Flotation Enrichment of Micro- and Nanosilica Formed During the Production of Silicon and Ferrosilicon

Abstract

This paper presents the results of experiments conducted on the flotation separation of cyclone dust particles. The flotation process was conducted using a laboratory flotation apparatus comprising three chambers. Experimental tests supported theoretical results of the theoretical reasoning and justification for the choice of parameters that the flotation process should have in order to extract particles of such small sizes. Furthermore, this work elucidates the concept of “nanobubbles” and substantiates their viability for use in the flotation of nanoparticles, given that bubbles of such a magnitude are firmly affixed to the hydrophobic surface of particles. Bubbles of a larger size than nanoparticles will float both hydrophobic and hydrophilic particles. The effective flotation of cyclone dust from the gas cleaning of silicon and ferroalloy production provided two materials as a result. The experiments yielded insights into the rational technological parameters of the flotation mode for obtaining new products. These insights were gleaned from the preliminary conditioning (conditioning time from 0.5 to 1.5 h) of wet cyclone dust (dry dust weight of 4 kg) with liquid glass (1.4 g per 1 dm³ of pulp) in a cavitation unit at a pH value of 8.5. The flotation process was conducted in a three-chamber flotation apparatus with a volume of 0.02 m³ for a duration of 90 min, utilizing a pneumohydraulic aerator with air suction from the atmosphere. In this instance, the pulp was conveyed via a pump at a pressure of 0.4 MPa from the initial cleansing chamber into the aerator. During the flotation process, kerosene (1 mg per 1 dm³ of pulp) and pine oil (2 mg per 1 dm³ of pulp) were added as additives. The resulting products were silicon dioxide (95%) and carbon nanoparticles (94%).

1. Introduction

The gas cleaning dust produced as a by-product of silicon and ferroalloy production is currently classified as class 4 hazardous waste. The silicon production facilities in Shelekhov and Kamensk-Uralsky, situated within the Russian Federation, are responsible for the generation of a total dust accumulation volume of approximately 35 thousand tons on a year-to-year basis. The majority of the waste is stored in open-air sludge fields, with only a minor proportion undergoing processing.

Currently, the technology of introducing silicon gas purification bag dust into special concrete is in use [1,2,3,4,5,6,7,8,9,10]. Cyclone dust contains up to 40% carbon in its composition and therefore cannot be used for construction purposes. During annealing, too much environmental damage is caused and the economic costs of calcination losses are high. In this regard, the problem of recycling these wastes is acute in enterprises. A solution is required to separate the main components of silicon gas purification dust into nanoscale silica and carbon in order to further use them in various industries. Industrial technologies that solve this problem have not been developed yet.

Occurring as amorphous silica (SiO₂) in the form of spherical particles, microsilica is a by-product of silicon or ferrosilicon production [1,2,3,4,5,6,7,8,9,10]. Used extensively in the construction industry, fumed silica serves as a modifying component in concrete manufacture, utilizing its chemical composition and physical properties as a highly active pozzolanic agent [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Crushed volcanic and sedimentary rocks, diatomites, volcanic ash, and tuffs are included in the natural additives. These include fly ash, blast furnace slags, and microsilica [25,26,27,28,29]. Ferroalloy wastes contain elevated SiO₂ contents of 85%–95%, with reduced contents down to 65%–75%. The specific surface area of microsilica ranges from 15,000 to 25,000 m²/kg, which is significantly higher than the specific surface area of Portland cement, which is between 300 and 400 m²/kg. Microsilica acts effectively as a micro-filler [30,31,32,33,34,35], with high pozzolanic activity.

It is known that high-purity silica (92% or more) has wide application in the construction field. Pure micro- and nanoscale carbon also has wide applications in rubber products, metallurgy, and other industries.

Cyclone dust is composed of up to 40% carbon, which precludes its use in construction applications [35,36]. The process of annealing results in significant environmental damage and substantial economic costs due to the loss incurred during the calcination stage. In this regard, enterprises are confronted with a significant challenge in terms of waste management. This necessitates the identification of an effective method for the separation of the primary constituents of silicon gas cleaning dust into nanosized silica and carbon, with the objective of subsequently utilizing these components in a range of industrial applications [37,38,39,40,41,42,43,44,45,46]. To date, no industrial technologies have been developed to address this issue. One potential method of separation is flotation; however, there are currently a number of challenges associated with this approach.

The principal reason for the low flotation rate of small particles is their low efficiency of collisions with conventional flotation bubbles of a given size and speed [46,47,48,49,50,51]. A number of flotation technologies have been developed with the objective of enhancing the efficiency of particle–bubble collisions. This has been achieved either by reducing the size of the bubbles or by increasing the apparent size of the particles. A substantial quantity of experimental data demonstrate that the efficacy of collecting bubble–particles increases with decreasing bubble size [37,38,39,40]. It is, however, important to note that the use of small bubbles is also associated with certain disadvantages. The low growth rate of relatively small bubbles with attached particles results in prolonged flotation times.

A further disadvantage is that the buoyancy of small bubbles may be insufficient to guarantee the selectivity of the process. Furthermore, microbubbles have been demonstrated to result in high water recovery, which in turn increases the entrainment of gangue minerals [41,42]. A reduction in bubble size can be achieved through the utilization of a variety of techniques, which can be categorized into two principal groups: mechanical and physicochemical. Mechanical methods include the design of flotation cells, specifically the shape of the rotor and stator, as well as the size of the gap between the rotor and stator. These modifications facilitate the dispersion of gas bubbles produced at the bottom of the flotation cell into smaller bubble sizes. Additionally, microporous materials may be utilized at the base of the cell, facilitating the formation of gas bubbles.

Additionally, the authors of [43,44,45] put forth a methodology for the generation of minute bubbles through the process of hydrodynamic cavitation. This process entails the formation and expansion of gas bubbles within a liquid due to the rupture of the liquid–liquid or liquid–solid interface under the influence of external forces. The generation of bubbles on the surface of particles by cavitation results in their spontaneous attachment to the particle, thereby eliminating the collision and attachment process that is often the rate-determining step in flotation [46,47,48]. Furthermore, the flotation efficiency of coarse particles is enhanced by cavitation, which reduces the probability of breakaway as bubble–particle aggregates increase in the liquid.

The primary objective of the aeration process is to create the requisite characteristics of the gas–liquid system that would facilitate the rational course of flotation separation of mineral particles in the pulp. One such characteristic is the specific surface area of the bubbles. It can be reasonably deduced that an increase in surface area will result in an enhanced probability of a hydrophobic mineral particle meeting and being fixed to an air bubble [49]. The specific surface area is dependent upon the dispersed composition of the bubbles and is subject to change over time, contingent upon the degree of stability of the aerated liquid (foam). A reduction in the stability of the latter results in a rapid decline in the specific surface area of the bubbles, an expansion in their dimensions, and an increase in mixing. This ultimately leads to a decline in the technological parameters of flotation. Conversely, high stability of the aerated liquid (foam) can impede the process of bubble mineralization or prevent the formation of sufficiently floating flotation complexes due to low bubble coalescence. Accordingly, for the optimal functioning of flotation, the two-phase gas–liquid system must be inherently stable. However, when mineral particles are introduced, the stability must not impede the attachment of bubbles to the particles and the formation of the requisite buoyancy.

This paper will examine the methods of increasing the stability of an aerated liquid (foam). The use of various types of foaming agents that prevent the merging of bubbles with each other has been found to be the most effective method for increasing stability. However, an increase in the concentration of the foaming agent in the pulp not only enhances the stability of the gas–liquid system but also leads to an exacerbation of the merging of bubbles with mineral particles [50]. Therefore, an alternative method of increasing the stability of the system must be employed, without an accompanying increase in the concentration of the foaming agent. One potential avenue for exploration is the alteration of the dispersed composition of the initial air bubbles obtained from the aerator.

In [51], a dependence analogous to that observed in [52] was obtained, namely that a reduction in the initial bubble size resulted in an increase in the equilibrium foam column height. This can be explained by the following considerations. The ascent rate of small bubbles is low, resulting in a longer “packing” time into foam. This, in turn, enhances the stability of the foam. The deformation of small air bubbles is less pronounced than that of larger bubbles due to the higher excess pressure within them. Consequently, the smaller bubbles are more effectively “packed” into a more hydrated foam, which also contributes to enhanced stability.

It is established that the lifespan of small bubbles on the foam surface is longer than that of larger bubbles. The rate of change in the column and, consequently, the volume of foam is slower when small bubbles rupture than when large bubbles rupture [53,54]. In a finely dispersed foam, the duration of the coalescence process is prolonged.

The invention [55] outlines a methodology for the separation of foams. The feedstock is subjected to conditioning with reagents, a foam layer is prepared by introducing gas and a foaming agent into the pulp in a flotation machine, the feedstock is conveyed to the foam, and the separation products are removed. The introduction of gas into the pulp is achieved through the formation of bubbles of equal diameter. This is accomplished through the utilization of a combination of pneumatic–hydraulic aerators, perforated chambers with calibrated holes, and a system of calibration capillaries. The ratio of foam volume to liquid volume in this foam is 3 to 8, which achieves a maximum output of mineral particles through the drain sill. In all cases, a foam layer multiplicity of between three and eight yielded the greatest output of mineral particles through the drain sill. The experimental data obtained demonstrate that a reduction in collector consumption is possible without compromising the process indicators. In accordance with the conditions previously outlined, the implementation of the proposed method resulted in the removal of mineral particles at a rate of less than 3%. This phenomenon can be attributed to the fact that the water content of the foam layer in the proposed method increases at a more pronounced rate with depth of immersion in the foam.

In industrial conditions, bubbles of the same size can be created using a variety of techniques, including pneumatic–hydraulic aerators operating in specially selected modes, perforated chambers with calibrated holes, and a system of calibrated capillaries. In [51], experiments were conducted to determine the dependence of the height of the equilibrium foam column on the degree of polydispersity of the specified air bubbles. A mathematical expectation of the bubble distribution function by a diameter of 1 mm, an air flow rate of 35 L/min, and an OPSB foaming agent concentration of 0.100 kg/m³ were used to determine the dependence of the height of the equilibrium foam column on the standard deviation. The standard deviation increased from 0.2 mm to 1 mm, accompanied by a corresponding decrease in foam height from 10 cm to 3 cm. The dimensions of the air bubbles were determined using a photoelectric device [56]. The obtained dependence demonstrated that an increase in polydispersity resulted in a reduction in the height of the foam layer. It is established that during the process of foam aging, its polydispersity increases. This phenomenon has been observed to result in a reduction in foam stability, particularly when the initial bubble size distribution is altered [57]. This phenomenon can be attributed to the following physical laws:

• The rise rates of bubbles of varying diameters differ, increasing the probability of their collision and coalescence.
• The greater the difference in diameter between the bubbles, the more actively they coalesce [58].
• The presence of bubbles of varying sizes within a less water-saturated foam results in a collapse that occurs at a significantly faster rate than that observed in a more water-saturated foam [59].
• The formation of a relatively small number of large bubbles with a high rise rate can result in significant mixing in a gas–liquid dispersed system [49].

In this regard, a topic of significant scientific interest is the separation of dust from the gas cleaning of silicon and ferrosilicon, and the subsequent creation of modifiers that are competitive in terms of both properties and cost. Additionally, there is a need to address the environmental issue of sludge accumulation.

2. Materials and Methods

2.1. Equipment and Methods of Flotation Enrichment

Figure 1 illustrates a visual representation of a laboratory-based flotation apparatus, specifically designed for the separation of cyclone dust into nanosized particles of carbon and silica. Figure 2 presents a diagrammatic view of the three-chamber configuration of the working volume of the flotation machine.

During the prolonged operation of the flotation machine, a reduction in foam thickness was observed across all chambers. As the greatest quantity of foaming agent fog was generated in the final cleaning chamber, air was drawn into the ejector from the upper portion of the aforementioned chamber. In the absence of this fog suction, the foam layer in all cleaning chambers exhibited a marked reduction in a few minutes of flotation. In order to create bubbles of the requisite size, it is essential to generate bubbles emanating from the aerator by adjusting the liquid–air ratio. Furthermore, it is necessary to utilize vortices within the cleaning chambers to separate air bubbles within the flotation machine and to separate the foaming agent within these chambers.

The initial step involved the flotation of the wet sludge, after which the samples were subjected to drying and preparation. Prior to flotation, the material was subjected to a conditioning process for a period of ten minutes. The elemental composition of the resulting products exhibited minimal variation compared to that of the initial sludge. It can be reasonably deduced that this phenomenon can be attributed to the hydrophobic nature of the particles in the dried sludge, which enables them to float within the foam product. In order to depress silicon dioxide particles, a longer conditioning period is necessary (steaming with liquid glass), during which their hydrophilization will occur.

A new series of experiments enabled us to ascertain several significant findings. At a pH value of 6, no separation of silicon dioxide and carbon particles was observed. The loading of sludge and reagents at the outset of flotation resulted in a change in the composition of the foam product, specifically in the content of carbon and silicon. It is evident that nanoparticles must be floated with a continuous supply of the initial material into the chamber volume. Furthermore, the concentration of the foaming agent must be maintained at an optimal level to ensure the maximum extraction of valuable components.

The formation of pulp–air vortices by the aerator–ejector resulted in the sedimentation of the most hydrophilic particles in the lower part of the flotation machine. Additionally, bubbles of varying sizes were separated in different chambers of the flotation machine, leading to the formation of flows that influenced the speed of liquid movement and the washing away of hydrophilic silica particles through the interbubble channels.

The pneumatic–hydraulic aerator (Figure 3) was responsible for providing the initial supply of pulp into the chamber volume, as well as for the dispersion of air bubbles in the pulp and the supply of the foaming agent emulsion into the flotation machine chamber.

Upon passing through the aerator’s vacuum zone, the pulp releases dissolved gas in the hydrophobic regions of the sludge (pressure flotation). Additionally, in the enclosed volume of the aerator, there is a high probability of particle–air bubble contact.

In selecting the foaming agent concentration from 20 to 150 mg per 1 L of pulp, a decrease in bubble size was observed when moving from the first, larger chamber to the last, due to the aerator in a four-chamber machine. This was accompanied by a natural increase in the thickness of the foam layer, which reached a maximum of 3 cm in the last chamber.

In a three-chamber flotation apparatus, the foam layer in the third chamber exhibited a thickness of 10 to 15 cm. Concurrently, the concentration of the foaming agent occurred in accordance with the aforementioned scenario. A fog of the foaming agent was generated within the chamber prior to the overflow threshold being reached. A chamber with the most water-saturated foam and the smallest bubbles was formed. In each cleaning chamber, the foam layer was progressively reduced in size until it reached the minimum thickness in the final chamber. The removal of the foam layer in this chamber occurred at the lowest speed. The elevated concentration of the foaming agent in the final cleaning chamber facilitated the stabilization of bubbles through its molecular interactions, thereby preventing the adhesion (film flotation) of hydrophilic silica particles. The flow of water, which was generated during the destruction of the foam, transported the silica particles into the flotation machine volume. The vortex generated by the aerator transported these particles to the lower section of the flotation machine. As a consequence of the low density of the foam, a proportion of the silica microspheres and other particles, including coke, were transported into the chamber product.

The measurements were obtained through direct observation with a measuring tape, taken through the transparent wall of the flotation machine. The foaming agent was introduced in a stepwise manner, with all other variables remaining constant.

In this instance, the pulp was conveyed via a pump at a pressure of 0.4 MPa from the initial cleansing chamber into the aerator. During the flotation process, kerosene (1 mg per 1 dm³ of pulp) and pine oil (2 mg per 1 dm³ of pulp) were added as additives.

2.2. Materials

Cyclone dust is a black mechanical mixture comprising predominantly large lumped pieces prior to sifting, which are subsequently reduced in size following the process (Figure 4).

The data obtained from the X-ray structural and X-ray fluorescence analyses, in addition to the SEM analysis, enabled us to ascertain the average composition of the cyclone dust generated during the production of silicon and ferroalloys (

Table 1. The data obtained from the X-ray structural and X-ray fluorescence analyses.

Component	SiO2	C	SiC	CaO	Al2O3	K2O	Fe2O3	MgO	Na2O	Other
%	67.37	23.47	5.03	1.19	0.24	0.24	0.21	0.059	0.059	2.132

).

The free carbon content was determined in accordance with the methodology outlined in reference [60].

3. Results

3.1. Flotation Kinetics of Cyclone Dust

This section presents the findings of the research into the kinetics of cyclone dust flotation in the flotation machine of the newly developed design. A series of experiments were conducted for the purposes of this research, during which the froth product was collected and weighed at 10 min intervals. The data obtained from the series of experiments were used to construct the flotation kinetics curve, as shown in Figure 5, and the specific flotation rate curve, as shown in Figure 6.

The aforementioned dependencies illustrate that the flotation rate remains consistent throughout the initial 50 min of the experiment, before exhibiting a notable speed decline by the 50th minute [51,61,62,63,64,65,66,67,68]. The majority of the carbon is extracted during the initial 50 min of the experiment, after which the rate of extraction declines. This is due to the fact that the most characteristic carbon particles are rapidly fixed on the bubbles and transported into the foam, while particles that are less susceptible to flotation due to their structure are extracted at a much slower rate.

The quality of the froth product remains stable for the first 50 min of flotation, after which point a deterioration in quality is observed. This is due to the fact that in the second half of the experiment, particles representing the remaining heterogeneous conglomerates of carbon and silicon dioxide begin to float. The analyses of foam product samples taken in the first 50 min practically do not contain particles of carbon and silicon conglomerate. Foam product samples taken after 50 min contain more and more carbon and silicon conglomerates.

The height of the froth layer exerts a significant influence on the flotation process. The height and structure of the froth are dependent on a number of factors, including the temperature and pH of the pulp, the intensity of aeration, and the design of the flotation machine. Nevertheless, the most straightforward method for controlling the height of the froth layer is to maintain the requisite concentration of the frothing agent in the pulp.

Figure 7 illustrates the measured thickness of the foam layer.

The alterations in the carbon and silicon composition can be attributed to the presence of microspheres with a lower density than the surrounding pulp, as well as the formation of silicon dioxide and carbon spheres with a considerable specific hydrophobic surface area. These entities contribute to an elevated carbon content in the initial foam product. Subsequently, in the following minutes, carbon particle–air flotation complexes predominantly float, resulting in an elevated carbon concentration in the foam product. The subsequent decline in the carbon content of the foam product can be attributed to two primary factors: firstly, a reduction in the concentration of the foaming agent, which facilitates the flotation of hydrophilic silicon dioxide nanoparticles; and secondly, a decrease in the height of the foam layer, whereby the hydrophilic and hydrophobic particles are unable to separate.

It would be erroneous to assume that the dependence illustrated in Figure 7 is universal; nevertheless, it is possible to derive certain conclusions based on it. Up to a certain concentration (in this case, up to 20 g/m³), the foaming agent does not accumulate at the air–liquid interface in an amount sufficient to form stable bubbles on the surface and grow the foam layer. Upon further increase in concentration (exceeding 20 g/m³ as illustrated in Figure 7), a notable rise in foam layer thickness is observed. In a certain range, the foam layer is observed to remain at its maximum thickness (in the measured case, 40–70 g/m³). This concentration range not only permits the formation of the highest possible foam layer, but also ensures that the foam remains sufficiently mobile and water-saturated, thereby facilitating effective separation within the foam layer.

The experiments yielded insights into the rational technological parameters of the flotation mode for obtaining new products. These insights were gleaned from the preliminary conditioning (conditioning time from 0.5 to 1.5 h) of wet cyclone dust (dry dust weight of 4 kg) with liquid glass (1.4 g per 1 dm³ of pulp) in a cavitation unit at a pH value of 8.5. The flotation process was conducted in a three-chamber flotation apparatus with a volume of 0.02 m³ for a duration of 90 min, utilizing a pneumohydraulic aerator with air suction from the atmosphere.

3.2. Result of Flotation Separation of Cyclone Dust

This section presents the findings of the study of froth and chamber flotation products obtained through the utilization of the aforementioned technology.

The foam product is a carbon concentrate with inherent hydrophobic properties that have been enhanced by flotation reagents. The carbon content of the foam product, as reported in reference [61], is within the range of 93–95 percent. Figure 8 illustrates the appearance of the foam product following the flotation of cyclone dust waste. An accompanying electron photo is also provided for reference.

The powder mixture is characterized by a black coloration. The structure comprises carbon nanotubes. The composition of the crystalline phase of the foam product, as determined by the X-ray diffraction (XRD) method, is presented in

Table 2. The composition of the crystalline phase of the foam product.

No.	Phase Identification	Content, Mass. %
1	C (Graphite)	55
2	SiO2 (Cristobalite)	33
3	SiC (Moissanite)	12

. The diffraction pattern is illustrated in Figure 9. The elemental composition is predominantly amorphous carbon and carbon nanotubes (94%), with residuals of silicon dioxide and silicon carbide (4%), unreacted residues of graphite anodes, and other impurities (2% in total).

It is important to note that despite the relatively high content of silicon compounds in the crystalline phase, the amount of total crystalline phase in the cyclone dust and, consequently, in the flotation froth product is in fact very small. Figure 10 and

Table 3. The granulometric composition of the froth product resulting from the flotation of cyclone dust.

X < Microns	0.1	0.2	0.4	0.8	1.5	3	6	12	25	45	100	200
Foam Product	0	0	0.6	1.9	4.1	9.2	16.3	26.5	35	45.6	79.8	99.4

illustrate the granulometric composition of the flotation froth product.

The particle size is up to 200 µm, with approximately 80 percent of the particles falling within the range of 0.4 µm to 100 µm. These properties indicate that this product can be used in the modification of cast iron and steel, as well as in any other area of application of carbon modifiers.

In addition to the carbon concentrate that is transferred into the froth product, the chamber flotation product is of practical utility. The chamber product is primarily composed of amorphous silicon dioxide, representing approximately 95% of its total composition. The resulting powder product is characterized by a grey coloration (Figure 11). The chamber flotation product contains between 2 and 5% free carbon and other impurities, as indicated in reference [61]. The diffraction pattern of the chamber flotation product is illustrated in Figure 12, and the composition of the crystalline phase is presented in

Table 4. Composition of the crystalline phase of the foam product according to XRD results.

No.	Phase Identification	Content, Mass. %
1	SiO2 (Quartz)	50
2	SiC (Moissanite)	35
3	SiO2 (Cristobalite)	10
4	C (Graphite)	5

. Figure 13 and

Table 5. The granulometric composition of the chamber product resulting from flotation of cyclone dust.

X < Microns	0.1	0.2	0.4	0.7	1	2.5	4	8	15	30	55	100
Foam Product	0	1.3	4.8	10.7	15.6	28.8	36.7	49.6	65.5	87.7	99.4	100

illustrate the granulometric composition of the chamber product resulting from cyclone dust flotation.

The chamber product of cyclone dust flotation is a concentrate of amorphous silicon dioxide with a particle size of up to 100 μm, with approximately 90% of the particles being smaller than 30 μm (Figure 13). The concentration of free carbon does not exceed 5%. These characteristics permit the utilization of the chamber product as a silicate modifier in a multitude of industrial sectors.

4. Discussion

4.1. Separation of Sludge Particles by Flotation Method

The results of our calculations demonstrate that the influence of surface forces on hydrophilic SiO₂ nanoparticles is greater than that of gravity and hydrostatics [68]. The contact angle α at the air–water interface can be calculated for a fixed SiO₂ particle in the form of a cylinder with a radius of r and a height of 2 r (see Figure 14).

The following equations were derived:

$$F_{г}-F_a=2\pi r^3({\rho }_{Si{O}_2}-{\rho }_{H_{2}O})\cdot g=2\cdot 3.14\cdot {10}^{-21}\cdot 1600\cdot 10=1.00531\cdot {10}^{-16}H,$$

(1)

$$F_{п.н}=\sigma \cdot 2\pi \cdot r=72\cdot {10}^{-3}\cdot 2\cdot 3.14\cdot {10}^{-7}=4.52\cdot {10}^{-8}H,$$

(2)

$$\sin \alpha ={\displaystyle \frac{F_{г}-F_a}{F_{п.н}}}={\displaystyle \frac{1.00531\cdot {10}^{-16}}{4.52\cdot {10}^{-8}}}=2.22222\cdot {10}^{-9},$$

(3)

$$\alpha =\arcsin \left(\sin \alpha \right)=2.22222\cdot {10}^{-9}$$

(4)

In this context, the following variables are defined: F_г represents the weight of the SiO₂ cylinder, F_a denotes the Archimedes force, and F_п.н is the surface tension force. The surface tension force (ρ_SiO2 = 2600 kg/m³) is the density of SiO₂, while the density of water (ρ_H2O = 1000 kg/m³) is that of water. The radius of the ball (r = 10⁻⁷ m) is also a constant. The surface tension at the air–water interface (σ = 72·10⁻³ N/m) is a variable, as is the acceleration of gravity (g = 10 m/s²).

It can be seen that, as a consequence of the usual dispersed composition of the initial bubbles during flotation, the size of the bubbles is significantly larger than that of the hydrophilic nanoparticles. Consequently, in the case of large bubbles, the size of which is considerably larger than that of the nanoparticles, the latter will be more effectively fixed under conditions of film flotation. As illustrated in Figure 15, an electron microscope photograph reveals the presence of micron-sized bubbles coated with SiO₂ nanospheres. It is a well-established fact that the contact angle of quartz is within the range of 0° to 10°. Therefore, when α is not significantly greater than zero, quartz nanoparticles can be readily floated. Accordingly, the flotation of nanosized particles should be conducted with bubbles of a similar size [62,63,64].

4.2. Theoretical Analysis of Obtaining Nanobubbles During Flotation of Silicon Production Waste

Particle–bubble interactions represent a central issue in a number of fields within the broader domain of physical chemistry, including hydrodynamics, colloidal hydrodynamics, surface forces, surface rheology, wetting film dynamics, colloidal stability, heterocoagulation, and adsorption dynamics at liquid interfaces. The mechanisms of particle–bubble interactions are of great consequence in determining flotation selectivity and collection efficiency. The latter is the probability that a colliding bubble and particles form a permanent aggregate, a process that is referred to as orthokinetic heterocoagulation in modern flotation theory [65,66,67]. The question of the permanent stability of the particle–bubble aggregate is essentially one of flotation thermodynamics, whereas the probability of a colliding bubble and particle forming an aggregate is controlled by flotation dynamics. It is crucial to recognize that flotation is fundamentally a dynamic process, where kinetics and energetics are inextricably linked. To illustrate, once a colliding bubble and a large particle have formed an aggregate, the lifetime of this structure is constrained by turbulent forces that disrupt the particle–bubble contact within the flotation cell. The current issue of determining the equilibrium state of a particle at the liquid–air interface has its historical origins in the publications of Frumkin and Kabanov [68]. In [68,69], the case of a gas bubble attached to a flat solid surface of infinite length was considered. This model served as a basis for understanding particle–bubble adhesion during flotation.

In [70], the process of wetting perimeter formation is analyzed based on capillarity theory, with consideration of line energy when a spherical particle makes contact with a flat liquid surface. The minimum particle size, which may vary, is calculated, and an accurate estimate of the minimum time of contact between the particle and the surface is made. The kinetic energy of the collision between the air bubble and the particles is employed in calculating the maximum size of the particles that can remain attached to the flotation. The data pertaining to flotation indicate that the minimum size of isolated particles is approximately 1 μm. A comparison of this radius with the theoretical limit yields a value for the line energy of the wetting perimeter ranging from 10⁻⁴ to 10⁻⁵ dyn. Concurrently, there are publications that present micrographs (Figure 16), which demonstrate the attachment of CO₂ nanobubbles with a size of 5–80 nm to a hydrophobic surface [71,72].

Micro- and nanobubbles (referred to as “MNBs”) represent a novel class of entities that differentiate them from conventional bubbles (macrobubbles) due to their reduced diameter. The specific area (surface area per volume) and degree of stagnation in the liquid phase of MNBs are advantageous, as they facilitate gas dissolution. Furthermore, it has been documented that free radicals are produced during the collapse of microbubbles as a consequence of the elevated ion concentration at the gas–liquid interface immediately preceding collapse [72].

In the case of a bubble diameter of 4 mm or less, provided that it is fixed on a solid substrate with a wetting angle greater than 60° and that the contact area diameter with the solid surface is of the same order of magnitude (50%–100% of the bubble diameter), the hydrostatic force can be considered to be negligible. This is the aforementioned discrepancy between the Archimedes force acting on the bubble and the Archimedes force acting on the liquid cylinder. Prior to determining the ratio of the bubble and particle sizes, it is first necessary to estimate the range of diameters of the most stable bubbles in an aqueous medium. The stability of the bubble shape is contingent upon the ratio of the hydrostatic and capillary pressures. This can be estimated based on the difference between the aforementioned pressures:

$${\mathrm{F}}_{\mathrm{k}}-{\mathrm{F}}_{г}={\displaystyle \frac{2\sigma }{R}}-\rho \mathit{gh},$$

(5)

In the following equation, F_k is the Archimedes’ force acting on the bubble, F_г is the Archimedes force acting on a liquid cylinder, R represents the radius of curvature of the bubble dome, h denotes the height of the bubble, ρ signifies the difference in densities of the liquid and gas within the bubble, g stands for the acceleration due to gravity, and σ represents the surface tension at the gas–liquid interface.

For h ≈ 2R, g = 980 cm/s², σ = 72 dyn/cm, ρ = 1000 kg/m³, a graph of the pressure difference function versus the bubble radius is plotted (Figure 17).

The resulting graph demonstrates that bubbles with a diameter of less than 1 mm exhibit the greatest stability in shape and are almost perfectly spherical. An increase in the diameter of the bubble ultimately results in its destruction into smaller bubbles.

In this section, we will consider the equation of motion of the bubble–mineral particle system with a cylindrical shape and a contact diameter of a, as illustrated in Figure 18. The system is moving with acceleration A. We note the existence of several forces acting upon the flotation complex, including T, which represents the tension force (fastening) acting on the bubble–particle system; Fч, which represents the force acting upon the mineral particle; and Fп, which represents the force acting upon the bubble. In light of the aforementioned considerations, we proceed to formulate a system of equations that encapsulate the balance of forces acting on the flotation complex.

The following equations were derived:

$$\left\{\begin{array}{l}{\mathrm{F}}_{ч}-\mathrm{T}=\mathrm{mA}\\ \mathrm{T}-{\mathrm{F}}_{п}={\displaystyle \frac{1}{\mathrm{k}}}\mathrm{mA}\end{array}\right.$$

(6)

In this context, the symbol m represents the mass of the mineral particle, while m/k denotes the relative mass of the bubble. The coefficient k determines the number of times the mass of the gas within the bubble is less than that of the particle.

From System (6), we derive the following equation:

F_ч − T = k(T − F_п)

(7)

In order to proceed, it is necessary to express Equation (7) as follows:

T = F_ч/(1 + k) + (k/(k + 1))F_п

(8)

Given that 1 + k is greater than 1000, it can be reasonably assumed that the first term on the right-hand side can be neglected, and thus that k/(k + 1) = 1. Consequently, the equation can be rewritten as follows:

T = F_п

(9)

Equation (7) can be rearranged to yield the following result:

$${\mathrm{F}}_{п}=V\rho g-{\displaystyle \frac{\pi a^2}{4}}h\rho g+{\displaystyle \frac{\pi a^2\sigma }{2R}}$$

(10)

T = πaσsinθ

(11)

To illustrate, the equilibrium state of a bubble on a mineral particle in a liquid can be described by the same Equation (7).

In this section, we will examine the ratio of forces in Equation (7) for a specific ratio of the particle and bubble diameters. Let us assume that the angle is 90°, that the density of the particles is 1000 kg/m³, that the acceleration of gravity is 10³ cm/s², that the surface stress is 0.50 dyn/m, and that the diameter of the particle–bubble contact is a = 0.1R. In this instance, it is reasonable to assume that the bubble will be approximately spherical in shape, and its volume can be calculated using the formula V = (4/3)πR³, with h = 2R. The objective is to ascertain the radius of the bubble at which the force in question will be equal to zero. It can be concluded that the bubble–particle system will be the most stable in this configuration.

$$f\left(R\right)=V\rho g-{\displaystyle \frac{\pi a^2}{4}}h\rho g+{\displaystyle \frac{\pi a^2\sigma }{2R}}-\pi a\sigma sin\theta ={\displaystyle \frac{4}{3}}\pi R^3·{10}^3-{\displaystyle \frac{\pi ·0.01R^3·{10}^3}{2}}+{\displaystyle \frac{\pi ·0.01R50}{2}}-\pi 0.1R50=0$$

(12)

From this equation, we can ascertain that the equilibrium state will be at R ≈ 0.00062 m. This demonstrates that there is a clear correlation between the size of the particles being floated and the size of the bubbles sitting on them. The equilibrium size of the bubbles will lie in a narrow range (from several nanometers to several millimeters) when other ratios of the diameter of the bubble contact with the mineral particle and the bubble diameter, other air–liquid surface tension, and wetting angle are taken into account. In light of the aforementioned evidence, it is pertinent to consider and analyze the nature of the bubbles that should be created during flotation.

The results of the aforementioned experiments demonstrate that the utilization of a pneumohydraulic aerator facilitates the generation of minute and consistently sized bubbles, which enhance the stability of the aerated liquid (foam). This, in turn, leads to a reduction in the consumption of foaming agents.

5. Conclusions

The following outcomes were attained as a consequence of this research project:

• The fundamental possibility was established and the rational conditions that facilitate the separation of nanosilica and nanocarbon from waste produced during silicon and ferroalloy production were determined.
• Technological modes for the efficient flotation of nanosized dust particles from the gas cleaning of silicon and ferrosilicon were established.
• The relationship between the particle size and the equilibrium size of the bubbles necessary for their fixation on the particle was determined through flotation, with the equation of motion of the bubble–particle flotation complex being considered. It was demonstrated that a range of ultra-dispersed materials with a size spanning from 0.1 nm to several millimeters can be formed, contingent on the ratio of the bubble diameter to the mineral particle size, the value of the surface tension, and the contact angle.
• The characteristics of the flotation separation of nanosized structures, namely, conglomerates of carbon nanotubes and spheres of silicon dioxide, have been identified. These structures are created by bubbles that are proportionate to the final particles (0.01–100 nm).
• The experiments yielded insights into the rational technological parameters of the flotation mode for obtaining new products. These insights were gleaned from the preliminary conditioning (conditioning time from 0.5 to 1.5 h) of wet cyclone dust (dry dust weight of 4 kg) with liquid glass (1.4 g per 1 dm³ of pulp) in a cavitation unit at a pH value of 8.5. The flotation process was conducted in a three-chamber flotation apparatus with a volume of 0.02 m³ for a duration of 90 min, utilizing a pneumohydraulic aerator with air suction from the atmosphere. In this instance, the pulp is conveyed via a pump at a pressure of 0.4 MPa from the initial cleansing chamber into the aerator. During the flotation process, kerosene (1 mg per 1 dm³ of pulp) and pine oil (2 mg per 1 dm³ of pulp) were added as additives.
• Two novel ultra-dispersed materials have been synthesized, comprising nanoscale silica- and carbon-containing materials (amorphous carbon and carbon nanotubes).