Moments Of Size Distribution Research Articles

Solving crystallization/precipitation population balance models in CADET, Part II: Size-based Smoluchowski coagulation and fragmentation equations in batch and continuous modes

A particle size-based Smoluchowski coagulation and fragmentation equation was solved in the free and open source process modeling package CADET. The WFV and MCNP schemes were selected to discretize the internal particle size coordinate. Weights in these schemes were modified to preserve and conserve the zeroth and third moments for size-based equations. Modified propositions and proofs for the scheme are provided. Analytical Jacobians were derived and implemented to reduce the solver’s runtime. A two-dimensional Smoluchowski coagulation and fragmentation equation with axial position as external coordinate was formulated and discretized to support simulations of continuous particulate processes in dispersive plug flow reactors. Five 1D and four 2D test cases were used to validate the implementation and benchmark the solver’s performance. The runtime, L1 error norm, L1 error rate, particle size distribution moments up to sixth order and several scalar metrics were analyzed in detail.

Hurricane Laura (2020): A Comparison of Drop Size Distribution Moments Using Ground and Radar Remote Sensing Retrieval Methods

AbstractHurricane Laura was the strongest hurricane to make landfall in Louisiana since 1969 with maximum sustained winds of 130 knots. One University of Oklahoma Shared Atmospheric Mobile and Teaching Polarmetric Radar (SR1‐P), and four portable in situ precipitation stations (PIPSs) equipped with parsivel disdrometers were spatially and temporally collocated with two NASA Global Precipitation Measurement Mission Dual‐frequency Precipitation Radar overpasses. The combined retrieval methods were able to quantify and compare drop size distribution moments and radar‐inferred precipitation processes before, during, and after the storm center made landfall. It was found that the magnitude of collision‐coalescence dominant precipitation decreased from before to after landfall. Further, the presence of a bright‐band becomes more evident across all percentiles in the post‐landfall overpass, indicating an increase in stratiform precipitation compared to convective precipitation after Laura moved inland. The PIPS showed an increase in mean drop size from 1.0 mm before landfall to as high as 4.0 mm in the eyewall, while decreasing to below 1.0 mm as Laura continued to move inland with a decrease in maximum echo top height of 0.5–1.0 km. Last, the Dual‐frequency Precipitation Radar (DPR) algorithm overestimated the normalized intercept parameter by 0.5–1.0 m−3 mm−1 compared to the PIPS implying differences in measured drop number concentration, potentially due to differences in measurement footprint or assumptions in the DPR retrieval algorithm. These findings can potentially be used to improve the DPR particle size distribution algorithm in tropical cyclones.

Microphysical Structure of the Marine Boundary Layer under Strong Wind and Sea Spray Formation as Seen from a 2D Explicit Microphysical Model. Part III: Parameterization of Height-Dependent Droplet Size Distribution

AbstractThis paper completes a series of studies using a 2D hybrid Lagrangian–Eulerian model to investigate the effect of sea spray on the thermodynamics and microphysics of the hurricane mixed layer. The evolution of the mixed layer was simulated by mimicking the motion of an air volume (in a Lagrangian sense) toward a tropical cyclone eyewall along a background airflow. During the radial motion, sea surface temperature, pressure, background wind speed, sea spray production rate, and turbulence intensity were altered according to the available observations. Analysis of the interaction between the hurricane mixed layer and the upper layers in terms of entrainment heat and moisture fluxes gives a new insight into the role of sea spray in the thermodynamics and microphysics of the mixed layer. The evaporation of sea spray leads to an increase in the relative humidity by 10%–15% and to a decrease in temperature by about 1–1.5 K, as compared to cases where sea spray is excluded. Sea spray leads to formation of drizzling clouds with the cloud base at the height of about 250 m. Taking the sea spray effect into account provides a good agreement between the thermodynamics of a simulated mixed layer and the observation data.A parameterization of droplet mass and size distributions as functions of height and wind speed is proposed. The horizontally averaged size distributions are approximated by a sum of lognormal distributions. The moments of size distributions and other integral properties are parameterized as functions of 10-m wind speed by means of simple polynomial expressions.

Development of a Moment Model for Condensational Obliteration of Nanoparticle Aggregates

A new method of simulating condensational obliteration in the framework of the moment method is presented. The nanoparticle size distribution was assumed to be represented by a time-evolving log-normal function during the particle growth. The rate of change in the primary particle size due to obliteration was expressed as a function of the surface growth rate and three size distribution moments. Model validation was carried out by comparing with sectional model simulations. Compared to the sectional model, the new moment model was shown to provide similar results for primary and aggregate particle growth with much less computing time.

Numerical modelling of diesel spray auto-ignition and combustion

This work presents the development and implementation of a spray combustion model based on the spray size distribution moments approach to spray modelling. The governing equations for both the gas and liquid phases are solved by the finite volume method based on an Eulerian framework. The model used in the auto-ignition and combustion analysis is a combination of the probability density function and eddy break-up approaches. This scheme is developed in order to effectively validate both ignition and combustion phases. It is designed to compromise between the chemical reaction rate when dealing with the auto-ignition mode, and the turbulence reaction rate which dominates the combustion mode using a reaction progress variable which represents the reaction level. These average reaction rate source terms in transport equations for species are evaluated by a probability density function averaging approach. The developed ignition and combustion model is shown to have significant potential as a numerical tool for predicting the phenomena involved in spray combustion in diesel engines.

Development and application of the drop number size moment modelling to spray combustion simulations

This work presents the development and implementation of spray combustion modelling based on the spray size distribution moments. In this spray model, the droplet size distribution of spray is characterised by the first four moments related to number, radius, surface-area and volume of droplets, respectively. The governing equations for gas phase and liquid phase employed are solved by the finite volume method based on an Eulerian framework. These constructed equations and source terms are derived based on the moment-average quantities which are the key concept for this work. The sub-model employed for ignition and combustion is the coupling reaction rate between Arrhenius model and Eddy Break-Up model (EBU) via a reaction progress variable. The results obtained from simulation are compared with the experimental and simulation data in the literature in order to assess the accuracy of present model. Comparing with the experimental results, present approach is capable to provide a qualitatively reasonable prediction for auto-ignition. In addition, the flame area developed during the combustion progress corresponds with the experimental data.

A Note on Moment Preservation of Finite Volume Schemes for Solving Growth and Aggregation Population Balance Equations

In this paper we improve the numerical schemes presented in a recent publication by Qamar and Warnecke [Chem. Eng. Sci., 31 (2007), pp. 1576-1589]. It is observed that the proposed numerical schemes do not preserve certain required moments of size distributions. In this paper, slightly different formulations are given that preserve these moments. The new formulations are compared with the old formulations by the application of three analytically solvable problems. It is shown that the new formulations are very accurate for predicting number density distributions as well as certain moments. While the old formulations are accurate enough for predicting the number density distribution, they show peculiar behavior for certain moments.

Stable Colloidal Dispersions of a Lipase−Perfluoropolyether Complex in Liquid and Supercritical Carbon Dioxide

The technique of hydrophobic ion pairing was used to solubilize the lipase from Candida rugosa in a fluorinated solvent, perfluoromethylcyclohexane (PFMC), in complex with a perfluoropolyether (PFPE) surfactant, KDP 4606. The enzyme-surfactant complex was determined to have a hydrodynamic diameter of 6.5 nm at atmospheric pressure by dynamic light scattering (DLS), indicating that a single lipase molecule is stabilized by surrounding surfactant molecules. The complex formed a highly stable colloidal dispersion in both liquid and supercritical carbon dioxide at high CO2 densities (>0.92 and 0.847 g/mL, respectively), with 4% by volume PFMC as a cosolvent, yielding a fluid that was orange, optically translucent, and very nearly transparent. DLS demonstrated aggregation of the enzyme-surfactant complexes in CO2 at 25 and 40 degrees C and various pressures (2000-5000 psia) with hydrodynamic diameters ranging from 50 to 200 nm. The mechanism by which the enzyme-surfactant particles aggregate was shown to be via condensation due to very low polydispersities as characterized by the size distribution moments. Interparticle interactions were investigated with respect to density and temperature, and it was shown that on decreasing the CO2 density, the particle size increased, and the stability against settling decreased. Particle size also decreased as the temperature was increased to 40 degrees C, at constant CO2 density. Nanoparticle aggregates of an enzyme-surfactant complex in CO2, which are nearly optically transparent and stable to settling, are a promising new alternative to previous types of dispersions of proteins in CO2 that either required water/CO2 microemulsions or were composed of large particles unstable to settling.

Construcción de un Simulador Dinámico para Procesos de Cristalización por Lotes a Vacío

A dynamic simulator for vacuum batch crystallization processes using a non-linear model is presented. The model is based on the equation of balance population, on mass and energy balances inside the crystallizer, on the steam continuity equation for the heating jacket and on constitutive relations. Using the technique of crystal size distribution moments, the equation of balance population is converted into a set of differential equations. The construction of simulator was done using Visual Basic. An example of its use is proposed comparing results with those of another simulator and with experimental information obtained in a pilot equipment pilot, obtaining similar results.

Solution of population balance equation using quadrature method of moments with an adjustable factor

For description of dispersed phases in many practical applications, population balance equations (PBE) of entities under investigation are coupled with the thermo-fluid dynamics of the surrounding fluid. Hence, solution of PBE needs to be implemented in a computational fluid dynamics (CFD) code, which leads to additional computational cost. The excess computational demand has limited the applicability of numerical techniques such as class method (CM) or Monte Carlo method (MCM). Although quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) have been shown to be accurate and computationally efficient when used with CFD codes, numerical difficulties can arise for cases where there is a large variation of moments. To circumvent this problem, the standard QMOM was modified by incorporating an adjustable factor, which allows the moments of size distribution to be adjusted, in order to improve the accuracy or reduce CPU time. The performance of this method for solving PBE has been evaluated by case studies involving pure aggregation and breakage, agglomeration and breakup, as well as particle growth, which have analytical solutions or exact solutions from CM or finite element method (FEM). The results demonstrate that the modified QMOM is capable of achieving high accuracy at a low CPU cost if an appropriate adjustable factor is chosen. An interesting feature is that different adjustable factors can be assigned to different processes depending on the balance between accuracy requirement and CPU cost.

Kinetic Model for Transformation from Nanosized Amorphous TiO2 to Anatase

We propose a kinetic model for the transformation of nanosized amorphous TiO2 to anatase with associated coarsening by coalescence. Based on population balance (distribution kinetics) equations for the size distributions, the model applies a first-order rate expression for transformation combined with Smoluchowski coalescence for the coarsening particles. Size distribution moments (number and mass of particles) lead to dynamic expressions for extent of reaction and average anatase particle diameter, which are compared with experimental data [Chem. Mater. 2002, 14, 4145]. Data at different temperatures yield activation energies for coalescence and transformation rate coefficients, as well as an overall endothermic transformation energy for combined transformation and dehydration.

Survival and relaxation time, pore size distribution moments, and viscous permeability in random unidirectional fiber structures

Computer simulation results are presented for the mean survival time, principal relaxation time, mean pore size, and mean square pore size, for random porous structures consisting of parallel nonoverlapping or partially overlapping fibers. The numerical procedure is based on a discrete step-by-step random walk mechanism simulating the Brownian diffusion trajectories of molecules in the porous media. Numerical results on the viscous permeability of these structures are computed with a method based on electrical conduction principles and compared to a variational bound derived from the mean survival time. The results show that nonoverlapping fiber structures exhibit lower values of the dimensionless mean survival time, principal relaxation time, mean pore size, and mean square pore size than randomly overlapping fiber structures of the same porosity, while partially overlapping fiber structures show behavior intermediate to those of the two extreme cases. The mean square pore size (second moment of the pore size distribution) is found to be a very good predictor of the mean survival time for non-, partially, and randomly overlapping fiber structures. Dimensionless groups representing the deviation of variational bounds from our simulation results vary in practically the same range as the corresponding values reported earlier for beds of spherical particles. A universal scaling expression of the literature relating the mean survival time to structural properties [S. Torquato and C. L. Y. Yeong, J. Chem. Phys. 106, 8814 (1997)] agrees very well with our results for all examined fiber structures, thus validated for the first time for porous media formed by partially overlapping particles. The permeability behavior of partially overlapping fiber structures resembles that of nonoverlapping fiber structures for flow parallel to the fibers, but not for transverse flow, where percolation phenomena prevail. The permeability results for beds of unidirectional partially overlapping fibers of moderate and low hard-core porosity compliment successfully earlier numerical data on the permeability of similar structures originating from high-porosity beds of nonoverlapping fibers.

Equivalence of two different integral representations of droplet distribution moments in condensing flow

It is proved that two different and independently derived integral representations of droplet size distribution moments encountered in the literature are equivalent and, moreover, consistent with the general dynamic equation that governs the droplet size distribution function. One of these representations consists of an integral over the droplet radius while the other representation consists of an integral over time. The proof is based on analytical solution of the general dynamic equation in the absence of coagulation but in the presence of both growth and nucleation. The solution derived is explicit in the droplet radius, which is in contrast with the literature where solutions are presented along characteristics. This difference is essential for the equivalence proof. Both the case of nonconvected vapor as well as the case of convected vapor are presented.

Internally consistent approach for modeling solid-state aggregation. I. Atomistic calculations of vacancy clustering in silicon

A computational framework is presented for describing the nucleation and growth of vacancy clusters in crystalline silicon. The overall approach is based on a parametrically consistent comparison between two representations of the process in order to provide a systematic method for probing the details of atomic mechanisms responsible for aggregation. In this paper, the atomistic component of the overall framework is presented. First, a detailed set of targeted atomistic simulations are described that characterize fully the thermodynamic and transport properties of vacancy clusters over a wide range of sizes. It is shown that cluster diffusion is surprisingly favorable because of the availability of multiple, almost degenerate configurations. A single large-scale parallel molecular dynamics simulation is then used to compute directly the evolution of the vacancy cluster size distribution in a supersaturated system initially containing 1000 uniformly distributed vacancies in a host lattice of 216 000 Si atoms at 1600 K. The results of this simulation are interpreted in the context of mean-field scaling theory based on the observed power-law evolution of the size distribution moments. It is shown that the molecular dynamics results for aggregation of vacancy clusters, particularly the evolution of the average cluster size, can be very well represented by a highly simplified mean-field model. A direct comparison to a detailed continuum model is made in a subsequent article.

RETRIEVAL OF OPTICAL PROPERTIES OF ATMOSPHERIC AEROSOLS FROM MOMENTS OF THE PARTICLE SIZE DISTRIBUTION

A technique is described for efficient retrieval of families of smooth model distributions, such as lognormals or modified gammas, from the lower moments of the particle size distribution from which aerosol optical properties can be accurately computed. The Multiple Isomomental Distribution Aerosol Surrogate (MIDAS) technique, along with the quadrature technique of McGraw et al. (1995 Geophys. Res. Lett. 22, 2929–2932), is evaluated by computing the extinction efficiency, asymmetry parameter, backscatter fraction, 180° backscattering cross section, upscatter fraction, mass scattering efficiency, and a direct shortwave forcing at 8 wavelengths for 28 test distributions derived from field observations of marine, continental, urban and stratospheric aerosols. For the 224 single wavelength evaluations with retrieved modified gammas the average magnitude of error for each of the computed optical properties was 2% or less, with the exception of the 180° backscattering cross section (4%), establishing the accuracy of the technique. It is concluded that this approach is useful for obtaining aerosol optical properties from the first 6 moments of the size distribution, permitting confident determination of these properties from models in which aerosol evolution is represented by evolution of the lower-order moments.

Quasi‐Self‐Preserving Log‐Normal Size Distributions in the Transition Regime

AbstractA log‐normal model, based on the extended SIMPLER‐algorithm utilizing the Modal Aerosol Dynamics Modelling Technique, and a sectional model are compared for Brownian coagulation in the transition regime. The models are in good agreement with respect to the calculated particle size distribution moments. Applying the log‐normal model a simple coagulation model consisting of a closed set of five equations is developed that does not require the solution of ordinary diferential equations.

A kinetic tree‐like model of non‐linear alternating copolymerization of monomers with non‐equal reactivities of functional groups and substitution effect

AbstractA kinetic model is presented that can be used to predict moments of size distribution of polymer species in the system where a tri‐functional monomer with groups of different reactivity (such as glycerol) reacts irreversibly with a bi‐functional monomer with no cycle formation. A Smoluchowski‐like equation is used to follow the time evolution of the size distribution of polymer molecules up to the gel point. The results are confronted with calculations made by using the statistical cascade theory. The two methods of calculations give slightly different results. The reasons for the differences are discussed.

Retrieval of aerosol size distribution moments from multiwavelength particulate extinction measurements

Two methods for inferring aerosol size distribution moments (for example, particle area and volume) from multiwavelength particulate extinction measurements have been investigated. The first is an analytic, or eigenvalue, technique that approximates an appropriate moment‐weighting function by a linear combination of kernel functions (Mie extinction cross sections). It then uses the same linear combination of the measured particulate extinction coefficients to infer the size distribution moment. The second technique is a conversion ratio approach that uses the ratio of the particulate extinction measurements at two wavelengths to choose a model moment‐to‐extinction conversion ratio. For each method, expressions are derived that estimate the uncertainty in retrieved moments. The eigenvalue technique has been tested by application to synthetic particulate extinction measurements calculated at wavelengths corresponding to those of the satellite‐borne Stratospheric Aerosol and Gas Experiment (SAGE) II Sun photometer. The synthetic measurements were calculated from model stratospheric aerosol size distributions and from stratospheric aerosol size distributions measured by a wire impactor before and after the eruption of El Chichón. Moments have been derived for two‐ and three‐wavelength retrievals. In general, it was found that retrieved moments agreed well with the analytical model values, when random experimental uncertainties of a few percent or less were included in the synthetic extinction measurements. Typical retrieval errors were between 10 and 20%, with a few exceptions. When random measurement uncertainties with a rms mean value of 5% were included in the synthetic extinction coefficients, retrieval errors increased somewhat, but they rarely exceeded 40%. When applied to synthetic particulate extinction measurements that were calculated from the wire impactor‐measured stratospheric aerosol size distributions and which included 30% rms random experimental uncertainties, the eigenvalue retrieval technique successfully reproduced the large changes in particle volume, effective radius, and distribution width that followed the eruption of El Chichón. Both the eigenvalue and conversion ratio moment retrieval methods have been used to infer the third moment, or volume, of the aerosol size distribution from actual particulate extinction measurements taken by SAGE II during an April 1985 correlative measurement experiment in Brazil. Corresponding inferences are found to be in good agreement with each other.

A stochastic model and the moment dynamics of the growth and size distribution in plant populations

A stochastic model for growth and size distribution in plant populations is proposed, which is described by the Kolmogorov forward equation (diffusion equation). The moments of size distribution are calculated based on the diffusion process. Theoretical analyses are made of the mechanisms and the dynamics of size distribution pattern of an even-aged plant population based on the experiemntal data of Impatiens balsamina L. Plant height, stem diameter and individual plant weight have their specific size-dependent growth pattern, and show almost normal, positively skewed and more positively skewed size distribution, respectively. The stochastic model incorporating size-dependence of individual growth explains these phenomena theoretically. The ecological meaning of the growth and size distribution patterns is discussed. The hypothesis that dry matter production by photosynthesis is first allocated to height growth and then to diameter growth is proposed.

Nucleation and growth of ice crystals in secondary refrigerant freezing

A mathematical model of the freezing-crystallization step of the secondary refrigerant freezing desalination process has been proposed which takes into account the influence of the refrigerant phase on the nucleation rate. The moments of the crystal size distribution have been estimated assuming crystal growth occurs by either of two ways: by heat transfer limited growth, and by growth accompanied by agglomeration. Empirical constants for the nucleation rate and for agitator power efficiency were determined from data of several pilot plant tests. The agreement between observed and calculated size distribution moments indicated that the crystal growth model assuming agglomeration may be the more realistic.