346 CHINESE OPTICS LETTERS / Vol. 6, No. 5 / May 10, 2008
Application of multi-phase particle swarm optimization
technique to retrieve the particle size distribution
Hong Qi (
ààà
÷÷÷
), Liming Ruan (
___
ááá
²²²
), Shenggang Wang (
fff
),
Meng Shi (
¤¤¤
), and Hui Zhao (
ëëë
)
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001
Received August 11, 2007
The multi-phase particle swarm optimization (MPPSO) technique is applied to retrieve the particle size
distribution (PSD) under dependent model. Based on the Mie theory and the Lambert-Beer theory, three
PSDs, i.e., the Rosin-Rammer (R-R) distribution, the normal distribution, and the logarithmic normal
distribution, are estimated by MPPSO algorithm. The results confirm the potential of the proposed
approach and show its effectiveness. It may provide a new technique to improve the accuracy and reliability
of the PSD inverse calculation.
OCIS codes: 290.2200, 120.5820, 300.6360.
Particle siz e distribution (PSD) plays an important role
in the field of production processes, product quality, a nd
energy consumption, so it is highly required to on-line
monitor the granularity to provide real-time measure-
ments of both s ize distribution and particle concentra-
tion in the industrial fields. The retrieval of PSD with
non-intrusive optical measurement has shown broad de-
velopment space and huge potential gradually. Based
on the absorption and scattering characteristics of the
particle cloud, the PSD measurement by optical tech-
niques has a lot of advantages, such as high measure -
ment speed, well-repeated implement, wide measurement
size range, easy automatization, etc.. The development
trend of PSD measurement is to improve the meas ure-
ment accuracy and modify the inverse algorithm. Nowa-
days, various optical techniques have been used to de-
termine particle size, such as diffr action light scattering
method, total light scattering method, angle light scat-
tering method, dynamic light scattering method, and
transmittance method
[1]
. Among them, the diffraction
light s c attering method and the total light transmittance
method are the two most common experimental tech-
niques in practice. However, in many cases o f practical
interest, the ass umed PSD by these methods is nec e s-
sarily inaccurate because of limitations imposed by the
imperfect signa l-to-noise ratio (SNR) of the raw data,
coupled with the “ill-conditioned” nature of the deconvo-
lution algorithms which need to calculate the first Fred-
holm integral equation. Theoretically speaking, the PSD
inverse problem is actually a first Fredholm integral equa-
tion problem which is typically ill-po sed and difficult
to be solved directly. Thus, many random search in-
telligent algorithms have been introduced to inverse the
PSD problems, such as g e netic alg orithm (GA), simu-
lated annealing (SA), evolution strategies (ES), and ar-
tificial neural networks (ANN)
[2−4]
. Co mpared with the
traditional gradient methods, the intelligent optimiza-
tions have some outstanding characteristics. Firstly, both
linear and nonlinear or ill and non-ill inverse problems
could be solved. Secondly, the inverse problem with com-
plicated direct oper ator or without analytic expression
could be solved. Thirdly, only the functional value is
needed for the o bjective function, without explicit ex-
pressions. Fourthly, since the evaluation is carrie d out
by the fitness value, the gradient information and the
prior information about the unknown function are not
needed. Reference [3] applied the particle swarm opti-
mization (PSO) algorithm to solve the inverse problem
for determining the PSD from a light transmittance tech-
nique, and obtained some reasonable results. In this pa-
per, we apply the multi-phase particle swarm optimiza-
tion (MPPSO) algorithm, which can guarantee the con-
vergence of the global optimization solution with high
accuracy, to the inverse problem of particle distribution
under dependent model.
Among the optica l measurement methods, the light
transmittance technique is simple in principle and con-
venient for the optical ar rangement and is a more use-
ful dia gnostic tool for spatially and temporally resolved
measurement of PSD in a wide range o f applications. The
theoretical details of the transmittance technique ar e dis-
cussed in the following.
When a collimated laser beam passes through a suspen-
sion of particles, the transmitted light will be attenuated
due to the absorbing and scattering of the particles. Ac-
cording to the Lamber t- Beer theory, if the suspensions of
particle cloud are polydispers e spheres and the multiple
scattering and interaction effects can be neglected, the
transmitted light intensity I may be expressed as
I = I
0
exp
"
−
π
4
Z
D
2
D
1
N
0
f(D)D
2
LQ
ex
(x, m)dD
#
, (1)
where I
0
is the incident light intensity, L is the mean
distance through which the laser passes, N
0
f(D) is the
number co nce ntration of the particles with diameter D
which is the PSD function to be measured, N
0
is the to-
tal number density of the particles, Q
ex
(x, m) is the ex-
tinction efficiency factor which is dependent on the size
parameter x = πλ/D and the complex refractive index
m and can be calculated by the Mie scattering theory
[5]
.
For a particle cloud with fixed PSD function, as suming
the amount of particles with diameter D
i
is N
i
, if there
are several incident laser light be ams with different wave-
length λ
j
(j = 1, 2, ··· , k), the following form equation
1671-7694/2008/050346-04
c
2008 Chinese Optics Letters
资源评论
