A flotation kinetics framework for treating intrinsic floatability as a measurable state variable in batch and continuous flotation systems

Flotation engineers often rely on measured flotation rates to evaluate circuit performance. However, these rates combine intrinsic particle floatability with system-level factors such as pulp concentration, gas dispersion, aeration, froth transport and hydrodynamic conditions. As a result, observed flotation rates do not reveal why recovery changes from cell to cell or why certain particles float more readily than others. Traditional kinetic models fit flotation data successfully but do not distinguish between intrinsic particle floatability and operating conditions, leaving floatability embedded within empirical rate parameters. In reality, intrinsic floatability, governed by hydrophobicity, reagent coverage, mineral liberation, surface oxidation state and particle size, is fundamentally distinct from the system-level constraints that control bubble–particle encounters, bubble surface availability and solids transport. The framework introduced here explicitly separates these influences, allowing intrinsic floatability to be determined directly from flotation data and treated as an independently measurable state variable of the flotation pulp. Derivation of the model Flotation systems operate on three distinct time scales: (1) hydrodynamic collisions occur in milliseconds, (2) bubbles live for seconds and (3) overall flotation kinetics evolve over minutes. This natural separation of time scales permits a rate expression for each particle size class [1]:(1) This rate expression shows that mass transfer of particles in size class i to the froth depends on three factors: bubble–particle encounter frequency, κk, bubble surface availability, χk, and the intrinsic floatability of that size class, ϕP,ik.is the mass transfer rate of particles of size class i to the froth during time interval Δtk, andis their average concentration in the pulp. The index k denotes either a froth collection interval in a batch flotation test or a flotation cell in a continuous circuit (Fig. 1). Summing over all particle size classes yields the overall flotation rate for the pulp:(2) Fig. 1 Framework for decoupling intrinsic floatability φfrom system-level factors κ, χ and µ. where ϕP,k represents the mean intrinsic floatability of the entire pulp. The formulation also introduces a transport-capacity parameter, μ, representing the maximum solids load bubbles can carry under given operating conditions.When the mass transfer rate is small, bubble coverage is negligible and χk ≈ 1, reducing the model to the familiar first-order rate law. Conversely, as approaches μk, bubble surfaces saturate and the rate transitions toward a transport-limited, zero-order regime with μk being the rate constant. Thus, the same formulation bridges dilute and concentrated flotation conditions without requiring ad hoc distribution assumptions. Importantly, intrinsic floatability can be calculated directly from routine froth removal data and known operating parameters. For each size fraction:(3) Summation over all size classes gives the intrinsic floatability of the overall pulp: (4) For flotation engineers, the key point is that intrinsic floatability is no longer hidden within a fitted rate constant. It becomes a measurable property of the pulp, size by size and cell by cell, enabling direct diagnosis of flotation selectivity. Results and discussion The framework was evaluated using both laboratory batch flotation tests and industrial flotation circuit data. For brevity, only results for the Bilfer copper rougher circuit are presented here (Table 1). The bubble–particle encounter rate, κk, decreases slightly from 61.8 to 53.3 s−¹ across the rougher bank, indicating a gradual reduction in flotation opportunity as the more floatable particles are recovered. The maximum transport capacity μk declines from 3.92 to 2.87 kgm−³s−¹, consistent with progressive depletion of chalcopyrite-rich particles and increasing gangue content in the pulp. χk stays close to unity because the ratio is very small, indicating minimal utilization of available bubble transport capacity. The approximately 40-fold drop in across the circuit further confirms that flotation operates well below the transport-limited regime and therefore behaves effectively as a first-order process throughout the rougher bank.] The table also presents the intrinsic floatability profiles of individual size fractions and of the overall pulp. In every cell, intrinsic floatability increases with decreasing particle size, consistent with improved mineral liberation and reduced particle locking in the finer fractions. More importantly, intrinsic floatability decreases progressively for every size fraction as the pulp advances through the circuit, directly revealing the preferential removal of highly floatable particles. By the fifth rougher cell, intrinsic floatability approaches negligible values for all size fractions, indicating that little selectively floatable material remains in the pulp. A key consequence of this approach is that flotation selectivity can be quantified directly rather than inferred from recovery trends alone. The extracted floatability profiles identify where floatable particles are being recovered, where floatability is being depleted and which cells continue to contribute meaningfully to separation performance. Such information is not available from conventional recovery or rate data alone. Conclusion This study presents a flotation framework that separates intrinsic particle floatability from transport and hydrodynamic effects. As a result, floatability is no longer treated as an apparent kinetic fitting parameter but as a measurable property of the flotation pulp. By incorporating bubble–particle encounter frequency, bubble surface availability and transport-capacity limitations, the framework provides a unified description of flotation behavior under both dilute and concentrated operating conditions and is directly applicable to laboratory and industrial flotation systems.The methodology requires only standard flotation measurements and readily available operating data, making it practical for routine application. Most importantly, it enables direct determination of intrinsic floatability profiles for individual size fractions and for the overall pulp, either throughout the entire duration of a batch flotation kinetic test or across the complete bank of flotation cells. For engineers, this provides a new way to diagnose flotation performance, quantify flotation selectivity, identify where floatability is being depleted within a circuit and evaluate the effectiveness of reagent schemes and operating conditions. In this way, the proposed approach transforms floatability from an inferred parameter into a measurable process variable that can be used to better understand and optimize flotation performance. Selected reference Polat M, Polat H (2023) A phenomenological kinetic flotation model: Distinct time-variant floatability distributions for the pulp and froth materials. Miner Eng 201:108217

A flotation kinetics framework for treating intrinsic floatability as a measurable state variable in batch and continuous flotation systems

Authored by: Mehmet Polat, Veli Guzel, Muammer Kobas and Hurriyet Polat

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