JPH032251B2 - - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention relates to a method for measuring particle size distribution using a laser diffraction image. Measuring the size of particles and their distribution state is useful not only for particle engineering, cytology, and air pollution measurement, but also for quality control, manufacturing process control, etc. in the manufacturing industry of iron ore, food, medicine, etc. This has become an extremely important issue.

Conventionally, methods for measuring particle size distribution using laser light include light scattering spectroscopy and methods for determining particle size distribution from a diffraction image of laser light by matrix calculation.
It was theoretically impossible to derive a continuous particle size distribution over a wide range.

The present invention has been made in view of the above, and an object of the present invention is to provide a method for measuring continuous particle size distribution using a laser diffraction image.

The present invention will be explained below with reference to the accompanying drawings. FIG. 1 is an example of an apparatus used to carry out the measurement method of the present invention, and shows an optical system and a signal processing system for obtaining a diffraction image of a particle to be measured.

1 is a light source that emits coherent light, such as a laser device; 2 is an optical system that converts light into parallel light; 3 is a surface to be measured on which particles are randomly distributed; 4 is a Fourier transform lens; and 5 is a particle This is the surface from which a diffraction image (Fourier transform image) is obtained.
6 is a photoelectric conversion device that converts the light intensity of the obtained diffraction image into an electrical signal, such as a diode array, a TV
A camera, etc., and 7 a device for processing electrical signals, such as a computer.

The object of the present invention is achieved by the following steps.

Laser light is projected onto randomly distributed particles to create a diffraction image of the particles.

Since the particles are considered to be spherical particles, a rotationally symmetrical diffraction image is created, so the vector radius ρ on the diffraction image plane can be selected at an arbitrary position.

Diffraction image intensity I ₀ created by one spherical particle with radius a
(ρ) can be calculated as follows using Fourier transform.

I ₀ (ρ)=C・a ⁴・{J ₁ (aρ)/aρ} ² ...(1) Here, J ₁ (x) is a first-order Betzel function of the first kind. c is a constant.

Therefore, the diffraction image intensity I(ρ) of a particle group having the particle size distribution N(a) can be written as follows.

I (ρ) = K∫ ^∞ ₀ I ₀ (ρ) N (a) da = K∫ ^∞ ₀ a ⁴ {J ₁ (aρ) / aρ} ² N (a) da ... (2) a is the particle radius , ρ is the spatial coordinate on the diffraction image plane,
N(a) is the particle size distribution, and k is a constant.

Here, by placing the photodetector on the radial axis ρ,
When the light intensity is detected, an electrical signal proportional to I(ρ) can be obtained (FIG. 2).

The following is a mathematical problem in which N(a) is determined by knowing I(ρ), so the functions appearing individually have no physical meaning and are simply data realized within a computer.

Consider a new function K(ρ,t)=ρ ² J ₁ (Tρ) (Figure 3). The signal I obtained from this K(ρ, t)
(ρ) and calculate the integral (total sum of data) with respect to ρ at [0, ∞), a new function F
(t) can be found (Figure 4).

F(t)=∫ ^∞ ₀ I(ρ)K(ρ,t)dρ=
K∫ ^∞ ₀ da∫ ^∞ ₀ dρ・a ⁴ {J ₁ (aρ)／aρ} ²・ρ ² J ₁ (t
ρ)Na =K∫ ^∞ ₀ a ² N(a)da∫ ^∞ ₀
J ² ₁ (aρ) J ₁ (tρ) dρ……(3) From the mathematical formula collection Therefore, equation (3) is F(t) = K′∫ ^∞ ₀ √ ² −(2) ² N(a)da
...(4) becomes. K' is a constant.

Therefore, a function ρ ² J ₁ (tρ) is created in the computer, and the sum of the products of data and I(ρ) is calculated to obtain a new function F(t).

Next, by performing coordinate transformation such that a ² = A and (t/2) ² = T (Figure 5), equation (4) becomes becomes. However, when A<T, it becomes 0. constant
K″ is ignored.

Looking at this equation (5), we see that the function √U(A) [U(A)
is the unit step function illustrated in Figure 6,
√U(A) is a function (FIG. 7) that extends the value of A to the negative region. ] and function N(√)/
It becomes a cross-correlation function with √. Therefore,
Let's find the Fourier transform of both sides of equation (5). F
Let the Fourier transform of (T) be (ω), and √U
The Fourier transform of (A) is H(ω), N(√)/√
If that of A is (ω), then from equation (5), (ω)=H(ω)・(ω)...(6). From this formula where constant = 1, (ω) = (ω)・1/H(ω), and the inverse filter 1/H(ω) is (ω)
will be multiplied by the inverse Fourier transform. √
The Fourier transform H(ω) of AU(A) is It can be calculated as 1/H(ω)=H(ω)・4/π・2(iω) ³
...(8) can be written. Multiplying a function on the Fourier surface by (iω) ³ corresponds to differentiating it three times on the real function surface (Figure 8). Therefore, rewriting equation (6), (ω)=(ω)・{(H(ω)(iω) ³ }...(9
), and by inverse Fourier transform of both sides of equation (9), we get By calculating the cross-correlation function of F(T) and h(T) (Figure 9), N(√)/√
is found.

h(T) is the inverse Fourier transform of H(ω)√
It is obtained by differentiating U(T) three times. Therefore, h(T)=∂ ³ /∂T ³ {√U(T)}. When h(T) is calculated on a computer, the 10th
It will look like the figure.

Finally, transform the coordinate A so that A=a ² ,
Multiply by √ to find N(a) (11th
figure).

Now, the particle size distribution N(a) of particles with radius a in their natural state can be calculated using the logarithmic normal distribution (logarithmic normal distribution) of the following formula, which is generally used in powder and granular material engineering.
distribution), then N(a)=1/a・exp(−0.5[2log a] ² )
...(11) The electric signal I(ρ) representing the light intensity of the obtained diffraction image of the particle is I(ρ)=k∫ ^∞ ₀ a ⁴ {J ₁ (aρ)/aρ} ² N(a)da …
...(12) (where ρ is a spatial coordinate on the diffraction image plane, and k is a constant), which results in the relationship curve shown in FIG.

A new function F(T) is created by multiplying this electrical signal I(ρ) by ρ ² J ₁ (2√ρ) and integrating it in the range [0, ∞].
is the following formula F(T)=∫ ^∞ ₀ √−N√／√dA
...(13) (However, A=a ² ) This is expressed as the relationship curve shown in FIG.

Figure 14 shows the operator h(T) obtained by differentiating √U(T) (where U(T) is a step function) three times, and h(T)=∂ ³ {√U(T )]/∂T ³ ...(14)

Then, by determining the cross-correlation function of F(T) and h(T), that is, the cross-correlation function between FIGS. 13 and 14, and determining the particle size distribution N(a), the particle size distribution curve shown in FIG. 15 is obtained. is obtained. In the figure, A is the distribution curve of equation (11) assumed at the beginning, and B is the distribution curve obtained by the present invention comprising the above processing. The two distribution curves coincide with each other except for a portion, and it is understood that according to the present invention, a continuous particle size distribution can be measured with extremely high accuracy.

The above is an expression of calculation results assuming that the particle size distribution is a lognormal distribution.

Although the particle size distribution expressed by equation (11) has been explained as an example, the case where the apparatus shown in FIG. 1 is used will be explained. First, a laser device 1 and an optical system 2 project a laser beam onto randomly distributed particles, and a diffraction image of the particles obtained on a surface 5 via a Fourier transform lens 4 is converted into a light intensity by a photoelectric conversion device 6. is converted into an electrical signal I(ρ) representing . Next, using the processing device 7, the electric signal I(ρ) is converted into the above (12),
Process according to equations (13) and (14), and obtain the particle size distribution N(a) by taking the cross-correlation function of F(T) and h(T) expressed by equations (13) and (14). Bye.

As explained above, in order to use the diffraction image of laser light, the present invention allows particles whose diameters range from about half the wavelength of light to several centimeters by changing the optical systems 2 and 4 in FIG. can measure particle size distribution.

In addition, since no special processing is required on the sample, it can be easily introduced into the manufacturing process, and online measurement is possible, making real-time measurement possible due to faster computer processing. Therefore, the present invention, together with the ability to continuously and accurately determine the particle size distribution, is useful for speeding up and increasing the accuracy of quality control and process control, such as detecting abnormalities in manufactured products and determining optimal parameters for raw materials. can contribute.

[Brief explanation of the drawing]

FIG. 1 shows an example of an optical system and a signal processing system of an apparatus used in the present invention. Figures 2 to 11 are graphs showing functions in the process of developing mathematical formulas; Figure 12 is a graph showing the intensity of the diffraction image of particles having the assumed particle size distribution used to explain the present invention;
Figure 3 is a graph showing the function obtained by multiplying Figure 12 by ρ ² J ₁ (2√ρ) and integrating it. FIG. 14 is a graph showing operator h(T) used in the present invention. FIG. 15 is a graph showing the particle size distribution N(a) obtained by calculating the cross-correlation function between FIG. 13 and FIG. 14. Codes in the figure: 1... Laser device, 2... Optical system, 3... Surface to be measured, 4... Fourier transform lens, 5... Surface from which a diffraction image is obtained, 6... Photoelectric conversion device, 7... ...Electrical signal converter.

Claims (1)

[Claims] 1. A laser beam is projected onto distributed particles to create a diffraction image of the particles, and an electric signal representing the spatial distribution of light intensity of this diffraction image I(ρ)=k∫ ^∞ ₀ a ⁴ {J ₁ (aρ)/aρ} ² N(a)da (where a is the particle radius, ρ is the primary variable of the space in which the diffraction image appears, N(a) is the particle size distribution, and k is a constant). A particle size distribution measuring method using a laser diffraction image, characterized in that the electric signal is processed according to the following formula to determine the particle size distribution N(a). Multiply I(ρ) by ρ ² J ₁ (2√ρ) [0, ∞]
Integrate over the range of , perform variable transformation such that a ² = A, derive [formula], and derive the correlation function between operator [formula] (here U (T) is a step function) and F (T). Find N(a).