A method for predicting the dispersion characteristics and settling behavior of multi-size particles in turbulent square ducts

Abstract

The application discloses a method for predicting the dispersion characteristics and the sedimentation behavior of multi-particle diameters in a turbulent square pipe, and relates to the technical field of fluid mechanics, and comprises the following specific steps: establishing a fluid mathematical model and solving a flow field; a mathematical model of fluid flow in the turbulent square pipe is established based on the Reynolds average Navier-Stokes equation combined with a k-epsilon turbulent model, square pipe geometry, fluid physical and boundary parameters are input, and the finite volume method is used for discrete solving to obtain flow field data of the distribution of velocity, pressure and turbulent intensity in the pipe; when the fluid flow mathematical model is established, the double equation k-epsilon turbulent model is combined and wall function correction is introduced, the wall function is used to associate the velocity gradient and the turbulent stress in the near-wall region, the boundary calculation of the turbulent kinetic energy and the dissipation rate is corrected, the numerical error caused by direct discretization is avoided, the flow field simulation in the near-wall region is more in line with the actual situation, and accurate flow field basic data are provided for subsequent particle motion prediction.

Description

Technical Field

[0001] This invention relates to the field of fluid mechanics, specifically to a method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube. Background Technology

[0002] In numerous industrial sectors such as chemical engineering, energy, and environmental protection, the flow of multi-sized particles within turbulent square tubes is extremely common. For example, in pneumatic conveying systems, various solid particulate materials are transported through pipelines; in liquid-solid circulating fluidized beds, solid particles and liquids circulate within square tubes to achieve specific reaction or separation processes; and in flue gas dust removal equipment, dust-laden gas flows within square tubes, where particles need to be effectively separated. The dispersion characteristics and sedimentation behavior of particles play a crucial role in these industrial processes, directly affecting the equipment's transmission efficiency, reaction effect, and separation performance. Accurately predicting the dispersion characteristics and sedimentation behavior of multi-sized particles in turbulent square tubes is of significant engineering importance for optimizing industrial equipment design, controlling operating parameters, and improving overall industrial production efficiency and product quality.

[0003] Currently, the main methods for predicting particle flow behavior in turbulent square tubes are experimental measurement and numerical simulation. Experimental measurement involves setting up experimental devices and using technologies such as high-speed photography and particle image velocimetry to obtain particle motion parameters. However, this method has many drawbacks: high experimental costs, long experimental cycles, and difficulty in measuring under complex working conditions. It cannot accurately capture particle flow under some special environments or conditions, especially for multi-size particles, where the motion information of particles of different sizes is difficult to obtain simultaneously and accurately, limiting its application in complex industrial scenarios. Existing numerical simulation methods also have significant shortcomings. Single-particle trajectory models ignore the interaction between particles and are only applicable to low particle concentrations, failing to meet the prediction needs of high-concentration particle flow in actual industry. Two-fluid models treat particles as a continuous medium, which can consider the overall motion of particles, but cannot accurately describe the differences between multi-size particles and the discrete characteristics of particles, resulting in large deviations between the prediction results and the actual situation. This leads to low prediction accuracy for the dispersion characteristics and sedimentation behavior of multi-size particles, making it difficult to meet actual industrial needs. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in turbulent square tubes. This method establishes a mathematical model of fluid flow within the turbulent square tube based on the Reynolds-averaged Navier-Stokes equations combined with a k-ε turbulence model, and introduces wall function corrections to accurately simulate the flow field distribution. It comprehensively considers the differences in physical properties of multi-size particles, constructs particle motion equations incorporating various forces, numerically solves these equations, and calculates them in groups by particle size to accurately account for the interactions between particles of different sizes. Finally, by calculating two key parameters—dispersion coefficient and sedimentation velocity—it achieves accurate prediction of the dispersion characteristics and sedimentation behavior of multi-size particles, providing a reliable theoretical basis for the optimized design and operational parameter control of related industrial equipment.

[0005] To solve the above-mentioned technical problems, the present invention provides the following technical solution: a method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube, the method comprising the following specific steps:

[0006] A mathematical model of the fluid was established and the flow field was solved: A mathematical model of the fluid flow in the turbulent square tube was established based on the Reynolds-averaged Navier-Stokes equations and the k-ε turbulence model. The geometric, fluid physics and boundary parameters of the square tube were input. To address the distortion of turbulence parameters in the near-wall region caused by the roughness of the square tube wall, a standard wall function was introduced for targeted correction. The flow field data of velocity, pressure and turbulence intensity distribution in the tube were obtained by discretization using the finite volume method.

[0007] Constructing particle motion equations: Obtain the physical property parameters of particles with multiple sizes, and combine them with the various forces acting on the particles in the tube to construct the motion equations of particles with multiple sizes.

[0008] Solving the particle motion equation to obtain motion information: Inputting the flow field data as boundary conditions into the particle motion equation, using the discrete element method and solving it in groups according to particle size, while considering inter-group collisions and agglomeration, to obtain the spatial position, velocity and concentration distribution of particles at different times.

[0009] Calculate dispersion and sedimentation parameters: Based on the spatial location, velocity and concentration distribution of particles, calculate the dispersion coefficient by statistically analyzing the variance of the distance from the particle to the tube axis, and calculate the sedimentation velocity by statistically analyzing the average vertical velocity of the particles, so as to predict the dispersion characteristics and sedimentation behavior of particles.

[0010] Furthermore, in the step of establishing a fluid mathematical model and solving the flow field, a mathematical model of fluid flow in a turbulent square tube is established based on the Reynolds-averaged Navier-Stokes equations and combined with a two-equation k-ε turbulence model. To address the distortion of turbulence parameters in the near-wall region caused by the roughness of the square tube wall, a standard wall function is introduced for targeted correction: the near-wall region is divided by dimensionless distance, and the correction value of turbulent kinetic energy in the near-wall region is derived using empirical correlations. Then, the dissipation rate correction value is calculated by correlating the correction value of turbulent kinetic energy in the near-wall region, the Karman constant, and the actual distance from the particle to the wall. By determining the geometric parameters of the turbulent square tube, the physical parameters of the fluid, and the boundary conditions, the parameters are input into the mathematical model. The mathematical model is discretized using the finite volume method, and the discretized equations are solved to obtain the flow field distribution data at different locations within the turbulent square tube at different times, including velocity distribution, pressure distribution, and turbulence intensity distribution.

[0011] Furthermore, in the step of establishing the fluid mathematical model and solving the flow field, the algorithm formula for the Reynolds-averaged Navier-Stokes equations is as follows: ,in, It is the density of the fluid flowing inside the turbulent square tube. , Is it fluid in , The time-averaged velocity component in the direction, , These are spatial coordinate components, corresponding to the spatial positions within the square tube, used to locate the spatial distribution of flow field parameters. It is the average pressure of the fluid. It is the fluid dynamic viscosity. yes .

[0012] Furthermore, in the step of establishing the fluid mathematical model and solving the flow field, for both equations... - Turbulence model, its turbulent kinetic energy The equation is ,in, It is turbulent kinetic energy, which characterizes the intensity of turbulent fluctuations. It is turbulent viscosity. It is the Prandtl number corresponding to turbulent kinetic energy. It is the turbulent kinetic energy generated by the average velocity gradient. It is the turbulent kinetic energy dissipation rate, and its calculation formula is: ,in, It is the dissipation rate Prandtl number. , It is an empirical constant.

[0013] Furthermore, in the step of establishing the fluid mathematical model and solving the flow field, the correction value of the turbulent kinetic energy in the near-wall region is derived using empirical correlations, and the formula is as follows: ,in, It is the correction value for turbulent kinetic energy in the near-wall region. It is an empirical constant. The wall friction velocity is used as the basis for calculating the dissipation rate correction value by correlating it with the near-wall turbulent kinetic energy correction value, the Karman constant, and the actual distance between the particle and the wall. The formula is as follows: ,in, This is the correction value for the near-wall dissipation rate. It is the Kármán constant. It is the actual distance from the particle to the wall.

[0014] Furthermore, in the step of constructing the particle motion equation, the physical property parameters of multi-size particles are obtained, specifically including the density, particle size, particle shape coefficient and surface charge density of each particle size. Combined with the various forces acting on the particles in the tube, the motion equation of the multi-size particles is constructed. These various forces include gravity, buoyancy, fluid drag, additional mass force, Magnus force, Saffman force, collision force and agglomeration force between particles, and collision rebound force between particles and the square tube wall.

[0015] Furthermore, in the step of constructing the particle motion equation, the motion equation of multi-diameter particles is constructed by combining the various forces acting on the particles inside the tube. The formula for this motion equation is as follows: ,in, It is particle mass. It's particle velocity. It is fluid drag. It is particle gravity. It is fluid buoyancy. yes , It is the Magnus force. It's Saffman force. It is the force of collision between particles. It is the cohesive force between particles. It is the collision and rebound force between the particles and the wall.

[0016] Furthermore, in the step of solving the particle motion equation to obtain motion information, the obtained flow field distribution data is used as the boundary condition input to construct the particle motion equation. The discrete element method is used to numerically solve the particle motion equation. During the solution process, multi-size particles are divided into different particle groups according to their particle size, and each group of particles is calculated separately. At the same time, the collision and aggregation effects between different particle size groups are considered through the interaction model between particles. Through numerical solution, the spatial position coordinates, velocity vectors and concentration distribution of multi-size particles at different times in the turbulent square tube are obtained.

[0017] Furthermore, in the step of calculating dispersion and sedimentation parameters, the dispersion coefficient is calculated by statistically analyzing the variance of the distance from the particles to the pipe axis. The calculation formula is as follows: ,in, It is the particle dispersion coefficient. It is the total number of particles. It is the first The distance from each particle to the central axis of the square tube. It is the average distance of all particles from the central axis.

[0018] Furthermore, in the step of calculating dispersion and sedimentation parameters, the sedimentation velocity is calculated by statistically analyzing the average vertical velocity of the particles. The calculation formula is as follows: ,in, It is the particle settling velocity. It is the first The vertical velocity component of each particle.

[0019] Compared with existing technologies, this method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in turbulent square tubes has the following advantages:

[0020] I. This invention, when establishing a mathematical model of fluid flow, combines a two-equation k-ε turbulence model and introduces wall function correction. By associating the near-wall velocity gradient with turbulent stress through the wall function, the boundary calculations of turbulent kinetic energy and dissipation rate are corrected, avoiding numerical errors caused by direct discretization. This makes the near-wall flow field simulation more realistic, providing accurate flow field basis data for subsequent particle motion prediction. At the same time, the invention introduces the differences in the physical properties of particles, enabling it to realistically and meticulously reflect the motion state of particles of various sizes. This effectively overcomes the problem of low prediction accuracy caused by traditional methods ignoring differences between particles and complex interactions, laying a solid foundation for accurate prediction of particle behavior.

[0021] Second, this invention, through an interaction model, fully considers the collision and aggregation effects between particle groups of different sizes, thereby improving the prediction accuracy of the dispersion characteristics and sedimentation behavior of multi-size particles. At the same time, by calculating the two key parameters of dispersion coefficient and sedimentation velocity, it can quantitatively describe the dispersion characteristics and sedimentation behavior of multi-size particles, providing a reliable theoretical basis for the optimized design, operation parameter control and fault diagnosis of related industrial equipment.

[0022] Other advantages, objectives and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination or study, or may be learned from the practice of the invention. Attached Figure Description

[0023] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0024] Figure 1 A flowchart of a method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube;

[0025] Figure 2 This is a flowchart illustrating the steps of a method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube. Detailed Implementation

[0026] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below.

[0027] Example 1

[0028] In the industrial sand transport scenario, a standard square pipe with an inner diameter of 0.5m and a length of 10m is selected as the transport pipeline. The transport medium is clean water at room temperature. First, the fluid physical parameters are defined. The density of water is determined according to the standard value at room temperature, and the dynamic viscosity is also adopted as the inherent parameter of water at room temperature. In terms of boundary conditions, considering the actual industrial sand transport requirements, the water flow velocity at the pipeline inlet is set to 1.2m / s, and the outlet end is connected to the atmosphere to maintain atmospheric pressure.

[0029] like Figure 1As shown, a mathematical model of fluid flow is established based on the Reynolds-averaged Navier-Stokes equations combined with a two-equation k-ε turbulence model. The algorithm formula for the Reynolds-averaged Navier-Stokes equations is as follows: ,in, It is the density of the fluid flowing inside the turbulent square tube. , Is it fluid in , The time-averaged velocity component in the direction, , These are spatial coordinate components, corresponding to the spatial positions within the square tube, used to locate the spatial distribution of flow field parameters. It is the average pressure of the fluid. It is the fluid dynamic viscosity. yes In the two-equation k-ε turbulence model, the turbulent kinetic energy k equation is: ,in, It is turbulent kinetic energy, which characterizes the intensity of turbulent fluctuations. It is turbulent viscosity. It is the Prandtl number corresponding to turbulent kinetic energy. It is the turbulent kinetic energy generated by the average velocity gradient. It is the turbulent kinetic energy dissipation rate, and its calculation formula is: ,in, It is the dissipation rate Prandtl number. , This is an empirical constant. To address the distortion of near-wall turbulence parameters caused by the roughness of the square tube wall, a standard wall function is introduced for targeted correction: the near-wall region is divided by a dimensionless distance, and the correction value for near-wall turbulent kinetic energy is derived using empirical correlations. The formula is as follows: ,in, It is the correction value for turbulent kinetic energy in the near-wall region. It is an empirical constant. The wall friction velocity is used as the basis for calculating the dissipation rate correction value by correlating it with the near-wall turbulent kinetic energy correction value, the Karman constant, and the actual distance between the particle and the wall. The formula is as follows: ,in, This is the correction value for the near-wall dissipation rate. It is the Kármán constant. The distance from the particle to the wall is the actual distance. After inputting all the determined geometric parameters of the square tube, the physical parameters of the water, and the boundary conditions into the model, the finite volume method is used to discretize the model, transforming the complex continuous equations into a set of discrete equations that can be numerically calculated. Solving the discretized set of equations yields detailed flow field data such as velocity distribution, pressure distribution, and turbulence intensity distribution at different locations inside the tube at different times. These data provide an accurate fluid environment basis for subsequent analysis of particle motion.

[0030] The sand and gravel particles involved in industrial sand transportation are divided into two groups: large-diameter (2-3mm) and small-diameter (0.5-1mm). The physical property parameters of the two groups of particles are obtained through experimental measurement or industry standard data. The density is determined according to the material of the sand and gravel, the particle size is accurately measured by sieving, the particle shape coefficient is referenced to the typical value of similar sand and gravel, and the surface charge density is detected by professional charge measuring instruments.

[0031] During transport within the square tube, sand and gravel particles are subjected to various forces. Besides basic gravity and buoyancy, they are also subjected to fluid drag from the water flow, and additional mass forces, Magnus forces, and Saffman forces generated by the relative motion between the particles and the fluid. Simultaneously, collisions occur between particles of different sizes, generating interparticle collision forces. Some particles may also agglomerate due to surface adsorption, forming agglomeration forces. When particles contact the square tube wall during their movement, they experience impact rebound forces from the wall. Taking all these forces into account, a motion equation suitable for multi-size sand and gravel particles in this scenario is constructed, and its motion formula is: ,in, It is particle mass. It's particle velocity. It is fluid drag. It is particle gravity. It is fluid buoyancy. yes , It is the Magnus force. It's Saffman force. It is the force of collision between particles. It is the cohesive force between particles. It is the collision and rebound force between the particles and the wall.

[0032] The detailed flow field data obtained is used as boundary conditions, including velocity, pressure, and turbulence intensity at each location and time. These are substituted into the constructed multi-size sand and gravel particle motion equation. Since the particle size varies, their motion patterns will differ. Therefore, when using the discrete element method to numerically solve the motion equation, the sand and gravel particles are first clearly divided into large-size and small-size groups according to their particle size. Each group of particles is calculated independently. During the calculation, the collision and aggregation processes between different particle size groups are accurately simulated through the particle interaction model. After multiple iterations, the specific spatial coordinates, velocity vectors, and concentration distribution of the two groups of sand and gravel particles in the pipe at different times are finally obtained, clearly showing the motion trajectory and distribution of the particles in the pipe.

[0033] Based on the obtained particle spatial location, velocity, and concentration distribution, dispersion and sedimentation parameters are calculated. The dispersion coefficient is calculated by statistically analyzing the variance of the distance from the particle to the pipe axis. The calculation formula is as follows: ,in, It is the particle dispersion coefficient. It is the total number of particles. It is the first The distance from each particle to the central axis of the square tube. This is the average distance of all particles from the central axis. The dispersion coefficient is calculated by statistically analyzing the distances of all sand and gravel particles (including large and small particle groups) from the central axis of the square tube. The magnitude of the dispersion coefficient directly reflects the uniformity of particle dispersion within the tube. A smaller coefficient indicates a more concentrated particle distribution, potentially leading to localized accumulation; a larger coefficient indicates more uniform particle dispersion and a more stable conveying process. The settling velocity is calculated by statistically analyzing the average vertical velocity of the particles. The formula is as follows: ,in, It is the particle settling velocity. It is the first The vertical velocity component of each particle can be used to predict the settling trend of sand and gravel particles during transportation. If the settling velocity is a large negative value, it indicates that the particles settle quickly and are prone to depositing at the bottom of the pipe, which may lead to pipe blockage. In this case, parameters such as the water flow rate need to be adjusted. If the settling velocity is close to zero or a small positive value, it indicates that the particles have good suspension in the pipe and the transportation state is ideal. This provides key data support for flow control, pipeline layout optimization, and anti-blockage measures for sand transportation in industrial pipelines.

[0034] Example 2

[0035] In the environmental protection flue gas dust removal scenario, the flue gas conveying pipeline adopts a square pipe with an inner diameter of 1.2m and a length of 20m to meet the flue gas emission requirements of industrial boilers or incinerators. The conveying medium is the flue gas emitted by the boiler, and its physical parameters need to be determined according to the actual flue gas composition. The density and dynamic viscosity refer to the standard parameters of flue gas under similar working conditions. In terms of boundary condition setting, combined with the flue gas emission pressure and flow velocity requirements, the flue gas flow velocity at the pipeline inlet is set to 1.8m / s. The outlet end is connected to the subsequent equipment of the dust removal system to maintain the standard atmospheric pressure.

[0036] like Figure 2As shown, a mathematical model of fluid flow is established based on the Reynolds-averaged Navier-Stokes equations combined with a two-equation k-ε turbulence model. To address the distortion of turbulence parameters in the near-wall region caused by the roughness of the square tube wall, a standard wall function is introduced for targeted correction. The near-wall region is divided by dimensionless distance, and the correction value for turbulent kinetic energy in the near-wall region is derived using empirical correlations. Then, the dissipation rate correction value is calculated by correlating the near-wall turbulent kinetic energy correction value, the Karman constant, and the actual distance from the particle to the wall. After inputting the square tube geometric parameters, flue gas physical parameters, and boundary conditions into the model, the finite volume method is used to discretize the model, transforming the three-dimensional flow field problem into a discrete grid node computation problem. The discretized equations are then solved to obtain flow field data such as velocity distribution, pressure distribution, and turbulence intensity distribution at different locations within the tube at different times. These data provide an accurate basis for analyzing the motion environment of dust particles in flue gas.

[0037] Dust particles in environmental flue gas dust removal are divided into two groups: coarse dust and fine dust. The physical property parameters of the two groups of dust particles are obtained through experimental analysis or environmental monitoring data. The density is determined according to the main components of the dust, the particle size is accurately measured using a laser particle size analyzer, the particle shape coefficient is determined with reference to the morphological characteristics of similar dust particles, and the surface charge density is detected by an electrostatic measuring instrument. During the flue gas flow, dust particles are subjected to a variety of forces. In addition to gravity and buoyancy, they are also subjected to the fluid drag generated by the flue gas flow, as well as the additional mass force and Magnus force generated by the relative motion and rotation of the particles and flue gas. In areas with a large flue gas velocity gradient, the particles are also subjected to Saffman force. Meanwhile, collisions occur between coarse and fine dust particles, as well as between dust particles of the same size, generating interparticle collision forces. Some fine dust particles, due to their strong surface adsorption, will agglomerate, forming agglomeration forces. When particles come into contact with the square tube wall during their movement, they will be subjected to the collision rebound force of the wall. Taking all these forces into account, a motion equation suitable for multi-size dust particles in this scenario is constructed to ensure that the equation can fully reflect the force situation of dust particles in flue gas, providing an accurate model for subsequent solutions to the particle motion law.

[0038] The obtained flue gas flow field data was used as boundary conditions and substituted into the constructed multi-size dust particle motion equation. Due to the large difference in particle size between coarse and fine dust, their motion characteristics in the flue gas are different. Therefore, when using the discrete element method to numerically solve the motion equation, the dust particles were first clearly divided into coarse dust group and fine dust group according to particle size. The calculation was performed separately for each group of particles. During the calculation, the collision and aggregation processes between different particle size groups were accurately simulated by using the interaction model between particles. Through multiple iterations, the specific spatial coordinates, velocity vectors (including horizontal flow velocity and vertical settling velocity) and concentration distribution (the number of dust particles per unit volume, with a focus on the particle concentration change in the dust removal area) of the two groups of dust particles in the pipe at different times were finally obtained. This clearly shows the motion trajectory, distribution pattern, and aggregation of dust particles in the flue gas and provides data support for subsequent dust removal efficiency analysis.

[0039] Based on the obtained spatial location, velocity, and concentration distribution of dust particles, dispersion and settling parameters are calculated. When calculating the dispersion coefficient, the distances of all dust particles (including coarse and fine dust groups) to the central axis of the square tube are first counted. These distance values ​​are summed and divided by the total number of particles to obtain the average distance. Then, the difference between the distance of each particle to the central axis and the average distance is calculated. The difference is squared, and finally, all squared values ​​are summed and divided by the total number of particles to obtain the dust particle dispersion coefficient. The dispersion coefficient reflects the dispersion of dust particles in flue gas. A larger coefficient indicates that the dust particles are more evenly distributed in the pipe, which is conducive to sufficient contact between the dust removal equipment and the particles, improving dust removal efficiency. A smaller coefficient indicates that the particles tend to accumulate in local areas, which may lead to excessive local load on the dust removal equipment and affect the overall dust removal effect. When calculating the settling velocity, the vertical velocity components of all dust particles are extracted, and the sum of these velocity components is divided by the total number of particles to obtain the dust particle settling velocity. The settling velocity can be used to predict the settling pattern of dust particles in flue gas. If the settling velocity of coarse dust is large, it indicates that it tends to settle at the bottom of the pipe, and a dust collection device can be considered to be installed at the bottom of the pipe. If the settling velocity of fine dust is small, it indicates that it has a long suspension time in the flue gas and needs to be captured by high-efficiency dust removal equipment. Based on the calculation results of the dispersion coefficient and settling velocity, the installation position of the dust removal equipment can be optimized and the flue gas flow parameters can be adjusted to improve the overall dust removal efficiency and meet the requirements of environmental emission standards.

[0040] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube, characterized in that, The method includes the following specific steps: A mathematical model of the fluid was established and the flow field was solved: Based on the Reynolds-averaged Navier-Stokes equations and combined with the two-equation k-ε turbulence model, a mathematical model of fluid flow in a turbulent square tube was established. To address the distortion of turbulence parameters in the near-wall region caused by the roughness of the square tube wall, a standard wall function was introduced for targeted correction: The near-wall region was divided by dimensionless distance, and the correction value of turbulent kinetic energy in the near-wall region was derived using empirical correlations. Then, the dissipation rate correction value was calculated by correlating the correction value of turbulent kinetic energy in the near-wall region, the Karman constant, and the actual distance from the particle to the wall. By determining the geometric parameters of the turbulent square tube, the physical parameters of the fluid, and the boundary conditions, the parameters were input into the mathematical model. The mathematical model was discretized using the finite volume method, and the discretized equations were solved to obtain the flow field distribution data at different locations and times in the turbulent square tube, including velocity distribution, pressure distribution, and turbulence intensity distribution. Constructing particle motion equations: Obtain the physical property parameters of particles with different particle sizes, and combine them with the various forces acting on the particles in the tube to construct the motion equations of particles with different particle sizes. The various forces include gravity, buoyancy, fluid drag, added mass force, Magnus force, Saffman force, interparticle collision force and cohesion force, and the collision rebound force between particles and the square tube wall. Solving the particle motion equation to obtain motion information: The flow field data is used as the boundary condition input to the particle motion equation. The discrete element method is used and the solution is performed by grouping particles by size. The collision and aggregation between particle groups of different sizes are considered through the interaction model between particles to obtain the spatial position, velocity and concentration distribution of particles at different times. Calculate dispersion and sedimentation parameters: Based on the spatial location, velocity and concentration distribution of particles, calculate the dispersion coefficient by statistically analyzing the variance of the distance from the particle to the tube axis, and calculate the sedimentation velocity by statistically analyzing the average vertical velocity of the particles, so as to predict the dispersion characteristics and sedimentation behavior of particles.

2. The method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube according to claim 1, characterized in that, In the step of establishing the fluid mathematical model and solving the flow field, the algorithm formula for the Reynolds-averaged Navier-Stokes equations is as follows: ,in, It is the density of the fluid flowing inside the turbulent square tube. , Is it fluid in , The time-averaged velocity component in the direction, , These are spatial coordinate components, corresponding to the spatial positions within the square tube, used to locate the spatial distribution of flow field parameters. It is the average pressure of the fluid. It is the fluid dynamic viscosity. yes .

3. The method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube according to claim 2, characterized in that, In the step of establishing a fluid mathematical model and solving the flow field, for both equations... - Turbulence model, its turbulent kinetic energy The equation is ,in, It is turbulent kinetic energy, which characterizes the intensity of turbulent fluctuations. It is turbulent viscosity. It is the Prandtl number corresponding to turbulent kinetic energy. It is the turbulent kinetic energy generated by the average velocity gradient. It is the turbulent kinetic energy dissipation rate, and its calculation formula is: ,in, It is the dissipation rate Prandtl number. , It is an empirical constant.

4. The method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube according to claim 1, characterized in that, In the step of constructing the particle motion equation, the physical property parameters of particles of different sizes are obtained, including the density, particle size, particle shape coefficient and surface charge density of each particle size. Combined with the various forces acting on the particles in the tube, the motion equation of the particles of different sizes is constructed. The various forces include gravity, buoyancy, fluid drag, additional mass force, Magnus force, Saffman force, collision force and cohesion force between particles, and collision rebound force between particles and the square tube wall.

5. The method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube according to claim 4, characterized in that, In the step of constructing the particle motion equation, the motion equation of multi-diameter particles is constructed by combining the various forces acting on the particles inside the tube. The formula for the motion equation is as follows: ,in, It is particle mass. It's particle velocity. It is fluid drag. It is particle gravity. It is fluid buoyancy. yes , It is the Magnus force. It's Saffman force. It is the force of collision between particles. It is the cohesive force between particles. It is the collision and rebound force between the particles and the wall.

6. The method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube according to claim 1, characterized in that, In the step of solving the particle motion equation to obtain motion information, the obtained flow field distribution data is used as the boundary condition input to construct the particle motion equation. The discrete element method is used to numerically solve the particle motion equation. During the solution process, multi-size particles are divided into different particle groups according to their size, and each group of particles is calculated separately. At the same time, the collision and aggregation between different particle size groups are considered through the interaction model between particles. Through numerical solution, the spatial position coordinates, velocity vectors and concentration distribution of multi-size particles at different times in the turbulent square tube are obtained.

7. The method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube according to claim 1, characterized in that, In the step of calculating dispersion and sedimentation parameters, the dispersion coefficient is calculated by statistically analyzing the variance of the distance from the particle to the pipe axis. The calculation formula is as follows: ,in, It is the particle dispersion coefficient. It is the total number of particles. It is the first The distance from each particle to the central axis of the square tube. It is the average distance of all particles from the central axis.

8. The method for predicting the dispersion characteristics and sedimentation behavior of multi-size particles in a turbulent square tube according to claim 1, characterized in that, In the step of calculating dispersion and sedimentation parameters, the sedimentation velocity is calculated by statistically analyzing the average vertical velocity of the particles. The calculation formula is as follows: ,in, It is the particle settling velocity. It is the first The vertical velocity component of each particle.

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