Numerical simulation of distribution of pollutant in channel

. The paper presents the problem of the propagation of a pollutant (phenol) emission in a channel through which pure water flows. The mathematical model of the presented problem is based on the continuity equations. The distributions of the main parameters of the system are constructed. It is shown that in the absence of ledges, the pollutant flow moves in the channel in the form of a jet with a clearly defined core. In this case, the presence of ledges, due to the turbulence of the flow, leads to the emergence of rather large areas of pollution along the height of the channel. The dynamics of changes in the concentration of a pollutant along the height of the channel has been studied. It is shown that in the case of a working pollution source, the highest concentration is observed near the free surface of the channel. In the case when the source is not working, the highest concentration is observed at the bottom.


Introduction
Currently, one of the main problems of ecology is the problem of pollution of river basins with various pollutants [1,2].Failure to take measures to prevent pollution of water bodies can significantly affect the state of the coastal ecosystem.In addition, the problem of water pollution also affects the health of people who use water for domestic and agricultural purposes.It should be noted that the problem of water pollution is most acute for regions with a highly developed chemical industry, as well as in regions with a densely developed network of oil pipelines [3,4].At the same time, as noted in [5], pollutants can both dissolve and be transported in the water flow in the form of suspended particles.
Mathematical models of the distribution of pollutants in water bodies are considered, for example, in [6][7][8][9][10][11].At the same time, the presented works are mainly devoted to the study of oil spills in coastal waters, taking into account various factors.Thus, in [7] in a threedimensional setting, two different scenarios of an oil spill into the sea are studied, which assume both low and high intensity of its outflow.At the same time, the model under consideration also took into account meteorological conditions, as well as sea water parameters (temperature, salinity).The study of migration of oil droplets from a damaged underwater pipeline is considered in [8].It should be noted that the model presented in the paper takes into account the influence of such factors as the speed of oil outflow from a damaged pipeline, its density, the depth of the pipeline, the speed and direction of water flow.A mathematical model of the distribution of pollutants in the riverbed in a two-dimensional formulation is presented in [9].The definitions of the velocity, concentration, and temperature fields in a channel during the propagation of an impurity in it, taking into account the occurrence of recirculation zones, are considered in [10].Numerical-analytical methods for solving the problems of migration of impurities in the atmosphere and the aquatic environment are considered in [11].In particular, the result of the presented work is the construction of the spatial and temporal distribution of pollutants in the air and water basins, as well as the study of the main patterns of their distribution.The construction of a numerical model for predicting the dispersion of pollutants in watercourses, taking into account their geometric shape, and the application of this model to solve the predictive problem of watercourse pollution in the event of an accidental leakage of radioactive water from a repository located on an island, is presented in [12].Modeling of the process of distribution of wastewater and pollutants when they are discharged into swamps is presented in [13].Features of thermal pollution of river basins are considered in [14].

Results and discussion
We will consider a flat vertical section of a channel through which water flows.Let us assume that in some of its area there is a volley emission of a pollutant (phenol), which is completely dissolved in water.It is required to determine the distributions of the concentration and velocity fields.
When describing the process of impurity propagation in water, we will make the following assumptions.We will assume that the process of water flow is turbulent; water and pollutant are incompressible and their densities are the same.The system of basic equations describing the distributions of the main parameters (components of the velocity vector, concentration) can be represented by a generalized differential equation of the form [15,6]: where t is time;  � (x, y) are spatial coordinates;  � (u, v) are the velocity vector components; ρ is the density; Φ is a generalized dependent variable; Γ Φ is the mass transfer coefficient; S Φ is the source term.The specific values of the listed quantities are given in the Table 1 [6].
It should be noted that the following boundary conditions are used to solve system (1) [12]:  on the left boundary of the channel, the inflow condition u = u 0 ;  on the right boundary of the channel (leakage) -constant pressure р = р 0 (р 0 is atmospheric pressure);  on solid walls, the impermeability condition u n = 0 (u n is the normal velocity component).For the parameters characterizing the system, the following values are accepted: µ is 10-3 Pa•s; u is 2 m/s and p 0 is 105 Pa.The diffusion coefficient was assumed to be D = 1.09•10-9 m 2 /s.When solving it was assumed that the pollution source operates until the time t = 500 s; then it turns off.In this case, the value of the concentration of the pollutant at the pollution source is c = 20 mol/m 3 .
The numerical solution was carried out in the "Comsol Multiphysics" software package.At the first stage, in a stationary formulation, the equations associated with the fluid flow in the channel are solved.Based on the velocity fields found in the non-stationary setting, the pollutant concentration field is found.Note: µ and µt are the molecular and turbulent viscosities, respectively, with µt=(cµρk^2)/ε; gi are the components of the gravitational acceleration vector; p is pressure; Sc and Sct are the molecular and turbulent Schmidt numbers; σk and σε, cµ, cε1, cε2, cε3 are the empirical constants of the turbulent model.
Figures 1 and 2 show, respectively, the velocity and pollutant concentration distributions in the channel for the time t = 120 s.It can be seen that in its narrowing part, corresponding to a shallow depth, the velocity increases.It follows from Figure 2 that the pollutant flow has a pronounced core moving along the canal water flow.It should be noted that before the ledge, the bulk of the pollutant was concentrated in the jet.After the ledge, due to the turbulence of the flow and strong mixing, it can be seen that the pollutant is concentrated in a rather large area of the channel in height.In Figure 4 are presented the dependences of the pollutant concentration on the height of the channel directly behind its ledge for the time moments t = 120 s and 900 s.In the first case, the concentration of the pollutant increases with height, and in the second, it decreases.This is explained by the fact that at t = 120 s the source continues to operate, while at t = 900 s there is no pollution source.It should be noted that on the free surface of the channel, the concentration of the pollutant is zero.

Conclusion
Numerical simulation of the pollutant propagation in the channel has been carried out.It is shown that in the absence of ledges, the pollutant flow moves in the channel in the form of a jet with a clearly defined core.In this case, the presence of ledges, due to the turbulence of the flow, leads to the emergence of rather large areas of pollution along the height of the channel.The dynamics of changes in the concentration of a pollutant along the height of the channel has been studied.It is shown that in the case of a working pollution source, the highest concentration is observed near the free surface of the channel.In the case when the source is not working, the highest concentration is observed at the bottom.
The work was carried out within the framework of the state task FEUR -2023 -0006, project "Development and creation of low-tonnage products and reagents (corrosion and scale inhibitors, antioxidants, biocides, additives, etc.) for petrochemical processes and water purification from pollution, replacing imported substances and materials.Theoretical and experimental approaches».

Fig. 1 .
Fig. 1.Velocity field for water flow in a channel.

Fig. 2 .
Fig. 2. Pollutant concentration field in the channel for t = 120 s.

Figure 3
Figure3also shows the pollutant concentration field for the time t = 900 s, which corresponds to the absence of the pollution source (the source is inactive).In this case, as follows from Figure3, a sufficiently large area along the entire height of the channel is occupied by pollutants.

Fig. 3 .
Fig. 3. Pollutant concentration field in the channel for t = 900 s.

Fig. 4 .
Fig. 4. Dependence of the pollutant concentration on the height of the channel for different moments of time.

Table 1 .
The specific values of the listed quantities.