A discrete solution method for particle coalescence and breakage process population balance

Abstract

A discrete solution method for particle number balance in the process of particle coalescence and breakage is provided, total extinction items and modified new generation items on the characteristic points of discrete intervals are obtained, then the particle number distribution of the updated discrete interval is obtained, finally the particle number density and order particle number moment at the moment are obtained, and the discrete solution for particle number balance in the process of particle coalescence and breakage is completed. By discretizing the particle size into several intervals, multiple characteristic points are set in each interval to reconstruct the particle size distribution and particle number moment, that is, to simultaneously predict the particle size distribution and particle number moment; the new generation items of each characteristic point are distributed according to the new generation particle moment, so as to ensure the conservation of high-order moment; the particle number is updated through the relative distance and threshold value in the discrete interval, so as to ensure the physical validity of the particle number.

Description

Technical Field

[0001] This invention relates to the field of particle process simulation technology, and in particular to a discrete solution method for particle number balance calculation in particle agglomeration and crushing processes. Background Technology

[0002] Particle agglomeration and crushing processes are widely present in engineering fields such as granulation, crystallization, crushing, precipitation, and aerosol processing. In these applications, the particle number balance equation, as the core mathematical model describing the evolution of particle size distribution over time in a particle system, is fundamental to simulating particle processes. However, the agglomeration and crushing terms in the particle number balance equation cause it to exhibit partial integral-differential characteristics, making it difficult to solve analytically. Therefore, numerically solving the particle number balance equation is crucial for modeling the mechanisms of particle systems.

[0003] Discrete algorithms are the mainstream numerical methods for solving particle number balance equations. They mainly rely on the mean value theorem to concentrate particles in an interval onto a single feature point, thereby transforming the particle number balance equation into a series of easily solvable differential equations. In recent years, the cell averaging method (Kumar, Peglow et al. (2008)) has been widely used. It can predict particle distribution under any discrete grid and ensure the conservation of the total number and total mass of particles.

[0004] Currently, discrete algorithms mainly focus on particle size distribution and can only reliably provide the zeroth and first moments of the particle number distribution. They cannot accurately obtain higher-order moments and related key particle properties, such as weighted average particle size and particle size distribution variation coefficient. Summary of the Invention

[0005] This invention discloses a discrete solution method for particle number balance calculation in particle agglomeration and crushing processes, in order to overcome the above-mentioned technical problems.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows:

[0007] A discrete solution method for particle number balance calculation in particle agglomeration and crushing processes includes the following steps:

[0008] S1: Based on the initial particle number density, the range of the particles is divided into multiple discrete intervals; wherein, the discrete intervals include multiple feature points;

[0009] S2: Obtain the number of eliminated particles at the feature point at the current sampling time under the action of aggregation or fragmentation, so as to obtain the total elimination term at the feature point in the discrete interval;

[0010] S3: Obtain the newly formed particles within the discrete interval at the current sampling time under the influence of aggregation or fragmentation. r The order of grain moments is used to obtain the new terms assigned to the feature points in the discrete interval;

[0011] S4: Based on the total disappearance term and the new generation term at the feature points of the discrete interval, obtain the rate of change of the number of particles at the feature points over time, or the rate of change of the number of particles within the discrete interval. The rate of change of the order particle number moment over time is used to obtain the particle number distribution in the discrete interval at the next sampling time.

[0012] S5: Based on the particle number distribution in the discrete interval at the next sampling time, obtain the particle number density and total number of particles in the discrete interval at the next sampling time. The number of particles is determined by discrete solutions to balance the number of particles during the agglomeration and crushing process.

[0013] Furthermore, S2 includes:

[0014] S21: The formula used to obtain the number of extinct particles at a feature point under the effect of aggregation is as follows:

[0015]

[0016] In the formula: For the first under the effect of aggregation The first discrete interval The number of extinct particles at each feature point; For the first Feature point index of a discrete interval; The total number of feature points within the discrete interval; Index the feature points in all discrete intervals; For the first The first discrete interval Feature points With all discrete intervals the first Feature points The probability of particles merging together; For all discrete intervals, the first... One feature point; For the first The first discrete interval One feature point; For the first The first discrete interval The number of particles corresponding to each feature point; For all discrete intervals, the first... The number of particles corresponding to each feature point; Let be the particle aggregation probability function;

[0017] S22: Obtain the number of destroyed particles at a feature point under the action of fragmentation;

[0018]

[0019] In the formula: For the first under the action of crushing The first discrete interval The number of extinct particles at each feature point; For the first The first discrete interval Feature points The probability of the corresponding particles breaking; Let the particle breakage probability function be used.

[0020] S23: Obtain the total disappearance term at the feature point:

[0021] In the process of pure aggregation, the first The first discrete interval The total number of terms that disappear at each feature point is: ;

[0022] In the pure crushing process, the first The first discrete interval The total number of terms that disappear at each feature point is: ;

[0023] During the coupling process of aggregation and breakup, the first The first discrete interval The total number of terms that disappear at each feature point is: ;

[0024] In the formula: For the first The first discrete interval The total extinction term at the nth feature point, i.e. the nth feature point The first discrete interval The total number of extinct particles at each feature point.

[0025] Furthermore, S3 includes:

[0026] S31: Obtain newly formed particles within a discrete interval under the effect of aggregation. r The number of grains in order;

[0027]

[0028] in,

[0029]

[0030] In the formula: For the first under the effect of aggregation Newly formed particles in discrete intervals The number of grains in order; , 2 satisfies this condition in all discrete intervals. Feature point index, ; The first of all discrete intervals 1 feature point and One feature point; For the first The lower bound of a discrete interval; For the first The upper bound of a discrete interval; It is the Dirac function; The first of all discrete intervals The number of particles corresponding to each feature point and The number of particles corresponding to each feature point;

[0031] S32: Obtain newly formed particles within a discrete interval under crushing action. r The number of grains in order;

[0032]

[0033]

[0034] In the formula: For the first under the action of crushing Newly formed particles in discrete intervals The number of grains in order; 3 satisfies the following conditions in all discrete intervals. Feature point index; For all discrete intervals, the first... 3 feature points; For all discrete intervals, the first... The number of particles corresponding to the three feature points; These are intermediate calculation parameters; For the first Feature points After the particles on the surface are broken, the resulting particle size is The density of sub-particles; Particle size; Let f(x) be the probability density function for breakage.

[0035] S33: Obtain the total number of newly formed particles at the feature points. r The number of grains in order;

[0036] The method for obtaining the total number of newly formed particles at feature points is as follows:

[0037] During pure agglomeration, the total number of newly formed particles is: ;

[0038] During the pure crushing process, the total number of newly formed particles is: ;

[0039] During the aggregation and breakup coupling process, the total number of newly formed particles is: ;

[0040] In the formula: For the first Total number of newly generated particles in a discrete interval The number of grains in order;

[0041] S34: Based on the total number of newly formed particles at the feature points r The number of particles is used to obtain the average particle size of newly formed particles. and the average representative particle size of existing particles The formula used is as follows:

[0042]

[0043]

[0044] In the formula: For the first The average particle size of newly formed particles in each discrete interval; For the first The first-order particle number moment of the total number of newly formed particles in a discrete interval; For the first The zero-order particle number moment of the total number of newly formed particles in a discrete interval; For the first The average representative particle size of existing particles in a discrete interval; For the first The first discrete interval One feature point; For the first The first discrete interval The number of particles corresponding to each feature point;

[0045] S35: Based on the total number of newly formed particles at the feature point r The number of particles in order, combined with the average particle size of newly formed particles. and the average representative particle size of existing particles The number of newly generated particles assigned to feature points is obtained, which is the number of newly generated items assigned to feature points in the discrete interval.

[0046] Furthermore, the method for obtaining the number of newly generated particles assigned to feature points is as follows:

[0047] when At that time, the newly generated particles are allocated to the first... Discrete intervals:

[0048]

[0049] In the formula: To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The number of newly formed particles at the first feature point of each discrete interval; To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The discrete interval of the nth time The number of newly formed particles at each feature point; This represents the transpose of the Vandermonde matrix with respect to a vector; T For transpose; For the first The total number of newly formed particles in a discrete interval Step moment;

[0050] when At that time, the newly generated particles are allocated to the first... ~ Discrete intervals:

[0051]

[0052] In the formula: To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The number of newly formed particles at the first feature point of each discrete interval; To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The discrete interval of the nth time The number of newly formed particles at each feature point.

[0053] Furthermore, S4 includes:

[0054] S41: Obtain the... Relative distance within discrete intervals and threshold :

[0055]

[0056]

[0057] In the formula: For the first The average value of existing particles within a discrete interval represents the relative distance between the particle size and the midpoint of that interval; Indicates the first The center particle size of each discrete interval; For the first The average representative particle size of existing particles in a discrete interval; Relative distance The threshold; express The total number of discrete intervals;

[0058] S42: Obtain the... The total correction term for each feature point in a discrete interval includes:

[0059] First, when If at that time, then ,make If it is zero, ,make It is zero;

[0060] Then, the formula used to obtain the total corrected neologism at the feature points is as follows:

[0061]

[0062] In the formula: For the first In the nth discrete interval Total modified neologisms at each feature point; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; Represents the Heaviside step function; For the first The average particle size of newly formed particles in each discrete interval; For the first The average representative particle size of existing particles in a discrete interval; For the first The average particle size of newly formed particles in each discrete interval; For the first The average representative particle size of existing particles in a discrete interval;

[0063] S43: According to the... In the nth discrete interval The total revised term at each feature point is used to obtain the updated particle number distribution in the discrete interval, including:

[0064] when At that time, the rate of change of the number of particles at the feature point over time is obtained to obtain... The moment The first discrete interval Number of particles corresponding to each feature point ;

[0065] The formula used to obtain the rate of change of the number of particles at a feature point over time is as follows:

[0066]

[0067] In the formula: For the first In the nth discrete interval Total modified neologisms at each feature point; For the first The total number of eliminated terms at each feature point; t For time; For the first The first discrete interval The number of particles corresponding to each feature point;

[0068] when At that time, obtain the first Within each discrete interval The rate of change of the number of particles over time is used to obtain The moment discrete intervals Order number moment to obtain The moment The first discrete interval Feature points and the corresponding number of particles ;

[0069] Among them, obtaining the first Within each discrete interval The formula used for the rate of change of the number of particles over time is as follows:

[0070]

[0071] In the formula: For the first discrete intervals The number of grains in order; For the first Total number of newly generated particles in a discrete interval Step moment;

[0072] Get The moment The first discrete interval Feature points and the corresponding number of particles The formula used is as follows:

[0073] .

[0074] Furthermore, the particle number density and total number of particles in the discrete interval at the next sampling time are obtained. The formula used for the number of particles per order is as follows:

[0075]

[0076]

[0077] In the formula: For the first The average representative particle size of existing particles in each discrete interval The corresponding particle number density; For the total The number of grains in order; For the index of the discrete interval; The total number of discrete intervals; The total number of feature points in the discrete interval; For the first Feature point index of the interval; Index the feature points for all discrete intervals.

[0078] Beneficial effects: The present invention provides a discrete solution method for particle number balance calculation in particle agglomeration and crushing processes. It obtains the total extinction term and the corrected new generation term at the feature points of the discrete interval at the current sampling time, thereby obtaining the updated particle number distribution in the discrete interval, and finally obtaining the particle number density at the next sampling time. The higher-order moment is used to perform discrete solutions for particle number balance calculations during particle agglomeration and crushing processes. By discretizing particle size into several intervals, with multiple feature points set in each interval, the particle size distribution and particle number moments are reconstructed, i.e., particle size distribution and particle number moments are predicted simultaneously. New terms are allocated to each feature point based on the new particle moment to ensure the conservation of higher-order moments. The particle number is updated using relative distances and thresholds within the discrete intervals to ensure the physical validity of the particle count. Attached Figure Description

[0079] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0080] Figure 1 This is a flowchart of the discrete solution method for particle number balance calculation in particle agglomeration and crushing process according to the present invention.

[0081] Figure 2 This is a schematic diagram of the discrete solution method for particle number balance calculation in the particle agglomeration and crushing process in an embodiment of the present invention.

[0082] Figure 3 This is a schematic diagram illustrating the principle of new particle allocation in an embodiment of the present invention;

[0083] Figure 4 This is a schematic diagram of the cloud droplet size distribution at the final moment in an embodiment of the present invention;

[0084] Figure 5 This is a schematic diagram of moment prediction error in an embodiment of the present invention;

[0085] Figure 6 This is a schematic diagram of the floc particle size distribution curves at different times in an embodiment of the present invention;

[0086] Figure 7 This is a graph showing the variation of the average particle size D43 of the flocs in an embodiment of the present invention. Detailed Implementation

[0087] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0088] This embodiment introduces a discrete solution method for particle number balance calculation in particle agglomeration and crushing processes, including the following steps: Figure 1 and Figure 2 As shown:

[0089] S1: Determine the sampling time interval Based on the initial particle number density at the current sampling time, the range where the particles are located is divided into... Each discrete interval is initialized with a set of discrete intervals, wherein each initialized discrete interval includes... p One feature point;

[0090] Specifically, determine the prediction time. and sampling time interval And, based on the initial particle number density and grid division at the current sampling time, initialize the feature points and particle number of the discrete interval, including:

[0091] (1) Divide the range of the particles There are discrete intervals, and within each discrete interval, set In this embodiment, there are 10 feature points. Choose 2 or 3.

[0092] (2) Calculate the number of particles based on the initial particle number density, the number of characteristic points, and the interval boundaries. discrete intervals Order number moment And solve using the product difference method. Determine feature points and corresponding number of particles ,in , , ; For the index of the discrete interval; The total number of discrete intervals; The order of the number of particles; For the first Feature point index of a discrete interval; For the first The total number of feature points in each discrete interval is set to be the same in this embodiment. For the first The first discrete interval 1 feature point; For the first The first discrete interval The number of particles corresponding to each feature point;

[0093] S2: Obtain the number of eliminated particles at the feature point at the current sampling time under the action of aggregation or fragmentation, so as to obtain the total elimination term at the feature point in the discrete interval;

[0094] Specifically, the particle number density is expressed in Dirac delta. Substituting the functional form into the particle number balance equation, the particle extinction term is numerically discretized, and the calculation is performed. The particle extinction term at any given moment.

[0095] S21: The formula used to obtain the number of extinct particles at a feature point under the effect of aggregation is as follows:

[0096] (1)

[0097] In the formula: For the first under the effect of aggregation The first discrete interval The number of extinct particles at each feature point; For the first Feature point index of a discrete interval; The total number of feature points within the discrete interval; Index the feature points in all discrete intervals; For the first The first discrete interval Feature points With all discrete intervals the first Feature points The probability of particles merging together; For all discrete intervals, the first... One feature point; For the first The first discrete interval One feature point; For the first The first discrete interval The number of particles corresponding to each feature point; For all discrete intervals, the first... The number of particles corresponding to each feature point; Let be the particle aggregation probability function, which describes the probability of particles merging together.

[0098] S22: Obtain the number of destroyed particles at a feature point under the action of fragmentation;

[0099] (2)

[0100] In the formula: For the first under the action of crushing The first discrete interval The number of extinct particles at each feature point; For the first The first discrete interval Feature points The probability of the corresponding particles breaking; Let be the particle breakage probability function, describing the probability that the particles will break.

[0101] S23: Obtain the total disappearance term at the feature point:

[0102] In the process of pure aggregation, the first The first discrete interval The total number of terms that disappear at each feature point is: ;

[0103] In the pure crushing process, the first The first discrete interval The total number of terms that disappear at each feature point is: ;

[0104] During the coupling process of aggregation and breakup, the first The first discrete interval The total number of terms that disappear at each feature point is: ;

[0105] In the formula: For the first The first discrete interval The total extinction term at the nth feature point, i.e. the nth feature point The first discrete interval The total number of extinct particles at each feature point.

[0106] S3: Obtain the newly formed particles within the discrete interval at the current sampling time under the influence of aggregation or fragmentation. r The order of grain moments is used to obtain the new terms assigned to the feature points in the discrete interval;

[0107] Specifically, the particle number density is expressed as a Dirac function and substituted into the particle number balance equation. The particle generation term is then numerically discretized, and the calculation is performed. At this moment Newly formed particles within a discrete interval The number of new particles is calculated by using a new particle redistribution strategy to determine the number of new particles distributed to each feature point.

[0108] Preferably, S3 includes:

[0109] S31: Obtain newly formed particles within a discrete interval under the effect of aggregation. r The number of grains in order;

[0110] (3)

[0111] in,

[0112]

[0113] In the formula: For the first under the effect of aggregation Newly formed particles in discrete intervals The number of grains in order; , 2 satisfies this condition in all discrete intervals. Feature point index, ; The first of all discrete intervals Feature points and One feature point; For the first The lower bound of a discrete interval; For the first The upper bound of a discrete interval; It is the Dirac function; The first of all discrete intervals The number of particles corresponding to each feature point and The number of particles corresponding to each feature point;

[0114] S32: Obtain the r-order particle number moment of newly formed particles in the discrete interval under crushing action;

[0115] (4)

[0116]

[0117] In the formula: For the first under the action of crushing Newly formed particles in discrete intervals The number of grains in order; 3 satisfies the following conditions in all discrete intervals. Feature point index; For all discrete intervals, the first... 3 feature points; For all discrete intervals, the first... The number of particles corresponding to the three feature points; These are intermediate calculation parameters; For the first Feature points After the particles on the surface are broken, the resulting particles have a diameter of [missing information]. The density of sub-particles; Particle size; Let f be the breakage probability density function, which describes the density of sub-particles produced after a particle breaks.

[0118] S33: Obtain the total number of newly formed particles at the feature points. r The number of grains in order;

[0119] The method for obtaining the total number of newly formed particles at feature points is as follows:

[0120] During pure aggregation, the total number of newly formed particles is: ;

[0121] During pure crushing, the total number of newly formed particles is: ;

[0122] During the aggregation and breakup coupling process, the total number of newly formed particles is: ;

[0123] In the formula: For the first Total number of newly generated particles in a discrete interval Step moment;

[0124] S34: Based on the total number of newly formed particles at the feature points r The number moment of new particles, specifically, is the average particle size of new particles obtained from the zero-order and first-order number moments of the total number of new particles. and the average representative particle size of existing particles The formula used is as follows:

[0125]

[0126] (5)

[0127] In the formula: For the first The average particle size of newly formed particles in each discrete interval; For the first The first-order particle number moment of the total number of new particles in a discrete interval; For the first The zero-order particle number moment of the total number of newly formed particles in a discrete interval; For the first The average representative particle size of existing particles in a discrete interval; For the first The first discrete interval One feature point; For the first The first discrete interval The number of particles corresponding to each feature point.

[0128] S35: Based on the total number of newly formed particles at the feature point r The number of particles is determined by the average particle size of the newly formed particles. and the average representative particle size of existing particles To obtain the number of newly generated particles assigned to feature points, i.e., the number of newly generated items assigned to feature points in a discrete interval;

[0129] Specifically, through comparison and The newly generated particles are assigned to adjacent feature points, such as... Figure 3 As shown:

[0130] ①When At that time, the newly formed particles at the feature points Moments assigned to the first The system calculates the number of newly formed particles assigned to each feature point within a discrete interval. :

[0131] (6)

[0132] In the formula: To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The number of newly formed particles at the first feature point of each discrete interval; To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The discrete interval of the nth time The number of newly formed particles at each feature point; This represents the transpose of the Vandermonde matrix with respect to a vector; T For transpose; For the first The total number of newly formed particles in a discrete interval Step moment;

[0133] ②When At that time, the newly generated particle item is assigned to the first... ~ The system calculates the number of particles assigned to each feature point within a discrete interval. :

[0134] (7)

[0135] In the formula: To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The number of newly formed particles at the first feature point of each discrete interval; To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The discrete interval of the nth time The number of newly formed particles at each feature point.

[0136] S4: Based on the total disappearance term and the new generation term at the feature points of the discrete interval, obtain the rate of change of the number of particles at the feature points over time, or the rate of change of the number of particles within the discrete interval. The rate of change of the order particle number moment over time is used to obtain the particle number distribution in the discrete interval at the next sampling time.

[0137] Specifically, to ensure the non-negativity of the particle number, the relative distance between the average representative particle size of existing particles within the discrete interval and the midpoint of the interval is considered. The threshold is used to determine and update the particle number distribution.

[0138] S41: Obtain the... Relative distance within discrete intervals and threshold :

[0139]

[0140] (8)

[0141] In the formula: For the first The average value of existing particles within a discrete interval represents the relative distance between the particle size and the midpoint of that interval; Indicates the first The center particle size of each discrete interval; For the first The average representative particle size of existing particles in a discrete interval; Relative distance The threshold; express The total number of discrete intervals;

[0142] S42: Obtain the... In the nth discrete interval The total modified neologisms at each feature point include:

[0143] Relative distance within discrete intervals Threshold determination, comparing in each discrete interval. and ,like Then when season When it is zero, season The result is zero; based on the corrected neologisms, the total corrected neologisms at each feature point are obtained as follows:

[0144] (9)

[0145] In the formula: For the first In the nth discrete interval Total modified neologisms at each feature point; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; Represents the Heaviside step function; For the first The average particle size of newly formed particles in each discrete interval; For the first The average representative particle size of existing particles in a discrete interval; For the first The average particle size of newly formed particles in each discrete interval; For the first The average representative particle size of existing particles in a discrete interval.

[0146] S43: According to the... In the nth discrete interval The total correction term at each feature point, combined with the relative distance within the discrete interval. The threshold determination is used to obtain the updated particle number distribution in the discrete interval, including:

[0147] Specifically, to ensure the non-negativity of the particle number, according to the first... In the nth discrete interval The correction term at each feature point, combined with the relative distance between the average representative particle size of existing particles within the discrete interval and the midpoint of the interval, is used. The threshold judgment, switching between different strategies (i.e., nonnegativity test and switching strategy) to obtain the updated particle number distribution in the discrete interval, including:

[0148] when At that time, obtain the first The first discrete interval Feature points The rate of change of the number of particles over time is used to obtain The moment The first discrete interval Number of particles corresponding to each feature point ( t + h ); whereby, feature points are obtained. The formula used to calculate the rate of change of the number of particles over time is as follows:

[0149] (10)

[0150] In the formula: For the first In the nth discrete interval Total modified neologisms at each feature point; For the first The total number of eliminated terms at each feature point; t For time; For the first The first discrete interval The number of particles corresponding to each feature point;

[0151] Specifically, the above differential equations are solved using classical numerical methods such as the Runge-Kutta method and the Euler method, and the calculations are performed. Number of particles at time .

[0152] when At that time, obtain the first Within each discrete interval The rate of change of the number of particles over time is used to obtain The moment discrete intervals Order number moment to obtain The moment The first discrete interval Feature points and the corresponding number of particles ;

[0153] Among them, obtaining the first Within each discrete interval The formula used for the rate of change of the number of particles over time is as follows:

[0154] (11)

[0155] In the formula: For the first discrete intervals The number of grains in order; For the first Total number of newly generated particles in a discrete interval Step moment.

[0156] Get The moment The first discrete interval Feature points and the corresponding number of particles The formula used is as follows:

[0157] (12)

[0158] Specifically, by solving equation (11) using the Runge-Kutta method, Euler method, etc., the calculation is performed. At present Order number moment By combining the product difference method to solve equation (12), the feature points are updated. and corresponding number of particles .

[0159] S5: Based on the particle number distribution in the discrete interval at the next sampling time, obtain the particle number density and total number of particles in the discrete interval at the next sampling time. The number of particles is determined by a discrete solution for the particle number balance in the particle agglomeration and crushing process.

[0160] (13)

[0161] (14)

[0162] in,

[0163] ,

[0164] like Then let If the calculation fails, return to S2 and continue the calculation; otherwise, stop and output the particle size distribution and particle number moments at each sampling time.

[0165] In the formula: For the first The average representative particle size of existing particles in each discrete interval The corresponding particle number density; For the total The number of grains in order; For the index of the discrete interval; The total number of discrete intervals; The total number of feature points in the discrete interval; For the first Feature point index of the interval; Index the feature points for all discrete intervals.

[0166] Example 1: Cloud Droplet Coalescence Process

[0167] Within clouds, smaller water droplets, under the influence of updrafts, collide with other droplets and merge into larger ones. This coalescence process plays a crucial role in cloud droplet growth. If we neglect the competitive effect of droplets on water vapor, the evolution of the cloud droplet spectrum over time can be described by the following droplet number balance equation:

[0168] (15)

[0169] In the formula, Indicates time, Indicates the particle size of cloud droplets, i.e. , Indicates the radius of the cloud droplet; This represents the aggregation rate. Based on known experiments in the field (Shafrir and Neiburger (1964)), , . Let be the particle agglomeration probability function, describing the particle size as With particle size The probability of particles merging together. for One of the independent variables of the function; s is the time unit, second. is a constant parameter in the clustering probability function.

[0170] Initially, there were 10 per cubic centimeter of air. -6 The total water content is 1 cubic centimeter of water, with an average cloud droplet radius of 10 μm. Average cloud droplet volume , Let be the number of cloud droplets. The corresponding cloud droplet number density is: In this embodiment, the particle number balance equation has an analytical solution:

[0171] (16)

[0172] In the formula, , ; Intermediate variables introduced to simplify expressions; precise values ​​of each moment. This can be achieved by modifying equation (16). Analytical integration yields a solution that can be obtained analytically. The corresponding prediction error is obtained as follows: . for The precise value of the step moment; The algorithm estimates the results. The number of grains in order; To take the absolute value;

[0173] In this embodiment, the prediction time Sampling interval At the final moment, the degree of cohesion Therefore, the predicted time is reasonable. The degree of aggregation; This is the exact value of the zeroth moment;

[0174] The equivalent particle size coordinates of cloud droplets are discretized as follows: Given an interval, the minimum point of the interval. , and ( ); each interval is set 1 feature point. For the first The lower bound of the interval, For the first The upper bound of the interval;

[0175] Figure 4 The cloud droplet spectra at the final moment are presented. As shown in the figure, the prediction results agree well with the exact solution.

[0176] Figure 5 The prediction errors for particle number moments from zero to fifth order are given, showing that the maximum prediction error does not exceed 0.6 when the aggregation degree reaches 0.6. (Right now The prediction error is much smaller than the relative error generally allowed in the industry. Therefore, this method can accurately solve the particle number balance equation, precisely predict the particle size distribution and higher-order particle number moments of cloud droplets, which helps improve the accuracy of cloud microphysical process simulation and provides scientific basis and technical support for climate simulation, weather forecasting, etc.

[0177] Example 2: Activated sludge flocculation process

[0178] Activated sludge can generally be considered as being composed of flocs of different particle sizes. During the flocculation process, the flocs aggregate and break up. The evolution of the floc particle size distribution over time can be represented by the following particle number balance equation:

[0179]

[0180] In the formula, Indicates time, Indicates the particle size of the flocculents. Represents the probability of clustering. / Let be the probability function for particle breakage, describing the particle size as / The probability of particles breaking. and Let be the independent variable of the function. Let floc breakage probability density function be used to describe particle size. After the flocs are broken, the resulting particles have a diameter of The density of flocs, and These are the two independent variables of the function. It is usually assumed that the flocs are uniformly broken into two sub-flocs, i.e. .

[0181] Based on known experiments in this field (Nopens, Biggs et al. (2002)), and , and , in In this embodiment, the average shear rate is represented. .

[0182] Initially, the average particle size of the activated sludge flocs was 5.5 μm. Based on the sludge mass in the suspension, the total number of flocs was calculated to be... The initial particle size distribution is as follows: ,in , .

[0183] In this embodiment, the prediction time Sampling interval The particle size essentially reaches a steady state before the final moment, therefore the prediction time is reasonable.

[0184] Floc particle size coordinates discretized as Given an interval, the minimum point of the interval. , and ( ); each interval is set 1 feature point.

[0185] Figure 6 and Figure 7 The floc particle size distribution and mass-average particle size D43 at different time points were predicted, among which... .Depend on Figure 6 It can be seen that the floc particle size gradually shifts towards larger sizes, indicating that flocculation is occurring; this shift almost reaches a steady state near the end of the experiment. This trend can also be clearly seen from the change in the mass-average particle size D43 over time, such as... Figure 7 As shown. Furthermore, the prediction results of this embodiment are in good agreement with the reference values, while the prediction error of the traditional discrete method (cell averaging method) increases significantly over time, demonstrating that this embodiment can accurately predict higher-order particle moments.

[0186] In summary, this embodiment can accurately solve the particle number balance equation, precisely predict the particle size distribution and higher-order particle number moments of sludge flocs, which helps improve the accuracy of the simulation of activated sludge flocculation process and provides key technical support for improving wastewater treatment efficiency.

[0187] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A discrete solution method for particle number balance calculation in particle agglomeration and crushing processes, characterized in that, Includes the following steps: S1: Based on the initial particle number density, the range of the particles is divided into multiple discrete intervals; wherein, the discrete intervals include multiple feature points; the particles are cloud droplets or flocculent particles; S2: Obtain the number of eliminated particles at the feature point at the current sampling time under the action of aggregation or fragmentation, so as to obtain the total elimination term at the feature point in the discrete interval; including: S21: Obtain the number of annihilated particles at a feature point under the effect of aggregation; S22: Obtain the number of destroyed particles at a feature point under the action of fragmentation; S23: Obtain the total extinction term at the feature point; S3: Obtain the newly formed particles within the discrete interval at the current sampling time under the influence of aggregation or fragmentation. r The order grain number moment is used to obtain the neologisms assigned to feature points in a discrete interval; including: S31: Obtain newly formed particles within a discrete interval under the effect of aggregation. r The number of grains in order; S32: Obtain newly formed particles within a discrete interval under crushing action. r The number of grains in order; S33: Obtain the total number of newly formed particles at the feature points. r The number of grains in order; S34: Based on the total number of newly formed particles at the feature points r The order particle number moment is used to obtain the average particle size of newly formed particles and the average representative particle size of existing particles. S35: Based on the total number of newly formed particles at the feature point r The number of new particles is obtained by combining the average particle size of new particles and the average representative particle size of existing particles to obtain the number of new particles assigned to the feature points, i.e., the new term assigned to the feature points of the discrete interval. S4: Based on the total disappearance term and the new generation term at the feature points of the discrete interval, obtain the rate of change of the number of particles at the feature points over time, or the rate of change of the number of particles within the discrete interval. The rate of change of the order particle number moment over time is used to obtain the particle number distribution in the discrete interval at the next sampling time. S5: Based on the particle number distribution in the discrete interval at the next sampling time, obtain the particle number density and total number of particles in the discrete interval at the next sampling time. The number of particles is determined by discrete solutions to balance the number of particles during the agglomeration and crushing process.

2. The discrete solution method for particle number balance calculation in particle agglomeration and crushing process according to claim 1, characterized in that, S2 includes: S21: The formula used to obtain the number of extinct particles at a feature point under the effect of aggregation is as follows: In the formula: For the first under the effect of aggregation The first discrete interval The number of extinct particles at each feature point; For the first Feature point index of a discrete interval; The total number of feature points within the discrete interval; Index the feature points in all discrete intervals; For the first The first discrete interval Feature points With all discrete intervals the first Feature points The probability of particles merging together; For all discrete intervals, the first... One feature point; For the first The first discrete interval One feature point; For the first The first discrete interval The number of particles corresponding to each feature point; For all discrete intervals, the first... The number of particles corresponding to each feature point; Let be the probability function for particle aggregation; S22: Obtain the number of destroyed particles at a feature point under the action of fragmentation; In the formula: For the first under the action of crushing The first discrete interval The number of extinct particles at each feature point; For the first The first discrete interval Feature points The probability of the corresponding particles breaking; Let be the probability function for particle breakage; S23: Obtain the total disappearance term at the feature point: In the process of pure aggregation, the first The first discrete interval The total number of terms that disappear at each feature point is: ; In the pure crushing process, the first The first discrete interval The total number of terms that disappear at each feature point is: ; During the coupling process of aggregation and breakup, the first The first discrete interval The total number of terms that disappear at each feature point is: ; In the formula: For the first The first discrete interval The total extinction term at the nth feature point, i.e. the nth feature point The first discrete interval The total number of extinct particles at each feature point.

3. The discrete solution method for particle number balance calculation in particle agglomeration and crushing process according to claim 1, characterized in that, S3 includes: S31: Obtain newly formed particles within a discrete interval under the effect of aggregation. r The number of grains in order; in, In the formula: For the first under the effect of aggregation Newly formed particles in discrete intervals The number of grains in order; , 2 satisfies this condition in all discrete intervals. Feature point index, ; The first of all discrete intervals 1 feature point and One feature point; For the first The lower bound of a discrete interval; For the first The upper bound of a discrete interval; It is the Dirac function; The first of all discrete intervals The number of particles corresponding to each feature point and The number of particles corresponding to each feature point; S32: Obtain newly formed particles within a discrete interval under crushing action. r The number of grains in order; In the formula: For the first under the action of crushing Newly formed particles in discrete intervals The number of grains in order; 3 satisfies the following conditions in all discrete intervals. Feature point index; For all discrete intervals, the first... 3 feature points; For all discrete intervals, the first... The number of particles corresponding to the three feature points; These are intermediate calculation parameters; For the first Feature points After the particles on the surface are broken, the resulting particle size is The density of sub-particles; Particle size; Let f(x) be the probability density function for breakage. S33: Obtain the total number of newly formed particles at the feature points. r The number of grains in order; The method for obtaining the total number of newly formed particles at feature points is as follows: During pure agglomeration, the total number of newly formed particles is: ; During the pure crushing process, the total number of newly formed particles is: ; During the aggregation and breakup coupling process, the total number of newly formed particles is: ; In the formula: For the first Total number of newly generated particles in a discrete interval The number of grains in order; S34: Based on the total number of newly formed particles at the feature points r The number of particles is used to obtain the average particle size of newly formed particles. and the average representative particle size of existing particles The formula used is as follows: In the formula: For the first The average particle size of newly formed particles in each discrete interval; For the first The first-order particle number moment of the total number of new particles in a discrete interval; For the first The zero-order particle number moment of the total number of newly formed particles in a discrete interval; For the first The average representative particle size of existing particles in a discrete interval; For the first The first discrete interval One feature point; For the first The first discrete interval The number of particles corresponding to each feature point; S35: Based on the total number of newly formed particles at the feature point r The number of particles in order, combined with the average particle size of newly formed particles. and the average representative particle size of existing particles The number of newly generated particles assigned to feature points is obtained, which is the number of newly generated items assigned to feature points in the discrete interval.

4. The discrete solution method for particle number balance calculation in particle agglomeration and crushing process according to claim 3, characterized in that, The method for obtaining the number of newly generated particles assigned to a feature point is as follows: when At that time, the newly generated particles are allocated to the first... Discrete intervals: In the formula: To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The number of newly formed particles at the first feature point of each discrete interval; To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The discrete interval of the nth time The number of newly formed particles at each feature point; This represents the transpose of the Vandermonde matrix with respect to a vector; T For transpose; For the first The total number of newly formed particles in a discrete interval Step moment; when At that time, the newly generated particles are allocated to the first... ~ Discrete intervals: In the formula: To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The number of newly formed particles at the first feature point of each discrete interval; To be allocated The number of newly formed particles on the surface, i.e., the number allocated to the first... The discrete interval of the nth time The number of newly formed particles at each feature point.

5. The discrete solution method for particle number balance calculation in particle agglomeration and crushing process according to claim 4, characterized in that, S4 includes: S41: Obtain the... Relative distance within discrete intervals and threshold : In the formula: For the first The average value of existing particles within a discrete interval represents the relative distance between the particle size and the midpoint of that interval; Indicates the first The center particle size of each discrete interval; For the first The average representative particle size of existing particles in a discrete interval; Relative distance The threshold; express The total number of discrete intervals; S42: Obtain the... The total correction term for each feature point in a discrete interval includes: First, when If at that time, then ,make If it is zero, ,make It is zero; Then, the formula used to obtain the total corrected neologism at the feature points is as follows: In the formula: For the first In the nth discrete interval Total modified neologisms at each feature point; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; for Time allocation The number of newly formed particles on the surface; Represents the Heaviside step function; For the first The average particle size of newly formed particles in each discrete interval; For the first The average representative particle size of existing particles in a discrete interval; For the first The average particle size of newly formed particles in each discrete interval; For the first The average representative particle size of existing particles in a discrete interval; S43: According to the... In the nth discrete interval The total revised term at each feature point is used to obtain the updated particle number distribution in the discrete interval, including: when At that time, the rate of change of the number of particles at the feature point over time is obtained to obtain... The moment The first discrete interval Number of particles corresponding to each feature point ( t + h ); The formula used to obtain the rate of change of the number of particles at a feature point over time is as follows: In the formula: For the first In the nth discrete interval Total modified neologisms at each feature point; For the first The total number of eliminated terms at each feature point; t For time; For the first The first discrete interval The number of particles corresponding to each feature point; when At that time, obtain the first Within each discrete interval The rate of change of the number of particles over time is used to obtain The moment discrete intervals Order number moment to obtain The moment The first discrete interval Feature points and the corresponding number of particles ; Among them, obtaining the first Within each discrete interval The formula used for the rate of change of the number of particles over time is as follows: In the formula: For the first discrete intervals The number of grains in order; For the first Total number of newly generated particles in a discrete interval Step moment; Get The moment The first discrete interval Feature points and the corresponding number of particles The formula used is as follows: 。 6. The discrete solution method for particle number balance calculation in particle agglomeration and crushing process according to claim 5, characterized in that, Obtain the particle number density and total number of particles in the discrete interval at the next sampling time. The formula used for the number of particles per order is as follows: In the formula: For the first The average representative particle size of existing particles in each discrete interval The corresponding particle number density; For the total The number of grains in order; For the index of the discrete interval; The total number of discrete intervals; The total number of feature points in the discrete interval; For the first Feature point index of the interval; Index the feature points for all discrete intervals.

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