Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm

Applied Thermal Engineering xxx (2014) 1e9

Contents lists available at ScienceDirect

Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm Zhenzong He, Hong Qi*, Yuchen Yao, Liming Ruan* School of Energy Science and Engineering, Harbin Institute of Technology, 92, West Dazhi Street, Harbin 150001, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 June 2014 Accepted 26 August 2014 Available online xxx

The Fruit Fly Optimization Algorithm (FOA) is applied to retrieve the particle size distribution (PSD) for the first time. The direct problems are solved by the modified Anomalous Diffraction Approximation (ADA) and the LamberteBeer Law. Firstly, three commonly used monomodal PSDs, i.e. the Rosin eRammer (ReR) distribution, the normal (NeN) distribution and the logarithmic normal (LeN) distribution, and the bimodal RosineRammer distribution function are estimated in the dependent model. All the results show that the FOA can be used as an effective technique to estimate the PSDs under the dependent model. Then, an optimal wavelength selection technique is proposed to improve the retrieval results of bimodal PSD. Finally, combined with two general functions, i.e. the Johnson's SB (J-SB) function and the modified beta (M-b) function, the FOA is employed to recover actual measurement aerosol PSDs over Beijing and Hangzhou obtained from the aerosol robotic network (AERONET). All the numerical simulations and experiment results demonstrate that the FOA can be used to retrieve actual measurement PSDs, and more reliable and accurate results can be obtained, if the J-SB function is employed. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Fruit Fly Optimization Algorithm Aerosol Particle size distribution Inverse problem

1. Introduction The particle size distribution (PSD) has an important effect on the production quality control of industry, environmental pollution detection, and human health protection etc., so it is highly required to on-line monitor the granularity [1e3]. However, the PSD is inconvenient to be measured directly. Usually, the PSD is retrieved by some experimental data and corresponding inverse technologies. The retrieval of the PSD with non-contact optical measurements has shown broad development space and huge potential advantage in recent years [2,4]. During the past several years, light scattering particle sizing techniques (e.g. spectral extinction, angle light scattering, diffraction light scattering and dynamic light scattering) have been widely utilized owing to offering a useful and effective approach to characterize a large number of industrial production processes. Their measurements vary from nanometer to millimeter [5,6]. Among these techniques, the spectral extinction method is regarded as the most viable for only requiring a simple optical layout and a commercial spectrophotometer [7,8]. The calculation of the extinction efficiency factor is vitally important in the spectral extinction

* Corresponding authors. Tel.: þ86 0451 86412638. E-mail addresses: [email protected] (H. Qi), [email protected] (L. Ruan).

method and usually calculated by the Mie theory [2,9]. However, the Mie theory can only be applied to the homogeneous spherical particle. The measurement of PSDs for non-spherical particles remains to be a challenging problem, and many rigorous numerical methods were introduced to calculate the radiative properties of the non-spherical particles, e.g., the T-matrix method (TMM), the separation of variables method (SVM), the finite-difference timedomain (FDTD) method etc. [10e12]. Nevertheless, based on these methods, the reproduction of the PSD of non-spherical particles will be computationally expensive and require large memory resources. Additionally, some of these algorithms have the numerical divergence problems when the size parameter, the asphericity, or the refractive index of the particle is extreme [13]. To circumvent these issues, Zhao and coworkers [14] considered the edge effects of particles and modified the traditional Anomalous Diffraction Approximation (ADA), which was proposed by Van de Hulst [15] as a means of computing the extinction cross section for large, optically soft spheres with c << 1 and jm  1j > 1, to calculate the absorption and scattering properties of particles in various shapes, e.g. spheres, spheroids, cuboids [13]. Both the numerical simulations and experimental results indicate that the modified ADA can be successfully applied to retrieve the PSD for spherical or spheroidal particles with high stability [16,17]. Generally speaking, the inverse methods of the PSD can be divided into three different categories, i.e. the analytic inverse

http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057 1359-4311/© 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

2

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9

Nomenclature

Qad

c0 D D

Qedge Qext Rand t

D1 ; D2 f(D) Fobj I I0 k L m n n0 N0 Nl

a function of the refractive index in general the particle diameter, mm the characteristic diameter of the monomodal distribution function, mm the characteristic diameters of the bimodal distribution function, mm the volume frequency distribution of the particle system with diameter D the objective function the intensity of the laser, W/(m2 sr) the total intensity of the laser, W/(m2 sr) the imaginary part of the complex refractive index the geometric thickness of the particle system, m the complex refractive index the real part of the complex refractive index the weight coefficient between the two peaks in the bimodal distribution the total number density of the particle system, m3 the number of the wavelengths

model, the independent model and the dependent model. For most particle systems can often be approximately regarded as obeying certain kind of the PSDs (e.g. the urban aerosol system usually satisfies the logarithmic normal distribution), the dependent model has been extensively studied and applied in many fields [16]. However, sometimes the particle systems do not have any priori assumption about the exact shape of the PSD. To solve this problem, Yu et al. and Popplewell et al. introduced the Johnson's SB (J-SB) distribution function and the modified beta (M-b) function as the general function to reproduce monomodal PSD, respectively [18,19]. Tang and coworkers [8,20] also employed the J-SB and M-b function to retrieve three commonly used distribution functions of the spherical particles with satisfactory results. During the past decades, many intelligent optimization algorithms have been applied to retrieve the PSD, e.g. the Particle Swarm Optimization (PSO) [2], the Genetic Algorithm (GA) [16], the General Regression Neural Networks (GRNN) [21], and the Ant Colony Optimization (ACO) [9,22,23]. More recently, a new effective swarm intelligent optimization, named Fruit Fly Optimization Algorithm (FOA) and stemmed from imitating fruit fly's foraging behavior, was put forward by Pan [24]. For updating its solution's location by randomly generating the direction and distance nearby the global best location, the FOA is simpler than the PSO and ACO. With the attractive characteristics, such as simple computational process, ease of transforming such concept into program code, ease of understanding and a very good convergence, the FOA has drawn widespread attention and been popularized to tackle some practical optimization applications all over the world [25e27]. To the best of the authors' knowledge, there have been no reports about the application of the FOA in the PSDs inverse problem. In the present article, based on the modified ADA and the LamberteBeer Law, the FOA is firstly applied to estimate the PSD of the spherical particle in dependent model. The remainder of the paper is organized as follows. Firstly, three commonly used monomodal PSDs, i.e. the RosineRammer (ReR) distribution function, the normal (NeN) distribution function and the logarithmic normal (LeN) distribution function, and the bimodal RosineRammer (ReR) distribution function are retrieved in the dependent model. Then, an optimal wavelength selection

the extinction efficiency calculated by the standard ADA the edge effective of extinction efficiency the extinction efficiency the random number the convergence time, s

Greeks symbols c the size parameter of particles s the dispersion ratio of the monomodal PSDs s1, s2 the narrowness indices of the bimodal PSDs l the incident wavelength of the laser, mm d the relative deviation of the PSDs ε the tolerance for minimizing the objective function Subscripts est the ext the max the mea the min the true the

estimated value extinction efficiency maximum value measurement value minimum value true value

technique is investigated. The FOA is also applied to retrieve the actual measurement aerosol PSDs obtained from the aerosol robotic network (AERONET). Finally, the main conclusions and ideas for further research are provided.

2. Methodologies 2.1. The principle of the spectral extinction method The spectral extinction method is based on the light scattering theory. When a beam of the collimated monochromatic laser, whose intensity is I0, passes through a suspended particle system, whose geometrical thickness is L, the transmitted light is scattered and absorbed by the particles, which causes the attenuation of the light. If the optical thickness is thin and the independent scattering dominates, the influence of the multiple scattering and interaction among the particles can be ignored [28]. According to the LamberteBeer Law, the intensity I of the transmitted light with the incident wavelength l for a spherical particle system is described by the following integral equation [2]:

ln

IðlÞ 3 ¼ L I0 ðlÞ 2

D Zmax

  N0 f ðDÞ Qext l; m; D dD D

(1)

Dmin

where I0(l) is the intensity of incident light, and I(l)/I0(l), obtained by actual measurement, is the transmissivity at wavelength l; L is the mean geometrical thickness of the particle system; f(D) is the unknown volume frequency distribution of the particle system with diameter D; N0 is total number density in particles per cubic centimeter; Dmax and Dmin (in mm) denote the upper and lower limits of the integration; Qext(l,m,D) is the extinction efficiency factor of a single particle which is a complex function of the particle diameter D, the wavelength l of incident light and the relative refractive m (the ratio between the particle and medium refractive index). According to the standard ADA, the extinction efficiency for a spherical particle with a diameter D can be expressed as [13]:

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9

  1 expð  irÞ 1  expð  irÞ þ Qad ðl; m; D Þ ¼ 4  Re  i  2 r r2

3

(2)

where pffiffiffiffiffiffiRe ffi denotes the real part of the complex; r ¼ pD(m  1)/l; i ¼ 1. The edge effect of spherical particle introduced by Zhao is depicted as [13]:

Qedge ¼

2c0 c2=3

(3)

where c0 is a function of the complex refractive index in general and is approximately equal to 0.99163 for optically soft particles; c ¼ pD/l is the size parameter of the particle. Considering the edge effect, the fixed expression for the extinction efficiency of large particles is written as [13]:

 .  Qext ¼ Qad 1 þ Qedge Qad :

(4)

For small particles, the extinction efficiency is expressed as [13]:

( Qext ¼ Qad

) 1 . ; 1þ 2 Qedge þ 1=½c1 ðQad þ 1Þ

Fig. 2. The food finding iterative process of fruit fly swarm.

c1 ¼ jm  1j: (5)

the information of the best fruit fly will be shared with the whole swarm during every iteration, and the next iteration will be only based on the information of previous best fruit fly. Fig. 3 displays fruit fly swarm flying routes for parameter optimization. The calculation repeats four times, and the corresponding flying routes are shown in Fig. 3. It can be found that the fruit fly swarm flying routes are relatively stable and the fruit fly swarm can find the best position at last, although there is a slight fluctuation during the random searching process. The detail management of the FOA can be divided into several steps, just as follows [24]:

Fig. 1 shows the curves of extinction efficiency calculated by modified ADA and Mie theory for a single sphere particle with different refractive indices, respectively. From this figure, it is obvious that there is a satisfactory agreement between the modified ADA and the Mie theory, especially for the particles with larger size parameter. In other words, the modified ADA can provide more satisfactory results for solving the light transport problem in large size parameter. Moreover, it can also be seen that the extinction efficiency fluctuates around 2, and the smaller the size parameter is, the stronger the fluctuation will be. When the size parameter is large enough, the value of extinction efficiency almost keeps constant, and this phenomenon will affect the retrieval results of PSDs.

Step 1 Initialize the main parameters of the FOA, i.e. the maximum iterative number maxgens, the population size npopsize, the tolerance of the objective function ε and the random fruit fly location (X_axis, Y_axis). Step 2 Give the random location (Xi, Yi) and distance for the search of food by every fruit fly. Randx and Randy are random vectors that are sampled from a uniform distribution [low, high]. If the new position exceeds the search space, make it equal to the low or high limit of the search space.

2.2. The principle of the FOA For being superior to other species in osphresis and visual sense, the fruit fly can make full use of its instinct to find food. The olfactories of fruit fly can pick up various smells of the food which disperse in the air, even if the food source is 40 km away. When getting close to the food location, the fruit flies find food and the company's flocking location with the help of the sensitive visual organs, and then fly towards that direction [24]. Fig. 2 shows the food finding iterative process of fruit fly swarm. It is obvious that

Xi ¼ X axis þ Randx

5

5 Mie theory Modified ADA

m=1.35+0.02i 4

Mie theory Modified ADA

m=1.8+0.04i 4 3 Qext

Qext

3 2

2

1

1

0

(6)

0

10

20

30

40 χ

50

60

70

80

0

0

10

20

30

40 χ

50

60

70

80

Fig. 1. The curves of extinction efficiency calculated by modified ADA and Mie theory with different refractive indices.

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

4

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9

Step 7 Loop to Step 2, if the program matches one of the following two stop criteria. (i) The value of the objective function is less than the tolerance ε, Fobj  ε; (ii) The number of the iteration reaches the user-defined iteration limit maxgens, iter > maxgens.

0.40 Route 1 Route 2 Route 3 Route 4

0.35

Best position

Y_axis

0.30 0.25

3. Retrieval of particle size distribution

0.20

3.1. Numerical simulations

0.15

In this study, three commonly used monomodal PSD functions, i.e. the ReR distribution, the NeN distribution, and the LeN distribution, are retrieved, respectively. Moreover, the bimodal ReR distribution is also estimated. All the studies above are under the dependent model. The mathematical representations of the monomodal volume frequency distribution functions are as follows [2]:

0.10 0.10

0.15

0.20

0.25 X_axis

0.30

0.35

0.40

Fig. 3. The fruit fly swarm flying routes for parameter optimization.

fRR ðD Þ ¼ Yi ¼ Y axis þ Randy

(7)

Step 3 For the food location is unknown, the distance to the origin is thus estimated first (Dist), then the smell concentration judgment value (S), which is the reciprocal of distance, is calculated.

Disti ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2i þ Y 2i

Si ¼ 1=Disti

(8) (9)

 s1   s  s D D   exp  D D D

"

2 # DD 1 fNN ðD Þ ¼ pffiffiffiffiffiffiffi  exp  2s2 2ps

(16)

"

2 # ln D  lnD 1 fLN ðD Þ ¼ pffiffiffiffiffiffiffi  exp  2pD ln s 2ðln sÞ2

(17)

where D is the characteristic diameter parameter; s is the dispersion ratio of the distribution. The bimodal ReR distribution function is expressed as [29]:

  s1  D þ ð1  n0 Þ  exp  D1 D1 D1  s2 1   s2  s D D  exp   2 D2 D2 D2 (18)

fRR ðD Þ ¼ n0  Step 4 Replace smell concentration judgment value (S) with the smell concentration judgment function (or called Fitness function) in order to find the smell concentration (Smelli) of the individual location of the fruit fly.

Smelli ¼ FuncitonðSi Þ

(10)

Step 5 Find the value bestSmell and corresponding location bestIndex of the fruit fly, which is with maximal smell concentration (finding the maximal value) among the fruit fly swarm, for different dimensions of smell vectors.

½bestSmell; bestIndex ¼ maxðSmellÞ

(11)

Step 6 Judge if the bestSmell is superior to the previous Smellbest. If so, update the value of the Smellbest and corresponding location, and go to Step 7; if not, go to Step 7 directly. The aim of this step is to keep the best smell concentration value and corresponding position, and make sure that the fruit fly swarm will use vision to fly towards that location.

Smellbest ¼ bestSmell

(12)

X axis ¼ XðbestIndexÞ

(13)

Y axis ¼ YðbestIndexÞ

(14)

(15)

s1





D

s1 1

where D1 and D2 denote the geometric mean diameter for the bimodal distribution function (in mm); s1 and s2 are the geometric standard deviation of the distribution; n0 is the weight coefficient 0 between the two peaks and is limited as 0  n  1. The complex refraction index of the spherical particle employed in this article refers to the practical situation. The complex refraction index of the coal ash particle was studied by Ruan et al. [30], and its real part n and imaginary part k are in the range of n ∊ [1.18, 1.92] and k ∊ [0.01, 1.13], respectively. The complex refractive index in this article is set as 1.35 þ 0.02i, which is assumed to be constant at different wavelengths for the sake of simplicity. To estimate the monomodal PSD functions which contain two parameters (D, s) to be retrieved, two measurement wavelengths in spectral extinction method are needed at least. Here, the wavelengths of the incident light are set as l ¼ 0.45, 0.50 mm in the two wavelengths model and l ¼ 0.45, 0.50, 0.55, 0.60 mm in the four wavelengths model. Unlike the monomodal PSD functions, there 0 are five parameters (D1 , s1, D2 , s2, n ) that need to be retrieved in the bimodal ReR distribution function, so the five wavelengths model, l ¼ 0.45, 0.50, 0.55, 0.60, 0.65 mm, and the seven wavelengths model, l ¼ 0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75 mm, are chose as the measurement wavelengths. The size measurement range of the particles is selected from 0.1 mm to 10 mm, which is the optimal measurement range in the spectral extinction particle size technique. For the monomodal ReR distribution, the true values are

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9

feasibility of this optimization algorithm, some characteristic parameters are used to evaluate the quality of inversion results are studied.

10 PSO, t=3.672s FOA, t=3.385s

Objective Function, Fobj

0.1

(1) The relative deviation of PSD d means the sum of the deviation error between the probability distribution estimated from the inverse calculation and the true PSD in every subinterval, and its mathematic expression is described as:

1E-3 1E-5 1E-7

( d¼ 0

200

400

600 Iteration

800

Parameters

npopsize

maxgens

D

s

Low

High

ε

Value

50

3000

0.01e10

0.01e10

0.002

0.002

108

 ½IðlÞ=I0 ðlÞmea ½IðlÞ=I0 ðlÞmea

i¼1

 i2 PN h ~ i¼1 ftrue Di

)1=2 (20)

)1=2

(2) The average computational time for every iteration convergence t is expressed in second.

set as (1.9, 1.8); for the monomodal NeN distribution, the true values are set as (2.0, 0.6); for the monomodal LeN distribution, the true values are set as (2.05, 1.85). The true values of the bimodal ReR distribution are fixed as (1.5, 4.9, 6.1, 6.85, 0.68). The retrieval of the spherical PSD is solved by minimizing the objective function Fobj, which is the sum of the square residual between the estimated and measured transmittance ratios. In the FOA, the objective function is described as the fitness function (Smellbest) and expressed as follow:

The performance of the FOA is studied through comparison with the standard PSO. Fig. 4 displays comparison of the objective function values of the FOA and standard PSO, when the LeN distribution is estimated under four wavelengths model. In the standard PSO, the social cognitive confidence coefficients are set as 1.0 and the inertia weight is set as 0.5. The detail description of the PSO refers to Ref. [31]. In the FOA, the low and high are set as 0.002 and 0.002, respectively. Both the algorithms use a population size of 50. The termination criteria are set as: (1) when the iteration accuracy is below 1012 and (2) when the maximum generation number of 1000 is reached. It can be found that the FOA can arrive at lower objective function value, compared with the standard PSO, while the objective function value of the PSO reduces faster than that of the FOA. Moreover, the required convergence time of the FOA is less than that of the standard PSO. As a whole, both the FOA and PSO are the effective optimization algorithm to retrieve the monomodal and bimodal PSD functions of spherical particles. In

2

est

   i2 ~  ftrue D ~ fest D i i

where N denotes the number of subinterval that the particle size ~ is the midpoint of the ith subinrange [Dmax, Dmin] is divided; D i ~ Þ is the true volume frequency distribution terval [Di, Diþ1]; ftrue ðD i ~ Þ is the estimated volume frequency in the ith subinterval; fest ðD i distribution in the ith subinterval. It is easy to find that the smaller the value d is, the more accurate the estimated results will be.

Table 1 The control parameters of the FOA for different monomodal distributions.

Nl  X ½IðlÞ=I0 ðlÞ

(

1000

Fig. 4. The comparison of objective function values of standard PSO and FOA.

Fobj ¼

PN h i¼1

1E-9 1E-11

5

(19)

where Nl is the number of the wavelengths. Considering the fact that the FOA is a stochastic optimization method and all optimizations have certain randomness, the calculation is repeated 50 times. Moreover, for the purpose of investigating the reliability and

Table 2 The control parameters of the FOA for the bimodal ReR distribution. Parameters

npopsize

maxgens

D1

s1

D2

s2

n0

Low

High

ε

Value

50

5000

0.01e10

0.01e10

0.01e10

0.01e10

0.5e1.0

0.002

0.002

108

Table 3 Retrieval results of the monomodal PSD functions by the FOA under two or four wavelengths models. Functions

Error

Two wavelengths

R  R ðD; sÞ ¼ ð1:9; 1:8Þ

N  N ðD; sÞ ¼ ð2:0; 0:6Þ

L  N ðD; sÞ ¼ ð2:05; 1:85Þ

0% 5% 10% 0% 5% 10% 0% 5% 10%

1.90 1.86 1.92 2.00 1.92 1.88 2.05 2.26 2.52

Four wavelengths s

D ± ± ± ± ± ± ± ± ±

7.29 3.38 2.67 1.89 3.85 6.03 1.14 1.04 2.33

        

4

10 102 101 103 102 102 103 101 101

1.80 1.96 2.29 0.60 0.64 0.98 1.85 2.11 2.64

d ± ± ± ± ± ± ± ± ±

1.45 8.38 1.67 7.59 7.45 6.32 1.49 1.42 4.28

        

3

10 102 101 103 102 101 103 101 101

0.00021 0.08581 0.14206 0.00790 0.10527 0.37604 0.00232 0.17460 1.13051

t (s) 5.912 10.469 13.581 6.431 12.228 15.334 4.404 9.226 13.363

s

D 1.90 1.89 1.87 2.00 2.01 2.05 2.05 2.03 2.01

± ± ± ± ± ± ± ± ±

1.96 1.31 3.50 5.58 5.79 2.56 1.27 4.93 5.03

        

4

10 103 102 104 103 102 105 103 102

1.80 1.85 1.90 0.60 0.62 0.73 1.85 1.83 1.82

± ± ± ± ± ± ± ± ±

4.56 3.99 2.12 2.31 2.92 2.17 2.50 8.82 1.38

        

4

10 103 102 103 102 101 105 103 102

d

t (s)

0.00006 0.01403 0.06961 0.00239 0.02812 0.16984 0.00003 0.01330 0.02055

6.221 12.442 16.502 7.546 15.091 18.125 6.749 12.281 16.442

*The mean and standard deviation of the 50 times retrieval results are shown as the form of x ± y in the table.

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

6

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9

0.45

0.6 0.50

0.3

0.35

0.30

0.30 0.5

0.25

1.0

1.5

2.0

2.5

2

0.15

Nλ=2,err=5%

Nλ=4,err=5%

0.10

Nλ=2,err=10%

Nλ=4,err=10%

0.05

4

6

8

Diameter (μm)

10

Fig. 5. Reproducibility of the monomodal ReR distribution by the FOA.

this study, the FOA is employed and the control parameters of the FOA for monomodal distributions and the bimodal distribution are shown in Table 1 and Table 2, respectively. Table 3 shows the reproducibility of the spherical particle swarm diameter distribution for the monomodal PSDs with different random measurement errors added to the extinction data, and the corresponding inverse curves are depicted in Figs. 5e7. The retrieval results of the FOA for the bimodal ReR distribution are given in Table 4, and the corresponding inverse curves are shown in Fig. 8. From Table 3 and Figs. 5e7, it can be seen that the retrieval results of the monomodal distributions are more accurate when the random error is not beyond 5% for both two and four wavelengths models. While the error rises up to 10% under two wavelengths model, the estimated results deteriorate and the convergence time increases, especially for the LeN distribution. However, the results retrieved under four wavelengths still show acceptable accuracy and robustness even with 10% error, although more convergence time is required. In other words, the investigations show that the results estimated using four wavelengths are more accurate and reliable than those using two wavelengths, although the four wavelengths model needs more convergence time. The reason may be that two wavelengths offer less transmitted light information which will lead to the multivalued retrieval results more easily compared with four wavelengths. From Table 4 and Fig. 8, the

0.8

0.7 0.6

0.6

0.5

fD

0.7

0.4

f(D)

0.5

0.3 1.2

1.6

2.4

2.8

Monomodal N-N distribution True value Nλ=2,err=2% Nλ=4,err=2%

0.3 0.2 0.1 0.0

2.0

Diamter (μm)

0.4

0

2

4

Nλ=2,err=5%

Nλ=4,err=5%

Nλ=2,err=10%

Nλ=4,err=10%

6

Diamter (μm)

8

0.20 0.5

1.0

1.5

2.0

Diamter (μm)

2.5

Monomodal L-N distribution True value Nλ=2,err=2% Nλ=4,err=2%

0.20

0.00

0

0.30 0.25

Monomodal R-R distribution True value Nλ=2,err=2% Nλ=4,err=2%

0.1

f(D)

0.35

Diameter (μm)

0.2

0.0

0.40

f(D)

fD

f(D)

0.4

0.35

0.40

0.45

0.5

0

2

4

Nλ=2,err=5%

Nλ=4,err=5%

Nλ=2,err=10%

Nλ=4,err=10%

6

Diameter (μm)

8

10

Fig. 7. Reproducibility of the monomodal LeN distribution by the FOA.

similar conclusion can also be drawn. Moreover, it is obvious that the major peak value of the PSD retrieved by the FOA is in good agreement with the original value under seven incident wavelengths, even in the present of 5% random error. Nevertheless, the deviation between the inverse result and the true value of the other peak is unsatisfactory when there is 5% error. That is to say, the major peak value shows more robust and insensitive to the measurement errors, compared with the other peak value. Since the spectral extinction data contain some important information about the particle system, it is necessary to study the influence of the spectrum on the retrieval of the PSDs. For improving the estimated results of the PSD, an optimum choice of the incident laser wavelengths is proposed. Table 5 lists the retrieval results of bimodal ReR distribution by the FOA with different choice incident laser wavelengths under 2% and 5% random errors, respectively. The original values of the bimodal ReR distribution are also fixed as (1.5, 4.9, 6.1, 6.85, 0.68). The corresponding inverse curves are depicted in Figs. 9 and 10. From Table 5 and Fig. 9, it is easy to find that when the number of the incident wavelengths is fixed, the larger value of the interval between incident laser wavelengths Dl is, the more reasonable agreement between the estimated results and true values will be. The similar conclusion can be drawn, when the bimodal ReR distribution is retrieved under 5% error (see Fig. 10). The reasons may be that for certain suspension particle system, if the incident laser wavelengths selected are close to each other, the spectral extinction data will contain less effective information which may weaken the benefits of the multi-spectrum measurement and lead to worse inverse results. Moreover, it can also be found that the inverse results of the smaller peak fluctuate more seriously than those of the major peak, when the interval Dl becomes smaller. In other words, the smaller peak value is more sensitive to wavelength selection, compared with the major peak value. Therefore, when the spectral extinction technique is applied to estimate the PSD, the interval of incident laser wavelengths should be selected large enough to avoid non-effective measured information and improve the accuracy of the inverse results. 3.2. Experiment results

10

Fig. 6. Reproducibility of the monomodal NeN distribution by the FOA.

The reliability of the FOA is also verified by retrieving the actual measurement PSDs of the aerosols, which are usually regarded as spherical particle. For the specific distribution function of the actual measurement aerosols is unknown beforehand, the bimodal J-SB and M-b functions are employed as the general function to estimate

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9

7

Table 4 Retrieval results of the bimodal ReR distribution by the FOA under five or seven wavelengths models with (1.5, 4.9, 6.1, 6.85, 0.68). Multiple wavelength models

Error

D1

Five wavelengths model

0% 2% 5% 0% 2% 5%

1.50 1.49 1.45 1.50 1.50 1.50

Seven wavelengths model

s2 ± ± ± ± ± ±

1.42 1.41 9.27 1.54 5.54 7.37

     

103 102 101 103 102 101

4.85 4.94 4.67 4.91 4.92 5.04

± ± ± ± ± ±

5.35 5.93 9.46 7.86 2.66 1.01

     

103 102 101 103 102 101

5.96 6.46 6.77 6.11 6.45 6.42

n0

s2

D2 ± ± ± ± ± ±

2.47 9.26 3.20 2.39 2.97 7.05

     

102 102 101 103 102 102

6.56 8.85 9.70 6.76 8.42 9.26

± ± ± ± ± ±

1.19 2.32 1.60 4.01 3.86 1.16

     

101 101 100 102 101 101

0.66 0.65 0.60 0.68 0.68 0.66

± ± ± ± ± ±

5.35 6.44 1.29 1.57 9.12 1.04

     

103 102 101 103 103 102

d

t (s)

0.04638 0.11698 0.25976 0.00375 0.06568 0.10940

16.48 32.24 35.15 24.22 41.09 41.49

**The mean and standard deviation of the 50 times retrieval results are shown as the form of x ± y in the table.

the actual aerosol PSDs. The mathematic expressions of the J-SB function and the M-b function are written as [20]:

Fig. 13, which are also available on AERONET [32]. The skewness of the aerosol PSDs over Beijing and Hangzhou are 0.457 and 1.446,

( 2      ) s01 s0 Dmax  Dmin D  Dmin M1  Dmin 2 ffi exp  1 ln  ln fJSB ðD Þ ¼ n0  pffiffiffiffiffiffi þ ð1  n0 Þ 2 Dmax  D Dmax  M1 2p ðD  Dmin ÞðDmax  DÞ ( 2      ) s02 s02 Dmax  Dmin D  Dmin M2  Dmin 2 exp  ln  ln  pffiffiffiffiffiffi 2 Dmax  D Dmax  M2 2p ðD  Dmin ÞðDmax  DÞ

0

0

fMb ðD Þ ¼ n  Z

ðD  Dmin Þa1 m1 ðDmax  DÞm1 Dmax

Dmin

0

þ ð1  n Þ

0

ðD  Dmin Þa1 m1 ðDmax  DÞm1 dD 0

Z

ðD  Dmin Þa2 m2 ðDmax  DÞm2 Dmax

Dmin

0

ðD  Dmin Þa2 m2 ðDmax  DÞm2 dD (22) 0

where s01 , M1, s02 , M2, a01 , m1, a02 , m2 and n are the characteristic parameters, 0  n0  1. The system control parameters of the FOA are kept the same as the dependent model (see Table 2). The aerosol volume size distributions in Beijing on 18 May 2014 and in Hangzhou on 11 February 2009, which are depicted in Figs. 11 and 12 with red bar, respectively, were provided by NASA's AERONET's Level 1.5 Version 2 inversion algorithm [32]. The corresponding complex refractive indices of aerosols are depicted in

1.2

Nλ =5,err=2%

f(D)

0.8 0.6

0.70

Nλ =5,err=5%

0.65

Nλ =7,err=0% Nλ =7,err=2%

0.60 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.20

Nλ =7,err=5%

0.15

f(D)

1.0

f(D)

Bimodal R-R distribution 0.80 True value Nλ =5,err=0% 0.75

0.2

0.00 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 Diameter (μm)

2

4

Table 5 Retrieval results of the bimodal ReR distribution by the FOA under different wavelengths.

Dl/mm (l1, l2, l3, l4, l5, l6, l7)/mm

0.05

0

respectively, which mean that this two distributions are positive skewness. The measurement ranges of the aerosol radius vary from 0.05 mm to 15 mm, and all the experiment data are measured at l ¼ 0.440, 0.675, 0.870, 1.020 mm. Combined with bimodal J-SB and M-b functions, the retrieval results by the FOA are listed in Table 6. The incident wavelengths are set as l ¼ 0.440, 0.5, 0.675, 0.8, 0.870, 0.9, 1.020 mm, and their corresponding inverse results are also illustrated in Figs. 11 and 12. From Fig. 11, it is obvious that there is a reasonable agreement between the true distribution and retrieval results by both two general functions, especially the J-SB function, although the estimated results of the major peak deviates slightly from the original ones. Similarly, in Fig. 12, it can be found that the deviation of the estimated results by M-b function is slightly serious, while the estimated results of J-SB function are reasonable and acceptable. It also can be found that the peak values positions of the distribution can be well estimated by J-SB function, compared with the M-b function. The retrieval results of the actual measurement aerosol PSDs show a little more discrepant than the estimated results in Section 3.1. The reasons may be summarized as follows: (1) For the exact type of the PSD function is unknown beforehand, the bimodal J-SB and M-b functions can only be employed as a kind of approximated distribution function, and the discrepancy between the inverse results and the original distribution is inevitable. Meanwhile, the volume size distribution obtained from the AERONET

0.10

0.4

0.0

(21)

Relative deviation d Error ¼ 2% Error ¼ 5%

6

Diameter (μm)

8

Fig. 8. Reproducibility of the bimodal ReR distribution by the FOA.

10

No. No. No. No. No. No.

1 2 3 4 5 6

0.02 0.04 0.06 0.08 0.1 0.2

(0.45, (0.45, (0.45, (0.45, (0.45, (0.45,

0.47, 0.49, 0.51, 0.53, 0.55, 0.65,

0.49, 0.53, 0.57, 0.61, 0.65, 0.85,

0.51, 0.57, 0.63, 0.69, 0.75, 1.05,

0.53, 0.61, 0.69, 0.77, 0.85, 1.25,

0.55, 0.65, 0.75, 0.85, 0.95, 1.45,

0.57) 0.69) 0.81) 0.93) 1.05) 1.65)

0.096110 0.072907 0.058346 0.050863 0.044092 0.039337

0.145541 0.110170 0.064813 0.061880 0.048217 0.042716

*The mean inverse results and relative deviation of the 50 times calculations are shown in the tables.

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

8

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9

1.2

Volume size distribution, N0 f(D)

0.80 0.75 0.70 0.65 0.60 1.1 1.2 1.3 1.4 1.5 1.6 1.7

0.6

0.15

f ( D)

f(D)

0.8

0.4

0.10 0.05

0.2 0.0

0.00 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Diameter (μm)

0

2

4

Hangzhou, China at 08:03:31 on 11 Feb. 2009 Retrieved by M-β functon Retrieved by J-SB functon

0.85

f(D)

1.0

0.20

0.90

Bimodal R-R distribution err=2% True value No.1 No.4 No.2 No.5 No.3 No.6

6

8

0.15

0.10

0.05

0.00

10

0.1

1 Radius (μm)

Diameter (μm) Fig. 9. Reproducibility of bimodal ReR distribution by the FOA with 2% random error.

1.2

0.8

Bimodal R-R distribution 0.95 0.90 err=5% 0.85 True value 0.80 No.1 No.4 0.75 No.2 No.5 0.70 No.3 No.6 0.65

0.6

0.15

f ( D)

0.4

0.0

0.10

4. Conclusions

0.05

0.2

0.00 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Diameter (μm)

0

2

4

Fig. 12. Reproducibility of aerosol volume size distribution in Hangzhou by the FOA.

AERONET's Level 1.5 Version 2 inversion algorithm may contain uncertain inverse error which will also influence the accuracy of the estimated PSD. However, as a whole, the FOA can still be used to retrieve actual measurement PSD, and both the bimodal J-SB and M-b functions, especially the J-SB function, can be employed as a kind of approximated distribution function to study the actual measurement aerosol PSD. And the research also demonstrate that the FOA algorithm can also be employed to estimate the PSD with skewness.

0.60 1.1 1.2 1.3 1.4 1.5 1.6 1.7

f(D)

f(D)

1.0

10

6

Diameter (μm)

8

10

Fig. 10. Reproducibility of bimodal ReR distribution by the FOA with 5% random error.

contains the volume frequency distribution f(D) and the total number concentration N0,which is different from the study in Section 3.1 (only the f(D) retrieved) and will deteriorate the inverse results. In addition, the refractive indices retrieved by NASA's

The FOA combined with the modified ADA and the LamberteBeer Law is employed to estimate the monomodal and bimodal spherical PSDs in light scattering particle sizing techniques for the first time. Three commonly used monomodal PSDs, i.e. the ReR distribution, the NeN distribution, and the LeN distribution, and the bimodal ReR distribution are retrieved under the dependent model. All the results show that the parameters of the monomodal and bimodal distribution functions can be estimated accurately by the FOA. Then, an optimization wavelength selection technique is studied. The research reveals that for the bimodal distribution, larger interval between the incident wavelengths will lead to more accurate results, and the estimated result of the major peak shows better robustness than that of the other peak. Finally, -2

0.15

0.10

0.05

1.50 5

1.45 1.40 1.35 1.30 1.25

Hangzhou, China Beijing, China at 22:51:24 on 18 May 2014 at 08:03:31 on 11 Feb. 2009 Real part Real part Imaginary part Imaginary part

1.20

4 3 2

1.15 1.10

1

1.05 1.00 400

0.1

1 Radius (μm)

10

Fig. 11. Reproducibility of aerosol volume size distribution in Beijing by the FOA.

500

600

700 800 900 Wavelength, λ (nm)

1000

0 1100

Imaginary part of the refractive index

Beijing, China at 22:51:24 on 18 May 2014 Retrieved by M-β functon Retrieved by J-SB functon

0.20

0.00

x10 6

1.55 Real part of the refractive index

Volume size distribution, N0 f(D)

0.25

Fig. 13. The refractive indices of aerosols in Beijing and Hangzhou available on AERONET [32].

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057

Z. He et al. / Applied Thermal Engineering xxx (2014) 1e9 Table 6 Retrieval results of aerosol volume size distributions by the FOA. Inverse results Aerosol volume size M-b ða01 ; m1 ; a02 ; m2 ; n0 Þ distribution Beijing Hangzhou

J-SB ðM1 ; s01 ; M2 ; s02 ; n0 Þ

ð0:008; 180:15; 0:31; 5:11; 0:061Þ ð0:14; 3:59; 4:02; 1:09; 0:015Þ ð0:012; 261:57; 0:41; 2:05; 0:035Þ ð0:28; 1:82; 5:52; 1:12; 0:04Þ

*The mean inverse results and relative deviation of the 50 times calculations are shown in the tables.

combined with the J-SB and M-b functions, the FOA is utilized to study the actual measurement aerosol PSDs obtained from the AERONET. The investigation reveals that the FOA can be used to estimate the actual measurement PSDs, and the retrieved results using J-SB function are more accurate than those using M-b function. Further study will focus on performance improvement of the FOA-based methodology as well as the applications of the FOA in the retrieval of PSDs of the non-spherical particles under the independent model.

Acknowledgements The supports of this work by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51121004), the Major National Scientific Instruments and Equipment Development Special Foundation of China (No. 51327803), and the National Natural Science Foundation of China (No. 51476043) are gratefully acknowledged. The author would also like to thank all the investigators of AERONET for their free provision of the aerosol properties over Beijing and Hangzhou. A very special acknowledgement is made to the editors and referees who make important comments to improve this paper.

References [1] H. Tang, X.G. Sun, G.B. Yuan, Calculation method for particle mean diameter and particle size distribution function under dependent model algorithm, Chin. Opt. Lett. 5 (2007) 31e33. [2] H. Qi, L.M. Ruan, S.G. Wang, M. Shi, H. Zhao, Application of multi-phase particle swarm optimization technique to retrieve the particle size distribution, Chin. Opt. Lett. 6 (2008) 346e349. [3] J.K. Du, W.C. Sun, X.D. Wang, G.L. Li, M.Z. Tan, L.Y. Fan, Experimental study on combustion and particle size distribution of a common rail diesel engine fueled with GTL/diesel blends, Appl. Therm. Eng. 70 (2014) 430e440. [4] Y.M. Wang, G.B. Liang, Z.D. Pan, Inversion of particle size distribution from light-scattering data using a modified regularization algorithm, Particuology 8 (2010) 365e371. [5] X.G. Sun, H. Tang, G.B. Yuan, Determination of the particle size range in the visible spectrum from spectral extinction measurements, Meas. Sci. Technol. 18 (2007) 3572. [6] J.Q. Shen, X.S. Cai, Optimized inversion procedure for retrieval of particle size distributions from dynamic light-scattering signals in current detection mode, Opt. Lett. 35 (2010). [7] M.X. Su, F. Xu, X.S. Cai, K.F. Ren, J.Q. Shen, Optimization of regularization parameter of inversion in particle sizing using light extinction method, China Particuology 5 (2007) 295e299.

9

[8] X.G. Sun, H. Tang, J.M. Dai, Inversion of particle size distribution from spectral extinction data using the modified beta function, Powder Technol. 190 (2009) 292e296. [9] H. Qi, B. Zhang, Y.T. Ren, L.M. Ruan, H.P. Tan, Retrieval of spherical particle size distribution using ant colony optimization algorithm, Chin. Opt. Lett. 11 (2013), 112901(1e5). [10] N.V. Voshchinnikov, V.G. Farafonov, Optical properties of spheroidal particles, Astrophys. Space Sci. 204 (1993) 19e86. [11] M.I. Mishchenko, G. Videen, V.A. Babenko, N.G. Khlebtsov, T. Wriedt, T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database, J. Quant. Spectrosc. Radiat. Transf. 88 (2004) 357e406. [12] T. Wriedt, Light scattering theories and computer codes, J. Quant. Spectrosc. Radiat. Transf. 110 (2009) 833e843. [13] J.Q. Zhao, Y.Q. Hu, Bridging technique for calculating the extinction efficiency of arbitrary shaped particles, Appl. Opt. 42 (2003) 4937e4945. [14] J.Q. Zhao, J.G. Li, Analytical transform techniques to retrieve non-spherical particle size distribution, J. Quant. Spectrosc. Radiat. Transf. 129 (2013) 287e297. [15] H.C. Hulst, H.C. Van de Hulst, Light Scattering: by Small Particles, Courier Dover Publications, 1957. [16] X.G. Sun, H. Tang, G.B. Yuan, Anomalous diffraction approximation method for retrieval of spherical and spheroidal particle size distributions in total light scattering, J. Quant. Spectrosc. Radiat. Transf. 109 (2008) 89e106. [17] H. Tang, J.Z. Lin, Retrieval of spheroid particle size distribution from spectral extinction data in the independent mode using PCA approach, J. Quant. Spectrosc. Radiat. Transf. 115 (2012) 78e92. [18] L.M. Popplewell, O.H. Campanella, M.D. Normand, M. Peleg, Description of normal, logenormal and RosineRammler particle populations by a modified version of the beta distribution function, Powder Technol. 54 (1988) 119e125. [19] A.B. Yu, Johnson's SB distribution function as applied in the mathematical representation of particle size distributions. Part 1: theoretical background and numerical simulation, Part. Part. Syst. Charact. 11 (1994) 291e298. [20] H. Tang, G.W. Liang, Inversion of particle size distribution from spectral extinction data using the bimodal Johnson's SB function, Powder Technol. 198 (2010) 330e336. [21] L.M. Gugliotta, G.S. Stegmayer, L.A. Clementi, V.D. Gonzalez, R.J. Minari, J.R. Leiza, J.R. Vega, A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements, Part. Part. Syst. Charact. 26 (2009) 41e52. [22] Z.Z. He, H. Qi, Y.Q. Wang, L.M. Ruan, Inverse estimation of the spheroidal particle size distribution using Ant Colony Optimization algorithms in multispectral extinction technique, Opt. Commun. 328 (2014) 8e22. [23] B. Zhang, H. Qi, Y.T. Ren, S.C. Sun, L.M. Ruan, Application of homogenous continuous Ant Colony Optimization algorithm to inverse problem of onedimensional coupled radiation and conduction heat transfer, Int. J. Heat Mass Transf. 66 (2013) 507e516. [24] W.T. Pan, A new fruit fly optimization algorithm: taking the financial distress model as an example, Knowledge-Based Syst. 26 (2012) 69e74. [25] Y.W. Zhang, C. Fang, Combinatorial optimization using FOA and GA in futures market technical analysis, in: 2012 12th International Conference on Control, Automation and Systems (ICCAS), October 17e21, 2012 in ICC, Jeju Island, Korea, 2012, pp. 2074e2078. [26] H.Z. Li, S. Guo, H.R. Zhao, C.B. Su, B. Wang, Annual electric load forecasting by a least squares support vector machine with a fruit fly optimization algorithm, Energies 5 (2012) 4430e4445. [27] S.M. Lin, Analysis of service satisfaction in web auction logistics service using a combination of fruit fly optimization algorithm and general regression neural network, Neural Comput. Appl. 22 (2013) 783e791. [28] S.X. Qin, X.S. Cai, Indirect measurement of the intensity of incident light by the light transmission fluctuation method, Opt. Lett. 36 (2011) 4068e4070. [29] N. Wang, Optic Measurement Technology of Particle Size and Its Application, Atomic Energy Press, Beijing, 2000. [30] L.M. Ruan, Q.Z. Yu, H.P. Tan, A transmission method for the determination of the radiation properties of small ash particles, J. Harbin Inst. Technol. (1994) 10e14. [31] E.W.C. Lim, Application of particle swarm optimization to Fourier series regression of non-periodic data, Ind. Eng. Chem. Res. 50 (2010) 2307e2322. [32] Aerosol robotic network (AERONET), http://aeronet.gsfc.nasa.gov/.

Please cite this article in press as: Z. He, et al., Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.08.057