Scaling and Similarity

As pointed out at the beginning of the chapter, when designing a new polymer processing operation to produce a product or to blend or compound a new material, it is often desirable or necessary to work on a smaller scale such as a laboratory extruder, internal batch mixer, stirring tank, etc. The evolving model must then be scaled up or down to the actual operation. 

When scaling a process, similarity between the various sizes and processes is sought. As a rule, a perfectly scalable prototype is one that is perfectly similar to its scaled system. A perfectly similar set of systems is one where all the dimensionless numbers or n-groups 

Screw and die characteristic curves for a 45 mm diameter extruder with an PE-LD. 

In most cases, scale-up by similarity is not always fully achieved. A process may be geometrically similar, but not thermally similar. Depending on the type of process involved, one or several kinds of similarities may be required. These may be geometric, kinematic, dynamic, thermal, kinetic or chemical similarities. 

Single screw extruder. Let us take the case of a single screw extruder section that works well when dispersing a liquid additive within a polymer matrix. The single screw extruder was already discussed in the previous section. However, the effect of surface tension, which is important in dispersive mixing, was not included in that analysis. Hence, if we also add surface tension as a relevant physical quantity, it would add one more column on the dimensional matrix. To find the additional dimensionless group associated with surface tension, as, and size of the dispersed phase, R, two new columns to the matrix in eqn. (4.32) must be added resulting in  

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It might be emphasised again that this whole procedure rests on the assumption that each factor is increasing in equal steps, for example, 10°C, 15°C, 20°C, 25°C. However, if we prefer, we can make the steps equal on a logarithmic scale, for example 1%, 2%, 4%, 8%. In this case, Xl will measure the fit to linearity on this scale, and similarly Xq and the higher terms deviations from linearity on this scale. 

Compressed air shall not be used for cleaning purposes except where reduced to less than 30 p.s.i. and then only with effective chip guarding and personal protective equipment which meets the requirements of Subpart E of this part. The 30 p.s.i. requirement does not apply for concrete form, mill scale and similar cleaning purposes. 

H. Chanson, S. Aoki and M. Maruyama, Unsteady air bubble entrainment and detrainment at plunging breaker Dominant time scales and similarity of water level variation, Coastal Eng. 46, 139-157 (2002). 

An additional potential problem is the tectonic instability of the rocks along the corridor. A 1,000 km (621 mi) long scarp developed after a recent earthquake in the Kunlun Shan (14 November 2001 measuring 8.1 on the Richter scale) and similar but more extensive fault scarps can be seen cutting across alluvial fans in the southern part of the corridor. It is probable that the rail bed could be damaged from time to time by a major earthquake, with resultant disruption in service. 

The Se spin-rotation constant can thus be obtained, establishing the Se shielding scale on the basis of the scale, and similarly for Te. 

The Cams Hquid-phase oxidation process is similar in principle however, it is operated continuously, its oxidation reaction vessels are of a much larger scale, and the separation of the manganate intermediate from the caustic melt is accompHshed without dilution by means of filtration (qv) (121—123). 

In cases where a large reactor operates similarly to a CSTR, fluid dynamics sometimes can be estabflshed in a smaller reactor by external recycle of product. For example, the extent of soflds back-mixing and Hquid recirculation increases with reactor diameter in a gas—Hquid—soflds reactor. Consequently, if gas and Hquid velocities are maintained constant when scaling and the same space velocities are used, then the smaller pilot unit should be of the same overall height. The net result is that the large-diameter reactor is well mixed and no temperature gradients occur even with a highly exothermic reaction. 

Fastness to Crocking. Crocking is defined as the transfer of color from the surface of a dyed fabric to another surface by mbbing. AATCC test method no. 8 is a method by which a colored test fabric swatch is fastened to the base of a Crockmeter and mbbed against a white crock test cloth under controlled conditions. Color transfer to the white cloth is evaluated by comparison with the AATCC Chromatic Transference Scale. A similar method, AATCC 116, uses a Rotary Vertical Crockmeter, which requires a smaller area of test fabric than the Crockmeter. 

Organophosphonates are similar to polyphosphates in chelation properties, but they are stable to hydrolysis and replace the phosphates where persistence in aqueous solution is necessary. They are used as scale and corrosion inhibitors (52) where they function via the threshold effect, a mechanism requiring far less than the stoichiometric amounts for chelation of the detrimental ions present. Threshold inhibition in cooling water treatment is the largest market for organophosphonates, but there is a wide variety of other uses (50). 

A chemical will be a solvent for another material if the molecules of the two materials are compatible, i.e. they can co-exist on the molecular scale and there is no tendency to separate. This statement does not indicate the speed at which solution may take place since this will depend on additional considerations such as the molecular size of the potential solvent and the temperature. Molecules of two different species will be able to co-exist if the force of attraction between different molecules is not less than the forces of attraction between two like molecules of either species. If the average force of attraction between dissimilar molecules A and B is and that between similar molecules of type B Fbb and between similar molecules of type A F a then for compatibility Fab - bb and AB - P/KA- This is shown schematically in Figure 5.5 (a). 

It is only po.ssible to obtain similar solutions in situations where the governing equations (Eqs. (12.40) to (12.44)) are identical in the full scale and in the model. This tequirement will be met in situations where the same dimensionless numbers are used in the full scale and in the model and when the constants P(i, p, fj.Q,.. . have only a small variation within the applied temperature and velocity level. A practical problem when water is used as fluid in the model is the variation of p, which is very different in air and in water see Fig. 12.27. Therefore, it is necessary to restrict the temperature differences used in model experiments based on water. 

Frequencies computed with methods other than Hartree-Fock are also scaled to similarly eliminate known systematic errors in calculated frequencies. The followng table lists the recommended scale factors for frequencies and for zero-point energies and for use in computing thermal energy corrections (the latter two items are discussed later in this chapter), for several important calculation types ... 

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