CN101430268B - Integral inverse-calculation method for grain diameter measurement - Google Patents

Abstract

The invention relates to a particle size measurement integral inversion arithmetic which is characterized by comprising the steps as follows: (1) particle group diffraction light distribution I(theta) with the particle size parameter distribution of f(x) is obtained by measuring; (2) an inversion expression of a double-integral form is obtained by analytic solutions of Hankel transform and Scholmilch equation; (3) the equation obtained in step (2) is carried out discretization treatment by a Gauss interpolation method; and (4) Gauss interpolation coefficient and interpolation node are substituted in a formula after the discretization treatment in step (3) to obtain the particle size distribution f(x). The integral inversion arithmetic is adopted, and internal integral is equal to a lowpass, thereby reducing the influence of noise on a measurement signal; and the new arithmetic only contains one Bessel function, so that the oscillation is reduced obviously.

Description

A kind of integral inverse-calculation method of grain diameter measurement

Technical field

The present invention relates to a kind of integral inverse-calculation method of grain diameter measurement, belong to the grain graininess fields of measurement.

Background technology

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

$\mathrm{xf}\left(x\right)=-\frac{8{\mathrm{\&pi;}}^3{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="H2008102401580E00011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{{\mathrm{\&lambda;}}^2}{\&Integral;}_0^{\&infin;}{J}_1\left(\mathrm{\&theta;x}\right)Y_1\left(\mathrm{\&theta;x}\right)\mathrm{\&theta;}\frac{d}{\mathrm{d\&theta;}}\left[{\mathrm{\&theta;}}^3\frac{I\left(\mathrm{\&theta;}\right)}{I_0}\right]\mathrm{d\&theta;}---\left(1\right)$

X=2 π a/ λ wherein, a 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.

This algorithm carries out numerical differentiation because measured value I (θ) has noise inevitably to the data that contain noise, not only can amplify noise, and be 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.

Summary of the invention

The object of the present invention is to provide a kind of integral inverse-calculation method of grain diameter measurement, to reduce noise to the influence of inversion result and obtain within the specific limits size distribution more accurately.

The integral inverse-calculation method of a kind of grain diameter measurement provided by the present invention comprises the following steps:

Step 1: at first by measuring the particle swarm diffraction intensity distribution I (θ) that a particle size parameter distribution is f (x);

Step 2:, obtained a kind of inverting expression formula of double integrator form by the analytic solution of Hankel conversion and Scholmilch equation

Step 3: utilize Gauss interpolation method discretize to handle above equation (2) and obtain

$f\left(x\right)=\frac{{\mathrm{\&pi;}}^3{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="H2008102401580E00011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{4{\mathrm{\&lambda;}}^2{I}_{0}x}\underset{n=0}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&theta;}}_n{J}_2\left({\mathrm{\&theta;}}_{n}x\right)\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{l}_m[\mathrm{\&alpha;}({\mathrm{\&theta;}}_{n+1},t_m)-\mathrm{\&alpha;}({\mathrm{\&theta;}}_n,t_m)]---\left(3\right)$

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.

$\mathrm{\&alpha;}({\mathrm{\&theta;}}_{n+1},t)=I[{\mathrm{\&theta;}}_{n+1}\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}t)/2]{\mathrm{\&theta;}}_{n+1}^3{\sin }^3(\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}t)---\left(4\right)$

Step 4: Gauss interpolation coefficient and interpolation knot are brought in the formula (3) after discretize is handled, thereby obtained size distribution f (x).

The integral inverse-calculation method of a kind of grain diameter measurement of the present invention, its advantage and effect are: owing to adopted above-mentioned integral inversion algorithm, internal integral is equivalent to a low-pass filter, thereby reduced the influence of noise to measuring-signal, and only contain a Bei Saier function in the new algorithm, then 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 integral inverse-calculation method of grain diameter measurement comprises the following steps:

Step 1: at first by measuring the particle swarm diffraction intensity distribution I (θ) that a particle size parameter distribution is f (x);

$I\left(\mathrm{\&theta;}\right)=\frac{{\mathrm{\&lambda;}}^2}{4{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="H2008102401580E00011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2{\mathrm{\&theta;}}^2}{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="H2008102401580E00012.png,H2008102401580E00021.png"\; state="\{\{state\}\}">I</figure-callout>}}_0{\&Integral;}_0^{\&infin;}{x}^2{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2008102401580E00021.png,H2008102401580E00022.png"\; state="\{\{state\}\}">J</figure-callout>}}_1^2\left(\mathrm{x\&theta;}\right)f\left(x\right)\mathrm{dx}---\left(1\right)$

Can be rewritten as

$\frac{4{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="H2008102401580E00011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2{\mathrm{\&theta;}}^2}{{\mathrm{\&lambda;}}^2{I}_0}I\left(\mathrm{\&theta;}\right)={\&Integral;}_0^{\&infin;}{x}^2{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2008102401580E00021.png,H2008102401580E00022.png"\; state="\{\{state\}\}">J</figure-callout>}}_1^2\left(\mathrm{x\&theta;}\right)f\left(x\right)\mathrm{dx}$

$={\&Integral;}_0^{\&infin;}{x}^2{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2008102401580E00021.png,H2008102401580E00022.png"\; state="\{\{state\}\}">J</figure-callout>}}_1^2\left(\mathrm{x\&theta;}\right)f\left(x\right)\mathrm{dx}$

Order

(3)

Promptly

(3’)

$={\&Integral;}_0^{\&infin;}x{J}_2\left(\mathrm{tx}\right)\mathrm{xf}\left(x\right)\mathrm{dx}$

And

$L\left(\mathrm{\&theta;}\right)=\frac{{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="H2008102401580E00011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2{\mathrm{\&theta;}}^2}{{\mathrm{\&lambda;}}^2{I}_0}I\left(\frac{\mathrm{\&theta;}}{2}\right)---\left(4\right)$

Then (2) formula can be written as

$L\left(2\mathrm{\&theta;}\right)={\&Integral;}_0^{\&infin;}{x}^2{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2008102401580E00021.png,H2008102401580E00022.png"\; state="\{\{state\}\}">J</figure-callout>}}_1^2\left(\mathrm{x\&theta;}\right)f\left(x\right)\mathrm{dx}$

Promptly

(6)

Step 2: according to the method for solving of Schlomilch equation, shape as

Equation, its continuous solution is

So

According to (3) and Hankel conversion, the n rank Hankel transform definition of f (r) is

$F_n\left(\mathrm{\&lambda;}\right)=H_n\left\{f\right\}={\&Integral;}_0^{\&infin;}r{J}_n\left(\mathrm{\&lambda;r}\right)f\left(r\right)\mathrm{dr}---\left(8\right)$

(9)

Here

Promptly

(10)

$={\&Integral;}_0^{\&infin;}x{J}_2\left(\mathrm{tx}\right)\mathrm{xf}\left(x\right)\mathrm{dx}$

Contravariant is changed to

$f\left(r\right)=H^{-1}\left\{F_n\right\}={\&Integral;}_0^{\&infin;}t{J}_n\left(\mathrm{tr}\right)F_n\left(t\right)\mathrm{dt}---\left(11\right)$

According to (9),

$\mathrm{xf}\left(x\right)=H^{-1}\left\{T\right\}={\&Integral;}_0^{\&infin;}\mathrm{\&theta;}{J}_2\left(\mathrm{\&theta;x}\right)T\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}$

So just, obtained a kind of inverting expression formula of double integrator form by the analytic solution of Hankel conversion and Scholmilch equation

Step 3: utilize Gauss interpolation method discretize to handle above equation (12) and obtain

$f\left(x\right)=\frac{{\mathrm{\&pi;}}^3{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="H2008102401580E00011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{4{\mathrm{\&lambda;}}^2{I}_{0}x}\underset{n=0}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&theta;}}_n{J}_2\left({\mathrm{\&theta;}}_{n}x\right)\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{l}_m[\mathrm{\&alpha;}({\mathrm{\&theta;}}_{n+1},t_m)-\mathrm{\&alpha;}({\mathrm{\&theta;}}_n,t_m)]---\left(13\right)$

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.

$\mathrm{\&alpha;}({\mathrm{\&theta;}}_{n+1},t)=I[{\mathrm{\&theta;}}_{n+1}\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}t)/2]{\mathrm{\&theta;}}_{n+1}^3{\sin }^3(\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}t)---\left(14\right)$

Step 4: Gauss interpolation coefficient and interpolation knot are brought in the formula (13) after discretize is handled, thereby obtained size distribution f (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 is mainly by semiconductor laser, beam expanding lens, sample cell, fourier lense, the diffraction image acquisition system, computer system is formed.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 spherical particle that floats through liquid, the diffraction pattern that on the focal plane of fourier lense, will present these population, with the diffraction image acquisition system signal is sent into data collecting card, send into computing machine again, 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 spectrum has been arranged, according to inversion algorithm, the designing and calculating program just can obtain the population size distribution structure 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 is greater than 1 o'clock, and the scattered light angular spectrum of (less than 6 degree) distributes to be similar to and thinks and satisfy Fraunhofer diffraction in the low-angle zone of forward scattering:

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

Wherein θ is an angle of diffraction, and a is a particle diameter, I ₀Be incident intensity, parameter x=2 π a/ λ, J ₁It is single order shellfish plug youngster function.

If consider that a particle size parameter distribution is the particle swarm of f (x), under uncorrelated single scattering situation, have

$I\left(\mathrm{\&theta;}\right)=\frac{{\mathrm{\&lambda;}}^2}{4{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="H2008102401580E00011.png"\; state="\{\{state\}\}">F</figure-callout>}}^2{\mathrm{\&theta;}}^2}{I}_0{\&Integral;}_0^{\&infin;}{x}^2{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2008102401580E00021.png,H2008102401580E00022.png"\; state="\{\{state\}\}">J</figure-callout>}}_1^2\left(\mathrm{x\&theta;}\right)f\left(x\right)\mathrm{dx}$

Generally speaking, dust size distribution f (x) meets Rosin-Rammler and distributes, as shown in Figure 2.

Then the diffraction light angular spectrum of this particle swarm distributes as shown in Figure 3.

Suppose in the emulation experiment that laser instrument sends the ruddiness that wavelength is 650nm, scattered light converges on the diffraction image collector through fourier lense, the focal length of fourier lens is 1m, incident intensity is 1cd, and the CCD pixel size is 50 μ m, and the valid pixel number is 1024.The particle diameter of sample particle is the R-R distribution from 0.09 μ m ~ 100 μ m.The interpolation knot number is 800 in the algorithm.Inversion result as shown in Figure 4.

Claims (1)

1. the integral inverse-calculation method of a grain diameter measurement, it is characterized in that: it comprises the following steps:

Step 1: at first by measuring the particle swarm diffraction intensity distribution I (θ) that a particle size parameter distribution is f (x);

Step 2:, obtained a kind of inverting expression formula of double integrator form by the analytic solution of Hankel conversion and Scholmilch equation

Wherein, x=2 π a/ λ, a 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;

Step 3: utilize Gauss interpolation method discretize to handle above equation (2) and obtain

$f\left(x\right)=\frac{{\mathrm{\&pi;}}^3{F}^2}{4{\mathrm{\&lambda;}}^2{I}_{0}x}\underset{n=0}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&theta;}}_n{J}_2\left({\mathrm{\&theta;}}_{n}x\right)\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}{l}_m[\mathrm{\&alpha;}({\mathrm{\&theta;}}_{n+1},t_m)-\mathrm{\&alpha;}({\mathrm{\&theta;}}_n,t_m)]---\left(3\right)$

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

$\mathrm{\&alpha;}({\mathrm{\&theta;}}_{n+1},t)=I[{\mathrm{\&theta;}}_{n+1}\sin (\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}t)/2]{\mathrm{\&theta;}}_{n+1}^3{\sin }^3(\frac{\mathrm{\&pi;}}{4}+\frac{\mathrm{\&pi;}}{4}t)---\left(4\right)$

Step 4: Gauss interpolation coefficient and interpolation knot are brought in the formula (3) after discretize is handled, thereby obtained size distribution f (x).

Images