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Particle sizing with improved genetic algorithm by ultrasound attenuation spectroscopy
Particle sizing with improved genetic algorithm by ultrasound attenuation spectroscopy Huinan Yang, Mingxu Su, Xue Wang, Jianfei Gu, Xiaoshu Cai PII: DOI: Reference:
S0032-5910(16)30507-1 doi: 10.1016/j.powtec.2016.08.027 PTEC 11864
To appear in:
Powder Technology
Received date: Revised date: Accepted date:
30 November 2015 18 May 2016 10 August 2016
Please cite this article as: Huinan Yang, Mingxu Su, Xue Wang, Jianfei Gu, Xiaoshu Cai, Particle sizing with improved genetic algorithm by ultrasound attenuation spectroscopy, Powder Technology (2016), doi: 10.1016/j.powtec.2016.08.027
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ACCEPTED MANUSCRIPT Particle sizing with improved genetic algorithm by
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ultrasound attenuation spectroscopy
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YANG Huinan, SU Mingxu*, WANG Xue, GU Jianfei, CAI Xiaoshu
Institute of Particle and Two-phase Flow Measurement / Shanghai Key Laboratory of Multiphase Flow and Heat
Transfer in Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
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*Corresponding author, E-mail: [email protected]
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Abstract: As a global optimization algorithm, genetic algorithm is an advantageous tool due to its global convergence, great robustness, suitable for parallel computing and so on. Selection of optimal
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parameters, e.g., maximum generations, population size, and genetic operators (crossover fraction
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and mutation fraction), is extremely crucial for particle size characterization by ultrasound
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attenuation spectroscopy with genetic algorithm. A series of particle system with different distribution functions were numerically investigated in this work. It revealed that the simulated results were
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consistent with the given particle size distribution when the maximum generations, population size crossover fraction and mutation fraction were appropriate chosen. Furthermore, the anti-noise performance of genetic algorithm and its three improved forms applied in particle system with bimodal distribution were also studied. Two groups of samples (micron-sized glass beads-glycerol suspension and aqueous polystyrene suspension) were investigated experimentally by ultrasound attenuation spectroscopy, and it showed a good agreement between Improved Genetic Algorithm 3
(IGA3) and microscope image analysis (MIA).
Keywords: Ultrasound spectroscopy; Particle sizing; Inversion; Genetic algorithm; Optimized parameter
1. Introduction
ACCEPTED MANUSCRIPT Particle sizing in two-phase flow is of great importance with the development of energy, petrochemical, mechanical, material preparation, biological and pharmaceutical industries. In
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recent years, ultrasound technique is widely used in particle sizing, compared to optical and
opaque,
undiluted
and
high/non-conducting
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electrical methods, ultrasound technique is advantageous due to its availability to optically dispersions
[1].
Ultrasound
attenuation
spectroscopy for particle sizing mainly includes three aspects: establishment of mathematical
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model, measurement of ultrasound attenuation spectroscopy, and the retrieval of particle size
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distribution. And the accuracy of the retrieval results would be influenced by selection of inappropriate inversion algorithms and unreasonable parameters.
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Several inversion algorithms have been developed in the previous researches, e.g.,
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Broyden-Fletcher-Goldfarb-Shanno
(BFGS)
algorithm,
Davidon-Fletcher-Powell
(DFP)
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algorithm, and Levenberg-Marquard (LM) algorithm, Twomey method and its improved form [2,3], Optimum-Regularization-Technique(ORT) [4], Chahine algorithm [5-7], Particle Swarm
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Optimization (PSO) [8-9], Ant Colony Optimization (ACO) [10], Fruit Fly Optimization (FOA) [11], and Genetic Algorithm (GA) [12]). Among them, the characteristics of genetic algorithm like universality, robustness, and feasible for parallel computing, etc., promote more and more attention and application in particle sizing [13-16]. In the present work, genetic algorithm with global optimal characteristics on particle sizing by ultrasound attenuation spectroscopy was investigated. Four optimal parameters (maximum generations, population size, crossover fraction and mutation fraction) were chosen with three different forms of distribution functions (Rosin-Rammer(R-R), Gaussian or Log-Normal). And the anti-noise performance of genetic algorithm was also discussed with numerical simulation.
ACCEPTED MANUSCRIPT Then, three improved forms of genetic algorithm applied in particle systems with bimodal distribution were studied. A series of experiments were performed with two groups of samples
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(micron-sized glass beads-glycerol suspension and aqueous polystyrene suspension) to obtain
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the particle size distribution, and the deduced results were compared with the data measured by microscope image analysis.
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2. Methods 2.1The inversion problem
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For the dependent model algorithm in inversion problems, an optimization procedure is carried
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out by minimizing the simulated attenuation spectrum and measured one. For example, the
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expression of R-R distribution function can be written as: (1)
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where dV/dR is the frequency distribution of particle volume, it is a function of radius R.
is the
characteristic radius, k is the distribution parameter, which is a dimensionless quantity (the
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larger the distribution parameter, the narrower the distribution). This optimization function is unique if
and k are determined. Therefore, an objective function about
and k should be
established for optimization in inversion problems. Here, Epstein-Carhart-Allegra-Hawley (ECAH) model [17, 18] is employed to predict the ultrasound attenuation spectroscopy. Considering the interaction between ultrasonic wave and dispersed spherical particle, three wave modes could be derived as a compression wave, a transverse wave and a thermal wave that could exist in both phases (dispersed particle and continuous fluid) of the system. The abovementioned three wave modes indicate that the ultrasound attenuation prediction could be a complicated and multi-parameters related process, which can be deduced by solving a linear
ACCEPTED MANUSCRIPT equation of six orders derived from the wave equations after applying the boundary condition on interface between the suspended spherical particle and continuous fluid. Thus for a
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polydisperse particle system containing the particle with various sizes, the overall attenuation
every single particle, namely:
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∞
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and phase velocity could be evaluated from the algebraic superposition of contribution from
where
(2)
is the theoretically predicted ultrasound attenuation spectrum. f is the
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ultrasound frequency, φ is the concentration of particle system,
is the wavenumber of the
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incident wave. The summation from i=1 to M represents the total contribution of each size bin, , where
is the scattering coefficient related to
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with a dispersed phase volume fraction
the model. The root mean square error describing the difference between the simulated and the
(3)
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measured spectra is:
where ameas, j is the measured spectrum, N is the number of selected frequencies in all the ultrasound attenuation spectrum. Dependent model algorithms, such as BFGS algorithm, DFP algorithm and LM algorithm are usually employed to solve Eq. (3). However, when the initial value or the particle size range is inappropriately chosen, it would lead to an inaccurate particle size distribution. And this situation can be avoided with global optimization genetic algorithm since multiple search points are utilized to search information simultaneously. Fig.1 showed the simulated particle size distribution with four kinds of algorithms (BFGS
ACCEPTED MANUSCRIPT algorithm, DFP algorithm, LM algorithm and Standard Genetic-Algorithm (SGA)) for the particles with radius 5 μm, the particle range was assumed between 0 and 100 μm, and initial
and k in Eq. (1), and the particle size distribution can be then
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algorithms to determine
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radius was 50 μm. The objective function was minimized with four different optimization
obtained. As shown in Fig.1, the median particle radius determined by BFGS, DFP and LM algorithm were 3.23 μm, 37.59 μm and 41.41 μm, respectively. While the radius calculated by
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SGA was 5.72 μm, it was in good agreement with given parameter.
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2.2 Genetic algorithm and related parameters Genetic algorithm is an efficient optimization method based on the principles of natural selection,
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searching global optimization solution without any initialization information [19-23]. As shown in
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Fig.1, the distribution parameter by SGA was different from the given one because of the
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randomness of genetic algorithm. In order to improve the measurement accuracy of SGA, Optimized-Genetic-Algorithm (OGA) which combined genetic algorithm and local optimization
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algorithm (BFGS, DFP, LM algorithm) is presented here. To improve the converging efficiency and accuracy of OGA, the influence of four related parameters, i.e., maximum generation, population size, crossover fraction and mutation fraction on the retrieval results were investigated and appropriate value or ranges of these parameters for particle sizing by ultrasound attenuation spectroscopy were found. When computational generation reaches to its maximum value (i.e., maximum generation), the calculation process ends. The operation efficiency of retrieval calculation would be greatly reduced, if the maximum generation is too large. However, if it is too small, the calculation process ends even when an optimum value is not found. Thus, it is critical to select an
ACCEPTED MANUSCRIPT appropriate value of maximum generation for the inverse problem. For a 500 times iteration optimization of objective function (i.e., the error function of Eq. (3)) with genetic algorithm, it
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revealed that when the computational generation was up to about 280, the optimal fitness value
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was almost constant.
Population size refers to the size of each generation, and it affects the search capability and running efficiency of the retrieval process. If the population size is larger, more models would be
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covered by one evolution, the diversity of the population can be guaranteed and the searching
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ability would be improved. But the operation efficiency is reduced due to more population chromosomes. Here, the retrieval calculation was performed when population size was chosen
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from 20 to 100 with different distribution parameters with three kinds of distribution functions
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(R-R, Gaussian and Log-Normal) and particle sizes (5 μm, 10 μm and 20 μm). It showed that
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the optimal fitness value was almost constant when the population size was larger than 50. The average numbers of population chromosomes in mating in the process of evolution is
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determined by the crossover fraction. When the crossover fraction is larger, the producing speed of new individual is faster and each generation would be fully crossed. However, the possibilities of breaking the fine pattern in population are increased and the structure of the superior individual will be destroyed. It would result in larger generation gap and randomization. While the crossover fraction is too small, the retrieval process would be stopped, if many individuals are directly copied to the next generations. Here, the retrieval calculation was carried out when crossover fraction was chosen between 0 and 1 for different step lengths with three kinds of distribution functions and particle size. It was found that the appropriate range of crossover fraction should between 0.4 and 0.85 for 50 times averaged calculation results.
ACCEPTED MANUSCRIPT The average numbers of the mutated population in the process of evolution is determined by the mutation fraction. If the mutation fraction is larger, more new individuals would be generated
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and the diversity of the population is increased. However, many fine patterns would be
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damaged. If the mutation fraction is too small, the solution would be much stable, but the abilities of generating new individual and restraining premature phenomenon would be poor. Here, the retrieval calculation was performed when mutation fractions were chosen from 0.0001
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to 0.1 for different distribution parameters with three kinds of distribution functions and particle
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size. It showed that the appropriate range of mutation fraction should be between 0.045 and 0.08 for 50 times averaged calculation results.
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Therefore, the maximum generation, population size, crossover fraction and mutation fraction
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were chosen to be 300, 60, 0.5 and 0.065 in the present work, respectively. The retrieval
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calculation was carried out with OGA and SGA for different distribution parameters with three kinds of distribution functions (R-R, Gaussian and Log-Normal) and particle sizes (5 μm, 10 μm
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and 20 μm), and the deduced 20 times averaged results were listed in Table 1. The relative error of distribution parameters determined by OGA was less than 5%, it revealed that the results were more accurate than SGA. Although the calculation time with OGA was 10 times longer than with SGA, the accuracy of retrieval calculation was greatly improved. Thus, OGA is available for retrieval calculation problem on particle sizing by ultrasound attenuation spectroscopy.
2.3 The anti-noise performance of OGA To investigate the anti-noise performance of OGA, the retrieval calculation was carried out with OGA by adding the measured ultrasound attenuation spectrum with WGN (White Gaussian
ACCEPTED MANUSCRIPT Noise) of 20/10/8/5/3 dB (i.e. with 5%~40% random error) with three kinds of distribution functions (R-R, Gaussian and Log-Normal) and particle characteristic size of 10 μm. The
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retrieved particle size distributions with adding different noises were compared with no noise
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and the given particle size distribution in Fig.2. As shown in Fig.2a, for the unimodal R-R distribution (distribution parameter k=7), the errors of radius and distribution parameters retrieved with OGA were less than 5% and 8%, respectively, when adding the ultrasound
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attenuation spectrums with WGN of 20/10/8/5/3 dB. For the unimodal Gaussian distribution
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(distribution parameter σ=3E-6), the radius retrieved with OGA was larger than 10 μm with noise of 3 dB and the errors were less than 8% with noise of 20/10/8/5 dB (Fig.2b).The errors of
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distribution parameters retrieved with OGA were less than 10% if adding the ultrasound
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attenuation spectrums with WGN of 20/10/8/5/3 dB. For the unimodal Log-Normal distribution
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(distribution parameter σ=0.5), the radius inversed by OGA was larger than 10 μm (Fig.2c) with noise of 3 dB and the errors were less than 10% when adding the WGN of of 20/10/8/5 dB. And
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the errors of distribution parameters retrieved by OGA were less than 10% when adding the noise of 20/10/8/5/3 dB. Therefore, it was found that the R-R distribution function should be employed due to the best anti-noise performance of OGA.
3 Three improved forms of genetic algorithm In addition to the unimodal distribution form mentioned above, there are bimodal and multimodal distribution forms in particle systems, and the bimodal condition is more commonly employed in practical application. Therefore, the genetic algorithms of three improved forms for bimodal distribution were presented here: Improved Genetic Algorithm 1 (IGA1), Improved Genetic Algorithm 2 (IGA2) and Improved Genetic Algorithm 3 (IGA3). A flowchart of these three
ACCEPTED MANUSCRIPT improved forms of genetic algorithms (IGA1, IGA2, and IGA3) was shown in Fig.3.
and distribution parameter k1 of the first
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For IGA1, five parameters (characteristic radius
and distribution parameter k2 of the second mode, and the
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mode, characteristic radius
distribution (
: 20 μm; k1:10;
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proportion value α of the first mode) were determined by OGA. A particle system with bimodal : 60 μm; k2:25; α=0.5) was investigated in the work. The
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comparison between the retrieval results of bimodal R-R distribution system determined with IGA1 and the given distributions were shown in Fig. 4a. It revealed that retrieved
was 23.47
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μm, it was apparently larger than 20 μm, distribution parameter of the second mode was 16.95,
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it was obviously smaller than 25. Therefore, it was found that IGA1 was not optimal for the
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retrieval problem of bimodal distribution situation. IGA2 which combined genetic algorithm with local optimization algorithm was utilized to retrieve bimodal distribution too. With IGA2, three ,
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important parameters (
, α)) were determined by genetic algorithm, and they then served
as the initial values of local optimization algorithm. These five parameters were then determined
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by the local optimization algorithm. The comparison between the retrieval results of bimodal R-R distribution system determined with IGA2 and the given distributions were shown in Fig. 4b. It was found that the retrieved parameters
with IGA2 were 20.84 μm, it was more accurate
than with IGA1, but k2 was 10.00, it was obviously smaller than 25. As both the retrieval results with IGA1 and IGA2 were not ideal, a kind of step-by-step optimization algorithm named IGA3 has been developed here: three parameters (
,
, α)
obtained by genetic algorithm, only k1 and k2 determined by the subsequent local optimization algorithm. The comparison between the retrieval results of bimodal R-R distribution system determined with IGA3 and the known parameters were shown in Fig. 4c. It was found that
,
ACCEPTED MANUSCRIPT k1,
k2, and α were 20.78 μm, 11.63, 59.84 μm, 24.19 and 0.49, respectively. It revealed that
the retrieved five parameters by IGA3 were improved relative to the former two algorithms,
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4. Experimental results and discussion
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4.1 Experimental setup and measurement method
An ultrasonic measurement system shown in Fig.5 developed in the previous work [24] was
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employed in the present work. A double-sample-cell method is used to obtain the ultrasound attenuation spectrum. The constant-temperature trough is filled with water (the absorption ) and the temperature is kept at 20 °C. Ultrasound waves generated by a
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coefficient is
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broadband ultrasonic transducer 1 (M310-SU) with an ultrasonic transmitter/receiver (PR-5800,
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Parametric) are transmitted through sample cell 1 (l1=10 mm), and then received by transducer 2. The signal is received by a two-channel high-speed data acquisition card (PCI-5114, NI) and
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recorded in a personal computer. The amplitude A1 is determined by fast Fourier transform with the original narrow pulse time-domain signals through transducer 2. The ultrasound waves
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generated by transducer 1 are then transmitted through sample cell 2 (l2=20 mm), and received by transducer 2. The amplitude A2 can be determined. And attenuation coefficient can be then obtained by:
(3) 4.2. Micron-sized glass beads-glycerol suspension samples Two kinds of micron-sized glass beads-glycerol suspensions with different particle size distributions (density is 4.19 g/cm3, volume concentration is 1% and sample numbers are SB040103 and SB040107, respectively) were investigated. The samples were mechanically stirred for three minutes before measurement by ultrasonic system. Fig.6 showed the
ACCEPTED MANUSCRIPT comparison of the particle size distribution for these two kinds of samples measured by Microscope Image Analysis (MIA) (PIP8.1, OMEC) and obtained with OGA (a), IGA1 (b), IGA2
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(c) and IGA3 (d) by ultrasonic attenuation spectroscopy (UAS), respectively. The radiuses
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obtained by MIA were 9.62 and 27.09 μm. As shown in Fig.6a, the retrieval results by OGA were 9.51 and 28.42 μm. It showed that two kinds of glass beads-glycerol suspension samples could be clearly distinguished. And the relative error for retrieved particle size distribution between
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OGA and MIA for sample SB040107 was 7.7% larger than that of sample SB040103. Because
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particle size of sample SB040107 is larger, it resulted in relatively higher speed of particle falling during measurement process. As shown in Fig.6b, the retrieval result of sample SB040107 by
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IGA1 was conformed to a bimodal distribution, although it should be a unimodal distribution.
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And there was a small peak at ～70 μm for sample SB040107 by IGA2 depicted in Fig.6c. And
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the retrieval result of sample SB040107 by IGA3 (Fig.6d) was a unimodal distribution. They were all unimodal distributions for the retrieval results of sample SB040103 by three improved
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algorithms (IGA1, IGA2, and IGA3) and the retrieved distribution parameter by IGA3 was the most consistent with the measured results by MIA.
4.3. Aqueous polystyrene suspension samples Three kinds of aqueous polystyrene suspensions (density is 1.05 g/cm3, volume concentration is 10% and the sample numbers are named as 1, 2 and 3, respectively) with different particle size distributions were investigated. Before being measured by ultrasonic measurement system, the samples were mechanically stirred for three minutes and dispersed for five minutes with ultrasonic dispersion instrument (FS-250, Shanghai Sonxi Ultrasonic Instrument) to eliminate part of the aggregated particles. The comparison of the particle size distributions for these
ACCEPTED MANUSCRIPT samples measured by MIA, OGA and other three improved algorithms (IGA1, IGA2, and IGA3) was shown in Fig.7. It revealed that the measured radiuses by MIA were 4.01, 10.83 and 31.79
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μm, respectively. The curves in Fig.7a indicated the retrieval results with OGA and the radiuses
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were 3.48, 11.07 and 28.36 μm, respectively. The three kinds of aqueous polystyrene suspension samples can be obviously distinguished. And the relative error of particle radius and frequency distribution for sample 3 was the largest, because the particle size of sample 3 were
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the largest which caused the higher speed of particle falling during measurement process. And
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acoustic scattering would be enhanced rapidly if the particle size and frequency increased, which led to the narrow of effective frequency band of ultrasound signals and the measurement
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accuracy was then influenced. For sample 3, the relative error between the retrieval results with
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IGA3 (Fig.7d) and MIA was 4.9%, which was better compared to IGA1 (7.6%, Fig.7b) and IGA 2
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(13.1%, Fig.7c), respectively. It showed that the retrieval results by IGA3 were the most consistent with the MIA measurements.
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5. Conclusions
The retrieval calculation with genetic algorithm on particle size characterization by ultrasound attenuation spectroscopy was presented in this paper. The global optimization genetic algorithm can well avoid converging to local optimal solution because of the global convergence. The optimal parameters of retrieval calculation problem on particle sizing by ultrasound attenuation spectroscopy based on the classical ECAH mathematical model were chosen: maximum generation was 300, population size was 60, crossover fraction was 0.5 and mutation fraction was 0.065. In order to verify the performance of the developed algorithms, both numerical investigation and experimental verification were employed. The unimodal and
ACCEPTED MANUSCRIPT bimodal distribution particle system were studied in the work. It was found that the performance of OGA3 was the best by comparing the retrieval results and given parameter or the data
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determined by MIA.
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Acknowledgments
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The authors gratefully acknowledge the support from the National Natural Science Foundation (Project No: 51176128, 51076106, 51306123), Shanghai Science and Technology
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Commission (Grant No: 13DZ2260900) and Project of Shanghai Science and Technology
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Commission (Grant No: 12ZZ142).
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ACCEPTED MANUSCRIPT [24] Zhang Wei, Su Mingxu, Cai Xiaoshu. Particle size distribution measurement based on
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ACCEPTED MANUSCRIPT Tables Table 1. Comparison of inverse results with three kinds of distribution functions by OGA and SGA
Input parameters (μm) =5, k=7
5.21
6.94
=10, k=7
9.96
6.97
=20, k=12
20.06 5.16
=10, σ=5
10.24
=20, σ=10
20.52
=10, σ=0.5
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=20, σ=0.8
9.95
12.95
8.54
17.66
10.37 σ
6.79
1.99
4.73
13.58
4.10
9.51
22.21
7.21
σ
σ
4.51
0.55
7.74
0.38
9.62
0.53
13.62
0.33
20.22
0.78
24.33
0.66
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=5, σ=0.5
k
2.56
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=5, σ=2.5
(μm) Log-Normal
12.08
SGA
4.17
σ
(μm),σ(E-6) Gaussian
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R-R
k
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OGA
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Distribution function
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25
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20
BFGS DFP LM SGA
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15
10
5
0 20
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0
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Volume frequency distribution (%)
Figures
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60
80
100
R (μm)
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Fig.1 Simulated results with four different algorithms for the particles (R = 5 μm)
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20
15
10
5
(a)
15
10
5
0
0
5
10
15
20
25
30
Given distribution Without noise 20dB 10dB 8dB 5dB 3dB
10
15
20
25
30
R (μm)
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25
5
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R (μm)
30
(b)
0
0
20
15
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10
5
0 0
5
10
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Volume frequency distribution (%)
20
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25
Given distribution Without noise 20dB 10dB 8dB 5dB 3dB
25
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Volume frequency distribution (%)
Given distribution Without noise 20dB 10dB 8dB 5dB 3dB
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Volume frequency distribution (%)
30 30
15
20
(c) 25
30
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R (μm)
Fig.2. The comparison of retrieval results without noise, adding with 20/10/8/5/3 dB noise and the given particle
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size distribution: (a) unimodal R-R distribution ( =10 μm, k=7); (b) unimodal Gaussian distribution ( =10 μm, σ=3E-6); (c) unimodal Log-Normal distribution ( =10 μm, σ=0.5)
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Fig.3. The flowchart of three improved forms of genetic algorithms (IGA1, IGA2, and IGA3)
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12
8
6
4
2
(a) 0
8
6
4
2
(b)
0
0
20
40
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80
12
20
40
60
80
R (μm)
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Given distribution Retrieval results
0
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R (μm)
10
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6
4
2
0 0
20
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Volume frequency distribution (%)
Given distribution Retrieval results
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10
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Given distribution Retrieval results
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Volume frequency distribution (%)
Volume frequency distribution (%)
12
40
60
(c) 80
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Fig.4. Comparison between the retrieval results of bimodal R-R distribution system determined with (a) IGA1 (b)
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IGA2 (c) IGA3 and the given distributions (
=20μm, k1=10,
=60μm, k2=25)
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Fig.5 Schematic drawing of experimental setup with ultrasonic attenuation spectroscopy (UAS)
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Fig.6 Comparison of the particle size distribution for two kinds of glycerol-glass beads suspension samples
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(volume fraction: 1%) measured by Microscope Image Analysis (MIA) and obtained with OGA (a), IGA1(b),
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IGA2 (c) and IGA3 (d) by ultrasonic attenuation spectroscopy (UAS)
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Fig.7 Comparison of the particle size distribution for three kinds of aqueous polystyrene suspension samples
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(volume fraction: 10%) measured by Microscope Image Analysis (MIA) and obtained with OGA (a), IGA1(b),
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IGA2 (c) and IGA3 (d) by ultrasonic attenuation spectroscopy (UAS)
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Graphical abstract
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Comparison of the particle size distribution for three kinds of aqueous polystyrene suspension samples (volume fraction: 10%) measured by Microscope Image Analysis (MIA) and IGA3 by
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ultrasonic attenuation spectroscopy (UAS)
ACCEPTED MANUSCRIPT Highlights
The optimal parameters of retrieval problem on particle size distribution measured by ultrasound attenuation spectroscopy based on the classical ECAH
The anti-noise performance of Optimized-Genetic-Algorithm (OGA) was
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mathematical model were chosen
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investigated
Three improved forms of genetic algorithm (IGA1, IGA2, and IGA3) were presented for the retrieval of particle system with bimodal distribution
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The performance of OGA, IGA1, IGA2 and IGA3 was compared with
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microscope image analysis experimentally
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S0032-5910(16)30507-1 doi: 10.1016/j.powtec.2016.08.027 PTEC 11864
To appear in:
Powder Technology
Received date: Revised date: Accepted date:
30 November 2015 18 May 2016 10 August 2016
Please cite this article as: Huinan Yang, Mingxu Su, Xue Wang, Jianfei Gu, Xiaoshu Cai, Particle sizing with improved genetic algorithm by ultrasound attenuation spectroscopy, Powder Technology (2016), doi: 10.1016/j.powtec.2016.08.027
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Particle sizing with improved genetic algorithm by
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ultrasound attenuation spectroscopy
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YANG Huinan, SU Mingxu*, WANG Xue, GU Jianfei, CAI Xiaoshu
Institute of Particle and Two-phase Flow Measurement / Shanghai Key Laboratory of Multiphase Flow and Heat
Transfer in Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
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*Corresponding author, E-mail: [email protected]
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Abstract: As a global optimization algorithm, genetic algorithm is an advantageous tool due to its global convergence, great robustness, suitable for parallel computing and so on. Selection of optimal
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parameters, e.g., maximum generations, population size, and genetic operators (crossover fraction
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and mutation fraction), is extremely crucial for particle size characterization by ultrasound
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attenuation spectroscopy with genetic algorithm. A series of particle system with different distribution functions were numerically investigated in this work. It revealed that the simulated results were
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consistent with the given particle size distribution when the maximum generations, population size crossover fraction and mutation fraction were appropriate chosen. Furthermore, the anti-noise performance of genetic algorithm and its three improved forms applied in particle system with bimodal distribution were also studied. Two groups of samples (micron-sized glass beads-glycerol suspension and aqueous polystyrene suspension) were investigated experimentally by ultrasound attenuation spectroscopy, and it showed a good agreement between Improved Genetic Algorithm 3
(IGA3) and microscope image analysis (MIA).
Keywords: Ultrasound spectroscopy; Particle sizing; Inversion; Genetic algorithm; Optimized parameter
1. Introduction
ACCEPTED MANUSCRIPT Particle sizing in two-phase flow is of great importance with the development of energy, petrochemical, mechanical, material preparation, biological and pharmaceutical industries. In
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recent years, ultrasound technique is widely used in particle sizing, compared to optical and
opaque,
undiluted
and
high/non-conducting
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electrical methods, ultrasound technique is advantageous due to its availability to optically dispersions
[1].
Ultrasound
attenuation
spectroscopy for particle sizing mainly includes three aspects: establishment of mathematical
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model, measurement of ultrasound attenuation spectroscopy, and the retrieval of particle size
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distribution. And the accuracy of the retrieval results would be influenced by selection of inappropriate inversion algorithms and unreasonable parameters.
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Several inversion algorithms have been developed in the previous researches, e.g.,
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Broyden-Fletcher-Goldfarb-Shanno
(BFGS)
algorithm,
Davidon-Fletcher-Powell
(DFP)
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algorithm, and Levenberg-Marquard (LM) algorithm, Twomey method and its improved form [2,3], Optimum-Regularization-Technique(ORT) [4], Chahine algorithm [5-7], Particle Swarm
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Optimization (PSO) [8-9], Ant Colony Optimization (ACO) [10], Fruit Fly Optimization (FOA) [11], and Genetic Algorithm (GA) [12]). Among them, the characteristics of genetic algorithm like universality, robustness, and feasible for parallel computing, etc., promote more and more attention and application in particle sizing [13-16]. In the present work, genetic algorithm with global optimal characteristics on particle sizing by ultrasound attenuation spectroscopy was investigated. Four optimal parameters (maximum generations, population size, crossover fraction and mutation fraction) were chosen with three different forms of distribution functions (Rosin-Rammer(R-R), Gaussian or Log-Normal). And the anti-noise performance of genetic algorithm was also discussed with numerical simulation.
ACCEPTED MANUSCRIPT Then, three improved forms of genetic algorithm applied in particle systems with bimodal distribution were studied. A series of experiments were performed with two groups of samples
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(micron-sized glass beads-glycerol suspension and aqueous polystyrene suspension) to obtain
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the particle size distribution, and the deduced results were compared with the data measured by microscope image analysis.
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2. Methods 2.1The inversion problem
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For the dependent model algorithm in inversion problems, an optimization procedure is carried
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out by minimizing the simulated attenuation spectrum and measured one. For example, the
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expression of R-R distribution function can be written as: (1)
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where dV/dR is the frequency distribution of particle volume, it is a function of radius R.
is the
characteristic radius, k is the distribution parameter, which is a dimensionless quantity (the
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larger the distribution parameter, the narrower the distribution). This optimization function is unique if
and k are determined. Therefore, an objective function about
and k should be
established for optimization in inversion problems. Here, Epstein-Carhart-Allegra-Hawley (ECAH) model [17, 18] is employed to predict the ultrasound attenuation spectroscopy. Considering the interaction between ultrasonic wave and dispersed spherical particle, three wave modes could be derived as a compression wave, a transverse wave and a thermal wave that could exist in both phases (dispersed particle and continuous fluid) of the system. The abovementioned three wave modes indicate that the ultrasound attenuation prediction could be a complicated and multi-parameters related process, which can be deduced by solving a linear
ACCEPTED MANUSCRIPT equation of six orders derived from the wave equations after applying the boundary condition on interface between the suspended spherical particle and continuous fluid. Thus for a
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polydisperse particle system containing the particle with various sizes, the overall attenuation
every single particle, namely:
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∞
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and phase velocity could be evaluated from the algebraic superposition of contribution from
where
(2)
is the theoretically predicted ultrasound attenuation spectrum. f is the
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ultrasound frequency, φ is the concentration of particle system,
is the wavenumber of the
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incident wave. The summation from i=1 to M represents the total contribution of each size bin, , where
is the scattering coefficient related to
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with a dispersed phase volume fraction
the model. The root mean square error describing the difference between the simulated and the
(3)
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measured spectra is:
where ameas, j is the measured spectrum, N is the number of selected frequencies in all the ultrasound attenuation spectrum. Dependent model algorithms, such as BFGS algorithm, DFP algorithm and LM algorithm are usually employed to solve Eq. (3). However, when the initial value or the particle size range is inappropriately chosen, it would lead to an inaccurate particle size distribution. And this situation can be avoided with global optimization genetic algorithm since multiple search points are utilized to search information simultaneously. Fig.1 showed the simulated particle size distribution with four kinds of algorithms (BFGS
ACCEPTED MANUSCRIPT algorithm, DFP algorithm, LM algorithm and Standard Genetic-Algorithm (SGA)) for the particles with radius 5 μm, the particle range was assumed between 0 and 100 μm, and initial
and k in Eq. (1), and the particle size distribution can be then
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algorithms to determine
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radius was 50 μm. The objective function was minimized with four different optimization
obtained. As shown in Fig.1, the median particle radius determined by BFGS, DFP and LM algorithm were 3.23 μm, 37.59 μm and 41.41 μm, respectively. While the radius calculated by
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SGA was 5.72 μm, it was in good agreement with given parameter.
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2.2 Genetic algorithm and related parameters Genetic algorithm is an efficient optimization method based on the principles of natural selection,
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searching global optimization solution without any initialization information [19-23]. As shown in
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Fig.1, the distribution parameter by SGA was different from the given one because of the
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randomness of genetic algorithm. In order to improve the measurement accuracy of SGA, Optimized-Genetic-Algorithm (OGA) which combined genetic algorithm and local optimization
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algorithm (BFGS, DFP, LM algorithm) is presented here. To improve the converging efficiency and accuracy of OGA, the influence of four related parameters, i.e., maximum generation, population size, crossover fraction and mutation fraction on the retrieval results were investigated and appropriate value or ranges of these parameters for particle sizing by ultrasound attenuation spectroscopy were found. When computational generation reaches to its maximum value (i.e., maximum generation), the calculation process ends. The operation efficiency of retrieval calculation would be greatly reduced, if the maximum generation is too large. However, if it is too small, the calculation process ends even when an optimum value is not found. Thus, it is critical to select an
ACCEPTED MANUSCRIPT appropriate value of maximum generation for the inverse problem. For a 500 times iteration optimization of objective function (i.e., the error function of Eq. (3)) with genetic algorithm, it
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revealed that when the computational generation was up to about 280, the optimal fitness value
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was almost constant.
Population size refers to the size of each generation, and it affects the search capability and running efficiency of the retrieval process. If the population size is larger, more models would be
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covered by one evolution, the diversity of the population can be guaranteed and the searching
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ability would be improved. But the operation efficiency is reduced due to more population chromosomes. Here, the retrieval calculation was performed when population size was chosen
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from 20 to 100 with different distribution parameters with three kinds of distribution functions
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(R-R, Gaussian and Log-Normal) and particle sizes (5 μm, 10 μm and 20 μm). It showed that
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the optimal fitness value was almost constant when the population size was larger than 50. The average numbers of population chromosomes in mating in the process of evolution is
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determined by the crossover fraction. When the crossover fraction is larger, the producing speed of new individual is faster and each generation would be fully crossed. However, the possibilities of breaking the fine pattern in population are increased and the structure of the superior individual will be destroyed. It would result in larger generation gap and randomization. While the crossover fraction is too small, the retrieval process would be stopped, if many individuals are directly copied to the next generations. Here, the retrieval calculation was carried out when crossover fraction was chosen between 0 and 1 for different step lengths with three kinds of distribution functions and particle size. It was found that the appropriate range of crossover fraction should between 0.4 and 0.85 for 50 times averaged calculation results.
ACCEPTED MANUSCRIPT The average numbers of the mutated population in the process of evolution is determined by the mutation fraction. If the mutation fraction is larger, more new individuals would be generated
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and the diversity of the population is increased. However, many fine patterns would be
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damaged. If the mutation fraction is too small, the solution would be much stable, but the abilities of generating new individual and restraining premature phenomenon would be poor. Here, the retrieval calculation was performed when mutation fractions were chosen from 0.0001
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to 0.1 for different distribution parameters with three kinds of distribution functions and particle
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size. It showed that the appropriate range of mutation fraction should be between 0.045 and 0.08 for 50 times averaged calculation results.
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Therefore, the maximum generation, population size, crossover fraction and mutation fraction
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were chosen to be 300, 60, 0.5 and 0.065 in the present work, respectively. The retrieval
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calculation was carried out with OGA and SGA for different distribution parameters with three kinds of distribution functions (R-R, Gaussian and Log-Normal) and particle sizes (5 μm, 10 μm
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and 20 μm), and the deduced 20 times averaged results were listed in Table 1. The relative error of distribution parameters determined by OGA was less than 5%, it revealed that the results were more accurate than SGA. Although the calculation time with OGA was 10 times longer than with SGA, the accuracy of retrieval calculation was greatly improved. Thus, OGA is available for retrieval calculation problem on particle sizing by ultrasound attenuation spectroscopy.
2.3 The anti-noise performance of OGA To investigate the anti-noise performance of OGA, the retrieval calculation was carried out with OGA by adding the measured ultrasound attenuation spectrum with WGN (White Gaussian
ACCEPTED MANUSCRIPT Noise) of 20/10/8/5/3 dB (i.e. with 5%~40% random error) with three kinds of distribution functions (R-R, Gaussian and Log-Normal) and particle characteristic size of 10 μm. The
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retrieved particle size distributions with adding different noises were compared with no noise
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and the given particle size distribution in Fig.2. As shown in Fig.2a, for the unimodal R-R distribution (distribution parameter k=7), the errors of radius and distribution parameters retrieved with OGA were less than 5% and 8%, respectively, when adding the ultrasound
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attenuation spectrums with WGN of 20/10/8/5/3 dB. For the unimodal Gaussian distribution
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(distribution parameter σ=3E-6), the radius retrieved with OGA was larger than 10 μm with noise of 3 dB and the errors were less than 8% with noise of 20/10/8/5 dB (Fig.2b).The errors of
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distribution parameters retrieved with OGA were less than 10% if adding the ultrasound
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attenuation spectrums with WGN of 20/10/8/5/3 dB. For the unimodal Log-Normal distribution
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(distribution parameter σ=0.5), the radius inversed by OGA was larger than 10 μm (Fig.2c) with noise of 3 dB and the errors were less than 10% when adding the WGN of of 20/10/8/5 dB. And
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the errors of distribution parameters retrieved by OGA were less than 10% when adding the noise of 20/10/8/5/3 dB. Therefore, it was found that the R-R distribution function should be employed due to the best anti-noise performance of OGA.
3 Three improved forms of genetic algorithm In addition to the unimodal distribution form mentioned above, there are bimodal and multimodal distribution forms in particle systems, and the bimodal condition is more commonly employed in practical application. Therefore, the genetic algorithms of three improved forms for bimodal distribution were presented here: Improved Genetic Algorithm 1 (IGA1), Improved Genetic Algorithm 2 (IGA2) and Improved Genetic Algorithm 3 (IGA3). A flowchart of these three
ACCEPTED MANUSCRIPT improved forms of genetic algorithms (IGA1, IGA2, and IGA3) was shown in Fig.3.
and distribution parameter k1 of the first
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For IGA1, five parameters (characteristic radius
and distribution parameter k2 of the second mode, and the
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mode, characteristic radius
distribution (
: 20 μm; k1:10;
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proportion value α of the first mode) were determined by OGA. A particle system with bimodal : 60 μm; k2:25; α=0.5) was investigated in the work. The
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comparison between the retrieval results of bimodal R-R distribution system determined with IGA1 and the given distributions were shown in Fig. 4a. It revealed that retrieved
was 23.47
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μm, it was apparently larger than 20 μm, distribution parameter of the second mode was 16.95,
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it was obviously smaller than 25. Therefore, it was found that IGA1 was not optimal for the
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retrieval problem of bimodal distribution situation. IGA2 which combined genetic algorithm with local optimization algorithm was utilized to retrieve bimodal distribution too. With IGA2, three ,
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important parameters (
, α)) were determined by genetic algorithm, and they then served
as the initial values of local optimization algorithm. These five parameters were then determined
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by the local optimization algorithm. The comparison between the retrieval results of bimodal R-R distribution system determined with IGA2 and the given distributions were shown in Fig. 4b. It was found that the retrieved parameters
with IGA2 were 20.84 μm, it was more accurate
than with IGA1, but k2 was 10.00, it was obviously smaller than 25. As both the retrieval results with IGA1 and IGA2 were not ideal, a kind of step-by-step optimization algorithm named IGA3 has been developed here: three parameters (
,
, α)
obtained by genetic algorithm, only k1 and k2 determined by the subsequent local optimization algorithm. The comparison between the retrieval results of bimodal R-R distribution system determined with IGA3 and the known parameters were shown in Fig. 4c. It was found that
,
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k2, and α were 20.78 μm, 11.63, 59.84 μm, 24.19 and 0.49, respectively. It revealed that
the retrieved five parameters by IGA3 were improved relative to the former two algorithms,
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4. Experimental results and discussion
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4.1 Experimental setup and measurement method
An ultrasonic measurement system shown in Fig.5 developed in the previous work [24] was
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employed in the present work. A double-sample-cell method is used to obtain the ultrasound attenuation spectrum. The constant-temperature trough is filled with water (the absorption ) and the temperature is kept at 20 °C. Ultrasound waves generated by a
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coefficient is
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broadband ultrasonic transducer 1 (M310-SU) with an ultrasonic transmitter/receiver (PR-5800,
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Parametric) are transmitted through sample cell 1 (l1=10 mm), and then received by transducer 2. The signal is received by a two-channel high-speed data acquisition card (PCI-5114, NI) and
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recorded in a personal computer. The amplitude A1 is determined by fast Fourier transform with the original narrow pulse time-domain signals through transducer 2. The ultrasound waves
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generated by transducer 1 are then transmitted through sample cell 2 (l2=20 mm), and received by transducer 2. The amplitude A2 can be determined. And attenuation coefficient can be then obtained by:
(3) 4.2. Micron-sized glass beads-glycerol suspension samples Two kinds of micron-sized glass beads-glycerol suspensions with different particle size distributions (density is 4.19 g/cm3, volume concentration is 1% and sample numbers are SB040103 and SB040107, respectively) were investigated. The samples were mechanically stirred for three minutes before measurement by ultrasonic system. Fig.6 showed the
ACCEPTED MANUSCRIPT comparison of the particle size distribution for these two kinds of samples measured by Microscope Image Analysis (MIA) (PIP8.1, OMEC) and obtained with OGA (a), IGA1 (b), IGA2
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(c) and IGA3 (d) by ultrasonic attenuation spectroscopy (UAS), respectively. The radiuses
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obtained by MIA were 9.62 and 27.09 μm. As shown in Fig.6a, the retrieval results by OGA were 9.51 and 28.42 μm. It showed that two kinds of glass beads-glycerol suspension samples could be clearly distinguished. And the relative error for retrieved particle size distribution between
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OGA and MIA for sample SB040107 was 7.7% larger than that of sample SB040103. Because
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particle size of sample SB040107 is larger, it resulted in relatively higher speed of particle falling during measurement process. As shown in Fig.6b, the retrieval result of sample SB040107 by
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IGA1 was conformed to a bimodal distribution, although it should be a unimodal distribution.
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And there was a small peak at ～70 μm for sample SB040107 by IGA2 depicted in Fig.6c. And
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the retrieval result of sample SB040107 by IGA3 (Fig.6d) was a unimodal distribution. They were all unimodal distributions for the retrieval results of sample SB040103 by three improved
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algorithms (IGA1, IGA2, and IGA3) and the retrieved distribution parameter by IGA3 was the most consistent with the measured results by MIA.
4.3. Aqueous polystyrene suspension samples Three kinds of aqueous polystyrene suspensions (density is 1.05 g/cm3, volume concentration is 10% and the sample numbers are named as 1, 2 and 3, respectively) with different particle size distributions were investigated. Before being measured by ultrasonic measurement system, the samples were mechanically stirred for three minutes and dispersed for five minutes with ultrasonic dispersion instrument (FS-250, Shanghai Sonxi Ultrasonic Instrument) to eliminate part of the aggregated particles. The comparison of the particle size distributions for these
ACCEPTED MANUSCRIPT samples measured by MIA, OGA and other three improved algorithms (IGA1, IGA2, and IGA3) was shown in Fig.7. It revealed that the measured radiuses by MIA were 4.01, 10.83 and 31.79
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μm, respectively. The curves in Fig.7a indicated the retrieval results with OGA and the radiuses
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were 3.48, 11.07 and 28.36 μm, respectively. The three kinds of aqueous polystyrene suspension samples can be obviously distinguished. And the relative error of particle radius and frequency distribution for sample 3 was the largest, because the particle size of sample 3 were
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the largest which caused the higher speed of particle falling during measurement process. And
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acoustic scattering would be enhanced rapidly if the particle size and frequency increased, which led to the narrow of effective frequency band of ultrasound signals and the measurement
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accuracy was then influenced. For sample 3, the relative error between the retrieval results with
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IGA3 (Fig.7d) and MIA was 4.9%, which was better compared to IGA1 (7.6%, Fig.7b) and IGA 2
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(13.1%, Fig.7c), respectively. It showed that the retrieval results by IGA3 were the most consistent with the MIA measurements.
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5. Conclusions
The retrieval calculation with genetic algorithm on particle size characterization by ultrasound attenuation spectroscopy was presented in this paper. The global optimization genetic algorithm can well avoid converging to local optimal solution because of the global convergence. The optimal parameters of retrieval calculation problem on particle sizing by ultrasound attenuation spectroscopy based on the classical ECAH mathematical model were chosen: maximum generation was 300, population size was 60, crossover fraction was 0.5 and mutation fraction was 0.065. In order to verify the performance of the developed algorithms, both numerical investigation and experimental verification were employed. The unimodal and
ACCEPTED MANUSCRIPT bimodal distribution particle system were studied in the work. It was found that the performance of OGA3 was the best by comparing the retrieval results and given parameter or the data
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determined by MIA.
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Acknowledgments
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The authors gratefully acknowledge the support from the National Natural Science Foundation (Project No: 51176128, 51076106, 51306123), Shanghai Science and Technology
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Commission (Grant No: 13DZ2260900) and Project of Shanghai Science and Technology
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Commission (Grant No: 12ZZ142).
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ACCEPTED MANUSCRIPT [24] Zhang Wei, Su Mingxu, Cai Xiaoshu. Particle size distribution measurement based on
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ACCEPTED MANUSCRIPT Tables Table 1. Comparison of inverse results with three kinds of distribution functions by OGA and SGA
Input parameters (μm) =5, k=7
5.21
6.94
=10, k=7
9.96
6.97
=20, k=12
20.06 5.16
=10, σ=5
10.24
=20, σ=10
20.52
=10, σ=0.5
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=20, σ=0.8
9.95
12.95
8.54
17.66
10.37 σ
6.79
1.99
4.73
13.58
4.10
9.51
22.21
7.21
σ
σ
4.51
0.55
7.74
0.38
9.62
0.53
13.62
0.33
20.22
0.78
24.33
0.66
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=5, σ=0.5
k
2.56
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=5, σ=2.5
(μm) Log-Normal
12.08
SGA
4.17
σ
(μm),σ(E-6) Gaussian
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R-R
k
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OGA
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Distribution function
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25
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20
BFGS DFP LM SGA
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15
10
5
0 20
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0
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Volume frequency distribution (%)
Figures
40
60
80
100
R (μm)
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Fig.1 Simulated results with four different algorithms for the particles (R = 5 μm)
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20
15
10
5
(a)
15
10
5
0
0
5
10
15
20
25
30
Given distribution Without noise 20dB 10dB 8dB 5dB 3dB
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R (μm)
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Given distribution Without noise 20dB 10dB 8dB 5dB 3dB
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Given distribution Without noise 20dB 10dB 8dB 5dB 3dB
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Volume frequency distribution (%)
30 30
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(c) 25
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R (μm)
Fig.2. The comparison of retrieval results without noise, adding with 20/10/8/5/3 dB noise and the given particle
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size distribution: (a) unimodal R-R distribution ( =10 μm, k=7); (b) unimodal Gaussian distribution ( =10 μm, σ=3E-6); (c) unimodal Log-Normal distribution ( =10 μm, σ=0.5)
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Fig.3. The flowchart of three improved forms of genetic algorithms (IGA1, IGA2, and IGA3)
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(a) 0
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Given distribution Retrieval results
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Volume frequency distribution (%)
Given distribution Retrieval results
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Given distribution Retrieval results
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Volume frequency distribution (%)
Volume frequency distribution (%)
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(c) 80
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R (μm)
Fig.4. Comparison between the retrieval results of bimodal R-R distribution system determined with (a) IGA1 (b)
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IGA2 (c) IGA3 and the given distributions (
=20μm, k1=10,
=60μm, k2=25)
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Fig.5 Schematic drawing of experimental setup with ultrasonic attenuation spectroscopy (UAS)
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Fig.6 Comparison of the particle size distribution for two kinds of glycerol-glass beads suspension samples
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(volume fraction: 1%) measured by Microscope Image Analysis (MIA) and obtained with OGA (a), IGA1(b),
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IGA2 (c) and IGA3 (d) by ultrasonic attenuation spectroscopy (UAS)
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Fig.7 Comparison of the particle size distribution for three kinds of aqueous polystyrene suspension samples
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(volume fraction: 10%) measured by Microscope Image Analysis (MIA) and obtained with OGA (a), IGA1(b),
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IGA2 (c) and IGA3 (d) by ultrasonic attenuation spectroscopy (UAS)
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Graphical abstract
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Comparison of the particle size distribution for three kinds of aqueous polystyrene suspension samples (volume fraction: 10%) measured by Microscope Image Analysis (MIA) and IGA3 by
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ultrasonic attenuation spectroscopy (UAS)
ACCEPTED MANUSCRIPT Highlights
The optimal parameters of retrieval problem on particle size distribution measured by ultrasound attenuation spectroscopy based on the classical ECAH
The anti-noise performance of Optimized-Genetic-Algorithm (OGA) was
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mathematical model were chosen
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investigated
Three improved forms of genetic algorithm (IGA1, IGA2, and IGA3) were presented for the retrieval of particle system with bimodal distribution
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The performance of OGA, IGA1, IGA2 and IGA3 was compared with
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microscope image analysis experimentally
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