Discrete-sectional Model Research Articles

Study of Numerical Diffusion in a Discrete-Sectional Model and Its Application to Aerosol Dynamics Simulation

ABSTRACT The discrete-sectional model is an important tool used to study aerosol dynamics, although it suffers inherent numerical diffusion errors that are dependent on the model parameters including section spacing, numerically conserved aerosol property, and number of discrete sizes. An analytical evaluation of the errors due to the deviations of the derived aerosol properties in the sectional formulation and the interchange between discrete-size and sectional aerosols is carried out. The errors are larger if the difference in the aerosol property function index (ξ) between the derived and conserved properties is larger. Results of parametric studies are reported for condensation and coagulation systems. The results for condensation systems show that the υ-model is better than the n-model and the υ2-model in predicting the integral properties of the size distribution. However, the deviation in the results between the n-model, υ-model, and υ2-model reduces as the section spacing is decreased. A finite va...

A Two-Dimensional Discrete-Sectional Model for Metal Aerosol Dynamics in a Flame

ABSTRACT Aerosol nucleation and growth in a laminar flame or a gas-phase reaction jet were considered in the free-molecular regime. A two-dimensional, axisymmetric, discrete-sectional model was developed based on a species equation. Parabolic conservation equations resulted from the formulation of the governing equations in boundary layer form, neglecting axial diffusion and axial thermophoretic velocities. The source terms of the equations included monomer formation by gas-phase chemical reaction, and growth or reduction by coagulation and condensation among all aerosol particles. A binomial expansion was used to approximate the collision rate. The equations were linearized by Newton's method and solved with a block tridiagonal solver. The results of the model exhibited reasonable agreement with an aerosol size distribution measured by dynamic light scattering. Spatial transport processes that could influence the aerosol dynamics included convection, diffusion, and thermophoresis. Their impact on the dev...

Characterization of iron oxide-silica nanocomposites in flames: Part II. Comparison of discrete-sectional model predictions to experimental data

A discrete-sectional model accounting for particle formation by chemical reaction and growth by coagulation and condensation is developed to predict the evolution of the nanocomposite aerosol size distribution in a multicomponent iron-silicon system in a flame. Particle formation by nucleation of the vapor is represented by an Arrhenius-type rate expression, with the rate constant being obtained from experiments and simulation results reported in the literature. Precursor vapor concentrations and the second aerosol volume moment predictions are compared to laser-induced fluorescence (LIF) and light scattering intensity measurements from experiments described in Part I20 of the paper. The results elucidate the important formation and growth mechanisms of nanocomposite ferric oxide-silica particles in flame reactors. The role of operating parameters such as precursor characteristics and temperature profiles on the final product characteristics is established.

An Approximate Method to Calculate the Collision Rates of a Discrete—Sectional Model

An approximate method is proposed to improve the computational efficiency of discrete-sectional representations of aerosol dynamics. The expense that is associated with the numerical calculation of intersectional conversion rates is reduced by using a binomial expansion of the nondimensional collision frequency function in the summand so that an analytical integral form for the inner collision summation is obtained. A considerable saving in computational expense is achieved in comparison with a double summation with a very small sacrifice in accuracy.

Modeling Small Cluster Deposition on the Primary Particles of Aerosol Agglomerates

ABSTRACT Dense particles, such as the primary particles in aerosol agglomerates, are formed by coagulation with rapid coalescence, or by condensation-like growth. This latter mechanism is explored in a new model. It uses the discrete-sectional model to describe the evolution of the particle mass distribution. In addition, new equations are used to describe the formation of primary particles. Primary particles are “born” when a new particle is formed above a critical size D melt. Particles smaller than D melt are assumed to be liquid-like, sintering rapidly in the available time. Changes in the final primary particle size Dp can be attributed to growth from small particle deposition. This growth increases as the monomer source rate decreases. For conditions representative of TiO2 formation from chloride in a tube reactor, Dp /D melt ranges from 1.5 to 3. When D meit decreases, as would be expected if temperature decreases, Dp /D melt increases. The result is that Dp is much less sensitive to temperature than would be expected from simple sintering models.

Time-Lag for Attainment of the Self-Preserving Particle Size Distribution by Coagulation

The time needed (time-lag) for aerosols and hydrosols to reach the self-preserving state by coagulation is investigated. A discretesectional model is used to calculate new, accurate, self-preserving distributions in the free-molecular and continuum regimes. Correlations for the corresponding time-lags for arbitrary broad, initially log-normal distributions are presented in dimensionless form and compared with literature data. Practical applications are given indicating that industrial aerosols rapidly attain the self-preserving state.

A discrete-sectional model for particulate production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime

A universal correlation (self-preserving map) for gas-phase production of powders is derived from first principles. A dimensionless discrete-sectional model for free-molecular aerosol coagulation is developed and systematically validated by examining the effects of section spacing and numerically conserved aerosol property (particle number, volume, and volume squared concentration) used in computing the evolution of the particle size distribution. Discrete-sectional models that numerically conserve the highest moment of the distribution (here, volume squared) most accurately predict the particle size distribution. This υ 2-based model is used to develop a self-preserving map in the process parameter space of dimensionless time for chemical reaction, θ, and dimensionless process residence time, X. The self-preserving map defines a priori conditions for generation of powders having their asymptotic self-preserving size distribution. If the reaction is nearly instantaneous (θ ⩽ 0.25), the time required for the distribution to become self-preserving is four times the characteristic time for monomer coagulation ( X ≈ 4); however, as the reaction becomes slower (θ ⩾ 0.25), the time for the distribution to become self-preserving is approximately 15 times the characteristic time for chemical reaction ( X ≈ 15 θ). The predictions of the present discrete-sectional model are compared against moment models for aerosol coagulation that assume that the particle size distribution remains monodisperse, self-preserving, or lognormally shaped throughout the process.

Time-Dependent Aerosol Models and Homogeneous Nucleation Rates

Two types of numerical models for homogeneous nucleation and particle growth are compared: models in which the time rate of change in the stable aerosol population is given by an analytical expression for the nucleation rate, and discrete models, in which an expression for the nucleation rate is not required nor explicitly calculated. The classical expression for the homogeneous nucleation rate, coupled to a moment model, is found to produce poor agreement with a discrete-sectional model except for very low values of dimensionless surface tension. A new expression for the homogeneous nucleation rate is proposed, which when coupled to a moment model produces excellent agreement with a discrete-sectional model over a wide range of dimensionless surface tensions. The new expression is also consistent with experimental data on homogeneous nucleation of dibutylphthalate.

A discrete-sectional solution to the aerosol dynamic equation

A discrete-sectional model is presented that can simulate the simulate the aerosol evolution associated with fast chemical reactions. The mechanisms of coagulation, evaporation, and chemical reactions are treated in this model. Test calculations were performed to examine the sensitivity of the resulting number concentration and size distribution to the breakpoint between the discrete and the sectional regime, k, and the number of sections, M, used. The numerical diffusion problem in this model due to the condensational growth was also examined. Simulations of refractory particle formation and growth in aerosol reactors by gas-phase chemical reactions were in good agreement with previous experimental observations.

A method for the synthesis of submicron particles

Powder synthesis by thermally induced vapor-phase reactions in aerosol reactors is described. The powders generated by this technique consist of high-purity, spherical, nonagglomerated, and uniformly sized particles in the 0.1-µm size range. Most of the particles are crystalline spheres. A theoretical examination based on the discrete-sectional model is presented. The comparison shows that the calculated size distribution of the final product from the aerosol reactor is fairly consistent with that observed experimentally.