Two-dimensional distribution prediction method for concentration of pollutants in iced river with absorption boundary

Abstract

The invention belongs to the technical field of water conservancy projects, and particularly relates to a two-dimensional distribution prediction method for the pollutant concentration of an iced river channel with an absorption boundary, and the method comprises the steps: S1, carrying out the analysis of the section size of a target river channel; s2, establishing a Cartesian coordinate system according to the target river channel, and establishing a calculation model according to a time dimension; s3, carrying out nondimensionalization on the longitudinal time-average flow velocity, the control equation of pollutant transportation and the boundary conditions of the longitudinal time-average flow velocity and the control equation; and S4, carrying out progressive analysis by utilizing a homogenization method, and finally obtaining a predicted value of the pollutant concentration of the target riverway in the freezing period. The analytical solution provided by the invention solves the problem that the distribution condition of rectangular river pollutants in the cold region in the river is difficult to predict after the initial transient state, and has important practical significance for water pollution treatment and ecological restoration.

Description

Technical Field

[0001] The invention belongs to the technical field of water conservancy engineering, and in particular relates to a two-dimensional distribution prediction method for pollutant concentration in an ice-covered river channel with an absorption boundary. Background Art

[0002] Urbanization, population growth, and industrial expansion have led to increased river pollution, posing a serious challenge to the global environment. However, more than half of the world's rivers are seasonally covered by ice and snow, most of which are located in high-latitude cold regions, with ice cover time exceeding 150 days per year. The presence of ice sheets adds a rough boundary while reducing atmospheric deposition and wind disturbance. Therefore, during the ice cover period, river runoff decreases and dissolved oxygen levels drop. On the other hand, ice sheets greatly reduce solar radiation and maintain a cold environment, thereby reducing the metabolic activity of microorganisms. In ice-covered rivers, low temperatures and low dissolved oxygen conditions weaken the degradation of microorganisms, thereby reducing the self-purification capacity of water bodies, making river pollution problems in cold regions particularly prominent. However, in previous studies on pollutant concentration distribution, researchers have focused on rivers without ice cover, and methods for predicting pollutant concentration distribution in ice-covered rivers have rarely been mentioned. However, due to the interaction between the ice cover boundary and the riverbed absorption effect, the migration of pollutants in ice-covered rivers becomes more complicated. Therefore, given the importance of studying pollutant migration for water pollution treatment and river management, it is crucial to thoroughly understand the pollutant diffusion process in ice-covered rivers.

[0003] In previous studies, researchers have extensively studied the average concentration distribution, while the true spatial concentration distribution has rarely been involved. However, in many diffusion processes, the actual concentration distribution may be more important than the average concentration distribution due to complex boundary conditions that may cause significant changes in the concentration distribution in different vertical sections. However, to the best of the author's knowledge, researchers have paid limited attention to the migration of pollutants in ice-covered rivers, and there is currently no clear solution. Therefore, considering the importance of understanding the concentration distribution of pollutants in ice-covered rivers for sewage treatment and ecological restoration in cold regions, this paper proposes a concise and applicable analytical solution for estimating the concentration distribution of pollutants in ice-covered rivers after experiencing an initial transient. Summary of the invention

[0004] The technical problem to be solved by the present invention is to overcome the defects of the prior art and provide a method for predicting the two-dimensional distribution of pollutant concentration in an ice-covered river channel with an absorption boundary.

[0005] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0006] In a first aspect of the present invention, a method for predicting the two-dimensional distribution of pollutant concentration in an ice-covered river with an absorbing boundary is provided, which specifically comprises the following steps:

[0007] S1. Analyze the cross-section dimensions of the target river channel, establish an analysis model of the river channel cross section, calculate and obtain the longitudinal time-averaged flow velocity of the cross section based on the length and depth of the cross section, and form the preliminary data collection of the analysis model;

[0008] S2. A Cartesian coordinate system is established with the target river channel, and a partial differential equation for calculating pollutant concentration is formed with the time dimension to establish a calculation model. The control equation for pollutant transport can be expressed as follows:

[0009]

[0010] Where C* is the pollutant concentration; t is the time dimension; u is the longitudinal time-averaged velocity of the target section; D is the molecular diffusion coefficient of the solute; x and z represent the longitudinal and vertical directions of the Cartesian coordinate system, respectively; and then the boundary conditions are set for the pollutant transport control equation;

[0011] S3. Non-dimensionalize the longitudinal time-averaged velocity, the control equations of pollutant transport and their boundary conditions to simplify the data of step S2;

[0012] S4. Using the homogenization method to perform asymptotic analysis, the two-dimensional distribution expression of pollutant concentration in the ice-covered rectangular river with an absorbing boundary is obtained, and the analytical solution based on the model in step S2 is obtained, and finally the predicted value of pollutant concentration in the target river during the ice-covered period is obtained.

[0013] Preferably, in step S4, the dimensionless concentration C is converted into three different time variables, namely, a fast time variable, a medium time variable and a slow time variable, and the average concentration function is obtained by combining the control equation of pollutant transport and its boundary conditions. <c>.

[0014] Preferably, in step 1, the vertical distribution of the longitudinal time-averaged flow velocity of the target section is calculated by the following formula:

[0015]

[0016] Where H is half of the channel depth; U is the maximum time-averaged velocity, and the calculation formula is U = gsinH 2 / 2ν, where ν is the kinematic viscosity coefficient.

[0017] Preferably, the dimensionless parameter is specifically:

[0018]

[0019] where η and ξ are dimensionless longitudinal and vertical coordinates, respectively, and L is the characteristic longitudinal length of the solute cloud. is the dimensionless longitudinal time-averaged velocity, τ is the dimensionless time variable, Pe is the Peclet number, C is the dimensionless concentration; β is the dimensionless bed absorption parameter.

[0020] In a second aspect of the present invention, a two-dimensional distribution prediction system for pollutant concentration in an ice-covered river with an absorption boundary is provided, which specifically comprises:

[0021] A data acquisition unit is used to measure the distance of the river structure and collect the depth of the target river;

[0022] A model operation unit, the model unit is composed of the method of steps S1-S4, and the model includes a data input port and an output port, the input port corresponds to the target river data one by one, and is used to input the data obtained by the data acquisition unit and then operate to obtain the predicted value of the two-dimensional distribution of pollutant concentration during the ice-covered period of the river;

[0023] The control unit is respectively linked to the data acquisition unit and the model operation unit, adjusts the computing power of the model operation unit through the central processing unit, and includes a hard disk module to store the output prediction data.

[0024] In a third aspect of the present invention, a computer device is provided, comprising a processor, a memory, and a computer program stored in the memory and executable on the processor, wherein the processor implements the method steps of the first aspect or the system functions of the second aspect when executing the computer program.

[0025] In a fourth aspect of the present invention, a computer-readable storage medium is provided, on which a computer program is stored, and when the computer program is executed by a processor, the method steps of the first aspect or the system functions of the second aspect are implemented.

[0026] Compared with the prior art, the present invention has the following beneficial effects:

[0027] The analytical solution proposed in the present invention solves the problem that the distribution of pollutants in rectangular rivers in cold regions after the initial transient state is difficult to predict, which has important practical significance for water pollution treatment and ecological restoration. The method proposed in the present invention has the following advantages:

[0028] (1) The method proposed in the present invention can not only predict the average concentration distribution of pollutants in a rectangular river during the ice-covered period, but also predict the spatial concentration distribution;

[0029] (2) The analytical solution proposed in the present invention has a concise expression. In engineering practice, Excel tables or programming programs can be used to achieve rapid calculations, greatly reducing manpower and material resources.

[0030] (3) In addition to proposing a prediction model, the present invention also optimizes its calculation model and constructs corresponding calculation schemes for a variety of different river environments, thereby increasing the robustness and versatility of the calculation model. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0032] Figure 1 This is a schematic diagram of a cross section of an ice-covered rectangular river channel according to the method of the present invention;

[0033] Figure 2 The average concentration distribution of Example 1 when the method proposed by the present invention is used;

[0034] Figure 3 The average concentration distribution of Example 1 when the method proposed by the present invention is used;

[0035] Figure 4 The longitudinal distribution of the average concentration and actual concentration of different Pe values ​​of Example 1 deduced by the method proposed by the present invention;

[0036] Figure 5 To use the method proposed by the present invention to deduce β * =0.5 is the vertical distribution of the dimensionless concentration CPe at different longitudinal positions of Example 1. DETAILED DESCRIPTION

[0037] The preferred embodiments of the present invention are described below in conjunction with the accompanying drawings. It should be understood that the preferred embodiments described herein are only used to illustrate and explain the present invention, and are not used to limit the present invention.

[0038] Example 1

[0039] The present invention provides a two-dimensional distribution expression for pollutant concentration in an ice-covered rectangular river channel with an absorbing boundary, wherein:

[0040] Step 1: Geometric dimension analysis of the target section, which is implemented as follows:

[0041] Obtain cross-sectional measurement data X and Z of the target cross section, where X is the length of the target cross section and Z is the depth of the target cross section;

[0042] The vertical distribution of the longitudinal time-averaged velocity of the target section is calculated by the following formula:

[0043]

[0044] Where u is the longitudinal time-averaged velocity of the target section; H is half of the channel depth; U is the maximum time-averaged velocity, and the calculation formula is U = gsinH 2 / 2ν, where ν is the kinematic viscosity coefficient.

[0045] Step 2: Analysis of pollutant migration process during the rectangular river ice-covered period, specifically implemented as follows:

[0046] The governing equation for pollutant transport can be expressed as follows:

[0047]

[0048] Where C* is the pollutant concentration; t is the time dimension; u is the longitudinal time-averaged flow velocity of the target section; D is the molecular diffusion coefficient of the solute; x and z represent the longitudinal and vertical directions of the Cartesian coordinate system, respectively; and then boundary conditions are established for the pollutant transport control equation.

[0049] In order to solve the transport equation, the boundary conditions are set as follows:

[0050]

[0051] Where Q is the mass of pollutants released instantaneously; δ(·) is the Dirac function; β′ is the bed absorption parameter.

[0052] Step 3: Make the transport equation, boundary conditions and velocity distribution dimensionless, which is implemented as follows:

[0053] Several dimensionless parameters are introduced, which are:

[0054]

[0055] where η and ξ are dimensionless longitudinal and vertical coordinates, respectively, and L is the characteristic longitudinal length of the solute cloud. is the dimensionless longitudinal time-averaged velocity, τ is the dimensionless time variable, Pe is the Peclet number, C is the dimensionless concentration; β is the dimensionless bed absorption parameter.

[0056] Combined with the dimensionless parameters, the dimensionless forms of the transport equations, boundary conditions and velocity distribution are expressed as:

[0057]

[0058] Where ε is the perturbation parameter, expressed as

[0059] Step 4: Use the homogenization method to perform asymptotic analysis to derive the two-dimensional distribution expression of pollutant concentration in the rectangular river during the ice-covered period. The specific implementation is as follows:

[0060] Combined with the perturbation parameters, the concentration C is converted into multiple scales as follows:

[0061] C=C0+ωC1+ε 2 C2+ε 3 C3+O(ε 4 ); (14)

[0062] Three time variables are introduced, namely the fast time variable (t0 = τ), the medium time variable (t1 = ετ) and the slow time variable (t2 = ε 2 τ). Therefore, applying the chain rule, the original time derivative can be expressed as:

[0063]

[0064] The governing equations for solute transport and boundary conditions incorporating concentration scales and time variables can be transformed into:

[0065]

[0066] Combining the transport equations and boundary conditions, we can obtain:

[0067]

[0068] In the formula T is a new time variable, defined as T = τ; D c =1+Pe 2 D c0 is an effective discrete factor, where For the depth average function <f>The explanation is as follows: the depth average function of a function f in the z direction <f>Defined as: F1-F9 are unknown coefficients. For rectangular ice-covered river channels, F1-F9 are calculated using the following formulas:

[0069]

[0070]

[0071] Therefore, the average concentration distribution of the rectangular ice-covered river channel can be expressed as:

[0072]

[0073] In the formula The constant term on the right side (i.e. 1) represents the effect of longitudinal diffusion. In practical engineering, Pe is usually large (Pe>100), so the solute diffusion is mainly affected by convection, so the constant 1 can be ignored, i.e. Furthermore, a new coordinate system independent of Pe can be obtained The average concentration expression is:

[0074]

[0075] By bringing back C1, C2 and C3, we can get the two-dimensional distribution expression of concentration:

[0076]

[0077] Where β * is a new dimensionless variable, defined as β * (≥0)=εβ.

[0078] The dimensionless concentration CPe is calculated by the formula:

[0079]

[0080] According to the above process steps, the corresponding model was established, the actual target river was directly tested, and numerical experiments were carried out according to the pollutant transport equation (2) combined with the initial conditions and boundary conditions (3)-(6);

[0081] The specific implementation is as follows:

[0082] By using dimensionless time T and space variable ζ, equation (8) can be rewritten as:

[0083]

[0084] For the longitudinal diffusion term on the right side of the equation, when Pe is large, the effect on the calculation result is not significant. However, in actual engineering, Pe is generally greater than 100, so the longitudinal diffusion term can be ignored. Therefore, equation (36) can be rewritten as:

[0085]

[0086] Numerical simulation in two-dimensional plane (H z ×L x , where H z is the vertical depth, L x The control equation of pollutant migration (i.e., formula (37)) is discretized using the finite volume method and the finite difference method. It is worth noting that this study assumes that the pollutant concentration is zero within a sufficiently large longitudinal distance upstream and downstream. Specifically, for the numerical experiment considered, it is expressed as In addition, the initial condition for pollutant transport is that the time step is set to ΔT = 0.2 × 10 -4 s, the spatial step lengths in the longitudinal and vertical directions are set to Δ(ζ / Pe) = 0.02L x , Δξ=0.05H z .

[0087] The time term in formula (37) The forward difference method is used to discretize into the following form:

[0088]

[0089] The first-order derivative term in formula (37) The central difference method is used to discretize into the following form:

[0090]

[0091] The second-order derivative term in formula (37) The discretization is performed using the second-order differential central difference method into the following form:

[0092]

[0093] The boundary conditions are discretized as:

[0094]

[0095] where j m It is the last node in the vertical direction.

[0096] Figure 2 and Figure 3 The average concentration distribution of Example 1 is deduced by the method proposed in the present invention, and the calculation results are verified by numerical experimental data.

[0097] Figure 4 and Figure 5 The longitudinal and vertical concentration distributions of Example 1 are respectively deduced using the method proposed in the present invention.

[0098] In another embodiment, the present invention provides a prediction system for two-dimensional distribution of pollutant concentration in an ice-covered river channel with an absorption boundary, which is used to implement the method of embodiment 1, and specifically includes:

[0099] A data acquisition unit is used to measure the distance of the river structure and collect the depth of the target river;

[0100] A model operation unit, the model unit is composed of the method of steps S1-S4, and the model includes a data input port and an output port, the input port corresponds to the target river data one by one, and is used to input the data obtained by the data acquisition unit and then operate to obtain the predicted value of the two-dimensional distribution of pollutant concentration during the ice-covered period of the river;

[0101] The control unit is respectively linked to the data acquisition unit and the model operation unit, adjusts the computing power of the model operation unit through the central processing unit, and includes a hard disk module to store the output prediction data.

[0102] In another embodiment, the present invention provides a computer device, comprising a processor, a memory, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the above method or the above system functions when executing the computer program.

[0103] In another embodiment, the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program implements the steps of the above method or the above system functions when executed by a processor.

[0104] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art can still modify the technical solutions described in the aforementioned embodiments or replace some of the technical features therein by equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.< / f> < / f> < / c>

Claims

1. A method for predicting the two-dimensional distribution of pollutant concentration in a rectangular river with an absorbing boundary during the ice-covered period, characterized in that: The specific steps include: S1. Analyze the cross-section dimensions of the target river channel, establish an analysis model of the river channel cross section, calculate and obtain the longitudinal time-averaged flow velocity of the cross section based on the length and depth of the cross section, and form the preliminary data collection of the analysis model; S2. A Cartesian coordinate system is established with the target river channel, and a partial differential equation for calculating pollutant concentration is formed with the time dimension to establish a calculation model. The control equation for pollutant transport can be expressed as follows: Where C* is the pollutant concentration; t is the time dimension; u is the longitudinal time-averaged velocity of the target section; D is the molecular diffusion coefficient of the solute; x and z represent the longitudinal and vertical directions of the Cartesian coordinate system, respectively; and then the boundary conditions are set for the pollutant transport control equation; S3. The longitudinal time-averaged velocity, the control equations of pollutant transport and their boundary conditions are dimensionless, thereby simplifying the formula of step S2; S4. Using the homogenization method to perform asymptotic analysis, the two-dimensional distribution expression of pollutant concentration in the ice-covered rectangular river with an absorbing boundary is obtained, and the analytical solution based on the model in step S2 is obtained, and finally the predicted value of pollutant concentration in the target river during the ice-covered period is obtained.

2. The method for predicting the two-dimensional distribution of pollutant concentration in an ice-covered river with an absorbing boundary according to claim 1 is characterized in that: In step S4, the dimensionless concentration C is converted into three different time variables, namely, fast time variable, medium time variable and slow time variable, and the average concentration function is obtained by combining the control equation of pollutant transport and its boundary conditions. <c> 。< / c> 3. The method for predicting the two-dimensional distribution of pollutant concentration in an ice-covered river with an absorbing boundary according to claim 1 is characterized in that: In step 1, the vertical distribution of the longitudinal time-averaged velocity of the target section is calculated by the following formula: Where H is half of the channel depth; U is the maximum time-averaged velocity, and the calculation formula is U = gsinH 2 / 2ν, where ν is the kinematic viscosity coefficient.

4. The method for predicting the two-dimensional distribution of pollutant concentration in an ice-covered river with an absorbing boundary according to claim 3 is characterized in that: The dimensionless parameters are: where η and ξ are dimensionless longitudinal and vertical coordinates, respectively, and L is the characteristic longitudinal length of the solute cloud. is the dimensionless longitudinal time-averaged velocity, τ is the dimensionless time variable, Pe is the Peclet number, C is the dimensionless concentration; β is the dimensionless bed absorption parameter.

5. A two-dimensional distribution prediction system for pollutant concentration in an ice-covered river with an absorption boundary, used to implement the method described in any one of claims 1 to 4, characterized in that: Specifically include: A data acquisition unit is used to measure the distance of the river structure and collect the depth of the target river; A model operation unit, the model unit is composed of the method of steps S1-S4, and the model includes a data input port and an output port, the input port corresponds to the target river data one by one, and is used to input the data obtained by the data acquisition unit and then operate to obtain the predicted value of the two-dimensional distribution of pollutant concentration during the ice-covered period of the river; The control unit is respectively linked to the data acquisition unit and the model operation unit, adjusts the computing power of the model operation unit through the central processing unit, and includes a hard disk module to store the output prediction data.

6. A computer device, comprising a processor, a memory, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of any one of the methods of claims 1 to 4 or the functions of the system of claim 5 when executing the computer program.

7. A computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the steps of the method according to any one of claims 1 to 4 or the functions of the system according to claim 5.

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