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Inversion of particle size distribution from spectral extinction data using the modified beta function
Powder Technology 190 (2009) 292–296
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / p o w t e c
Inversion of particle size distribution from spectral extinction data using the modified beta function Xiaogang Sun ⁎, Hong Tang, Jingmin Dai Department of Automation Measurement and Control Engineering, P.O. Box 308, Harbin Institute of Technology, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 20 December 2007 Received in revised form 19 June 2008 Accepted 5 August 2008 Available online 23 August 2008 Keywords: Spectral extinction Inversion Particle size distribution Dependent model Modified beta function
a b s t r a c t In spectral extinction particle sizing technique, an inversion method in the dependent model is proposed, in which the commonly used particle size distribution functions are retrieved using the modified beta function as the assumed type of distribution function. The feasibility and reliability of this inversion method are investigated, and the inversion using the Johnson's SB function as the assumed type of distribution function is also discussed for comparison. Simulation experiments indicate that the commonly used monomodal particle size distribution functions can be retrieved well using the modified beta function as a versatile function in the dependent model with the spectral extinction data. This inversion method not only overcomes the difficulty caused by the inability to determine the type of specified distribution function accurately beforehand, but also allows the description of different types of distributions by a single model. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Light scattering particle sizing techniques have been widely used in recent years. Among these techniques, the spectral extinction method is probably the most attractive one because it requires a simple optical layout and only a commercial spectrophotometer [1–4]. The particle size distribution to be determined can be retrieved by inversion methods, using the extinction data at multiple wavelengths in the spectral extinction particle sizing technique. The inversion methods can be divided into two categories. The first category is the independent model algorithm, while the second category is the dependent model algorithm. In the dependent model, some prior information about the function type of particle size distribution must be known beforehand, and then the particle size distribution is retrieved with some optimization algorithms. However, sometimes, the function type of particle size distribution may not be known exactly. So if the function type of the assumed particle size distribution has an obvious deviation from the true distribution, the inversion error will be very large. To overcome this difficulty, researchers had applied some versatile functions to fit the commonly used particle size distribution functions [5–7]. But few studies were conducted on using the versatile functions to retrieve the particle size distribution in the spectral extinction particle sizing technique. In this paper, the particle size distributions to be determined actually conforms to Rosin–Rammler function, log–normal function, and normal function, respectively, and then they are retrieved according to the function types of modified beta function and Johnson's ⁎ Corresponding author. Tel.: +86 45186413151 8007. E-mail address: [email protected] (X. Sun). 0032-5910/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2008.08.017
SB function. In so doing, we can find a proper versatile function, which can be commonly used with a good accuracy, to retrieve the monomodal particle size distributions. 2. Methods When a beam of parallel monochromatic radiation light of intensity I0 passes through a suspension of particle system (thickness L) with a refraction index different from that of the dispersed medium, the scattering and absorption will lead to an attenuation of the transmitted light intensity. According to the Lambert–Beer law, if the suspension particles of particle system are polydisperse spherical, and the multiple scattering and interaction effects are neglectable, the transmitted light intensity I is expressed as follows [8,9]: ln
D max Q IðλÞ 3 ext ðλ; m; DÞ f ðDÞdD = − L ND ∫ I0 ðλÞ 2 D D min
ð1Þ
where the extinction value I(λ)/I0(λ) at wavelength λ is obtained by actual measurements; Qext(λ,m,D) is the Mie extinction efficiency of a single particle which is a complex function of particle diameter D (in μm), wavelength λ (in μm) in the medium and relative refractive index m; ND is the total particle number of particle system; the lower and upper integration limits are denoted by Dmin and Dmax(in μm); f (D) is the volume frequency distribution of particle system, which is the particle size distribution to be determined [10–13]. The particle size distribution can be retrieved by some inversion methods based on the extinction data at multiple wavelengths in Eq. (1). Eq. (1) is the Fredholm integral equation of the first kind. This is a classical ill-conditioned problem, which means that different
X. Sun et al. / Powder Technology 190 (2009) 292–296
293
distributions can fit the extinction data with the same level of accuracy, so a proper inversion method must be used to obtain an acceptable particle size distribution. Actually, many particle systems can often be approximately described by a specified size distribution, therefore the dependent model algorithm is used in this work [14–16]. Among these well-behaved distribution functions available, the most widely used distribution functions are the Rosin–Rammler(R–R) function, log– normal(L–N) function and normal(N–N) function. The mathematical representations of their volume frequency distributions are as follows: k−1 k ! k D D fR–R ðDÞ= exp − D D D ! 2 1 ðln D− ln uÞ exp − fL–N ðDÞ=pffiffiffiffiffiffi 2 2πD lnσ 2ðln σ Þ ! ðD−u1 Þ2 1 fN–N ðDÞ= pffiffiffiffiffiffi exp − 2σ 21 2π σ 1
ð2Þ ð3Þ ð4Þ
¯¯ (in μm), k (dimensionless), u (in μm), σ (dimensionless), u1 where D (in μm), σ1 (dimensionless) are the characteristic parameters; ln u is the mean of ln D; lnσ is the standard deviation of ln D; u1 is the mean of D; σ1 is the standard deviation of D. It has been generally accepted that no unique distribution function can represent all the particle size distributions encountered in reality, but still several versatile functions exist to represent the most commonly used particle size distributions. Yu et al. suggested that the Johnson's SB distribution function could be used as a general function for the representation of monomodal particle size distributions. The function is explicitly given by [5]: ( ) σ2 σ2 Dmax −Dmin D−Dmin M−Dmin 2 exp − 2 ln − ln fSB ðDÞ=pffiffiffiffiffiffi 2 Dmax −D Dmax −M 2π ðD−Dmin ÞðDmax −DÞ
ð5Þ where M and σ2 are the characteristic parameters. Popplewell et al. proposed the modified beta function to represent the monomodal particle size distribution. The function is described by [7]: fmbeta ðDÞ=
ðD−Dmin Þαm1 ðDmax −DÞm1 ∫
Dmax
Dmin
ðD−Dmin Þαm1 ðDmax −DÞm1 dD
Fig. 1. Inversion results of original R–R distributions with modified beta function and Johnson's SB function.
ð6Þ
where α and m1 are the characteristic parameters. 3. Simulation experiments The intention of this work is to validate that the monomodal R–R, L–N and N–N particle size distributions can be retrieved well by the modified beta function as a versatile function in the spectral extinction particle sizing technique. The inversion of particle size distributions using the Johnson's SB function as the versatile function is also carried out for comparison. To validate the feasibility and reliability of this inversion method, we perform different computer simulations by using different particle size distributions (including both wide and narrow distributions). Fig. 1 shows the inversion results of original R–R distributions using the modified beta function and the Johnson's SB function, respectively as the versatile function. The relative refractive index m = 1.235, and the genetic algorithm is used as the optimization algorithm. In the dependent model, the inversion of particle size distribution is the solution to obtaining the characteristic parameters of particle size distribution by certain optimization algorithm. The genetic algorithm is superior to those simplex optimization methods in the search of global optimal value of multi-object function. The overall particle size measurement range is limited from 0.1 μm to 10 μm in diameter, which is the optimal measurement range in
spectral extinction particle sizing technique. The number of subinterval is equal to 100 in the range. Since the Mie extinction efficiency Qext(λ,m,D) is an oscillatory function of particle diameter D, which needs more than two equations (that is at least two equations such as ¯¯ and Eq. (1) at different wavelengths) to obtain the two parameters D k. Here four incident wavelengths (0.4 μm, 0.55 μm, 0.65 μm, 0.8 μm) are selected in the visible spectrum region. In order to examine the effect of random noise on the inversion of particle size distributions, ±2% random noise is also added to the extinction data. As can be seen from Fig. 1, the inversion results using the modified beta function and Johnson's SB function are all acceptable for both narrow and wide distributions when 0% or ±2% random noise is added to the extinction data, implying that it is possible to retrieve the monomodal R–R particle size distribution using both functions as the versatile function. In order to investigate the reliability of this inversion method, we also use the inversion error ξ to characterize the quality of inversion results: N h ~ i2 1=2 ~ ∑ frec Di −fori Di n=
i=1
N h i 1=2 ~ 2 ∑ fori Di
ð7Þ
i=1
where N represents that the particle size range [Dmin,Dmax] is divided into N subintervals; D˜ i is the midpoint of the ith subinterval [Di,Di + 1];
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Table 1 Reproducibility of original R–R distributions with modified beta function (mbeta) and ¯¯ ,k) = (2,7)) Johnson's SB function (SB) ((D
Table 4 Reproducibility of original L–N distributions with modified beta function and Johnson's SB function ((u,σ) = (6,1.15))
Inversion Inversion Inversion Inversion Random Inversion results results errors errors noise results ¯¯ ,k) (R–R) ξ (R–R) (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (D
Inversion errors ξ (SB)
Inversion Inversion Inversion Inversion Random Inversion results results errors errors noise results (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (u,σ) (L–N) ξ (L–N)
Inversion errors ξ (SB)
0%
0.1638
0%
0.1690
0.1632
0%
0.1950
±2%
0.1823
±2%
0.2204
±2%
0% ±2% ±2% ±2%
(2.0067, 7.0078) (1.9992, 7.0563) (1.9503, 7.1059) (1.9578, 7.1569) (2.0626, 7.2303)
0.0159 0.0073 0.1248 0.1085 0.1446
(0.2263, 113.7443) (0.2361, 111.7860) (0.2257, 105.9897) (0.2260, 111.7447) (0.2241, 130.8512)
0.1445 0.1428 0.1704 0.1620 0.1822
(1.8807, 5.0113) (1.8810, 4.9974) (1.8697, 4.4287) (1.8733, 4.5621) (1.8506, 5.1872)
Table 2 Reproducibility of original R–R distributions with modified beta (mbeta) function and ¯¯ ,k) = (7,10)) Johnson's SB (SB) function ((D Inversion Random Inversion errors noise results ¯¯ ,k) (R–R) ξ (R–R) (D 0% 0% ±2% ±2% ±2%
(6.9999, 10.0046) (7.0031, 10.0116) (6.8553, 11.0996) (6.9037, 10.4058) (6.9015, 10.4307)
4.15 × 10− 4 0.0031 0.1913 0.1092 0.1124
Inversion Inversion Inversion results results errors (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB)
Inversion errors ξ (SB)
(2.2254, 9.6515) (2.2253, 9.6508) (2.0654, 13.5734) (2.0343, 13.1126) (2.0690, 13.2196)
0.0696
0.1020 0.1019 0.1345 0.1504 0.1262
(6.8064, 2.7556) (6.8065, 2.7550) (6.6782, 3.1449) (6.6421, 3.0788) (8.8170, 1.2265)
0.0697 0.1407 0.1546 0.2665
fori(D˜ i) is the original volume frequency distribution in the ith ˜ i) is the retrieved volume frequency distribution in subinterval; frec(D the ith subinterval. The reproducibility of this inversion method for original R–R distribution functions are given in Table 1 and Table 2. The characteristic parameters of original R–R distribution functions are the same as those in Fig. 1, and the inversion results using the R–R function as the assumed type of particle size distribution are also included in Tables 1 and 2. From the results listed in Tables 1 and 2, we can see that the reproducibility using the modified beta function as the assumed type of particle size distribution is superior to that using the Johnson's SB function when ±2% random noise is added to the extinction data. Furthermore, the inversion errors using the modified beta function as the assumed type of particle size distribution are of
(5.9469, 1.1444) (5.9982, 1.1497) (6.0880, 1.1336) (6.1359, 1.1338) (5.8223, 1.1317)
0.0535 0.0022 0.1223 0.1518 0.1810
(1.4905, 18.3084) (1.4906, 18.3092) (1.4032, 17.1281) (1.4032, 17.1273) (1.4814, 19.1736)
0.1591 0.1591 0.1779 0.1779 0.1754
(5.9428, 3.3887) (5.9305, 3.4605) (5.7985, 3.2475) (5.7985, 3.2474) (5.7034, 3.2475)
0.1855 0.2531 0.2531 0.2630
the same order magnitude as those using the R–R function. So the modified beta function is completely reliable to retrieve the monomodal R–R particle size distributions. Table 3 shows the inversion results of original R–R distributions function with different subintervals in the particle size range from – 0.1 μm to 10 μm. The set parameter of original R–R distribution is (D,k) = (2,7), and the random noise is equal to 0%. From the results listed in Table 3, we can see that the inversion errors with 20 subintervals and 50 subintervals are greater than those with 200 subintervals and 100 subintervals. But the calculation is time-consuming when the number of subinterval is equal to 200. In addition, the inversion results with 100 subintervals are as good as those with 200 subintervals. So we choose 100 subintervals to retrieve the monomodal R–R particle size distribution in the particle size range from 0.1 μm to 10 μm. Table 4 shows the reproducibility of original L–N distributions with the modified beta function and Johnson's SB function as the assumed type of particle size distribution, and their inversion curves are shown in Fig. 2. The incident wavelengths and the relative refractive index are the same as that in Fig. 1. It can be seen from Table 4 that there is reasonable agreement between the inversion results using the modified beta function and the original L–N function when ±2% random noise is added to the extinction data. The reproducibility using the modified beta function as the assumed type of particle size distribution is still superior to that using the Johnson's SB function
Table 3 ¯¯ ,k) = (2,7), Inversion results of original R–R distributions with different subintervals ((D random noise = 0%) Inversion Inversion Number of Inversion Inversion Inversion results errors results subintervals results errors ¯¯ ,k) (R–R) ξ (R–R) (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (D
Inversion errors ξ (SB)
200
0.1615
100 50 20
(1.9998, 7.0148) (1.9992, 7.0563) (1.9966, 7.2506) (1.9843, 9.0695)
0.0019 0.0073 0.0322 0.1624
(0.2362, 115.3185) (0.2361, 111.7860) (0.2271, 138.2249) (0.2253, 116.7065)
0.1378 0.1428 0.1563 0.1653
(1.8823, 4.9981) (1.8810, 4.9974) (1.8638, 4.5637) (1.8552, 4.5437)
0.1632 0.1938 0.2063
Fig. 2. Inversion results of original L–N distributions with modified beta function and Johnson's SB function ((u,σ) = (6,1.15)).
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Table 5 Reproducibility of original N–N distributions with modified beta function and Johnson's SB function ((u1,σ1) = (5,1.2)) Inversion Inversion Inversion Inversion Random Inversion results results errors errors noise results (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (u1,σ1) (N–N) ξ (N–N)
Inversion errors ξ (SB)
0%
0.2740
0% ±2% ±2% ±2%
(5.0023, 1.1993) (5.0024, 1.1994) (4.8174, 0.9559) (4.8137, 1.0352) (4.6944, 0.9437)
0.0014 0.0015 0.2425 0.1797 0.3040
(1.1706, 9.1881) (1.1706, 9.1882) (1.2006, 8.3721) (1.3065, 8.7556) (1.6214, 8.3835)
0.2591 0.2590 0.2767 0.4142 0.7242
(5.3699, 2.3594) (5.3701, 2.3595) (6.8130, 2.6991) (6.2985, 2.2413) (6.7097, 2.5873)
0.2741 1.1944 0.8016 1.1131
when ±2% random noise is added to the extinction data in the inversion of monomodal L–N particle size distribution. Tables 5 and 6 show the reproducibility of original N–N distributions using the modified beta function and Johnson's SB function, and their inversion curves are shown in Fig. 3. The incident wavelengths and the relative refractive index are also the same as that in Fig. 1. It is possible to see that the retrieved distributions using the modified beta function are slightly different from the original N–N distributions. But from the practical point of view, the results can be considered satisfactory. On the other hand, there is an obvious discrepancy between the retrieved distributions using Johnson's SB function and the original N–N distribution when ±2% random noise is added to the extinction data. In other words, the modified beta function is more suitable for the inversion of monomodal N–N particle size distributions. Meanwhile, the width of original particle size distribution certainly has an effect on the inversion error. From the inversion results listed in Table 6, we can see that the increase of width can increase the inversion error, but the inversion errors are still acceptable for the wide particle size distribution. So it is feasible to retrieve the commonly used particle size distribution (for wide or narrow distributions) using the modified beta function as the assumed type of particle size distribution. From all the simulation results mentioned above, it can be inferred that it is feasible and reliable to use the modified beta function to retrieve the original R–R, L–N, and N–N particle size distributions in the dependent model. Meanwhile, the inversion results are still satisfactory when ±2% random noise is added to the extinction data. The modified beta function has a high flexibility and general
Table 6 Reproducibility of original N–N distributions with modified beta function and Johnson's SB function ((u1,σ1) = (4.5,1.7)) Inversion Random Inversion errors noise results (u1,σ1) (N–N) ξ (N–N) 0% 0% ±2% ±2% ±2%
(4.4999, 1.7994) (4.4982, 1.7936) (4.8323, 1.9502) (4.9752, 1.9864) (4.8069 1.9316)
2.89 × 10− 4 0.0032 0.1683 0.2162 0.1569
Inversion Inversion Inversion results results errors (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB)
Inversion errors ξ (SB)
(1.0185, 3.1643) (1.0342, 3.1935) (1.0897, 3.2348) (0.4053, 2.1854) (1.4367, 3.1732)
0.2264
0.2247 0.2382 0.2854 0.4397 0.5433
(4.2744, 1.7251) (4.9756, 1.7197) (5.1319, 1.7398) (5.7148, 1.6597) (6.7253, 1.6788)
Fig. 3. Inversion results of original N–N distributions with modified beta function and Johnson's SB function.
applicability to both narrow and wide distribution functions which are commonly used. 4. Conclusions In spectral extinction particle sizing technique, the modified beta function is used as the versatile function to retrieve the commonly used monomodal particle size distributions in the dependent model. The feasibility and validity of this inversion method are researched, and the inversion using the Johnson's SB function as the versatile function is also performed for comparison. Simulation results indicate that a reasonable representation of the commonly used particle size distribution can be obtained by using the modified beta function as the assumed type of particle size distribution. This inversion method has the advantages of simplicity, rapidity. It overcomes the difficulty which arises when the type of specified distribution function cannot be determined exactly beforehand in the dependent model. More work needs to be done, and the inversion of multimodal distributions using a more flexible versatile function will be considered in further studies.
0.2789 0.3473 0.5738 1.0115
Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 50336010). The authors thank M.D. Normand (University of Massachusetts, USA) for useful discussions on Johnson's SB function.
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[8] E. Marioth, B. Koenig, H. Krause, S. Loebbecke, Ind. Eng. Chem. Res. 339 (2000) 4853–4857. [9] A.K. Roy, S.K. Sharma, J. Opt. A: Pure Appl. Opt. 7 (2005) 675–684. [10] C.F. Bohrend, R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley &Sons INC, New York, 1998, pp. 146–167. [11] D. Rosskamp, F. Truffer, S. Bolay, M. Geiser, Opt. Express 15 (2007) 2683–2690. [12] W. Liang, Y. Xu, Y.Y. Huang, A. Yariv, J.G. Fleming, S.W. Yu, Opt. Express 12 (2004) 657–669. [13] K. Nakamura, T. Kawabata, Y. MoriSize, Powder Technol. 131 (2003) 120–128. [14] M.L. Arias, G.L. Frontini, Part. Part. Syst. Charact. 23 (2007) 374–380. [15] A.P. Nefedov, O.F. Petrov, O.S. Vaulina, Appl. Opt. 36 (1997) 1357–1366. [16] B. Manohar, B.S. Sridhar, Powder Technol. 120 (2001) 292–297.
Contents lists available at ScienceDirect
Powder Technology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / p o w t e c
Inversion of particle size distribution from spectral extinction data using the modified beta function Xiaogang Sun ⁎, Hong Tang, Jingmin Dai Department of Automation Measurement and Control Engineering, P.O. Box 308, Harbin Institute of Technology, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 20 December 2007 Received in revised form 19 June 2008 Accepted 5 August 2008 Available online 23 August 2008 Keywords: Spectral extinction Inversion Particle size distribution Dependent model Modified beta function
a b s t r a c t In spectral extinction particle sizing technique, an inversion method in the dependent model is proposed, in which the commonly used particle size distribution functions are retrieved using the modified beta function as the assumed type of distribution function. The feasibility and reliability of this inversion method are investigated, and the inversion using the Johnson's SB function as the assumed type of distribution function is also discussed for comparison. Simulation experiments indicate that the commonly used monomodal particle size distribution functions can be retrieved well using the modified beta function as a versatile function in the dependent model with the spectral extinction data. This inversion method not only overcomes the difficulty caused by the inability to determine the type of specified distribution function accurately beforehand, but also allows the description of different types of distributions by a single model. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Light scattering particle sizing techniques have been widely used in recent years. Among these techniques, the spectral extinction method is probably the most attractive one because it requires a simple optical layout and only a commercial spectrophotometer [1–4]. The particle size distribution to be determined can be retrieved by inversion methods, using the extinction data at multiple wavelengths in the spectral extinction particle sizing technique. The inversion methods can be divided into two categories. The first category is the independent model algorithm, while the second category is the dependent model algorithm. In the dependent model, some prior information about the function type of particle size distribution must be known beforehand, and then the particle size distribution is retrieved with some optimization algorithms. However, sometimes, the function type of particle size distribution may not be known exactly. So if the function type of the assumed particle size distribution has an obvious deviation from the true distribution, the inversion error will be very large. To overcome this difficulty, researchers had applied some versatile functions to fit the commonly used particle size distribution functions [5–7]. But few studies were conducted on using the versatile functions to retrieve the particle size distribution in the spectral extinction particle sizing technique. In this paper, the particle size distributions to be determined actually conforms to Rosin–Rammler function, log–normal function, and normal function, respectively, and then they are retrieved according to the function types of modified beta function and Johnson's ⁎ Corresponding author. Tel.: +86 45186413151 8007. E-mail address: [email protected] (X. Sun). 0032-5910/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2008.08.017
SB function. In so doing, we can find a proper versatile function, which can be commonly used with a good accuracy, to retrieve the monomodal particle size distributions. 2. Methods When a beam of parallel monochromatic radiation light of intensity I0 passes through a suspension of particle system (thickness L) with a refraction index different from that of the dispersed medium, the scattering and absorption will lead to an attenuation of the transmitted light intensity. According to the Lambert–Beer law, if the suspension particles of particle system are polydisperse spherical, and the multiple scattering and interaction effects are neglectable, the transmitted light intensity I is expressed as follows [8,9]: ln
D max Q IðλÞ 3 ext ðλ; m; DÞ f ðDÞdD = − L ND ∫ I0 ðλÞ 2 D D min
ð1Þ
where the extinction value I(λ)/I0(λ) at wavelength λ is obtained by actual measurements; Qext(λ,m,D) is the Mie extinction efficiency of a single particle which is a complex function of particle diameter D (in μm), wavelength λ (in μm) in the medium and relative refractive index m; ND is the total particle number of particle system; the lower and upper integration limits are denoted by Dmin and Dmax(in μm); f (D) is the volume frequency distribution of particle system, which is the particle size distribution to be determined [10–13]. The particle size distribution can be retrieved by some inversion methods based on the extinction data at multiple wavelengths in Eq. (1). Eq. (1) is the Fredholm integral equation of the first kind. This is a classical ill-conditioned problem, which means that different
X. Sun et al. / Powder Technology 190 (2009) 292–296
293
distributions can fit the extinction data with the same level of accuracy, so a proper inversion method must be used to obtain an acceptable particle size distribution. Actually, many particle systems can often be approximately described by a specified size distribution, therefore the dependent model algorithm is used in this work [14–16]. Among these well-behaved distribution functions available, the most widely used distribution functions are the Rosin–Rammler(R–R) function, log– normal(L–N) function and normal(N–N) function. The mathematical representations of their volume frequency distributions are as follows: k−1 k ! k D D fR–R ðDÞ= exp − D D D ! 2 1 ðln D− ln uÞ exp − fL–N ðDÞ=pffiffiffiffiffiffi 2 2πD lnσ 2ðln σ Þ ! ðD−u1 Þ2 1 fN–N ðDÞ= pffiffiffiffiffiffi exp − 2σ 21 2π σ 1
ð2Þ ð3Þ ð4Þ
¯¯ (in μm), k (dimensionless), u (in μm), σ (dimensionless), u1 where D (in μm), σ1 (dimensionless) are the characteristic parameters; ln u is the mean of ln D; lnσ is the standard deviation of ln D; u1 is the mean of D; σ1 is the standard deviation of D. It has been generally accepted that no unique distribution function can represent all the particle size distributions encountered in reality, but still several versatile functions exist to represent the most commonly used particle size distributions. Yu et al. suggested that the Johnson's SB distribution function could be used as a general function for the representation of monomodal particle size distributions. The function is explicitly given by [5]: ( ) σ2 σ2 Dmax −Dmin D−Dmin M−Dmin 2 exp − 2 ln − ln fSB ðDÞ=pffiffiffiffiffiffi 2 Dmax −D Dmax −M 2π ðD−Dmin ÞðDmax −DÞ
ð5Þ where M and σ2 are the characteristic parameters. Popplewell et al. proposed the modified beta function to represent the monomodal particle size distribution. The function is described by [7]: fmbeta ðDÞ=
ðD−Dmin Þαm1 ðDmax −DÞm1 ∫
Dmax
Dmin
ðD−Dmin Þαm1 ðDmax −DÞm1 dD
Fig. 1. Inversion results of original R–R distributions with modified beta function and Johnson's SB function.
ð6Þ
where α and m1 are the characteristic parameters. 3. Simulation experiments The intention of this work is to validate that the monomodal R–R, L–N and N–N particle size distributions can be retrieved well by the modified beta function as a versatile function in the spectral extinction particle sizing technique. The inversion of particle size distributions using the Johnson's SB function as the versatile function is also carried out for comparison. To validate the feasibility and reliability of this inversion method, we perform different computer simulations by using different particle size distributions (including both wide and narrow distributions). Fig. 1 shows the inversion results of original R–R distributions using the modified beta function and the Johnson's SB function, respectively as the versatile function. The relative refractive index m = 1.235, and the genetic algorithm is used as the optimization algorithm. In the dependent model, the inversion of particle size distribution is the solution to obtaining the characteristic parameters of particle size distribution by certain optimization algorithm. The genetic algorithm is superior to those simplex optimization methods in the search of global optimal value of multi-object function. The overall particle size measurement range is limited from 0.1 μm to 10 μm in diameter, which is the optimal measurement range in
spectral extinction particle sizing technique. The number of subinterval is equal to 100 in the range. Since the Mie extinction efficiency Qext(λ,m,D) is an oscillatory function of particle diameter D, which needs more than two equations (that is at least two equations such as ¯¯ and Eq. (1) at different wavelengths) to obtain the two parameters D k. Here four incident wavelengths (0.4 μm, 0.55 μm, 0.65 μm, 0.8 μm) are selected in the visible spectrum region. In order to examine the effect of random noise on the inversion of particle size distributions, ±2% random noise is also added to the extinction data. As can be seen from Fig. 1, the inversion results using the modified beta function and Johnson's SB function are all acceptable for both narrow and wide distributions when 0% or ±2% random noise is added to the extinction data, implying that it is possible to retrieve the monomodal R–R particle size distribution using both functions as the versatile function. In order to investigate the reliability of this inversion method, we also use the inversion error ξ to characterize the quality of inversion results: N h ~ i2 1=2 ~ ∑ frec Di −fori Di n=
i=1
N h i 1=2 ~ 2 ∑ fori Di
ð7Þ
i=1
where N represents that the particle size range [Dmin,Dmax] is divided into N subintervals; D˜ i is the midpoint of the ith subinterval [Di,Di + 1];
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Table 1 Reproducibility of original R–R distributions with modified beta function (mbeta) and ¯¯ ,k) = (2,7)) Johnson's SB function (SB) ((D
Table 4 Reproducibility of original L–N distributions with modified beta function and Johnson's SB function ((u,σ) = (6,1.15))
Inversion Inversion Inversion Inversion Random Inversion results results errors errors noise results ¯¯ ,k) (R–R) ξ (R–R) (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (D
Inversion errors ξ (SB)
Inversion Inversion Inversion Inversion Random Inversion results results errors errors noise results (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (u,σ) (L–N) ξ (L–N)
Inversion errors ξ (SB)
0%
0.1638
0%
0.1690
0.1632
0%
0.1950
±2%
0.1823
±2%
0.2204
±2%
0% ±2% ±2% ±2%
(2.0067, 7.0078) (1.9992, 7.0563) (1.9503, 7.1059) (1.9578, 7.1569) (2.0626, 7.2303)
0.0159 0.0073 0.1248 0.1085 0.1446
(0.2263, 113.7443) (0.2361, 111.7860) (0.2257, 105.9897) (0.2260, 111.7447) (0.2241, 130.8512)
0.1445 0.1428 0.1704 0.1620 0.1822
(1.8807, 5.0113) (1.8810, 4.9974) (1.8697, 4.4287) (1.8733, 4.5621) (1.8506, 5.1872)
Table 2 Reproducibility of original R–R distributions with modified beta (mbeta) function and ¯¯ ,k) = (7,10)) Johnson's SB (SB) function ((D Inversion Random Inversion errors noise results ¯¯ ,k) (R–R) ξ (R–R) (D 0% 0% ±2% ±2% ±2%
(6.9999, 10.0046) (7.0031, 10.0116) (6.8553, 11.0996) (6.9037, 10.4058) (6.9015, 10.4307)
4.15 × 10− 4 0.0031 0.1913 0.1092 0.1124
Inversion Inversion Inversion results results errors (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB)
Inversion errors ξ (SB)
(2.2254, 9.6515) (2.2253, 9.6508) (2.0654, 13.5734) (2.0343, 13.1126) (2.0690, 13.2196)
0.0696
0.1020 0.1019 0.1345 0.1504 0.1262
(6.8064, 2.7556) (6.8065, 2.7550) (6.6782, 3.1449) (6.6421, 3.0788) (8.8170, 1.2265)
0.0697 0.1407 0.1546 0.2665
fori(D˜ i) is the original volume frequency distribution in the ith ˜ i) is the retrieved volume frequency distribution in subinterval; frec(D the ith subinterval. The reproducibility of this inversion method for original R–R distribution functions are given in Table 1 and Table 2. The characteristic parameters of original R–R distribution functions are the same as those in Fig. 1, and the inversion results using the R–R function as the assumed type of particle size distribution are also included in Tables 1 and 2. From the results listed in Tables 1 and 2, we can see that the reproducibility using the modified beta function as the assumed type of particle size distribution is superior to that using the Johnson's SB function when ±2% random noise is added to the extinction data. Furthermore, the inversion errors using the modified beta function as the assumed type of particle size distribution are of
(5.9469, 1.1444) (5.9982, 1.1497) (6.0880, 1.1336) (6.1359, 1.1338) (5.8223, 1.1317)
0.0535 0.0022 0.1223 0.1518 0.1810
(1.4905, 18.3084) (1.4906, 18.3092) (1.4032, 17.1281) (1.4032, 17.1273) (1.4814, 19.1736)
0.1591 0.1591 0.1779 0.1779 0.1754
(5.9428, 3.3887) (5.9305, 3.4605) (5.7985, 3.2475) (5.7985, 3.2474) (5.7034, 3.2475)
0.1855 0.2531 0.2531 0.2630
the same order magnitude as those using the R–R function. So the modified beta function is completely reliable to retrieve the monomodal R–R particle size distributions. Table 3 shows the inversion results of original R–R distributions function with different subintervals in the particle size range from – 0.1 μm to 10 μm. The set parameter of original R–R distribution is (D,k) = (2,7), and the random noise is equal to 0%. From the results listed in Table 3, we can see that the inversion errors with 20 subintervals and 50 subintervals are greater than those with 200 subintervals and 100 subintervals. But the calculation is time-consuming when the number of subinterval is equal to 200. In addition, the inversion results with 100 subintervals are as good as those with 200 subintervals. So we choose 100 subintervals to retrieve the monomodal R–R particle size distribution in the particle size range from 0.1 μm to 10 μm. Table 4 shows the reproducibility of original L–N distributions with the modified beta function and Johnson's SB function as the assumed type of particle size distribution, and their inversion curves are shown in Fig. 2. The incident wavelengths and the relative refractive index are the same as that in Fig. 1. It can be seen from Table 4 that there is reasonable agreement between the inversion results using the modified beta function and the original L–N function when ±2% random noise is added to the extinction data. The reproducibility using the modified beta function as the assumed type of particle size distribution is still superior to that using the Johnson's SB function
Table 3 ¯¯ ,k) = (2,7), Inversion results of original R–R distributions with different subintervals ((D random noise = 0%) Inversion Inversion Number of Inversion Inversion Inversion results errors results subintervals results errors ¯¯ ,k) (R–R) ξ (R–R) (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (D
Inversion errors ξ (SB)
200
0.1615
100 50 20
(1.9998, 7.0148) (1.9992, 7.0563) (1.9966, 7.2506) (1.9843, 9.0695)
0.0019 0.0073 0.0322 0.1624
(0.2362, 115.3185) (0.2361, 111.7860) (0.2271, 138.2249) (0.2253, 116.7065)
0.1378 0.1428 0.1563 0.1653
(1.8823, 4.9981) (1.8810, 4.9974) (1.8638, 4.5637) (1.8552, 4.5437)
0.1632 0.1938 0.2063
Fig. 2. Inversion results of original L–N distributions with modified beta function and Johnson's SB function ((u,σ) = (6,1.15)).
X. Sun et al. / Powder Technology 190 (2009) 292–296
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Table 5 Reproducibility of original N–N distributions with modified beta function and Johnson's SB function ((u1,σ1) = (5,1.2)) Inversion Inversion Inversion Inversion Random Inversion results results errors errors noise results (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB) (u1,σ1) (N–N) ξ (N–N)
Inversion errors ξ (SB)
0%
0.2740
0% ±2% ±2% ±2%
(5.0023, 1.1993) (5.0024, 1.1994) (4.8174, 0.9559) (4.8137, 1.0352) (4.6944, 0.9437)
0.0014 0.0015 0.2425 0.1797 0.3040
(1.1706, 9.1881) (1.1706, 9.1882) (1.2006, 8.3721) (1.3065, 8.7556) (1.6214, 8.3835)
0.2591 0.2590 0.2767 0.4142 0.7242
(5.3699, 2.3594) (5.3701, 2.3595) (6.8130, 2.6991) (6.2985, 2.2413) (6.7097, 2.5873)
0.2741 1.1944 0.8016 1.1131
when ±2% random noise is added to the extinction data in the inversion of monomodal L–N particle size distribution. Tables 5 and 6 show the reproducibility of original N–N distributions using the modified beta function and Johnson's SB function, and their inversion curves are shown in Fig. 3. The incident wavelengths and the relative refractive index are also the same as that in Fig. 1. It is possible to see that the retrieved distributions using the modified beta function are slightly different from the original N–N distributions. But from the practical point of view, the results can be considered satisfactory. On the other hand, there is an obvious discrepancy between the retrieved distributions using Johnson's SB function and the original N–N distribution when ±2% random noise is added to the extinction data. In other words, the modified beta function is more suitable for the inversion of monomodal N–N particle size distributions. Meanwhile, the width of original particle size distribution certainly has an effect on the inversion error. From the inversion results listed in Table 6, we can see that the increase of width can increase the inversion error, but the inversion errors are still acceptable for the wide particle size distribution. So it is feasible to retrieve the commonly used particle size distribution (for wide or narrow distributions) using the modified beta function as the assumed type of particle size distribution. From all the simulation results mentioned above, it can be inferred that it is feasible and reliable to use the modified beta function to retrieve the original R–R, L–N, and N–N particle size distributions in the dependent model. Meanwhile, the inversion results are still satisfactory when ±2% random noise is added to the extinction data. The modified beta function has a high flexibility and general
Table 6 Reproducibility of original N–N distributions with modified beta function and Johnson's SB function ((u1,σ1) = (4.5,1.7)) Inversion Random Inversion errors noise results (u1,σ1) (N–N) ξ (N–N) 0% 0% ±2% ±2% ±2%
(4.4999, 1.7994) (4.4982, 1.7936) (4.8323, 1.9502) (4.9752, 1.9864) (4.8069 1.9316)
2.89 × 10− 4 0.0032 0.1683 0.2162 0.1569
Inversion Inversion Inversion results results errors (α,m1) (mbeta) ξ (mbeta) (M,σ2) (SB)
Inversion errors ξ (SB)
(1.0185, 3.1643) (1.0342, 3.1935) (1.0897, 3.2348) (0.4053, 2.1854) (1.4367, 3.1732)
0.2264
0.2247 0.2382 0.2854 0.4397 0.5433
(4.2744, 1.7251) (4.9756, 1.7197) (5.1319, 1.7398) (5.7148, 1.6597) (6.7253, 1.6788)
Fig. 3. Inversion results of original N–N distributions with modified beta function and Johnson's SB function.
applicability to both narrow and wide distribution functions which are commonly used. 4. Conclusions In spectral extinction particle sizing technique, the modified beta function is used as the versatile function to retrieve the commonly used monomodal particle size distributions in the dependent model. The feasibility and validity of this inversion method are researched, and the inversion using the Johnson's SB function as the versatile function is also performed for comparison. Simulation results indicate that a reasonable representation of the commonly used particle size distribution can be obtained by using the modified beta function as the assumed type of particle size distribution. This inversion method has the advantages of simplicity, rapidity. It overcomes the difficulty which arises when the type of specified distribution function cannot be determined exactly beforehand in the dependent model. More work needs to be done, and the inversion of multimodal distributions using a more flexible versatile function will be considered in further studies.
0.2789 0.3473 0.5738 1.0115
Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 50336010). The authors thank M.D. Normand (University of Massachusetts, USA) for useful discussions on Johnson's SB function.
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