Particle Size Distribution Reconstruction Research Articles

Population balance modelling and reconstruction by quadrature method of moments for wet granulation

Population balance methods utilised in multiphase flow simulations mark a significant advancement in computational fluid dynamics. However, existing approaches exhibit shortcomings, such as being prone to inaccuracies or being computationally prohibitive. Addressing these challenges, a recent innovation in closure for the method of moments is the introduction of quadrature based moments methods (QBMM). Discretising a distribution by a number of discrete elements, QBMM facilitate efficient and accurate tracking of density distributions, particularly for particle size distributions (PSD). However, obtaining the full particle size distribution information using these methods requires reconstructing the distribution from a finite set of moments, which is not a trivial step.This study introduces a novel combination of the maximum entropy reconstruction (MER) and QBMM, establishing a robust and rapid framework for the time evolution and reconstruction of PSDs. As proof of concept for this framework, we focus on the direct quadrature method of moments (DQMOM) with spatially homogeneous and monovariate distributions. We show that coupling of MER with DQMOM has numerous advantages. To verify the framework, special cases of constant growth, aggregation, and breakage are considered for which analytical solutions can be found. Furthermore, we show the advantage of using DQMOM with volume-based over length-based distributions, and address numerical as well as theoretical issues.Application of the framework is successfully conducted on the evolution of the PSD from a twin-screw wet granulation dataset, considering all active primary physical mechanisms in a wet granulation process, namely growth, aggregation, and breakage. This showcases the consistency of the proposed framework and underscores its applicability to real-world scenarios.

Reconstruction of particle size distribution from cross-sections

Reconstruction of particle size distribution from cross-sections

Soot particle size distribution reconstruction in a turbulent sooting flame with the split-based extended quadrature method of moments

The Method of Moments (MOM) has largely been applied to investigate sooting laminar and turbulent flames. However, the classical MOM is not able to characterize a continuous particle size distribution (PSD). Without access to information on the PSD, it is difficult to accurately take into account particle oxidation, which is crucial for shrinking and eliminating soot particles. Recently, the Split-based Extended Quadrature Method of Moments (S-EQMOM) has been proposed as a numerically robust alternative to overcome this issue [Salenbauch et al., “A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth, and oxidation,” J. Aerosol Sci. 128, 34–49 (2019)]. The main advantage is that a continuous particle number density function can be reconstructed by superimposing kernel density functions (KDFs). Moreover, the S-EQMOM primary nodes are determined individually for each KDF, improving the moment realizability. In this work, the S-EQMOM is combined with a large eddy simulation/presumed-probability density function flamelet/progress variable approach for predicting soot formation in the Delft Adelaide Flame III. The target flame features low/high sooting propensity/intermittency and comprehensive flow/scalar/soot data are available for model validation. Simulation results are compared with the experimental data for both the gas phase and the particulate phase. Good quantitative agreement has been obtained especially in terms of the soot volume fraction. The reconstructed PSD reveals predominantly unimodal/bimodal distributions in the first/downstream portion of this flame with particle diameters smaller than 100 nm. By investigating the instantaneous and statistical sooting behavior at the flame tip, it has been found that the experimentally observed soot intermittency is linked to mixture fraction fluctuations around its stoichiometric value that exhibits a bimodal probability density function.

A joint moment projection method and maximum entropy approach for simulation of soot formation and oxidation in diesel engines

A joint moment projection method and maximum entropy approach for treating the soot population balance equations is developed and presented in this work. The moment projection method is used to solve the population balance equations and generate moments that are supplied to the maximum entropy approach as a post-processing technique to reconstruct the soot particle size distribution. The particle size range required by the maximum entropy for particle size distribution reconstruction is determined based on the weighted particles generated in the moment projection method. The performance of the joint approach is first evaluated by solving a set of simplified population balance equations in MatLab, then it is implemented into a Stochastic Reactor Model engine code to simulate the formation and oxidation of soot particles in a single-cylinder direct injection diesel engine. Results suggest that the joint approach has the advantages of ease of implementation, high accuracy and low computational cost. It enables a detailed analysis on the soot formation and oxidation processes in diesel engines. Complete information on the soot particle size distribution can be provided with little CPU cost induced.

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.

On the Information Content of Coarse Data with Respect to the Particle Size Distribution of Complex Granular Media: Rationale Approach and Testing

The particle size distribution (PSD) of complex granular media is seen as a mathematical measure supported in the interval of grain sizes. A physical property characterizing granular products used in the Andreasen and Andersen model of 1930 is re-interpreted in Information Entropy terms leading to a differential information equation as a conceptual approach for the PSD. Under this approach, measured data which give a coarse description of the distribution may be seen as initial conditions for the proposed equation. A solution of the equation agrees with a selfsimilar measure directly postulated as a PSD model by Martín and Taguas almost 80 years later, thus both models appear to be linked. A variant of this last model, together with detailed soil PSD data of 70 soils are used to study the information content of limited experimental data formed by triplets and its ability in the PSD reconstruction. Results indicate that the information contained in certain soil triplets is sufficient to rebuild the whole PSD: for each soil sample tested there is always at least a triplet that contains enough information to simulate the whole distribution.

Comparisons of methods for reconstructing particle size distribution from its moments

The reconstruction of particle size distribution (PSD) from a finite number of its moments is extremely important for many industrial operations, especially in dynamic multi-phase flows in chemical engineering where the composition of discrete phases evolves with time and space. For example, control of the fuel droplet PSD in diesel engines has a significant influence on the flame structure. Different reconstruction methods have been proposed in the literature during the last decades. However, most of them have only been tested with particular and simple cases. This paper describes and evaluates three typical reconstruction methods: parameter fitting method with prescribed distributions, extended quadrature method of moments (EQMOM) and maximum entropy (ME). The corresponding mathematical formulations of these methods are described. A number of test cases are devised where different types of PSDs are adopted so that a comprehensive evaluation of these methods (advantages and limitations) can be achieved. In this paper, a new approach for estimating the particle size range is proposed to improve the reconstruction performance of the ME method. Furthermore, a coupled moment projection method (MPM) and the revised ME approach for particle dynamic simulation is introduced and compared with EQMOM for a combined particle process of inception, growth, coagulation and fragmentation. Results suggest that the revised ME method is very efficient with a wide range of applicability. High accuracy can be achieved with this revised ME method as long as sufficient moments are adopted. By contrast, the parameter fitting method has a narrow applicability and EQMOM suffers from the oscillation problem when more than one kernel function are used. The findings observed in this work can be used as a good guide for choosing a proper PSD reconstruction technique.

Detailed modeling of soot particle formation and comparison to optical diagnostics and size distribution measurements in premixed flames using a method of moments

Recently improved knowledge about key processes for the formation and growth of soot particles lead to very extensive soot models describing several kinetic pathways for particle nucleation and growth in detail. One of these models has been proposed by D’Anna and coworkers (D’Anna et al., 2010, Sirignano et al., 2010). In these original studies, the multivariate formulation was solved with a sectional approach, while a more recent study (Salenbauch et al., 2017) also showed the suitability of moment methods such as the Conditional Quadrature Method of Moments (CQMOM) (Yuan et al., 2011). However, being a moment method, CQMOM does not allow to directly access the soot particle size distribution (PSD). This prevents the consistent comparability of CQMOM results to soot measurements based on a scanning mobility particle sizer (SMPS). Furthermore, the comparison to laser-induced fluorescence (LIF) experiments is also limited, as the LIF signal only represents small particles and their concentration is only extractable from the simulation results if the PSD shape is known.The aim of this study is to extend the previously developed CQMOM soot model by a PSD reconstruction step applying the concept of entropy maximization. Maintaining the efficiency of the CQMOM moment inversion algorithm to close the moment equations, the model extension enables to evaluate the diameter-based PSD in a post-processing step without prescribing a specific shape as input. The updated algorithm is applied to simulate soot formation in two different burner-stabilized premixed C2H4/O2/N2 flames with a very lightly sooting (C/O = 0.67) and a heavily sooting (C/O = 0.77) character. Numerical results are compared to recently published LIF, laser-induced incandescence (LII) and SMPS measurement data. The analysis investigates the model’s capability to predict phenomena such as uni- or bimodal distribution shapes and the transition of small nanostructures to agglomerates in the two target flames both of which exhibit very different sooting behaviour.

Estimation of nucleation and growth parameters from in situ Raman spectroscopy in carbonate systems

This paper presents a mathematical model to describe precipitation in carbonate systems. The model combines the population balance equation with geochemistry, a multivariate kinetics model, and the particle size distribution reconstruction technique. In situ Raman spectroscopy data recorded during unseeded magnesite precipitation were used to validate the model. Nucleation and growth rates were described by empirical equations based on classical nucleation theory and the birth-and-spread growth mechanism. The parameters were estimated through the inversion of the model on Raman spectroscopy data without calibration. The model agrees very well with the experiments and upon optimization the particle size distribution of the produced crystals was determined through Laplace transformation technique. In situ Raman spectroscopy is a fundamental tool to monitor precipitation and solution composition. The model presented here allows to extract quantitative information from a Raman spectra recorded in a chemically complex solution.

Reconstruction of particle size distribution and optical constant based on time-resolved information

Reconstruction of particle size distribution and optical constant based on time-resolved information

Particle Size Distribution Reconstruction Using a Finite Number of Its Moments through Artificial Neural Networks: A Practical Application

An artificial neural network (ANN) approach to reconstruct the particle size distribution (PSD) is proposed in this paper. This novel technique has been applied for acetaminophen crystallization in ethanol. Several experimental PSDs taken in different operating conditions, such as temperature and agitation degree, at different stages of the process have been considered, in order to ensure a wide range of different distributions for the system taken into account. The first stage of the ANN modeling is represented by the structure definition and the network training through a backpropagation algorithm in which the experimental PSDs and their associated vector of moments have been used. The second stage is represented by the feedforward application and validations of the proposed model estimating the PSDs using a finite set of experimental moments compared with their associated PSDs and then, using a set of time-dependent moments, obtaining the transitory PSD. The proposed approach represents a more suitable way to reconstruct the PSD in full, for the first time without assuming any reference distribution or knowing in advance the shape of the experimental PSD, leading to a generalized characterization of the PSDs with possible implementations in other multiphase unit operations.

Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements

We present further results of the simulation research on the constrained regularized least squares (CRLS) solution of the ill-conditioned inverse problem in spectral extinction (turbidimetric) measurements, which we originally presented in this journal [Appl. Opt. 49, 4591 (2010)]. The inverse problem consists of determining the particle size distribution (PSD) function of a particulate system on the basis of a measured extinction coefficient as a function of wavelength. In our previous paper, it was shown that under assumed conditions the problem can be formulated in terms of the discretized Fredholm integral equation of the first kind. The CRLS method incorporates two constraints, which the PSD sought will satisfy: nonnegativity of the PSD values and normalization of the PSD to unity when integrated over the whole range of particle size, into the regularized least squares (RLS) method. This leads to the quadratic programming problem, which is solved by means of the active set algorithm within the research. The simulation research that is the subject of the present paper is a continuation and extension of the research described in our previous paper. In the present research, the performance of the CRLS method variants is compared not only to the corresponding RLS method variants but also to other regularization techniques: the truncated generalized singular value decomposition and the filtered generalized singular value decomposition, as well as nonlinear iterative algorithms: The Twomey algorithm and the Twomey-Markowski algorithm. Moreover, two methods of selecting the optimum value of the regularization parameter are considered: The L-curve method and the generalized cross validation method. The results of our simulation research provide even stronger proof that the CRLS method performs considerably better with reconstruction of PSD than other inversing methods, in terms of better fidelity and smaller uncertainty.

Use of a Kalman filter to reconstruct particle size distributions from FBRM measurements

The reconstruction of particle size distributions from chord length distributions is well known to be a tricky task. The usual approach is to first set up a model that describes the relation between a given particle size distribution and the resulting chord length distribution and then try to invert this model. The difficulty is that the inversion turns out to be an ill-posed problem. The idea of this work is to use not only information about the measurement procedure, but also about the crystallisation process. Process knowledge and measurement knowledge are combined by a Kalman filter with a constrained update step. By this approach, a direct inversion of the measurement equation is no longer necessary.

The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions

In this paper, the reconstruction of particle size distributions (PSDs) using particle swarm optimization (PSO) techniques from dynamic light scattering (DLS) data was established. Three different objective functions containing non-smooth constrained objective function, smooth functional objective function of Tikhonov regularization and L objective function, were employed. Simulated results of unimodal, bimodal and bi-dispersed particles show that the PSO technique with non-smooth constrained objective function produces narrower PSDs focusing on peak position in the presence of random noise, the PSO technique with smooth functional of Tikhonov regularization creates relative smooth PSDs, which could be successfully applied to the broad particles inversion, and the PSO technique with L objective function yields smooth PSDs, which saves calculation amount. Experimental results as well as comparisons with CONTIN algorithm and Cumulants method demonstrate the performance of our algorithms. Therefore, the PSO techniques employing the three different objective functions, which only require objective function and need a few initial guesses, may be applied to the reconstruction of PSDs from DLS data.

Improved regularized solution of the inverse problem in turbidimetric measurements

We present results of simulation research on the constrained regularized least-squares (RLS) solution of the ill-conditioned inverse problem in turbidimetric measurements. The problem is formulated in terms of the discretized Fredholm integral equation of the first kind. The inverse problem in turbidimetric measurements consists in determining particle size distribution (PSD) function of particulate system on the basis of turbidimetric measurements. The desired PSD should satisfy two constraints: nonnegativity of PSD values and normalization of PSD to unity when integrated over the whole range of particle size. Incorporating the constraints into the RLS method leads to the constrained regularized least-squares (CRLS) method, which is realized by means of an active set algorithm of quadratic programming. Results of simulation research prove that the CRLS method performs considerably better with reconstruction of PSD than the RLS method in terms of better fidelity and smaller uncertainty.

Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods

By means of a numerical study we show particle-size distributions retrieved with the Chin-Shifrin, Phillips-Twomey, and singular value decomposition methods. Synthesized intensity data are generated using Mie theory, corresponding to unimodal normal, gamma, and lognormal distributions of spherical particles, covering the size parameter range from 1 to 250. Our results show the advantages and disadvantages of each method, as well as the range of applicability for the Fraunhofer approximation as compared to rigorous Mie theory.

Measurement of particle-size distribution and volume fraction in concentrated suspensions with photon migration techniques

The determination of the particle size distribution and the volume fraction in concentrated suspensions from the multiwavelength measurement of isotropic-scattering coefficients by use of frequency-domain photon migration techniques is demonstrated for three different polydisperse polystyrene suspensions. When a Newton-type inverse algorithm is used, the successful recovery of the particle size distribution, in the form of a Weibull function, and the volume fraction of polystyrene suspensions is achieved. Our results are in excellent agreement with dynamic light-scattering size distribution measurements. On consideration of the particle mass conservation as an additional constraint penalty term in the inverse algorithm, it is shown that the quality of the particle size distribution reconstruction can be improved. Because no calibration is needed, photon migration techniques are especially suited for on-line measurement of the particle size distribution and the volume fraction in the chemical- and the pharmaceutical-based industries.

EM algorithm reconstruction of particle size distributions from diffusion battery data

Abstract The EM (Expectation-Maximization) algorithm is presented as a novel method of reconstructing ultrafine particle size distributions from diffusion battery data. The algorithm is an iterative technique for finding maximum-likelihood estimates for grouped particle size distributions. Diffusion battery data were simulated with noise and reconstructions were performed using the EM algorithm and three other methods. In each case the goal was to retrieve known test distributions. Comparison of the reconstructed distributions demonstrated the superiority of the EM algorithm for retrieving particle size information from Poisson-distributed data. The EM algorithm is statistically sound, mathematically simple, and does not require smoothing parameters nor does it impose unnecessary constraints on the solution vector. Due to the algorithm's inherent characteristics, non-negativity and unique convergence of the solution vector are guaranteed.

Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion

The approximations of Penndorf and Shifrin-Punina to the Mie solution at forward scattering angles are extended to smaller size parameter values. The present approximation, Eq. (7), is found to represent accurately the Mie result down to x ~ 0.5-1.0 for refractive index m = 1.33, and to x ~ 2.0 for much larger index values. The implications of this result are discussed relative to the reconstruction of particle size distributions utilizing the Shifrin-Fymat analytical inversion formula of forward scattered intensities.