1,713 research outputs found
Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in R 3
No full textLet X be a smooth bordered surface in R 3 with a smooth boundary and σ̂ a smooth anisotropic conductivity on X. If the genus of X is given, then starting from the Dirichlet-to-Neumann operator Λ σ̂ on ∂X, we give an explicit procedure to find a unique Riemann surface Y (up to a biholomorphism), an isotropic conductivity σ on Y and a quasiconformal diffeomorphism F:X→Y which transforms σ̂ into σ.As a corollary, we obtain the following uniqueness result: if σ 1 and σ 2 are two smooth anisotropic conductivities on X with Λ σ1= Λ σ2, then there exists a smooth diffeomorphism Φ:X̄ → X̄ such that Φ|∂X=Id and Φ*σ 1=σ 2. © The Author(s) 2011
Deterministic modeling and stochastic simulation of poly-alkoxylation reactions
No full textThe main scope of this work is to show the feasibility and the advantage of using a stochastic approach to describe the poly-alkoxylation kinetics of different substrates. For this purpose, the reactions of ethylene and propylene oxides with respectively ethylene glycol, 1-octanol, and 2-octanol were considered. Two kinetic models were used for interpreting all the kinetic runs available in the literature, one deterministic and another one stochastic, for a useful comparison between the two different approaches. As the adopted reaction mechanism, rate laws, and related kinetic parameters were the same for both the kinetic models, the obtained results for what concerns the substrate consumption, and the oligomers distribution profiles were the same in both cases. In the case of the stochastic kinetic approach, the calculations must be made on a small volume of the reacting mixture containing a sufficiently high number of molecules that is suitable for the statistical analysis but as small as possible for reducing the calculation time. The calculations made have allowed individuating this optimal volume. This study is propaedeutic to the application of a stochastic kinetic approach to the study of ethylene-propylene oxides copolymerization that cannot be faced with a deterministic model for the extremely long or impracticable calculation time due to the great number of material balance differential equations that must be integrated
Monte Carlo Approach to the Simulation of Ethylene Oxide and Propylene Oxide Polymerization and Copolymerization
No full textThe simulation of block or random ethylene oxide (EO) and propylene oxide (PO) copolymerization with a deterministic model requires the solution of many thousands of differential and algebraic equations. This approach, therefore, is not practicable, requiring too long computer calculation time. The use of a stochastic model allows overcoming this drawback. In this work, a Monte Carlo based model has been developed considering, for a given substrate such as ethylene glycol, only the reaction of the overall consumption of the alkoxides. The addition of new units of EO or PO, respectively, to the hydroxyls of the substrate or the terminal hydroxyls of the growing chains is ruled by the stochastic probability of the two mentioned events. The developed model considers also the effect of the side reaction of PO conversion to allyl alcohol with the formation of new growing polymeric chains. To do the calculations a microscopic volume, assumed to be a well-mixed system, is considered containing about 5000 or a few more molecules, which is a number significant for statistical validity. Kinetic laws and related parameters have been estimated from data published in the literature. Examples of simulations of high-molecular-weight EO and PO homopolymers, of an EO-PO random copolymer, and a triblocks copolymer of the type (EO)a-(PO)b-(EO)a will be presented
Polyethoxylation and polypropoxylation reactions: Kinetics, mass transfer and industrial reactor design
No full textEthoxylation and propoxylation reactions are performed in the industry to produce mainly non-ionic surfactants and ethylene oxide (EO)–propylene oxide (PO) copolymers. Both the reactions occur in gas–liquid reactors by feeding gaseous EO, PO or both into the reactor containing a solution of an alkaline catalyst (KOH or NaOH).
Non-ionic surfactants are produced by using liquid starters like fatty alcohols, fatty acids or alkyl-phenols, while when the scope is to prepare EO–PO copolymers the starter can be a mono- or multi-functional alcoholof low molecular weight. Both reactions are strongly exothermic, and EO and PO, in some conditions, can give place to runaway and also to explosive side reactions. Therefore, the choice of a suitable reactor is a key factor for operating in safe conditions. A correct reactor design requires: (i) the knowledge of the kinetic laws governing the rates of the occurring reactions; (ii) the role of mass and heat transfer in affecting the reaction rate; (iii) the solubility of EO and PO in the reacting mixturewith the non-ideality of the reacting solutions considered; (iv) the density of the reacting mixture. All these aspects have been studied by our research group for different starters of industrial interest, and the data collected by using semibatch well stirred laboratory reactors have been employed for the simulation of industrial reactors, in particular Gas–Liquid Spray Tower Loop Reactors
Special Issue on “Industrial Chemistry Reactions: Kinetics, Mass Transfer and Industrial Reactor Design”
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Influence of the vapor-liquid equilibria (VLE) on the kinetics in gas-liquid and gas-liquid-solid systems
No full textGas-liquid and gas-liquid-solid reactors are frequently characterized by the presence of components, reagents or products, that are partitioned between liquid and vapor phase. In this case we have a difficulty in describing the kinetic behaviour of such type of reactors when it occurs that: (i) the amount of partitioned component in the two phases is never negligible in both phases; (ii) the vapor-liquid equilibrium could have a non-ideal behaviour. We need therefore, to introduce in the kinetic model the mass balance equations for describing the partition of the components affecting their concentration in liquid phase also considering, when necessary, the non-ideal behaviour of the involved phases. Vapor-liquid equilibria (VLE) in reactive systems are poorly treated in the literature, especially for reactions occurring at high pressures. In the present work we will examine the different possibilities occurring in practice and the methods to solve them. Some practical examples are reported for better explaining both the theoretical and practical approach. © 2003 Elsevier Science B.V. All rights reserved
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