J.C. de la Cal

Bernstein-type operators diminish the φ-variation

We show that a large class of Bernstein-type operators usually considered in approximation theory diminish the total and the fine φ-variation, thus extending a classical result on 1-variation diminution concerning the Bernstein polynomials. Also, the closely related problem of approximation in φ-variation is thoroughly discussed. For these purposes, we use a probabilistic approach in which coupling methods play a fundamental role.

Bernstein-Durrmeyer operators☆

Abstract In this paper, we consider the Durrmeyer-type modifications of the classical Bernstein, Szasz and Baskakov operators, and deal with two different kind of problems. Firstly, we obtain new results concerning preservation of shape properties, Lipschitz constants and global smoothness, as well as monotonic convergence under convexity. To do this, we use a probabilistic approach based on representations of these operators in terms of stochastic processes having a.s. nondecreasing paths and satisfying a suitable martingale-type condition. Secondly, we show that the Szasz-Durrmeyer operator is the limit, in an appropriate sense, of both the Bernstein-Durrmeyer and the Baskakov-Durrmeyer operators. We provide rates of convergence which are derived from the bounds for the total variation distance between the probability measures involved.

Effect of the preparation on the activity and selectivity of supported nickel catalysts

Abstract The influence of the method of preparation and type of support on the activity and selectivity of supported nickel catalysts was investigated. Silica and silica alumina were used as supports. Catalysts were prepared over these supports by incipient wetness impregnation and by precipitation-deposition. In each instance two different temperatures of reduction (300 and 500°C) were used. Physico-chemical characterization included total nickel content, total surface area, metallic surface area, degree of reduction, mean particle size and temperature-programmed reduction. The reactions of n-pentane in the presence of hydrogen and the hydrogenation of benzene were studied. The former was found to be structure sensitive with a well defined maximum of activity as the particle size increased. On the other hand, owing to the scatter of the data, the results obtained for the hydrogenation of benzene are open to discussion. The selectivity of the reaction of n-pentane in the presence of hydrogen was not affected by the type of support but it was significantly influenced by the method of preparation.

High solids content miniemulsion polymerization of vinyl acetate in a continuous stirred tank reactor

Abstract The dynamics of high solids content miniemulsion polymerization of vinyl acetate produced in a continuous stirred tank reactor (CSTR), under different emulsification processes and hydrophobes were studied. In order to gain more insight into the process, a previous kinetic study in a batch reactor was performed. In addition, the rheological behavior of latexes produced by miniemulsion was compared with those obtained by conventional emulsion. From the experiments made in batch mode, it was observed that poly(vinyl acetate) or polystyrene as the sole hydrophobes were able to create miniemulsions as far as the time elapsed between sonication and polymerization was small. The same behavior was observed in the CSTR, where the feasibility of those systems to eliminate the oscillatory performance of CSTR reactors, also at high solids content was demonstrated. The sonication itself or the sole presence of hydrophobe — without sonication — was not enough to avoid the oscillatory behavior. Further, it was shown that the viscosity of miniemulsions at the beginning of the CSTR process is notably lower than emulsions, thus preventing the initial mixing and heating problems.

Kinetics and properties in metallocene catalysed propene polymerisations

Abstract The kinetic behaviour of propene polymerisation in heptane using bis(cyclopentadienyl)zirconium dichloride/methylaluminoxane as catalyst system was studied. Propene polymerisations were performed in both homogeneous and heterogeneous systems. The polymerisation rate was measured by continuously monitoring the monomer consumption. The effect of the process variables such as temperature, pressure, Al/Zr ratio and catalyst concentration on the catalytic performance (activity and polymer properties) were investigated. The experimental results of both the homogeneous and heterogeneous systems were compared, and a kinetic model was proposed to fit the experimental evolution of catalyst activity at different Al/Zr ratios and catalyst concentrations for the homogeneous system.

Inverse microemulsion polymerization of MADQUAT initiated with sodium metabisulfite

The kinetics of the inverse microemulsion polymerization of MADQUAT initiated with sodium metabisulfite in both batch and semi-continuous reactors was studied using reaction calorimetry. It was found that the metabisulfite was able to initiate the polymerization at low temperature (20°C) and that in batch reactor the polymerization stopped well before the complete conversion of the monomer due to complete consumption of the initiator, which disappeared according to first-order kinetics. High conversions were reached in short periods of time when the initiator was continuously fed during the process. In the semi-continuous process, the transport of metabisulfite from the droplets of the initiator solution to the particles seemed to be diffusionally limited and was accelerated by the availability of the emulsifier. The polymerization rate showed a complex evolution during the process owing to the interplay between transport phenomena, polymerization and initiation rates, and nucleation rate.

The Loop Process

Continuous reactors, specially continuous stirred tank reactors (CSTRs), are used to manufacture high production emulsion polymers [1]. The use of CSTRs provides constant quality products and easiness of on-line control. However, in these reactors the heat removal is not effective due to their low heat transfer area/volume ratio, therefore only small conversion increments can be achieved in a single CSTR, and a series of CSTRs has to be used. Continuous loop reactors, CLRs, are becoming and attractive alternative for the production of emulsion polymers. A CLR, whose employment for the manufacture of emulsion polymers was first time described by Lanthier [2,3], consists of a tubular loop that connects the inlet and the outlet of a recycle pump. Reactants are continuously fed into the reactor and the product is also continuously withdrawn from the reactor. The history of this development can be found in ref. [4]. The CLR presents several advantages for the emulsion polymerization process. Its large heat transfer area/reactor volume ratio allows high conversions (98%) in short residence times (8 min.) to be achieved [5]. This results in a substantial reduction of the reactor volume. Geddes [6] reported that a 5,000 liter batch tank reactor can be replaced in terms of production rate by a 50 liter CLR. Because of the small volume and the short residence time at which this reactor is operated, the CLR can be used with great flexibility and minimum losses in the manufacture of different emulsion polymers. The small volume and the absence of head space make the process intrinsically safe. The loop reactor presents also some disadvantages for the manufacture of emulsion polymers: i) core-shell particles cannot be produced, ii) the particle size distribution is difficult to modify and small particle sizes require substantial amounts of surfactants, iii) recipes with high mechanical stability are required to prevent shear induced coagulation; iv) It may suffer from cyclic behavior due to intermittent nucleations, also found in CSTRs; and v) very little has been published on this process in both open and patent literature. In this paper, a review will be given of the designs and flow pattern of the CLRs, polymerization studies carried out in these reactors, and mathematical modeling of CLRs.

Surfactant-free poly(methyl methacrylate)/poly(vinylamine) (PMMA/PVAm) amphiphilic core-shell polymer particles

Surfactant-free emulsion polymerization for the synthesis of poly(methyl methacrylate)/poly(vinylamine) (PMMA/PVAm) amphiphilic core-shell polymer particles is investigated in this work. Particularly, the effect of the PVAm molar mass and the tert-butyl hydroperoxide (TBHP)/PVAm molar ratio on the kinetics of reaction and particle nucleation is studied, and the morphology of the polymer particles carefully analyzed. Interestingly, decreasing the TBHP/PVAm molar ratio leads to faster kinetics of the polymerization reaction. Decreasing the molar mass of PVAm also yields faster kinetics. The morphology of the PMMA/PVAm polymer particles is core-shell like, and the PVAm chains are located at the surface of the polymer particles acting as stabilizer. Thus, the pH affects their morphology and the PVAm chains are fully extended at low pH and hence are swollen by water and have large size, whereas at high pH, the PVAm chains shrink and particle sizes are smaller.

Two Phase Flow through Packed Bed Static and Dynamic Accessibilities of the Packing

We present a simple description of a gas-liquid flow through a packed bed under trickle flow conditions. A careful examination of the physics of the phenomena leads us to model the flow as a directed percolation process on a random network. Two accessibilities are introduced: the static accessibility resulting from the random distribution of the geometric properties of the packing and the dynamic accessibility resulting from energetical requirements. Under usual flow conditions (very far from the critical region) and for usual packing (no correlation between the two accessibilities) a simple model leads to the flow rate distribution. It is compared with experimental results obtained on a pilot scale trickling filter.