Taylor-expansion moment method for agglomerate coagulation due to Brownian motion in the entire size regime

Through applying the Taylor-expansion technique to the particle general dynamic equation, the newly proposed Taylor-expansion moment method (TEMOM) is extended to solve agglomerate coagulation due to Brownian motion in the entire size regime. The TEMOM model disposed by Dahneke's solution (TEMOM–Dahneke) is proved to be more accurate than by harmonic mean solution (TEMOM–harmonic) through comparing their results with the reference sectional model (SM) for different fractal dimensions. In the transition regime, the TEMOM–Dahneke gives the more accurate results than the quadrature method of moments with three nodes (QMOM3). The mass fractal dimension is found to play an important role in determining the decay of agglomerate number and the spectrum of agglomerate size distribution, but the effect decreases with decreasing agglomerate Knudsen number. The self-preserving size distribution (SPSD) theory and linear decay law for agglomerate number are only applicable to be in the free molecular regime and continuum plus near-continuum regime, but not perfectly in the transition regime.     

Extension of the Smoluchowski Theory to Transitions from Dilute to Dense Regime of Brownian Coagulation: Triple Collisions

In order to study the transition from dilute (controlled by binary collisions) to dense (controlled by multiple collisions) regime of coagulation of colloidal or aerosol suspensions, the Smoluchowski equation is generalized by consideration of triple collisions in the kinetic approach, recently proposed by the authors for coagulation of comparable size particles. A good agreement of the new model predictions with more general results of the direct numerical simulations by Langevin dynamics (from the literature) is attained in a relatively wide range of the fractional volume, corresponding to the transition from dilute to dense regime of coagulation dynamics, in which multiple collisions among more than three particles can be neglected.

Copyright 2014 American Association for Aerosol Research     

PHYSICAL ANALOGS AND PERFORMANCE OF A BOX MODEL FOR COMPOSITION AND GROWTH OF H2SO4/H2O AND HNO3/H2SO4/H2O AEROSOL IN THE STRATOSPHERE

A physical box model simulating the aerosol particle evolution along air mass trajectories is developed to provide a tool for interpreting the local observations of stratospheric aerosols (i.e., polar stratospheric clouds). The model calculates the composition and the size distributions of H₂SO₄/H₂O and HNO₃/H₂SO₄/H₂O liquid droplets. The parameterization of the physical processes affecting the dynamics of HNO₃ and H₂SO₄ solid hydrates and ice particle size distributions is also included, but not used. This work is restricted to some speculations about the liquid to solid transition, according to existing theories. The evolution of liquid particles is simulated taking into account nucleation, diffusive condensation/evaporation and coagulation. This paper reports the physical and numerical details of the model, which are discussed within the framework of the current understanding of the stratospheric aerosol physics. Performance and limitations of the model are discussed on the basis of the evolution of particle size, and composition along synthetic air mass thermal histories. Size distributions and size-dependent acid weight fractions of the liquid stratospheric aerosols consisting of HNO₃/H₂SO₄/H₂O are calculated in the cases of air mass thermal histories with different cooling rates and with rapid temperature fluctuations.     

A new approach to the Brownian coagulation theory

An overview of the new approach to the Brownian coagulation theory developed in the author’s previous papers, is presented. The traditional diffusion approach is critically analysed and shown to be valid only in the particular case of coalescence of small particles with large ones, R₁?R₂. It is shown that coalescence of comparable size particles occurs in the kinetic regime, realised under condition of homogeneous spatial distribution of particles, in the two modes, continuum and free molecular. However, the expression for the collision frequency function in the continuum mode of the kinetic regime formally coincides with the standard expression derived in the diffusion regime for the particular case of large and small particles. Transition from the continuum to the free molecular mode can be described by the interpolation expression derived within the new analytical approach with fitting parameters that can be specified numerically, avoiding semi-empirical assumptions of the traditional models.     

Modeling and experimental evaluation of an aerosol generator for very high number currents based on a free turbulent jet

In this paper we report on theoretical and experimental work on aerosol formation in a free turbulent jet. A hot DEHS vapor issues through a circular nozzle into slowly moving cold air. Vapor concentration and temperatures are such that particles are formed via homogeneous nucleation close to the nozzle upon mixing with the surrounding air. The vapor is completely quenched in the nucleation regime so that further particle growth is controlled by coagulation. A simple growth dynamics model is presented and the theory is used to design a generation system that produces liquid aerosols at a very high number current [up to 10¹² particles (s)]. The aerosol properties can be controlled by two easily adjustable parameters. The aerosol properties are related to these parameters by simple scaling laws. The results of measurements of the number current and the average particle size support these scaling laws.     

Dynamics of ESIPT fluorescent methylmethacrylate-benzazole dye copolymers

The dynamics of fluorescent methylmethacrylate-benzazole dye copolymers were investigated in different concentration regimes by dynamic light scattering. In the dilute regime the polymer behaves as typical polydisperse linear chains in good solvent with a dynamics dominated by a single fast mode. In the semidilute regime, the cooperative diffusion coefficient, D_coop and the correlation length, ξ could be obtained. Above the semidilute regime the intensity autocorrelation functions show two-step decays, indicating the existence of low range correlations. The dye incorporation, even though small, affects the copolymer dynamics behavior in concentrated solutions if compared to PMMA, which is probably ascribed to a polymer-solvent interaction.     

A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion

A new numerical approach for solving coagulation equation, TEMOM model, is first presented. In this model, the closure of the moment equations is approached using the Taylor-series expansion technique. Through constructing a system of three first-order ordinary differential equations, the most important indexes for describing aerosol dynamics, including particle number density, particle mass and geometric standard deviation, are easily obtained. This approach has no prior requirement for particle size spectrum, and the limitation existing in the log-normal distribution theory automatically disappears. This new approach is tested by comparing it with known accurate solutions both in the free molecular and the continuum regime. The results show that this new approach can be used to solve the particle general dynamic equation undergoing Brownian coagulation with sufficient accuracy, while less computational cost is needed.     

The Effect of van der Waals and Viscous Forces on Aerosol Coagulation

An analysis is carried out to determine the combined effect of van der Waals and viscous fluid forces on coagulation of spherical aerosol particles in the free molecular, transition, and continuum regimes. The effect of viscous forces is taken into account by modifying the particle diffusion coefficient. An asymptotic solution is substituted for the classical formulation of viscous forces. The results of free molecular and continuum regimes are then extended to the transition size range by an interpolation formula.     

Asymptotic Solution of Moment Approximation of the Particle Population Balance Equation for Brownian Agglomeration

In the present study, the asymptotic solutions for particle moment approximation of population balance equation for Brownian agglomeration have been obtained analytically. At long time, the dimensionless particle moment (M_C) is an explicit monotonic decreasing function of fractal dimension in the free molecule regime, while it is a constant in the continuum regime. The asymptotic agglomerate growth rate is consistent with previous qualitative analysis and numerical solution, but the present asymptotic solution is more concise and straightforward.

Copyright 2015 American Association for Aerosol Research     

Evaluating the Mobility of Nanorods in Electric Fields

The mobility of a nonspherical particle is a function of both particle shape and orientation. In turn, the higher magnitude of electric field causes nonspherical particles to align more along the field direction, increasing their mobility or decreasing their mobility diameter. In previous works, Li et al. developed a general theory for the orientation-averaged mobility and the dynamic shape factor applicable to any axially symmetric particles in an electric field, and applied it to the specific cases of nanowires and doublets of spheres. In this work, the theory for a nanowire is compared with experimental results of gold nanorods with known shape determined by TEM images. We compare the experimental measured mobility sizes with the theoretical predicted mobility in the continuum, free molecular, and the transition regime. The mobility size shift trends in the electric fields based on our model, expressed both in the free molecular regime and in the transition regime, are in good agreement with the experimental results. For rods of dimension: width d_r = 17 nm and length L_r = 270 nm, where one length scale is smaller than the mean free path and one larger, the results clearly show that the flow regime of a slender rod is mostly controlled by the diameter of the rod (i.e., the smallest dimension). In this case, the free molecule transport properties best represented our nanorod. Combining both theory and experiment we show how, by evaluating the mobility as a function of applied electric field, we can extract both rod length and diameter.

Copyright 2013 American Association for Aerosol Research     

Development of an Aerosol Mass Spectrometer for Size and Composition Analysis of Submicron Particles

The importance of atmospheric aerosols in regulating the Earth's climate and their potential detrimental impact on air quality and human health has stimulated the need for instrumentation which can provide real-time analysis of size resolved aerosol, mass, and chemical composition. We describe here an aerosol mass spectrometer (AMS) which has been developed in response to these aerosol sampling needs and present results which demonstrate quantitative mea surement capability for a laboratory-generated pure component NH₄ NO₃ aerosol. The instrument combines standard vacuum and mass spectrometric technologies with recently developed aerosol sampling techniques. A unique aerodynamic aerosol inlet (developed at the University of Minnesota) focuses particles into a narrow beam and efficiently transports them into vacuum where aerodynamic particle size is determined via a particle time-of-flight (TOF) measurement. Time-resolved particle mass detection is performed mass spectrometrically following particle flash vaporization on a resistively heated surface. Calibration data are presented for aerodynamic particle velocity and particle collection efficiency measurements. The capability to measure aerosol size and mass distributions is compared to simultaneous measurements using a differential mobility analyzer (DMA) and condensation particle counter (CPC). Quantitative size classification is demonstrated for pure component NH₄ NO₃ aerosols having mass concentrations 0.25_mu g m -3. Results of fluid dynamics calculations illustrating the performance of the aerodynamic lens are also presented and compared to the measured performance. The utility of this AMS as both a laboratory and field portable instrument is discussed.     

Thermophoretic interactions of aerosol particles with constant temperatures

This paper presents an analytical study of the thermophoretic motion of two free aerosol spheres with constant temperatures by using a method of reflections. The particles are allowed to differ in radius, in temperature, and in surface properties. The Knudsen numbers are assumed small so that a continuum model describes the fluid flow with a thermal creep and a hydrodynamic slip at the particle surfaces. The method of reflections is based on an analysis of the thermal and hydrodynamic disturbances produced by a single sphere with constant temperature placed in an arbitrarily varying temperature field. The results for two-sphere interactions are correct to O(r₁₂⁻⁷), where r₁₂ is the distance between the particle centers. For the special situation of two identical spheres, the effect of particle interactions will drive the pair system approaching each other if the particle temperature is less than the temperature of the surrounding. While the temperature of the particles is higher than the surrounding temperature, the thermophoretic force obtains a repulsive effect between the particles. Based on a microscopic model the results for two-particle interactions are applied to find the effect of particle concentration on the average thermophoretic velocity in a bounded suspension. In general, the effect of interactions on thermophoretic coagulation of particles with constant temperatures can be stronger than that on sedimentation.     

Determination of the Scalar Friction Factor for Nonspherical Particles and Aggregates Across the Entire Knudsen Number Range by Direct Simulation Monte Carlo (DSMC)

The friction factor of an aerosol particle depends upon the Knudsen number (Kn), as gas molecule–particle momentum transfer occurs in the transition regime. For spheres, the friction factor can be calculated using the Stokes–Millikan equation (with the slip correction factor). However, a suitable friction factor relationship remains sought-after for nonspherical particles. We use direct simulation Monte Carlo (DSMC) to evaluate an algebraic expression for the transition regime friction factor that is intended for application to arbitrarily shaped particles. The tested friction factor expression is derived from dimensional analysis and is analogous to Dahneke's adjusted sphere expression. In applying this expression to nonspherical objects, we argue for the use of two previously developed drag approximations in the continuum (Kn → 0) and free molecular (Kn → ∞) regimes: the Hubbard–Douglas approximation and the projected area (PA) approximation, respectively. These approximations lead to two calculable geometric parameters for any particle: the Smoluchowski radius, R _S, and the projected area, PA. Dimensional analysis reveals that Kn should be calculated with PA/πR _S as the normalizing length scale, and with Kn defined in this manner, traditional relationships for the slip correction factor should apply for arbitrarily shaped particles. Furthermore, with this expression, Kn-dependent parameters, such as the dynamic shape factor, are readily calculable for nonspherical objects. DSMC calculations of the orientationally averaged drag on spheres and test aggregates (dimers, and open and dense 20-mers) in the range Kn = 0.05–10 provide strong support for the proposed method for friction factor calculation in the transition regime. Experimental measurements of the drag on aggregates composed of 2–5 primary particles further agree well with DSMC results, with differences of less than 10% typically between theoretical predictions, numerical calculations, and experimental measurements.

Copyright 2012 American Association for Aerosol Research     

An experimental and theoretical investigation of rarefied gas flow through circular tube of finite length

The conductance of nitrogen gas through circular tube of finite length was measured in the continuum and transition regimes for length to diameter ratios L/D ranging from 0.045 to 33.4 and pressure ratios across the tubes P₁/P₂ from 1.1 to 23. A numerical analysis was carried out to estimate the conductance using the continuum approach in the continuum regime and transition regime at low Knudsen number, and using direct simulation Monte Carlo (DSMC) method in the transition regime at high Knudsen number. The observed conductances were compared with the simulation and an empirical equation derived by Hanks-Weissberg. Both the experimental and simulation results show that the conductances at a constant gas flow rate increases linearly with increasing arithmetic mean pressure across the tube P_av=(P₁+P₂)/2, irrespective of the P₁/P₂ ratio. The observed conductances were smaller than those predicted by the Hanks-Weissberg's equation. The deviation increases with increasing gas flow rate, and with decreasing L/D ratio. It was confirmed that the deviation occurs due to the increase in the effects of inertia and expansion in the flowing gas with increasing flow rate and decreasing L/D. A semi-empirical equation was derived by substituting the Poiseuille term in the Bernoulli formula with Hank-Weissberg's equation under the condition of isothermal expansion. The proposed equation was found to be valid in the range of the continuum regime to the transition regime at low Knudsen number.     

Analytical expression for the friction coefficient of DLCA aggregates based on extended Kirkwood–Riseman theory

We use a self-consistent field method, which we have previously validated, to calculate the translational friction coefficient of fractal aerosol particles formed by diffusion-limited cluster aggregation (DLCA). Our method involves solving the Bhatnagar–Gross–Krook model for the velocity around a sphere in the transition flow regime. The velocity and drag results are then used in an extension of Kirkwood–Riseman theory to obtain the drag on the aggregate. Our results span a range of primary sphere Knudsen numbers from 0.01 to 100 for clusters with up to N = 2000 primary spheres. Calculated friction coefficients are in good agreement with experimental data and approach the correct continuum and free molecule limits for small and large Knudsen numbers, respectively. Results show that particles exhibit more continuum-like behavior as the number of primary spheres increase, even when the primary particle is in the free molecule regime; as an illustrative example, the friction coefficient for aggregates with primary sphere Kn = 1 is approximately equal to the continuum friction coefficient for N > 500. We estimate that our calculations are within 10% of the true values of the friction coefficients for the range of Kn and N presented here. Finally, we use our results to develop an analytical expression (Equation (38)) for the friction coefficient over a wide range of aggregate and primary particle sizes.

Copyright © 2017 American Association for Aerosol Research     

A coupled CFD-Monte Carlo method for simulating complex aerosol dynamics in turbulent flows

A coupled computational fluid dynamics (CFD)-Monte Carlo method is presented to simulate complex aerosol dynamics in turbulent flows. A Lagrangian particle method-based probability density function (PDF) transport equation is formulated to solve the population balance equation (PBE) of aerosol particles. The formulated CFD-Monte Carlo method allows investigating the interaction between turbulence and aerosol dynamics and incorporating individual aerosol dynamic kernels as well as obtaining full particle size distribution (PSD). Several typical cases of aerosol dynamic processes including turbulent coagulation, nucleation and growth are studied and compared to the sectional method with excellent agreement. Coagulation in both laminar and turbulent flows is simulated and compared to demonstrate the effect of turbulence on aerosol dynamics. The effect of jet Reynolds (Re_j) number on aerosol dynamics in turbulent flows is fully investigated for each of the studied cases. The results demonstrate that Re_j number has significant impact on a single aerosol dynamic process (e.g., coagulation) and the simultaneous competitive aerosol dynamic processes in turbulent flows. This newly modified CFD-Monte Carlo/PDF method renders an efficient method for simulating complex aerosol dynamics in turbulent flows and provides a better insight into the interactions between turbulence and the full PSD of aerosol particles.

Copyright © 2017 American Association for Aerosol Research     

Simulation of Aerosol Dynamics: A Comparative Review of Algorithms Used in Air Quality Models

A comparative review of algorithms currently used in air quality models to simulate aerosol dynamics is presented. This review addresses coagula tion, condensational growth, nucleation, and gas particle mass transfer. Two major approaches are used in air quality models to represent the particle size distribution: (1) the sectional approach in which the size distribution is discretized into sections and particle properties are assumed to be constant over particle size sections and (2) the modal approach in which the size distribution is approxi mated by several modes and particle properties are assumed to be uniform in each mode. The sectional approach is accurate for coagulation and can reproduce the major characteristics of the evolution of the particle size distribution for condensa tional growth with the moving-center and hybrid algorithms. For coagulation and condensational growth, the modal approach provides more accurate results when the standard deviations of the modes are allowed to vary than it does when they are fixed. Predictions of H₂SO₄ nucleation rates are highly sensitive to environ mental variables and simulation of relative rates of condensation on existing particles and nucleation is a preferable approach. Explicit treatment of mass transfer is recommended for cases where volatile species undergo different equilib rium reactions in different particle size ranges (e.g., in the presence of coarse salt particles). The results of this study provide useful information for use in selecting algorithms to simulate aerosol dynamics in air quality models and for improving the accuracy of existing algorithms.     

Correction for sampling errors due to coagulation and wall loss in laminar and turbulent tube flow: direct solution of canonical ‘inverse’ problem for log-normal size distributions

Aerosol sampling from industrial environments (e.g. combustion engines) or natural environments (e.g. the troposphere) frequently involves conveying the sample to a downstream (‘sheltered’) instrument via an upstream tube or duct. While the instrument may be capable of characterizing, say, the particle size distribution (PSD) of the aerosol actually presented to it, the investigator is, of course, usually more interested in the PSD of the aerosol entering the upstream sampling tube. Invariably, this differs from that measured because of several systematic phenomena—perhaps the two most obvious of which are particle size-dependent losses to the tube walls (i.e. incomplete ‘penetration’) and PSD distortion due to suspended particle-particle coagulation when the particle concentrations are sufficiently high. We show here how recent research on the use of ‘moment methods’ to predict the effects of size-dependent walls loss and/or Brownian coagulation in flow systems can now be brought to bear to conveniently solve this ‘inverse’ problem by numerically integrating the quasi-one-dimensional coupled moment equations in the upstream direction, using downstream (measured) aerosol properties in the definitions of all dimensionless dependent variables and parameters. Illustrative ‘universal’ graphs are presented here for the sampling of log-normally distributed ‘inertialess’ (Brownian) aerosols in long straight adiabatic ducts for both commonly encountered extremes of particle Knudsen number Kn_p 1(free molecule) or Kn_p 1 (continuum), as well as convenient rational approximations derived from the leading terms of a Taylor series expansion of the above-mentioned dimensionless moment equations.     

A Moment Method of the Log-Normal Size Distribution with the Critical Size Limit in the Free-Molecular Regime

The present work proposes new formulations of the moment in the free-molecular size regime involving (1) boundary equations at the critical size for evaporation, condensation, and nucleation, and (2) b coefficient functions for coagulation that are improved by two parameters (standard deviation and nondimensional critical size). Using these formulations based on the error function, the critical particle size is readily introduced into the log-normal moment method for applications in general aerosol dynamics. In the situation that the particle size distribution is located near the critical size, the proposed moment method (which considers the critical size limit) improves predictions of total particle number and particle volume concentrations as compared with previously well-used log-normal moment methods for sizes ranging from 0 to ∞. However, as the size distribution approaches to the continuum size regime, the influence of the critical size becomes smaller. Thus, the new formulations are expected to improve microphysical parameterization in the free-molecular regime in aerosol-transport models.

Copyright 2014 American Association for Aerosol Research