A numerical simulation method for population equilibrium of cohesive sediment flocculation based on stochastic method

Abstract

The present invention discloses a method for numerical simulation of cohesive sediment flocculation population equilibrium based on a random method, including (1) generating a quasi-random number sequence; (2) calculating the probability of particle aggregation and fragmentation according to water body environmental conditions and the properties of sediment and biomass particles; (3) selecting aggregation events and fragmentation events according to the number of excessive flocs; (4) using the acceptance-rejection method to test flocs to simulate the floc aggregation and fragmentation process; (5) allocating mass after the event occurs, and calculating the simulation time according to the occurrence rate of the event; (6) repeating steps (3) to (5) to complete the calculation of the entire time period; (7) calculating the particle size distribution and characteristic particle size of the cohesive sediment after flocculation. The present invention simulates the flocculation process between cohesive sediment and biomass based on a random method, calculates the evolution of the particle size distribution and characteristic particle size of the cohesive sediment, solves the problem of cohesive sediment flocculation that is simplified or ignored in the water-sand model, and provides an important technical means for the accurate simulation of cohesive sediment flocculation.

Description

Technical Field

[0001] The invention relates to the technical field of numerical simulation of sediment flocculation, and in particular to a numerical simulation method of viscous sediment flocculation population equilibrium based on a random method. Background Art

[0002] The viscous sediment in the natural water and land environment has a small particle size. Under the influence of complex physical, chemical and biological processes in the water body, it continuously aggregates and breaks, resulting in flocculation. It mainly exists in the water body in the form of biological-mineral flocs, which in turn changes the particle size distribution and sedimentation rate of the viscous sediment. The understanding of the flocculation mechanism of viscous sediment restricts the research on engineering and environmental issues such as channel dredging and pollutant transport.

[0003] How to accurately predict the particle size distribution of flocs is a key problem in revealing their movement characteristics. Current cohesive sediment flocculation models often focus on their mineral composition and physical properties, and lack understanding of the impact of biological flocculation and the simulation of particle size distribution. This approach, to a certain extent, restricts the prediction accuracy and application of large-scale sediment transport models.

[0004] Therefore, in order to more fully consider the flocculation characteristics of cohesive sediment, it is urgent to establish a numerical model that couples the influence of biological particles and includes the flocculant aggregation and fragmentation processes. Summary of the invention

[0005] Purpose of the invention: The present invention aims to provide a numerical simulation method for flocculation population equilibrium of viscous sediment based on a stochastic method to improve the accuracy of simulation results.

[0006] Technical solution: The numerical simulation method of the flocculation population equilibrium of viscous sediment based on a random method of the present invention comprises the following steps:

[0007] (1) Generate a quasi-random number sequence;

[0008] (2) Calculate the probability of particle aggregation and fragmentation based on water environmental conditions, properties of viscous sediment, and properties of biomass particles;

[0009] (3) Selecting coalescence events and breakup events based on the number of excessive flocs;

[0010] (4a) If the coalescence event is selected in step (3), randomly select floc particles i and j, and use the acceptance-rejection method to test their coalescence probability. If the test passes, the two particles coalesce. If not, randomly select again until the coalescence event successfully occurs.

[0011] (4b) If the fragmentation event is selected in step (3), randomly select floc particle i, and use the acceptance-rejection method to test its fragmentation probability. If the test passes, the particle is fragmented. If it fails, randomly select again until the fragmentation event successfully occurs.

[0012] (5) calculating and updating the floc particle properties after step (4a) or (4b), including particle size, fractal dimension, coalescence probability, and breakage probability;

[0013] (6) Calculate the advancement simulation time. The simulation will repeat steps (3) to (6) until the calculation period is completed;

[0014] (7) After the simulation time meets the calculation period, the calculation includes particle size distribution, characteristic particle size, and floc properties of the biological component.

[0015] Furthermore, the steps (2) to (5) are based on a two-variable population balance equation, which is:

[0016]

[0017] Where: V and V b are the floc volume and the biomass volume, n(V,V b ,t) is the volume (V,V b ) of flocs at time t; β(V,V b ; V′, V′ b ) and α(V,V b ; V′, V′ b ) represent the volumes respectively (V, V b ) and (V′,V′ b ) of two flocs; a(V,V b ) is the volume (V,V b ) of floc breakage probability function; b(V,V b |V′,V′ b ) is the volume (V′, V′ b ) is broken into flocs with a volume of (V, V b ) Distribution function of flocs.

[0018] Furthermore, in step (2), the water environment conditions include turbulent shear rate, water density, water dynamic viscosity, and temperature; and the properties of viscous sediment and biomass particles include initial particle size distribution and particle density.

[0019] Furthermore, in step (2), the coalescence probability is the product of the collision frequency and the coalescence efficiency, and the collision frequency takes into account Brownian motion, unequal speed sedimentation, turbulent shearing effect and biological viscosity effect.

[0020] Furthermore, the collision frequency calculation formula is:

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] Where: β ij is the floc collision frequency, β1 and β2 are the particle size-related collision frequency and the biological component-related collision frequency, respectively, and D i and D j are the particle sizes of flocs i and j, respectively, K is the Boltzmann constant, T is the absolute temperature, μ is the dynamic viscosity of the water, G is the turbulent shear rate, g is the gravitational acceleration, ω i and ω j are the settling velocities of particles i and j, and are the biomass components of flocs i and j, respectively, η MB is the parameter affecting the collision frequency due to mineral-biological components.

[0027] Furthermore, in step (3), the particle size of the oversized flocs is set to be proportional to the Kolmogorov microscale, and the effect of biological viscosity on the growth of the maximum particle size of the flocs is considered; the probability of a breakage event is:

[0028]

[0029] Where: n b is the number of oversized flocs; when P frag If it is greater than the generated quasi-random number, the fragmentation event is selected to occur, otherwise the merging event is selected.

[0030] Furthermore, in step (4a), if the ratio of the probability of merging the two selected flocs to the maximum probability of merging among all possible floc pairs is greater than the generated quasi-random number, the two selected flocs are successfully merged, and a floc is randomly copied in the system to fill the particle vacancy caused by the merging, and the number of particles is kept unchanged during the simulation process. Otherwise, a floc pair is randomly searched again until the event successfully occurs; the technical formula for the maximum probability of merging is:

[0031]

[0032] Where: n a and n rare the number of successful and failed merging, respectively.

[0033] Furthermore, in step (4b), if the ratio of the breakage probability of the selected flocs to the maximum breakage probability of all possible flocs is greater than the generated quasi-random number, the selected flocs are successfully broken, and the additional new flocs generated will randomly replace an equal number of flocs in the system, keeping the number of particles unchanged during the simulation. Otherwise, flocs will be randomly searched again until the event successfully occurs.

[0034] Furthermore, the time step is solved and advanced according to the occurrence rate of different events. The calculation formula of the time step is:

[0035]

[0036] Where: Δt is the time step, ij > is the average coalescence probability of all particle pairs, C0 is the initial concentration of particles, and are the average mass of the system before and after the event, is the initial system average mass, ij > is the average fragmentation probability of all particles, and κ is the number of successful events.

[0037] Furthermore, in step (7), if it is necessary to calculate the floc particle size characteristics in the equilibrium state, the results need to be averaged after reaching the pseudo-equilibrium state.

[0038] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: significantly improving the accuracy of simulation results, based on the flocculation mechanism of viscous sediment, introducing the volume of biological components, expanding the traditional flocculation population equilibrium model into a two-variable population equilibrium model, and using a random method to solve it, thereby achieving accurate simulation of the properties of viscous sediment-biological flocs, providing an important technical means for in-depth research on the mineral-biological flocculation mechanism and accurate simulation of viscous sediment movement, and making up for some deficiencies in the previous flocculation model that ignored the biological component. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 is a flow chart of the present invention;

[0040] Figure 2 This is the univariate model validation result of the present invention;

[0041] Figure 3 It is the verification result of the bivariate model of the present invention. DETAILED DESCRIPTION

[0042] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0043] ​​The present invention constructs a mathematical model of mineral-biological viscous sediment flocculation based on the flocculation mechanism of viscous sediment, simulates the particle flocculation process of solid-solid two-phase and solid-liquid two-phase interactions between sediment / biological particles and between flocs and fluid environment, and calculates the evolution of floc properties such as particle size distribution, fractal dimension, and biological components of viscous sediment flocs over time; the mathematical model of mineral-biological viscous sediment flocculation of the present invention, that is, the two-variable population balance equation is as follows:

[0044]

[0045] Where: V and V b are the floc volume and the biomass volume, n(V,V b ,t) is the volume (V,V b ) of flocs at time t; β(V,V b ; V′, V′ b ) and α(V,V b ; V′, V′ b ) represent the volumes respectively (V, V b ) and (V′,V′ b ) of two flocs; a(V,V b ) is the volume (V,V b ) of floc breakage probability function; b(V,V b |V′,V′ b ) is the volume (V′, V′ b ) is broken into flocs with a volume of (V, V b ) The distribution function of the flocs; the first term on the right side of the equation The volume due to aggregation is (V,V b ) of the floc formation rate, the second term The volume due to aggregation is (V,V b ) is the rate at which the flocs decrease, and the third term -n(V,V b ,t)·a(V,V b ) is the volume due to crushing (V, V b ) is the rate at which the flocs decrease, the fourth term The volume due to crushing is (V,V b ) of floc formation rate.

[0046] Specifically, Figure 1 As shown, the numerical simulation method of the viscous sediment flocculation population equilibrium based on the random method of the present invention comprises the following steps:

[0047] (1) Generate a sequence of quasi-random numbers (low variance numbers);

[0048] (2) According to the water environment conditions, including turbulent shear rate, water density ρ w , water body dynamic viscosity, temperature; sediment particle density ρ s , biomass particle properties, including initial particle size distribution, particle density, and calculation of particle aggregation probability and breakage probability;

[0049] Among them, the probability of coalescence is the collision frequency β ij The product of the collision frequency and the aggregation efficiency, the collision frequency takes into account Brownian motion, unequal speed sedimentation, turbulent shear effect and biological viscosity effect, and the aggregation efficiency is generally taken as a constant of 1.

[0050] Specifically, the collision frequency function β ij The calculation formula is:

[0051]

[0052]

[0053]

[0054]

[0055]

[0056] Where: β ij is the floc collision frequency, β1 and β2 are the particle size-related collision frequency and the biological component-related collision frequency, respectively, and D i and D j are the particle sizes of flocs i and j, respectively, K is the Boltzmann constant, T is the absolute temperature, μ is the dynamic viscosity of the water, G is the turbulent shear rate, g is the gravitational acceleration, ω i and ω j are the settling velocities of particles i and j, and are the biomass components of flocs i and j, respectively, η MB is the parameter affecting the collision frequency due to mineral-biological components.

[0057] Among them, the probability function of floc breakage adopts the calculation formula related to its intensity, and the probability function of breakage a i The calculation formula is:

[0058]

[0059] Where: a i is the probability function of breaking, C1 is the correction coefficient, μ is the dynamic viscosity of water, G is the turbulent shear rate, F y is the floc strength, nf is the floc fractal dimension, D iis the particle size of floc i, D p is the initial particle size. After the flocs are broken, a binary breakup distribution is adopted, that is, the flocs are broken into two sub-flocs, and the mass ratio is determined by a quasi-random number.

[0060]

[0061] Where: D f is the floc particle size, and δ is the fitting constant.

[0062] (3) Selecting coalescence events and breakup events based on the number of oversized flocs;

[0063] The particle size of oversized flocs is proportional to the Kolmogorov microscale, considering the effect of biological viscosity on the growth of the maximum particle size of flocs; the probability of a breakup event P frag The calculation formula is:

[0064]

[0065] Where: n b is the number of oversized flocs. frag If it is greater than the generated quasi-random number, the fragmentation event is selected to occur, otherwise the merging event is selected.

[0066] (4) The acceptance-rejection method was used to test the floc collision frequency, aggregation efficiency, and fragmentation probability, and to simulate the floc aggregation and fragmentation process;

[0067] (4a) If the coalescence event is selected in step (3), randomly select floc particles i and j, and use the acceptance-rejection method to test their coalescence probability. If the test passes, the two particles coalesce. If not, randomly select again until the coalescence event successfully occurs.

[0068] Specifically, in the coalescence event, if the ratio of the coalescence probability of the two selected flocs to the maximum coalescence probability of all possible floc pairs is greater than the generated quasi-random number, the two selected flocs are successfully coalesced, and a floc is randomly copied in the system to fill the particle vacancy caused by the coalescence, and the number of particles is kept unchanged during the simulation. Otherwise, floc pairs are randomly searched again until the event successfully occurs. When calculating the maximum probability, the product of the approximate coefficient and the average coalescence probability can be used to approximate the maximum coalescence probability to improve the model operation efficiency;

[0069] The formula for calculating the maximum probability of aggregation is:

[0070]

[0071] Where: n a and n r are the number of successful and failed merging, respectively.

[0072] (4b) If the fragmentation event is selected in step (3), randomly select floc particle i, and use the acceptance-rejection method to test its fragmentation probability. If the test passes, the particle is fragmented. If it fails, randomly select again until the fragmentation event successfully occurs.

[0073] Specifically, in a breakup event, if the ratio of the breakup probability of the selected floc to the maximum breakup probability of all possible flocs is greater than the generated quasi-random number, the selected floc is successfully broken, and the generated additional new flocs will randomly replace an equal number of flocs in the system, keeping the number of particles unchanged during the simulation. Otherwise, flocs will be randomly searched again until the event occurs successfully.

[0074] (5) calculating and updating the floc particle properties after step (4a) or (4b), including particle size, fractal dimension, coalescence probability, and breakage probability;

[0075] (6) Calculate the advancement simulation time. The simulation will repeat steps (3) to (6) until the calculation period is completed;

[0076] Specifically, the simulation time step of each event is implicit, that is, the time step is solved and advanced according to the occurrence rate of different events, and the time is calculated according to the event-driven Monte Carlo method. The time step Δt in the coalescence and fragmentation events is:

[0077]

[0078] Where: Δt is the time step, ij > is the average coalescence probability of all particle pairs, C0 is the initial concentration of particles, and are the average mass of the system before and after the event, is the initial system average mass, ij > is the average fragmentation probability of all particles, and κ is the number of successful events.

[0079] (7) After the simulation time meets the calculation period, the floc properties such as particle size distribution, characteristic particle size, and biological components are calculated.

[0080] Among them, if it is necessary to calculate the floc particle size characteristics under equilibrium, the results need to be averaged after reaching the quasi-equilibrium state to reduce the error caused by the random method.

[0081] Model Validation:

[0082] (1) Univariate model validation (not considering the volume component of biological particles)

[0083] The data in the literature were used for verification. In the experiment, the particle size of the single sediment mineral was 0.1-5 μm, the sediment concentration was 0.5 g / L, and the turbulent shear rate of the water body was G = 5, 10, 20, 40 s​​-1 .

[0084] The initial particle size in the model is the same as the particle size distribution of the sediment mineral particles used in the experiment. The sediment concentration and the turbulent shear rate of the water body are consistent with the experimental conditions. The particle size distribution and median particle size of the sediment flocs are used for verification. The relevant parameter values ​​used are shown in Table 1.

[0085] Table 1 Parameters related to verification of single variable simulation example

[0086]

[0087] The verified particle size distribution and median particle size results are as follows Figure 2 As shown, this model can better simulate the changes in floc particle size distribution and median particle size during flocculation under single variable (pure sediment) conditions.

[0088] (2) Bivariate model validation (considering the volume component of biological particles)

[0089] The data of the microalgae-kaolin test were used for verification. The microalgae used in the test was Cyclotella meni, the particle size of the kaolin mineral was 4 μm, the particle size of the microalgae cell was 13 μm, the sediment concentration was 0.1 g / L, and the microalgae concentration was 3×10 9 cells / m 3 , the turbulent shear rate of water is G = 40, 60, 80s -1 .

[0090] The environmental conditions and particle properties used in the model are consistent with the experimental conditions. The oversized floc size is proportional to the Kolmogorov microscale, and the proportionality coefficient is 3.1. The particle size distribution and median particle size of the sediment flocs are used for verification. The relevant parameter values ​​used are shown in Table 2.

[0091] Table 2. Related parameters for verification of dual variable simulation examples

[0092]

[0093] The particle size distribution and median particle size results of the bivariate model validation are shown in Figure 2. Figure 3 As shown, this model can also simulate the changes of floc particle size distribution and median particle size during flocculation under the dual variable (silt + microalgae) condition.

Claims

1. A numerical simulation method for the flocculation equilibrium of viscous sediment based on a stochastic method, characterized in that: The numerical simulation method comprises the following steps: (1) Generate a quasi-random number sequence; (2) The probability of particle aggregation and fragmentation is calculated based on the water environment conditions, the properties of viscous sediment and the properties of biological particles. The probability of aggregation is the product of the collision frequency and the aggregation efficiency. The collision frequency takes into account Brownian motion, unequal velocity sedimentation, turbulent shear and biological viscosity. The collision frequency calculation formula is: Where: β ij is the floc collision frequency, β1 and β2 are the particle size-related collision frequency and the biological component-related collision frequency, respectively, and D i and D j are the particle sizes of flocs i and j, respectively, K is the Boltzmann constant, T is the absolute temperature, μ is the dynamic viscosity of the water, G is the turbulent shear rate, g is the gravitational acceleration, ω i and ω j are the settling velocities of particles i and j, and are the biomass components of flocs i and j, respectively, η MB is the parameter affecting the collision frequency by mineral-biological components; (3) Selecting coalescence events and breakup events based on the number of oversized flocs; (4a) If the coalescence event is selected in step (3), randomly select floc particles i and j, and use the acceptance-rejection method to test their coalescence probability. If the test passes, the two particles coalesce. If not, randomly select again until the coalescence event successfully occurs. (4b) If the fragmentation event is selected in step (3), randomly select floc particle i, and use the acceptance-rejection method to test its fragmentation probability. If the test passes, the particle is fragmented. If it fails, randomly select again until the fragmentation event successfully occurs. (5) calculating and updating the floc particle properties after step (4a) or (4b), including particle size, fractal dimension, coalescence probability, and breakage probability; Wherein, the steps (2) to (5) are based on a two-variable population balance equation, which is: Where: V and V b are the floc volume and the biomass volume, n(V,V b ,t) is the volume (V,V b ) of flocs at time t; β(V,V b ; V′, V′ b ) and α(V,V b ; V′, V′ b ) represent the volumes respectively (V, V b ) and (V′,V′ b ) of two flocs; a(V,V b ) is the volume (V,V b ) of floc breakage probability function; b(V,V b |V′,V′ b ) is the volume (V′, V′ b ) is broken into flocs with a volume of (V, V b ) Distribution function of flocs; (6) Calculate the advancement simulation time. The simulation will repeat steps (3) to (6) until the calculation period is completed; (7) After the simulation time meets the calculation period, the calculation includes particle size distribution, characteristic particle size, and floc properties of the biological component.

2. The method for numerical simulation of flocculation population equilibrium of viscous sediment based on stochastic method according to claim 1 is characterized in that: In the step (2), the water environment conditions include turbulent shear rate, water density, water dynamic viscosity, and temperature; the properties of viscous sediment and biological particles include initial particle size distribution and particle density.

3. The method for numerical simulation of flocculation population equilibrium of viscous sediment based on stochastic method according to claim 1 is characterized in that: In the step (3), the particle size of the oversized flocs is set to be proportional to the Kolmogorov microscale, and the effect of biological viscosity on the growth of the maximum particle size of the flocs is considered; the probability of the breakage event is: Where: n b is the number of oversized flocs; when P frag If it is greater than the generated quasi-random number, the fragmentation event is selected to occur, otherwise the merging event is selected.

4. The method for numerical simulation of flocculation population equilibrium of viscous sediment based on stochastic method according to claim 1 is characterized in that: In the step (4a), if the ratio of the probability of merging the two selected flocs to the maximum probability of merging among all possible floc pairs is greater than the generated quasi-random number, the two selected flocs are successfully merged, and a floc is randomly copied in the system to fill the particle vacancies caused by the merging, and the number of particles is kept unchanged during the simulation process. Otherwise, a floc pair is randomly searched again until the event successfully occurs; the technical formula for the maximum probability of merging is: Where: n a and n r are the number of successful and failed merging, respectively.

5. The method for numerical simulation of flocculation population equilibrium of viscous sediment based on stochastic method according to claim 1, characterized in that: In the step (4b), if the ratio of the probability of breaking of the selected flocs to the maximum probability of breaking of all possible flocs is greater than the generated quasi-random number, the selected flocs are successfully broken, and the additional new flocs generated will randomly replace an equal number of flocs in the system, keeping the number of particles unchanged during the simulation. Otherwise, flocs will be randomly searched again until the event successfully occurs.

6. The method for numerical simulation of flocculation population equilibrium of viscous sediment based on stochastic method according to claim 1, characterized in that: The time step is solved and advanced according to the occurrence rate of different events. The calculation formula of the time step is: Where: Δt is the time step, ij > is the average coalescence probability of all particle pairs, C0 is the initial concentration of particles, and are the average mass of the system before and after the event, is the initial system average mass, ij > is the average fragmentation probability of all particles, and κ is the number of successful events.​​ 7. The method for numerical simulation of flocculation population equilibrium of viscous sediment based on stochastic method according to claim 1, characterized in that: In the step (7), if it is necessary to calculate the floc particle size characteristics in the equilibrium state, the results need to be averaged after reaching the pseudo-equilibrium state.

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