Mathematical Modelling for Polymer Processing: by Antonio Fasano (auth.), Vincenzo Capasso (eds.)

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By Antonio Fasano (auth.), Vincenzo Capasso (eds.)

Polymers are components made up of macromolecules shaped through millions of atoms equipped in a single (homopolymers) or extra (copolymers) teams that repeat themselves to shape linear or branched chains, or lattice buildings. the concept that of polymer strains again to the years 1920's and is without doubt one of the most important rules of final century. It has given nice impulse to indus­ attempt but in addition to basic examine, together with existence sciences. Macromolecules are made from sm all molecules referred to as monomers. the method that brings monomers into polymers is named polymerization. A basic contri­ bution to the commercial creation of polymers, relatively polypropylene and polyethylene, is because of the Nobel prize winners Giulio Natta and Karl Ziegler. the information of Ziegler and Natta date again to 1954, and the method has been stronger continually through the years, really in regards to the layout and shaping of the catalysts. bankruptcy 1 (due to A. Fasano ) is dedicated to a assessment of a few effects in regards to the modelling of the Ziegler- Natta polymerization. the categorical ex am pie is the construction of polypropilene. the method is very advanced and all reports with proper mathematical contents are relatively contemporary, and several other difficulties are nonetheless open.

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106) (using the more convenient form in Lagrangian coordinates). 104) for T. 106) for the expansion velo city field. 87), with the corresponding changes. For the purpose of proving existence it is convenient to introduce at t = 0 a small layer of already penetrated aggregate, so that the initial condition for

57) is Q DC r ="2. 61) Here Q is a function of time on r < a or r > a, but with a possible jump at r = a. Regularity of c at r = 0 forces Q == 0 on r < a. 58) now reads a[c] = > a. The conservation jump condition ~ a . 62) This is a differential equation for a(t), on ce the dependence of [cl and Q upon the radius are determined. 60) determine c(a-) and c(a+), hence [cl as a function of a. 60). 14 Fig. 2. Geometrical construction of c(a-) and c(a+). 61) in r > a is required. Let h(c) be a function such that h'(c) = D(c).

Ricei, Existence of solutions for a continuous multigrain model of polymerization in polymers, to appear on M3AS. D. Andreueci, R. Rieei, Mathematical problems in the Ziegler-Natta polymerization process. In "Complex Flows in Industrial Proeesses", A. , MSSET Birkhäuser (2000), 215-238. G. Collina, P. Sgarzi, G. Baruzzi, Recenti sviluppi dei concetto di Reactor Granule Technology: Multicatalysts Reactor Granule Technology, in "Simposio Montel! 96", S. o, G. , SATE, Ferrara (1996), 41-59. J. Bear, Dynamics of Fluids in Porous Media, America Elsevier, New York, 1972.

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