The measurement principle of lar particle size
(Dandong Bettersize Instrument Ltd. Wangyongquan)
Introduction
At prent, lar diffraction particle size tester has been applied widely, especially in abroad, it has been recognized conformably. The remarkable features are: high measurement precision, fast respon speed, good repetitiveness, wide measurable particle diameter range and touchless measurement etc.
The rearch 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 veral years, we have developed multifold model lar particle size tester successfully, the main performance is alike with foreign same products.
The measurement principle of lar diffraction particle size tester
The operational principle of lar diffraction particle size tester we studied bas on Fraunhofer diffracti
on 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 lar diffraction particle size tester es attached drawing 1. Low energy lar nds monochromic light of wavelength 0.635 um, and the light pass 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 receivin
g lens λ：the wavelengt
h of incidence light
()()()12162
12
24
20
⎥⎦
⎤
⎢⎣⎡=X X J f d I I λπθ
J1 ：first order Besl function θ：scattering angle
When the diffracted light intensity distribution of the lar lays upon the No n ring of photoelectric detector (ring radius is from Sn to Sn+1, corresponding scattering angle is from θn to θn+1), the light energy is:
The equation (1) is substitute into I （θ）, then we get:
J0 ：zero order Besl function
If there are N quantity particles of diameter d, the received light energy on No n light
ring is N times （N ·en ）more 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 u W to reprent dimension distribution, the relation between W and N is:
In the equation ：ρ is particle density ，above equation is substitute into equation （4）,we get ：
The equation (6) t 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 u, so we divide the diameter into 96 ctions, the geometric shape of photoelectric detector e 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. Choo particle diameter ction according to the following formula calculating: in the formula ：
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θπsin d X =()()()
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==⎰
+n dS S I e n n
S S n πθ()()()()[]
()34
121120212
02
++--+=
n n n n n X J X J X J X J I d e π()()()()[]()
44
1
,21
1
,
20
,21
,20
20
++--+=
∑n i n i n
i n
i i
i n X
J X
J X J X J d N I e π()563
i i
i d W N πρ=
()()()()[]
()6231,211,20,21,2
00 ++--+=
∑n i n i n i n i i
i n X J X J X J X J d W I e ρ
97
20977.11
to i S S i i =*=-079536
.01=S 97
137.1to i f
S D i
i ==λπ
f ：receivin
g lens of focal lengt
h 180mm λ：miconductor lar of wavelength 0.635
Above formula shows the ction upper limit is Dn and ction lower limit is Dn+1 of No n particle grade in 96 particles grade.
The geometric mean value can be chon as the reprentative 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 p
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article dimension. To be convenient, we u 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 veral distribution function restrictive method, and assuming ction weights of 96 particles grade are W1、W2、W3…W95、W96, light intensity values of all rings are E1、E2、E3…E95、E96. Free distribution method:
First step: assuming initial value of every particle grade ction wight Wi is 1, institute
into formula (6), working out light intensity values of all rings e1、e2、e3…e95、e96, then we calculate light intensity variance by formula (7):
The variance is in variable χ, 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 ctions according to formula (9):
Second step: updated weight values Wi of all partic le grade ctions are institute into
formula (6), working out diffracted light intensity of all rings e1、e2、e3…e95、e96, and calculating light intensity variance by formula (7), comparing this variance with last variance, if σ2 is greater than χ, turn to third step; or:
96
11
to i D D d i i i =*=+()
()
72
96
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2 ∑=-=i i i E e σ()
8 i
学期末个人总结i i e
E =κ()
9 i
i i W W *=κ
Update χ 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 ctions according to formula (9).
Repeat cond step.
Third step: the value Wi is our want final all particle grade ctions weight. The ction 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 ：
()
10 ∑=
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i i W W f 11f R =()
1196
21 to i f R R i
i i =+=-()()()
122exp 21
22高中自评
⎥⎦
⎤⎢⎣⎡--==σπσϕu x dx d x f ()1
=⎰∞
∞
-dx x f ⎪⎭
⎫ ⎝⎛-=σu x t dx
dt =σ⎪⎪⎭
⎫ ⎝⎛-=2exp 21
2t dt d πϕ()
132exp 21
2
dt t d u
x ⎰⎰-∞-⎪⎪⎭
⎫
⎝⎛-=
σ
ϕ
πϕ
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 u least square method to fit percentage, work out Ri and corresponding t I in turn by interpolation in integral table, then solve coefficients σand u by the following formula.
Convert coefficients σand 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 lar 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 6.97-7.75μm. 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 1μm and 10μm, 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%.
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961961 ⎪⎪⎪⎪⎪⎩
⎪⎪
⎪⎪⎪
⎨
⎧⎪⎭⎫ ⎝⎛-⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛-⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛=⎪⎭⎫
⎝⎛-⎪⎭⎫ ⎝⎛⎪⎭⎫
⎝⎛⎪⎭⎫ ⎝⎛-=∑∑∑∑∑∑∑∑∑∑∑===========i i i i i i i i i i i i i i i i i i i i i i i i t t t x t t x u t t t x t x σ()
n
bx R -=exp 100()
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n b R +=⎪⎭⎫ ⎝
⎛
