CN109145252B - Particle size distribution function reconstruction method and device based on PSD-LIR - Google Patents

Particle size distribution function reconstruction method and device based on PSD-LIR Download PDF

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CN109145252B
CN109145252B CN201810871439.XA CN201810871439A CN109145252B CN 109145252 B CN109145252 B CN 109145252B CN 201810871439 A CN201810871439 A CN 201810871439A CN 109145252 B CN109145252 B CN 109145252B
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distribution function
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谢明亮
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Huazhong University of Science and Technology
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Abstract

The invention discloses a particle size distribution function reconstruction method and device based on PSD-LIR, and belongs to the field of particle swarm size characteristic detection. Taking 7 moments M according to the definition of the moments of the particle volumekK is 0, 1/3, 2/3, 1, 4/3, 5/3 and 2, corresponds to 7 points on the distribution function, adds zero points at two ends of the distribution function, and inverts a distribution function curve of a particle scale by interpolation and optimal approximation; zero order moment M0Is the population concentration at a given time and place; 1/3 order moment M1/3Proportional to the average diameter of the population of particles; 2/3 order moment M2/3Proportional to the surface area of the particle; 1 order moment M1Proportional to the total volume of the particles; 4/3 order moment M4/3Proportional to the particle settling terminal velocity; 5/3 order moment M5/3Proportional to the mass flux of the particles; 2 order moment M2Proportional to the total scattered light of the population. The method and the equipment have the characteristics of simplicity, rapidness and high efficiency.

Description

Particle size distribution function reconstruction method and device based on PSD-LIR
Technical Field
The invention belongs to the field of Particle swarm Size characteristic detection, and particularly relates to a Particle Size Distribution function Reconstruction method based on a linear interpolation theory and finite moment quantity (PSD-LIR).
Background
The particle size distribution plays an important role in many products, not only affecting the properties of the product, such as porosity, density and strength, but also many process properties depend on the particle size distribution, such as viscosity, aggregation morphology, etc. As the most important physical feature of the multispectral system, the particle size spectral distribution has two forms: one is a discrete distribution and the other is a continuous distribution. Here, the continuous distribution function of the particles is reconstructed using the volume as a characteristic parameter. Because the use of volume is easy to define and even irregular particles compared to diameter and surface area. Moreover, the volume is kept constant when two particles collide and adhere to each other.
In the detection of an actual particle system, general characteristic parameters of a particle group, such as the number of particles, the specific surface area of the particles, the total mass or volume of the particles, the mass flow rate of the particles and the like, are often obtained. Alternatively, the resulting particle distribution function is less reproducible and even requires off-line measurements, such as a laser particle sizer.
In particle dynamics simulations, due to the complexity of the system, it is also the evolution of the moment of the particle distribution function that is often tracked. Theoretically, the distribution function of the particles is equivalent to the infinite moment of the particles, but under the practical condition, the infinite moment cannot be calculated, so that the calculation of the example distribution function cannot be realized according to the infinite moment.
Disclosure of Invention
Aiming at the defects or the improvement requirements of the prior art, the invention provides a particle size distribution function reconstruction method based on PSD-LIR, and aims to select a specific moment parameter and carry out linear interpolation to carry out inversion reconstruction on the particle distribution function so as to realize low-order inversion of the particle distribution function.
To achieve the above object, according to one aspect of the present invention, there is provided a PSD-LIR-based particle size distribution function reconstruction method based on the definition of the moment quantity of the particle volume
Figure BDA0001752249520000021
Taking k as 0, 1/3, 2/3, 1, 4/3, 5/3 and 2 to obtain 7 moments MkAdding zero points at two ends of the distribution function corresponding to 7 points on the distribution function to total 9 points; inverting a particle size distribution function curve through interpolation and an optimal approximation method based on the 9 points; wherein v represents the volume of the particle, x represents the position of the particle, t represents time, and n represents the number density of the particle; 7 moments MkThe physical meanings are as follows: zero order moment M0: is the population concentration at a given time and place; 1/3 order moment M1/3: proportional to the average diameter of the population of particles; 2/3 order moment M2/3: proportional to the surface area of the particle; 1 order moment M1: proportional to the total volume of the particles; 4/3 order moment M4/3: proportional to the particle settling terminal velocity; 5/3 order moment M5/3: proportional to the mass flux of the particles; 2 order moment M2: proportional to the total scattered light of the population.
In order to achieve the above object, in another aspect, the present invention further provides a particle size distribution function reconstruction method based on PSD-LIR, including the following steps:
step 1: predicting the interval [ a, b ] of the distribution function of the particles, equally dividing the interval [ a, b ] into 8 segments, wherein 9 nodes are provided, and the coordinates of each node are respectively as follows:
Figure BDA0001752249520000022
step 2: the system of equations AY-B for the coefficients of the reconstructed distribution function is established, wherein,
Figure BDA0001752249520000023
Figure BDA0001752249520000035
j. i are subscript numbers;
and step 3: solving for AY ═ B, i.e. Y ═ A-1B, obtaining the coefficient Y of the distribution function as f (x)i) A linear reconstruction distribution function p (x) is obtained:
Figure BDA0001752249520000031
wherein the content of the first and second substances,
Figure BDA0001752249520000032
Figure BDA0001752249520000033
Figure BDA0001752249520000034
and 4, step 4: adjusting distribution function intervals [ a, b ]]And then, repeating the steps 1-3 to obtain a distribution function coefficient Y1＝f1(xi) And corresponding P1(x) Calculating f (x) and f1(x) Or P (x) and P1(x) If less than a given threshold, output P (x)Or P1(x) As a result of the reconstruction.
To achieve the above object, in another aspect, the present invention further provides a computer-readable storage medium having a computer program stored thereon, where the computer program is executed by a processor to implement any one of the methods as described above.
In order to achieve the above object, in another aspect, the present invention further provides a PSD-LIR based particle size distribution function reconstruction apparatus, including the computer-readable storage medium as described above and a processor for calling and processing a computer program stored in the computer-readable storage medium.
In general, compared with the prior art, the above technical solution contemplated by the present invention can obtain the following beneficial effects:
1. because the 7 specific moment parameters selected by the method only focus on the results of all moment parameters rather than the obtaining process, the moment parameter results obtained by various detection technologies and methods can be integrated to further obtain the size distribution function of the particles, the moment of the reconstructed distribution function is consistent with the detected moment, and the main characteristics of the particle size distribution, such as the mean value, the variance and the like, can be reflected;
2. the invention can integrate various detection techniques and methods, thus reducing the deviation caused by a single measurement method and accidental factors;
3. the order of the highest moment involved in the calculation solving process is second order, and the particle size distribution function can be inverted only by second-order calculation, so that the phenomenon of high error caused by high-order moment calculation is avoided;
4. the reconstruction method is simple, rapid and efficient, and has extremely high application prospect in the field of particle size distribution function reconstruction based on PSD-LIR on line.
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FIG. 1 is a principal method flow diagram of the preferred embodiment of the present invention;
FIG. 2 is a graph of selected raw distribution functions for an example reconstruction of the present invention;
fig. 3 is a result and comparison of the inversion of the original distribution function of fig. 2 according to the method of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The moment of the particle size distribution function has important physical significance, and the moment based on the particle volume is generally defined as:
Figure BDA0001752249520000041
wherein v represents the volume of the particle, x represents the position of the particle, t represents time, and n represents the number density of the particle; moment MkWherein k is 0, 1/3, 2/3, 1, 4/3, 5/3 and 2, and the physical meanings are as follows: zero order moment M0: is the population concentration at a given time and place;
1/3 order moment M1/3: proportional to the average diameter of the population of particles;
2/3 order moment M2/3: proportional to the surface area of the particle;
1 order moment M1: proportional to the total volume of the particles;
4/3 order moment M4/3: proportional to the particle settling terminal velocity;
5/3 order moment M5/3: proportional to the mass flux of the particles;
2 order moment M2: proportional to the total scattered light of the population;
these moments can be detected by various experimental methods in the prior art, such as a particle counter, a laser particle analyzer (or a malvern particle analyzer), a mike ASAP2460 physical adsorption apparatus, a filter membrane gravimetric method, a differential centrifugation method, a rayleigh scattering method, and the like.
The basic method of the invention for the reconstruction of the distribution function is as follows: the above 7 important and detectable moments correspond to 7 points on the distribution function, plus the zeros at both ends of the distribution function, for a total of 9 points. By means of interpolation theory and optimal approximation principle, distribution function curve may be drawn.
The calculation flow of the inversion reconstruction particle distribution function is shown in fig. 1, and the specific steps are as follows:
step 1: and (3) predicting an interval [ a, b ] where the distribution function is located, equally dividing the interval into 8 sections, wherein the coordinate of each node is as follows:
Figure BDA0001752249520000051
step 2: according to the definition of moments and the principle of linear interpolation, an equation set AY as B of reconstruction function coefficients is established, and the matrix form is as follows:
Figure BDA0001752249520000061
wherein the content of the first and second substances,
Figure BDA0001752249520000062
j. i are subscript numbers;
and step 3: solving the equation in step 2 to obtain the coefficient f (x) of the distribution functioni) The distribution function can be constructed as:
Figure BDA0001752249520000063
wherein the content of the first and second substances,
Figure BDA0001752249520000064
Figure BDA0001752249520000065
Figure BDA0001752249520000066
4. adjusting the distribution function interval, and repeating the process of 1-3 to obtain a distribution function coefficient Y1＝f1(xi) And corresponding P1(x) Calculating f (x) and f1(x) Or P (x) and P1(x) If less than a given threshold, outputting P (x) or P1(x) As a result of the reconstruction
The method of the invention is described below with reference to a practical case:
[ example of reconstruction ]
For the original distribution function F (x) of FIG. 2, the interval of the distribution function is estimated to be [0,0.001 ]]. Obtaining coordinate value x by 8 equally dividing0～x8And linear interpolation can construct the matrix a as follows:
Figure BDA0001752249520000071
and moment of each order Mk：
Figure BDA0001752249520000072
It should be noted that, since the matrix a is generally a pathological matrix, Y is equal to a-1In the process of B, a method for inversion by using a pathological matrix, such as pseudo-inversion (pinv), QR decomposition and other technologies, is required. Taking the interval [0,0.9]Obtaining the coefficient f (x) of the reconstructed distribution functioni) Comprises the following steps:
Figure BDA0001752249520000073
the adjustment interval is [0,0.81 ]]The obtained coefficient f of the reconstructed distribution function1(xi) The following were used:
Figure BDA0001752249520000074
the 2-norm of the difference is 167.8993, which is an acceptable solution relative to a peak value of 3000. By adding two end zeros, the distribution function p (x) can be reconstructed, and as shown in fig. 3, the distribution characteristics of the reconstructed distribution function p (x) and the original distribution function f (x) can be consistent compared, which illustrates the reliability of the method. In general, P obtained after adjustment can be selected1(x) As an output, however, in the present embodiment, since 167.8993 is small, P (x) or P is selected1(x) With little difference, the present embodiment selects p (x) as the output.
In other embodiments, P (x) and P may also be used1(x) Is compared with the previous f (x) by thresholdi) And f1(xi) With 7 points calculation, the norm would be smaller. The threshold value can be set according to actual requirements, and the smaller the threshold value is, the higher the precision is.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (4)

1. A particle size distribution function reconstruction method based on PSD-LIR is characterized in that the method is based on the definition of the moment of the particle volume
Figure FDA0001752249510000011
Taking k as 0, 1/3, 2/3, 1, 4/3, 5/3 and 2 to obtain 7 moments MkAdding zero points at two ends of the distribution function corresponding to 7 points on the distribution function to total 9 points; inverting a particle size distribution function curve through interpolation and an optimal approximation method based on the 9 points; wherein v represents the volume of the particle, x represents the position of the particle, t represents time, and n represents the number density of the particle; 7 moments MkThe physical meanings are as follows:
zero order moment M0: is the population concentration at a given time and place;
1/3 order moment M1/3: proportional to the average diameter of the population of particles;
2/3 order moment M2/3: proportional to the surface area of the particle;
1 order moment M1: proportional to the total volume of the particles;
4/3 order moment M4/3: proportional to the particle settling terminal velocity;
5/3 order moment M5/3: proportional to the mass flux of the particles;
2 order moment M2: proportional to the total scattered light of the population.
2. A particle size distribution function reconstruction method based on PSD-LIR is characterized by comprising the following steps:
step 1: predicting the interval [ a, b ] of the distribution function of the particles, equally dividing the interval [ a, b ] into 8 segments, wherein 9 nodes are provided, and the coordinates of each node are respectively as follows:
Figure FDA0001752249510000012
step 2: the system of equations AY-B for the coefficients of the reconstructed distribution function is established, wherein,
Figure FDA0001752249510000021
Figure FDA0001752249510000022
j. i are subscript numbers;
and step 3: solving for AY ═ B, i.e. Y ═ A-1B, obtaining the coefficient Y of the distribution function as f (x)i) A linear reconstruction distribution function p (x) is obtained:
Figure FDA0001752249510000023
wherein the content of the first and second substances,
Figure FDA0001752249510000024
Figure FDA0001752249510000025
Figure FDA0001752249510000026
and 4, step 4: adjusting distribution function intervals [ a, b ]]And then, repeating the steps 1-3 to obtain a distribution function coefficient Y1＝f1(xi) And corresponding P1(x) Calculating f (x) and f1(x) Or P (x) and P1(x) If less than a given threshold, outputting P (x) or P1(x) As a result of the reconstruction.
3. A computer-readable storage medium, characterized in that a computer program is stored on the computer-readable storage medium, which computer program, when being executed by a processor, carries out the method of claim 1 or 2.
4. A PSD-LIR based particle size distribution function reconstruction device comprising a computer readable storage medium according to claim 3 and a processor for invoking and processing a computer program stored in the computer readable storage medium.
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CN104458512A (en) * 2014-12-01 2015-03-25 华中科技大学 Method for measuring fractal dimension of particle group
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