LIU Zhigang，F(xiàn)ANG Hongyi，ZHU Ruihan，HE Zhenzong*，MAO Junkui

1.AECC Sichuan Gas Turbine Establishment，Mianyang 621000，P.R.China；2.College of Energy and Power Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

Abstract：Particles，including soot，aerosol and ash，usually exist as fractal aggregates. The radiative properties of the particle fractal aggregates have a great influence on studying the light or heat radiative transfer in the particle medium. In the present work，the performance of the single-layer inversion model and the double-layer inversion model in reconstructing the geometric structure of particle fractal aggregates is studied based on the light reflectancetransmittance measurement method. An improved artificial fish-swarm algorithm（IAFSA）is proposed to solve the inverse problem. The result reveals that the accuracy of double-layer inversion model is more satisfactory as it can provide more uncorrelated information than the single-layer inversion model. Moreover，the developed IAFSA show higher accuracy and better robustness than the original artificial fish swarm algorithm（AFSA）for avoiding local optimization problems effectively. As a whole，the present work supplies a useful kind of measurement technology for predicting geometrical morphology of particle fractal aggregates.

Key words：inversion radiative problem；artificial fish swarm algorithm；radiative property；particle fractal aggregate；geometrical morphology

0 Introduction

Particles，including soot，aerosol and ash，usu?ally exist in the form of fractal aggregates，which are suspended in the atmosphere and in industrial equipment，such as combustion chambers and fur?naces. The scattering and absorption properties of particles are important factors in the study of heat ra?diation transfer in industrial equipment and light radi?ation transfer in the atmosphere. The size distribu?tion，complex refractive index，and morphology of particles have great influence on the scattering and absorption characteristics［1-2］. The optical properties of particles and their interaction with radiation can be reflected by the complex refractive index. As the basic properties of particles，the complex refractive index and particle size distribution are usually con?sidered to be invariant. Many scholars have investi?gated on the size distribution and complex refractive index of particles using several types of methods［3-9］.

The geometrical form of particle and its aggre?gates is non-essential and easy to change. However，it also has an obvious effect on the radiation proper?ties of particle［10-11］. Accurate inversion of geometric feature parameters becomes more difficult as the soot is aging and evoluting along the direction of flame height，as shown in Fig.1.

Fig.1 Formation and morphology of soot aggregate in a dif?fusion flame height[12]

The non-invasive measurement method can be realized by optical means，which avoids these prob?lems and proves to be more accurate and effective.According to the theory of radiation inverse prob?lem，the geometric characteristic parameters of par?ticles can be estimated based on the external radia?tive transfer signals obtained from the particle medi?um. In view of the fact that the optical radiation transmission signal can provide a wealth of measure?ment information such as angle，spectrum and posi?tion，different scholars have proposed a variety of methods to reconstruct the geometric characteristics of particles and their fractal aggregates［13-16］. Howev?er，the description of the geometric characteristics of the particle fractal aggregates is still a problem that needs to be further studied to improve the accu?racy.

The purpose of this paper is to compare two in?version models in reconstructing the geometry of particle aggregates. The improved artificial fish swarm algorithm（IAFSA）is applied to improve the accuracy of the inverse problem. Firstly，the re?lated principles of positive problem，such as fractal aggregate theory，are introduced. Then，AFSA and IAFSA are used to reconstruct the geometric char?acteristics of fractal aggregates. Finally，the results of this paper are analyzed and the conclusion and prospect are given.

1 Mathematical Model for Direct Problem

1.1 Fractal aggregate theory

As described by the fractal theory（Fig.2），there are several important parameters in describing the morphology and structure of fractal aggregates，i.e.，fractal dimension Df，mean radius of the mono?mers a，total number of primary monomers Np，root mean square radius Rg，and fractal prefactor kf.The relationship between these parameters can be described by［17］

Fig.2 Schematic of fractal-like aggregates

where riis the distance from the i th sphere to the center of the aggregate mass. In this paper，Rgcon?formed to the log-normal（L-N）distribution con?structed by Zhang et al.［18］and the volume frequency distributions can be described as

where Rg，avdenotes the characteristic radius；σ the narrowness index. Our previous work［11］reported that Np，a，Dfand Rghad an important influence on the prediction of the radiation characteristics，while the fractal prefactor had a little one，which can be ig?nored.

1.2 Light reflectance?transmittance measure?ment method

When a collimated monochromatic laser beam impinges on the system with particle aggregates at room temperature（Fig.3），the radiative transfer in one-dimension particle system can be described as［1］

where Iλ(z，s) denotes the spectral radiative intensi?ty in the direction s at location z；l the wavelength of incident laser；αλand σλthe spectral absorption and scattering coefficient，respectively；Φλ(si，s)the scattering phase function；and Ωithe solid angle.αλ，σλand Φλ(si，s)can be described as

Fig.3 Schematic model of light-scattering measurement methods

where N(Rg，i) denotes the number concentration of the sample，N(Rg，i)=Ntot×fL?N(Rg，i)；Ntotthe to?tal number concentration of the sample；and Φp，ithe scattering cross-section，the ab?sorption cross-section，and the scattering phase function of fractal aggregate. In this paper，the RDG-FA method is applied to estimate the radiative properties of the particle aggregates as it is easy to program.The calculation efficiency is high and the precision is almost the same as that of GMM. The corresponding mathematical expression is［15，19-20］

where k=2π/λ denotes the wave number in the vacuum；the vertical（for incident radiation）and vertical（for scattered radiation）polarized differen?tial scattering cross-section of aggregates；andt he scattering cross-section and absorption crosssection of monomers；E(m) and F(m) the functions of the complex refractive index m of the monomer；G(kRg)a generalization function，and its mathemati?cal description is

where S(qRg) is the aggregate structure factor. As shown in Fig.4，there is a satisfactory agreement be?tween the results obtained by the RDG-FA method and those by the GMM model in predicting the radi?ation characteristics of fractal aggregates［20-21］. In this paper，the RDG-FA method is used to predict the radiation characteristics of fractal aggregates.

Fig.4 Radiative properties of fractal aggregates predicted by the GMM and RDG-FA methods

The mathematical expression of the boundary condition is

where I0denotes the total incident light intensity；the light intensity incident to the internal medium from the light incident side and the light output side of the sample，respectively；L the geometrical thickness of the medium；and θ the polar angle. In this paper，the finite volume method（FVM）is used to simulate the radiation equation because of its good performance in precision and cal?culation time［22］. The specific implementation pro?cess can be referred in Ref.［23］. The mathematical expressions of hemispherical reflectance R and trans?mittance τ are

2 Mathematical Model for Inverse Problem

In water areas，fish can find the place where nutrients are abundant by themselves or by follow?ing other fish，so the place where fish usually live most is the place where nutrients are abundant in the water area. Inspired by this phenomenon，the artifi?cial fish-swarm algorithm （AFSA） is applied to solve different problems. There are four operators in the AFSA：preying behavior，swarming behavior，follwing behavior and random behavior，and the state of the individual in the fish group is the vector to be solved. Through the cooperation between the individual fish in the fish group，the problem can be solved. In the algorithm，a bulletin board is usually set up to record the current optimal individual state.

2.1 Standard AFSA

Assuming that the current state of artificial fish individual i is Xi=(xi1，xi2，…，xin)，the food con?centration of the fish group is set as Yi=f (Xi)，that is，the objective function to be solved，the dis?tance between fish groups can be expressed as di，j=visual is the visual field of fish，δ is the crowding factor，step is the moving step of fish group，and try_number is the number of repeated at?tempts in foraging behavior. The flow of AFSA is as follows［24-27］.

（1）Prey：Assuming that the current state of the fish is Xi，a new state is randomly selected with?in its field of vision visual（di，j＜dvisual），as shown in Eq.（18）. If f (Xj)＜f (Xi)，the fish moves one step to the state according to Eq.（19）；otherwise，choose a new Xjand try again. If the fish maintain its position after trying try_number times，it moves the fish one step at random. nRand1and nRand2are ran?dom numbers in the range of［0，1］.

（2）Swarm：Assuming that the current state of the fish is Xi，the central position Xcand the number of partners nfare searched in its field of vision visu?al. If Yc/nf＞dYi，which means that there are not a lot of fish in the center and there is still a large amount of food there. The fish will take a step in this direction，as shown in Eq.（20）. Otherwise，preying behavior will be carried out. nRand3is a ran?domly generated number between 0 and 1.

（3）Follow：Assuming that the fish’s current state is Xi，searching partner Xjwhose Yjis the maximum in its field of vision. If Yj/nf＞Yi，it indi?cates that there is more food and less fish around the partner Xj，and the fish will thus move forward to that position，otherwise they will continue to prey.

（4）Update the bulletin board：The state of the best fish in history is recorded on a bulletin board.All the artificial fish check their own state every iter?ation，and if f (Xj)＜f (Xbest)，changes Xbestto Xi.

（5）Random：Assuming that the current state of the fish is Xi，the fish swarm randomly in their own field of vision visual without performing any other behavior.

2.2 Improved AFSA fusing differential evolu?tionary algorithm

Differential evolutionary algorithm is a kind of evolutionary algorithm proposed by Storn and Price in 1997［27］to solve continuous global optimization problems. Its basic idea is achieved by mutation，crossover and evolution to produce new individuals.Among them，mutation is to weigh the difference between the vectors of two individuals in the popula?tion and then sum them with the third individual ac?cording to certain rules to produce new individuals.Then，crossover is to combine the new individual with the target individual to produce the probing in?dividual，as shown in Eq.（21）. Finally，the probing individual and the original target individual are com?pared. The original individual will be replaced if its target value is lower than that of the new one；other?wise， it will still be preserved， as shown in Eq.（22）

where F is a mutation factor between［0，2］，i，k，l and m the different individuals in the fish group，CRa cross factor in the range of［0，1］，and Q(i)a ran?domly selected fish group. The flow of AFSA based on differential evolutionary algorithm is as fol?lows：

Step 1Initialize parameters such as artificial fish size，field of view visual，moving step of fish group step，maximum number of repeated attempts try_number，and the maximum number of iterations.

Step 2Calculate the objective function of in?dividual fish. Compare it with the value of the bulle?tin board. Choose the better one to assign to the bul?letin board.

Step 3Each individual fish executes the prey behavior，swarm behavior and follow behavior.

Step 4Compare the objective functions of the three behaviors and select the optimal values.

Step 5On the basis of the optimal value se?lected by Step 4，the difference approximation is carried out，and the objective function of the prob?ing individual and the target individual is calculated.Compare values with that of the bulletin board.Choose the best one to assign to the bulletin board.

Step 6Check the termination conditions，if a predetermined number of evolutions or a predeter?mined objective value is reached，then output the optimal solution（artificial fish state and function val?ue in bulletin board）. The algorithm terminates，otherwise，turn to Step 3.

3 Numerical Simulation

3.1 Single?layer inverse model

The objective function value Fobjis defined as the sum of the squared residuals of the ratio between the estimated signals ratios and the measured sig?nals ratios. The lower the objective function value is，the closer the result is to the real value. There?fore，the geometric parameters are inversed by mini?mizing Fobj.

In order to reduce the certain randomness of the stochastic optimization，all the inversion results have been calculated for N=30. The reliability and feasibility of the optimization algorithm is evaluated by the following characteristic parameters：

（1）The relative deviation ξ，which means the sum of the deviation between the probability distri?bution predicted by the IAFSA and the true distribu?tion of Rg，can be expressed as

where N′is the number of subintervals that the size rangethe midpoint of the i th subintervalthe true distribution in the i th subinterval；andthe predicted distribution in the i th subinterval.

（2）The standard deviation η and the relative error δ here are defined as

3.1.1 Comparison of standard AFSA and IAF?SA

The performance of IAFSA can be investigated by comparing with that of AFSA. Table 1 lists the parameters of different algorithms. In this paper，when one of the following conditions are met，the al?gorithm is terminated：the maximum number of gen?erations is equal to 1 000 or the iteration accuracy is less than 10—16.Table 2 lists the results of different algorithms when（Df，Rg)=(1.8，80).Table 2 shows that the results obtained by IAFSA are more accu?rate than that of the AFSA in the inversion of Dfand Rg，regardless of adding random measurement er?rors. And the performance of the two algorithms in inversion of fractal dimension Rgis more satisfactory than that in inversion of radius of revolution Df.However，even with 5% measurement error added，the inverse results of Dfand Rgare acceptable. The values of the objective function between AFSA and IAFSA are compared in Fig.5. It is easy to find that the convergence properties of the IAFSA algorithm are better than those of the AFSA algorithm，which means that the IAFSA is of higher efficiency and ac?curacy in terms of the application of inversion.

Table 1 System control parameters of the AFSA algo?rithms

Table 2 Results retrieved by different AFSA algorithms when (Df,Rg)=(1.8,80)

Fig.5 Comparison of objective function values of AFSA and IAFSA

The accuracy of inverse results is affected to a certain extent by the multi-value of the results. It means there are more than one result satisfying the convergence conditions at the end of inversion calcu?lation. Thus，the accuracy of inversion results is re?duced. As far as this paper is concerned，multi-val?ue includes the same experimental hemispherical re?flectance R and transmittance τ corresponding to many couples of(Df，Rg)，which means a unique so?lution may not be found in the inverse problem.Fig.6 depicts the distribution of objective functions in the single-layer model. As can be seen from the graph of the objective function，multiple points on a curve are in the minimum region. Comparing the in?verse results of Dfand Rg，it can be found that the inversion accuracies of Dfand Rgare different even without random measurement error. As can be seen from Fig.6，the range of Dfthat meets the conver?gence conditions covers the whole inversion range［1，3］，while the range of Rgthat meets the conver?gence conditions is only in the range of［75，90］. It means the retrieval results for Dfare worse than those for Rgas the multi-value characteristics are more serious.

Fig.6 Distribution of objective function value under singlelayer inverse model

3.1.2 Retrieval of geometrical characteristic pa?rameters of particle fractal aggregates

Table 3 lists the results retrieved by the singlelayer model when Rgobeying L-N distribution.Their true values are set as (Np，Df，Rg，av，σ，a)=(60，1.7，117，2.1，10). The inversion accuracy de?creases with the increase of random measurement er?ror. And the relative errors δ and standard errors η increase accordingly. Fig.7 depicts the retrieval curves of the probability density distribution fL?N(Rg) of the root mean square radius Rgin differ?ent cases. The black dot symbols represent the real value curve，the red box represents the retrieval curve for inversion of three parameters，the blue tri?angle represents the retrieval curve for inversion of four parameters，and the green triangle represents the retrieval curve for inversion of five parameters.It can be found that the inversion accuracy decreases as the number of inversion parameters increases.

Table 3 Results retrieved by single?layer model when Rg obeying L?N distribution

Fig.7 Retrieval curves of root mean square radius using different measurement methods

When only three parameters are inverted，the inverse accuracies of results are acceptable even with 3% random measurement error，and the maxi?mum relative error is only 7.1%. The inversion ac?curacy decreases with more parameters retrieved.Under 3% random measurement error，the maxi?mum relative errors of Np，Df，Rg，avand σ inversion results are 15.4%，13.8%，7.3% and 6.8%，re?spectively. At the same time，the inversion results deteriorate gradually when five parameters are re?trieved. When 1% random measurement error is added，the relative errors of Npand Dfare large，be?ing 18.9% and 13.7%，respectively. The inversion results of Rg，av，σ，and a are better，and the stan?dard deviation is controlled within 11%.

3.2 Double?layer inverse model

The predicted results of the geometrical charac?teristic parameters are satisfactory without random errors. However，the accuracy of inversion decreas?es gradually as random errors are added. The reason may be that only a couple of the reflectance and transmittance signals can be obtained，while more than two characteristic parameters，i.e.the mean ra?dius of the monomers a，total number of primary monomers Np，and the root mean square radius Rg，need to be retrieved. To solve this problem and im?prove the inverse accuracy，the double-layer inverse model，which can provide more useful information about the particle medium，is used in this paper.Two thicknesses of the layers are used （L1=0.02 m，L2=0.04 m）. The true values of them are set as (N，Df，Rg，av，σ，a)=(60，1.7，117，2.1，10).And the objective function Fobjis

Table 4 lists the results retrieved by using the double-layer inverse model when Rgobeying L-N distribution. Fig.8 shows the distribution of the dis?tribution of objective functions in the double-layer model. Fig.9 shows that the overall trend of the esti?mation results using the double-layer model is con?sistent with that of the single-layer model. When the random measurement errors are not added，the cal?culation results are better. While inverse results be?come worse when the random measurement error added increase. As the number of simultaneous in?version parameters increases，the accuracy of calcu?lation results decreases.

Table 4 Results retrieved by double?layer inverse model when Rg obeying L?N distribution

Fig.8 Distribution of objective function value under doublelayer inverse model

Fig.9 Retrieval curves of root mean square radius using the measurement angles with different intervals

Compared Fig.6 with Fig.8，it can be found that when the objective function value arrives at 10e-16in the single-layer-inverse model，the regions of the objective function value tend to different points，which means the retrieval results are not unique. While the double-layer inverse model is ap?plied，the multiplicity of the inverse results will be decreased，and just a few points satisfy the objective function value equal to 10e-16. When the random er?rors added to the measurement signals，the objec?tive function value may not arrive at the conver?gence limit. For example，with 3% random error added to the measurement signals，the objective function value can only reach 10e-12. From Figs.6，8，it can be found that the region of the objective function value less than 10e-12in single-layer-in?verse model is larger than that in double-layer in?verse model，which means more couples of charac?teristic parameters in single-layer-inverse model that meet objective function value equal to 10e-12. The phenomenon can explain why more satisfactory re?trieval results are obtained in double-layer inverse model. Specifically，the inversion results of five pa?rameters are satisfactory even when the random measurement noise is increased by 3%. Its relative error is generally less than 10%. Only the inversion result of Npis slightly worse，its relative error is 11.7%. Compared with the results obtained by the single-layer model，it has a considerable improve?ment. This is because using the double-layer model can provide more useful information about fractal ag?gregates than using the single-layer one.

4 Conclusions

Based on the IAFSA，this paper investigates the robustness and reliability of two inverse model in reconstructing the geometrical characteristic pa?rameters of fractal aggregates. The conclusions are as follows：

（1）When retrieving the geometric parameters of fractal aggregates，IAFSA is more accurate than AFSA，and its calculation speed is faster.

（2）With the increase of the number of simulta?neous inversion parameters，the inversion accuracy gradually decreases. However，the inversion results are still satisfactory even with 3% random measure?ment error.

（3） Compared with the single-layer inverse model，the retrieval results obtained by the doublelayer inverse model show better convergence accura?cy and robustness as the double-layer inverse model is more effective to avoid the multi value characteris?tics of retrieval results and improve the accuracy of inversion results.

In conclusion，the IAFSA and the double-layer inverse model are effective and reliable in recon?structing the geometric structure of fractal aggre?gates.