Model theory —> Scale

A technique which can assist in the scale-up of commercial plants designs is the use of scale models. A scale model is an experimental model which is smaller than the hot commercial bed but which has identical hydrodynamic behavior. Usually the scale model is fluidized with air at ambient conditions and requires particles of a different size and density than those used in the commercial bed. The scale model relies on the theory of similitude, sometimes through use of Buckingham s pi theorem, to design a model which gives identical hydrodynamic behavior to the commercial bed. Such a method is used in the wind tunnel testing of small model aircraft or in the towing tank studies of naval vessels. 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

In this article, we have surveyed typical properties of isotropic and liquid crystal solutions of liquid-crystalline stiff-chain polymers. It had already been shown that dilute solution properties of these polymers can be successfully described by the wormlike chain (or wormlike cylinder) model. We have here concerned ourselves with the properties of their concentrated solutions, with the main interest in the applicability of two molecular theories to them. They are the scaled particle theory for static properties and the fuzzy cylinder model theory for dynamical properties, both formulated on the wormlike cylinder model. In most cases, the calculated results were shown to describe representative experimental data successfully in terms of the parameters equal or close to those derived from dilute solution data. 

The concentration blob model of scaling theory has been introduced in [DCF+75, Far7G], where it has been shown to qualitatively explain a lot of experiments, A comprehensive review is given in [dG79]. 

Both the Flory-Stockmayer mean-field theory and the percolation model provide scaling relations for the divergence of static properties of the polymer species at the gelation threshold. 

Computational methods have accompanied the development of the Polarizable Continuum Model theory throughout its history. In the building of the molecular cavity and its sampling together with the resolution of the BEM equations we nowadays have a large choice of alternative algorithms, suitable for all kinds of molecular calculations. Linear scaling both in time and space is achieved in both fields. 

The model theory requires that in the scale-up from model- to the full-scale not only the geometric similarity (V/l3 = idem) is maintained, but also that all the other pi-numbers describing the problem keep the same numerical values (pi= idem). This implies that, for example, in the measurements with the boat and ship models both process numbers... 

Reliable scaling-up of the desired operating conditions from the model to the full-scale plant. This is based on the scale invariance of the pi-space. According to the model theory, two processes may be considered to be similar if they take place under geometrically similar conditions and all dimensionless numbers which describe the process have the same numerical value. 

After a review of the customary calculation methods [97] for both essential classes of dryers (convective and contact dryers), the dimensioning methods for spray dryers [98, 99], fluidized and spouted bed dryers [100, 101, 102], cascading rotary dryers [103], pneumatic conveying dryers [104], conductive-heating agitated dryers [105] and layer dryers [106] were presented. They all confirmed the initially made conclusion that the scaling up of dryers is still made today without dimensional analysis and the model theory based thereupon. 

Some particular aspects of mixing systems should be considered in relation to the general principles of model theory. First, if a general correlation is to be used for scale-up, it must be certain that the variables used in the correlation are the only ones acting in the given situation. Because of the complexity of mixing systems, we often lack... 

This name covers all polymer chains (diblocks and others) attached by one end (or end-block) at ( external ) solid/liquid, liquid/air or ( internal ) liquid/liq-uid interfaces [226-228]. Usually this is achieved by the modified chain end, which adsorbs to the surface or is chemically bound to it. Double brushes may be also formed, e.g., by the copolymers A-N, when the joints of two blocks are located at a liquid/liquid interface and each of the blocks is immersed in different liquid. A number of theoretical models have dealt specifically with the case of brush layers immersed in polymer melts (and in solutions of homopolymers). These models include scaling approaches [229, 230], simple Flory-type mean field models [230-233], theories solving self-consistent mean field (SCMF) equations analytically [234,235] or numerically [236-238]. Also first computer simulations have recently been reported for brushes immersed in a melt [239]. 

These questions touch the fundamentals of model theory, which is based on dimensional analysis. Although they have been used in the field of fluid dynamics and heat transfer for more than a century - cars, aircraft, vessels, and heat exchangers were scaled up according to these principles they have only been applied to a limited extent in most areas of process technology, although stirring technology represents a notable exception. 

It is a difficult problem to fit the results in highly concentrated polymer solutions to the existing theories of polymer solutions. However, deGennes mesh model and scaling concepts [16] might provide a qualitative explanation to these phenomena. 

Statistical handling of data and statistical methods (3) cyber infrastructure (data preservation, processing, access) (9) theory and modeling of scaling (4)... 

The aim of model theories [51], which predict band offsets through the macroscopic parameters of the materials, is to find or specify a reference energy level ( r) for the metals and semiconductors. Upon formation of the heterojunction, one merely hues up the reference levels in the two materials. In this case, the valence band offset between two materials is simply obtained as the difference between the two valence band maxima, measured relatively to such a reference level. enables one to define an absolute scale for all semiconductors or metals. Once is calculated, we need to know only the semiconductor s valence band maximum Ev (or the highest occupied molecular orbital (HOMO) levels in molecules) relative to for each material, which following Tersoff s approach [51] we denote as ... 

The expressions are an outcome of the terminal model theory with several steady-state assumptions related to free-radical fiux (14,23). Based on copolymerization studies and reactivity ratios, chloroprene monomer is much more reactive than most vinyl and diene monomers (Table 1). 2,3-Dichloro-l,3-butadiene is the only commercially important monomer that is competitive with chloroprene in the free-radical copolymerization rate. 2,3-Dichlorobutadiene or ACR is used commercially to give crystallization resistance to the finished raw polymer or polymer vulcanizates. a-Cyanoprene (1-cyano-l,3-butadiene) and /3-cyanoprene (2-cyano-1,3-butadiene) are also effective in copolymerization with chloroprene but are difficult to manage safely on a commercial scale. Acrylonitrile and methacrylic acid comonomers have been used in limited commercial quantities. Chloroprene-isoprene and chloroprene-styrene copolymers were marketed in low volumes during the 1950s and 1960s. Methyl methacrylate has been utilized in graft polymerization particularly for vinyl adhesive applications. A myriad of other comonomers have been studied in chloroprene copolymerizations but those copolymers have not been used with much commercial success. 

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