Simulation Method for Distribution and Gyration Radius of Silt Flocs Based on Langevin Model

Abstract

The present invention relates to a simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model, belonging to the technical field of molecular thermodynamic motion simulation. It solves the problem in the prior art that it is difficult to accurately simulate the random motion of flocs, and includes: Step S1, obtaining the parameters of sediment particles in the sludge flocs and establishing a Langevin dynamics model for the sludge flocculation motion; Step S2, splitting the established Langevin dynamics model for the sludge flocculation motion into two subsystems; Step S3, discretizing the two subsystems to obtain the numerical simulation values of the velocities and positions of the sediment particles in the flocs; Step S4, obtaining the numerical simulation of the change of the sediment particle distribution density function of the flocs with time as the long-time distribution density of the flocs; Step S5, obtaining the gyration radius of the flocs; Step S6, applying the obtained long-time distribution density and gyration radius of the flocs to the sewage treatment system and the water conservancy management system, and optimizing the design and operation parameters of the system.

Description

Technical Field

[0001] The present invention belongs to the technical field of molecular thermodynamics motion simulation, and particularly relates to a simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model. Background Art

[0002] The sludge flocculation motion is an important process for sediment settlement, transportation and distribution in water bodies, and has important applications in fields such as river dredging, water conservancy projects, sewage treatment and chemical production. During the sludge flocculation motion process, effective simulation of the floc distribution density and gyration radius helps to deeply understand the motion law of the flocs, calculate the sedimentation velocity and distribution law of the flocs, which provides a scientific basis and technical support for preventing water pollution, controlling soil erosion and controlling siltation. Thus, it plays an important role in optimizing engineering design, reducing construction costs and improving engineering efficiency.

[0003] During the motion process, sediment particles randomly collide with liquid molecules and are affected by thermodynamic random noise, so their motion is random. Due to the influence of complex factors such as frictional force, gravity, intermolecular attraction, thermal force and the potential force of the molecule itself, the motion environment of sediment particles shows high non-linearity. At present, the methods in this technical field are difficult to accurately simulate the random motion distribution law of flocs and predict physical laws or physical quantities such as the gyration radius of flocs.

[0004] There are at least the following problems in the prior art:

[0005] (1) The existing numerical simulation methods for the gyration radius of flocs cannot handle the simulation problem of the gyration radius of flocs in the flocculation motion under high non-linear conditions;

[0006] (2) The existing evaluation methods for numerically simulating the gyration radius of flocs have large errors or low calculation efficiency on a long time scale, and cannot meet the requirements of high-precision long-time numerical calculation of the gyration radius of flocs. In addition, the existing methods cannot effectively simulate the long-time numerical distribution density of flocs. Summary of the Invention

[0007] In view of this, the present invention provides a new method for efficiently numerically simulating the floc distribution density and floc gyration radius of sludge flocculation motion for the Langevin dynamics equation capable of simulating the high non-linearity of sludge flocculation motion.

[0008] The simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model provided by the embodiment of the present invention includes the following steps:

[0009] Step S1, obtaining the parameters of sediment particles in the sludge flocs, and establishing a Langevin dynamics model for the sludge flocculation motion based on the parameters of the sediment particles;

[0010] Step S2: Split the established Langevin kinetic model of sludge flocculation motion into two subsystems;

[0011] Step S3: Discretize the two subsystems obtained by splitting to obtain the numerical simulation values of the velocity and position of sediment particles in the flocs;

[0012] Step S4: Through the numerical simulation values of the position of sediment particles in the flocs obtained in S3, obtain the numerical simulation of the change of the sediment particle distribution density function of the flocs with time as the long-time distribution density of the flocs;

[0013] Step S5: Through the numerical simulation values of the position of sediment particles in the flocs in S3, obtain the gyration radius of the flocs;

[0014] Step S6: Apply the obtained long-time distribution density and gyration radius of the flocs to the sewage treatment system and water conservancy management system to accurately predict the movement trajectory and aggregation behavior of the flocs during the treatment process, and optimize the design and operation parameters of the system.

[0015] Optionally, step S1 specifically includes:

[0016] Obtain the parameters of sediment particles in the sludge flocs, including obtaining the damping force of the sludge solution on the sediment particles and the thermal external force on the sediment particles;

[0017] Establish the Langevin kinetic model of sludge flocculation motion, expressed as

[0018]

[0019] where t represents the time, Q(t) represents the position of the sediment particle, P(t) represents the movement velocity of the sediment particle, α is a positive integer, and the -(Q(t)) 2α+1 term represents the potential term carried by the sediment particle, v represents the damping force of the sludge solution on the sediment particle and is obtained through actual measurement, 1 / v>0 is the Lagrangian time scale, {W(t), t≥0} represents the thermal external force on the sediment particle and is obtained through actual measurement, is a positive constant, c0 is the Kolmogorov constant, and ∈ is the dissipation rate.

[0020] Optionally, step S2 specifically includes:

[0021] Split the Langevin kinetic model (1) of sludge flocculation motion to obtain the following two subsystems: For t∈T n ,

[0022]

[0023] and

[0024]

[0025] Among them, the superscript D represents the deterministic subsystem, and the superscript S represents the stochastic subsystem. Given a time step τ ∈ (0, 1), a uniform partition is made for the interval [0, +∞): 0 ≤ t1 < t2 < … < t n <…, and denote the nth time point t n = nτ, and the nth time interval T n = [t n , t n+1 ), n ∈ N, where N represents the set of natural numbers, represents the velocity of the deterministic subsystem at time t, represents the velocity of the deterministic subsystem at time t n ; represents the velocity of the stochastic subsystem at time t, represents the velocity of the stochastic subsystem at time t n ; represents the position of the deterministic subsystem at time t, represents the position of the deterministic subsystem at time t n ; represents the position of the stochastic subsystem at time t, represents the position of the stochastic subsystem at time t n ;

[0026] Optionally, step S3 specifically includes:

[0027] Calculating the numerical solutions of subsystems (2) and (3) at time t n+1 as the numerical simulation values of the velocity and position of the sediment particles in the floc, as follows:

[0028]

[0029] where n ∈ N, P0 = P(0), Q O = Q(0), and {P n , Q n : n ∈ N} is the numerical simulation solution of the velocity and position of the floc particles, θ is a real number representing the integration variable, and here is the numerical solution of the deterministic subsystem, representing the numerical simulation value of the velocity of the sediment particles in the floc, is the numerical solution of the deterministic subsystem, representing the numerical simulation value of the position of the sediment particles in the floc.

[0030] Optionally, step S4 specifically includes: Substituting the mth sediment particle at time t k into step S3 to obtain the position of the mth sediment particle at time t k ​The position distribution of sediment particles in the state space is counted by the built-in functions of Matlab and normalized to obtain the numerical simulation of the sediment particle distribution density function of the floc at time t k as the long-time distribution density of the floc.

[0031] Optionally, step S5 specifically includes:

[0032] Analyze the sediment particle content in the sample by using the method of random sampling, so as to determine the value of the number M of sediment particles in the floc, and calculate the approximate value of the gyration radius of the floc by using the Monte Carlo method:

[0033]

[0034] where represents the simulated values of the velocity and position of the m-th sediment particle driven by Brownian motion obtained by substituting the m-th sediment particle at time t k into step S3, K is a positive integer and represents the maximum time node of the calculation, φ1(·, ·) and φ2(·, ·) are test functions, and the initial position and the initial velocity value x = (P(0), Q(0)) T The approximate value of the gyration radius of the floc obtained is used as the gyration radius of the floc.

[0035] Compared with the prior art, the simulation method of the distribution and gyration radius of silt flocs based on the Langevin model provided by the embodiments of the present invention proposes an efficient numerical simulation method for the floc distribution density and the gyration radius, which can effectively simulate the physical quantities at steady state; in addition, the method has long-time numerical calculation stability and convergence, can improve the accuracy of long-time floc physical quantity simulation, and can provide reliable simulation for efficiently predicting the floc distribution density and the gyration radius of the Langevin dynamics model of silt flocculation movement, realizing long-time stable and efficient numerical simulation of the floc distribution law of silt flocculation movement in a highly non-linear complex environment, and being used for efficiently simulating the gyration radius of the floc, so as to provide an effective treatment decision-making scheme when used in fields such as river dredging, water conservancy projects, sewage treatment, and chemical production. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings used in the embodiments. By referring to the drawings, the features and advantages of the present invention can be more clearly understood. The drawings are schematic and should not be construed as imposing any limitation on the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0037] Figure 1Flowchart of the simulation method for sludge floc distribution and gyration radius based on the Langevin model provided according to an embodiment of the present invention.

[0038] Figure 2 Convergence order diagram of the simulation method for sludge floc distribution and gyration radius based on the Langevin model provided according to an embodiment of the present invention.

[0039] Figure 3 Simulation accuracy diagram of the floc gyration radius obtained from the embodiment of the simulation method for sludge floc distribution and gyration radius based on the Langevin model provided according to an embodiment of the present invention.

[0040] Figure 4 Long-term evolution comparison diagram of the calculation error between the floc gyration radius obtained from the embodiment of the simulation method for sludge floc distribution and gyration radius based on the Langevin model provided according to an embodiment of the present invention and the floc gyration radius obtained by the prior art method.

[0041] Figures 5 to 8 Long-term evolution diagram of the floc distribution density obtained from the embodiment of the simulation method for sludge floc distribution and gyration radius based on the Langevin model provided according to an embodiment of the present invention.

[0042] Figure 9 Long-term evolution diagram of the floc gyration radius obtained from the embodiment of the simulation method for sludge floc distribution and gyration radius based on the Langevin model provided according to an embodiment of the present invention. Detailed implementation mode

[0043] In order to more clearly understand the above objects, features and advantages of the present invention, the present invention will be further described in detail below with reference to the drawings and specific implementation modes. It should be noted that, without conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other.

[0044] Many specific details are set forth in the following description in order to provide a thorough understanding of the present invention, but the present invention may be practiced in other ways different from those described herein, and therefore, the scope of protection of the present invention is not limited by the specific embodiments disclosed below.

[0045] The following details the simulation method for sludge floc distribution and gyration radius based on the Langevin model provided according to an embodiment of the present invention.

[0046] Symbol description:

[0047] Q(t) Position of sediment particles;

[0048] P(t) Movement speed of sediment particles;

[0049] The damping force of the silt particles in the silt solution is obtained through actual measurement;

[0050] The thermal external force on the silt particles, W(t), is obtained through actual measurement;

[0051] σ is a positive constant;

[0052] α is a positive integer;

[0053] c0 is the Kolmogorov constant;

[0054] ∈ is the dissipation rate;

[0055] E is the expectation;

[0056] Test function;

[0057] x1 is the initial velocity;

[0058] x2 is the initial position;

[0059] x = (x1, x2) is the initial vector with the initial velocity and initial position as components;

[0060] (P x (t), Q x (t)) T The exact solution at time t with x as the initial vector;

[0061] T is the termination time;

[0062] π is the stationary distribution;

[0063] τ is the time step;

[0064] t is the time;

[0065] t n The nth time point, n ∈ N, where N represents natural numbers;

[0066] T n The nth time interval, n ∈ N, where N represents natural numbers;

[0067] D is the deterministic subsystem label;

[0068] S is the stochastic subsystem label;

[0069] The velocity of the deterministic subsystem at time t The deterministic subsystem at time t n The velocity at this point;

[0070] The velocity of the stochastic subsystem at time t;

[0071] The velocity of the stochastic subsystem at time t n ;

[0072] The position of the deterministic subsystem at time t;

[0073] The position of the deterministic subsystem at time t n ;

[0074] The position of the stochastic subsystem at time t;

[0075] The position of the stochastic subsystem at time t n ;

[0076] P n Denotes the numerical simulation solution of the velocity of the floc particles;

[0077] Q n Denotes the numerical simulation solution of the position of the floc particles;

[0078] θ is a real number, representing the integration variable;

[0079] Preliminarily obtain the intermediate value of the velocity simulation value of the sediment particles in the floc; Preliminarily obtain the intermediate value of the position simulation value of the sediment particles in the floc; p represents the variable of the particle velocity;

[0080] q represents the variable of the particle position;

[0081] υ is the variable of the damping force of the silt solution on the sediment particles;

[0082] C H A positive constant related to the energy function H;

[0083] H is the energy function;

[0084] L 2 (R 2 ; π)R 2 The space of functions that are square-integrable with respect to the measure π on (R; π)R;

[0085] R is the real number space;

[0086] k is a sequence of positive integers;

[0087] K is the maximum time node for calculation;

[0088] C is a positive constant;

[0089] The velocity of the sediment particles at time t with x as the initial vector k ;

[0090] The position of the sediment particle at time t with x as the initial vector; k

[0091] The long - time distribution density of the floc mass ρ(q);

[0092] The test function φ1(·, ·);

[0093] The test function φ2(·, ·);

[0094] P n The numerical simulation solution of the velocity of the floc particle;

[0095] Q n The numerical simulation solution of the position of the floc particle;

[0096] M The number of sediment particles in the floc;

[0097] R f (P, Q) The approximate value of the gyration radius of the floc;

[0098] The simulation values of the velocity and position of the m - th sediment particle driven by Brownian motion.

[0099] According to the embodiment of the present invention, a simulation method for the distribution and gyration radius of silt flocs based on the Langevin model includes the following steps:

[0100] Step S1: Obtain the parameters of the sediment particles in the silt flocs, and establish a Langevin kinetic model for the silt flocculation motion based on these parameters of the sediment particles. The obtained parameters of the sediment particles may include the damping force of the silt solution on the sediment particles, the thermal external force on the sediment particles, etc. The above parameters can be obtained through actual measurement of the silt flocs.

[0101] Establish a Langevin kinetic model for the silt flocculation motion, expressed as

[0102]

[0103] Among them, the equation satisfies the deterministic initial value (the values of the initial position and initial velocity) condition. Denote the values of the initial position and initial velocity as x = (P(0), Q(0)) T . Here, t represents the time, Q(t) represents the position of the sediment particle, P(t) represents the motion velocity of the sediment particle, α is a positive integer, -(Q(t)) 2α+1 ​The term represents the force term carried by the sediment particles, \(v\) represents the damping force of the sediment particles in the silt solution and is obtained through actual measurement, \(1 / v>0\) is the Lagrangian time scale, \(\{W(t),t\geq0\}\) is a one-dimensional standard Brownian motion, representing the thermal external force acting on the sediment particles and obtained through actual measurement. is a positive constant, \(c_0\) is the Kolmogorov constant, and \(\epsilon\) is the dissipation rate.

[0104] The solution distribution of the Langevin kinetic model (1) of silt flocculation motion will tend to a stationary distribution in the long run, which is also called the ergodicity of the Langevin kinetic model of silt flocculation motion. That is, in the space \(L^2\) 2 (\mathbb{R} 2 ;\pi), the following convergence holds:

[0105]

[0106] Among them, in order to emphasize the dependence of the solution on the values of the initial position and initial velocity, the exact solution at time \(t\) with \(x\) as the initial position and initial velocity is denoted as \((P x (t),Q x (t)) T , \(E\) represents the expectation, represents the test function, \(\mathbb{R}\) represents the real number space, \(T\) represents the termination time, and \(\pi\) represents the stationary distribution. That is to say, during the silt flocculation process, the motion of floc particles will reach a steady state after a long time, manifested as the distribution density of the flocs tending to a Gibbs probability distribution, that is, \(\pi\). At this time, the gyration radius of the flocs will tend to a stationary constant value, which can be obtained from the above convergence limit.

[0107] Step S2: Split the established Langevin kinetic model (1) of silt flocculation motion into two subsystems. Given the time step \(\tau\in(0,1)\), make a uniform partition of the interval \([0,+\infty)\): \(0\leq t_1<t_2<\cdots<t n <\cdots\), and denote \(t n =n\tau\), \(T n =[t n ,t n+1 ), \(n\in\mathbb{N}\), and \(\mathbb{N}\) represents the set of natural numbers. Split the Langevin kinetic model (1) of silt flocculation motion to obtain the following two subsystems: For \(t\in T n ,

[0108]

[0109] and

[0110]

[0111] Among them, the superscript \(D\) represents the deterministic subsystem, and the superscript \(S\) represents the stochastic subsystem. denotes the velocity of the deterministic subsystem at time t, denotes the deterministic subsystem at time t n at the velocity of, denotes the velocity of the stochastic subsystem at time t, denotes the stochastic subsystem at time t n at the velocity of, denotes the position of the deterministic subsystem at time t, denotes the deterministic subsystem at time t n at the position of, denotes the position of the stochastic subsystem at time t, denotes the stochastic subsystem at time t n at the position of.

[0112] Step S3, discretize the two subsystems obtained by splitting to obtain the numerical simulation values {P n , Q n : n ∈ N} of the velocities and positions of the sediment particles in the flocs, simply referred to as the numerical solution, where P n represents the numerical simulation solution of the velocity of the floc particles, and Q n represents the numerical simulation solution of the position of the floc particles. Specifically, calculate the numerical solutions of subsystems (2) and (3) at t n+1 at the moment, as follows:

[0113]

[0114] where n ∈ N, P0 = P(0), Q0 = Q(0), and θ is a real number representing the integration variable. Here represents the numerical solution of the deterministic subsystem, that is, the intermediate value of the simulation values of the velocities and positions of the sediment particles in the flocs obtained initially.

[0115] Property 1: The above numerical solution tends to a stationary distribution in the long run. Thus, the distribution density and gyration radius of the floc motion can be obtained from the stationary distribution and the numerical solution. For the energy function the numerical solution satisfies: This shows that the energy of the discrete system has uniform dissipation with respect to time. Thus, the numerical solution will reach a steady state after a long time: that is, the following convergence holds in L 2 (R 2 ; π τ ) where p represents the variable of the particle velocity, q represents the variable of the particle position, H represents the energy function, and C HDenote a positive constant related to the energy function H. Therefore, the simulation method of sludge floc distribution and gyration radius based on the Langevin model provided by the embodiments of the present invention has the ability to simulate the floc distribution density and gyration radius for a long time.

[0116] Property 2: The simulation method of sludge floc distribution and gyration radius based on the Langevin model provided by the embodiments of the present invention has a first-order calculation accuracy. The polynomial moments of the method have long-term stability, and the exponential moments have an exponential growth law: Furthermore, it is obtained that the method has a first-order calculation accuracy, that is: where, denotes the velocity of the sediment particle at time t with the initial vector x k and, denotes the position of the sediment particle at time t with the initial vector x k C is a positive constant. Thus, it shows that the simulation method has the ability to numerically simulate physical quantities with high accuracy for a long time.

[0117] Step S4, obtain a numerical simulation of the sediment particle distribution density function of the floc as the long-term distribution density of the floc. Calculate an approximate value of the sediment particle distribution density function of the floc: Use the calculation method in the above step S3 to obtain the position k of the m-th sediment particle at time t Through the built-in function of Matlab, statistically analyze the position distribution of sediment particles in the state space and perform normalization processing to obtain the numerical simulation of the sediment particle distribution density function of the floc at time t k .

[0118] Generate a definite initial position and initial velocity value through an initial random distribution. As time increases, the floc particle distribution gradually tends to be stable, and the theoretical distribution density in its stable state can be expressed as

[0119]

[0120] where, q represents the variable of the particle position, υ represents the variable of the damping force of the sediment particle by the sludge solution, and R represents the real number space.

[0121] Step S5, calculate the gyration radius of the floc. Set a definite initial position and initial velocity value x, and test functions φ1(x1, x2) = x2 2 and φ2(x1, x2) = x2. The gyration radius of the floc at the equilibrium state is where where, x = (x1, x2), x1 represents the velocity variable, and x2 represents the position variable.

[0122] Obtain the value of the number M of flocculated sediment particles through a random sampling method, and use the Monte Carlo method to calculate the approximate value of the flocculation gyration radius:

[0123]

[0124] Among them, m represents the m-th sediment particle, represents the simulated value of the velocity and position of the m-th sediment particle driven by Brownian motion. K is a positive integer and represents the maximum time node of the calculation. φ1(p,q) = q 2 and φ2(p,q) = q are test functions. Obtain the approximate value of the flocculation gyration radius as the gyration radius of the floc.

[0125] Step S6: Apply the obtained long-time distribution density and gyration radius of the flocs to sewage treatment systems, water conservancy management systems, etc. It can not only accurately predict the movement trajectories and aggregation behaviors of the flocs during the treatment process, but also optimize the design and operating parameters of the system, and improve the pollutant removal efficiency. In addition, it can also be used to improve the dosing strategy of flocculants, enhance the effect of sewage purification, and help monitor and control the distribution of suspended solids and sediments in water bodies in water conservancy management to ensure the sustainable management of the water environment and ecological balance.

[0126] The following provides a detailed description of an embodiment of a simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model according to the embodiments of the present invention.

[0127] Example 1

[0128] Step S1: Establish the Langevin kinetic equation for the sludge flocculation movement affected by non-linear potential energy. The potential energy of the sediment particle is Therefore, take α = 1, and the equation has an external force F = -(Q(t)) 3 . Assume that the Lagrangian time scale is 0.1, that is, take v = 10, and assume that the influence intensity of the random force is Then the following non-linear Langevin equation for the sediment particle movement is obtained:

[0129]

[0130] Assume that the initial position value and the initial velocity value are P0 = 1, Q0 = 1 respectively, and the end time: T = 1.

[0131] Step S2: The consistent partition step size is taken as: τ = 2 -12 , and the Langevin kinetic model (1) of the sludge flocculation movement is split to obtain the following two systems: For t ∈ T n , n = 1,..., 2 12 ,

[0132]

[0133] and

[0134]

[0135] Step S3: Discretize the system (2) to obtain the numerical simulation values of the sediment particle velocity and position as follows:

[0136]

[0137] Figure 1 is the flowchart for implementing the method (4) of the present invention. By processing according to the Figure 1 process, the velocity and position of the flocculated sediment particles at different times {P n , Q n : n = 0,..., 2 12} can be obtained.

[0138] Next, numerically calculate and comparatively analyze the convergence order, floc distribution density, calculation accuracy of the gyration radius, and time evolution of the calculation error in the examples of the method provided by the embodiments of the present invention.

[0139] Analysis 1: The exact solution of the system (1) can be generated from the solution obtained by the above formula (4) with a step size τ = 2 -1 and is denoted as {P n , Q n : n = 0,..., 2 15}. Assume there are 3000 sediment particles in the floc, and the calculation error: Here, the superscript i with different values (i.e., (P i , Q i ) T and represent the motion velocities and positions of different sediment particles. Uniform step sizes of 2 -9 , 2 -10 , 2 -11 are respectively taken to generate the simulation values and strong errors of the sediment particle velocity and position, and a loglog curve graph is plotted to obtain Figure 2 , thus numerically verifying that the velocities and positions of the sediment particles obtained by the method of the present invention can have an optimal convergence order of 1st order. Figure 2 In

[0140] Analysis 2: Further, calculate the approximate error between the numerical simulation value and the true value of the gyration radius of the floc: A loglog curve graph can be plotted to obtain Figure 3, Thus, the accuracy of calculating the gyration radius of the flocs by the method described in the present invention is verified to be 1 by this value.

[0141] Analysis 3: In order to compare the long-term evolution of the calculation error of the gyration radius of the flocs, another method considered for comparison with the method provided by the embodiment of the present invention is the following splitting algorithm (SAVF1):

[0142]

[0143] This method has a strong convergence order of 1, but it does not have ergodicity and cannot calculate the gyration radius of the flocs in a stable state for a long time. Taking the end time as T = 1000, the step size of the exact solution is 1.25×2 -6 , and the step size of the numerical solution is 2 -5 , and the number of sediment particles is taken as 50000. Calculating the strong error numerically according to the foregoing method, the comparison graph obtained is Figure 4 , where the legend SAVF is the error curve of the gyration radius of the flocs obtained by the method provided by the embodiment of the present invention; SAVF1 represents the error curve of the gyration radius of the flocs calculated by other existing methods. Figure 4 It shows that the algorithm described in the present invention not only has the characteristic of long-term calculation stability, but also significantly improves the calculation accuracy.

[0144] Step S4: Simulate the long-term distribution density of the flocs in the silt flocculation process.

[0145] Calculate the density maps of the floc position distributions at t = 0, t = 2, t = 256, and t = 512 respectively, count the sediment particle content in the uniformly divided intervals and normalize it to obtain the distribution histogram, and approximately obtain the empirical density function of the sediment particles in space, and obtain Figures 5 - 8 . From Figures 5 - 8 , it can be seen that as time increases, the empirical density function of the floc particles tends to the density function of a Gibbs measure, which reflects that the method provided by the embodiment of the present invention has the ability to calculate the distribution density of the floc particles for a long time.

[0146] Step S5: Calculate the gyration radius of the flocs.

[0147] Given the values of 5 initial positions and initial velocities,

[0148] which are x = (1, 1) T , (2, 2) T , (3, 3) T , (4, 4) T , (5, 5) T , the end time is T = 500, the step size of the numerical solution is 0.005, the number of sediment particles is M = 50000, and K = 10 5, the approximate value of the gyration radius of the flocs is calculated as

[0149]

[0150] Figure 9 is the evolution diagram of the gyration radius of the flocs obtained from the above 5 values of different initial positions and initial velocities with respect to time t n , and the evolution diagram when n ≤ K.

[0151] Step S6, apply the simulation method of the sediment particle floc distribution density and gyration radius to the sewage treatment system and the water conservancy management system. Calculating the spatial distribution density of the flocs in the sewage treatment tank or the river channel water body can show which areas have sedimentation or blockage problems, so as to optimize the flow path of the fluid and the distribution area of the flocs in the reaction tank, adjust the size and shape of the reaction tank or regularly clean the river channel to improve the sewage treatment efficiency and prevent river channel siltation. In addition, the size of the gyration radius of the flocs determines the rotation and migration behavior of the flocs in the flow field, which can be used to predict the behavior of the suspended solids in different treatment stages, so as to determine the best placement position and dosage of the flocculant to ensure that the flocs can quickly aggregate and effectively settle, and achieve reasonable regulation of water quality.

[0152] Combining the above steps can show that the present invention can efficiently and highly accurately simulate the long-time distribution density of the flocs in the silt flocculation process based on the Langevin dynamics model and calculate the size of the gyration radius of the flocs. At the same time, it also provides a method basis for the long-time high-precision numerical simulation requirements in other practical applications of the nonlinear Langevin equation in the fields of physics, biology, etc.

[0153] All the above optional technical solutions can be combined arbitrarily to form the optional embodiments of the present application, which will not be elaborated here one by one.

[0154] It should be understood that the order of the numbers of the steps in the above embodiments does not mean the order of execution. The order of execution of each process should be determined by its function and internal logic, and should not constitute any limitation to the implementation process of the embodiments of the present invention.

[0155] The above is only the preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

Claims

1. A simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model, characterized in that, It includes the following steps: Step S1: Obtain the parameters of the sediment particles in the silt flocs, and establish a Langevin dynamics model for the silt flocculation movement based on the parameters of the sediment particles; Step S2: Split the established Langevin dynamics model for the silt flocculation movement into two subsystems; Step S3: Discretize the two subsystems obtained by splitting to obtain the numerical simulation values of the velocities and positions of the sediment particles in the flocs; Step S4: Through the numerical simulation values of the positions of the sediment particles in the flocs obtained in S3, obtain the numerical simulation of the change of the sediment particle distribution density function of the flocs with time as the long-term distribution density of the flocs; Step S5: Through the numerical simulation values of the positions of the sediment particles in the flocs in S3, obtain the gyration radius of the flocs; Step S6: Apply the obtained long-term distribution density and gyration radius of the flocs to the sewage treatment system and the water conservancy management system to accurately predict the movement trajectories and aggregation behaviors of the flocs during the treatment process, and optimize the design and operation parameters of the system; Step S2 specifically includes: Split the Langevin dynamics model for the silt flocculation movement to obtain the following two subsystems: For and where \(t\) represents the time, the superscript \(D\) represents the deterministic subsystem, and the superscript \(S\) represents the stochastic subsystem. Given the time step \(\tau\in(0,1)\), a uniform partition of the interval \([0,+\infty)\) is made: \(0\leq t_1 < t_2<\cdots<t\) n <\cdots, and denote the \(n\)th time point \(t\) n =n\tau\), the \(n\)th time interval \(T\) n =[t n ,t n+1 ), \(n\in N\), where \(N\) represents the set of natural numbers. denotes the velocity of the deterministic subsystem at time \(t\in T\) n ; denotes the velocity of the deterministic subsystem at time \(t\) n ; denotes the velocity of the stochastic subsystem at time \(t\); denotes the velocity of the stochastic subsystem at time \(t\) n ; denotes the position of the deterministic subsystem at time \(t\); denotes the position of the deterministic subsystem at time \(t\) n ; denotes the position of the stochastic subsystem at time \(t\); denotes the position of the stochastic subsystem at time \(t\) n ; \(\alpha\) is a positive integer, \(v\) represents the damping force of the silt solution on the sediment particles and is obtained through actual measurement, \(\{W(t),t\geq0\}\) represents the thermal external force on the sediment particles and is obtained through actual measurement. is a positive constant, \(c_0\) is the Kolmogorov constant, and \(\epsilon\) is the dissipation rate.

2. The simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model according to claim 1, characterized in that Step S1 specifically includes: Obtain the parameters of the sediment particles in the silt flocs, including obtaining the damping force of the silt solution on the sediment particles and the thermal external force on the sediment particles; Establish a Langevin dynamics model for the silt flocculation movement, expressed as Among them, Q(t) represents the position of sediment particles, P(t) represents the movement speed of sediment particles, and the term -(Q(t)) 2α+1 represents the force term carried by sediment particles, and 1 / v>0 is the Lagrangian time scale.

3. The simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model according to claim 1, characterized in that, Step S3 specifically includes: The numerical solutions of the calculation subsystems (2) and (3) at time t n+1 are used as the numerical simulation values of the velocity and position of the sediment particles in the flocs, as follows: where \(n\in N\), \(P_0 = P(0)\), \(Q_0 = Q(0)\), \(\{P n ,Q n : n\in N\}\) is the numerical simulation solution of the velocity and position of the floc particles, \(\theta\) is a real number representing the integration variable. Here is the numerical solution of the deterministic subsystem, representing the numerical simulation value of the velocity of the sediment particles in the floc, is the numerical solution of the deterministic subsystem, representing the numerical simulation value of the position of the sediment particles in the floc.

4. The simulation method for sludge floc distribution and gyration radius based on the Langevin model according to claim 3, characterized in that Step S4 specifically includes: Substitute the m-th sediment particle at time t k into step S3 to obtain the position of the m-th sediment particle at time t k Use the built-in functions in Matlab to statistically analyze the position distribution of sediment particles in the state space and perform normalization processing to obtain the numerical simulation of the sediment particle distribution density function of the floc at time t k as the long-term distribution density of the floc​ 5. The simulation method for the distribution and gyration radius of sludge flocs based on the Langevin model according to claim 4, wherein Step S5 specifically includes: Analyze the sediment particle content in the sample by using the random sampling method to determine the value of the number M of sediment particles in the flocs, and calculate the approximate value of the gyration radius of the flocs by using the Monte Carlo method: Among them, represents substituting the m-th sediment particle at time t k into the simulation values of the velocity and position of the m-th sediment particle driven by Brownian motion obtained in step S3. K is a positive integer and represents the maximum time node for calculation. φ1(·, ·) and φ2(·, ·) are test functions, and the initial position and the value of the initial velocity x = (P(0), Q(0)) T , and the approximate value of the floc gyration radius obtained is used as the floc gyration radius.

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