CN113218825A - High-robustness particle size inversion method - Google Patents

High-robustness particle size inversion method Download PDF

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CN113218825A
CN113218825A CN202110557491.XA CN202110557491A CN113218825A CN 113218825 A CN113218825 A CN 113218825A CN 202110557491 A CN202110557491 A CN 202110557491A CN 113218825 A CN113218825 A CN 113218825A
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particle
particle size
particles
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extinction
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于雪莲
罗如芝
房鑫宇
潘立辉
吴佳哲
宋良峰
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Harbin University of Science and Technology
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    • G—PHYSICS
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    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N15/00—Investigating characteristics of particles; Investigating permeability, pore-volume, or surface-area of porous materials
    • G01N15/02—Investigating particle size or size distribution
    • G01N15/0205—Investigating particle size or size distribution by optical means, e.g. by light scattering, diffraction, holography or imaging
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
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Abstract

The invention relates to a high-robustness particle size inversion method, which can obtain the particle size distribution of a measured particle system by measuring the light intensity ratio of a light beam before and after the light beam passes through the particle system according to the classical Mie scattering theory and the Lambert beer theorem under the conditions of known wavelength, relative refractive index and the like. Aiming at the ill-conditioned property of the traditional target function, the regularization parameters calculated by an L curve method are adopted to construct the target function with penalty factors, so that the ill-conditioned property of the function can be reduced. According to the inversion calculation method, a good inversion result can be obtained by combining with a quantum particle group optimization algorithm. Under the condition of insufficient prior information, compared with the traditional inversion algorithm, the method has obvious advantages in the aspects of measurement result accuracy, calculation process stability, running time and robustness.

Description

High-robustness particle size inversion method
Technical Field
The invention belongs to the field of extinction method microparticle signal detection, and particularly relates to a high-robustness particle size inversion method.
Background
Studies have shown that when humans are exposed to high PM2.5 concentrations for a long period of time, the risk of lung cancer and cardiopulmonary disease increases dramatically, and mortality increases with increasing concentrations. Due to the characteristic that PM2.5 is difficult to settle, the PM can float in the air for a long time, so that the visibility and the climate of the atmosphere are influenced, and the traffic safety and the travel safety are seriously damaged, so that the research on the particulate matters becomes an important part for protecting the environment and the human safety.
Particle size measurement modes are abundant, and particle detection methods are diversified under different detection environments. Currently, common particle size measurement methods include mechanical methods, electrical induction methods, wave characteristic methods, and the like. The mechanical method can be divided into a screening method and a sedimentation method, but the mechanical method needs to be contacted with a measured object, so that the possibility of influencing the measured object exists; although the electric induction method has high detection precision, the electric induction method has the defects of particle blockage and poor noise resistance; the wave characteristic method includes: the microscopic method and the light scattering method, wherein the microscopic method is a result obtained by observation with human eyes, and is difficult to avoid the influence of subjective consciousness of a person to be detected.
The extinction method, total light scattering method or turbidity method, is one of the static light scattering methods. The principle of the method is that when a beam of light passes through the particles to be measured, the particles scatter and absorb incident light, so that transmitted light intensity is attenuated, and the particle size distribution of the particles can be obtained by utilizing the attenuation degree of the light intensity under different wave bands. The theories adopted by this method are mie scattering theory and lambert beer's law. Compared with other detection methods, the light scattering method has: (1) the application range is wide: the light scattering method is suitable for measuring solid, liquid and air; (2) the particle size measurement range is wide: the diffraction scattering method is generally used for detecting large particles, the upper limit can reach millimeter level, the dynamic scattering method is generally used for detecting small particles, and the lower limit can reach nanometer level. The particle size measurement range of the light scattering method is between millimeters and nanometers; (3) the measuring speed is fast: the propagation speed of light is high, and the rapid detection can be realized; (4) the online measurement can be realized: the light transmission is transmission and non-contact, and is suitable for on-line measurement. The extinction method has great prospect in the fields of on-line detection of smoke dust emission concentration, measurement and monitoring of high molecular polymer process and industrial on-line monitoring of particle granularity and concentration.
The inverse problem of particle size distribution is the first type of Fredholm (Fredholm) integral problem, and at present, no mature method exists for theoretical solution of the equation. In the actual calculation process, the condition number of the extinction coefficient matrix T is often very large, so that the ill-conditioned character is very serious, an abnormal number solution is often generated, and the actual distribution situation is not met. Inversion algorithms have strict requirements on running time, robustness and accuracy of calculation results. Therefore, in order to improve the stability and accuracy of particle size inversion, it is important to select a proper inversion algorithm.
Disclosure of Invention
Aiming at the defects of the existing extinction method technology, the invention aims to provide a particle size inversion method with optimal combination in the aspects of robustness, running time, stability of a calculation process, accuracy of a measurement result and the like.
In order to solve the technical problems, the invention adopts the following technical scheme:
a high robustness particle size inversion method is characterized in that:
obtaining a normalized light energy distribution calculation formula according to the Mie scattering theory and the Lambert beer theorem, discretizing the light energy distribution calculation formula and then inverting the particle size distribution, and the method comprises the following steps:
the method comprises the following steps: mie scattering is a solution for uniform spherical particles, and the mie scattering model is shown in fig. 1.
Incident light is transmitted along the positive direction of a Z axis, an electric vector E is transmitted along the positive direction of an X axis, r is the distance between a scattered light reference point p and a scattering center, a plane which is formed by the reference point p and the Z axis and is vertical to an XOY plane is called a scattering plane and is represented by POZ, the included angle between the reference point and the scattering center is theta, and the included angle between a vibration plane formed by the electric vector of the incident light and the POZ is theta
Figure BDA0003077709540000021
The wavelength is λ.
Wherein E is⊥SComponent parallel to the POZ, E//SPerpendicular to the POZ component; s1And S2Respectively representing the complex amplitudes of the scattered light in the vertical and horizontal directions, said S1And S2Orthogonal to each other and related to the scattering angle theta.
Scattered intensity I perpendicular to the POZ⊥SAnd the scattered intensity I parallel to the POZ//SIs expressed as
Figure BDA0003077709540000022
Figure BDA0003077709540000023
And the total scattered light intensity ISIs expressed as
Figure BDA0003077709540000024
In formula (3): i.e. i1、i2The intensity function parallel to the POZ and perpendicular to the POZ is shown.
Scattering intensity function relation i1、i2Can be expressed as
Figure BDA0003077709540000025
Figure BDA0003077709540000026
In formulas (4) and (5): n is related to the property of the particle, is a positive integer and is used for characterizing the order of the partial wave of the particle. The key to this relationship is mie scattering, which translates the scattered intensity function into a combination of coefficients.
an、bnCalled the Mie scattering coefficient, expressed as
Figure BDA0003077709540000027
Figure BDA0003077709540000031
Distribution function pi of Mie scattering coefficient and scattering angle thetanAnd τnIs expressed as the sum of infinite series
Figure BDA0003077709540000032
Figure BDA0003077709540000033
Wherein psinAnd xinN-th order first and second class Bessel functions, psin' and xin' is the derivative of the Bessel function, Pn (1)Is a Legendre function of order n.
The scattering coefficient and extinction coefficient of particles of any size and any refractive index can be derived by using the Mie scattering theory, and the expression is
Figure BDA0003077709540000034
As can be seen from equation (10), the extinction coefficient and the scattering coefficient are related to anAnd bnAs a function of (c).
The accurate solution of the extinction coefficient is the key point for obtaining a high-precision solution by the inversion of the particle size distribution of the extinction method, so that the extinction coefficient is calculated by selecting a Mie algorithm under the condition of acceptable time consumption.
Step two: the extinction method is shown in the schematic diagram 1, I in the schematic diagram 20For the incident light intensity, I denotes the transmitted light intensity through the particle system. When the measured particle system satisfies the uniform spherical monodispersed particle system of uncorrelated single scattering, the expression of extinction value for the wavelength of incident light is λ
Figure BDA0003077709540000035
In formula (11): wherein L is the optical path; I. i is0Respectively representing the light intensity before and after passing through the particle system; λ represents the wavelength of the light beam; m is the relative refractive index; d represents the particle size of the particle system to be detected; k is a radical ofext(λ, m, D) are extinction coefficient values based on Mie scattering theory;
in actual measurement, the particle system to be measured is mostly polydisperse system composed of different particles. Therefore, the extinction value expression of the polydispersion system based on the Mie scattering theory can be deduced
Figure BDA0003077709540000036
In formula (12): dmax、DminRespectively representing the upper limit and the lower limit of the particle size of the particle series particles to be detected; n (D) represents a particle diameter of [ D, D + dD ]]The total number of particles in the range of the particle system to be measured. Discretizing the formula (12), and recording an extinction matrix E ═ In (I/I)0)1,In(I/I0)2,...,In(I/I0)n)T(ii) a Extinction coefficient matrix T is 1/Dikext(λiM, D) and converting the number distribution n (D) into a weight frequency distribution w (D), writing a matrix form:
E＝TW (13)
wherein E is an extinction matrix; t is an extinction coefficient matrix; w is the particle size distribution of the particles to be obtained; as can be seen, equation (13) is a conventional objective function, and has serious ill-conditioned behavior.
Step three: constructing an objective function with a penalty factor by adopting regularization parameters calculated by an L curve method:
Figure BDA0003077709540000041
step four: the method of combining quantum particle populations constitutes an optimization problem.
At the t +1 generation, the position evolution equation of the ith particle in the jth dimension can be changed into
Figure BDA0003077709540000042
Wherein i is 1, 2.. m, m is the particle population size, j is 1, 2.. d, d is the dimension, β is the contraction and expansion factor, u is the contraction and expansion factorij(t) is a random number in the range of (0,1), pij(t) satisfies
Figure BDA0003077709540000043
In the formula (16), r1,r2Is a random number in the range of (0,1), c1,c2As a learning factor, pbest is a local extremum individual, gbest is a global extremum individual, and therefore,
Figure BDA0003077709540000044
can also be regarded as a random number;
mbest is the average optimal position of the particles, and the calculation expression is
Figure BDA0003077709540000045
Step five: assuming that the particle size distribution follows a R-R monomodal distribution
Figure BDA0003077709540000046
In formula (18): w (D) is the weight frequency distribution to be determined, X and N are the characteristic parameters to be determined, X is a parameter which characterizes the particle size and N is a parameter which characterizes the particle distribution. The larger N is, the narrower the distribution curve is, and the smaller N is, the wider the distribution curve is; assuming that the state of the particle is two distribution parameters X ═ X, N of equation (18), the fitness function is
Figure BDA0003077709540000051
And particle size inversion based on quantum particle swarm optimization:
firstly, initializing parameters such as population scale M, spatial dimension d, learning factors c and r and maximum iteration number M, and randomly generating M particle positions X-X (X, N);
calculating a fitness function value corresponding to each position of the initial particle swarm, taking the position of the particle with the minimum fitness function as an optimal position, and storing the current optimal position and the fitness function value;
updating the positions of the particles, namely updating the positions of the particles according to formulas (15) to (17), calculating a fitness function value, and updating the optimal position and the fitness function value;
and finally, outputting a result, namely obtaining an optimal solution and finishing optimization when the iteration number reaches M, otherwise, adding 1: t to t +1 to the current iteration number, and repeating the step three until the algorithm iterates to the maximum iteration number.
Step six: substituting two distribution parameters X ═ (X, N) of quantum particle swarm optimization into the originally assumed R-R unimodal distribution to obtain final w (d).
Compared with the prior art, the invention has the beneficial effects that: the invention provides a high-robustness particle size inversion method aiming at the problems that a search calculation result is unstable, the particle size inversion problem is easy to fall into local optimum, an extinction coefficient matrix is ill-conditioned and the like in particle size measurement, and the method is accurate in measurement result, short in running time and stable in calculation process.
Drawings
FIG. 1 is a schematic view of the Mie scattering model of the present invention;
FIG. 2 is a schematic diagram of the extinction principle method of the present invention;
FIGS. 3(a), (b), and (c) are L-plots of 0%, 5%, and 10% noise according to the present invention;
FIG. 4 is a flow chart of a quantum-behaved particle swarm processing procedure of the present invention;
FIGS. 5(a), (b), (c) are graphs showing the inversion results of a spherical particle system in which the particle diameters of the particles according to the present invention obey R-R unimodal distribution;
FIG. 6 is a schematic diagram of the whole implementation process of the present invention.
Detailed Description
The invention will be described in detail and completely by taking the attached drawings of the invention and listing simulation examples.
The invention provides a high-robustness particle size inversion method, which comprises the following steps of obtaining a normalized light energy distribution calculation formula according to the Mie scattering theory and the Lambert beer theorem, discretizing the light energy distribution calculation formula, and inverting the particle size distribution, wherein the method comprises the following steps:
the method comprises the following steps: mie scattering is a solution for uniform spherical particles, and is the most common method for analyzing particle scattering at present, and a mie scattering model is shown in fig. 1.
Incident light is transmitted along the positive direction of a Z axis, an electric vector E is transmitted along the positive direction of an X axis, r is the distance between a scattered light reference point p and a scattering center, a plane which is formed by the reference point p and the Z axis and is vertical to an XOY plane is called a scattering plane and is represented by POZ, the included angle between the reference point and the scattering center is theta, and the included angle between a vibration plane formed by the electric vector of the incident light and the POZ is theta
Figure BDA0003077709540000061
The wavelength is λ.
Wherein E is⊥SComponent parallel to the POZ, E//SPerpendicular to the POZ component; s1And S2Respectively representing the complex amplitudes of the scattered light in the vertical and horizontal directions, said S1And S2Orthogonal to each other and related to the scattering angle theta.
Scattered intensity I perpendicular to the POZ⊥SAnd the scattered intensity I parallel to the POZ//SIs expressed as
Figure BDA0003077709540000062
Figure BDA0003077709540000063
And the total scattered light intensity ISIs expressed as
Figure BDA0003077709540000064
In formula (3): i.e. i1、i2The intensity function parallel to the POZ and perpendicular to the POZ is shown.
Scattering intensity function relation i1、i2Can be expressed as
Figure BDA0003077709540000065
Figure BDA0003077709540000066
In formulas (4) and (5): n is related to the property of the particle, is a positive integer and is used for characterizing the order of the partial wave of the particle. The key to this relationship is mie scattering, which translates the scattered intensity function into a combination of coefficients.
an、bnCalled the Mie scattering coefficient, expressed as
Figure BDA0003077709540000067
Figure BDA0003077709540000068
Distribution function pi of Mie scattering coefficient and scattering angle thetanAnd τnIs expressed as the sum of infinite series
Figure BDA0003077709540000069
Figure BDA0003077709540000071
Wherein psinAnd xinN-th order first and second class Bessel functions, psin' and xin' is the derivative of the Bessel function, Pn (1)Is aOrder n Legendre function.
The scattering coefficient and extinction coefficient of particles of any size and any refractive index can be derived by using the Mie scattering theory, and the expression is
The extinction coefficient of a particle is one of the important parameters for inverting the particle size distribution by an extinction method. The scattering coefficient and extinction coefficient of particles of any size and any refractive index can be derived theoretically by using the Mie scattering, and the expression is
Figure BDA0003077709540000072
The extinction coefficient and scattering coefficient are with respect to anAnd bnAs a function of (c). Therefore, the particle diameter, refractive index, and wavelength of incident light affect the magnitude of the extinction coefficient to different extents.
The accurate solution of the extinction coefficient is the key point for obtaining a high-precision solution by the inversion of the particle size distribution of the extinction method, so that the extinction coefficient is calculated by selecting a Mie algorithm under the condition of acceptable time consumption.
Step two: FIG. 2 shows the principle of extinction measurement, I in FIG. 20For the incident light intensity, I denotes the transmitted light intensity through the particle system. When the measured particle system satisfies the uniform spherical monodispersed particle system of uncorrelated single scattering, the expression of extinction value for the wavelength of incident light is λ
Figure BDA0003077709540000073
In formula (11): wherein L is the optical path; I. i is0Respectively representing the light intensity before and after passing through the particle system; λ represents the wavelength of the light beam; m is the relative refractive index; d represents the particle size of the particle system to be detected; k is a radical ofext(λ, m, D) are extinction coefficient values based on Mie scattering theory;
in actual measurement, the particle system to be measured is mostly polydisperse system composed of different particles. Therefore, the extinction value expression of the polydispersion system based on the Mie scattering theory can be deduced
Figure BDA0003077709540000074
In formula (12): dmax、DminRespectively representing the upper limit and the lower limit of the particle size of the particle series particles to be detected; n (D) represents a particle diameter of [ D, D + dD ]]The total number of particles in the range of the particle system to be measured. Discretizing the formula (12), and recording an extinction matrix E ═ In (I/I)0)1,In(I/I0)2,...,In(I/I0)n)T(ii) a Extinction coefficient matrix T is 1/Dikext(λiM, D) and converting the number distribution n (D) into a weight frequency distribution w (D), writing a matrix form:
E＝TW(13)
wherein E is an extinction matrix; t is an extinction coefficient matrix; w is the particle size distribution of the particles to be obtained; as can be seen, equation (13) is a conventional objective function, and has serious ill-conditioned behavior.
Step three: constructing an objective function with a penalty factor by adopting regularization parameters calculated by an L curve method:
Figure BDA0003077709540000081
FIGS. 3(a), (b), and (c) are graphs of L with 0%, 5%, and 10% noise added, respectively;
step four: the method of combining quantum particle populations constitutes an optimization problem.
Fig. 4 shows a flow chart of a quantum particle population.
At the t +1 generation, the position evolution equation of the ith particle in the jth dimension can be changed into
Figure BDA0003077709540000082
Wherein i is 1, 2.. m, m is the particle population size, j is 1, 2.. d, d is the dimension, β is the contraction and expansion factor, u is the contraction and expansion factorij(t) is a random number in the range of (0,1), pij(t) satisfies
Figure BDA0003077709540000083
In the formula (16), r1,r2Is a random number in the range of (0,1), c1,c2As a learning factor, pbest is a local extremum individual, gbest is a global extremum individual, and therefore,
Figure BDA0003077709540000084
can also be regarded as a random number;
mbest is the average optimal position of the particles, and the calculation expression is
Figure BDA0003077709540000085
Step five: assuming that the particle size distribution follows a R-R monomodal distribution
Figure BDA0003077709540000086
In formula (18): w (D) is the weight frequency distribution to be determined, X and N are the characteristic parameters to be determined, X is a parameter which characterizes the particle size and N is a parameter which characterizes the particle distribution. The larger N is, the narrower the distribution curve is, and the smaller N is, the wider the distribution curve is; assuming that the state of the particle is two distribution parameters X ═ X, N of equation (18), the fitness function is
Figure BDA0003077709540000091
Then, the particle size inversion based on the quantum particle swarm algorithm comprises the following steps:
firstly, initializing parameters such as population scale M, spatial dimension d, learning factors c and r and maximum iteration number M, and randomly generating M particle positions X-X (X, N);
then calculate the fitness function for the current particle location: calculating a fitness function value corresponding to each position of the initial particle swarm, taking the position of the particle with the minimum fitness function as an optimal position, and storing the current optimal position and the fitness function value;
next, updating the particle position: updating the positions of the particles according to formulas (15) to (17), calculating a fitness function value, and updating the optimal position and the fitness function value;
and finally, outputting a result, namely obtaining an optimal solution and finishing optimization when the iteration number reaches M, otherwise, adding 1: t to t +1 to the current iteration number, and repeating the step three until the algorithm iterates to the maximum iteration number.
In order to verify the feasibility of the quantum particle swarm algorithm, a simulation experiment is carried out on a uniform spherical particle swarm which obeys R-R unimodal distribution, and the two conditions of inverting a regularized target function and an non-regularized target function based on the quantum particle swarm algorithm are compared. In the simulation process, two actual characteristic parameters of the R-R distribution function are assumed to be narrow distribution with X being 3.1, N being 9, particle size range being 0.1-10 μm, relative complex index m being (1.59-0.01i)/1.332, and two wavelengths being 632.8nm and 532nm, respectively.
And (3) substituting two parameters X and N optimized by the quantum particle swarm optimization algorithm into the originally assumed R-R unimodal distribution to invert the final particle size distribution.
In order to detect the noise resistance of the algorithm, 5% and 10% of random noise is added to the transmitted light energy value respectively, the maximum iteration number is 100, the program runs for 3 times, and the average value is taken. The inversion result is shown in fig. 5, where w (d) represents the weight frequency distribution of the particle size. From fig. 5(a), (b), and (c), it can be found that the particle size distribution of the particles can be successfully inverted by introducing the quantum particle swarm optimization algorithm on the basis of the dependent mode and the regularized objective function. A stable result can be obtained by running the program for 3 times, and the quantum particle swarm is not easy to fall into a local optimal solution, so that a more accurate result can be obtained. When 5% and 10% of noise is added to the extinction value respectively, the inversion distribution after regularization can still be highly matched with the theoretical distribution, and the quantum particle swarm optimization has outstanding advantages in the aspect of robustness. The whole inversion process takes no more than 2s, and can be better used in particle inversion. The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (2)

1. A high-robustness particle size inversion method is characterized by comprising the following steps:
step one, obtaining an extinction value expression of a polydisperse particle system according to the Mie scattering theory and the Lambert beer theorem
Figure FDA0003077709530000011
Wherein L is the optical path; I. i is0Respectively representing the light intensity before and after passing through the particle system; λ represents the wavelength of the light beam; m is the relative refractive index; d represents the particle size of the particle system to be detected; dmax、DminRespectively representing the upper limit and the lower limit of the particle size of the particle series particles to be detected; k is a radical ofext(λ, m, D) are extinction coefficient values based on Mie scattering theory; n (D) represents a particle diameter of [ D, D + dD ]]The total number of particles in the range of the particle system to be detected;
discretizing the extinction value expression to obtain a traditional objective function E which is TW, wherein E is an extinction matrix; t is an extinction coefficient matrix; w is the particle size distribution of the particles to be obtained; the traditional objective function has ill-conditioned property;
step three, introducing a Tikhonov regularization theory to construct a new objective function
Figure FDA0003077709530000012
Figure FDA0003077709530000013
Represents a norm of order 2; alpha is a regularization parameter;
step four, solving alpha through an L curve method;
and fifthly, inverting W by using a quantum particle swarm optimization algorithm.
2. The method as claimed in claim 1, wherein T-1/D is the inverse method of particle sizeikext(λi,m,D)；DiIs the particle size of the particles; lambda [ alpha ]iIs the laser wavelength; m is the relative refractive index; d is the average particle diameter.
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贾茜媛;郭天太;曹丽霞;孔明;赵军;: "细菌觅食优化算法在光散射法颗粒粒度反演中的应用", 光学技术, vol. 43, no. 5, pages 0 - 4 *

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* Cited by examiner, † Cited by third party
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CN114544444A (en) * 2022-02-24 2022-05-27 上海理工大学 Method for measuring crystal grain size distribution in crystallization process
CN114544444B (en) * 2022-02-24 2023-12-01 上海理工大学 Crystal particle size distribution measuring method in crystallization process
CN115436237A (en) * 2022-09-22 2022-12-06 东莞理工学院 Particle group size distribution determining method, system, equipment and storage medium
CN115436237B (en) * 2022-09-22 2023-08-01 东莞理工学院 Particle swarm size distribution determining method, system, equipment and storage medium

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