Condensation/coagulation kinetics for mixture of liquid and solid particles: analytical solutions

Kinetic equations describing a simultaneous coagulation and condensation in a mixture of liquid and solid particles (for example, water drops and ice particles in supercooled cloud) are formulated. New analytical solutions to the equations for coagulation kernels, K(g,s)=2; K(g,s)=g+s; K(g,s)=2gs, and particle condensation rates, v(g)=βg, are obtained and analyzed. The results can be used for verification of numerical methods employed in simulation of microphysical processes in supercooled clouds.     

真空法动态制冰的非等温结晶动力学

以氯化钠水溶液为制冰溶液,采用真空法制备得到冰浆。从理论上论证非等温结晶动力学模型Jeziorny法和Mo法用于描述冰浆结晶行为的可行性。通过实验测定不同结晶时间下的冰浆含冰率,并建立了冰浆生成过程的非等温动力学方程。研究结果表明：Jeziorny法和Mo法可很好地描述冰浆的非等温结晶过程;Jeziorny模型指数n的值在0.473~0.525间,表明不同实验条件下的结晶机理基本一致,冰晶呈片状增厚生长;Jeziorny模型中的结晶动力学参数K_c、t_0.25和Mo模型中的冷却速率函数F（T）的变化规律均表明高冷却速率可促进冰浆的形成,同时氯化钠的添加会在一定程度上抑制冰晶生长。     

Coagulation of highly concentrated aerosols

A number of material synthesis processes such as flame, plasma and laser ablation have been developed for production of films and powders at low pressure and high temperature. At these conditions particle growth typically takes place by coagulation in the free molecule and transition regimes. As economic manufacturing of these materials favors operation at high particle concentrations, classic coagulation theory may not be sufficient to describe the ensuing aerosol dynamics, especially if fractal-like particles are formed. The coagulation rate of highly concentrated, polydisperse aerosols is investigated here from the free molecule to the continuum regime by solving the corresponding Langevin dynamics (LD) equations. The LD simulations are validated by monitoring the attainment of the self-preserving size distribution (SPSD) for dilute particle volume fractions, φ_s, below 0.1%. High particle concentrations in the free molecule regime lead to deviations of the aerosol dynamics from the kinetic theory of gases especially during instantaneous coalescence (completely inelastic particle–particle collisions) resulting in slower coagulation rates and slightly narrower SPSDs than in conventional dilute aerosols. In the transition regime, the coagulation rate of highly concentrated aerosols is progressively higher than that for dilute aerosols as growing particles enter the continuum regime where coagulation rates are 2–30 times higher than that of classic Smoluchowski theory. At high particle concentrations (φ_s>1%), a SPSD is approached (σ_g,n=1.42) that does not exhibit the characteristic minimum at the transition regime of dilute aerosols. A relationship is developed for the aerosol coagulation rate of highly concentrated aerosols from the free molecule to continuum regime.     

Kinetic simulation of methane hydrate formation and dissociation in porous media

The vast amount of hydrocarbon gas deposited in the earth's crust as gas hydrates has significant implications for future energy supply and global climate. A 3-D simulator for methane hydrate formation and dissociation in porous media is developed for designing and interpreting laboratory and field hydrate experiments. Four components (hydrate, methane, water and salt) and five phases (hydrate, gas, aqueous-phase, ice and salt precipitate) are considered in the simulator. The intrinsic kinetics of hydrate formation or dissociation is considered using the Kim-Bishnoi model. Water freezing and ice melting are tracked with primary variable switch method (PVSM) by assuming equilibrium phase transition. Mass transport, including two-phase flow and molecular diffusions, and heat transfer involved in formation or dissociation of hydrates are included in the governing equations, which are discretized with finite volume difference method and are solved in a fully implicit manner. The developed simulator is used here to study the formation and the dissociation of hydrates in laboratory-scale core samples. In hydrate formation from the system of gas and ice (G+I) and in hydrate dissociation systems where ice appears, the equilibrium between aqueous-phase and ice (A-I) is found to have a “blocking” effect on heat transfer when salt is absent from the system. Increase of initial temperature (at constant outlet pressure), introduction of salt component into the system, decrease of outlet pressure, and increase of boundary heat transfer coefficient can lead to faster hydrate dissociation.     

Design of packed towers for absorption and desorption of carbon dioxide using hot promoted K2CO3 solution

The mathematical model developed for the design of a packed column for the absorption of carbon dioxide in hot promoted potassium carbonate solution (Gas Sep Purif (1989) Vol 3 p 152; (1990) Vol 4 p 58) is extended to take into account the condensation of water present in the gas feed to the absorption column. The revised model is further extended to obtain the design of the desorption column where the feed gas is pure steam and the exit gas consists of carbon dioxide and steam. The integration of the relevant differential equations for the desorption tower is done by starting the calculations from the top conditions of the column where the exit flow rate of steam and exit gas temperature are guessed and verified later after completing the tower calculations with the help of the overall material and energy balance calculations.     

Modelling of phenol—formaldehyde polymerization reaction

In the condensation polymerization of phenol and formaldehyde, the ortho and para positions exhibit different reactivities. In addition, along a polymer chain, internal reactive sites have lower reactivity, possibly due to shielding phenomena^1,2. A detailed kinetic model for novalac formation has been proposed accounting for all these factors. Five different kinds of reactive sites have been shown to exist on a polymer chain and the reactivity of a given reaction has been assumed to be governed completely by the reactive site involved in it. A mass balance has been written for each site and the equations have been solved numerically to examine the effect of different parameters on the course of the condensation polymerization. The model predicts polymer chains with small levels of branching, which is consistent with experimental observations.     

Methylolation of melamine with incipient condensation. II. Mathematical modeling

A novel mathematical model is presented for the reaction between melamine (M) and formaldehyde (F) at pH = 9.0 and at temperatures between 38 and 90°C. It is based on a kinetic scheme that includes reversible methylolations, irreversible formation of (unsubstituted) methylene bridges, reversible formation of (unsubstituted) ether bridges, instantaneous dissolution of M, and instantaneous equilibrium for the hydration/dehydration of F. The model predicts the distributions of molecular weights and functionalities of the evolving MF resin. Arrhenius expressions were adjusted for the seven kinetic constants on the basis of measurements reported in the first part of this series. Even though the final products contain thousands of different molecular species, 21 of them constitute more than 90% of the total weights. During the initial period with negligible condensation, the undissolved M distorts the distributions of molecular weights and functionalities; but the reversibility of methylolations corrects for such distortion prior to the effective start of condensation. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

The log-normal size distribution theory of brownian aerosol coagulation for the entire particle size range: part II—analytical solution using Dahneke’s coagulation kernel

In this study an analytical solution of the Brownian coagulation equation utilizing the method given in Park et al. [(1999) J. Aerosol Sci. 30, 3–16] but using a more detailed coagulation coefficient instead of the simple approach—the harmonic mean—Park et al. [(1999) J. Aerosol Sci. 30, 3–16] is derived. Comparisons are made with the analytical solution given in Park et al. [(1999) J. Aerosol Sci. 30, 3–16], with a method of moments model, with a log-normal reference model as well as with a sectional model (Landgrebe and Pratsinis, (1990) J. Colloid Interface Sci. 139, 63–86).In order to make a fair comparison between the models, all of them should use the same coagulation coefficients except Park et al. [(1999) J. Aerosol Sci. 30, 3–16] which is already fixed to the harmonic mean method. The most frequently used coagulation coefficients to describe the process of Brownian coagulation in the entire size regime (Fuchs, Dahneke, harmonic mean) were compared and a universal enhancement function to the near-continuum coagulation kernel is given. Furthermore, Fuchs’ coagulation kernel is reviewed and modified for the use with different sized particles. It was shown that Dahneke’s coagulation kernel is very close to those of Fuchs and Wright with a relative error of less than 1%. Based on this finding, Dahneke’s coagulation kernel is used in the models developed as well as during the comparison.In order to develop fast and easy-to-use coagulation models we used the moment method assuming a log-normal size distribution function [Lee et al., (1997), J. Colloid Interface Sci. 188, 486–492]. In the first step the integrals over the particle size distribution are solved analytically and DGEAR (IMSL, 1980) is used to solve for the time dependence. In the second step we solved also the time dependence of the parameters of the size distribution analytically. The models developed are compared over a wide parameter range with a reference model based on the moment method of log-normal size distribution functions as well as a sectional reference model. For a worst case, the moment method shows a relative error of less than 4% for the number concentration; the mean geometric standard deviation and the volume square concentration and for the geometric standard deviation the relative error is less than 8% in the self-preserving state. The analytical solution developed shows a maximum relative error of less than 10% if the dimensionless time is limited to K_coN₀t=1.     

In situ study on kinetic behavior during asymmetric membrane formation via phase inversion process using Raman spectroscopy

Asymmetric membrane formation has been studied by using an in situ analysis developed with a Micro Raman spectroscopy, which emphasizes kinetic aspects of the phase inversion process. Changes in composition with time was successfully measured on the gelation bath‐side as well as inside the precipitated phase for the polymer/solvent/nonsolvent system of polysulfone/1‐methyl‐2‐pyrrolidinone/ethanol. The results shows that resulting relative mass transfer rates of solvent and nonsolvent during the phase inversion process strongly influence the final membrane morphologies. In addition, ternary compositions at which phase separation initiates were explored by analyzing the coagulation front. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 75: 135–141, 2000     

Nonisothermal crystallization behavior of melt‐intercalated polyethylene‐clay nanocomposites

The influence of nanoclay particles on the nonisothermal crystallization behavior of intercalated polyethylene (PE) prepared by melt‐compounding was investigated. It is observed that the crystallization peak temperature (T_p) of PE/clay nanocomposites is slightly but consistently higher than the neat PE at various cooling rates. The half‐time (t_0.5) for crystallization decreased with increase in clay content, implying the nucleating role of nanoclay particles. The nonisothermal crystallization data are analyzed using the approach of Avrami (Polymer 1971, 12, 150), Ozawa (Polym Eng Sci 1997, 37, 443), and Mo and coworkers (J Res Natl Bur Stand 1956, 57, 217), and the validity of the different kinetic models to the nonisothermal crystallization process of PE/clay nanocomposites is discussed. The approach developed by Mo and coworkers successfully explains the nonisothermal crystallization behavior of PE and PE/clay nanocomposites. The activation energy for nonisothermal crystallization of neat PE and PE/clay nanocomposites is determined using the Kissinger (J Res Natl Bur Stand 1956, 57, 217) method. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 102: 3809–3818, 2006     

Multicomponent Aerosol Dynamics of the Pb-O2 System in a Bench Scale Flame Incinerator

A study was carried out to understand the formation and growth of lead particles in a flame incinerator. A bench scale flame incinerator was used to perform controlled experiments with lead acetate as a test compound. A dilution probe in conjunction with real-time aerosol instruments was used to measure the evolution of the particle size distribution at different locations in the flame region. A multicomponent lognormal aerosol model is developed accounting for the chemistry of the lead-oxygen system, and various aerosol dynamic phenomena such as nucleation, coagulation, and condensation. Reasonable agreement is obtained between the predictions of the model using appropriate kinetic parameters and the experimental results.     

Arbitrary-Moment Internally Mixed Dynamic Equation

Although aerosol models usually endeavor to predict the number or mass distribution of particles, many applications require other constant moments, moments that vary with particle size, or multiple moments. Here we derive the equations governing multi-component condensation/evaporation for a general distribution function that may present an arbitrary moment (either integer or non-integer) or an arbitrary combination of such moments. We used analytical solution of the evolution equation instead of a commonly used operator splitting. A variant of Bott's positive definite advection scheme was employed to simulate advection processes. Several variants of this general distribution function were tested: sum of the mass (3rd moment) and number (0th moment) distributions; an intermediate (1.5th moment) distribution; and the sum of these mass, number and intermediate distributions. These functions were tested using different evolution scenarios (condensation or evaporation) and different size grid resolutions (fine and coarse). The overall improvement comparing to conventional one- or two-moment aerosol sectional models (e.g., TOMAS, Adams and Seinfeld 2002 Adams, P. J. and Seinfeld, J. H. 2002. Predicting Global Aerosol Size Distributions in General Circulation Models. J. Geophys. Res., 107: D19 [Google Scholar]) is several orders of magnitude. Besides, our approach involves the solution of half as many equations and storage of less information than two-moment schemes.     

Estimating nucleation rates from apparent particle formation rates and vice versa: Revised formulation of the Kerminen–Kulmala equation

Connections between observed particle formation rates (typically at diameter 3 nm or larger) and the actual nucleation rates have important applications in atmospheric science. First, nucleation theories can be evaluated and second, semi-empirical particle formation rates can be developed for large scale models that neglect the cumbersome initial steps of formation and growth. Kerminen and Kulmala, by estimating the particle formation rate, nucleation mode growth rate and scavenging rate onto background particles (coagulation sink) from measured size distribution evolution, derived a simple yet rather accurate formula for this purpose [Kerminen V.-M., Kulmala, M. (2002). Analytical formulae connecting the “real” and the “apparent” 25 nucleation rate and the nuclei number concentration for atmospheric nucleation events, Journal of Aerosol Science 33, 609–622]. The present work reformulates the original theory in a way that two drawbacks are eliminated: (1) the original expression was derived using a slightly inaccurate coagulation sink dependence on particle size and (2) was based on knowing the condensation sink which requires knowledge of the condensing vapors.     

On the Applicability of a Quasi-Stationary Solution of the Diffusion Equation for the Hydrate Layer Formed at the Gas–Ice (Water) Interface

The problem of methane hydrate formation when the process is controlled by gas diffusion in the hydrate layer formed at the gas–ice (or water) interface is solved. It is shown that an approximate quasi-stationary solution of the diffusion equation is in good agreement with its numerical solution over a wide range of the solubility of the gas in the hydrate, which is dependent on pressure. It is found that the time for the complete transition of the water (or ice) phase into the hydrate state decreases with an increase in the saturation concentration of the mobile gas in the hydrate. The kinetic equations derived based on a quasi-stationary solution of the diffusion equation, which are relationships for the intensity of hydrate formation in snow-containing (or water-containing) formations during the filtration of hydrate-forming gases, are used to describe the concentration fields of the diffusing gas and the dynamics of hydrate layer growth.     

PYROLYSIS AND CHLORINATION OF SMALL HYDROCARBONS

The bases of the kinetic modeling of Chlorine containing systems are addressed in this article

A kinetic scheme, involving more than 300 elementary reactions and 46 molecular and radicalic species, has been developed based on general thermochemical kinetic theories as well as the consolidated know-how in the field of hydrocarbon pyrolysis

Several comparisons with commercial and laboratory experimental data indicate a fair agreement in a wide area of lower and higher pressures (up to 30atm) also covering the high temperature range of methylchloride pyrolysis (1000°C). Mathematical model of EDC pyrolysis furnaces already allows to evaluate process performances and alternatives accounting for fouling rates and on-stream times

The kinetic scheme can be applied to study the methane chlorination reaction system and the dichloroethane (EDC) pyrolysis system. The kinetic scheme is coupled with a furnace model and appropriate mathematical techniques to solve the resulting system of material, energy and momentum conservation equations. A coking model has also been incorporated as fouling of coils determines the run length of the furnaces. Simulation of the pyrolysis of EDC to produce vinyl chloride shows that the results are in reasonable agreement with observed values and trends. Application of the model for the design of coils and evaluation of process alternatives is discussed.     

Emulsion polymerization of styrene: Simulation the effects of mixed ionic and non-ionic surfactant system in the presence of coagulation

A detailed model was developed for emulsion polymerization of styrene in batch reactor to predict the evolution of the product particle size distribution. The effect of binary surfactant systems (ionic/non-ionic surfactants) with different compositions was studied. The zero–one kinetics was employed for the nucleation rate, with the model comprising a set of rigorously developed population balance equations. The modeling incorporated particle formation by both nucleation and coagulation phenomena. The partial differential equations describing the particle population were discretized using finite volume elements. Binary surfactant systems, comprising sodium dodecyl sulfate (SDS) as anionic, and a commercial polyether polyol (Brij35^®) as non-ionic surfactants, were examined with different mass ratios. Increasing non-ionic surfactant mass fraction in binary surfactant system showed the decrease of particle number due to intensifying the coagulation between particles. Broader particle size distributions with greater average particle size were obtained with non-ionic surfactant comparing those obtained with anionic one.     

A study on the specific rate of breakage of cement materials in a laboratory ball mill

The specific rate of breakage (S_i) in the widely accepted first-order expression of grinding rate is one of the important factors required to evaluate a grinding process, particularly for the initial grinding stage in various mill types.In this study, the effects of ball diameter and feed size on the specific rate of breakage were investigated on limestone, trass and clinker samples at batch grinding conditions based on a kinetic model. Eight different monosize fractions were prepared between 1.7 and 0.106 mm, using a √2 sieve series. The specific rates of breakage (S_i) were determined from the size distributions at different grinding times, and the specific rates of breakage were compared for three different ball diameters (41, 25.4 and 9.5 mm).The results indicated that the variation of the specific rate of breakage with feed size of cement materials could be expressed. For the specific rate of breakage of each material, empirical equations were developed to express it as a function of feed size and ball diameter.     

Determination of the smallest particle size detectable in condensation nucleus counters by observation of the coagulation of SO2 photo-oxidation products

The method proposed by Walter et al. (1973) for determining the smallest particle size detectable in condensation nucleus counters (CNC) was conducted experimentally. A dynamic flow system was developed for observation of the particle concentration changes due to coagulation for particles produced by heteromolecular nucleation of H₂SO₄ and water vapor through the irradiation of SO₂ and carrier gas with u.v.-light. Experiments indicate the presence of thermal coagulation as well as other processes. Evidence was found to isolate the coagulation effects from other processes. A smallest particle size detectable in condensation nucleus counters with standard expansion ratio thus was determined to be r_L = 1.6 × 10⁻⁷ cm.     

Analysis of multiphase polycarbonate polymerization in a semibatch reactor

A physical and chemical model that describes the interfacial condensation of bisphenol-A to polycarbonate in a multiphase gas-liquid-liquid semi-batch reactor is presented. The particular polymerization kinetics model consists of initiation, propagation and termination reactions between dissolved bisphenol-A monomer, phosgene gas, a phenolic polymer chain growth stopper and various resulting polymer species. Using these kinetics, a physical model for a semi-batch stirred tank reactor is developed. The model accounts for finite liquid-liquid and gas-liquid mass transfer resistances and includes either plug-flow or complete backmixing of the gas bubbles as options for the gas flow pattern. The model equations are solved using the z-transform method from which the polymer MWD and the moments of the polymer MWD are obtained. The utility of the model as a quantitative means of assessing the effect of various kinetic and transport parameters on the polymer MWD is examined for a particular case.