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Abstract
The accuracy of the particle size distribution (PSD) recovered from laser diffraction measurements by the Chin-Shifrin integral transform algorithm is reduced unless the proper angular parameters, including the lower and upper angular integration limits (θmax and θmin) and the angular resolution (Δθ), are used. To determine the selection criteria for these parameters, we use two metrics: the inversion error of the non-negative PSD, ε1, and the fitting error of the scattered laser light, ε2. By studying the variation of the minimum θmax and ε2 with the particle size at different inversion errors, and by analyzing the inversion error, as θmin and Δθ are varied, the optimized selection criteria for the minimum θmax, θmin and Δθ are obtained respectively. The inversion errors of the Chin-Shifrin algorithm with different selection criteria are compared, the different PSDs are recovered, and the optimal pixel selection range of the linear charge-coupled-device (CCD) array is determined according to the optimized selection criteria. Simulation results show that the optimized selection criteria for the angular parameters make the PSDs retrieved with the Chin-Shifrin algorithm more accurate.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Integral transform algorithms [1–3] have been widely used for measuring particle size distributions (PSDs) by laser diffraction [4,5], because they are faster and more stable than the numerical quadrature methods [6,7]. By comparing the inversion results of five integral transform algorithms [1,2,8,9], Koo and Hirleman found that the Chin-Shifrin(C-S) integral transform algorithm has the better results [10]. The C-S algorithm requires the integrating over the angular range zero to infinity. However, given that the Fraunhofer diffraction approximation is limited to small forward, scattering angles and that the optical detector has a fixed size, only a finite range of angles can be selected in the actual inversion process, resulting in a decrease in the accuracy of recovered particle size distribution (rPSD) [11,12].
The accuracy of the C-S algorithm depends on the selection of angular parameters including the upper and lower limit angles of integration, θmax and θmin respectively, and the angular resolution Δθ. To obtain a more accurate PSD and to reduce the errors caused by improper selection of parameters, selection criteria are required to choose the optimal parameters. Knight et al. suggested that θmax should include 96% of the diffraction light intensity, so they proposed that θmax < 15.47/x (x is the particle size parameter) [13]. However, this criterion did not consider the angular limit of the diffraction approximation. Using the sampling principle and its extensions [14], Riley and Hoμmaiti showed that the angular resolution Δθ should be less than π/(2xmax) and π/[2(xmax-xmin)] respectively (xmax and xmin are the upper and the lower bounds of x respectively) [15,16]. Liu demonstrated that the selection of the three angular parameters, depends inverse-linearly on x and presented the general selection criteria for the angular parameters in the integral transform inversion algorithm [17]. Yang et al. determined θmax by limiting the value of the diffraction approximation error, and the Δθ selection criteria according to the θmax and the characteristics of the cosine transform in the integral equation [18]. However, by using the above criteria to determine the angular parameters, the error in the inversion results are still large. Therefore, it is necessary to further study the selection criteria of the angular parameters.
In this paper, the influences of three angular parameters on the inversion results of the C-S algorithm are discussed, and the optimized selection criteria for θmax, θmin and Δθ are proposed. The inversion accuracy of the C-S algorithm with optimized angular parameters is analyzed and the applicability of the optimized selection criteria is verified.
2. The C-S integral transform algorithm
2.1 The C-S integral transform algorithm theory
For a suspension of spherical particles illuminated by a collimated beam of light with wavelength λ, the scattered light, I(θ), from the suspension is given by the Fredholm integral equation of the first kind,
Here f(x) is the particle number distribution, x = πd/λ is the particle size parameter, d is the particle diameter, I(θ,x,m) is the scattered light kernel function, m is the relative refractive index, and θ is the scattering angle. The kernel function of the scattered light can be accurately determined by Mie theory [19]. When x >> 1, the angular variation of the intensity is independent of the relative refractive index, the Fraunhofer diffraction approximation is valid at forward scattering angle, and the diffracted light intensity is given bywhere I0 is the intensity of incident light, k = 2π/λ is the wave number of the incident light, F is the focal length of the Fourier lens, and J1 is the first-order Bessel function of first kind. Because is very small in the actual inversion process, the approximation sinθ≈θ is used and Eq. (1) becomesUsing the Bateman-Titchmarsh-Fox transformation, the number PSD is obtained asEquation (4) is the expression for the C-S integral transform inversion algorithm. Here Y1 is the first-order Bessel function of the second kind. The coefficient quadratic inversely proportional to x in Eq. (4) increases sharply as x decreases, which causes the value of f(x) to increase sharply. Equation (4) allows the PSD, f(d), to be written aswhere C = −8k2F2λ2/(3I0) is a constant which is not related to particle diameter or scattering angle, Hcs(πdθ/λ) = J1(πdθ/λ)Y1(πdθ/λ)(πdθ/λ) is the integral kernel function, and Ecs(πdθ/λ) = d[I(θ,πd/λ)θ3]/dθ is the differentiation function, which is related to scattered light data containing particle size information. In Eq. (5), the range of scattering angle is 0~∞. In practice, when θmin, θmax and Δθ are set, the PSD is given by2.2 The influence of different angular parameters on inversion results
θmin, θmax and Δθ are the three essential parameters of the C-S algorithm. The following discussion outlines the influence of these parameters on the inversion results.
The angular limitation of the Fraunhofer diffraction approximation, the limited size of the photodetector and the low signal-to-noise ratio at large scattering angle mean that θmax is finite. The effect of the finite θmax on the PSDs recovered by the C-S algorithm is shown in Fig. 1(a). It can be seen that there are many side peaks in the PSD and that the larger θmax is, the smaller the amplitude of the side peaks are and the closer the recovered PSD is to the true PSD.
Fig. 1 The influence of different angular parameters on the PSDs recovered via the C-S algorithm: (a) The influence of the upper limit of angular integration, θmax, on inverted PSDs. (b)The influence of the lower limit of angular integration, θmin, on inverted PSDs. (c) The influence of the angular resolution, Δθ, on inverted PSDs.
The Airy disk, which is located in the center of the diffraction pattern, contains most of the diffracted light and a small part of the incident light. Excessive diffracted light will make the photodetector saturate, so that θmin cannot be set very small. The influence of θmin on the PSDs recovered by the C-S algorithm is shown in Fig. 1(b) where it can be seen that the peak value of the PSD increases sharply as θmin increases, negative values appear to the right of the peak (large size) and the PSD becomes unstable.
When the θmax and θmin are determined, the influence of Δθ on the PSDs recovered by the C-S algorithm is shown in Fig. 1(c). It can be seen that the larger the Δθ, the more oscillation occurs in the side peaks.
From the consideration above we see that the parameters, θmax, θmin and Δθ, determine the amount of diffracted light data used for inversion, and thus affect the stability and accuracy of the PSD recovered by the C-S algorithm. Therefore, it is necessary to determine the optimized selection criteria for the three angular parameters.
3. Optimized criteria for angular parameter selection
To determine the optimized selection criteria, a series of inversion processes are then carried out by using simulated Fraunhofer light scattering data for the model PSDs. The simulated intensity patterns are calculated by Eq. (3). It is convenient to use the normal distribution as the assumed true PSDs. The number expression of the normal distribution is
where μ1 is the mean particle diameter, and σ1 is the standard deviation. A set of normal PSDs with different mean diameters and variance of 5 μm are chosen to be inverted to determine the criteria for angular parameter selection and for comparison of the inversion results with different criteria.To verify the performance of the proposed optimized selection criteria and to test the robustness of the C-S algorithm with the Fraunhofer approximation, three typical asymmetric distributions, Rosin-Rammler, gamma and lognormal distributions, were used. The Rosin-Rammler distribution is
where n and D0 are the shape and scale parameter respectively.The gamma distribution is defined as
where Γ(s) is the gamma function, s is the shape parameter and t is the scale parameter.The lognormal distribution is expressed as
Where the two parameters μ2 and σ2 are mean and standard deviation of the variables’ natural logarithm respectively. The mean particle size is .Simulated diffracted intensity patterns for a normal distribution and a Rosin-Rammler distribution are shown in Fig. 2.
Fig. 2 Simulated diffracted intensity patterns for the model PSDs: (a) Normal particle size distribution with μ = 80 μm and σ = 5 μm. (b) Rosin-Rammler particle size distribution with D0 = 60 μm and n = 13.
The C-S inversion algorithm does not guarantee the non-negativity of the inversion results, so a non-negativity constraint must be imposed on the recovered PSD to obtain the non-negative PSD (NNPSD). To evaluate quantitatively the influences of the angular parameters on the PSD recovered by the C-S algorithm and further verify the performance of optimized selection criteria, we define the inversion error of the NNPSD as
where N is the number of points in the NNPSD, fnncs(d) is the recovered NNPSD, and fin(d) is the true PSD.The scattered light is collected by the photodetector and then inverted by the C-S algorithm to obtain the PSD. The smaller the error between the best-fit light signal resulting from determining the NNPSD and the collected scattered light, the closer the inversion result should be to the true PSD. Therefore, the fitting precision to the scattered light is an important index for evaluating the inversion results. The fitting error can be defined as
Where Ifit,θ is the fitting scattered light of fnncs(d), Iin,θ is the true or collected scattered light.3.1 Optimized selection Criterion 1: determination of θmax
To reduce the effects of θmin and Δθ, these two angular parameters are given the small value of 2 × 10−5 rad, then the errors in the NNPSD are mainly caused by θmax. When the inversion error ε1 is 0.03, 0.05, 0.1 and 0.15 respectively, the fitting error, ε2, variation with the particle diameter is shown in Fig. 3(a). It can be seen that, despite the different inversion errors, ε1, the fitting errors, ε2, vary in the same way with particle size. For the PSD with larger particles, the fitting error, ε2, needs to be smaller to ensure the same error in the NNPSD. Further, as the particle size increases, ε2 changes slowly and tends gradually to zero.
Fig. 3 (a). The fitting error, ε2, of the scattered light for different inversion errors, ε1. (b). The minimum upper limit of angular integration, θmax, for different inversion errors, ε1.
θmax can be easily determined by studying its dependence on particle size for different values of inversion error ε1. For ε1 values of 0.03, 0.05, 0.10 and 0.15, the required θmax is shown in Fig. 3(b). As can be seen from this figure, the smaller the inversion error ε1, the larger θmax required. With the increase of the particle diameter, θmax tends to decrease and gradually stabilizes at a certain angle. The θmax required for inversion fluctuates sharply within a smaller particle size range. When ε1 is 0.05, θmax gradually stabilizes at 0.0355 rad, so the optimized Criterion 1 to determine θmax can be given by Eq. (13).
3.2 Optimized selection Criterion 2: determination of θmin
θmax can be determined by Criterion 1, and if Δθ is chosen as 2 × 10−5 rad, then the optimized lower limit of angular integration is obtained when the θmin minimizes the inversion error, ε1. To determine the optimized θmin, a normal particle size distribution with mean diameter of 100 μm and variance of 5 μm is chosen to be inverted. The variation of the inversion error ε1 with θmin is shown in Fig. 4(a). It can be seen that the curve is approximately V-shaped. The error ε1 increases when the value of θmin is larger or less than the optimized value. It changes slowly for small θmin, and sharply for larger θmin.
Fig. 4 (a). Variation of the inversion error, ε1, with θmin. (b). The relationship between the optimized θmin and size.
Figure 4(b) shows the optimized θmin determined for the various normal PSDs. It can be seen that the optimized θmin decreases with increasing particle diameter. Fitting this data over the range 5 μm ~200 μm, gives the following relationship between the optimized θmin and the particle diameter,
3.3 Optimized selection Criterion 3: determination of Δθ
θmin and θmax can be determined by Criterion 1 and Criterion 2, respectively. To minimize the error ε1, the ideal Δθ for different particle sizes can be obtained. This is shown in Fig. 5(a). As can be seen from the figure, the ideal Δθ is very small, especially for large particles, so it may be difficult to meet the requirement in the actual measurement process.
Fig. 5 The variation of Δθ with particle diameter: (a) Ideal angular resolution. (b) Optimized angular resolution.
When a linear CCD is used as a photodetector generally some data collected from adjacent pixels are averaged as one data point to reduce the influence of laser speckle noise [20,21]. This results in larger angular resolution and decreases the accuracy of the inversion results. To obtain more effective scattered light data, the angular resolution should not be set too large. The angular resolution is determined by setting ε1 = 0.06. With this condition, the angular resolution is larger than the ideal Δθ and the inversion error is not large. Figure 5(b) shows the variation of optimized Δθ with particle diameter. The optimized selection Criterion 3 to determine the optimized angular resolution is obtained by fitting the data to get
4. Performance analysis of the optimized selection criteria
4.1 Comparison of the different angular parameter selection criteria
Table 1 shows the C-S algorithm inversion error, ε1, obtained for various PSDs using different criteria to determine θmax. Here θmax1 is the value determined using our Criterion 1, 6/x is a value from the literature [17]. The resulting main peak sizes of the recovered NNPSDs are shown in Table 2. θmin = 2 × 10−5 rad was used in these calculations.
Table 1. The variation of inversion error ε1 with particle diameter for the different minimum θmax.
Table 2. Main peak particle sizes of the recovered NNPSDs for the different minimum θmax.
As can be seen from the tables, if θmax = 6/x, when the particle diameter exceeds 30 μm, the inversion error ε1 is lager than 0.1 and when the particle diameter exceeds 50 μm, the error of the peak particle diameter exceeds 2%. The larger the particle size, the larger the error. θmax = 6/x is too small, that is to say that a large amount of scattered light data are lost and an accurate PSD cannot be recovered. Using θmax determined with Criterion 1 gives an accurate peak particle diameter and, for particles larger than 30 μm, a smaller inversion error.
Given that the Fraunhofer diffraction approximation is valid only at small forward scattering angles, the maximum value for θmax needs to be restricted. In the literature [13], 15.47/x is proposed as the maximum value for θmax. One can also calculate a value θmaxY, for the maximum value for θmax when the diffraction approximation error [18] (the relative refractive index is m = 1.5) is 5%. The variation of inversion error ε1 with particle diameter for the two criteria are shown in Table 3. It can be seen from Table 3 that with θmax = 15.47/x, the inversion error becomes large as the particle size increases. When the particle diameter is > 70 μm, the error exceeds the error obtained using Criterion 1 to set θmax. In the small particle diameter range setting θmax = 15.47/x will exceed the effective angular range of the diffraction approximation [18]. The value θmax = θmaxY is obtained by considering the diffraction approximation error, and it also gives high inversion accuracy. So the appropriate θmax is between the value given by Criterion 1 and θmaxY.
Table 3. The variation of inversion error ε1 with particle diameter for the different maximum θmax.
The inversion error, ε1, of NNPSDs obtained by the C-S algorithm are shown in Table 4, when Δθ is chosen in various ways. Δθ = Δθ1 is from using Criterion 3, Δθ = π/(2xmax), Δθ = π/[2(xmax-xmin)], Δθ = 0.6/x and Δθ = ΔθY are proposed in the literature [15–18]. The determination of π/(2xmax), π/[2(xmax-xmin)] and ΔθY based on the same priciple, Nyquist-Shannon sampling theorem and its extensions. The value of 0.6/x is the maximum tolerable angular resolution and is produced by numerical experiments when the C-S algorithm is used. Here θmax is determined by Criterion 1, and θmin = 2 × 10−5 rad. As can be seen from Table 4, when Δθ is determined by the optimized selection Criterion 3, the inversion error ε1 is significantly smaller than the errors obtained by other selection criteria, and the variation of ε1 with the particle diameter is small. Also, the accuracy of the recovered NNPSD was significantly higher than for the other values of Δθ.
Table 4. The inversion error ε1 of recovered NNPSDs for different Δθ.
4.2 Application of the optimized selection criteria to different particle size distributions
To verify the proposed optimized selection criteria, a number of different PSDs are recovered using the C-S algorithm.
Two unimodal PSDs obeying the Rosin-Rammler distribution with D0 = 60 μm, n = 10 and D0 = 100 μm, n = 13 are used to generate the scattered light signals and the data are analyzed with the C-S algorithm to recover the PSD. The angular parameters are determined by the optimized selection criteria described above. The recovered PSD data are shown in Fig. 6. For the PSD in Fig. 6(a), the peak value of the true PSD is 59 μm, the peak value of the recovered NNPSD is 59.5 μm, and the relative error is 0.85%. The inversion error ε1 = 0.038. For the PSD in Fig. 6(b), the peak value of the true distribution is 99 μm. The peak value of the recovered NNPSD is 99.5 μm, the error is 0.5%, and the inversion error ε1 = 0.041.
Fig. 6 NNPSDs recovered by the C-S algorithm for particles obeying the Rosin-Rammler distribution: (a) D0 = 60 μm, n = 10. (b) D0 = 100 μm, n = 13.
To test the robustness of the C-S algorithm with the Fraunhofer approximation, the intensity patterns are generated by Mie theory with defined asymmetrical distributions in Section 3. The simulated and retrieved PSDs are shown in Fig. 7. The simulation parameters are Fig. 7(a) D0 = 60 μm, n = 5 and Fig. 7(b) D0 = 100 μm, n = 7 for the Rosin-Rammler distributions, Fig. 7(c) s = 10, t = 6 and Fig. 7(d) s = 20, t = 5 for the gamma distributions, and Fig. 7(e) μ2 = 4.06, σ2 = 0.25 and Fig. 7(f) μ2 = 4.57, σ2 = 0.22 for the lognormal distributions. The resulting ε1 are 0.063, 0.059 for Rosin-Rammler distributions, 0.067, 0.056 for gamma distributions, 0.078 and 0.087 for lognormal distributions respectively. It can be seen that the angular parameters obtained by the optimized selection criteria are well suited to recovering different asymmetric unimodal PSDs. These results show that the C-S algorithm with the Fraunhofer approximation is very robust.
Fig. 7 NNPSDs recovered via the C-S algorithm for different PSDs from intensity patterns by Mie theory with three types of asymmetrical PSDs.
A bimodal PSD composed of two unimodal Rosin-Rammler distributions in the ratio of 2:1 with the parameters D0 = 75 μm, n = 10 and D0 = 120 μm, n = 16 was generated. The optimized selection criteria and D0 = 60 μm give θmin = 1.38 × 10−3 rad, Δθ = 2.08 × 10−4 rad and θmax = 3.55 × 10−2 rad. With these parameters the recovered non-negative PSD is shown in Fig. 8(a) and the inversion error is ε1 = 0.089. If the angular parameters are determined based on D0 = 130 μm, then θmin = 8.92 × 10−4 rad, Δθ = 8.74 × 10−5 rad, and θmax = 3.55 × 10−2 rad are obtained. The recovered PSD is shown in Fig. 8(b), and the inversion error ε1 = 0.062.
Fig. 8 PSD results for the C-S algorithm for the multiple peak particle size distribution for different angular parameters: (a) PSD recovered with angular parameters determined based on D0 = 60 μm. (b) PSD recovered with angular parameters determined based on D0 = 130 μm.
According to the optimized selection criteria, θmin and Δθ determined by the larger peak particle diameter are smaller, and the difference between θmax values for different particle sizes is small. Therefore, the inversion error can be much smaller when the multiple peak PSD is inverted by the C-S algorithm and angular parameters are selected according based on the larger peak value of the particle diameters.
4.3 Determination of the optimal pixel range using the optimized selection criteria
It is of interest also to consider optimal range of pixels required in the Fraunhofer-based particle size measurement method. This can also be determined via the optimized selection criteria. Essentially Criterion 1 will give the last pixel, Criterion 2 will give the first pixel and Criterion 3 will give the pixel spacing.
The C-S algorithm requires a large amount of scattered light data to get the PSD, and the literature [17] suggests that the linear angular resolution is better than logarithmic angular resolution. Hence, to improve the inversion accuracy, a linear CCD is used as the photodetector. We use the linear CCD recommended in the literature [22], determine the optimal pixel range with the optimized selection criteria and calculate the inversion results. The pixel size of the CCD is L = 4.7 μm, the total number of pixels is 7450. If the focal length of the Fourier lens is F = 300 mm, the minimum angular resolution is Δθ = atan(L/F) = 1.57 × 10−5 rad, and the maximum integral angle is 0.116 rad.
The optimal sensing pixels determined by the above optimized criteria are shown in Fig. 9. According to the analysis of Criterion 1, when the inversion error of the NNPSD is the same, the change of θmax is very small for different particle sizes, and the value of θmax tends to be stable. Therefore, when the particle diameter is > 100 μm, the position of the end pixel hardly varies, as shown in Fig. 9(a). The charts in Figs. 9(b) and 9(c) show that the angular parameters determined by the optimized selection criterion are in accordance with the characteristics of the start pixel and the pixel spacing in the actual measurement.
Fig. 9 The optimal pixel range determined by the optimized selection criteria: (a) The end pixel determined by Criterion 1. (b) The start pixel determined by Criterion 2. (c) The pixel spacing determined by Criterion 3.
From a normal PSD with mean particle size 70 μm and variance 5 μm the light scattering signal was generated and inverted by the C-S algorithm. When F = 300 mm, the angular parameters determined by the optimal selection criteria are θmin = 1.28 × 10−3 rad, Δθ = 1.76 × 10−4 rad, and θmax = 3.55 × 10−2 rad. The pixel range is 82-2268, the pixel spacing is 11. Using 199 data points in the recovered PSD, the error in the NNPSD is ε1 = 0.058. The PSD is shown in Fig. 10. We can see that the inversion error of the C-S algorithm with the optimized pixel parameters obtained by the optimized selection criteria is small and the inversion result is accurate.
5. Conclusion
The amount of scattered light detected and used to recover the PSD using the Fraunhofer light scattering method depends on the selection of angular parameters, and affects the accuracy of the PSD recovery via the Chin-Shifrin integral transform algorithm. The accuracy of the recovered PSD can be improved by using the optimal angular parameters. We have outlined a procedure for determining the optimized selection criteria for the angular parameters. By analyzing simulated data we found that both the fitting error of the scattered light and the minimum upper limit of angular integration decrease with increasing particle diameter while the error in the recovered NNPSD remains unchanged. This provides the criterion to determine the minimum upper limit of angular integration. The relationship between the error of the recovered NNPSD and the lower limit of angular integration is seen to exhibit a V-shaped curve. This suggests an optimal value which makes the inversion error the smallest and gives the optimized selection criterion for the lower limit of angular integration. Furthermore, by considering the angular resolution requirements of a linear CCD, the optimized selection criterion for angular resolution is obtained. Using simulated light scattering data from common types of PSDs we have shown that the proposed optimized selection criteria can improve the accuracy and stability of the PSD recovered via the Chin-Shifrin inversion algorithm.
Funding
Natural Science Foundation of Shandong Province (Grant no. ZR2017LF026, ZR2016EL16, ZR2018MF032); Key Research and Development Program of Shandong Province (2017GGX10125).
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