt) | Secondary reagent (30% Wt) |
Grade (%) |
Recovery(%) | S.E (%) | Cost (USD/kg) | Access factor |
|---|---|---|---|---|---|---|---|
| 1 | TO6 | Tween 80 | 15.00 | 70.06 | 37.75 | 2.77 | 4 |
| 2 | TO6 | Span 80 | 15.00 | 66.69 | 35.93 | 2.98 | 4 |
| 3 | TO6 | ENP6 | 17.50 | 81.91 | 52.35 | 2.44 | 5 |
| 4 | TO6 | ENP10 | 16.70 | 87.64 | 53.49 | 2.71 | 5 |
| 5 | TO6 | EIT | 15.70 | 75.51 | 43.05 | 5.26 | 1 |
| 6 | TO6 | NOS | 15.00 | 74.92 | 40.36 | 3.85 | 2 |
| 7 | TO6 | Diesel | 22.20 | 91.11 | 69.85 | 1.97 | 5 |
| 8 | TO6 | Kerosene | 21.60 | 82.50 | 62.16 | 1.96 | 4 |
Table 4.
Normalized decision matrix and entropy-based weight calculation for evaluation criteria.
| Option | Normalization | ||||
|---|---|---|---|---|---|
| Grade | Recovery | S.E | Cost | Access factor | |
| 1 | 0.302 | 0.313 | 0.263 | 0.310 | 0.354 |
| 2 | 0.302 | 0.298 | 0.251 | 0.333 | 0.354 |
| 3 | 0.352 | 0.366 | 0.365 | 0.273 | 0.442 |
| 4 | 0.336 | 0.391 | 0.373 | 0.303 | 0.442 |
| 5 | 0.316 | 0.337 | 0.300 | 0.588 | 0.088 |
| 6 | 0.302 | 0.334 | 0.282 | 0.430 | 0.177 |
| 7 | 0.447 | 0.407 | 0.487 | 0.220 | 0.442 |
| 8 | 0.435 | 0.368 | 0.434 | 0.219 | 0.354 |
| Weights index | |||||
| 1 | 0.108 | 0.111 | 0.096 | 0.116 | 0.133 |
| 2 | 0.108 | 0.106 | 0.091 | 0.124 | 0.133 |
| 3 | 0.126 | 0.130 | 0.133 | 0.102 | 0.167 |
| 4 | 0.120 | 0.139 | 0.135 | 0.113 | 0.167 |
| 5 | 0.113 | 0.120 | 0.109 | 0.220 | 0.033 |
| 6 | 0.108 | 0.119 | 0.102 | 0.161 | 0.067 |
| 7 | 0.160 | 0.145 | 0.177 | 0.082 | 0.167 |
| 8 | 0.156 | 0.131 | 0.157 | 0.082 | 0.133 |
 |
0.994 | 0.998 | 0.987 | 0.974 | 0.960 |
 |
0.006 | 0.002 | 0.013 | 0.026 | 0.040 |
 |
0.068 | 0.028 | 0.145 | 0.298 | 0.462 |
Table 5.
Applying weight index and determining positive and negative ideal solution.
| Option | Grade | Recovery | S.E | Cost | Access factor |
|---|---|---|---|---|---|
| 1 | 0.020 | 0.009 | 0.038 | 0.092 | 0.163 |
| 2 | 0.020 | 0.008 | 0.036 | 0.099 | 0.163 |
| 3 | 0.024 | 0.010 | 0.053 | 0.081 | 0.204 |
| 4 | 0.023 | 0.011 | 0.054 | 0.090 | 0.204 |
| 5 | 0.021 | 0.009 | 0.043 | 0.175 | 0.041 |
| 6 | 0.020 | 0.009 | 0.041 | 0.128 | 0.082 |
| 7 | 0.030 | 0.011 | 0.071 | 0.066 | 0.204 |
| 8 | 0.029 | 0.010 | 0.063 | 0.065 | 0.163 |
 |
0.030 | 0.011 | 0.071 | 0.065 | 0.204 |
 |
0.020 | 0.008 | 0.036 | 0.175 | 0.041 |
Table 6.
Determining the distance of options and similarity index.
| Option |  |
 |
RC | Rank |
|---|---|---|---|---|
| 1 | 0.05957 | 0.14789 | 0.71288 | 5 |
| 2 | 0.06402 | 0.14408 | 0.69238 | 6 |
| 3 | 0.02469 | 0.18913 | 0.88452 | 2 |
| 4 | 0.03085 | 0.18494 | 0.85704 | 3 |
| 5 | 0.19886 | 0.00734 | 0.03558 | 8 |
| 6 | 0.14124 | 0.06235 | 0.30627 | 7 |
| 7 | 0.00033 | 0.19984 | 0.99834 | 1 |
| 8 | 0.04159 | 0.16686 | 0.80048 | 4 |
The entropy-based weights for the evaluation criteria show that access factor had the highest weight (0.462), followed by cost (0.298), separation efficiency (0.145), grade (0.068), and recovery (0.028). This indicates that availability and cost considerations dominated the TOPSIS evaluation, while technical performance metrics contributed less. As a result, the analysis revealed that the TO6–diesel combination achieved the highest RC score (0.9983), decisively ranking it as the optimal formulation. This result aligns with the experimental findings in Sect. “Bench floation experiments”, where TO6–diesel demonstrated the highest recovery, grade, and separation efficiency. Its high accessibility and well-established supply chain further enhanced its standing.
The second and third ranks were occupied by TO6–ENP6 (RC = 0.8845) and TO6–ENP10 (RC = 0.8570), both of which showed strong technical performance. However, their relatively lower positions may be influenced by possible environmental considerations and somewhat higher costs. The TO6–kerosene blend (RC = 0.8004) also performed well, presenting a feasible alternative to diesel in regions with kerosene abundance.
Lower-performing formulations included TO6–Tween 80 and TO6–Span 80, which had RC values of 0.7128 and 0.6923, respectively. The least favorable formulation was TO6–EIT, which scored an RC of only 0.0355, largely due to poor separation efficiency and very limited accessibility.
Conclusion
This study presents a robust integration of flotation with advanced multi-criteria decision-making (MCDM) techniques to address the complex challenge of optimizing collector systems for apatite flotation from low-grade phosphate ores. As part of the experiments, different reagents were used to perform micro-flotation and bench-flotation experiments. The purpose of selecting these reagents was to explore the most suitable collector for use in the apatite processing circuit on a technical and economically sound basis. This approach successfully demonstrates the superior efficacy of ethoxylated tall oil fatty acid-based collectors, particularly TO6, in alkaline environments that provides a structured and quantitative basis for reagent selection in mineral processing.
Ethoxylated TOFA-based collectors, specifically TO6 and TO9, demonstrate markedly enhanced flotation performance for apatite under alkaline conditions. Among the reagents evaluated, TO6 exhibited the highest recovery efficiency, exceeding 90% at approximately pH 9.5. This superior performance can be primarily ascribed to the presence of ethoxylated chains, which contribute to improved aqueous solubility, elevated surface activity, and greater affinity for the apatite mineral surface. The enhanced selectivity of TO6 is likely due to its optimal hydrophilic-lipophilic balance (HLB) value of 9.9, which promotes the formation of a favorable number of hydrogen bonds, thereby facilitating selective adsorption onto apatite surfaces. Conversely, oleic acid, a traditional fatty acid-based collector, consists of a long nonpolar hydrocarbon chain and a terminal carboxylate functional group. Its flotation activity primarily arises from the nonpolar hydrocarbon tail, which imparts surface activity. However, its comparatively lower selectivity and solubility in alkaline conditions limit its efficiency relative to ethoxylated TOFA derivatives.
The bench-scale flotation results clearly highlight that TO6–diesel is the optimal combination among the tested formulations. It provides the best balance of recovery, grade, and selectivity, confirming the findings from micro-flotation experiments and justifying its top ranking in the TOPSIS decision model (Sect. “TOPSIS decisison-making model”). The observed performance trend also supports the hypothesis that collector synergy especially between ethoxylated and hydrocarbon-based reagents can be a decisive factor in flotation efficiency. However, comprehensive economic assessments and detailed environmental evaluations are recommended for future studies.
The TOPSIS model effectively captured the multidimensional nature of collector selection, balancing laboratory performance with economic and logistic constraints. It identified TO6–diesel as the highest-ranked collector formulation based on the evaluated technical and economic criteria. This data-driven, quantitative approach strengthens decision-making in mineral processing, offering a scalable framework adaptable to other ore types and reagent systems.
Beyond the specific case study, this work demonstrates the applicability of an integrated MCDM-based framework for reagent assessment in flotation systems. It establishes a replicable approach for the integration of mineral process engineering, and decision sciences in mineral beneficiation. Further investigation is required to optimize adsorption conditions under pilot-scale and variable process environments. Furthermore, the economic feasibility of employing this collector for industrial Use must also be evaluated. It is also recommended that comprehensive life cycle assessments be conducted to further validate environmental sustainability.
Acknowledgement
The authors thank Miras-E-Kavir Mines Development Co., Ltd. and Amirkabir University of Technology for funding this research.
Author contributions
**Saeed Abdollahi:** Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Visualization, Writing – original draft **Sajjad Afraei:** Supervision, Project administration, Methodology, Validation, Writing – review & editing **Mehdi Irannajad:** Supervision, Project administration.
Funding
The authors declare that no funds, grants, or other financial support were received during the preparation of this paper.
Data availability
The datasets generated and analyzed during the current study have been uploaded as Supplementary Dataset files and are available in the online submission system. They can also be requested from the corresponding author.
Declarations
Competing interest
The authors declare no competing interests.
Footnotes
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets generated and analyzed during the current study have been uploaded as Supplementary Dataset files and are available in the online submission system. They can also be requested from the corresponding author.