Numerical and similarity solutions for reversible population balance equations with size-dependent rates

Population balance equations (PBEs) for reversible aggregation-fragmentation processes are important to particle agglomeration and dissolution, polymerization and degradation, liquid droplet coalescence and breakup, and floc coagulation and disintegration. Moment solutions provide convenient solutions to the PBEs, including steady state and similarity solutions, but may not be feasible for complex forms of size-dependent rate coefficients and stoichiometric kernels. Numeric solutions are thus necessary not only for applications, but also for the study of the mathematics of PBEs. Here we propose a numerical method to solve PBEs and compare the results to moment solutions. The numeric results are consistent with known steady state and asymptotic long-time similarity solutions and show how processes can be approximated by self-similar formulations.     

Temperature effects for isothermal polymer crystallization kinetics

We adopt the cluster size distribution model to investigate the effect of temperature on homogeneous nucleation and crystal growth for isothermal polymer crystallization. The model includes the temperature effects of interfacial energy, nucleation rate, growth and dissociation rate coefficients, and equilibrium solubility. The time dependencies of polymer concentration, number and size of crystals, and crystallinity (in Avrami plots) are presented for different temperatures. The denucleation (Ostwald ripening effect) is also investigated by comparing moment and numerical solutions of the population balance equations. Agreement between the model results and temperature-sensitive experimental measurements for different polymer systems required strong temperature dependence for the crystal-melt interfacial energy.     

Partial Results in Δ 3 0 -Categoricity in Linear Orderings and Boolean Algebras

The notion of ₃ ⁰ -categoricity in linear orderings and Boolean algebras is examined. We provide a proof for the fact that there are uncountably many relatively ₃ ⁰ -categorical linear orderings, and furnish a proof of another fact which suggests that the (unrelatively) ₃ ⁰ -categorical linear orderings may be very difficult to classify. In stark contrast to these results for linear orderings, a complete classification of the relatively ₃ ⁰ -categorical Boolean algebras is given.     

Experimentally determined oscillator strengths and linewidths for the Schumann-Runge band system of molecular oxygen—II. The (2-0) to (5-0) bands

Experimental oscillator strengths and predissociation linewidths have been measured at room temperature for the (2-0) to (5-0) Schumann-Runge bands of molecular oxygen using the Adelaide 6 m vacuum ultraviolet monochromator operated at a resolution of about 0.06 Å. Photoelectric detectors were used to measure the ultraviolet absorption at normally two different gas pressures for 37 groups of rotational lines, and the resulting data were interpreted using an equivalent width type of analysis. The variation with N″ of the oscillator strengths within each band was found to be smaller than that measured earlier for the higher vibrational bands. Measured oscillator strengths agree well with previously accepted values except for the (3-0) band, while the linewidths of the present work agree well with recent theoretical values for the (3-0) and (4-0) bands but are larger for the (2-0) and (5-0) bands. A modification was performed on the parameters of a recent theoretical predissociation model to produce excellent overall agreement between experimental and theoretical linewidths from ν' = 2 to 14. The present work provides results which can be used to construct accurate synthetic absorption profiles for studies of the atmospheric absorption of ultraviolet radiation by the Schumann-Runge bands.     

Densely Continuous Forms in Vietoris Hyperspaces

For countably paracompact normal spaces X and locally compact separable metric spaces Y, a characterization is given for the closure of the set of densely continuous forms from X to Y in the hyperspace of nonempty closed subsets of X × Y under the Vietoris topology. This shows that for such X having no isolated points, every closed subset of X × R that is dense over X can be Vietoris approximated by a semicontinuous function on X.     

Some Isoperimetric Inequalities in Electrochemistry and Hele Shaw Flows

Isoperimetric inequalities are applied to a moving-boundaryproblem for doubly-connected domains. This problem occurs forexample in electrochemistry, in which case the domains in questionare the electrolyte of an electrolytic cell. The two electrodessurrounding the electrolyte are assumed to grow or dissolve,at different rates in general, by electrochemical reaction.We obtain optimal estimates showing, for example, that the leastchange in volume of each electrode always occurs in sphericalsymmetry.