CN101793665A - Limited distribution integral inversion algorithm for grain diameter measurement - Google Patents

Abstract

The invention relates to a limited distribution integral inversion algorithm for grain diameter measurement. The invention is characterized in that the algorithm comprises the following steps: (1) obtaining the light distribution I (theta) of diffraction light of a grain group the grain size parameter distribution of which is fab (x) in the limited grain size interval [a,b] by measuring; (2) obtaining an inversion expression by Hankel conversion and the analytic solution of a Scholmilch equation; (3) carrying out discretization treatment on the equation obtained in step (2) by utilizing the Gaussian interpolation method; and (4) substituting Gaussian interpolation coefficients and interpolation nodes into the formula after by discretization treatment in step (3), thereby obtaining the grain diameter distribution fab (x). Since the integral inversion algorithm with a limited grain diameter distribution range is adopted, the invention is suitable for the condition that in actual measurement, the grain diameters are distributed in a limited interval. In the formula (2) of the claims, spline function derivative is directly adopted instead of difference, thereby decreasing theoretical errors. The algorithm only contains a Bessel function, thereby remarkably decreasing inversion result oscillation.

Description

A kind of limited distribution integral inversion algorithm of grain diameter measurement

[technical field]

The present invention relates to a kind of limited distribution integral inversion algorithm of grain diameter measurement, belong to the grain graininess fields of measurement.

[background technology]

The integral inversion algorithm that utilizes laser diffractometry to measure size distribution at present generally adopts Chin-Shifrin integral transformation method, establishes f (x) and is the mass distribution probability density function of population, then

$\mathrm{xf}\left(x\right)=-\frac{8{\mathrm{\&pi;}}^3{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{{\mathrm{\&lambda;}}^2}{\&Integral;}_0^{\&infin;}{J}_1\left(\mathrm{\&theta;x}\right)Y_1\left(\mathrm{\&theta;x}\right)\mathrm{\&theta;}\frac{d}{\mathrm{d\&theta;}}\left[{\mathrm{\&theta;}}^3\frac{I\left(\mathrm{\&theta;}\right)}{I_0}\right]\mathrm{d\&theta;}---\left(1\right)$

Particle parameter x=π D/ λ wherein, D is a particle diameter, and θ is an angle of diffraction, and λ is a wavelength, and F is the focal length of lens, I ₀Be incident intensity.

Because measured value I (θ) has noise inevitably, formula (1) carries out numerical differentiation to the data that contain noise, not only can amplify noise, and is ill-posed problem, thereby can cause very big error.Unnecessary excessive concussion appears in the inversion result that shows as in big particle diameter and small particle size distribution scope.

A kind of novel integral inversion algorithm (application number: 200810240158.0) has been proposed in patent " a kind of integral inversion algorithm of grain diameter measurement ", this algorithm has obtained a kind of double integrator form inverting expression formula of size-grade distribution by the analytic solution of Hankel conversion and Scholmilch equation

$f\left(x\right)=\frac{{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{{\mathrm{\&lambda;}}^2{I}_{0}x}{\&Integral;}_0^{\&infin;}\mathrm{\&theta;}{J}_2\left(\mathrm{\&theta;x}\right)\frac{d\left[{\&Integral;}_0^{\frac{\mathrm{\&pi;}}{2}}{\left(\mathrm{\&theta;}\sin \mathrm{\&phi;}\right)}^{3}I\left(\frac{\mathrm{\&theta;}\sin \mathrm{\&phi;}}{2}\right)d\right]\mathrm{\&phi;}}{\mathrm{d\&theta;}}\mathrm{d\&theta;}---\left(2\right)$

Owing to adopted the described integral inversion algorithm of formula (2), internal integral is equivalent to a low-pass filter, thereby has reduced the influence of noise to measuring-signal.In formula (2) limit of integration of particle diameter be [0 ,+∞), yet in the practical application particle size distribution range can not all be approximately [0 ,+therefore ∞), be necessary to study the limited integral inversion algorithm of particle size distribution range, i.e. limited distribution integral inversion algorithm.

[summary of the invention]

The object of the present invention is to provide a kind of limited distribution integral inversion algorithm of grain diameter measurement, to reduce noise to the influence of inversion result and obtain in limited particle size distribution range size distribution more accurately.

The limited distribution integral inversion algorithm of a kind of grain diameter measurement proposed by the invention comprises the following steps:

Step 1: be f at first by measuring the particle size parameter distribution _a ^b(x) the particle swarm diffraction intensity distribution I (θ) of particle swarm on limited grain graininess interval [a, b];

Step 2: the analytic solution by Hankel conversion and Scholmilch equation obtain the inverting expression formula

Step 3: utilize the Gauss interpolation method that formula (3) discretize is handled and obtain

$f_a^b\left(x\right)=\frac{1}{2{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\frac{{\mathrm{\&pi;}}^3{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{{\mathrm{\&lambda;}}^2{I}_{0}x}\underset{k}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{{\mathrm{\&lambda;}}_k}^2{J}_2\left({\mathrm{\&lambda;}}_{k}x\right)}{{\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">J</figure-callout>}}_3^2\left({\mathrm{\&lambda;}}_{k}b\right)}\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{l}_m[3{\left(\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)\right)}^{3}I\left(\frac{{\mathrm{\&lambda;}}_k\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)}{2}\right)+\frac{{\mathrm{\&lambda;}}_k{\left(\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)\right)}^4}{2}{I}^{\&prime;}\left(\frac{{\mathrm{\&lambda;}}_k\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)}{2}\right)]---\left(4\right)$

Wherein, l _mAnd t _mBe respectively interpolation coefficient and interpolation knot, N represents the sum of angle of diffraction subregion, and M represents the interpolation knot sum.

Step 4: Gauss interpolation coefficient and interpolation knot are brought in the formula (4) after discretize is handled, thereby obtained size distribution f _a ^b(x).

The limited distribution integral inversion algorithm of a kind of grain diameter measurement that the present invention provides, its advantage and effect are: owing to adopted the limited integral inversion algorithm of particle size distribution range, the present invention to be applicable to that size distribution is in the situation of finite interval in the actual measurement.(3) directly adopt the form of splines differential in the formula without difference, reduced original reason error, only contain a Bei Saier function in the new algorithm, then the inversion result concussion significantly reduces.

[description of drawings]

Figure 1 shows that grain graininess measurement mechanism synoptic diagram

Figure 2 shows that and meet the particle size distribution figure that R-R distributes

Figure 3 shows that the diffraction light angular spectrum distribution plan of particle swarm

Figure 4 shows that emulation experiment inversion result distribution plan

Concrete label is among the figure:

1, semiconductor laser 2, beam expanding lens 3, sample cell

4, fourier lense 5, diffraction image acquisition system 6, computer system

[embodiment]

The present invention, promptly a kind of limited distribution integral inversion algorithm of grain diameter measurement comprises the following steps:

Step 1: be f at first by measuring the particle size parameter distribution _a ^b(x) the particle swarm diffraction intensity distribution I (θ) of particle swarm on limited grain graininess interval [a, b];

$I\left(\mathrm{\&theta;}\right)=\frac{{\mathrm{\&lambda;}}^2}{4{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2{\mathrm{\&theta;}}^2}{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HSA00000062213700021.png"\; state="\{\{state\}\}">I</figure-callout>}}_0{\&Integral;}_a^b{x}^2{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="HSA00000062213700021.png"\; state="\{\{state\}\}">J</figure-callout>}}_1^2\left(\mathrm{x\&theta;}\right)f_a^b\left(x\right)\mathrm{dx}---\left(5\right)$

Step 2: method for solving and Hankel conversion according to the Schlomilch equation have

{ λ in the formula _kB} is J _n(x) positive zero point, promptly

J _n(λ _kb)＝0(k＝1，2，…N) (7)

The exchange of integration and derivative has just obtained the inverting expression formula in the formula (6)

Step 3: utilize the Gauss interpolation method that equation (8) is carried out the discretize processing and obtain

$f_a^b\left(x\right)=\frac{1}{2{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\frac{{\mathrm{\&pi;}}^3{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{{\mathrm{\&lambda;}}^2{I}_{0}x}\underset{k}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{{\mathrm{\&lambda;}}_k}^2{J}_2\left({\mathrm{\&lambda;}}_{k}x\right)}{{\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">J</figure-callout>}}_3^2\left({\mathrm{\&lambda;}}_{k}b\right)}\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{l}_m[3{\left(\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)\right)}^{3}I\left(\frac{{\mathrm{\&lambda;}}_k\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)}{2}\right)+\frac{{\mathrm{\&lambda;}}_k{\left(\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)\right)}^4}{2}{I}^{\&prime;}\left(\frac{{\mathrm{\&lambda;}}_k\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)}{2}\right)]---\left(9\right)$

Wherein, l _mAnd t _mBe respectively interpolation coefficient and interpolation knot, N represents the sum of angle of diffraction subregion, and M represents the interpolation knot sum.

Step 4: Gauss interpolation coefficient and interpolation knot are brought in the formula (9) after discretize is handled, thereby obtained size distribution f _a ^b(x).

Below in conjunction with drawings and Examples the present invention is described in further details.

In grain graininess measuring process of the present invention, applied device as shown in Figure 1, it mainly is made up of semiconductor laser, beam expanding lens, sample cell, fourier lense, diffraction image acquisition system, computer system.Traditional laser particle analyzer major part all adopts this device.

This test macro principle of work is: the light of semiconductor laser output expands bundle through beam expanding lens, the parallel sample cell that incides, because in the pond is the particulate that floats through liquid, the diffraction pattern that on the focal plane of fourier lense, will present these population, by the diffraction image acquisition system with the diffraction pattern digitizing and send into computing machine, computing machine at first carries out digital filtering to this digital picture, export each gray values of pixel points then and with this light intensity I (θ) as each point, after the measured value of diffraction pattern has been arranged, according to inversion algorithm, the designing and calculating program just can obtain the population size distribution in the sample cell.

This algorithm is based on the Fraunhofer diffraction theory.In light scattering method grain graininess measuring process, when particle diameter 4 times greater than wavelength, the relative index of refraction of particle was greater than 1 o'clock, and the scattered light angular spectrum of (less than 6 degree) distributes to be similar to and regards Fraunhofer diffraction as in the low-angle zone of forward scattering:

$I\left(\mathrm{\&theta;}\right)=I_0{\left[\frac{D{J}_1\left(\mathrm{x\&theta;}\right)}{2\mathrm{\&theta;}}\right]}^2$

Wherein θ is an angle of diffraction, and D is a particle diameter, I ₀Be incident intensity, particle parameter x=π D/ λ, J ₁It is single order shellfish plug youngster function.

If consider that the particle size parameter distribution is f on finite interval [a, b] _a ^b(x) particle swarm under uncorrelated single scattering situation, has

$I\left(\mathrm{\&theta;}\right)=\frac{{\mathrm{\&lambda;}}^2}{4{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HSA00000062213700011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2{\mathrm{\&theta;}}^2}{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HSA00000062213700021.png"\; state="\{\{state\}\}">I</figure-callout>}}_0{\&Integral;}_a^b{x}^2{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="HSA00000062213700021.png"\; state="\{\{state\}\}">J</figure-callout>}}_1^2\left(\mathrm{x\&theta;}\right)f_a^b\left(x\right)\mathrm{dx}$

Generally speaking, dust size distribution f _a ^b(x) meet Rosin-Rammler (brief note is R-R) and distribute, as shown in Figure 2.Then the diffraction light angular spectrum of this particle swarm distributes (diffraction intensity adds 1% random noise) as shown in Figure 3.

Suppose in the emulation experiment, laser instrument sends the ruddiness that wavelength is 632.8nm, and scattered light converges on the diffraction image collector through fourier lense, and the focal length of fourier lense is 1m, incident intensity is 1cd, and the R-R that the size distribution of sample particle is obeyed 1 μ m to 100 μ m distributes.The interpolation knot number is 100 in the algorithm, and the upper limit b of particle parameter x is 10000.Inversion result as shown in Figure 4.

Claims (1)

1. the limited distribution integral inversion algorithm of a grain diameter measurement is characterized in that: comprise the following steps:

Step 1: be f at first by measuring the particle size parameter distribution _a ^b(x) particle swarm is gone up diffraction intensity distribution I (θ) at limited grain graininess interval [a, b];

Step 2: the analytic solution by Hankel conversion and Scholmilch equation obtain the inverting expression formula

Wherein, particle parameter x=π D/ λ, D is a particle diameter, and particle size distribution range is [a, b], and a＜b, and θ is an angle of diffraction, and λ is a wavelength, F is the focal length of lens, I ₀Be incident intensity.

Step 3: utilize the Gauss interpolation method that equation (1) is carried out the discretize processing and obtain

$f_a^b\left(x\right)=\frac{1}{2b^2}\frac{{\mathrm{\&pi;}}^3{F}^2}{{\mathrm{\&lambda;}}^2{I}_{0}x}\underset{k}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{{\mathrm{\&lambda;}}_k}^2{J}_2\left({\mathrm{\&lambda;}}_{k}x\right)}{J_3^2\left({\mathrm{\&lambda;}}_{k}b\right)}\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{l}_m[3{\left(\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)\right)}^{3}I\left(\frac{{\mathrm{\&lambda;}}_k\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)}{2}\right)+\frac{{\mathrm{\&lambda;}}_k{\left(\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)\right)}^4}{2}{I}^{\&prime;}\left(\frac{{\mathrm{\&lambda;}}_k\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}{t}_m)}{2}\right)]---\left(2\right)$

Wherein, l _mAnd t _mBe respectively interpolation coefficient and interpolation knot, N represents the sum of angle of diffraction subregion, and M represents the interpolation knot sum;

Step 4: Gauss interpolation coefficient and interpolation knot are brought in the formula (2) after discretize is handled, thereby obtained size distribution f _a ^b(x).

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