Diameter of a collection of different-size spherical objects such as particles, droplets, and bubbles can be represented by a single value called the mean diameter. There are many ways of calculating and expressing the mean diameter of particulate matter. One of them is the Sauter mean diameter also called the surface-volume mean diameter. This diameter has been extensively used in numerous publications without paying attention to its physical meaning. In this article, the physical meaning of the Sauter mean diameter of spherical objects was derived and presented. The Sauter mean diameter of a collection of spherical objects of different sizes is equal to the diameter of equisized spherical objects forming a collection. The polysized and equisized systems have different numbers of spherical objects, identical total surface area, and identical total volume. If the surface energy of all spheres of both systems is the same, they can be called equienergetic.

Fluorite, as a scarce nonrenewable strategic non-metallic mineral resource, is the primary raw material for fluorine products used in diverse fields such as metallurgy, national defense, chemical and optical industries. With the increasing expansion of the related fields, the demand for high-quality fluorite continues to grow. Hence, the surge of interest in effectively utilizing fluorite resources has led to vast attention worldwide. So far, significant endeavors have been done to enhance the beneficiation of fluorite from relatively low-grade ores. It has been well appreciated that the froth flotation is of the most importance. However, to the best of the authors' knowledge, it lacks a thorough and critical review on the recent developments in fluorite flotation. This article begins with introducing the deposits and unique physical and chemical properties of fluorite from the perspective of the crystal structure. It is followed by a systematic review of common reagents involved in fluorite flotation, including collectors, depressants, regulators, modifiers, and frothers. Specifically, the synergistic effect of collectors and depressants on the recovery of fluorite is elaborated for the first time. Finally, the most widely seen fluorite-flotation cases, including separation of fluorite from quartz, calcite, barite, and sulfide, are summarized individually. The present review sheds new light on the deep understanding of fluorite flotation, the future synthesis of reagents, as well as their schemes in practical use. Meanwhile, such a novel rain of thought provided in this work has the potential to guide the flotation of other similar minerals extensively.

This paper reviews recent developments in the fundamental understating of ultrafine (nano) bubbles (NBs) and presents technological advances and reagent types used for their generation in flotation. The generation of NBs using various approaches including ultrasonication, solvent exchange, temperature change, hydrodynamic cavitation, and electrolysis was assessed. Most importantly, restrictions and opportunities with respect to the detection of NBs were comprehensively reviewed, focusing on various characterization techniques such as the laser particle size analyzer (LPSA), nanoparticle tracking (NTA), dynamic light scattering (DLS), zeta-phase light scattering (ZPALS), and zeta sizer. As a key feature, types and possible mechanisms of surfactants applied to stabilize NBs were also explored. Furthermore, flotation-assisted nano-bubbles was reported as an efficient method for recovering minerals, with a special focus on flotation kinetics. It was found that most researchers reported the existence and formation of NBs by different techniques, but there is not enough information on an accurate measurement of their size distribution and their commonly used reagents. It was also recognized that a suitable method for generating NBs, at a high rate and with a low cost, remains a technical challenge in flotation. The application of hydrodynamic cavitation based on a venturi tube and using the LPSA and NTA in laboratory scales were identified as the most predominant approaches for the generation and detection of NBs, respectively. In this regard, neither pilot- nor industrial-scale case studies were found in the literature; they were only highlighted as future works. Although the NB-stabilizing effects of electrolytes have been well-explored, the mechanisms related to surfactants remain the issue of further investigation. The effectiveness of the NB-assisted flotation processes has been mostly addressed for single minerals, and only a few works have been reported for bulk materials. Finally, we believe that the current review paves the way for an appropriate selection of generating and detecting ultrafine bubbles and shines the light on a profound understanding of its effectiveness.

In this paper, a scheme of flotation frothers classification is presented. The scheme first indicates the physical system in which a frother is present and four of them i.e., pure state, aqueous solution, aqueous solution/gas system and aqueous solution/gas/solid system are distinguished. As a result, there are numerous classifications of flotation frothers. The classifications can be organized into a scheme described in detail in this paper. The frother can be present in one of four physical systems, that is pure state, aqueous solution, aqueous solution/gas and aqueous solution/gas/solid system. It results from the paper that a meaningful classification of frothers relies on choosing the physical system and next feature, trend, parameter or parameters according to which the classification is performed. The proposed classification can play a useful role in characterizing and evaluation of flotation frothers.

Electrocoagulation (EC) is a promising method that has been effectively employed in the treatment of various types of industrial effluents.