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Microplastics (tiny plastic particles less than 5 mm in size) have become a global environmental problem due to their widespread presence in the environment and potentially harmful effects on marine organisms and ecosystems. One of the main sources of microplastics is wastewater, which contains a significant amount of it and enters natural water bodies (rivers, lakes, oceans), disrupting their ecosystems. Thus, the development of effective methods for the removal of microplastics from wastewater is important for the preservation of the health of aquatic ecosystems.

Let’s consider one of the effective ways to remove microplastics from wastewater - flotation. This process involves attaching polluting particles to air bubbles that rise to the surface of the water and can be mechanically removed. However, the effectiveness of microplastic flotation depends on various factors, such as the properties of the microplastic, the chemical composition of the water and the flotation conditions.

In this research paper, we aim to simulate the process of microplastic flotation using electrolysis production of gas bubbles and to investigate the factors affecting its effectiveness. Our results can help optimize the microplastic electroflotation process and develop more effective methods for its removal from wastewater. This is extremely important since the growing use of plastics worldwide has led to the appearance of an alarming amount of microplastics in the environment [1,2]. For example, in the study [3] it is estimated that the oceans alone contain more than 5 trillion microplastic particles. Another study showed that wastewater after treatment facilities can emit up to 4.2 million microplastic particles into the environment per day [4]. The data obtained as a result of the literature review emphasize the need to solve the problem of microplastic pollution of water bodies [5].

This paper considers B.S. Ksenofontov’s multi-stage model [6-9] of the electroflotation process. The model is presented in Figure 1. Glitter consisting of small plastic particles was chosen as polluting particles, the peculiarities of the material’s behavior in water were taken into account and two different types of bubbles arising during electroflotation were considered [10-12]: hydrogen bubbles and oxygen bubbles [13,14].

In the considered model, state A is the initial state of the system; B and C are the states of adhesion of particles to bubbles; E is the precipitation of particles; D is the state of particles in the foam layer.

The process can be described by a system of differential equations:

$\{\begin{array}{c}\frac{dt}{d}{C}_{A}=-{k}_{1}\xb7{C}_{A}-{k}_{2}\xb7{C}_{A}-{k}_{3}\xb7{C}_{A}-{k}_{6}\xb7{C}_{A};\\ \frac{dt}{d}{C}_{B}={k}_{1}\xb7{C}_{A}-{k}_{4}\xb7{C}_{B};\\ \frac{dt}{d}{C}_{C}={k}_{2}\xb7{C}_{A}-{k}_{5}\xb7{C}_{C};\\ \frac{dt}{d}{C}_{E}={k}_{3}\xb7{C}_{A};\\ \frac{dt}{d}{C}_{D}={k}_{4}\xb7{C}_{B}+{k}_{5}\xb7{C}_{C}+{k}_{6}\xb7{C}_{A},\end{array}\text{(1)}$

Where *C _{A}, C_{B}, C_{C}, C_{D}, C_{E}* – particle concentrations in states А, B, C, D, and E; state А – polluting particles in their original form, В – flotation complexes with oxygen bubbles, С – flotation complexes with hydrogen bubbles, E represents the state of precipitated particles, D is the state of particles in the foam layer;

Initial conditions for this system of equations at t = 0:

${C}_{A}\left(0\right)={C}_{A0}=1mg/l$

${C}_{B}\left(0\right)=0$

${C}_{C}\left(0\right)=0\text{(2)}$

${C}_{D}\left(0\right)=0$

${C}_{K}\left(0\right)=0$

The system of equations (1) was solved in the Mathcad software package.

The probability of the formation of a flotation complex is determined by the constants *k _{1,2}*, which can be calculated using the formulas [15,16]:

${k}_{1}=\frac{1,5\xb7{k}_{\u044d}\xb7j\xb7E}{{k}_{0}\xb7{D}_{{O}_{2}}\xb7{\rho}_{{O}_{2}}};\text{(3)}$

${k}_{2}=\frac{1,5\xb7{k}_{\u044d}\xb7j\xb7E}{{k}_{0}\xb7{D}_{{H}_{2}}\xb7{\rho}_{{H}_{2}}};\text{(4)}$

Where *k _{3}* is the electrochemical equivalent of the substance, kg/C;

*j* is current density, A/m^{2};

*E* is the efficiency of particle capture by a gas bubble during flotation, (DN);

*K _{0}* is bubbles polydispersity factor, (DN);

${\text{D}}_{{\text{H}}_{\text{2}}},{\text{D}}_{{\text{0}}_{2}}$ , are mean diameters of the bubbles in the flotation cell, m;

*ρ* is gas density, kg/m^{3};

*q* is the bubbling rate, m^{3}/(m^{2} . s).

Sedimentation of microplastic particles is represented in k3 coefficient. To determine the coefficient, an experiment was conducted to observe the deposition of microplastic particles. k3 the coefficient is determined by the formula:

${k}_{3}=\frac{{v}_{sed}}{H},\text{(5)}$

Where *v _{sed}* is sedimentation velocity of a microplastic particle, m/s;

*H* is the depth of the flotation chamber, m.

Sedimentation velocity is determined by the formula:

${v}_{sed}=\frac{S}{t},\text{(6)}$

Where *S* is the distance traveled by a microplastic particle, m;

*t* is the time it took for the particle to travel the distance S, s.

To determine the sedimentation velocity, 10 microplastic particles were observed. Figures 2 and 3 show an example of observation. Table 1 shows the results of observations.

Figures 2 and 3 show an example of a precipitating particle. Table 1 shows the results of observations. According to the results of the experiment, the coefficient *k _{3}* = 0,0036

The rise of the flotation complexes is characterized by coefficients *k _{4,5}*, which can be calculated by the formula:

${k}_{4,5}=\frac{{v}_{r4,5}}{H},\text{(7)}$

Where *v _{r4,5}* is rise velocity of flotation complex, m/s;

*H* is the depth of the flotation chamber, m.

The transition of the polluting particle into the foam layer is characterized by the *k _{6}* coefficient, which is calculated by the formula:

${k}_{6}=\frac{{v}_{f}}{H},\text{(8)}$

Where *vf* is the velocity of self-ascent of a microplastic particle, m/s;

*H* is the depth of the flotation chamber, m.

The determination of the coefficient *k _{6}* according to the formula (8) by the formula (8) is possible if the density of the microplastic is less than the density of water. Since part of the microplastics rose to the surface during the experiment, this assumption was accepted. The particle self-ascent velocity is determined similarly to the sedimentation velocity.

To determine the self-ascent velocity and the coefficient *k _{6}* 10 microplastic particles were observed. Figure 4 shows the initial state of the particle, Figure 5 shows the final state.

The results of the experiment are presented in Table 2. According to the results of the experiment, the value of the coefficient *k _{6}* is 0,0025 s

The efficiency of particle capture by gas bubbles is determined by the formula [11]:

$E=0,5\xb7\frac{{r}_{p}^{1,6}}{{r}_{b}^{2}}\xb7{A}^{1/6},\text{(9)}$

Where *r _{ų}* is particle radius, m;

*γ _{n}* is bubble radius, m;

*A* is the Hamaker constant, J.

In addition to electroflotation, the model takes into account the presence of Al(OH)3 coagulant. The value of the Hamaker constant for this case is given in Table 3.

The rise velocity of the flotation complex is determined by the formula [15]:

${v}_{r}=\frac{{\stackrel{-}{D}}^{2}\xb7g\xb7\left({\rho}_{w}-{\rho}_{g}\right)}{18\xb7\mu},\text{(10)}$

Where $\stackrel{-}{D}$ is the mean diameter of the bubbles in the flotation cell, m;

*ρw* is water density, kg/*m ^{3}*;

*ρg* is gas density, kg/*m ^{3}*;

*µ* is water viscosity, kg/m·s.

Thus, the presented multi-stage flotation model allows us to consider the influence of various parameters on the efficiency of the process, such as the size of bubbles (depending on the nature of the flotation process), current density, and bubbling rate. In addition, the multi-stage model takes into account various options for the location of pollution particles during the water treatment process.

For the calculation and graphical representation of the model, a similar process was carried out for the purification of oily wastewater [17,18]. The calculation was made using the initial data presented in Table 3. Reference data is taken from [15,16,18-66].

The parameters calculated by formulas (9), and (10) are presented in Table 4. The obtained values of the constants are presented in Table 5.

The solution of the system of differential equations in graphical form is shown in Figure 6.

Using the graphical solution, the flotation time was determined. To achieve a degree of purification of 80%, the required flotation time is 1900 seconds.

From the presented flotation process model, it can be concluded that despite the ability of microplastics to independently rise into the foam layer and precipitate other features of the material slow down the flotation time.

Thus, it is necessary to provide additional stages of finer purification, but the use of electroflotation with the ability to adjust the size of bubbles can significantly reduce the concentration of microplastics in water. In the future, the model will be tested experimentally and refined to obtain the final results.

The results obtained in this work provide a rationale for choosing the most efficient electroflotation apparatus for wastewater treatment from microplastics.

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