The self-preserving size distribution theory. I. Effects of the Knudsen number on aerosol agglomerate growth

Gas-phase synthesis of fine solid particles leads to fractal-like structures whose transport and light scattering properties differ from those of their spherical counterparts. Self-preserving size distribution theory provides a useful methodology for analyzing the asymptotic behavior of such systems. Apparent inconsistencies in previous treatments of the self-preserving size distributions in the free molecule regime are resolved. Integro-differential equations for fractal-like particles in the continuum and near continuum regimes are derived and used to calculate the self-preserving and quasi-self-preserving size distributions for agglomerates formed by Brownian coagulation. The results for the limiting case (the continuum regime) were compared with the results of other authors. For these cases the finite difference method was in good in agreement with previous calculations in the continuum regime. A new analysis of aerosol agglomeration for the entire Knudsen number range was developed and compared with a monodisperse model; Higher agglomeration rates were found for lower fractal dimensions, as expected from previous studies. Effects of fractal dimension, pressure, volume loading and temperature on agglomerate growth were investigated. The agglomeration rate can be reduced by decreasing volumetric loading or by increasing the pressure. In laminar flow, an increase in pressure can be used to control particle growth and polydispersity. For D(f)=2, an increase in pressure from 1 to 4 bar reduces the collision radius by about 30%. Varying the temperature has a much smaller effect on agglomerate coagulation.     

Thermodynamic properties of gaseous cerium phosphate studied by Knudsen effusion mass spectrometry

Gaseous CePO₂ has been identified by Knudsen effusion mass spectrometry during vaporization of CeO₂ and magnesium diphosphate from tungsten double, two‐temperature effusion cell. Structure and molecular parameters of gaseous cerium phosphate under study were determined using quantum chemical calculations. On the basis of equilibrium constants measured for gas‐phase reaction, standard formation enthalpy of CePO₂ was determined to be ?508 ± 41 kJ ? mol^?1 at the temperature 298 K.     

Knudsen effusion mass spectrometric determination of mixing thermodynamic data of liquid Ag–In–Sn alloy

The vaporisation of a liquid Ag–In–Sn system has been investigated at 1273–1473 K by Knudsen effusion mass spectrometry (KEMS) and the data fitted to a Redlich–Kister–Muggianu (RKM) sub-regular solution model. Nineteen different compositions have been examined at six fixed indium mole fractions, X_In = 0.10, 0.117, 0.20, 0.30, 0.40 and 0.50. The ternary L-parameters, the thermodynamic activities and the thermodynamic properties of mixing have been evaluated using standard KEMS procedures and from the measured ion intensity ratios of Ag⁺ to In⁺ and Ag⁺ to Sn⁺, using a mathematical regression technique described by us for the first time. The intermediate data obtained directly from the regression technique are the RKM ternary L-parameters. From the obtained ternary L-parameters the integral molar excess Gibbs free energy, the excess chemical potentials, the activity coefficients and the activities have been evaluated. Using the temperature dependence of the activities, the integral and partial molar excess enthalpies and entropies were determined. In addition, for comparison, for some compositions, also the Knudsen effusion isothermal evaporation method (IEM) and the Gibbs–Duhem ion intensity ratio method (GD-IIR) were used to determine activities and good agreement was obtained with the data obtained from fitting to the RKM model.     

Pore-scale modeling of rarefied gas flow in fractal micro-porous media,using lattice Boltzmann method (LBM)

Due to the widespread use of rarefied gas flow in micro-porous media in industrial and engineering problems, a pore-scale modeling of rarefied gas flow through two micro-porous media with fractal geometries is presented, using lattice Boltzmann method. For this purpose, square- and circular-based Sierpinski carpets with fractal geometries are selected due to their inherent behavior for real porous media. Diffusive reflection slip model is used and developed for these porous media through this study. With this respect, the planar Poiseuille flow is selected as a benchmark and validated with the literature. The effect of Knudsen number (Kn) on the permeability is investigated and compared in each geometry. It is shown that as Knudsen number increases, the permeability will increase due to the gas slippage effect on the solid blocks. In addition, it is observed that the permeability is more sensitive to the gaseous flow behavior at the slip and beginning of transition flow regimes. At last, the permeability relationship with Knudsen number is presented with a higher coefficient of determination for both fractal geometries, showing that this relation is logarithmic.

     

Eine neue Methode zur Messung von temperaturabhängigen Partialdrücken in geschlossenen Systemen. Die Bestimmung der Bildungsenthalpie und -entropie von PtI2(s)

Determination of Temperature Dependent Partial Pressures in Closed Systems – a New Method. The Heat of Formation for PtI₂(s) A new method to determine temperature dependent partial pressures of gaseous species in equilibria with condensed phases in closed systems (silica ampoules) at temperatures up to 1000 °C and pressures p_i 0.01 < p_i < 10 bar is presented. It is based on the determination of the change of mass in the gasphase caused by solid-gas transition at higher temperatures of substances which are deposited at one end of the ampoule. The results of the measurements give informations about reaction mechanisms, enthalpies and entropies. The reliability of the method is demonstrated at the example of the system Pt/I₂. The heat of formation and the entropy of PtI₂(s) (δ_BH°(PtI₂(s), 298) = –51.4 kJ · mol^–1, S°(PtI₂(s), 298) = 119.3 J · K^–1 · mol^–1) are computed from experimental results. The heat of thermal decomposition of PtI₂(s) was reconsidered by Knudsen Mass Spectrometry.     

The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

A Mass Spectrometric Study of the Sublimation of Lutetium Tribromide under Knudsen and Langmuir Conditions

The molecular and ionic sublimation of lutetium tribromide under thermodynamic equilibrium (Knudsen effusion) conditions and from the open surface of a LuBr₃ single crystal (Langmuir conditions) was studied by high-temperature mass spectrometry. Vapor contained the LuBr₃, Lu₂Br₆, Lu₃Br₉, and Lu₄Br₁₂ molecules and the Br^?, LuBr₄ ^?, Lu₂Br ₇ ^? , and Lu₃Br ₁₀ ^? negative ions. The partial pressures of the molecules in saturated vapor and the ratio between the sublimation coefficients of monomers and dimers under free vaporization conditions were determined. The degree of the electron impact-induced fragmentation of LuBr₃ molecules under Knudsen and Langmuir sublimation conditions was analyzed. The second and third laws of thermodynamics were used to calculate the enthalpies of sublimation in the form of monomers and oligomers (Knudsen vaporization) and the corresponding activation energies of sublimation (Langmuir vaporization). Ion—molecular equilibria with the participation of negative ions were studied. The enthalpies of formation of molecules and ions in the gas phase were obtained.     

Synergistic use of Knudsen effusion quadrupole mass spectrometry, solid-state galvanic cell and differential scanning calorimetry for thermodynamic studies on lithium aluminates

Three ternary oxides LiAl₅O₈(s), LiAlO₂(s) and Li₅AlO₄(s) in the system Li-Al-O were prepared by solid-state reaction route and characterized by X-ray powder diffraction method. Equilibrium partial pressure of CO₂(g) over the three-phase mixtures {LiAl₅O₈(s)+Li₂CO₃(s)+5Al₂O₃(s)}, {LiAl₅O₈(s)+5LiAlO₂(s)+2Li₂CO₃(s)} and {LiAlO₂(s)+Li₅AlO₄(s)+2Li₂CO₃(s)} were measured using Knudsen effusion quadrupole mass spectrometry (KEQMS). Solid-state galvanic cell technique based on calcium fluoride electrolyte was used to determine the standard molar Gibbs energies of formations of these aluminates. The standard molar Gibbs energies of formation of these three aluminates calculated from KEQMS and galvanic cell measurements were in good agreement. Heat capacities of individual ternary oxides were measured from 127 to 868 K using differential scanning calorimetry. Thermodynamic tables representing the values of Δ_fH⁰(298.15 K), S⁰(298.15 K) S⁰(T), C_p⁰(T), H⁰(T), {H⁰(T)-H⁰(298.15 K)}, G⁰(T), Δ_fH⁰(T), Δ_fG⁰(T) and free energy function (fef) were constructed using second law analysis and FACTSAGE thermo-chemical database software.     

Mass spectrometric determination of the ionisation cross-sections of BaO, Ba, BaF2 and BaI2 by electron impact

A Nier-type mass spectrometer coupled with a Knudsen cell was used for the ionisation cross-section measurements of Ba, BaO, BaF₂ and BaI₂, using appropriate internal standards with known ionisation cross-sections. The vapour pressure ratio of standard to investigated compound, needed for calculation was obtained via the chemical analysis of the effusate. The use of this approach eliminated several sources of uncertainties. The problems concerning the preferential loss of fragment ions was also discussed and partly eliminated. The results obtained were compared with predictions, usually used in the field of high temperature mass spectrometry and a more recent approach using a model for ionisation cross-sections of high temperature molecules (MCM).     

Kinetics of the uptake of SO2 on mineral oxides: Improved initial uptake coefficients at 298 K from pulsed Knudsen cell experiments

The uptake of SO₂ on γ‐Fe₂O₃, γ‐Al₂O₃, and Saharan dust has been studied at T = 298 K using a Knudsen cell reactor operated in a steady‐state as well as in a pulsed mode. The initial uptake coefficients determined in the steady‐state mode have been found to be affected by surface saturation as well as bulk diffusion of SO₂ resulting in an apparent dependence of the initial uptake coefficients on the sample mass of the mineral oxides. However, by operating the Knudsen cell in the pulsed mode with shorter response time, these effects could be suppressed. Initial uptake coefficients of γ_ini (Fe₂O₃) = (8.8 ± 0.4) × 10^?2, γ_ini (Al₂O₃) = (7.4 ± 0.9) × 10^?3, and γ_ini (Saharan dust) = (7.6 ± 0.5) × 10^?2 were derived. This suggests that Fe₂O₃ is an important component in determining the reactivity of mineral dusts. © 2006 Wiley Periodicals, Inc. Int J Chem Kinet 38: 242–249, 2006     

Mass Spectrometric Investigations of Thermodynamic Properties of the RbCl/GdCl3 System

The vaporization of pure RbCl, GdCl₃, and RbCl‐GdCl₃ samples of different phase compositions was investigated in the temperature range between 666 K and 982 K by use of the Knudsen effusion mass spectrometry. The gaseous species RbCl, Rb₂Cl₂, GdCl₃, and RbGdCl₄ were identified in the equilibrium vapours and their partial pressures were determined. The enthalpy of dissociation of RbGdCl₄(g), Δ_dissH°(859 K) = 263.1 ± 7.7 kJ mol^—1, was evaluated by second law treatment of the equilibrium partial pressures. The thermodynamic activities of RbCl and GdCl₃ were obtained at 800 K in the two‐phase fields {Rb₃GdCl₆(s) + liquid} and {RbGd₂Cl₇(s) + GdCl₃(s)}. The Gibbs free energies of formation of the pseudo‐binary phases Rb₃GdCl₆(s), Δ_fG°(800 K) = —75.1 ± 2.5 kJ mol^—1 and RbGd₂Cl₇(s), Δ_fG°(800 K) = —40.6 ± 1.2 kJ mol^—1, were evaluated from the thermodynamic activities of the components. The results are compared with the available literature data.     

Implementation of proficiency testing schemes for a limited number of participants

A metrological background for the selection and use of proficiency testing (PT) schemes for a limited number N of laboratories-participants (less than 20–30) is discussed. The following basic scenarios are taken into account: (1) adequate matrix certified reference materials (CRM) or in-house reference materials (IHRM) with traceable property values are available for PT use as test items; (2) no appropriate matrix CRM is available, but a CRM or IHRM with traceable property values can be applied as a spike or similar; (3) only an IHRM with limited traceability is available. The discussion also considers the effect of a limited population of PT participants N _p on statistical assessment of the PT results for a given sample of N responses from this population. When N _p is finite and the sample fraction N/N _p is not negligible, a correction to the statistical parameters may be necessary. Scores suitable for laboratory performance assessment in such PT schemes are compared. Presented at the 3rd International Conference on Metrology, November 2006, Tel Aviv, Israel.     

On the anti-Kekulé number and anti-forcing number of cata-condensed benzenoids

Previously introduced concepts the anti-Kekulé number and anti-forcing number (Vukicevic et al., J. Math. chem., in press) are applied to cata-condensed benzenoids. It is shown that all cata-benzenoids have anti-Kekulé number either 2 or 3 and both classes are characterized. The explicit formula for anti-forcing number of chain (unbranched) cata-benzenoids is given. It is also shown that anti-forcing number of any cata-benzenoid goes up to h/2 where h is the number of hexagons in a cata-benzenoid.      

Studying the volatility of pyrazolone complexes of rare-earth elements by means of Knudsen effusion

The temperature dependences of the pressure of saturated vapor of pyrazolone complexes of rare-earth elements Ln(PMIP)₃ (where Ln = Y, Ho, Er, Tm, Lu; PMIP = 1-phenyl-3-methyl-4-isobutyryl-5-pyrazolone) are studied via Knudsen effusion, and the enthalpy of their sublimation is determined. Mass spectra and differential scanning calorimetry data are obtained.     

Thermodynamic studies on lithium ferrites

Thermodynamic studies on ternary oxides of Li-Fe-O systems were carried out using differential scanning calorimetry, Knudsen effusion mass spectrometry, and solid-state electrochemical technique based on fluoride electrolyte. Heat capacities of LiFe₅O₈(s) and LiFeO₂(s) were determined in the temperature range 127-861 K using differential scanning calorimetry. Gibbs energies of formation of LiFe₅O₈(s) and LiFeO₂(s) were determined using Knudsen effusion mass spectrometry and solid-state galvanic cell technique. The combined least squares fits can be represented as

Δ_fG_m^o(LiFe₅O₈,s,T)/kJ mol⁻¹ (±6)=−2341+0.6764(T/K) (588≤T/K≤971)     

Study on sublimation of solid electrolyte (AgI)0.5‐(AgPO3)0.5 with Knudsen effusion mass spectrometry

Knudsen high‐temperature mass spectrometry was used to study the process of sublimation of a solid electrolyte (AgI)_0.5‐(AgPO₃)_0.5. Monoatomic iodine ions were found to be the dominant species in the electron ionization mass spectra below 600°C. With an increase in temperature, the relative content of phosphorus oxide ions, mainly [PO]⁺ and [P₄O₁₀]⁺, increased. Under further heating, we observed silver iodide and iodine dimer ions together with phosphorus dimer and tetramer ions and clusters [IPO₂]⁺, [IP₂]⁺, [I₃P]⁺. Using the experimental logarithmic dependencies of ion signal intensities versus the reciprocal absolute temperature of the effusion cell, the apparent sublimation enthalpies Δ_sH of the ions giving the most intense signals were estimated. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical Investigation of Brownian,Gradient, and Turbulent Coagulation Under Moving Vehicle Conditions in an Underground Garage

Particle coagulation behavior can affect particle formation and dispersion in vehicle plumes. An Eulerian particle transport model combining a Realizable k–? model, Brownian, gradient, and turbulent coagulation has been developed to analyze particle coagulation behavior in an underground garage under moving-vehicle conditions. The results show that prominent coagulation of nanoparticles occurs within a limited region, which is at a distance of less than 0.2 m in the vertical direction and no more than 0.4 m in the exhaust direction from a given vehicle. The coagulation of small particles with diameters of less than 130 nm is dominated by Brownian motion, while gradient and turbulent coagulation significantly affect the coagulation of particles with median diameters of 420 nm and larger diameters of 600–950 nm, respectively. The influence of turbulent coagulation increases as vehicle speed and particle size increase. The half-time due to coagulation is approximately two times and 10–20 times larger than the corresponding value due to dilution in the regions less than 0.2 and 1.5 m along the tailpipe centerline, respectively. It is demonstrated that coagulation has considerable influence on particle dispersion in the region less than 0.2 m from the tailpipe, compared with dilution.