OPTIMAL WAVELENGTH SELECTION ALGORITHM OF NON-SPHERICAL PARTICLE SIZE DISTRIBUTION BASED ON THE LIGHT EXTINCTION DATA

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1 THERMAL SCIENCE, Year 202, Vol. 6, No. 5, pp OPTIMAL WAVELENGTH SELECTION ALGORITHM OF NON-SPHERICAL PARTICLE SIZE ISTRIBUTION BASE ON THE LIGHT EXTINCTION ATA y Hong TANG * College of Metrology & Measurement Engineering, China Jiliang University, Hangzhou, China Short paper OI: /TSCI205353T Introduction In this paper, the anomalous diffraction approximation method is improved for calculating the extinction efficiency of non-spherical particles. Through this step, the range of the refractive index of particles can e enlarged, and the improved anomalous diffraction approximation method can e applied easily to the calculation of extinction efficiency for the most kinds of non-spherical particles. Meanwhile, an optimal wavelength selection algorithm is proposed for the inversion of non-spherical particle size distriution in the dependent mode. Through the improved anomalous diffraction approximation method, the computation time is sustantially reduced compared with the rigorous methods, and a more accurate inversion result of particle size distriution is otained using the optimal wavelength selection method. Key words: particle size distriution, light extinction, anomalous diffraction, wavelength Atmospheric aerosols are suspensions of small solid or liquid particles in the atmosphere. To understand the effect of atmospheric aerosol on the atmospheric visiility degradation phenomena require knowledge of aerosol optical properties and their particle size distriutions []. While particle size distriution is related to some processes such as nucleation [2], atomization [3], coagulation [4], deposition [5], synthesis [6], and reakage [7]. Light scattering particle sizing techniques have een widely used during the recent years since they provide an important tool for the characterization of a large numer of industrial production processes. These techniques mostly contain the total light scattering, angle light scattering, diffraction light scattering and dynamic light scattering, and the measurement range of them ranges from nanometer to millimeter. It can in fact e used for on-line monitoring of micron or su-micron particle systems thus providing real time measurements of oth particle size distriution and particle concentration [8, 9]. But the calculation of extinction efficiency ased on the classical Mie theory is complicated and time consug, what is important is that the Mie theory can only e applied to the homogeneous spherical particle, while most particle systems are non-spherical particles. A suitale approximation method is the anomalous diffraction approximation (AA), which can cal- Author s tanghong@cjlu.edu.cn

2 354 THERMAL SCIENCE, Year 202, Vol. 6, No. 5, pp culate the extinction efficiency of spherical and non-spherical particles [0-2]. Although some rigorous methods have een developed for calculating the extinction efficiency of randomly oriented non-spherical particle such as the extended oundary-condition method (T-matrix) and the finite-difference time-domain method (FT), the AA is still an effective alternative which could avoid the complexity in calculating the extinction efficiency. In this paper, we will use the improved AA method to calculate the extinction efficiency of non-spherical particles. Meanwhile, we will propose an optimal wavelength selection algorithm for the inversion of non-spherical particle size distriution in the dependent mode. Theory and computation When a eam of parallel monochromatic light passes through a particle system (thickness L) with its refractive index different from that of the dispersant medium, oth the scattering and the asorption lead to an attenuation of the transmitted light. According to the Lamert-Beer law, the transmitted light intensity I is defined as ln I(λ)/I 0 (λ) = τ(λ)l where I 0 (λ) is the incident light intensity at wavelength λ, I(λ)/I 0 (λ) is otained y actual measurement, and τ(λ) is the turidity. In the asence of the multiple scattering and interaction effects, the transmitted light intensity I for a spherical particle system is given y the integral equation: max I( λ) 3 Qext ( λ, m, ) f( ) ln LN d I ( λ ) = 2 () 0 where Q ext (λ, m, ) is the extinction efficiency of a single spherical particle which is a function of the particle diameter, the wavelength λ in the medium and the relative refractive index m can e calculated, N is the total numer of particles, the lower and upper integration limits are denoted y and max, and f() is the frequency distriution of a particle system. Within the framework of AA, the extinction efficiency of a spheroidal particle has the same form as a sphere. When the incident light propagates along the rotation axis of the spheroidal particle, the extinction efficiency of a spheroidal particle is given y [4]: Q ad exp( iw) exp( iw) = 4Re i w w where w = 2ka(m )/λ, a is the semiaxis length of the rotation axis, and k the wave numer. For a particle whose typical size is much larger than the incident wavelength or high asoring, the edge can not e treated as sharp and the effect of the curvature y the particle must e included. After considering the edge effect term, the domain of applicaility of AA can e extended. The edge effect term is derived as: Q edge 2c 0 a 2/3 /k 2/3 4/3 where is the semiaxis length of the other axis, r = a/ is the aspect ratio, for prolate spheroid r > and for olate spheroid r < and c 0 = We use the standard AA plus the edge term to calculate the extinction efficiency of spheroidal particles when the or axis length is in the range from ~0 μm, and the standard AA is used when the or axis length is in the range from 0.~ μm. According to the Lamert-Beer s law, the extinction caused y spheroidal particles with a particle size distriution f (, r) can e expressed as [5]: (2)

3 THERMAL SCIENCE, Year 202, Vol. 6, No. 5, pp max max I( λ) ln LN Qext(, λ mrpr,,) (,) f(,)/ r Vr (,)ddr I ( ) 0 r λ = (3) r where Q ext (λ, m,, r) is the extinction efficiency of a spheroidal particle; here we use the angular averaging extinction efficiency. P(, r) is the averaging projected area of the particle, f (, r) the frequency distriution function of particles in volume, and V (, r) the volume of a spheroidal particle. The extinction values are calculated y the improve AA method. The distriution of spheroidal particles is assumed to have R-R distriution with the expression: f R-R k k kr kr k kr r r (,) r = exp exp r r r where, k, r, and k r are the characteristic parameters of spheroidal R-R distriution. The visile extinction spectrums have relationship with the characteristic parameters of spheroidal R-R particle size distriution. If the visile extinction spectrum is a monotone ascending we have < μm; > 3.5 μm for the visile extinction spectrum is a monotone descending; > 2 μm for the visile extinction spectrum is curving curve. When the relative refractive index of particles is fixed, the visile extinction spectrum are changed from a monotone ascending to curving and then to monotone descending with the increasing (fig. ). We propose an optimal wavelength selection algorithm for the inversion of non-spherical particle size distriution in the dependent mode. The corresponding wavelengths were selected as the measurement wavelengths. Furthermore, the imum and the maximal wavelengths in the visile region were also selected as the measurement wavelengths. Tale gives the inversion results of spheroidal particles with the proposed optimal wavelength selection algorithm. In ta., No. denotes that optimal wavelength selection algorithm is used and No. 2~No. 3 denote that wavelengths are selected with random methods. It is clear to see that good inversion results can e otained with the optimal wavelength selection algorithm. Tale. Inversion results of spheroidal particles [(, k, r, k r ) = (3.5, 5,.5, 5)] No. No. 2 No. 3 No. 4 (λ, λ 2, λ 3, λ 4, λ 5 )/μm (0.4, 0.46, 0.55, 0.65, 0.8) (0.46, 0.55, 0.65, 0.7, 0.76) (0.4, 0.5, 0.6, 0.7, 0.8) (0.5, 0.53, 0.6, 0.7, 0.75) Inversion results 0% random noise % random noise (3.5004, 4.985,.5024, ) (3.4998, 5.078,.4880, ) (3.4998, ,.4926, ) (3.5006, ,.4838, 3.063) (3.4898, 4.0,.4886, ) (3.4749, ,.4807, 3.394) (3.4768, 6.25,.486, 3.08) (3.5909, ,.4794, ) Inversion error ς (0% random noise) (4) Conclusions An improved AA method is used for calculating the extinction efficiency of spheroidal particles, in which the complexity in calculating the extinction efficiency of non-sphe-

4 356 THERMAL SCIENCE, Year 202, Vol. 6, No. 5, pp (,3,3.5,5) (,7,3.5,5) (,5,3.5,5) 0.9 (,5,.5,5) (,5,.5,7) 0.8 (,5,.5,3) (2,3,3.5,5) (2,7,3.5,5) (2,5,3.5,5) (2,5,.5,5) (2,5,.5,7) (2,5,.5,3) (a) λ/μm = μm () λ/μm = 2 μm (c) λ/μm = 3.5 μm rical particles is diished, and the computation time is sustantially reduced. Meanwhile, the visile extinction spectrums of spheroidal particles are analyzed, and an optimal wavelength selection algorithm is proposed for the inversion of spheroidal particle size distriution. The visile extinction spectrums have relationship with the distriution of spheroidal particles, and the optimal wavelength selection algorithm can otain good results for the inversion of spheroidal particle size distriution. Acknowledgment This work was supported y the National Natural Science Foundation of China (No , No ) and Zhejiang Province Natural Science Funds (Y6047). References (3.5,3,3.5,5) (3.5,7,3.5,5) (3.5,5,3.5,5) (3.5,5,.5,5) (3.5,5,.5,7) (3.5,5,.5,3) [] Upadhyay, R. R., Ezekoye. O. A., Treatment of Size-ependent Aerosol Transport Processes Using Quadrature Based Moment Methods, J. Aerosol Sci., 37 (2006), 7, pp [2] Yu, M. Z., Lin, J. Z., Binary Homogeneous Nucleation and Growth of Water-Sulfuric Acid Nanoparticles Using a TEMOM Model, International Journal of Heat and Mass Transfer, 53 (200), 4, pp [3] Lin, J. Z., Qian, L. J., Xiong, H. B., Effects of Operating Conditions on the roplet eposition onto Surface in the Atomization Impinging Spray with an Impinging Plate, Surface and Coatings Technology, 203 (2009), 2, pp (7.5,3,3.5,5) (7.5,7,3.5,5) (7.5,5,3.5,5) (7.5,5,.5,5) (7.5,5,.5,7) (7.5,5,.5,3) (d) λ/μm = 7.5 μm Figure. Visile extinction spectrums of spheroidal particles with R-R distriutions (m =.235)

5 THERMAL SCIENCE, Year 202, Vol. 6, No. 5, pp [4] Yu, M. Z., Lin, J. Z., Chan, T. L., A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion, Aerosol Science and Technology, 42 (2008), 9, pp [5] Lin, J. Z., Lin, P. F., Chen, H. J., Research on the Transport and eposition of Nanoparticles in a Rotating Curved Pipe, Physics of Fluids, 2 (2009), 2200, pp. - [6] Yu, M. Z., Lin, J. Z., Chan, T. L., Numerical Simulation of Nanoparticle Synthesis in iffusion Flame Reactor, Powder Technology, 8 (2008),, pp [7] Gan, F. J., Lin, J. Z., Yu, M. Z., Particle Size istriution in a Planar Jet Flow Undergoing Shear- Induced Coagulation and Breakage, Journal of Hydrodynamics, 22 (200), 4, pp [8] Pedocchi, F., Garcia, M. H., Noise-Resolution Trade-off in Projection Algorithms for Laser iffraction Particle Sizing, Applied Optics, 45 (2006), 5, pp [9] Khletsov, B. N., et al., Studies of Phosphatidylcholine Vesicles y Spectroturidimetric and ynamic Light Scattering Methods, Journal of Quantitative Spectroscopy and Radiative Transfer, (2003),, pp [0] Van de Hulst, H. C., Light Scattering y Small Particles, John Wiley and Sons, New York, USA, 957 [] Mishchenko, M. I., Travis, L.., Light Scattering y Polydispersions of Randomly Oriented Spheroids with Sizes Comparale to Wavelengths of Oservation, Applied Optics, 33 (994), 30, pp [2] Zumer, S., Light Scattering from Nematic roplets: Anomalous-iffraction Approach, Physical Review A, 37 (988), 0, pp [3] Sun, X. G., Tang, H., Yuan, G. B., Anomalous iffraction Approximation Method for Retrieval of Spherical and Spheroidal Particle Size istriutions in Total Light Scattering, Journal of Quantitative Spectroscopy and Radiative Transfer, 09 (2008),, pp [4] Jalava, J. P., Taavitsainen, V. M., Haario, H., Lameerg, L., eteration of Particle and Crystal Size istriution from Turidity Spectrum of TiO 2 Pigments y Means of T-Matrix, Journal of Quantitative Spectroscopy and Radiative Transfer, 60 (998), 3, pp Paper sumitted: August, 202 Paper revised: Septemer 3, 202 Paper accepted: Septemer 2, 202