The measurement principle of laser particle size

The measurement principle of laser particle size

(Dandong Bettersize Instrument Ltd. Wangyongquan)

Introduction

At present, laser diffraction particle size tester has been applied widely, especially in abroad, it has been recognized conformably. The remarkable features are: high measurement precision, fast response speed, good repetitiveness, wide measurable particle diameter range and touchless measurement etc.

The research and production of the kind of instruments in China is comparatively short. The need of the instruments is at least 100 every year in home market, but the lowest price in foreign countries is 50,000 $ every machine, so our country pays foreign exchange 5000,000 $ in purchasing the kind of instruments at least every year. In latest several years, we have developed multifold model laser particle size tester successfully, the main performance is alike with foreign same products.

The measurement principle of laser diffraction particle size tester

The operational principle of laser diffraction particle size tester we studied bases on Fraunhofer diffraction and Mie scattering theory. We know from physics optics deduction that the scattering of incidence light vs particles accord with classical Mie theory. Mie scattering theory is rigorous mathematic root of Maxwell electromagnetic wave equation group, while Fraunhofer diffraction is only a kind of approximation of Mie scattering theory. Fraunhofer diffraction is applicable to the situation that particle diameter is far more than the incidence wave length, and is assumed that the light source and receiving screen are all boundlessly far away the diffracting screen. Considering from the theory, Fraunhofer diffraction is relatively simpler in application.

The basic device of laser diffraction particle size tester sees attached drawing 1. Low energy laser sends monochromic light of wavelength 0.635 um, and the light passes through space filtering and diffusing beam lens to filter miscellaneous light and form Max diameter 10mm parallel monochromic light beam. The light beam irradiates the particles in measurement area, and occurs light diffractive phenomenon. The intensity distribution of diffracted light follows to Fraunhofer diffraction theory. Fourier conversion lens at the back of measurement area is receiving lens (knowing lens range), scattered light forms far magnetic field diffractive graph on back focus surface. Multi-ring photoelectric detector on back focus surface of receiving lens can receive the energy of diffracted light and translate it into electric signal and output. The center hole of the detector measures the consistency of allowable sample volume. The diffractive graph of the particles is still and centralizes on light shaft range of the lens. So it does not matter that the particles are dynamic to pass through analyzing light beam, the diffractive graph is a constant to any lens distance. The lens conversion is optics, so it is very fast.

According to Fraunhofer diffraction theory, when a spherical particle of diameter d is within measurement area, its light intensity distribution of any angle is:

In the equation:

f the focal length of receiving lens

the wavelength of incidence light

J1 first order Bessel functionscattering angle When the diffracted light intensity distribution of the laser lays upon the No n ring of photoelectric detector (ring radius is from Sn to Sn+1, corresponding scattering angle is fromn ton+1), the light energy is:

The equation (1) is substitute into I, then we get:

J0 zero order Bessel function

If there are N quantity particles of diameter d, the received light energy on No n light ring is N timesNenmore than that of one particle. On the analogy of this, if there are Ni quantity particles of diameter di in the particles, total diffracted light energy in the particles is the sum of all particles diffracted light energy, that is,

If we use W to represent dimension distribution, the relation between W and N is:

In the equation is particle densityabove equation is substitute into equation4,we get

The equation (6) set up the corresponding relation between every ring diffracted light signal on photoelectric detector and particle diameter and distribution of measured particles.

In particle calculating, there are 96 effective rings on photoelectric detector we use, so we divide the diameter into 96 sections, the geometric shape of photoelectric detector see attached drawing 2, radius data of every ring is as follows (unit: mm):

Above formula shows inner radius is Sn and outer radius is Sn+1 of No n ring.

Choose particle diameter section according to the following formula calculating:

in the formula

f receiving lens of focal length 180mm

semiconductor laser of wavelength 0.635

Above formula shows the section upper limit is Dn and section lower limit is Dn+1 of No n particle grade in 96 particles grade.

The geometric mean value can be chosen as the representative value of particle diameter in every particle grade:

We can work out coefficient matrix by formula (6), once we mensurate light energy distribution E on 96 effective light rings, work out system of linear equations (6), we can get weight distribution W of particle dimension. To be convenient, we use least square method to process data. We assume that weight distribution W accords with some distribution rule (called distribution function restrictive method), or arbitrary initial value (called free distribution method), and calculate diffracted light energy of 96 rings on photoelectric detector, compare with real value one by one, until the error between two values is the least.

The following is the discussion on the solutions of free distribution method and several distribution function restrictive method, and assuming section weights of 96 particles grade are W1W2W3W95W96, light intensity values of all rings are E1E2E3E95E96.

Free distribution method:

First step: assuming initial value of every particle grade section wight Wi is 1, institute into formula (6), working out light intensity values of all rings e1e2e3e95e96, then we calculate light intensity variance by formula (7):

The variance is in variable , calculating proportionality coefficient between measured value and calculated value of every ring light intensity according to formula (8):

Update weight values Wi of all particle grade sections according to formula (9):

Second step: updated weight values Wi of all particle grade sections are institute into formula (6), working out diffracted light intensity of all rings e1e2e3e95e96, and calculating light intensity variance by formula (7), comparing this variance with last variance, if 2 is greater than, turn to third step; or:

Update value, calculate proportionality coefficient between measured value and calculated value of every ring light intensity according to formula (8), update weight values Wi of all particle grade sections according to formula (9).

Repeat second step.

Third step: the value Wi is our want final all particle grade sections weight. The section percentage can be calculated by formula (10):

The percentage greater than some a particle diameter (screen) is calculated by formula (11):

The normal distribution of distribution function restrictive method: the formula is

in the formula

assume

then

the formula (12) turn into

so

The formula (13) is a standard normal distribution function; there are integral tables in various statistics books.

The corresponding points of particle diameters and accumulative percentages on normal probability coordinate paper should appear like beeline. We can use least square method to fit percentage, work out Ri and corresponding tI in turn by interpolation in integral table, then solve coefficients and u by the following formula.

Convert coefficients and u into t value, then work out R by interpolation in integral table.

Rosin-Rammler distribution of distribution function restrictive method:

Assume the rest percentage on screen of hole diameter x is R, Rosin-Rammle educed from probability theory:

After simplification

The above formula is a linear equation; we can count two coefficients referring to formula (14), and return to get accumulative percentage.

Know about particles

Particle size distribution

To understand the meanings on output result of laser diffraction particle size tester, we need to explain some basic concepts.

First: The result is basis on volume. For example, result shows the distribution is 11% within the range 6.97-7.75m. That means total volume of all particles in this range is 11% of total volume of all particles in whole distribution. Simply, we assume the sample has two kinds of particles, their diameters are 1m and 10m, every kind is 50%, that is to say, the volume of a large particle is more 1000 times than that of small, so the volume of large particles is 99.9% of total volume. Certainly, for a kind of particle size distribution, the diameters of all particles are same, whether quantity or volume, the distribution is 100%.

Second: The result is indicated by equivalent sphere. Assuming the diameter of a columnar particle is20m, height is 60m, the volume is:

Convert it into a spherical volume, the diameter is between 20m and 60m:

Third: The derivation of distribution function. The distribution we analyze is particle size group indication of a set optical system with best resolution. All distribution parameters are educed from the basic distribution. In calculating, representative diameter of every particle size section means geometric mean of two-end values of this particle size section, there is a little different with arithmetic average.

Data, sample and background

The measuring output is the array of numerical value, including the test results of background and samples. If we want to get the practical scattering of sample particle its self, we must take out the background measurement from sample measurement, at the same time, must correct. After correction, background is:

In the formula

D is the passage No of data j

S is the sample measurement

B is the background measurement

Ob is opacity definition is

Ls is measured light intensity by center detector when a sample is put in the sample pool.

Lb is measured light intensity with only pure dispersing medium, but without sample.

Analyzing, deviation and data fit

Analyzing is that the measured data and other experiment parameter are processed with restrictive square method of iteration course and educed output.

Estimate an initial particle size distribution, and input a scattering matrix to optical model selected to predict forthcoming light scattering. It is showed in mathematics:

If vector Lj indicates measured data, vector Ri indicates result, and then they can connect by the following equation:

Lj=AijRi

Aij is scattering matrix included in optics correction, can be calculated according to scattering theory accurately.

Compare the calculated value with real value, and use a set of designed correct project to modify and adjust, repeat this process, until the calculated value and measured value tally with acceptable extent, at the time, output the particle size distribution as result.

Comparing the least square error of the calculated value with that of measured value, It is showed in mathematics:

R'is the result of present analyzing stage, with analyse going on, the present residual error is decreasing with the reducing of fit, the calculated value AijRi' is more closer to measured value Lj.

Optical Model

Laser diffraction particle size tester measures that the range of scattering angle is 0.01-15generally and the range of particle diameter is over 1m.

It is known that for such particle diameter and scattering angle, scattering character has no relation with the internal optical property of sample. Such instruments use usually Fraunhofer scattering theory, because it doesnt need any assumption to particle optical property.

Actually, when the particle diameter begins to reduce to below 10m and particles dip in liquid, or are transparency in optics, Fraunhofer scattering model occurs errors generally.

In order to measure the particle diameter of least 0.05m, the measure range of the instrument is broadened up to 135. The scattering of small particles has close relation with the optical property of scattering material under so big angle, and cannot be omitted. Mie Theory on scattering meets the case, which describes the scattering of optical uniform sphere and makes some assumptions to optical property of the particle. Mie Theory includes Fraunhofer scattering and anomalous scattering model entirely and they are conformable in proper range.

Choose the appropriate optical model for experiment from the instrument, and set it as standard value. Because of adopting the optical model, you neednt think of optical property of the sample when you use the instrument. However, to ensure high precise sample, the optical model is considered probably.

Derivative diameter and distribution statistics

The analysis outcome is the relative volume distribution of the particle among a series of particle size groups, and the distribution D[m,n] statistics is calculated on basis of the outcome, which is the recognized method by international. The particle outcome within two boundaries of particle size groups can be worked out by means of inner interpolation.

The definition of derivative diameter:

In the formula:

Vi are the relative volume among the groups, average diameter of the group is di. m and n is integerindicating the type of derivative diameter. D[43]volume weighted average diameterD[32]surface weighted average diameterD[10]arithmetic average diameter.

Distribution statistics

Average diameter

Standard deviation

Skew

Kurt

Specific surface area Specific surface areaSSAis the ratio of total area and gross weight of the particle.

Span and Uniformity

The definition of distribution span

Span is the distribution width and has no relation with middle particle diameter; uniformity is distribution pattern, also relation with middle particle diameter, shows the extent of distribution deviating from the middle. The definition is:

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Dandong Bettersize Instruments Ltd.

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