Download Improving of Measuring Method for Particle Size Distribution Based

A method of resolution increase and noise removal
in using shifrin-transform to measure particle size distribution
Han yue2 Yang Zong-Ling1 Qiao Xing1 Qian peng3 Yuan Yin-Nan1 Ding Si-Hong1 Dai bing1,2＊
1. School of Mechanical Engineering, Nantong University, Nantong, 226019, China; 2. School of Sciences,
Nantong University, Nantong, 226019, China; 3.School of Geography, Nantong University, Nantong,
226019, China)
Abstract: Based on laser diffraction and Shifrin-transform, the measurement method of particle size
distribution has been improved extensively. While, in real measurements, some noise peaks exist in the data
of inversion and are misread as particle distribution peaks. On the other hand, the improved method used a
truncation function as a filter is hard to distinguish adjacent peaks. In this paper, by introducing bimodal
resolution criterion, the filter function is optimized. And a quasi truncation function with the optimized
filter function is studied to achieve optimal bimodal resolution and remove noise peaks. This new quasi
truncation function fits multimode distribution very well. By combining the quasi truncation function with
Shifrin-transform, noise peaks are removed well and the adjacent peaks are distinguished clearly. Finally,
laser diffraction experiments are applied and the particle size distribution is analyzed by applying this
method. The results show that the quasi truncation function has better bimodal resolution than the
truncation function. Generally, by combining the quasi truncation function with Shifrin-transform, in
particle size distribution measurements with laser diffraction, the bimodal resolution is greatly increased
and the noise is removed well. And our results can restore perfectly the original distribution. Therefore, the
new method by combining the quasi truncation function with Shifrin-transform provides a feasible and
effective way to measure the multimode particle size distribution by laser diffraction.
Key Words: diffraction; particle size distribution; Shifrin-transform; quasi truncation function; inversion;
noise; resolution

This research was financially supported by the National Science Foundation Funds of China (Grant
No.51376095) and Jiangsu Province Environmental Research Projects (No.2014049).
1
In the measurement of particle size distribution
(PSD), the optical method has been widely used

I ( )  
0
because of its non-contact and quick testing
J 12 ( x ) x 2 f ( x )
dx
F 2 k 2 2
(1)
particles testing techniques, represented by Coulter
Where, k is the wave number, F is the focal
length,  is the scattering angle, and J 1 is the first
counter and Malvern particle analyzer, respectively,
level of the Bessel function of the first kind. And
characteristics [1-7]. Single particle and group
have been developed well. At the end of the last
f  x  is the distribution function, as a function of
century, due to the development of charge-coupled
particle size. Chin and Shifrin [8,10,11] obtained the
device (CCD) components, a method of PSD
analytic solution of the inverse problem as
measurement based on Shifrin-transform and laser
n( x )  x 3 f ( x ) 
diffraction was also developed quickly [8-12]. This
method has a high precision and the users don’t

2 xF 2 k 2  x J1 ( x )Y1 ( x )
0
require knowledge of the information such as the
d 3
[ I ( )]d
d
(2)
distribution model in advance [7,8,10,11]. However, it
has a significant defect [8,9,11,12]. That is, in actual
Where, n  x  is the distribution of particle quality
measurements, noise in the data of inversion can
as a function of size, and Y1 is the first level of the
easily be misread as PSD peaks, which blocks it
Bessel function of the second kind. Equation (2) is
applying widely. Recently，we proposed a filter that
called Shifrin-transform, which can invert the PSD
involved inserting a truncation function to avoid
using the angle distribution of the light scattering
misreading noise peaks [12]. However, in our recent
intensity.
experiments, the bimodal resolution was low because
Many previous reports [6,10-12] have showed that
of the adjacent peak overlap effect. In order to remove
the actual range of sampling angles needs to be
the noise peaks and increase its bimodal resolution, a
from  min to  max in equation (2) . So, data loss is
quasi truncation function, with an optimized filter
inevitable in actual sampling. This loss of data brings
function is proposed in this paper to increase the
some noise, which can be easily misread as
bimodal resolution. And then the quasi truncation
distribution peaks. In fact, this phenomenon makes
function is combined with Shifrin-transform to solve
shifrin method become a fatal flaw in the extended
the problem of noise and adjacent peak overlap. This
application quickly. After that, Dai Bing et al. [12]
method specifies an effective way for using
proposed a complete truncation function (CTF) based
Shifrin-transform and laser diffraction to obtain
on some characteristics of Fourier transform to
multimode PSD.
improve Shifrin method. The filter function was
defined as
1. Shortcoming of Shifrin-transform inversion
and CTF method
Particle size parameter is set to x ( x   d
,
 is the wavelength of incident light, and d is the
particle diameter), and x  1 . For a large number of
particles, if the optical thickness  is far less than 1,
A( )  (1  (
 3/2 3/2
) )
 max
(3)
In some experimental measurements [12], the
method can remove noise, perfect to restore the
original distribution. While, when two of the peaks are
then only single scattering can be considered. In fact,
closed, serious problems will be encountered. Fig.1
single scattering is appropriate assumptions for
shows the simulation inversion of PSD with three
practical
peaks using CTF and Shifrin method respectively.
problems
encountered
in
real
time
diffraction
Here, n  x  is the ratio of the particle quality with
approximation, the total scattering intensity on the
size parameter x and the total particle quality with all
focal plane of the objective lens is described as
sizes (the same as below), and defined as a positive
[10,11,13]
value according to its physics significance (the same
[1,3,10,11].
According
to
the
2
as below). In its original distribution, particle sizes at
In order to obtain the optimal filter function, based
three peaks are 25.0µm, 32.0µm, and 50.0µm,
on the influence of filter function on the noises and the
respectively, and the sampling angles are assumed to
peak
be within a range of 0.1 to 5.0 (seen also in Fig.2
and Fig.3). Due to the data collecting difficultly
performance of filter function, a quasi truncation
below 0.1 , the minimum of the sampling angle is
peak resolution than the complete truncation function
selected as 0.1 , and, due to the approximate
requirements of Fraunhofer diffraction, the maximum
and the other filter function. The quasi truncation
Especially, the noise peak at 42µm is easily misread as
distribution peak. For CTF method, the noises in
closed peaks become indistinguishable in CTF method.
It is fatal for restoring the original distribution.
the


   min
A( )  1  [
] 
  ( max   min ) 
1
1
 =1.15, = + ,    
2
2
Shifrin method are much larger than the CTF method.
the original distribution, overlap seriously. These two
adjusting
truncation
function (QCTF) is selected as follows
Fig.1, it is observed that the noises in inversion for
problem that the two peaks, at 25.0µm and 32.0µm in
by
function is found out, which has a better neighbor
of the sampling angle is selected as 5.0 . Seen from
inversion nearly disappear. But it causes another
resolution,

（4）
In order to explain the peak resolution problem,
here, a bimodal distinguish criterion, learning from the
light wave resolution method in wave optics [14,15],
is introduced here. When the distance (  ) of two peak
positions is half of the sum (L) of full-width at half
maximum of each peak, the two peaks can be
CTF method
2
can be
L
bimodal resolution,
appropriately distinguished. Therefore,
shfrin method
used
to
represent
the
2
2
 1 , two peaks can’t be
 2 . When
L
L
2
 1 , two peaks can be
distinguished. When
L
2
 1 , two peaks
appropriately distinguished. When
L
n(x)
and 0 
0
20
40
d/m
60
80
100
can be distinguished well. The larger is
Fig.1 Simulation inversion of PSD with three peaks
using CTF and shifrin methods. Particle sizes of peaks
are 25.0µm, 32.0µm and 50.0µm respectively.
2
, the higher
L
is the bimodal resolution.
Fig.2 is the simulation inversion of PSD with five
peaks using shifrin method, CTF method and QCTF
method. Peak particle sizes are 20.0µm, 26.5µm,
2. Proposal of a new method
40.0µm, 47.0µm, and 65.0µm respectively. Seen from
In order to restore the original distribution well, the
result of measurement needs to be small noise, high
resolution, and high accuracy. Reported works [8-12]
have
pointed
out
Shifrin
method
has
high
measurement precision. But it has also serious noises.
While CTF method has low noised, but low neighbor
peak resolution. Therefore, an optimal filter function
or the other methods to meet the requirements of both
the noise removal and high resolution are found out in
Fig.2, and according to our bimodal resolution criteria,
the neighbor peaks at 20.0µm and 26.5µm, as well the
ones at 40.0µm and 47.0µm, cannot be distinguished
for the CTF method, while, for the QCTF method,
they can be distinguished. Thus, the QCTF method
has higher bimodal resolution than the CTF method.
At the same time, the QCTF method, as well as the
CTF method, has better noise removal than the Shifrin
method, as shown in Fig.2.
the following section.
2.1 A quasi truncation function
3
QCTF method
CTF method
shfrin method
15.0µm, 23.0µm, 45.0µm, 51.5µm, 70.0µm, 75.0µm.
According to the QCTF inversion results, the new
method
is
redistricted
in
the
interval
of
0.0µm-100.0µm. And shifrin method is applied for
three intervals of 12.0µm-25.0µm, 42.0µm-53.0µm,
n(x)
and 68.0µm -78.0µm. The QCTF method is applied
for the other intervals.
current method
QCTF method
CTF method
20
40 d/m 60
80
100
Fig.2 Simulation inversion of PSD with five peaks
n(x)
0
using QCTF, CTF and shifrin method. Peak particle
sizes are respectively 20.0µm, 26.5µm, 40.0µm,
47.0µm and 65.0µm.
A lot of simulation experiments have indicated that
QCTF method has high bimodal resolution, and it is
0
20
40 d/m 60
80
100
suitable to simulate single, bimodal and multimodal
Fig.3 Simulation inversion of PSD with six peaks
particle distribution. However, QCTF method can
using CTF method, QCTF method and our current
reluctantly distinguish the two peaks at 20.0µm and
method. Peak particle sizes are 15.0µm, 23.0µm,
26.5µm in Fig.2. That is to say, on the basis of noise
45.0µm, 51.5µm, 70.0µm and 75.0µm, respectively.
removal, even the optimal filter function method is
difficult to solve fundamentally the adjacent peak
Seen from Fig.3, although the CTF method can
overlapping problem. The original reason is that,
remove noises, it is almost unable to restore the
besides the characteristics of the noise removal, the
original PSD because of the adjacent peak overlap
optimal filter function method stretches the peak
effect. Two peaks at 45.0µm and 51.5µm, as well as
width, which declines the bimodal resolution.
the ones at 70.0µm and 75.0µm, cannot be
2.2 The method of combination quasi truncation
distinguished. On the other hand, the bimodal
function with Shifrin-transform
resolution of the QCTF method is better the results of
Seen from Fig.1 and 2, Shifrin method has narrower
the CTF method, while the two peaks at 70.0µm and
peak width than the others. Then, a method of
75.0µm still cannot be distinguished. The method of
combination the quasi truncation function with
combination the QCTF and Shifrin-transform greatly
Shifrin-transform is proposed here to obtain better
increases the bimodal resolution. Seen from Fig.3, it
noise removal and narrower peak width. Firstly,
shows clearly that the six PSD peaks are separated
QCTF method is applied to do the inversion. Then the
well and, there has been almost no noise on the curve.
inversion interval is redistricted according to the
Thus, the current method combining the QCTF and
inversion results. After that, Shifrin method is applied
Shifrin-transform can perfectly solve both the noise
for those intervals with particle distribution peaks, and
obstacle and adjacent peak overlap effect.
QCTF method is applied for the other intervals. Fig.3
shows the simulation inversion of PSD with six peaks
3. Experimental verification
using CTF method, QCTF method and our current
method. The peak particle sizes are respectively
3.1 Equipment and method of experiment
4
As shown in Fig.4, the optical particle-sizing
measured by various techniques for the mixture of
instrument used in our study is mainly composed of a
GBW(E) 120006, GBW(E) 120042, GBW(E) 120045
He-Ne laser, an attenuation system, an extended
and GBW(E) 120046. Here, the laser wavelength is
filtering collimation system, a sample box, a lens, a
632.8 nm, the focal length of the objective lens was
wedge-shaped baffle, a line array CCD, a probe circuit,
300 mm. Then, the angle interval can be calculated
a data acquisition card, and a computer system. The
standard particle latex spheres were provided by the
as   11 m / 300mm  0.002101 . And the sampling
angle  min and  max are 0.3004 and 4.7941 ,
Nuclear Industry Beijing Institute of Chemical
respectively, and the total number of pixels is 2140.
Engineering.
Mean value of three tests was taken as the light
6
intensity distribution result. When the current method
was used to measure PSD with three peaks, according
to the results of QCTF method, Shifrin method was
1
applied for 17.0µm-30.0µm and 42.0µm-48.0µm
3
2
4
7
5
intervals, and QCTF method was applied for the other
intervals. When PSD with four peaks was measured,
comp-
data
probe
Shifrin method was applied for 17.0µm-30.0µm and
uter
card
circuit
42.0µm-53.0µm intervals, and QCTF method was
applied for the other intervals.
1-Laser 2- Attenuator 3-Extender filter 4-Sample
Seen from Fig.5 and Fig.6, results of Shifrin
5- Lense 6- Wedge baffle 7-CCD
method bring many noise peaks, while results of CTF
Fig. 4 Schematic of the optical particle-sizing
method result serious adjacent peak overlap effect.
instrument
Therefore, they cannot restore their original PSD.
The particles labeled GBW(E) 120006 (nominal
Results of QCTF method do not only remove the
peak size is 20.46±0.4µm), GBW(E) 120042 (nominal
noises, but also have certain bimodal resolution. The
peak size is 27.10±1.3µm) and GBW(E) 120045
neighbor peaks formed by GBW(E) 120006 and
(nominal peak size is 45.60±1.3µm) were suspended
GBW(E) 120042 are separated by QCTF method
in pure water, and then the mixture of PSD with three
(seen in Fig.5 and Fig.6). While the adjacent peaks
peaks was formed. The particles labeled GBW(E)
formed by GBW(E) 120045 and GBW(E) 120046 are
120046 (nominal peak size is 51.20±0.6µm) were
not distinguished by QCTF method (seen in Fig.6). On
dropped into the above suspension, and then the
the other hand, the method of combination the QCTF
mixture of PSD with four peaks was formed. By
with Shifrin-transform can clearly show all peaks of
adjusting the attenuation system, the scattered light
the original distribution on the basis of noise removal.
intensity distribution on the CCD was properly
As shown in Fig.5, and Fig.6, peak particle sizes are
obtained, and the obtained signal was converted to an
measured to be 20.3µm, 27.4µm, 45.7µm, and 51.5µm,
electric signal. The model of the Linear CCD was
respectively, which are agreement with the nominal
TCD103C (Toshiba) with 2592 pixels and the size of a
sizes of the samples. Hence, as analyzed from above
pixel was11.0µm. In order to reduce the influence of
results, the current method has not only high bimodal
the central spot, it was blocked by a wedge-shaped
resolution but also good effect of the noise removal.
baffle.
The
3.2 Results and analysis of experiment
Shifrin-transform can be applied widely to measure
Fig.5 is the results of PSD measured by various
method of combination
the QCTF with
the PSD of group particles by laser diffraction.
techniques, including the current method, QCTF
method, CTF method, and Shifrin method, for the
mixture of GBW(E) 120006, GBW(E) 120042, and
GBW(E) 120045. And fig.6 is the results of PSD
5
current method
QCTF method
CTF method
restore the original PSD.
(c).The current method fundamentally improves the
inversion method based on Shifrin-transform, and it
provides a feasible and effective way for measuring
shifrin method
n(x)
actual PSD by laser diffraction.
Acknowledgement
This research was financially supported by the
National Science Foundation Funds of China (Grant
No.51376095) and Jiangsu Province Environmental
Research Projects (No.2014049). We gratefully
acknowledge the help of Prof. Xiangdong Luo, who
0
20
40 d/m 60
80
100
give a lot of advisers in our work and English
polishing.
Fig.5 PSD obtained by various techniques for mixture
of GBW(E)120006, GBW(E)120042 and
Corresponding author
GBW(E)120045
Dai Bing, email:[email protected]
current method
QCTF method
CTF method
shifrin method
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Fig.6 PSD obtained by various techniques for mixture
Inversion
of GBW(E)120006, GBW(E)120042, GBW(E)120045
Determination for the Estimation of Particle Size
and GBW(E)120046
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在 Shifrin 变换测量粒度分布中
同时提高分辨率及去除噪声的方法研究
韩月 2 杨宗苓 1 乔星 1 钱鹏 3 袁银男 1 丁思红 1 戴兵 1,2
（1. 南通大学 机械工程学院，江苏 南通 226019；2.南通大学 理学院，江苏 南通 226019；
3. 南通大学 地理科学学院，江苏 南通 226019）
摘 要： 基于激光衍射和 Shifrin 变换测量颗粒尺寸分布的方法已经得到进一步改进，然而在
实际测量时，反演数据中存在一些易被误读为分布峰的噪声，另一方面，已有的以完全截断
函数为滤波函数的改进方法又存在不易分辨邻近双峰的缺陷。为此，文章引入双峰判据并优
化滤波函数，提出一种准截断函数，实现最优的双峰分辨率及去噪的目的，且能适应多种模
式的分布。通过将准截断函数与 Shifrin 变换相结合，既很好地去除噪声又使邻近峰可辩。最
后应用该法通过激光衍射实验对粒度分布进行了验证测量。结果表明：准截断函数方法较完
全截断函数方法的双峰分辨率有所提高，而将准截断函数与 Shifrin 变换相结合，可大大提高
双峰分辨率，并去除噪声，完美还原原分布。因此，该法为利用激光衍射反演测量多模式的
粒度分布提供了一条可行及有效的途径。
关键词：衍射；颗粒粒度；测量；Shifrin 变换；准截断函数；反演；噪声；分辨率

基金项目：国家自然科学基金资助项目，基金号：51376095。江苏省环保科研课题，项目号：2014049
作者简介：韩月（1990-），女，硕士研究生，从事颗粒的激光测试技术研究。
导师简介：戴兵（1964-），男，教授，从事颗粒的散射理论及测试技术研究。[email protected]
通讯联系人：戴兵，Email: [email protected]
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