Retrieval of particle size distribution in the dependent model using the moment method

Xiaogang Sun; Hong Tang; Jingmin Dai

doi:10.1364/OE.15.011507

1. Introduction

Light scattering particle sizing techniques have been widely used during the recent years since they provide an important tool for the characterization of a large number of industrial production processes. These techniques mostly contain the spectral extinction, angle light scattering, diffraction light scattering and dynamic light scattering [1–4]. The measurement range of them ranges form nanometer to millimeter. Among these techniques, the spectral extinction technique is probably the most attractive one because it requires a simple optical layout and can be realized by adaptation of a commercial spectrophotometer. It can in fact be used for in-line monitoring of micron or sub-micron particle systems thus providing real time measurements of both particle size distributions and particle concentration. With the urgent demand for in situ particle sizing, the spectral extinction technique has grown with rapid developments and broad potential applications.

The particle size distribution can be retrieved by some inversion methods based on the measurements of extinction values at multiple wavelengths. The developments of stable methods to retrieve the particle size distribution have long been a subject of research effort. These methods can be divided into three categories [5]. The first category is the analytic inversion method in which an approximate scattering kernel is used and the integral equation is retrieved analytically. The second category is the independent model algorithm in which no a prior information about the particle size distribution is available and the particle size distribution is retrieved by the discrete liner equations set. The third category is the dependent model algorithm in which some a prior information about the shape of the size distribution is assumed and the true particle size distribution is retrieved using a certain optimization algorithm [6, 7].

Actually, many particle systems can often been approximately described by a specified size distribution, therefore the dependent model algorithm is used in the work. In the dependent model, although several studies have been conducted on the retrieval of particle size distribution using the numerical integration method, there has been little study on the retrieval of particle size distribution using the moment method because the extinction efficiency is calculated from the Mie’s theory for spherical particle which requires the summation of an infinite series of complicated terms containing spherical Bessel functions. However, the origin moments of particle size distribution can be related to some important characteristics of the particle size distribution, which provide important insight into the particle system [8,9]. In this paper, we present the theoretical and experimental study of the retrieval of particle size distribution using the moment method in the spectral extinction technique. In doing so, we have obtained very simple and reasonably accurate mathematical relationships, which allow us to obtain a few relevant parameters for the construction of particle size distribution with a good accuracy.

2. Theory

The spectral extinction particle sizing technique is based on the light scattering theory. When a beam of parallel monochromatic radiation light of intensity I₀ passes through a suspension of particle system (thickness L) with an refraction index different from that of the dispersant medium, scattering and absorption lead to an attenuation of the transmitted light. According to the Lambert-Beer law, if the suspension of particle system is polydisperse spherical and the multiple scattering and interaction effects can be neglected, the transmitted light intensity I is defined as follows[10]:

\ln \frac{I (λ)}{I_{0} (λ)} = - \frac{3}{2} \times L \times N_{D} \times \int_{D_{min}}^{D_{max}} \frac{Q_{ext} (λ, m, D)}{D} \times f (D) d D .

where I₀(λ) is the incident light intensity at wavelength λ(i.e. the intensity of transmitted light through the suspending medium-in the absence of suspended particles), the extinction value I(λ)/I₀(λ) is obtained by actual measurements, Q_ext(λ,m,D) is the Mie extinction efficiency of a single particle which is a complex function of particle diameter D, wavelength λ in the medium and relative refractive index m (the ratio between the particle and medium refractive index), N_D is the total number of particles, the lower and upper integration limits are denoted by D _min and D _max, f(D) is the volume frequency distribution of particle system with a diameter between D and D+dD, which is the particle size distribution function to be determined[11–13].

Equation (1) is Fredholm integral equation of the first kind. This is a classic ill conditioned problem, which means that different distributions can fit the data I(λ)/I₀(λ). Therefore the retrieval of Eq. (1) is not a trivial task, because the retrieved distribution might be highly unstable. Hence, frequently one has to resort to approximation schemes. To retrieve the particle size distribution from spectral extinction data in the dependent model one need to solve the optimal problem. The optimization method proceeds to fit the spectral extinction data by varying the form of particle size distribution until the best characteristic parameters are achieved.

In the dependent model, different types of particle size distribution functions have been used such as the log-normal function, Rosin-Rammler function, gamma function and so on. Among these available well-behaved distribution functions, we have chosen to work with the log-normal and Rosin-Rammler function.

The monomodal log-normal(L-N) volume frequency distribution function f(D) is described by:

f_{L - N} (D) = \frac{1}{\sqrt{2 π} D \ln σ} \times \exp (- \frac{{(\ln D - \ln u)}^{2}}{{2 (\ln σ)}^{2}}) .

where D is the particle diameter(in µm), u is the geometric number mean diameter(in µm),σ is the geometric standard deviation.

In this work, the particle size distribution from the spectral extinction data is reconstructed in combination with the moments of particle size distribution in the dependent model algorithm. The moment method is a low order method that reduces the initial problem to a simpler one, and the computation time is substantially reduced compared with the numerical integration methods, which is especially suitable for in-line particle size measurement.

Conventional moment models are computationally less demanding, but have been restricted in their application to those systems for which the set of the moment equations is closed [14]. For a size distribution f(D) of spherical particles, the lth origin moment M_l is written as:

M_{l} = \int_{0}^{+ \infty} D^{l} f (D) d D .

where l is an arbitrary real number. If l becomes 0, M₀=1. When l is 1, M₁ is equivalent to the mean value of f(D), and when l is 2, the variance of f(D) is M₂-(M₁)².

Applying the monomodal L-N distribution to the moment relation, the relation between each moment is represented by the expression[14,15]:

M_{l} = u^{l} \exp (\frac{l^{2}}{2} {(\ln σ)}^{2}) .

The characteristic parameters u andσ of L-N distribution can be derived by the first-order and second-order moments:

u = \frac{{(M_{1})}^{2}}{\sqrt{M_{2}}} .

σ = \exp (\sqrt{\ln (\frac{M_{2}}{{(M_{1})}^{2}})}) .

Using the two ordinary equations in regards to the moment relation, the geometric mean diameter u and geometric standard deviation σ can be obtained. With the values of u and σ given by Eqs. (5) and (6), it is possible to construct the L-N size distribution function.

The monomodal Rosin-Rammler(R-R) volume frequency distribution function f(D) is defined as follows[16]:

f_{R - R} (D) = \frac{k}{\bar{D}} \times {(\frac{D}{\bar{D}})}^{k - 1} \times \exp (- {(\frac{D}{\bar{D}})}^{k}) .

where D is the particle diameter(in µm), D is the is the characteristic diameter of the distribution (in µm), k is narrowness index of the distribution.

The relation between each moment of the R-R distribution can be expressed as:

M_{l} = {\bar{D}}^{l} gamma ((k + l) ⁄ k) .

So the characteristic parameters of monomodal particle size distribution may be obtained by the first-order and second-order moments of the particle size distribution.

The original extinction efficiency Q _ext(λ,m,D) can be obtained from the complicated Mie’s solution to Maxell’s equation for spherical particle. However, the mathematical expression of Q _ext(λ,m,D) requires the summation of an infinite series of complicated terms containing spherical Bessel functions, and the integration equation cannot be evaluated as a moment formula in its present from. For this reason, Q _ext(λ,m,D) should be expressed as a function of the power of particle diameter in order to apply the moment method. In this study, the single particle extinction efficiency Q _ext(λ,m,D) which is derived theoretically using Mie’s theory, is approximated with a higher order polynomial. Subsequently, the extinction efficiency can be obtained as follows [8]:

Q_{ext} (λ_{j}, m, D) = \sum_{i = 0}^{P} A_{ij} D^{P - i} .

where i=0…P, j is the jth wavelength, A_ij is the approximated coefficient in regard to P order polynomials which is known beforehand and varied with λ_j, m, and the particle size range. The P value also depends on λ_j, m, and the particle size range.

By applying this moment operator to Eq. (1), the extinction value I(λ)/I₀(λ) can be developed with the general form:

\ln (\frac{I}{I_{O}})_{j} = - \frac{3}{2} \times L \times N \times \sum_{i = 0}^{P} A_{ij} M_{P - i - 1} .

Combination Eq. (10) with Eq. (4) or Eq. (8), one can derive the characteristic parameters of the L-N or R-R particle size distribution using a certain optimization algorithm.

Figure 1 shows the comparison of the approximated single spherical particle extinction efficiency with the one which is theoretically derived from the Mie theory. The relative refractive index m=1.235. The curve of the extinction efficiency is oscillatory as a function of the particle diameter, and the shape of the extinction efficiency curve is different in different particle size range. So the fitted polynomial is also different in different particle size range. For the case of (a) where the particles are in the range from 0.1~1 um in diameter, the third-order polynomial is used to fit the extinction efficiency. For the case of (b) where the particles are in the range from 1~5 um in diameter, the tenth-order polynomial is used. For the case of (c) and (d), the twentieth-order polynomial is used to fit the extinction efficiency. As shown in Fig. 1, both results show good agreement without a great loss of accuracy in different particle size range. Here, the overall particle size measurement range is limited from 0.1~10 um in diameter, which is the optimal measurement rang in spectral extinction technique. Thus, we can conclude that the approximated polynomial can be alternatively applied to the retrieval of the particle size distribution instead of the Mie solution for spherical particle.

Figure 2 shows the comparison of the approximated single spherical particle extinction efficiency with the one which is theoretically derived from the Mie theory with different relative refractive indices. For the case of (a), the twentieth-order polynomial is used to fit the extinction efficiency. For the case of (b), the curve of the extinction efficiency is smoother because the imaginary part of the relative refractive index is not zero. So the sixteenth-order polynomial is used. As shown in Fig. 2, both results also show good agreement especially for the case of (b), and the P values also vary with the varied relative refractive indices.

Fig. 1. Fitting curve of the extinction efficiency for spherical particle in different particle size range

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Fig. 2. Same as Fig.1 but with different relative refractive indices

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3. Computer simulations and experiments

Using the extinction values at several wavelengths in combination with the moment relation, the particle size distribution can be reconstructed. To valid the feasibility of the moment method, we perform different computer simulations by using different size distributions. Each simulation consists of two steps: the first step is creating a size distribution to generate the extinction intensity signals using the Mie extinction efficiency at multiple wavelengths as the input data. The total particle number N _D is set N _D=1, because N _D is just an arbitrary scaling factor for the integral properties of the particle size distribution, L=10 mm. The second step is that these data is processed and the particle size distribution is retrieved via the optimization algorithm. In order to examine the effect of noise on the retrieval of particle size distribution, 2% random noise is also added to the intensity signals by multiplying each extinction value by (1+0.02*R(0,1)), representing a 2% noise level, where R(0,1) represents a normally distributed random number with mean zero and standard deviation one. The genetic algorithm is superior to these simplex optimization techniques in the search of global optimal values of multi-object function instead of local ones. It is a heuristic combinatorial search technique that comes from the concept of natural genetics and the Darwinian theory of survival of the fittest, and it obtains its solution in a way after such genetic operations as selection, crossover, and mutation. The genetic algorithm has advantages of simplicity, globality, parallelism although it is a little time consuming. In our investigation, the initial population is generated randomly from a uniform distribution where each variable is chosen within the bounds. At the end of each current generation the elitist strategy is applied and the worst individual is replaced by the best one of the former generation to ensure the globe convergence.

Table 1 shows the inversion results of the monomodal R-R distributions in different particle size range using the moment method. The relative refractive index m=1.235. The order of the fitted polynomial is determined according to the different particle size range. Since the fitted polynomial of the extinction efficiency is still an oscillatory function. The oscillatory nature of the extinction efficiency determines that it needs more than two equations to solve the two parameters D and k. Here four incident wavelengths(0.4 um, 0.55 um, 0.65 um, 0.8 um) are selected in the visible spectrum for the monomodal distributions. It is possible to see that the retrieved size parameter D provides almost perfect agreement with the given one, but the distribution parameter k is slightly sensitive to the noise. From the practical point of view the results can be considered satisfactory in all the size range, especially for the case of putting 2% random noise in the extinction values. It is important to note that the accuracy of the inversion results is increased by using a higher order polynomial as the fitted extinction efficiency. However, further increasing the order of the polynomial leads to the fitted extinction efficiency become more and more oscillatory and thus the quality of the results cannot be improved.

Table 1. Inversion Results of the Monomodal R-R Distributions

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To further investigate the feature, we also use the numerical integration method to retrieve the particle size distribution as a comparison with the moment method, and the inversion results in 1~5 um and 0.1~10 um size range for the monomodal R-R distributions are shown in Fig. 3. The set parameters are listed in Table 1. Through comparison, one can see that the inversion curves coincide nearly with the set curves in the figure for both the moment and numerical integration method when the 0% random noise is added to the extinction values, suggesting that it is possible to retrieve the particle size distribution with good results using the moment method. When putting 2% random noise into the extinction values, the inversion results using the moment method are better than that using the numerical integration method, which turns out that the moment method is more insensitivity to the random noise. The comparison of reproducibility for the moment method and numerical integration method in 0.1~10 um range is given in Table 2, and the set parameters are also listed in Table 1. According to the results in Table 2, we can draw a conclusion that it is very feasible to retrieve the particle size distribution in the visible spectrum using the moment method, and the reproducibility for the moment method are superior to the numerical integration method, though the inversion results using the numerical integration method are better than that using the moment method under the ideal condition.

Fig. 3. Comparison of the inversion results for R-R distributions with moment and numerical integration method

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Table 2. Comparison of Reproducibility for the Moment Method and Numerical Integration Method with R-R Distributions

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Table 3. Inversion Results of the Monomodal L-N Distributions

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The second test is devised to study the monomodal L-N distributions using the moment method. The inversion results with different monomodal L-N distribution parameters are summarized in Table 3. The relative refractive index of particle m=1.235. The good features of retrieved results mentioned in Table 2 are still presented for L-N distribution function.

Figure 4 depicts the comparison of the inversion results for the L-N distributions with moment and numerical integration method in the 0.1~1 um and 1~10 um size range, respectively. In the range from 0.1~1 um, the retrieved distribution becomes somewhat noisier, while its shape is fairly close to the shape of the given one, which is due to the fact that the parameter σ is more sensitive to the random noise, but the retrieved result is still acceptable. Table 4 lists the monomodal L-N distributions inversion results varied with different random noise using the moment and numerical integration method, respectively. The retrieved results using the moment method are superior to that using the numerical integration method except in the ideal case.

Fig. 4. Comparison of the inversion results for L-N distributions with moment and numerical integration method

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Table 4. Comparison of Reproducibility for the Moment and Numerical Integration Method with L-N Distributions

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As a further test for investigating the reliability, we consider the bimodal distributions. The bimodal R-R volume frequency distribution function f(D) is defined as follows:

f (D) = n * (\frac{k_{1}}{{\bar{D}}_{1}} \times {(\frac{D}{{\bar{D}}_{1}})}^{k_{1} - 1} \times \exp (- {(\frac{D}{{\bar{D}}_{1}})}^{k_{1}})) + (1 - n) * (\frac{k_{2}}{{\bar{D}}_{2}} \times {(\frac{D}{{\bar{D}}_{2}})}^{k_{2} - 1}) \times \exp (- {(\frac{D}{{\bar{D}}_{2}})}^{k_{2}}))

where D̄₁, D̄₂ are the characteristic diameters of the distribution(in µm), k ₁, k ₂ are the narrowness indices of the distribution, n is the weight coefficient between the two peaks.

Table 5. Inversion Results of the Bimodal R-R Distributions

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The inversion results of bimodal R-R distributions in the range from 0.1~10 um in diameter are shown in Table 5. The relative refractive index of particle m=1.235. There are five parameters (D̄₁,k ₁,D̄,k ₂,n) to be retrieved and six incident wavelengths(0.4 um, 0.5 um, 0.6 um, 0.7 um, 0.75 um, 0.8 um) are selected for the bimodal distributions. In order to present the results clearly, the retrieved size distributions are also depicted in Fig. 5. The same conclusion can be drawn that it is very feasible to retrieve the particle size distribution using the moment method for the bimodal size distributions. The bimodal R-R distributions inversion results with different random noise for moment and numerical integration method are listed in Table 6.

From all the simulations results above mentioned, it can be inferred that it is very feasible and reliable to use the moment method to retrieve the particle size distribution in the dependent model. The reproducibility for the moment method are superior to the numerical integration method, especially for the case of putting 2% or 4% random noise in the extinction measurement values. Therefore, the size parameters can be retrieved with a good accuracy.

Fig. 5. Comparison of the inversion results for bimodal R-R distributions with moment and numerical integration method in the range from 0.1~10 um in diameter

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Table 6. Comparison of Reproducibility for the Moment and Numerical Integration Method with Bimodal R-R Distributions

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Table 7. Inversion Results of Standard Particles

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The reliability of the moment method is also verified by the retrieval of reference material. The experimental data of reference material is provided by Prof. X.S. Cai. The standard monodisperse spherical polystyrene particle samples with labeled diameter D=1.98 um and D=3.17 um are produced by Beijing Institute of Nuclear Engineering. The particles are dispersed in water, whose relative refractive indices are m=1.235. The incident wavelengths are selected as λ ₁=0.8 um, λ ₂=0.6728 um, λ ₃=0.5695 um and λ ₄=0.4098 um. In this experiment, the extinction intensity ratios at any two wavelengths are used as the input data, and the R-R distribution function is assumed, then the data are processed using the genetic algorithm in combination with the moment relation. The results are presented in Table 7. For a monodisperse or polydisperse particle system, the Sauter mean diameter D ₃₂ [defined in Eq. (12)] is the optimal expression of mean diameter for the particle system. As a comparison, we also use the data to retrieve the particle size distribution with numerical integration method. It is possible to see that the retrieved mean diameter D ₃₂ is nearly closed to the labeled one. The difference between the retrieved result and the labeled diameter is less than 10%, which satisfies the demand of the standard monodisperse spherical polystyrene particles.

D_{32} = \frac{\int_{D_{min}}^{D_{max}} D^{3} f (D) d D}{\int_{D_{min}}^{D_{max}} D^{2} f (D) d D}

4. Conclusion

In spectral extinction particle sizing, the moment method is applied to retrieve the particle size distribution in the dependent model algorithm. The relationships of the origin moments of the monomodal, bimodal R-R and L-N particle size distributions with the corresponding characteristic parameters are investigated to analyze the feasibility of retrieval using the moment method. The higher order polynomial with regard to the particle diameter is used to fit the single spherical particle extinction efficiency which is derived theoretically using the Mie’s solution to Maxwell’s equation in order to apply the moment method. Simulation and experimental results indicate that fairly reasonable results in different size range are obtained even in the case where the bimodal distributions are retrieved. The genetic algorithm is able to yield good results and has high stability in presence of random noise. The computation time is substantially reduced compared with the numerical integration methods, which is especially suitable for in-line particle size measurement.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.50336010). The authors thank X. S. Cai (Institute of Particle & Two phase Flow Measurement, University of Shanghai for Science & Technology) for the offer of the experimental data.

References and links

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Particle Size Range	Fitted Order of Qext	Set Parameters (D̄, k)	Inversion Parameters (D̄, k) 0%	Inversion Parameters (D̄, k) 2%
0.1~1um	3	(0.5,9)	(0.4926,17.3885)	(0.4806,17.7832)
1~5 um	10	(3.5,7)	(3.4996,6.9895)	(3.5157,6.9449)
1~10 um	17	(7,8)	(7.0079,8.1977)	(7.1163,8.4042)
1~10 um	20	(7,8)	(7.0056,8.1596)	(7.1156,8.2855)
1~10 um	23	(7,8)	(7.0161,8.2445)	(7.0790,7.7331)
0.1~10 um	18	(5,5)	(4.9083,4.9353)	(5.0438,5.3843)
0.1~10 um	20	(5,5)	(4.9102,4.9365)	(5.0435,5.3836)
0.1~10 um	23	(5,5)	(4.9092,4.9360)	(5.0443,5.3859)

Random Noise	Moment	Numerical Integration
0%	(4.9092,4.9360)	(5.0016,5.0042)
2%	(5.0424,5.3821)	(5.1090,5.5055)
2%	(5.3718,5.8309)	(5.3942,5.9788)
2%	(5.4251,6.2978)	(5.4360,6.4366)
4%	(5.0560,5.7423)	(5.0924,5.8307)
4%	(5.4497,6.3709)	(5.4567,6.4685)
4%	(5.4739,7.0960)	(5.4762,7.1983)

Particle Size Range	Fitted Order of Qext	Set Parameters (u,σ)	Inversion Parameters (u,σ) 0%	Inversion Parameters (u,σ) 2%
0.1~1um	3	(0.5,1.1)	(0.4962,1.0433)	(0.4925,1.0110)
1~5 um	10	(3,1.2)	(3.0123,1.1943)	(3.0263,1.1942)
1~10 um	17	(5,1.1)	(4.9976,1.1010)	(4.9740,1.0981)
1~10 um	20	(5,1.1)	(4.9988,1.1011)	(4.9755,1.0983)
0.1~10 um	18	(2,1.3)	(2.0010,1.3015)	(1.9858,1.3015)
0.1~10 um	20	(2,1.3)	(1.9997,1.2997)	(1.9839,1.2996)
0.1~10 um	23	(2,1.3)	(1.9994,1.2996)	(1.9823,1.2988)

Random Noise	Moment	Numerical Integration
0%	(4.9988,1.1011)	(4.9998,1.0995)
2%	(4.9755,1.0983)	(4.9766,1.0967)
2%	(5.0041,1.0985)	(5.0049,1.0969)
2%	(5.0069,1.0965)	(5.0077,1.0949)
4%	(4.9530,1.0952)	(4.9544,1.0937)
4%	(5.0090,1.0961)	(5.0096,1.0944)
4%	(5.0140,1.0920)	(5.0148,1.0903)

Particle Size Range	Fitted Order of Qext	Set Parameters (D̄₁,k₁, D̄2,k₂,n)	Inversion Parameters (D̄₁,k₁, D̄₂,k₂,n) 0%	Inversion Parameters (D̄₁,k₁, D̄₂,k₂,n) 2%
0.1~10 um	18	(0.8,5,6,10,0.5)	(0.7203,13.4576, 5.9107,6.9169,0.4403)	(0.7644,5.9886, 3.9225,6.1231,0.5746)
0.1~10 um	20	(0.8,5,6,10,0.5)	(0.7976,3.5603, 6.0213,10.4507,0.5204)	(0.8218,17.9391,6.3479,16.19610.5146)
0.1~10 um	23	(0.8,5,6,10,0.5)	(0.8016,4.7026,6.0092,10.1810,0.5032)	(0.8274,11.1365, 6.3359,15.7899,0.5134)

Retrieval of particle size distribution in the dependent model using the moment method

Abstract

1. Introduction

2. Theory

3. Computer simulations and experiments

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Tables (7)

Equations (12)

Optics Express

Random Noise	Moment	Numerical Integration
0%	(0.8016,4.7026, 6.0092,10.1810,0.5032)	(0.7903,10.7550, 6.0009,10.0021,0.4877)
2%	(0.8274,11.1365, 6.3359,15.7899,0.5134)	(0.8094,17.2143 6.3850,14.4943,0.5008)
2%	(0.8165,4.2432, 5.9731,16.0024,0.5492)	(0.7980,15.9388, 5.9903,15.5078,0.5234)
4%	(0.8168,3.9880, 5.9364,19.9806,0.5728)	(0.8396,9.1596, 6.4841,19.9787,0.5217)
4%	(0.8194,4.3609, 5.9373,19.9978,0.5676)	(0.8043,13.2944, 5.9201,19.9834,0.5474)

Labeled Diameter	Particle Size Range	Fitted Order of Q _ext	D ₃₂ (Moment)	D ₃₂ (Numerical Integration)
1.98 um	0.1~10 um	20	2.033 um	2.034 um
3.26 um	0.1~10 um	20	3.180 um	3.175 um