Inversion Of Particle Size Distribution Research Articles

Simultaneous Inversion of Particle Size Distribution, Thermal Accommodation Coefficient, and Temperature of In-Flame Soot Aggregates Using Laser-Induced Incandescence.

Measuring the size distribution and temperature of high-temperature dispersed particles, particularly in-flame soot, holds paramount importance across various industries. Laser-induced incandescence (LII) stands out as a potent non-contact diagnostic technology for in-flame soot, although its effectiveness is hindered by uncertainties associated with pre-determined thermal properties. To tackle this challenge, our study proposes a multi-parameter inversion strategy-simultaneous inversion of particle size distribution, thermal accommodation coefficient, and initial temperature of in-flame soot aggregates using time-resolved LII signals. Analyzing the responses of different heat transfer sub-models to temperature rise demonstrates the necessity of incorporating sublimation and thermionic emission for accurately reproducing LII signals of high-temperature dispersed particles. Consequently, we selected a particular LII model for the multi-parameter inversion strategy. Our research reveals that LII-based particle sizing is sensitive to biases in the initial temperature of particles (equivalent to the flame temperature), underscoring the need for the proposed multi-parameter inversion strategy. Numerical results obtained at two typical flame temperatures, 1100 K and 1700 K, illustrate that selecting an appropriate laser fluence enables the simultaneous inversion of particle size distribution, thermal accommodation coefficient, and initial particle temperatures of soot aggregates with high accuracy and confidence using the LII technique.

A regularization algorithm of dynamic light scattering for estimating the particle size distribution of dual-substance mixture in water

Dynamic light scattering (DLS) is a nondestructive, well-established technique for the size characterization of proteins, nanoparticles, polymers, and colloidal dispersions. However, current DLS techniques are only applied to particle groups of single composition due to the limitation of their inversion algorithm. In this study, we propose a particle size distribution inversion algorithm based on the Tikhnonov regularization method that can be applied to the dual-substance particle mixture. The algorithm retrieves the particle size distributions of two substances, respectively, by taking advantage of their refractive index differences. The simulation results reveal that the algorithm has excellent accuracy and stability when the scattering angle is 30°. Instead of the original identity matrix, the first-order difference matrix and second-order difference matrix are used as the regular matrix when utilizing the Tikhnonov algorithm, which obviously improves the anti-interference, accuracy, and stability of the algorithm. Furthermore, the inversion of particle size distribution is carried out at a 0.01%–1% noise level, which shows that the algorithm has an available antinoise ability. Finally, experimental particle size measurements for a mixture of polystyrene beads and toner particles demonstrate that the proposed algorithm is superior to the traditional Tikhnonov algorithm in applicability and accuracy.

Multi-wavelength method based on global optimization for particle size distribution

The particle size distribution (PSD) of particle medium plays an important role in the field of particle science, so the inversion of PSD is of great significance. To study spherical particle PSD, a multi-wavelength detection model based on a global optimization algorithm called OptQuest nonlinear programming (OQNLP) is established in this paper and an experiment to verify the reliability of the system is designed. The numerical results show that the selection of detection wavelength has great influence on the results of PSD inversion. The relative error of PSD parameters is minimized by choosing the wavelength at the peak of extinction coefficient curve of appropriate particle size. Both simulation and experimental results indicate that the five-wavelength method has the highest testing accuracy. When high accuracy is not required, choosing the four-wavelength method is the most suitable testing method. Furthermore, the universality of the model is also confirmed for the Rosin-Rammer (R-R) function, normal (N-N) function, and lognormal (L-N) function.

Particle size distribution measurement based on the angular scattering efficiency factor spectra inversion-simulation and experiment.

The quantification of the particle size distribution (PSD) within a particle system is significant to various domains, including atmospheric and environmental sciences, material science, civil engineering, and human health. The scattering spectrum reflects the PSD information of the particle system. Researchers have developed high-precision and high-resolution PSD measurements for monodisperse particle systems through scattering spectroscopy. However, for polydisperse particle systems, current methods based on light scattering spectrum and Fourier transform analysis can only obtain the information of the particle component, but cannot provide the relative content information of each component. In this paper, a PSD inversion method based on the angular scattering efficiency factors (ASEF) spectrum is proposed. By establishing a light energy coefficient distribution matrix, and then measuring the scattering spectrum of the particle system, PSD can be measured in conjunction with inversion algorithms. The simulations and experiments conducted in this paper substantiate the validity of the proposed method. Unlike the forward diffraction approach that measures the spatial distribution of scattered light I(θ) for inversion, our method uses the multi-wavelength distribution information of scattered light β(λ). Moreover, the influences of noise, scattering angle, wavelength, particle size range, and size discretization interval on PSD inversion are studied. The method of condition number analysis is proposed to identify the appropriate scattering angle, particle size measurement range, and size discretization interval, and it can reduce the root mean square error(RMSE) of PSD inversion. Furthermore, the method of wavelength sensitivity analysis is proposed to select the spectral band with higher sensitivity to particle size changes, thereby improving the computational speed and avoiding the problem of diminished accuracy caused by the reduction of the number of wavelengths used.

Particle size distribution inversion using theWeibull-distribution adaptive-parameters cuckoosearch algorithm.

A Weibull-distribution adaptive-parameters cuckoo search (WACS) algorithm is proposed, which can converge quickly and prevent falling into local optimal values, and thus improve the global search performance of a cuckoo search (CS) algorithm. In simulations, particle size inversions were performed using the proposed algorithm for unimodal and bimodal particle systems obeying Johnson's SB, Rosin-Rammler, and normal distribution, and the results were compared to the original CS algorithm, Weibull-distribution CS algorithm, and adaptive-parameters CS algorithm. Among them, the WACS algorithm has the best accuracy. The relative root mean squared error (RRMSE) was three to four orders of magnitude lower than the CS algorithm. The noise immunity of the algorithm was verified by comparing the particle size inversion error. Random noise [1%, 10%] was added to the scattered light energy of the target function, in 1% noise increments. The WACS algorithm prevailed, and the advantage became more obvious as the noise increased. A small-angle forward scattering experimental platform was built, and ferric tetroxide particles were selected as the measured particles. Experimental measurements were carried out on a unimodal particle system (50µm) and bimodal particle system (50 and 100µm), while the WACS algorithm was used on particle size distribution inversion. Compared to the CS algorithm, the RRMSE of the WACS algorithm was approximately 51% lower on unimodal and 66% lower on bimodal particle population inversions.

Particle size distribution inversion in dynamic light scattering by adaptive step-size non-negative least squares

The non-negative least squares algorithm is widely used in dynamic light scattering, however, it suffers the shortcomings of sparse solutions and poor anti-noise performance. It is found that the construction of the kernel matrix of the algorithm is of great significance to the accuracy of the final results. In this paper, we propose an adaptive strategy to find the optimal step-size to construct the kernel matrix. By calculating the L2-norm between the measured and reconstructed light intensity autocorrelation function, the optimal step-size is determined corresponding to the minimum difference error. Then the kernel matrix constructed by this optimal step-size is used in the inversion. The final result is obtained by fitting the solution to a predefined model, such as a Gaussian model. This strategy improves the sparsity of NNLS solution, and thus the accuracy and stability of NNLS inversion results. The simulation and experiments demonstrate that the adaptive step-size strategy improves the stability and accuracy of the non-negative least squares inversion algorithm for dynamic light scattering.

Influence of dispersion on particle size-distribution based on multi-wavelength scattering measurement

We demonstrate that the dispersion of the refractive index (RI) influences the retrieval accuracy of the particle size distribution (PSD) in the multi-wavelength angular scattering measurement. A numeric simulation was performed, which indicated that the dispersion results in a 10% to 50% error on scattering intensity measurement at certain scattering angles. In addition, although the dispersion showed little effect on the ill-posedness of the scattering kernel, the difference between the dispersion and no dispersion in the scattering kernel is so amplified in PSD inversion that accuracy decreases to such an extent that it becomes unacceptable. The accuracy of PSD retrieval, considering dispersion, was higher than that without dispersion. Finally, it was also proposed that variation of RI with incident wavelength must be considered in designing the scattering kernel to improve the retrieval accuracy in the multi-wavelength angular scattering measurement.

Inversion of particle size distribution based on iterative non-negative Philips-Twomey algorithm

The precision of particle size distribution (PSD) is a key indicator to evaluate inversion algorithms. In the light scattering particle sizing technology, constructed on the inversion method of the traditional non-negative Philips-Twomey (NNPT) algorithm, an iterative NNPT (INNPT) algorithm was proposed in this paper, applied for the inversion of PSD. In simulations, the inversion accuracy performance of NNPT and INNPT algorithms were compared and analyzed by using different widths of PSD that conform to Johnson’s SB unimodal and bimodal functions. A small angle forward scattering method of PSD measurement system using Charge Coupled Device (CCD) as a photodetector was constructed and the national standard particles were tested. Both the results of simulations and experiments show that the inverse accuracy and stability of INNPT algorithm are superior to NNPT algorithm especially in the conditions of narrow distribution.

Accuracy calculation for lidar ratio and aerosol size distribution by dual-wavelength lidar

This paper presents a method to obtain aerosol-lidar ratio and particle-size distribution simultaneously with a dual-wavelength lidar, when the aerosol particle-size distribution is to be approximated by a Junge distribution. The manuscript analyzed the relationship between the aerosol particle ratio, particle-size distribution, and particle complex refractive index. The results show that the particle properties have a great impact on the lidar ratio. If the selected lidar ratio is not suitable, the extinction coefficient obtained from the Fernald method will cause a great uncertainty in the inversion of particle-size distribution. A lidar-ratio iteration method was introduced to solve the influence of improper lidar-ratio selection on particle-size distribution inversion, and then conducted simulation calculations and analysis. The simulation results show that the lidar-ratio iteration method effectively solves the problem of uncomfortable lidar ratio, and can obtain relatively accurate lidar ratio and particle-size distribution parameters at the same time; the uncertainty of iteration method was discussed in the manuscript.

Research on bat algorithms for inversion of particle size distribution in spectral extinction method

The development of rapid and effective inversion algorithm is an important subject in the field of particle size distribution (PSD) measurement. An intelligent bat algorithm has been proposed to solve the problem of independent mode inversion in particle size distribution reconstruction based on spectral extinction method. The improved algorithm is more suitable for particle size distribution inversion under extinction method by enriching its behavior mode, improving local search mode and adding greedy strategy. In simulations, uniform spherical particles that obey the unimodal, bimodal and multimodal R-R distribution were used, and the scattering intensity values of the target function was added to the random noise of ±0%, ±3%, ±5%, ±8% respectively. In the inversion calculation of the improved bat algorithm, 400, 800 and 1000 iterations were performed for different distributions, respectively. Compared with modified pattern search method, the improved bat algorithm has strong noise resistance, moderate inversion time, independent of initial solution and high inversion accuracy for complex distribution. Compared with genetic algorithm, the improved bat algorithm has fast convergence speed, high accuracy and short time-consuming. Considering the practical application, the algorithm is suitable for field measurement and has a wide application prospect.

小角前向散射偏振比法颗粒粒度分布反演

小角前向散射偏振比法颗粒粒度分布反演

Inversion of particle size distribution using the artificial fish swarm algorithm

Abstract In the small-angle forward scattering technique, an inversion method of particle size distribution (PSD) using artificial fish swarm algorithm (AFSA) is proposed, which can acquire optimal characteristic parameters of the PSD. Simulations are performed to verify the effectiveness of AFSA, in which spheroidal particles of a unimodal distribution that conforms to Johnson’s S B {\mathrm{S}_{\mathrm{B}}} function is inversed at different levels of gaussian white noise. The comparison of the smooth objective function and non-smooth objective function of AFSA is conducted and discussed. The measurement system of the small-angle forward scattering based on Mie scattering theory is constructed and experiments are performed to verify the practicability of AFSA. A CCD sensor is adopted to receive the scattered light instead of the conventional photodetector. Pinhole is first proposed to replace standard particles and calibrate the measurement system, which can eliminate the influence of particle concentration. After the calibration experiments, the PSD of standardized polystyrene microspheres are measured and modified. Both the simulation and experiment results indicate that the PSD can be successfully inverted by AFSA with high reliability and stability in certain conditions, and the system calibration can further improve the inversion accuracy.

A Modified Landweber Algorithm for Inversion of Particle Size Distribution Combined With Tikhonov Regularization Theory

Particle size distribution (PSD) measurement based on the static light scattering method has been widely used in the environmental field and combustion diagnostics, such as PM2.5 measurement and combustion process monitoring. The PSD inversion is mathematically related to the Fredholm integral equation of the first kind. Although the Tikhonov regularization algorithm is one of the effective inversion methods to solve the ill-posed linear equation, it still has the disadvantages of excessive smoothness and low accuracy. Thus, a preconditioned Landweber algorithm combined with the Tikhonov regularization theory (the improved algorithm) is proposed in this paper. The Tikhonov regularization theory is used to pretreat the preconditioner B contained in the preconditioned Landweber algorithm. Simulations are conducted to compare the inversion results of the conventional Landweber algorithm, the preconditioned Landweber algorithm, the Tikhonov Chahine algorithm, and the improved algorithm at different signal-to-noise ratios. A CCD-based small-angle forward scattering measurement system is built. A standardized polystyrene microsphere with a diameter of $35.05~\mu \text{m}$ is used to evaluate the above algorithms. Both numerical and experimental results show that the improved algorithm improves the accuracy of the inversion results and is insensitive to the ring parameter of the detector. The experimental results of the standardized polystyrene microsphere reveal that the relative errors for the median diameter $50~\mu \text{m}$ are better than 3%. The improved algorithm can provide a highly reliable and stable inversion result.

Weighting inversion of dynamic light scattering based on particle-size information distribution character

In particle sizing with dynamic light scattering (DLS) technique, the determination of particle size distribution (PSD), via inversing the autocorrelation function (ACF) of scattering light, is usually limited by the inherently low particle size information in ACF data and, the lack of targeted inversion on the noise restriction and the particle size information utilization. For the ACF data in DLS measurement, most of particle size information is centrally contained in the decay section and the larger noise is contained in the larger delay section. However, no consideration of the particle size information distribution in the ACF data for the routine inversion method increases the difficulty of the accurate PSD inversion, especially the broad and bimodal PSDs. Until now, it is still a difficult problem to obtain an accurate recovery of the broad and bimodal PSDs, specifically the bimodal PSD with a peak position ratio less than 2:1 and containing large particles (350 nm). In this paper, a character-weighted constrained regularization (CW-CR) method is proposed, in which, the particle size information distribution in the ACF as the base and the adjustment parameter as the exponent are used to weight the ACF. By using the weighting coefficients corresponding to the particle size information distribution along the delay time in ACF, the CW-CR method can enhance the utilization of the particle size information in ACF data, and effectively weaken the effect of noise at large delay time. With this method, the closely spaced bimodal PSD (with nominal diameters of m 350 nm:500 nm in simulation, m 300 nm:502 nm in experiment) is recovered successfully at a high noise level of 0.01. It shows that the CW-CR method, combined with the multiangle DLS (MDLS) measurement, can effectively make the best use of the particle size information hiding in the noisy ACF data, and improve the resolution of bimodal PSD as well as the capability of noise suppression. So it can make the advantages of MDLS more highlighted than the routine method in the recovery of the broad and bimodal PSDs.

A hybrid artificial bee colony algorithm and pattern search method for inversion of particle size distribution from spectral extinction data

In this paper, a hybrid artificial bee colony (ABC) algorithm and pattern search (PS) method is proposed and applied for recovery of particle size distribution (PSD) from spectral extinction data. To be more useful and practical, size distribution function is modelled as the general Johnson’s function that can overcome the difficulty of not knowing the exact type beforehand encountered in many real circumstances. The proposed hybrid algorithm is evaluated through simulated examples involving unimodal, bimodal and trimodal PSDs with different widths and mean particle diameters. For comparison, all examples are additionally validated by the single ABC algorithm. In addition, the performance of the proposed algorithm is further tested by actual extinction measurements with real standard polystyrene samples immersed in water. Simulation and experimental results illustrate that the hybrid algorithm can be used as an effective technique to retrieve the PSDs with high reliability and accuracy. Compared with the single ABC algorithm, our proposed algorithm can produce more accurate and robust inversion results while taking almost comparative CPU time over ABC algorithm alone. The superiority of ABC and PS hybridization strategy in terms of reaching a better balance of estimation accuracy and computation effort increases its potentials as an excellent inversion technique for reliable and efficient actual measurement of PSD.

Use of non-negative constraint in Tikhonov regularization for particle sizing based on forward light scattering

The set of linear equations in the inversion of particle size distribution (PSD) based on forward light scattering is an ill-posed problem. In order to solve the inverse problem of this kind, a number of inversion algorithms have been proposed. The regularization algorithm can reconstruct the PSD, but in usual case, the solution may contain negative values and is strongly oscillating. Owing to the natural reason, the solution should be non-negative and smooth. In this paper, a simple non-negative constraint (NNC) is used with a combination of the Tikhonov regularization. Simulations and experiments show that the regularization with NNC can achieve more reasonable results.

Inversion of particle size distribution by spectral extinction technique using the attractive and repulsive particle swarm optimization algorithm

The particle size distribution (PSD) plays an important role in environmental pollution detection and human health protection, such as fog, haze and soot. In this study, the Attractive and Repulsive Particle Swarm Optimization (ARPSO) algorithm and the basic PSO were applied to retrieve the PSD. The spectral extinction technique coupled with the Anomalous Diffraction Approximation (ADA) and the Lambert-Beer Law were employed to investigate the retrieval of the PSD. Three commonly used monomodal PSDs, i.e. the Rosin-Rammer (R-R) distribution, the normal (N-N) distribution, the logarithmic normal (L-N) distribution were studied in the dependent model. Then, an optimal wavelengths selection algorithm was proposed. To study the accuracy and robustness of the inverse results, some characteristic parameters were employed. The research revealed that the ARPSO showed more accurate and faster convergence rate than the basic PSO, even with random measurement error. Moreover, the investigation also demonstrated that the inverse results of four incident laser wavelengths showed more accurate and robust than those of two wavelengths. The research also found that if increasing the interval of the selected incident laser wavelengths, inverse results would show more accurate, even in the presence of random error.

Application of Genetic Algorithms in Particle Inversion

In the general light scattering particle size measurement methods, the inversion error of results is nonuniform distribution. To solve this problem, genetic algorithms based on real number coding were proposed to be applied in the inversion of particle size distribution. In this method, the fitness function was further improved in order to reduce the individual error, it not only considered sum of all errors but also took into account the individual maximum error. With this improved fitness function, the error distribution is homogenized. The light spectrum about measurement and calculation are basic entirely fitting. The results show that the algorithm is accurate and reliable.

Inversion of particle size distribution from light-scattering data using a modified regularization algorithm

A modified regularization algorithm with a more proper operator was proposed for the inversion of particle size distribution (PSD) from light-scattering data in a laser particle sizer based on the Mie scattering principle. The Generalized Cross-Validation (GCV) method and the L-curve method were used for determining the regularization parameter. The Successive Over-Relaxation (SOR) iterative method was used to increase the exactness and stability of the converged result. The simulated results based on the modified algorithm are in a good agreement with the experimental data measured for nine standard particulate samples, their mixtures as well as three natural particulate materials with irregular shapes, indicating that this modified regularization method is not only feasible but also effective for the simulation of PSD from corresponding light-scattering data.

Inversion of particle-size distribution with improved conjugate gradient algorithm

In order to ameliorate convergence of the algorithm to invert the particle-size distribution (PSD) from laser diffraction data, an improved conjugate gradient algorithm (ICGA) is proposed. This method is independent of any given a priori information of the particle-size distribution. In the algorithm, each objective function is constructed according to an equation of the system of equations. Then iterations are carried out continuously between objective functions by choosing conjugate gradient directions, and thus the objective functions are tied up. An iteration step-adjusting parameter is introduced, which depends on the row index vectors of the matrix equation. Two narrowly distributed particulate-certified reference materials, their mixture, and a widely distributed particle plate are used as samples to verify the algorithm. Experimental results show that the ICGA is sufficiently convergent and that the convergence points are stable. The presented method can be used to invert unimodal and multimodal PSD with high precision.